math-g5-m4-full-module

New York State Common Core 

5 Mathematics Curriculum 

G R A D E 

X 

Table of Contents 

GRADE 5 • MODULE 4 

Multiplication and Division of Fractions and Decimal 

Fractions 


Module Overview ......................................................................................................... i 

Topic A: Line Plots of Fraction Measurements ........................................................ 4.A.1 

Topic B: Fractions as Division .................................................................................. 4.B.1 

Topic C: Multiplication of a Whole Number by a Fraction ...................................... 4.C.1 

Topic D: Fraction Expressions and Word Problems ................................................. 4.D.1 

Topic E: Multiplication of a Fraction by a Fraction ................................................. 4.E.1 

Topic F: Multiplication with Fractions and Decimals as Scaling and Word 

Problems…………………………………………………………………………………………....... 4.F.1 

Topic G: Division of Fractions and Decimal Fractions ..............................................4.G.1 

Topic H: Interpretation of Numerical Expressions ................................................... 4.H.1 

Module Assessments ............................................................................................. 4.S.1 

Module 4: Multiplication and Division of Fractions and Decimal Fractions 

Date: 11/10/13 i 

© 2013 Common Core, Inc. Some rights reserved. commoncore.org 

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NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview Lesson 5 

Grade 5 • Module 4 

Multiplication and Division of 

Fractions and Decimal Fractions 

OVERVIEW 

In Module 4, students learn to multiply fractions and decimal fractions and begin work with fraction division. 

Topic A begins the 38-day module with an exploration of fractional measurement. Students construct line 

plots by measuring the same objects using three different rulers accurate to 1/2, 1/4, and 1/8 of an inch 

(5.MD.2). 

Students compare the line plots and explain how changing the accuracy of the unit of measure affects the 

distribution of points. This is foundational to the understanding that measurement is inherently imprecise, as 

it is limited by the accuracy of the tool at hand. Students use their knowledge of fraction operations to 

explore questions that arise from the plotted data. The interpretation of a fraction as division is inherent in 

this exploration. To measure to the quarter inch, one inch must be divided into four equal parts, or 1 ÷ 4. 

This reminder of the meaning of a fraction as a point on a number line, coupled with the embedded, informal 

exploration of fractions as division, provides a bridge to Topic B’s more formal treatment of fractions as 

division. 

Interpreting fractions as division is the focus of Topic B. Equal sharing with area models (both concrete and 

pictorial) gives students an opportunity to make sense of division of whole numbers with answers in the form 

of fractions or mixed numbers (e.g., seven brownies shared by three girls; three pizzas shared by four people). 

Discussion also includes an interpretation of remainders as a fraction (5.NF.3). Tape diagrams provide a linear 

model of these problems. Moreover, students see that by renaming larger units in terms of smaller units, 

division resulting in a fraction is just like whole number division. 

Topic B continues as students solve real world problems (5.NF.3) and generate story contexts for visual 

models. The topic concludes with students making connections between models and equations while 

reasoning about their results (e.g., between what two whole numbers does the answer lie?). 


Date: 11/10/13 ii 





Module Overview Lesson 

NYS COMMON CORE MATHEMATICS CURRICULUM 5 

In Topic C, students interpret finding a fraction of a set (3/4 of 24) as multiplication of a whole number by a 

fraction (3/4 × 24) and use tape diagrams to support their understandings (5.NF.4a). This in turn leads 

students to see division by a whole number as equivalent to multiplication by its reciprocal. That is, division 

by 2, for example, is the same as multiplication by 1/2. Students also use the commutative property to relate 

fraction of a set to the Grade 4 repeated addition interpretation of multiplication by a fraction. This opens 

the door for students to reason about various strategies for multiplying fractions and whole numbers. 

Students apply their knowledge of fraction of a set and previous conversion experiences (with scaffolding 

from a conversion chart, if necessary) to find a fraction of a measurement, thus converting a larger unit to an 

equivalent smaller unit (e.g.,1/3 min = 20 seconds and 2 1/4 feet = 27 inches). 

Interpreting numerical expressions opens Topic D as students learn to evaluate expressions with parentheses, 

such as 3 × (2/3 –1/5) or 2/3 × (7 + 9) (5.OA.1). They then learn to interpret numerical expressions such as 3 

times the difference between 2/3 and 1/5 or two-thirds the sum of 7 and 9 (5.OA.2). Students generate word 

problems that lead to the same calculation (5.NF.4a), such as, “Kelly combined 7 ounces of carrot juice and 5 

ounces of orange juice in a glass. Jack drank 2/3 of the mixture. How much did Jack drink?” Solving word 

problems (5.NF.6) allows students to apply new knowledge of fraction multiplication in context, and tape 

diagrams are used to model multi-step problems requiring the use of addition, subtraction, and multiplication 

of fractions. 

Topic E introduces students to multiplication of fractions by fractions—both in fraction and decimal form 

(5.NF.4a, 5.NBT.7). The topic starts with multiplying a unit fraction by a unit fraction, and progresses to 

multiplying two non-unit fractions. Students use area models, rectangular arrays, and tape diagrams to 

model the multiplication. These familiar models help students draw parallels between whole number and 

fraction multiplication, and solve word problems. This intensive work with fractions positions students to 

extend their previous work with decimal-by-whole number multiplication to decimal-by-decimal 

multiplication. Just as students used unit form to multiply fractional units by wholes in Module 2 (e.g., 3.5 × 2 

= 35 tenths × 2 ones = 70 tenths), they will connect fraction-by-fraction multiplication to multiply fractional 

units‐by-fractional units (3.5 × 0.2 = 35 tenths × 2 tenths = 70 hundredths). Reasoning about decimal 

3 

of a foot = 3 × 12 inches 

Express 5 3 ft as inches. 

4 4 4 

1 foot = 12 inches 

5 3 ft = (5 × 12) inches + 4 (3 × 12) inches 

4 

= 60 + 9 inches 

= 69 inches 

3 

× 12 4 

placement is an integral part of these lessons. Finding fractional parts of customary measurements and 

measurement conversion (5.MD.1) concludes Topic E. Students convert smaller units to fractions of a larger 

unit (e.g., 6 inches = 1/2 ft). The inclusion of customary units provides a meaningful context for many 

common fractions (1/2 pint = 1 cup, 1/3 yard = 1 foot, 1/4 gallon = 1 quart, etc.). This topic, together with the 


Date: 11/10/13 iii 







fraction concepts and skills learned in Module 3, opens the door to a wide variety of application word 

problems (5.NF.6). 

Students interpret multiplication in Grade 3 as equal groups, and in Grade 4 students begin to understand 

multiplication as comparison. Here, in Topic F, students once again extend their understanding of 

multiplication to include scaling (5.NF.5). Students compare the product to the size of one factor, given the 

size of the other factor (5.NF.5a) without calculation (e.g., 486 × 1,327.45 is twice as large as 243 × 1,327.45, 

because 486 = 2 × 243). This reasoning, along with the other work of this module, sets the stage for students 

to reason about the size of products when quantities are multiplied by numbers larger than 1 and smaller 

than 1. Students relate their previous work with equivalent fractions to interpreting multiplication by n/n as 

multiplication by 1 (5.NF.5b). Students build on their new understanding of fraction equivalence as 

multiplication by n/n to convert fractions to decimals and decimals to fractions. For example, 3/25 is easily 

renamed in hundredths as 12/100 using multiplication of 4/4. The word form of twelve hundredths will then 

be used to notate this quantity as a decimal. Conversions between fractional forms will be limited to 

fractions whose denominators are factors of 10, 100, or 1,000. Students will apply the concepts of the topic 

to real world, multi‐step problems (5.NF.6). 

Topic G begins the work of division with fractions, both fractions and decimal fractions. Students use tape 

diagrams and number lines to reason about the division of a whole number by a unit fraction and a unit 

fraction by a whole number (5.NF.7). Using the same thinking developed in Module 2 to divide whole 

numbers, students reason about how many fourths are in 5 when considering such cases as 5 ÷ 1/4. They also 

reason about the size of the unit when 1/4 is partitioned into 5 equal parts: 1/4 ÷ 5. Using this thinking as a 

backdrop, students are introduced to decimal fraction divisors and use equivalent fraction and place value 

thinking to reason about the size of quotients, calculate quotients, and sensibly place the decimal in quotients 

(5.NBT.7). 

The module concludes with Topic H, in which numerical expressions involving fraction-by-fraction 

multiplication are interpreted and evaluated (5.OA.1, 5.OA.2). Students create and solve word problems 

involving both multiplication and division of fractions and decimal fractions. 

The Mid-Module Assessment is administered after Topic D, and the End-of-Module Assessment follows Topic 

H. 


Date: 11/10/13 iv 







Focus Grade Level Standards 

Write and interpret numerical expressions. 

5.OA.1 

Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with 

these symbols. 


Date: 11/10/13 v 







5.OA.2 

Write simple expressions that record calculations with numbers, and interpret numerical 

expressions without evaluating them. For example, express the calculation “add 8 and 7, then 

multiply by 2” as 2 × (8 +7). Recognize that 3 × (18932 + 921) is three times as large as 18932 

+ 921, without having to calculate the indicated sum or product. 

Perform operations with multi-digit whole numbers and with decimals to hundredths. 

5.NBT.7 

Add, subtract, multiply, and divide decimals to hundredths, using concrete models or 

drawings and strategies based on place value, properties of operations, and/or the 

relationship between addition and subtraction; relate the strategy to a written method and 

explain the reasoning used. 

Apply and extend previous understandings of multiplication and division to multiply and 

divide fractions. 

5.NF.3 

5.NF.4 

5.NF.5 

5.NF.6 

5.NF.7 

Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word 

problems involving division of whole numbers leading to answers in the form of fractions or 

mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 

For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 

equals 3, and that when 3 wholes are shared equally among 4 people each person has a share 

of size 3/4. If 9 people want to share a 50‐pound sack of rice equally by weight, how many 

pounds of rice should each person get? Between what two whole numbers does your answer 

lie? 

Apply and extend previous understandings of multiplication to multiply a fraction or whole 

number by a fraction. 

a. Interpret the product of (a/b) × q as a parts of a partition of q into b equal parts; 

equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual 

fraction model to show (2/3 × 4 = 8/3, and create a story context for this equation. Do the 

same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.) 

Interpret multiplication as scaling (resizing), by: 

a. Comparing the size of a product to the size of one factor on the basis of the size of the 

other factor, without performing the indicated multiplication. 

b. Explaining why multiplying a given number by a fraction greater than 1 results in a 

product greater than the given number (recognizing multiplication by whole numbers 

greater than 1 as a familiar case); explaining why multiplying a given number by a fraction 

less than 1 results in a product smaller than the given number; and relating the principle 

of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1. 

Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by 

using visual fraction models or equations to represent the problem. 

Apply and extend previous understandings of division to divide unit fractions by whole 

numbers and whole numbers by unit fractions. (Students able to multiple fractions in general 

can develop strategies to divide fractions in general, by reasoning about the relationship 

between multiplication and division. But division of a fraction by a fraction is not a 


Date: 11/10/13 vi 







requirement at this grade level.) 

a. Interpret division of a unit fraction by a non-zero whole number, and compute such 

quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction 

model to show the quotient. Use the relationship between multiplication and division to 

explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3. 

b. Interpret division of a whole number by a unit fraction, and compute such quotients. For 

example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the 

quotient. Use the relationship between multiplication and division to explain that 4 ÷ 

(1/5) = 20 because 20 × (1/5) = 4. 

c. Solve real world problems involving division of unit fractions by non‐zero whole numbers 

and division of whole numbers by unit fractions, e.g., by using visual fraction models and 

equations to represent the problem. For example, how much chocolate will each person 

get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 

cups of raisins? 

Convert like measurement units within a given measurement system. 

5.MD.1 

Convert among different-sized standard measurement units within a given measurement 

system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real 

world problems. 

Represent and interpret data. 

5.MD.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). 

Use operations on fractions for this grade to solve problems involving information presented 

in line plots. For example, given different measurements of liquid in identical beakers, find the 

amount of liquid each beaker would contain if the total amount in all the beakers were 

redistributed equally. 

Foundational Standards 

4.NF.1 

4.NF.2 

4.NF.3 

Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction 

models, with attention to how the number and size of the parts differ even though the two 

fractions themselves are the same size. Use this principle to recognize and generate 

equivalent fractions. 

Compare two fractions with different numerators and different denominators, e.g., by 

creating common denominators or numerators, or by comparing to a benchmark fraction 

such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the 

same whole. Record the results of comparisons with symbols >, =, or 1 as a sum of fractions 1/b. 

a. Understand addition and subtraction of fractions as joining and separating parts referring 

to the same whole. 


Date: 11/10/13 vii 







4.NF.4 

4.NF.5 

4.NF.6 

b. Decompose a fraction into a sum of fractions with the same denominator in more than 

one way, recording each decomposition by an equation. Justify decompositions, e.g., by 

using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 

+ 1 + 1/8 = 8/8 + 8/8 + 1/8. 

Apply and extend previous understandings of multiplication to multiply a fraction by a whole 

number. 

a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model 

to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 

× (1/4). 

b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply 

a fraction by a whole number. For example, use a visual fraction model to express 3 × 

(2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.) 

c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by 

using visual fraction models and equations to represent the problem. For example, if 

each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at 

the party, how many pounds of roast beef will be needed? Between what two whole 

numbers does your answer lie? 

Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and 

use this technique to add two fractions with respective denominators 10 and 100. (Students 

who can generate equivalent fractions can develop strategies for adding fractions with unlike 

denominators in general. But addition and subtraction with unlike denominators in general is 

not a requirement at this grade.) For example, express 3/10 as 30/100, and add 3/10 + 4/100 

= 34/100. 

Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 

62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. 

Focus Standards for Mathematical Practice 

MP.2 

MP.4 

Reason abstractly and quantitatively. Students reason abstractly and quantitatively as they 

interpret the size of a product in relation to the size of a factor, interpret terms in a 

multiplication sentence as a quantity and a scaling factor and then create a coherent 

representation of the problem at hand while attending to the meaning of the quantities. 

Model with mathematics. Students model with mathematics as they solve word problems 

involving multiplication and division of fractions and decimals and identify important 

quantities in a practical situation and map their relationships using diagrams. Students use a 

line plot to model measurement data and interpret their results in the context of the 

situation, reflect on whether results make sense, and possibly improve the model if it has not 

served its purpose. 

MP.5 Use appropriate tools strategically. Students use rulers to measure objects to the 1/2, 1/4 

and 1/8 inch increments recognizing both the insight to be gained and the limitations of this 

tool as they learn that the actual object may not match the mathematical model precisely. 


Date: 11/10/13 viii 







Overview of Module Topics and Lesson Objectives 

Standards Topics and Objectives 

5.MD.2 A Line Plots of Fraction Measurements 

Lesson 1: Measure and compare pencil lengths to the nearest 1/2, 1/4, 

and 1/8 of an inch, and analyze the data through line plots. 

5.NF.3 B Fractions as Division 

Lessons 2–3: 

Lesson 4: 

Lesson 5: 

Interpret a fraction as division. 

Use tape diagrams to model fractions as division. 

Solve word problems involving the division of whole numbers 

with answers in the form of fractions or whole numbers. 

5.NF.4a C Multiplication of a Whole Number by a Fraction 

Lesson 6: 

Lesson 7: 

Lesson 8: 

Lesson 9: 

Relate fractions as division to fraction of a set. 

Multiply any whole number by a fraction using tape diagrams. 

Relate fraction of a set to the repeated addition interpretation 

of fraction multiplication. 

Find a fraction of a measurement, and solve word problems. 

Days 

1 

4 

4 

5.OA.1 

5.OA.2 

5.NF.4a 

5.NF.6 

D 

Fraction Expressions and Word Problems 

Lesson 10: 

Compare and evaluate expressions with parentheses. 

Lesson 11–12: Solve and create fraction word problems involving addition, 

subtraction, and multiplication. 

3 

Mid-Module Assessment: Topics A–D (assessment ½ day, return ½ day, 

remediation or further applications 1 day) 

2 

5.NBT.7 

5.NF.4a 

5.NF.6 

5.MD.1 

5.NF.4b 

E 

Multiplication of a Fraction by a Fraction 

Lesson 13: 

Lesson 14: 

Lesson 15: 

Multiply unit fractions by unit fractions. 

Multiply unit fractions by non-unit fractions. 

Multiply non-unit fractions by non-unit fractions. 

8 

Lesson 16: 

Solve word problems using tape diagrams and fraction-byfraction 

multiplication. 

Lessons 17–18: Relate decimal and fraction multiplication. 

Lesson 19: 

Convert measures involving whole numbers, and solve multistep 

word problems. 


Date: 11/10/13 ix 







Standards Topics and Objectives 

Days 

Lesson 20: 

Convert mixed unit measurements, and solve multi-step word 

problems. 

5.NF.5 

5.NF.6 

F 

Multiplication with Fractions and Decimals as Scaling and Word Problems 

Lesson 21: 

Explain the size of the product, and relate fraction and decimal 

equivalence to multiplying a fraction by 1. 

4 

Lessons 22–23: Compare the size of the product to the size of the factors. 

Lesson 24: 

Solve word problems using fraction and decimal multiplication. 

5.OA.1 

5.NBT.7 

5.NF.7 

G 

Division of Fractions and Decimal Fractions 

Lesson 25: 

Lesson 26: 

Divide a whole number by a unit fraction. 

Divide a unit fraction by a whole number. 

7 

Lesson 27: 

Solve problems involving fraction division. 

Lesson 28: 

Write equations and word problems corresponding to tape and 

number line diagrams. 

Lessons 29: Connect division by a unit fraction to division by 1 tenth and 1 

hundredth. 

Lessons 30–31: Divide decimal dividends by non‐unit decimal divisors. 

5.OA.1 

5.OA.2 

H 

Interpretation of Numerical Expressions 

Lesson 32: 

Interpret and evaluate numerical expressions including the 

language of scaling and fraction division. 

2 

Lesson 33: 

Create story contexts for numerical expressions and tape 

diagrams, and solve word problems. 

End-of-Module Assessment: Topics A–H (assessment ½ day, return ½ day, 

remediation or further applications 2 days) 

3 

Total Number of Instructional Days 38 


Date: 11/10/13 x 







Terminology 

New or Recently Introduced Terms 

• Decimal divisor (the number that divides the whole and that has units of tenths, hundredths, 

thousandths, etc.) 

• Simplify (using the largest fractional unit possible to express an equivalent fraction) 

Familiar Terms and Symbols 1 

• Denominator (denotes the fractional unit, e.g., fifths in 3 fifths, which is abbreviated to the 5 in 3 ) 

• Decimal fraction 

• Conversion factor 

• Commutative Property (e.g., 4 × = × 4) 

• Distribute (with reference to the distributive property, e.g., in × 15 = (1 × 15) + ( × 15)) 

• Divide, division (partitioning a total into equal groups to show how many units in a whole, e.g., 

5 ÷ = 25) 

• Equation (statement that two expressions are equal, e.g., 3 × 4 = 6 × 2) 

• Equivalent fraction 

• Expression 

• Factors (numbers that are multiplied to obtain a product) 

• Feet, mile, yard, inch, gallon, quart, pint, cup, pound, ounce, hour, minute, second 

• Fraction greater than or equal to 1 (e.g., , , an abbreviation for 3 + ) 

• Fraction written in the largest possible unit (e.g., 3 3 3 

or 1 three out of 2 threes = ) 

• Fractional unit (e.g., the fifth unit in 3 fifths denoted by the denominator 5 in 3 ) 

• Hundredth ( or 0.01) 

• Line plot 

• Mixed number ( , an abbreviation for 3 + ) 

• Numerator (denotes the count of fractional units, e.g., 3 in 3 fifths or 3 in 3 ) 

• Parentheses (symbols ( ) used around a fact or numbers within an equation) 

• Quotient (the answer when one number is divided by another) 

• Tape diagram (method for modeling problems) 

1 These are terms and symbols students have seen previously. 


Date: 11/10/13 xi 







• Tenth ( or 0.1) 

• Unit (one segment of a partitioned tape diagram) 

• Unknown (the missing factor or quantity in multiplication or division) 

• Whole unit (any unit that is partitioned into smaller, equally sized fractional units) 

Suggested Tools and Representations 

• Area models 

• Number lines 

• Tape diagrams 

Scaffolds 2 

The scaffolds integrated into A Story of Units give alternatives for how students access information as well as 

express and demonstrate their learning. Strategically placed margin notes are provided within each lesson 

elaborating on the use of specific scaffolds at applicable times. They address many needs presented by 

English language learners, students with disabilities, students performing above grade level, and students 

performing below grade level. Many of the suggestions are organized by Universal Design for Learning (UDL) 

principles and are applicable to more than one population. To read more about the approach to 

differentiated instruction in A Story of Units, please refer to “How to Implement A Story of Units.” 

2 Students with disabilities may require Braille, large print, audio, or special digital files. Please visit the website, 

www.p12.nysed.gov/specialed/aim, for specific information on how to obtain student materials that satisfy the National Instructional 

Materials Accessibility Standard (NIMAS) format. 


Date: 11/10/13 xii 







Assessment Summary 

Type Administered Format Standards Addressed 

Mid-Module 

Assessment Task 

End-of-Module 

Assessment Task 

After Topic D Constructed response with rubric 5.OA.1 

5.OA.2 

5.NF.3 

5.NF.4a 

5.NF.6 

5.MD.1 

5.MD.2 

After Topic H Constructed response with rubric 5.OA.1 

5.OA.2 

5.NBT.7 

5.NF.3 

5.NF.4a 

5.NF.5 

5.NF.6 

5.NF.7 

5.MD.1 

5.MD.2 


Date: 11/10/13 xiii 






G R A D E 


Topic A 

Line Plots of Fraction Measurements 

5.MD.2 

Focus Standard: 5.MD.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 

1/8). Use operations on fractions for this grade to solve problems involving information 

presented in line plots. For example, given different measurements of liquid in identical 

beakers, find the amount of liquid each beaker would contain if the total amount in all 

the beakers were redistributed equally. 

Instructional Days: 1 

Coherence -Links from: G4–M5 Fraction Equivalence, Ordering, and Operations 

-Links to: G6–M2 Arithmetic Operations Including Dividing by a Fraction 

Topic A begins the 38-day module with an exploration of fractional measurement. Students construct line 

plots by measuring the same objects using three different rulers accurate to 1/2, 1/4, and 1/8 of an inch 

(5.MD.2). Students compare the line plots and explain how changing the accuracy of the unit of measure 

affects the distribution of points (see line plots below). This is foundational to the understanding that 

measurement is inherently imprecise as it is limited by the accuracy of the tool at hand. 

Students use their knowledge of fraction operations to explore questions that arise from the plotted data 

“What is the total length of the five longest pencils in our class? Can the half inch line plot be reconstructed 

using only data from the quarter inch plot? Why or why not?” The interpretation of a fraction as division is 

inherent in this exploration. To measure to the quarter inch, one inch must be divided into 4 equal parts, or 

1 4. This reminder of the meaning of a fraction as a point on a number line coupled with the embedded, 

informal exploration of fractions as division provides a bridge to Topic B’s more formal treatment of fractions 

as division. 

Pencils measured to 1 2 inch 

Pencils measured to 1 4 inch 

Topic A: Line Plots of Fraction Measurements 

Date: 11/9/13 4.A.1 



Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported.License.

NYS COMMON CORE MATHEMATICS CURRICULUM Topic A 5 

A Teaching Sequence Towards Mastery of Line Plots of Fraction Measurements 

Objective 1: Measure and compare pencil lengths to the nearest 1/2, 1/4, and 1/8 of an inch, and 

analyze the data through line plots. 

(Lesson 1) 

Topic A: Line Plots of Fraction Measurements 

Date: 11/9/13 4.A.2 




NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 5•4 

Lesson 1 

Objective: Measure and compare pencil lengths to the nearest 1/2, 1/4, 

and 1/8 of an inch and analyze the data through line plots. 

Suggested Lesson Structure 

•Fluency Practice 

•Application Problem 

•Concept Development 

•Student Debrief 

Total Time 

(11 minutes) 

(8 minutes) 

(31 minutes) 

(10 minutes) 

(60 minutes) 

Fluency Practice (11 minutes) 

• Compare Fractions 4.NF.2 

• Decompose Fractions 4.NF.3 

• Equivalent Fractions 4.NF.1 

(4 minutes) 

(4 minutes) 

(3 minutes) 

Compare Fractions (4 minutes) 

Materials: (S) Personal white boards 

Note: This fluency review prepares students for this lesson’s Concept Development. 

T: (Project a tape diagram labeled as one whole and partitioned into 2 equal parts. Shade 1 of the 

parts.) Say the fraction. 

S: 1 half. 

T: (Write to the right of the tape diagram. Directly below the tape diagram, project another tape 

diagram partitioned into 4 equal parts. Shade 1 of the parts.) Say this fraction. 

S: 1 fourth. 

T: (Write to the right of the tape diagrams.) On your boards, use the greater than, less than, 

or equal sign to compare. 

S: (Write .) 

Continue with the following possible suggestions: , , , , , , , 

, and . 

Lesson 1: Measure and compare pencil lengths to the nearest 1/2, 1/4, and 1/8 

of an inch and analyze the data through line plots. 

Date: 11/10/13 

4.A.3 





Decompose Fractions (4 minutes) 


Note: This fluency review prepares students for this lesson’s Concept Development. 

T: (Write a number bond with as the whole and as the part.) Say the whole. 

S: 2 thirds. 

T: Say the given part. 

S: 1 third. 

T: On your boards, write the number bond. Fill in the 

missing part. 

S: (Write as the missing part.) 

T: Write an addition sentence to match the number 

bond. 

S: . 

T: Write a multiplication sentence to match the number bond. 

S: 2 × = . 

Continue with the following possible suggestions: , , and . 

Equivalent Fractions (3 minutes) 

T: (Write .) 

T: Say the fraction. 

S: 1 half. 

T: (Write .) 

T: 1 half is how many fourths? 

S: 2 fourths. 

Continue with the following possible sequence: and . 

T: (Write .) 

T: Say the fraction. 

S: 1 half. 

T: (Write .) 

T: 1 half, or 1 part of 2, is the same as 2 parts of what unit? 

S: Fourths. 

Continue with the following possible sequence: and . 



Date: 11/10/13 

4.A.4 





Application Problem (8 minutes) 

The following line plot shows the growth of 10 bean plants on their second week after sprouting. 

a. What was the measurement of the shortest plant? 

b. How many plants measure inches? 

c. What is the measure of the tallest plant? 

d. What is the difference between the longest and shortest measurement? 

Note: This Application Problem provides an opportunity for a quick, formative assessment of student ability 

to read a customary ruler and a simple line plot. As today’s lesson is time-intensive, the analysis of this plot 

data is necessarily simple. 

Concept Development (31 minutes) 

Materials: (S) Inch ruler, Problem Set, 8½″ × 1″ strip of paper (with straight edges) per student 

Note: Before beginning the lesson, draw three number lines, one beneath the other, on the board. The lines 

should be marked 0–8 with increments of halves, fourths, and eighths, respectively. Leave plenty of room to 

put the three line plots directly beneath each other. 

Students will compare these line plots later in the lesson. 

T: Cut the strip of paper so that it is the same length as 

your pencil. 

S: (Measure and cut.) 

T: Estimate the length of your pencil strip to the nearest 

inch and record your estimate on the first line in your 

Problem Set. 

T: If I ask you to measure your pencil strip to the nearest 

half inch, what do I mean? 

NOTES ON 

MULTIPLE MEANS OF 

REPRESENTATION: 

Use colored paper for the pencil 

measurements to help students see 

where their pencil paper lines up on 

the rulers. 

S: I should measure my pencil and see which half-inch or whole-inch mark is closest to the length of my 

strip. When I look at the ruler, I have to pay attention to the marks that split the inches into 2 

equal parts. Then look for the one that is closest to the length of my strip. I know that I will give 

a measurement that is either a whole number or a measurement that has a half in it. 



Date: 11/10/13 

4.A.5 





Discuss with students what they should do if their pencil strip is between two marks (e.g., 6 and 

students that any measurement that is more than halfway should be rounded up. 

). Remind 

T: Use your ruler to measure your strip to the nearest half inch. Record your measurement by placing 

an X on the picture of the ruler in Problem 2 on your Problem Set. 

T: Was the measurement to the nearest half inch accurate? Let’s find out. Raise your hand if your 

actual length was on or very close to one of the half-inch marking on your ruler. 

T: It seems that most of us had to round our measurement in order to mark it on the sheet. Let’s 

record everyone’s measurements on a line plot. As each person calls out his or her measurement, I’ll 

record on the board as you record on your sheet. (Poll the students.) 

A typical class line plot might look like this: 

T: Which pencil measurement is the most common, or frequent, in our class? Turn and talk. 

Answers will vary by class. In the plot above, 

inches is most frequent. 

T: Are all of the pencils used for these measurements exactly the same length? (Point to the X’s above 

the most frequent data point— inches on the exemplar line plot.) Are they exactly inches long? 

MP.5 

S: No, these measurements are to the nearest half inch. The pencils are different sizes. We had to 

round the measurement of some of them. My partner and I had pencils that were different 

lengths, but they were close to the same mark. We had to put our marks on the same place on the 

sheet even though they weren’t really the same length. 

T: Now let’s measure our strips to the nearest quarter inch. How is measuring to the quarter inch 

different from measuring to the half inch? Turn and talk. 

S: The whole is divided into 4 equal parts instead of just 2 equal parts. Quarter inches are smaller 

than half inches. Measuring to the nearest quarter inch gives us more choices about where to put 

our X’s on the ruler. 

Follow the same sequence of measuring and recording the strips to the nearest quarter inch. Your line plot 

might look something like this: 

T: Which pencil measurement is the most frequent this time? 

Answers will vary by class. The most frequent above is 

inches. 

T: If the length of our strips didn’t change, why is the most frequent measurement different this time? 



Date: 11/10/13 

4.A.6 





Turn and talk. 

S: The unit on the ruler we used to measure and record 

was different. The smaller units made it possible for 

me to get closer to the real length of my strip. I 

rounded to the nearest quarter inch so I had to move 

my X to a different mark on the ruler. Other people 

probably had to do the same thing. 

T: Yes, the ruler with smaller units (every quarter inch 

instead of every half inch) allowed us to be more precise 

NOTES ON 



Write math vocabulary words on 

sentence strips and display as they are 

used in context (e.g., precise, accurate). 

with our measurement. This ruler (point to the plot on the board) has more fractional units in a 

given length, which allows for a more precise measurement. It’s a bit like when we round a number 

by hundreds or tens. Which rounded number will be closer to the actual? Why? Turn and talk. 

S: When we round to the tens place we can be closer to the actual number, because we are using 

smaller units. 

T: That's exactly what's happening here when we measure to the nearest quarter inch versus the 

nearest half inch. How did your measurements either change or not change? 

S: My first was 4 inches but my second was closer to 4 and quarter. My first measurement was 4 

and a half inches. My second was 4 and 2 quarter inches, but that’s the same as 4 and a half inches. 

When I measured with the half-inch ruler, my first was closer to 4 inches than to 3 and a half 

inches, but when I measured with the fourth-inch ruler, it was closer to 3 and 3 quarter inches, 

because it was a little closer to 3 and 3 quarter inches than 4 inches. 

T: Our next task is to measure our strips to the nearest eighth of an inch and record our data in a third 

line plot. Look at the first two line plots. What do you think the shape of the third line plot will look 

like? Turn and talk. 

S: The line plot will be flatter than the first two. There are more choices for our measurements on 

the ruler, so I think that there will be more places where there will only be one X than on the other 

rulers. The eighth-inch ruler will show the differences between pencil lengths more than the halfinch 

or fourth-inch rulers. 

Follow the sequence above for measuring and recording line plots. 

T: Let’s find out how accurate our measurements are. Raise your hand if your actual strip length was on 

or very close to one of the eighth-inch markings on the ruler. (It is likely that many more students will 

raise their hands than before.) 

S: (Raise hands.) 

T: Work with your partner to answer Problem 5 on your Problem Set. 

You may want to copy down the line plots on the board for later analysis with your class. 

Problem Set (10 minutes) 

Students should do their personal best to complete the Problem Set within the allotted 10 minutes. Some 

problems do not specify a method for solving. This is an intentional reduction of scaffolding that invokes 

MP.5, Use Appropriate Tools Strategically. Students should solve these problems using the RDW approach 

used for Application Problems. 



Date: 11/10/13 

4.A.7 





For some classes, it may be appropriate to modify the 

assignment by specifying which problems students should 

work on first. With this option, let the careful sequencing 

of the Problem Set guide your selections so that problems 

continue to be scaffolded. Balance word problems with 

other problem types to ensure a range of practice. Assign 

incomplete problems for homework or at another time 

during the day. 

Student Debrief (10 minutes) 

Lesson Objective: Measure and compare pencil lengths to 

the nearest 1/2, 1/4, and 1/8 of an inch and analyze the 

data through line plots. 

The Student Debrief is intended to invite reflection and 

active processing of the total lesson experience. 

Invite students to review their solutions for the Problem 

Set. They should check work by comparing answers with a 

partner before going over answers as a class. Look for 

misconceptions or misunderstandings that can be 

addressed in the Debrief. Guide students in a conversation 

to debrief the Problem Set and process the lesson. 

You may choose to use any combination of the questions 

below to lead the discussion. However, it is recommended 

that the first bullet be a focus for this lesson’s discussion. 

• How many of you had a pencil length that didn’t 

fall directly on an inch, half-inch, quarter-inch, or 

eighth-inch marking? 

• If you wanted a more precise 

measurement of your pencil’s length, what 

could you do? (Guide student to see that 

they could choose smaller fractional units.) 

• When someone tells you, “My pencil is 5 

and 3 quarters inches long,” is it 

reasonable to assume that his or her 

pencil is exactly that long? (Guide 

students to see that in practice, all 

measurements are approximations, even 

though we assume they are exact for the 

sake of calculation.) 

• How does the most frequent pencil length change with each line plot? How does the number of 

each pencil length for each data point change with each line plot? Which line plot had the most 



Date: 11/10/13 

4.A.8 





repeated lengths? Which had the fewest repeated lengths? 

• What is the effect of changing the precision of the ruler? What happens when you split the wholes 

on the ruler into smaller and smaller units? 

• If all you know is the data from the second line plot, can you reconstruct the first line plot? (No. An 

X at inches on the second line plot could represent a pencil as short as inches or as long as 4 

inches in the first line plot. However, if an X is on a half-inch mark—3, , 4, , etc.—on the second 

line plot, then we know that it is at the same half-inch mark in the first line plot.) 

• Can the first line plot be completely reconstructed knowing only the data from the third line plot? 

(No, in general, but more of the first line plot can be reconstructed from the third than the second 

line plot.) 

• High-performing student accommodation: Which points on the third line plot can be used and which 

ones cannot be used to reconstruct the first line plot? 

• Which line plot contains the most accurate measurements? Why? Why are smaller units generally 

more accurate? 

• Are smaller units always the better choice when measuring? (Lead students to see that different 

applications require varying degrees of accuracy. Smaller units do allow for greater accuracy, but 

greater accuracy is not always required.) 

Exit Ticket (3 minutes) 

After the Student Debrief, instruct students to complete the Exit Ticket. A review of their work will help you 

assess the students’ understanding of the concepts that were presented in the lesson today and plan more 

effectively for future lessons. You may read the questions aloud to the students. 



Date: 11/10/13 

4.A.9 




NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 Problem Set 5•4 

Name 

Date 

1. Estimate the length of your pencil to the nearest inch. ______________ 

2. Using a ruler, measure your pencil strip to the nearest inch and mark the measurement with an X above 

the ruler below. Construct a line plot of your classmates’ pencil measurements. 







Date: 11/10/13 

4.A.10 





5. Use all three of your line plots to answer the following. 

a. Compare the three plots and write one sentence that describes how the plots are alike and one 

sentence that describes how they are different. 

b. What is the difference between the measurements of the longest and shortest pencils on each of the 

three line plots? 

c. Write a sentence describing how you could create a more precise ruler to measure your pencil strip. 



Date: 11/10/13 

4.A.11 




NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 Exit Ticket 5•4 

Name 

Date 

1. Draw a line plot for the following data measured in inches: 

2. Explain how you decided to divide your wholes into fractional parts, and how you decided where your 

number scale should begin and end. 



Date: 11/10/13 

4.A.12 




NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 Homework 5•4 

Name 

Date 

1. A meteorologist set up rain gauges at various locations around a city, and recorded the rainfall amounts 

in the table below. Use the data in the table to create a line plot using inches. 

a. Which location received the most rainfall? 

Location 

Rainfall Amount 

(inches) 

b. Which location received the least rainfall? 

c. Which rainfall measurement was the most frequent? 

d. What is the total rainfall in inches? 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 



Date: 11/10/13 

4.A.13 






G R A D E 

Topic B 

Fractions as Division 

5.NF.3 


Focus Standard: 5.NF.3 Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve 

word problems involving division of whole numbers leading to answers in the form of 

fractions or mixed numbers, e.g., by using visual fraction models or equations to 

represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, 

noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally 

among 4 people each person has a share of size 3/4. If 9 people want to share a 50‐ 

pound sack of rice equally by weight, how many pounds of rice should each person get? 

Between what two whole numbers does your answer lie? 



G4–M6 

Decimal Fractions 


Interpreting fractions as division is the focus of Topic B. Equal sharing with area models (both concrete and 

pictorial) gives students an opportunity to make sense of the division of whole numbers with answers in the 

form of fractions or mixed numbers (e.g., seven brownies shared by three girls, three pizzas shared by four 

people). Discussion also includes an interpretation of remainders as a fraction (5.NF.3). Tape diagrams 

provide a linear model of these problems. Moreover, students see that by renaming larger units in terms of 

smaller units, division resulting in a fraction is just like whole number division. 

Topic B continues as students solve real world problems (5.NF.3) and generate story contexts for visual 

models. The topic concludes with students making connections between models and equations while 

reasoning about their results (e.g., between what two whole numbers does the answer lie?). 

Topic B: Fractions as Division 

Date: 11/9/13 4.B.1 




NYS COMMON CORE MATHEMATICS CURRICULUM Topic B 5 

A Teaching Sequence Towards Mastery of Fractions as Division 

Objective 1: Interpret a fraction as division. 

(Lessons 2–3) 

Objective 2: Use tape diagrams to model fractions as division. 

(Lesson 4) 

Objective 3: Solve word problems involving the division of whole numbers with answers in the form of 

fractions or whole numbers. 

(Lesson 5) 

Topic B: Fractions as Division 

Date: 11/9/13 4.B.2 




NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 5 

Lesson 2 

Objective: Interpret a fraction as division. 






Total Time 

(8 minutes) 

(12 minutes) 

(30 minutes) 

(10 minutes) 

(60 minutes) 


The line plot shows the number of miles run by Noland in his PE class last month, rounded to the nearest 

quarter mile. 

X 

X 

X 

X X X 

X X X X 

0 1/4 1/2 3/4 1 

(miles) 

a. If Noland ran once a day, how many days did he run? 

b. How many miles did Noland run altogether last month? 

c. Look at the circled data point. The actual distance Noland ran that day was at least ____mile and less 

than ____mile. 

Note: This Application Problem reinforces the work of yesterday’s lesson. Part (c) provides an extension for 

early finishers. 


• Factors of 100 4.NF.5 


• Decompose Fractions 4.NF.3 

• Divide with Remainders 5.NF.3 

(2 minutes) 

(4 minutes) 

(3 minutes) 

(3 minutes) 

Lesson 2: Interpret a fraction as division. 

Date: 11/10/13 4.B.3 





Factors of 100 (2 minutes) 

Note: This fluency prepares students for fractions with denominators of 4, 20, 25, and 50 in G5–M4–Topic G. 

T: (Write 50 × ____ = 100.) Say the equation filling in the missing factor. 

S: 50 × 2 = 100. 

Continue with the following possible suggestions: 25 × ___= 100, 4 × ____ = 100, 20 × ____ = 100, 50 × ____ = 

100. 

T: I’m going to say a factor of 100. You say the other factor that will make 100. 

T: 20. 

S: 5. 

Continue with the following possible suggestions: 25, 50, 5, 10, and 4. 



Note: This fluency reviews Grade 4 and Grade 5─Module 3 

concepts. 

T: (Project a tape diagram partitioned into 2 equal parts. 

Shade 1 of the parts.) Say the fraction. 

S: 1 half. 

T: (Write to the right of the tape diagram. Directly 

below the first tape diagram, project another tape 

diagram partitioned into 4 equal parts. Shade 3 of the 

parts.) Say this fraction. 

S: 3 fourths. 

T: What’s a common unit that we could use to compare 

these fractions? 

S: Fourths. Eighths. Twelfths. 

NOTES ON 



For English language learners or 

students who need to review the 

relative size of fractional units, folding 

square paper into various units of 

halves, thirds, fourths, and eighths can 

be beneficial. Allow students time to 

fold, cut, label and compare the units 

in relation to the whole and each 

other. 

T: Let’s use fourths. (Below write ___ , write .) On your boards, write in the unknown 

numerator and a greater than or less than symbol. 

S: (Write .) 

Continue with the following possible suggestions, comparing and , and , and , and and . 

Decompose Fractions (4 minutes) 


Note: This fluency reviews Grade 4 and Grade 5–Module 3 concepts. 


Date: 11/10/13 4.B.4 





T: (Write a number bond with as the whole and 3 missing parts.) On your boards, break apart 3 fifths 

into unit fractions. 

S: (Write for each missing part.) 

T: Say the multiplication equation for this bond. 

S: 3 × = . 

Continue with the following possible suggestions: , , and . 

Divide with Remainders (4 minutes) 


Note: This fluency prepares students for this lesson’s Concept Development. 

T: (Write 8 ÷ 2 = .) Say the quotient. 

S: 4. 

T: Say the remainder. 

S: There isn’t one. 

T: (Write 9 ÷ 2 = .) Quotient? 

S: 4. 

T: Remainder? 

S: 1. 

Continue with the following possible suggestions: 25 ÷ 5, 27 ÷ 5, 9 ÷ 3, 10 ÷ 3, 16 ÷ 4, 19 ÷ 4, 12 ÷ 6, and 

11 ÷ 6. 


Materials: (S) Personal white boards, 15 square pieces of paper per pair of students 

Problem 1 

2 ÷ 2 

1 ÷ 2 

1 ÷ 3 

2 ÷ 3 

T: Imagine we have 2 crackers. Use two pieces of your paper to represent 

the crackers. Share the crackers equally between 2 people. 

S: (Distribute 1 cracker per person.) 

T: How many crackers did each person get? 


Date: 11/10/13 4.B.5 





MP.4 

S: 1 cracker. 

T: Say a division sentence that tells what you just did with the cracker. 

S: 2 ÷ 2 = 1. 

T: I’ll record that with a drawing. (Draw the 2 ÷ 2 = 1 image on 

the board.) 

T: Now imagine that there is only 1 cracker to share between 

2 people. Use your paper and scissors to show how you 

would share the cracker. 

S: (Cut one paper into halves.) 

T: How much will each person get? 

S: 1 half of a cracker. 

T: Work with your partner to write a number sentence that 

shows how you shared the cracker equally. 

S: 1 ÷ 2 = . ÷ 2 = . 2 halves ÷ 2 = 1 half. 

T: I’ll record your thinking on the board with another drawing. 

(Draw the 1 ÷ 2 drawing, and write the number sentence 

beneath it.) 

Repeat this sequence with 1 ÷ 3. 

T: (Point to both division sentences on the board.) Look 

at these two number sentences. What do you notice? 


S: Both problems start with 1 whole, but it gets divided 

into 2 parts in the first problem and 3 parts in the 

second one. I noticed that both of the answers are 

fractions, and the fractions have the same digits in 

them as the division expressions. When you share 

the same size whole with 2 people, you get more than 

when you share it with 3 people. The fraction looks 

a lot like the division expression, but it’s the amount 

that each person gets out of the whole. 

NOTES ON 


EXPRESSION: 

Students with fine motor deficits may 

find the folding and cutting of the 

concrete materials taxing. Consider 

allowing them to either serve as 

reporter for their learning group 

sharing the findings, or allowing them 

to use online virtual manipulatives. 

T: (Point to the number sentences.) We can write the division expression as a fraction. 1 divided by 2 

is the same as 1 half. 1 divided by 3 is the same as 1 third. 

T: Let’s consider sharing 2 crackers with 3 people. Thinking about 1 divided by 3, how much do you 

think each person would get? Turn and talk. 

S: It’s double the amount of crackers shared with the same number of people. Each person should get 

twice as much as before, so they should get 2 thirds. The division sentence can be written like a 

fraction, so 2 divided by 3 would be the same as 2 thirds. 

T: Use your materials to show how you would share 2 crackers with 3 people. 

S: (Work.) 


Date: 11/10/13 4.B.6 





Problem 2 

3 ÷ 2 

T: Now, let’s take 3 crackers and share them equally with 2 

people. (Draw 3 squares on the board, underneath the 

squares, draw 2 circles.) Turn and talk about how you can 

share these crackers. Use your materials to show your 

thinking. 

S: I have 3 crackers, so I can give 1 whole cracker to both 

people. Then I’ll just have to split the third cracker into 

halves and share it. Since there are 2 people, we could 

cut each cracker into 2 parts and then share them equally 

that way. 

T: Let’s record these ideas by drawing. We have 3 crackers. I 

heard someone say that there is enough for each person to 

get a whole cracker. Draw a whole cracker in each circle. 

S: (Draw.) 

T: How many crackers remain? 

S: 1 cracker. 

T: What must we do with the remaining cracker if we want to 

keep sharing equally? 

S: Divide it in 2 equal parts. Split it in half. 

T: Add that to your drawing. How many halves will each person 

get? 

S: 1 half. 

T: Record that by drawing one-half into each circle. How many crackers 

did each person receive? 

S: 1 and crackers. 

T: (Write 3 ÷ 2 = 1 beneath the drawing.) How many halves are in 1 and 1 half? 

S: 3 halves. 

T: (Write next to the equation.) I noticed that some of you cut the crackers in 2 equal parts before 

you began sharing. Let’s draw that way of sharing. (Re-draw 3 wholes. Divide them in halves 

horizontally.) How many halves were in 3 crackers? 

S: 6 halves. 

T: What’s 6 halves divided by 2? Draw it. 

S: 3 halves. 


Date: 11/10/13 4.B.7 





Problem 3 

4 ÷ 2 

5 ÷ 2 

T: Imagine 4 crackers shared with 2 people. How many would 

each person get? 

S: 2 crackers. 

T: (Write 4 ÷ 2 = 2 on the board.) Let’s now imagine that all four 

crackers are different flavors, and both people would like to 

taste all of the flavors. How could we share the crackers equally 

to make that possible? Turn and talk. 

S: To be sure everyone got a taste of all 4 crackers, we would 

need to split all the crackers in half first, and then share. 

T: How many halves would we have to share in all? How many 

would each person get? 

S: 8 halves in all. Each person would get 4 halves. 

T: Let me record that. (Write 8 halves ÷ 2 = 4 halves.) Although 

the crackers were shared in units of one-half, what is the total 

amount of crackers each person receives? 

S: 2 whole crackers. 

Follow the sequence above to discuss 5 2 using 5 crackers of the same 

flavor followed by 5 different flavored crackers. Discuss the two ways 

of sharing. 

T: (Point to the division equations that have been recorded.) Look at all the division problems we just 

solved. Talk to your neighbor about the patterns you see in the quotients. 

S: The numbers in the problems are the same as the numbers in the quotients. The division 

expressions can be written as fractions with the same digits. The numerators are the wholes that 

we shared. The denominators show how many equal parts we made. The numerators are like 

the dividends, and the denominators are like the divisors. Even the division symbol looks like a 

fraction. The dot on top could be a numerator and the dot on the bottom could be a denominator. 

T: Will this always be true? Let’s test a few. Since 1 divided by 4 equals 1 fourth, what is 1 divided by 

5? 

S: 1 fifth. 

T: (Write 1 ÷ 5 = .) What is 1 ÷ 7? 

S: 1 seventh. 

T: 3 divided by 7? 

S: 3 sevenths. 

T: Let’s try expressing fractions as division. Say a division expression that is equal to 3 eighths. 

S: 3 divided by 8. 


Date: 11/10/13 4.B.8 





T: 3 tenths? 


T: 3 hundredths? 



Students should do their personal best to complete the 

Problem Set within the allotted 10 minutes. For some 

classes, it may be appropriate to modify the assignment by 

specifying which problems they work on first. Some 

problems do not specify a method for solving. Students 

solve these problems using the RDW approach used for 

Application Problems. 


Lesson Objective: Interpret a fraction as division. 







addressed in the Debrief. Guide students in a 

conversation to debrief the Problem Set and process the 

lesson. 


below to lead the discussion. 

• What did you notice about Problems 4(a) and 

4(b)? What were the wholes, or dividends, and 

what were the divisors? 

• What was your strategy to solve Problem 1(c)? 

• What pattern did you notice between 1(b) and 

1(c)? What was the relationship between the 

size of the dividends and the quotients? 

• Discuss the division sentence for Problem 2. 

What number is the whole and what number is 

the divisor? How is the division sentence 

different from 2 ÷ 3? 

• Explain to your partner the two sharing approaches in Problem 3. (The first approach is to give each 


Date: 11/10/13 4.B.9 





girl 1 whole then partition the remaining bars. The second approach is to partition all 7 bars, 21 

thirds, and share the thirds equally.) When might one approach be more appropriate? (If the cereal 

bars were different flavors, and each person wanted to try each flavor.) 

• True or false? Dividing by 2 is the same as 

multiplying by . (If needed, revisit the fact that 3 

÷ 2 = = 3 × .) 


After the Student Debrief, instruct students to complete 

the Exit Ticket. A review of their work will help you assess 

the students’ understanding of the concepts that were 

presented in the lesson today and plan more effectively 

for future lessons. You may read the questions aloud to 

the students. 


Date: 11/10/13 4.B.10 





Name 

Date 

1. Draw a picture to show the division. Write a division expression using unit form. Then express your 

answer as a fraction. The first one is done for you. 

a. 1 ÷ 5 = 5 fifths ÷ 5 = 1 fifth = 

b. 3 ÷ 4 

c. 6 ÷ 4 


Date: 11/10/13 4.B.11 





2. Draw to show how 2 children can equally share 3 cookies. Write an equation and express your answer as 

a fraction. 

3. Carly and Gina read the following problem in their math class. 

Seven cereal bars were shared equally by 3 children. How much did each child receive? 

Carly and Gina solve the problem differently. Carly gives each child 2 whole cereal bars and then divides 

the remaining cereal bar between the 3 children. Gina divides all the cereal bars into thirds and shares 

the thirds equally among the 3 children. 

a. Illustrate both girls’ solutions. 

b. Explain why they are both right. 


Date: 11/10/13 4.B.12 





4. Fill in the blanks to make true number sentences. 

a. 2 ÷ 3 = b. 15 ÷ 8 = c. 11 ÷ 4 = 

d. = ______ ÷ ________ e. = ______÷ _______ f. = ________ ÷ _______ 


Date: 11/10/13 4.B.13 





Name 

Date 

1. Draw a picture that shows the division expression. Then write an equation and solve. 

a. 3 ÷ 9 b. 4 ÷ 3 


a. 21 ÷ 8 = b. = ______ ÷ _______ c. 4 ÷ 9 = d. = _____ ÷ ______ 


Date: 11/10/13 4.B.14 





Name 

Date 

1. Draw a picture to show the division. Express your answer as a fraction. 

a. 1 ÷ 4 

b. 3 ÷ 5 

c. 7 ÷ 4 

2. Using a picture, show how six people could share four sandwiches. Then write an equation and solve. 


Date: 11/10/13 4.B.15 






a. 2 ÷ 7 = b. 39 ÷ 5 = c. 13 ÷ 3 = 

d. = ________ ÷ ________ e. = _______÷ ______ f. = ______ ÷ ________ 


Date: 11/10/13 4.B.16 





Lesson 3 

Objective: Interpret a fraction as division. 






Total Time 

(12 minutes) 

(5 minutes) 

(33 minutes) 

(10 minutes) 

(60 minutes) 


• Convert to Hundredths 4.NF.5 


• Fractions as Division 5.NF.3 

• Write Fractions as Decimals 4.NF.5 

(3 minutes) 

(4 minutes) 

(3 minutes) 

(2 minutes) 

Convert to Hundredths (3 minutes) 


Note: This fluency prepares students for decimal fractions later in the module. 

T: I’ll say a factor, and then you’ll say the factor you need to multiply it by to get 100. 50. 

S: 2. 

Continue with the following possible suggestions: 25, 20, and 4. 

T: (Write = .) How many fours are in 100? 

S: 25. 

T: Write the equivalent fraction. 

S: (Write = .) 

Continue with the following possible suggestions: = , = , = , = , = , = , and 

= . 


Date: 11/10/13 4.B.17 







Note: This fluency reviews Grade 4 and Grade 5–Module 3 concepts. 

T: (Write _ .) Write a greater than or less than symbol. 

S: (Write > .) 

T: Why is this true? 

S: Both have 1 unit, but halves are larger than sixths. 

Continue with the following possible suggestions: a a a a a 

Students should be able to reason about these comparisons without the need for common units. Reasoning 

such as greater or less than half or the same number of different sized units should be the focus. 

Fractions as Division (3 minutes) 


Note: This fluency reviews G5–M4–Lesson 2 content. 

T: (Write 1 ÷ 3.) Write a complete number sentence using the expression. 

S: (Write 1 ÷ 3 = .) 

Continue with the following possible sequence: 1 ÷ 4 and 2 ÷ 3. 

T: (Write 5 ÷ 2.) Write a complete number sentence using the expression. 

S: (Write 5 ÷ 2 = or 5 ÷ 2 = .) 

Continue with the following possible suggestions: 13 ÷ 5, 7 ÷ 6, and 17 ÷ 4. 

T: (Write .) Say the fraction. 

S: 4 thirds. 

T: Write a complete number sentence using the fraction. 

S: (Write 4 ÷ 3 = or 4 ÷ 3 = 1 .) 

Continue with the following possible suggestions: a . 

Write Fractions as Decimals (2 minutes) 

Note: This fluency prepares students for fractions with denominators of 4, 20, 25, and 50 in G5–M4–Topic G. 


S: 1 tenth. 

T: Say it as a decimal. 


Date: 11/10/13 4.B.18 





S: 0.1. 

Continue with the following possible suggestions: 

a 

T: (Write 0.1 =___.) Write the decimal as a fraction. 

S: (Write 0.1 = .) 

Continue with the following possible suggestions: 0.2, 0.4, 0.8, and 0.6. 


Hudson is choosing a seat in art class. He scans the 

room and sees a 4-person table with 1 bucket of art 

supplies, a 6‐perso table with 2 buckets of supplies, 

and a 5‐perso table with 2 buckets of supplies. 

Which table should Hudson choose if he wants the 

largest share of art supplies? Support your answer 

with pictures. 

Note: Students must first use division to see which fractional portion of art supplies is available at each table. 

Then students compare the fractions and find which represents the largest value. 



Problem 1 

A baker poured 4 kilograms of oats equally into 3 bags. What is 

the weight of each bag of oats? 

T: In our story, which operation will be needed to find 

how much each bag of oats weighs? 

S: Division. 

T: Turn and discuss with your partner how you know and 

what the division sentence would be. 

S: The total is 4 kg of oats being divided into 3 bags, so 

the division sentence is 4 divided by 3. The whole is 

4, and the divisor is 3. 

T: Say the division expression. 

S: 4 ÷ 3. 

T: (Write 4 ÷ 3 and draw 4 squares on the board.) Let’s 

represent the kilograms with squares like we used 

yesterday. They are easier to cut into equal shares 


Date: 11/10/13 4.B.19 





than circles. 

T: Tur a talk about how you’ll share the 4 kg of oats equally in 3 bags. Draw a picture to show your 

thinking. 

S: Every bag will get a whole kilogram of oats, and then we will split the last kilogram equally into 3 

thirds to share. So, each bag gets a whole kilogram and one-third of another one. I can cut all 

four kilograms into thirds. Then split them into the 3 bags. Each bag will get 4 thirds of kg. I 

know the answer is four over three because that is just another way to write 4 ÷ 3. 

T: As we saw yesterday, there are two ways of dividing the oats. Let me record your approaches. 

(Draw the approaches on the board and restate.) Let’s say the ivisio sentence with the quotient. 

S: 4 ÷ 3 = 4 thirds. 4 ÷ 3 = 1 and 1 third. 

T: (Point to the diagram on board.) When we cut them all into thirds and shared, how many thirds, in 

all, did we have to share? 

S: 12 thirds. 

T: Say the division sentence in unit form starting with 12 thirds. 

S: 12 thirds ÷ 3 = 4 thirds. 

T: (Write 12 thirds ÷ 3 = 4 thirds on the board.) What is 4 thirds as a mixed number? 

S: 1 and 1 third. 

T: (Write algorithm on board.) Let’s show how we divided the oats using the division algorithm. 

T: How many groups of 3 can I make with 4 kilograms? 

S: 1 group of three. 

T: (Record 1 in the quotient.) What’s 1 group of three? 

S: 3. 

T: (Record 3 under 4.) How many whole kilograms are left 

to share? 

S: 1. 

T: What did we do with this last kilogram? Turn and 

discuss with your partner. 

S: This one remaining kilogram was split into 3 equal 

parts to keep sharing it. I had to split the last 

kilogram into thirds to share it equally. The 

quotient is 1 whole kilogram and the remainder is 1. The quotient is 1 whole kilogram and 1 third 

kilogram. Each of the 3 bags get 1 and 1 third kilogram of oats. 

T: Let’s recor what you sai . (Point to the remainder of 1.) This remainder is 1 left over kilogram. To 

keep sharing it, we split it into 3 parts (point to the divisor), so each bag gets 1 third. I’ll write 1 third 

next to the 1 in the quotient. (Write next to the quotient of 1.) 

T: Use the quotient to answer the question. 

S: Each bag of oats weighs 1 kilograms. 

T: Let’s check our answer. How can we know if we put the right amount of oats in each bag? 

S: We can total up the 3 parts that we put in each bag when we divided the kilograms. The total we 


Date: 11/10/13 4.B.20 





get should be the same as our whole. The sum of the equal parts should be the same as our 

dividend. 

T: We have 3 groups of 1 . Say the multiplication sentence. 

S: 3 × 1 . 

T: Express 3 copies of using repeated addition. 

S: . 

T: Is the total the same number of kilograms we had before we shared? 

S: The total is 4 kilograms. It is the same as our whole before we shared. 3 ones plus 3 thirds is 3 

plus 1. That’s 4. 

T: We’ve see more tha o e way to write ow how to share 4 kilograms in 3 bags. Why is the 

quotient the same using the algorithm? 

S: The same thing is happening to the flour. It is being divided into 3 parts. We are just using 

another way to write it. 

T: Let’s use different strategies in our next problem as well. 

Problem 2 

If the baker doubles the number of kilograms of oats to be poured 

equally into 3 bags, what is the weight of each bag of oats? 

T: What’s the whole i this problem? Turn and share with 

your partner. 

S: 4 doubled is 8. 4 times 2 is 8. The baker now has 8 

kilograms of oats to pour into 3 bags. 

T: Say the whole. 

S: 8. 

T: Say the divisor. 

S: 3. 

T: Say the division expression for this problem. 

S: 8 ÷ 3. 

T: Compare this expression with the one we just did. What 

do you notice? 

S: The whole is twice as much as the problem before. 

The number of shares is the same. 

T: Using that insight, make a prediction about the quotient 

of this problem. 

S: Since the whole is twice as much shared with the same 

number of bags, then the answer should be twice as 

much as the answer to the last problem. Two times 

4 thirds is equal to 8 thirds. The answer should be 


Date: 11/10/13 4.B.21 





double. So it should be 1 + 1 and that is . 

T: Work with your partner to solve, and confirm the predictions you made. Each partner should use a 

different strategy for sharing the kilograms and draw a picture 

of his or her thinking. Then, work together to solve using the 

standard algorithm. 

Circulate as students work. 

T: How many kilograms are in each bag this time? Whisper and 

tell your partner. 

S: Each bag gets 2 whole kilograms and of another one. 

Each bag gets a third of each kilogram which would be 8 

thirds. 8 thirds is the same as 

kilograms. 

T: If we split all the kilograms into thirds before we share, how 

many thirds are in all 8 kilograms? 

S: 24 thirds. 

T: Say the division sentence in unit form. 

S: 24 thirds ÷ 3 = 8 thirds. 

T: (Set up the standard algorithm on the board and solve 

it together.) The quotient is 2 wholes and 2 thirds. 

Use the quotient to answer the question. 

S: Each bag of oats weighs kilograms. 

T: Let’s ow check it. Say the addition sentence for 3 

groups of . 

S: . 

T: So, 8 ÷ 3 = . How does this quotient compare with 

our predictions? 

S: This answer is what we thought it would be. It was 

double the last quotient which is what we predicted. 

T: Great. Let’s ow cha ge our whole one more time 

and see how it affects the quotient. 

Problem 3 

If the baker doubles the number of kilograms of oats again and 

they are poured equally into 3 bags, what is the weight of each 

bag of oats? 

Repeat the process used in Problem 2. When predicting the 

quotient, be sure students notice that this equation is two 

times as much as Problem 1, and four times as much as 

Problem 2. This is important for the scaling interpretation of multiplication. 

NOTES ON 



For those students who need the 

support of concrete materials, continue 

to use square paper and scissors to 

represent the equal shares along with 

the pictorial and abstract 

representations. 


Date: 11/10/13 4.B.22 





The closing extension of the dialogue, in which students realize the efficiency of the algorithm, is detailed 

below. 

T: Say the division expression for this problem. 

S: 16 ÷ 3. 

T: Say the answer as a fraction greater than 1. 

S: 16 thirds. 

T: What would be an easy strategy to solve this problem? Draw out 16 wholes to split into 3 groups, or 

use the standard algorithm? Turn and discuss with a partner. 

S: (Share.) 

T: Solve this problem independently using the standard algorithm. You may also draw if you like. 

S: (Work.) 

T: Let’s solve usi g the sta ar algorithm. (Set up sta ar algorithm a solve o the boar .) What 

is 16 thirds as a mixed number? 

S: 5 . 

T: Use the quotient to answer the question. 

S: Each bag of oats weighs 5 kilograms. 

T: Let’s check with repeate a itio . Say the e tire 

addition sentence. 

S: 5 5 5 16. 

T: So 16 ÷ 3 = 5 . How does this quotient compare with 

our predictions? 

S: This answer is what we thought it would be. It was 

quadruple the first quotient. We were right; it was 

double of the last quotient, which is what we 

predicted. 


NOTES ON 

MULTIPLE MEANS 

OF REPRESENTATION: 

Fractions are generally represented in 

student materials using equation 

editing software using horizontal line to 

separate numerator from denominator 

(e.g., ). However, it may be wise to 

expose students to other formats of 

notating fractions, such as those which 

use a diagonal to separate numerator 

from denominator (e.g., ⁄ or ). 

Students should do their personal best to complete the Problem Set within the allotted 10 minutes. For 

some classes, it may be appropriate to modify the assignment by specifying which problems they work on 

first. Some problems do not specify a method for solving. Students solve these problems using the RDW 

approach used for Application Problems. 


Lesson Objective: Interpret a fraction as division. 

The Student Debrief is intended to invite reflection and active processing of the total lesson experience. 

Invite students to review their solutions for the Problem Set. They should check work by comparing answers 


Date: 11/10/13 4.B.23 





with a partner before going over answers as a class. Look for misconceptions or misunderstandings that can 

be addressed in the Debrief. Guide students in a conversation to debrief the Problem Set and process the 

lesson. 

You may choose to use any combination of the questions below to lead the discussion. 

• What pattern did you notice between Problems 

1(b) and 1(c)? Look at the whole and the divisor. 

Is 3 halves greater than, less than, or equal to 6 

fourths? What about the answers? 

• What’s the relationship between the answers for 

Problems 2(a) and 2(b)? Explain it to your 

partner. ( Students should note that Problem 

2(b) is four times as much as Problem 2(a).) Can 

you generate a problem where the answer is the 

same as Problem (a), or the same as Problem (b)? 

• Explain to your partner how you solved for 

Problem 3(a)? Why do we need one more 

warming box than our actual quotient? 

• We expressed our remainders today as fractions. 

Compare this with the way we expressed our 

remainders as decimals in Module 2. How is it 

alike? How is it different? 




the stu e ts’ u ersta i g of the co cepts that were 

presented in the lesson today and plan more effectively for 

future lessons. You may read the questions aloud to the 

students. 


Date: 11/10/13 4.B.24 





Name 

Date 

1. Fill in the chart. The first one is done for you. 

Division 

Expression 

Unit Forms 

Improper 

Fraction 

Mixed 

Numbers 

Standard Algorithm 

(Write your answer in whole numbers and 

fractional units, then check.) 

a. 5 ÷ 4 

20 fourths ÷ 4 

= 5 fourths 

 

1 

4 5 

- 4 

1 

Check 

: 

4 × = + + + 

= 4 + 

= 4 + 1 

= 5 

___ halves ÷ 2 

b. 3 ÷ 2 

= ___ halves 

c. ___ ÷ ___ 

24 fourths ÷ 4 

= 6 fourths 

4 6 

d. 5 ÷ 2 


Date: 11/10/13 4.B.25 





2. A principal evenly distributes 6 reams of copy paper to 8 fifth-grade teachers. 

a. How many reams of paper does each fifth-grade teacher receive? Explain how you know using 

pictures, words, or numbers. 

b. If there were twice as many reams of paper and half as many teachers, how would the amount each 

teacher receives change? Explain how you know using pictures, words, or numbers. 

3. A caterer has prepared 16 trays of hot food for an event. The trays are placed in warming boxes for 

delivery. Each box can hold 5 trays of food. 

a. How many warming boxes are necessary for delivery if the caterer wants to use as few boxes as 

possible? Explain how you know. 

b. If the caterer fills a box completely before filling the next box, what fraction of the last box will be 

empty? 


Date: 11/10/13 4.B.26 





Name 

Date 

1. A baker made 9 cupcakes, each a different type. Four people want to share them equally. How many 

cupcakes will each person get? 

Fill in the chart to show how to solve the problem. 

Division 

Expression 

Unit Forms 

Fractions and 

Mixed numbers 


Draw to show your thinking: 


Date: 11/10/13 4.B.27 





Name 

Date 


Division 

Expression 

Unit Forms 

Improper 

Fractions 

Mixed 

Numbers 


(Write your answer in whole numbers 

and fractional units, then check.) 

a. 4 ÷ 3 

12 thirds ÷ 3 

= 4 thirds 

1 

3 4 

- 3 

1 

Check 

: 

3 × = + + 

= 3 + 

= 3 + 1 

= 4 

___ fifths ÷ 5 

b. ___ ÷ ___ 

= ___ fifths 

c. ___ ÷ ___ 

___ halves ÷ 2 

= ___ halves 

2 7 

d. 7 ÷ 4 


Date: 11/10/13 4.B.28 





2. A coffee shop uses 4 liters of milk every day. 

a. If they have 15 liters of milk in the refrigerator, after how many days will they need to purchase 

more? Explain how you know. 

b. If they only use half as much milk each day, after how many days will they need to purchase more? 

3. Polly buys 14 cupcakes for a party. The bakery puts them into boxes that hold 4 cupcakes each. 

a. How many boxes will be needed for Polly to bring all the cupcakes to the party? Explain how you 

know. 

b. If the bakery completely fills as many boxes as possible, what fraction of the last box is empty? How 

many more cupcakes are needed to fill this box? 


Date: 11/10/13 4.B.29 





Lesson 4 

Objective: Use tape diagrams to model fractions as division. 






Total Time 

(12 minutes) 

(7 minutes) 

(31 minutes) 

(10 minutes) 

(60 minutes) 





(4 minutes) 

(4 minutes) 

(4 minutes) 


Note: This fluency prepares students for G5–M4–Topic G. 


S: 1 tenth. 


S: Zero point one. 

Continue with the following possible suggestions: , , , and . 

T: (Write = ____.) Say the fraction. 

S: 1 hundredth. 


S: Zero point zero one. 


T: (Write 0.01 = ____.) Say it as a fraction. 


T: (Write 0.01 = .) 

Lesson 4: Use tape diagrams to model fractions as division. 

Date: 11/10/13 4.B.30 








Note: This fluency prepares students for G5–M4–Topic G. 

T: (Write: = .) Write the equivalent fraction. 


T: (Write = = ____.) Write 1 fourth as a decimal. 

S: (Write = = 0.25.) 

Continue with the following possible suggestions: , , , , , , , , , , and . 



Note: This fluency reviews G5–M6–Lessons 2 and 3 content. 

T: (Write 1 ÷ 2.) Solve. 

S: (Write 1 ÷ 2 = .) 

Continue with the following possible sequence: 1 ÷ 5 and 3 ÷ 4. 

T: (Write 7 ÷ 2.) Solve. 

S: (Write 7 ÷ 2 = or 7 ÷ 2 = .) 

Continue with the following possible suggestions: 12 ÷ 5, 11 ÷ 

6, 19 ÷ 4, 31 ÷ 8, and 49 ÷ 9. 

T: (Write .) Write the fraction as a whole number 

division expression. 

S: (Write 5 ÷ 3.) 

NOTES ON 


ACTION AND 

EXPRESSION: 

If students are comfortable with 

interpreting fractions as division, 

consider foregoing the written 

component of this fluency and ask 

students to visualize the fractions, 

making this activity more abstract. 

Continue with the following possible suggestions: , , 

and , 37 ÷ 8, and 40 ÷ 9. 


Four grade-levels need equal time for indoor recess, and the 

gym is available for three hours. 


Date: 11/10/13 4.B.31 





a. How many hours of recess will each grade level receive? Draw a picture to support your answer. 

b. How many minutes? 

c. If the gym could accommodate two grade-levels at once, how many hours of recess would each 

grade-level get? 

Note: Students practice division with fractional quotients, which leads into the day’s lesson. Note that the 

whole remains constant in (c) while the divisor is cut in half. Lead students to analyze the effect of this halving 

on the quotient as related to the doubling of the whole from previous problems. 



Problem 1 

Eight tons of gravel is equally divided between 4 dump trucks. How much gravel is in one dump truck? 

T: Say a division sentence to solve the problem. 

S: 8 ÷ 4 = 2. 

T: Model this problem with a tape diagram. (Pause as 

students work.) 

T: We know that 4 units are equal to 8 tons. (Write 4 

units = 8.) We want to find what 1 unit is equal to. 

T: (Write 1 unit = 8 ÷ 4.) 

T: How many tons of gravel is in one dump truck? 

S: 2. 

T: Use your quotient to answer the question. 

S: Each dump truck held 2 tons of gravel. 

Problem 2 

Five tons of gravel is equally divided between 4 dump trucks. How much gravel is in one dump truck? 

T: (Change values from previous problem to 5 tons and 

4 trucks on board.) How would our drawing be 

different if we had 5 tons of gravel? 

S: Our whole would be different, 5 and not 8. The 

tape diagram is the same except for the value of the 

whole. We’ll still partition it into fourths, because 

there are still 4 trucks. 

T: (Partition a new bar into 4 equal parts labeled with 5 

as the whole.) 

T: We know that these 4 units are equal to 5 tons. 

(Write 4 units = 5.) We want to find what 1 unit is 


Date: 11/10/13 4.B.32 





equal to. (Write a question mark beneath 1 fourth of the bar.) What is the division expression you’ll 

use to find what 1 unit is? 

S: 5 ÷ 4. 

T: (Write 1 unit = 5 ÷ 4.) 5 ÷ 4 is? 

S: Five-fourths. 

T: So each unit is equal to five-fourths tons of gravel. Can 

we prove this using the standard algorithm? 

T: What is 5 ÷ 4? 

S: One and one-fourth. 

T: (Write 5 ÷ 4 = .) Use your quotient to answer the 

question. 

S: Each dump truck held one and one-fourth tons of 

gravel. 

T: Visualize a number line. Between which two adjacent 

whole numbers is 1 and one-fourth? 

S: 1 and 2. 

T: Check our work using repeated addition. 

Problem 3 

NOTES ON 


ACTION AND 

EXPRESSION: 

Provide number lines with fractional 

markings for the students who still need 

support to visualize the placement of 

the fractions. 

A 3 meter ribbon is cut into 4 equal pieces to make flowers. What is the length of each piece? 

T: (Write 3 ÷ 4 on the board.) Work with a partner and draw a tape diagram to solve. 

T: Say the division expression you solved. 


T: Say the answer as a fraction. 

S: Three-fourths. 

T: (Write on the board.) In this case, does it make 

sense to use the standard algorithm to solve? Turn 

and talk. 

S: No, it’s just divided by 4, which is . I don’t 

think so. It’s really easy. We could, but the 

quotient of zero looks strange to me. It’s just 

easier to say 3 divided by 4 equals 3 fourths. 


S: Each piece of ribbon is m long. 

T: Let’s check the answer. Say the multiplication expression starting with 4. 

S: 4 × . . 3. 

T: Our answer is correct. If we wanted to place our quotient of on a number line, between what two 


Date: 11/10/13 4.B.33 





adjacent whole numbers would we place it? 

S: 0 and 1. 

Problem 4 

14 gallons of water is used to completely fill 3 fish tanks. If each tank holds the same amount of water, how 

many gallons will each tank hold? 

T: Let’s read this problem together. (All read.) Work 

with a partner to solve this problem. Draw a tape 

diagram and solve using the standard algorithm. 

T: Say the division equation you solved? 

S: 14 ÷ 3 = . 

T: Say the quotient as a mixed number? 

S: . 


S: The volume of each fish tank is gallons. 

T: So, between which two adjacent whole numbers 

does our answer lie? 

S: Between 4 and 5. 

T: Check your answer with multiplication. 

S: (Check their answers.) 










Lesson Objective: Use tape diagrams to model fractions 

as division. 






Date: 11/10/13 4.B.34 









lesson. 



• What pattern did you notice between Problem 

1(a) and Problems 1(b), 1(c), and 1(d)? What did 

you notice about the wholes or dividends and the 

divisors? 

• In Problem 2(c), can you name the fraction of 

using a larger fractional unit? In other words, can 

you simplify it? Is this the same point on the 

number line? 

• Compare Problems 3 and 4. What’s the division 

sentence for both problems? What’s the whole 

and divisor for each problem? (Problem ’s 

division expression is 4 ÷ 5, and Problem ’s 

division expression is 5 ÷ 4.) 

• Explain to your partner the difference between 

the questions asked in Problem 4(a) and 4(b). 

(Problem 4(a) is asking the fraction of the 

birdseeds, which is one-fourth and 4(b) is asking 

the number of pounds of birdseeds which is 1 

and one-fourth.) 

• How was our learning today built on what we 

learned yesterday? (Students may point out that 

the models used today were more abstract than 

the concrete materials of previous days or that 

they were able to see the fractions as division 

more easily as equations than in days previous.) 






for future lesson. You may read the questions aloud to the 

students. 


Date: 11/10/13 4.B.35 




NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 Problem Set 5 

Name 

Date 

1. Draw a tape diagram to solve. Express your answer as a fraction. Show the multiplication sentence to 

check your answer. The first one is done for you. 

a. 1 ÷ 3 = 

1 

0 

3 1 

- 0 

1 

Check: 

3 × 3 

= + + 3 3 3 

= 3 3 

? 

3 units = 1 

1 unit = 1 ÷ 3 

= 

= 

b. 2 ÷ 3 = 

c. 7 ÷ 5 = 

d. 14 ÷ 5 = 


Date: 11/10/13 4.B.36 






Division Expression 

Fraction 

Between which two 

whole numbers is your 

answer? 


a. 13 ÷ 3 

3 

3 

4 and 5 

4 

3 13 

-12 

1 

b. 6 ÷ 7 0 and 1 

7 6 

c. _____÷_____ 

d. ____÷_____ 

3 

40 32 


Date: 11/10/13 4.B.37 





3. Greg spent $4 on 5 packs of sport cards. 

a. How much did Greg spend on each pack? 

b. If Greg spent half as much money, and bought twice as many packs of cards, how much did he spend 

on each pack? Explain your thinking. 

4. Five pounds of birdseed is used to fill 4 identical bird feeders. 

a. What fraction of the birdseed will be needed to fill each feeder? 

b. How many pounds of birdseed are used to fill each feeder? Draw a tape diagram to show your 

thinking. 

c. How many ounces of birdseed are used to fill three birdfeeders? 


Date: 11/10/13 4.B.38 




NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 Exit Ticket 5 

Name 

Date 

Matthew and his 3 siblings are weeding a flower bed with an area of 9 square yards. If they share the job 

equally, how many square yards of the flower bed will each child need to weed? Use a tape diagram to show 

your thinking. 


Date: 11/10/13 4.B.39 




NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 Homework 5 

Name 

Date 

1. Draw a tape diagram to solve. Express your answer as a fraction. Show the addition sentence to support 

your answer. The first one is done for you. 

a. 1 ÷ 4 = 

1 

Check: 

× 

? 4 units = 1 

0 

4 1 

- 0 

1 

= + + + 

= 

1 unit = 1 ÷ 4 

= 

= 

b. 4 ÷ 5 = 

c. 8 ÷ 5 = 

d. 14 ÷ 3 = 


Date: 11/10/13 4.B.40 






Division Expression 

Fraction 

Between which two 

whole numbers is your 

answer? 


a. 16 ÷ 5 3 and 4 

3 

5 16 

-15 

1 

b. _____÷_____ 

3 

0 and 1 

c. _____÷_____ 

2 7 

d. _____÷_____ 


Date: 11/10/13 4.B.41 





3. Jackie cut a 2-yard spool into 5 equal lengths of ribbon. 

a. How long is each piece of ribbon? Draw a tape diagram to show your thinking. 

b. What is the length of each ribbon in feet? Draw a tape diagram to show your thinking. 

4. Baa Baa the black sheep had 7 pounds of wool. If he separated the wool into 3 bags, each holding the 

same amount of wool, how much wool would be in 2 bags? 

5. An adult sweater is made from 2 pounds of wool. This is 3 times as much wool as it takes to make a baby 

sweater. How much wool does it take to make a baby sweater? Use a tape diagram to solve. 


Date: 11/10/13 4.B.42 





Lesson 5 

Objective: Solve word problems involving the division of whole numbers 

with answers in the form of fractions or whole numbers. 





Total Time 

(12 minutes) 

(38 minutes) 

(10 minutes) 

(60 minutes) 


• Fraction of a Set 4.NF.4 

• Write Division Sentences as Fractions 5.NF.3 

• Write Fractions as Mixed Numbers 5.NF.3 

(4 minutes) 

(3 minutes) 

(5 minutes) 

Fraction of a Set (4 minutes) 


Note: This fluency prepares students for G5–M4–Lesson 6. 

T: (Write 10 × ) 10 copies of one-half is…? 

S: 5. 

T: (Write 10 × .) 10 copies of one-fifth is…? 

S: 2. 

Continue with the following possible sequence: 8 × , 8 × , 6 × , 30 × , 42 × , 42 × , 48 × , 54 × , and 

54 × . 

Write Division Sentences as Fractions (3 minutes) 


Note: This fluency reviews G5–M4–Lesson 4. 

T: (Write 9 ÷ 30 = ____.) Write the quotient as a fraction. 

Lesson 5: Solve word problems involving the division of whole numbers with 

answers in the form of fractions or whole numbers. 

Date: 11/9/13 

4.B.43 





S: (Write 9 ÷ 30 = ) 

T: Write it as a decimal. 

S: (Write 9 ÷ 30 = = 0.3.) 

Continue with the following possible suggestions: 28 ÷ 40, 18 ÷ 60, 63 ÷ 70, 24 ÷ 80, and 63 ÷ 90. 

Write Fractions as Mixed Numbers (5 minutes) 



T: (Write = ____ ÷ ____ = ____.) Write the fraction as a division problem and mixed number. 

S: (Write = 13 ÷ 2 = .) 

Continue with the following possible suggestions: , , , , , , , , , , , , , and . 


Materials: (S) Problem Set 

Suggested Delivery of Instruction for Solving Lesson 5’s Word Problems 

1. Model the problem. 

Have two pairs of students who can successfully model the 

problem work at the board while the others work 

independently or in pairs at their seats. Review the following 

questions before beginning the first problem: 

• Can you draw something? 

• What can you draw? 

• What conclusions can you make from your drawing? 

As students work, circulate. Reiterate the questions above. 

After two minutes, have the two pairs of students share only 

their labeled diagrams. For about one minute, have the 

demonstrating students receive and respond to feedback and 

questions from their peers. 

NOTES ON 

MULTIPLE MEANS 

OF ENGAGEMENT: 

Appropriate scaffolds help all students 

feel successful. Students may use 

translators, interpreters, or sentence 

frames to present their solutions and 

respond to feedback. Models shared 

may include concrete manipulatives. 

If the pace of the lesson is a 

consideration, allow presenters to 

prepare beforehand. 

2. Calculate to solve and write a statement. 

Give everyone two minutes to finish work on that question, sharing their work and thinking with a peer. All 

should write their equations and statements of the answer. 



Date: 11/9/13 

4.B.44 





3. Assess the solution for reasonableness. 

Give students one to two minutes to assess and explain the reasonableness of their solution. 

Problem 1 

A total of 2 yards of fabric is used to make 5 identical pillows. How much fabric is used for each pillow? 

This problem requires understanding of the whole and the divisor. The whole of 2 is divided by 5, which 

results in a quotient of 2 fifths. Circulate, looking for different visuals (tape diagram and the region models 

from G5–M4–Lessons 2–3) to facilitate a discussion as to how these different models support the solution of 

. 

Problem 2 

An ice-cream shop uses 4 pints of ice cream to make 6 sundaes. How many pints of ice cream are used for 

each sundae? 

This problem also requires the students’ understanding of the whole versus the divisor. The whole is 4, and it 

is divided equally into 6 units with the solution of 4 sixths. Students should not have to use the standard 

algorithm to solve, because they should be comfortable interpreting the division expression as a fraction and 

vice versa. Circulate, looking for alternate modeling strategies that can be quickly mentioned or explored 

more deeply, if desired. Students might express 4 sixths as 2 thirds. The tape diagram illustrates that larger 

units of 2 can be made. Quickly model a tape with 6 parts (now representing 1 pint), shade 4, and circle sets 

of 2. 



Date: 11/9/13 

4.B.45 





Problem 3 

An ice-cream shop uses 6 bananas to make 4 identical sundaes. How much banana is used in each sundae? 

Use a tape diagram to show your work. 

This problem has the same two digits (4 and 6) as the previous problem. However, it is important for 

students to realize that the digits take on a new role, either as whole or divisor, in this context. Six wholes 

divided by 4 is equal to 6 fourths or 1 and 2 fourths. Although it is not required that students use the 

standard algorithm, it can be easily employed to find the mixed number value of . 

Students may also be engaged in a discussion about the practicality of dividing the remaining of the 2 

bananas into fourths and then giving each sundae 2 fourths. Many students may clearly see that the bananas 

can instead be divided into halves and each sundae given 1 and 1 half. Facilitate a quick discussion with 

students about which form of the answer makes more sense given our story context (i.e., should the sundae 

maker divide all the bananas in fourths and then give each sundae 6 fourths, or should each sundae be given 

a whole banana and then divide the remaining bananas?). 

Problem 4 

Julian has to read 4 articles for school. He has 8 nights to read 

them. He decides to read the same number of articles each 

night. 

a. How many articles will he have to read per night? 

b. What fraction of the reading assignment will he read 

each night? 

In this problem Julian must read 4 articles over the course of 8 

nights. The solution of 4 eighths of an article each night might 

imply that Julian can simply divide each article into eighths and 

read any 4 articles on any of the 8 nights. Engage in a discussion 

that allows students to see that 4 eighths must be interpreted 

as 4 consecutive eighths or 1 half of an article. It would be most 

practical for Julian to read the first half of an article one night 

and the remaining half the following night. In this manner, he 

will finish his reading assignment in the 8 days. Part (b) 

NOTES ON 

MULTIPLE MEANS FOR 

ACTION AND 

EXPRESSION: 

Support English language learners as 

they explain their thinking. Provide 

sentence starters and a word bank. 

Examples are given below. 

Sentence starters: 

“I had ____ (unit) in all.” 

“ unit equals ____.” 

Word bank: 

fraction of divided by remainder 

half as much 

twice as many 



Date: 11/9/13 

4.B.46 





provides for deeper thinking about units being considered. 

Students must differentiate between the article-as-unit and assignment-as-unit to answer. While 1 half of an 

article is read each night, the assignment has been split into eight parts. Take the opportunity to discuss with 

students whether or not the articles are all equal in length. Since we are not told, we make a simplifying 

assumption in order to solve, finding that each night 1 eighth of the assignment must be read. Discuss how 

the answer would change if one article were twice the length of the other three. 

Problem 5 

Forty students shared 5 pizzas equally. How much pizza did each student receive? What fraction of the pizza 

did each student receive? 

As this is the fifth problem on the page, students may recognize the division expression very quickly and 

realize that 5 divided by 40 yields 5 fortieths of pizza per student, but in this context it is interesting to discuss 

with students the practicality of serving the pizzas in fortieths. Here, one might better ask, “How can I make 

40 equal parts out of 5 pizzas?” This question leads to thinking about making the least number of cuts to 

each pizza—eighths. Now the simplified answer of 1 eighth of a pizza per student makes more sense. The 

follow-up question points to the changing of the unit from how much pizza per student (1 eighth of a pizza) to 

what fraction of the total (1 fortieth of the total amount). Because there are so many slices to be made, 



Date: 11/9/13 

4.B.47 





students may use the dot, dot, dot format to show the smaller units in their tape diagram. Others may opt to 

simply show their work with an equation. 

Problem 6 

Lillian had 2 two-liter bottles of soda, which she distributed equally between 10 glasses. 

a. How much soda was in each glass? Express your answer as a fraction of a liter. 

b. Express your answer as a decimal number of liters. 

c. Express your answer as a whole number of milliliters. 

This is a three-part problem that asks students to find the amount of soda in each glass. Carefully guide 

students when reading the problem so they can interpret that 2 two-liter bottles are equal to 4 liters total. 

The whole of 4 liters is then divided by 10 glasses to get 4 tenths liters of soda per glass. In order to answer 

Part (b), students need to remember how to express fractions as decimals (i.e., = 0.1, = 0.01, and = 

0.001). For Part (c), students may need to be reminded about the equivalency between liters and milliliters 

(1 L = 1,000 mL). 

Problem 7 

The Calef family likes to paddle along the Susquehanna River. 

a. They paddled the same distance each day over the course of 3 days, traveling a total of 14 miles. 

How many miles did they travel each day? Show your thinking in a tape diagram. 



Date: 11/9/13 

4.B.48 





b. If the Calefs went half their daily distance each day, but extended their trip to twice as many days, 

how far would they travel? 

In Part (a), students can easily use the standard algorithm to solve 14 miles divided by 3 days is equal to 4 and 

2 thirds miles per day. Part (b) requires some deliberate thinking. Guide the students to read the question 

carefully before solving it. 


Lesson Objective: Solve word problems involving the division of whole numbers with answers in the form of 

fractions or whole numbers. 





lesson. 


• How are the problems alike? How are they different? 

• How was your solution the same as and different from those that were demonstrated? 

• Did you see other solutions that surprised you or made you see the problem differently? 

• Why should we assess reasonableness after solving? 

• Were there problems in which it made more sense to express the answer as a fraction rather than a 

mixed number and vice versa? Give examples. 







Date: 11/9/13 

4.B.49 





Name 

Date 

1. A total of 2 yards of fabric is used to make 5 identical pillows. How much fabric is used for each pillow? 

2. An ice-cream shop uses 4 pints of ice cream to make 6 sundaes. How many pints of ice cream are used for 

each sundae? 

3. An ice-cream shop uses 6 bananas to make 4 identical sundaes. How much banana is used in each sundae? 

Use a tape diagram to show your work. 



Date: 11/9/13 

4.B.50 





4. Julian has to read 4 articles for school. He has 8 nights to read them. He decides to read the same number 

of articles each night. 

a. How many articles will he have to read per night? 

b. What fraction of the reading assignment will he read each night? 

5. Forty students shared 5 pizzas equally. How much pizza will each student receive? What fraction of the 

pizza did each student receive? 

6. Lillian had 2 two-liter bottles of soda, which she distributed equally between 10 glasses. 

a. How much soda was in each glass? Express your answer as a fraction of a liter. 



Date: 11/9/13 

4.B.51 





b. Express your answer from as a decimal number of liters. 

c. Express your answer as a whole number of milliliters. 

7. The Calef family likes to paddle along the Susquehanna River. 

a. They paddled the same distance each day over the course of 3 days, travelling a total of 14 miles. 

How many miles did they travel each day? Show your thinking in a tape diagram. 

b. If the Calefs went half their daily distance each day, but extended their trip to twice as many days, 

how far would they travel? 



Date: 11/9/13 

4.B.52 





Name 

Date 

A grasshopper covered a distance of 5 yards in 9 equal hops. How many yards did the grasshopper travel on 

each hop? 

a. Draw a picture to support your work. 

b. How many yards did the grasshopper travel after hopping twice? 



Date: 11/9/13 

4.B.53 





Name 

Date 

1. When someone donated 14 gallons of paint to Rosendale Elementary School, the fifth grade decided to 

use it to paint murals. They split the gallons equally among the four classes. 

a. How much paint did each class have to paint their mural? 

b. How much paint will three classes use? Show your thinking using words, numbers, or pictures. 

c. If 4 students share a 30 square foot wall equally, how many square feet of the wall will be painted by 

each student? 

d. What fraction of the wall will each student paint? 



Date: 11/9/13 

4.B.54 





2. Craig bought a 3-foot long baguette, and then made 4 equally sized sandwiches with it. 

a. What portion of the baguette was used for each sandwich? Draw a visual model to help you solve 

this problem. 

b. How long, in feet, is one of Craig’s sandwiches? 

c. How many inches long is one of Craig’s sandwiches? 

3. Scott has 6 days to save enough money for a $45 concert ticket. If he saves the same amount each day, 

what is the minimum amount he must save each day in order to reach his goal? Express your answer in 

dollars. 



Date: 11/9/13 

4.B.55 






G R A D E 


Topic C 

Multiplication of a Whole Number by 

a Fraction 

5.NF.4a 

Focus Standard: 5.NF.4a Apply and extend previous understandings of multiplication to multiply a fraction or 

whole number by a fraction. 



equivalently, as the result of a sequence of operations a × q ÷ b. For example, use 

a visual fraction model to show (2/3 × 4 = 8/3, and create a story context for this 

equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.) 



In Topic C, students interpret finding a fraction of a set (3/4 of 24) as multiplication of a whole number by a 

fraction (3/4 × 24) and use tape diagrams to support their understandings (5.NF.4a). This in turn leads 

students to see division by a whole number as equivalent to multiplication by its reciprocal. That is, division 

by 2, for example, is the same as multiplication by 1/2. 

Students also use the commutative property to relate fraction of a set to the Grade 4 repeated addition 

interpretation of multiplication by a fraction. This opens the door for students to reason about various 

strategies for multiplying fractions and whole numbers. Students apply their knowledge of fraction of a set 

and previous conversion experiences (with scaffolding from a conversion chart, if necessary) to find a fraction 

of a measurement, thus converting a larger unit to an equivalent smaller unit (e.g., 1/3 min = 20 seconds and 

2 1/4 feet = 27 inches). 

Topic C: Multiplication of a Whole Number by a Fraction 

Date: 11/10/13 4.C.1 




NYS COMMON CORE MATHEMATICS CURRICULUM Topic C 5 

A Teaching Sequence Towards Mastery of Multiplication of a Whole Number by a Fraction 

Objective 1: Relate fractions as division to fraction of a set. 

(Lesson 6) 

Objective 2: Multiply any whole number by a fraction using tape diagrams. 

(Lesson 7) 

Objective 3: Relate fraction of a set to the repeated addition interpretation of fraction multiplication. 

(Lesson 8) 

Objective 4: Find a fraction of a measurement, and solve word problems. 

(Lesson 9) 

Topic C: Multiplication of a Whole Number by a Fraction 

Date: 11/10/13 4.C.2 





Lesson 6 

Objective: Relate fractions as division to fraction of a set. 






Total Time 

(6 minutes) 

(12 minutes) 

(32 minutes) 

(10 minutes) 

(60 minutes) 


Olivia is half the age of her brother, Adam. Olivia’s sister, 

Ava, is twice as old as Adam. Adam is 4 years old. How 

old is each sibling? Use tape diagrams to show your 

thinking. 

Note: This Application Problem is intended to activate 

students’ prior knowledge of half of in a simple context 

as a precursor to today’s more formalized introduction to 

fraction of a set. 


• Sprint: Divide Whole Numbers 5.NF.3 


(8 minutes) 

(4 minutes) 

Sprint: Divide Whole Numbers (8 minutes) 

Materials: (S) Divide Whole Numbers Sprint 

Note: This Sprint reviews G5–M4–Lessons 2–4. 




T: I’ll say a division sentence. You write it as a fraction. 4 ÷ 2. 

Lesson 6: Relate fractions as division to fraction of a set. 

Date: 11/10/13 4.C.3 





S: . 

T: 6 4. 

S: . 

T: 3 4. 

S: . 

T: 2 10. 

S: . 

T: Rename this fraction using fifths. 

S: . 

T: (Write .) Write the fraction as a division equation and solve. 

S: 56 ÷ 2 = 28. 

Continue with the following possible suggestions: 6 thirds, 9 thirds, 18 thirds, , 8 fourths, 12 fourths, 28 

fourths, and . 


Materials: (S) Two-sided counters, drinking straws, personal 

white boards 

Problem 1 

of 6 = ___ 

T: Make an array with 6 counters turned to the red side 

and use your straws to divide your array into 3 equal 

parts. 

T: Write a division sentence for what you just did. 

S: 6 ÷ 3 = 2. 

T: Rewrite your division sentence as a fraction and speak it as you write it. 

S: (Write ) 6 divided by 3. 

NOTES ON 


ACTION AND 

EXPRESSION: 

If students struggle with the set model 

of this lesson, consider allowing them 

to fold a square of paper into the 

desired fractional parts. Then have 

them place the counters in the sections 

created by the folding. 

T: If I want to show 1 third of this set, how many counters should I turn over to yellow? Turn and talk. 

S: Two counters. Each group is 1 third of all the counters, so we would have to turn over 1 group of 

2 counters. Six divided by 3 tells us there are 2 in each group. 

T: 1 third of 6 is equal to? 

S: 2. 


Date: 11/10/13 4.C.4 





T: (Write of 6 = 2.) How many counters should be turned over to 

show 2 thirds? Whisper to your partner how you know. 

S: I can count from our array. One third is 2 counters, then 2 thirds 

is 4 counters. Six divided by 3 once is 2 counters. Double 

that is 4 counters. I know 1 group out of 3 groups is 2 

counters, so 2 groups out of 3 would be 4 counters. Since 

of 6 is equal to 2, then of 6 is double that. Two plus 2 is 4. 

6 ÷ 3 2, but I wrote 6 ÷ 3 as a fraction. 

T: (Write of 6 = ___.) What is 2 thirds of 6 counters? 

S: 4 counters. 

T: (Write of 6 = ___.) What is 3 thirds of 6 counters? 

S: 6 counters. 

T: How do you know? Turn and discuss with your 

partner. 

S: I counted 2, 4, 6. is a whole, and our whole set is 6 

counters. 

T: Following this pattern, what is 4 thirds of 6? 

S: It would be more than 6. It would be more than the 

whole set. We would have to add 2 more counters. It 

would be 8. 6 divided by 3 times 4 is 8. 

Problem 2 

of 12 = ___ 

T: Make an array using 12 counters turned to the red side. Use 

your straws to divide the array into fourths. (Draw an array on 

the board.) 

T: How many counters did you place in each fourth? 

S: 3. 

T: Write the division sentence as a fraction on your board. 

S: = 3. 

T: What is 1 fourth of 12? 

S: 3. 

T: (Write of 12 = 3.) 1 fourth of 12 is equal to 3. Look at your 

NOTES ON 



It is acceptable for students to orient 

their arrays in either direction. For 

example, in Problem 2, students may 

arrange their counters in the 3 × 4 

arrangement pictured, or they may 

show a 4 × 3 array that is divided by 

the straws horizontally. 

array. What fraction of 12 is equal to 6 counters? Turn and discuss with your partner. 

S: I see 2 groups is equal to 6 so the answer is . Since 1 fourth is equal to 3, and 6 is double that 

much, I can double 1 fourth to get 2 fourths. 


Date: 11/10/13 4.C.5 





T: (Write of 12 = 6.) 2 fourths of 12 is equal to 6. What is another way to say 2 fourths? 

S: 1 half. 

T: Is 1 half of 12 equal to 6? 

S: Yes. 

Follow this sequence with of 9, of 12, and of 15, as necessary. 

Problem 3 

Mrs. Pham has 8 apples. She wants to give of the apples to her students. How many apples will her 

students get? 

T: Use your counters or draw an array to show how many apples Mrs. Pham has. 

S: (Represent 8 apples.) 

T: (Write of 8 = ___.) How will we find 3 fourths of 8? Turn and talk. 

S: I divided my counters to make 4 equal parts. Then I 

counted the number in 3 of those parts. I can 

draw 4 rows of 2 and count 2, 4, 6, so the answer is 6 

apples. I need to make fourths. That’s 4 equal 

parts, but I only want to know about 3 of them. 

There are 2 in 1 part and 6 in 3 parts. I know if 1 

fourth is equal to 2, then 3 fourths is 3 groups of 2. 

The answer is 6 apples. 

Problem 4 

In a class of 24 students, are boys. How many boys are in 

the class? 

T: How many students are in the whole class? 

S: 24. 

T: What is the question? 

S: How many boys are in the class? 

T: What fraction of the whole class of 24 are boys? 

S: . 

T: Will our answer be more than half of the class or less than half? How do you know? Turn and talk. 

S: 5 sixths is more than half, so the answer should be more than 12. Half of the class would be 12, 

which would also be 3 sixths. We need more sixths than that so our answer will be more than 12. 

T: (Write of 24 = ___ on the board.) Use your counters or draw to solve. Turn and discuss with a 

partner. 

S: We should draw a total of 24 circles, and then split them into 6 equal groups. We can draw 4 

groups of 6 circles. We will have 6 columns representing 6 groups, and each group will have 4 


Date: 11/10/13 4.C.6 





circles. We could draw 6 rows of 4 circles to 

show 6 equal parts. We only care how many are 

in 5 of the rows, and 5 × 4 is 20 boys. We 

need to find sixths, so we need to divide the set 

into 6 equal parts, but we only need to know 

how many are in 5 of the groups. That’s 4, 8, 12, 

16, 20. There are 20 boys in the class. 

T: (Point to the drawing on the board.) Let’s think of this another way. What is of 24? 

S: 4. 

T: How do we know? Say the division sentence. 

S: 24 6 = 4. 

T: How can we use of 24 to help us solve for of 24? 

Whisper and tell your partner. 

S: of 24 is equal to 4. of 24 is just 5 groups of 4. 4 + 4 + 

4 + 4 + 4 = 20. I know each group is 4. To find 5 groups, 

I can multiply 5 × 4 = 20. 

20. 

(24 divided by 6) times 5 is 

T: I’m going to rearrange the circles a bit. (Draw a bar 

directly beneath the array and label 24.) We said we 

needed to find sixths, so how many units should I cut the 

whole into? 

S: We need 6 units the same size. 

T: (Cut the bar into 6 equal parts.) If 6 units are 24, how many circles in one unit? How do you know? 

S: Four, because 24 ÷ 6 is 4. 

T: (Write of 24 = 4 under the bar.) Let me draw 4 counters into each unit. Count with me as I write. 

S: 4, 8, 12, 16, 20, 24. 

T: We are only interested in the part of the class that is boys. How many of these units represent the 

boys in the class? 

S: 5 units. 5 sixths. 

T: What are 5 units worth? Or, what is 5 sixths of 24? (Draw a bracket around 5 units and write of 24 

S: 20. 

=___.) 

T: (Write the answer on the board.) 

T: Answer the question with a sentence. 

S: There are 20 boys in the class. 


Date: 11/10/13 4.C.7 














Lesson Objective: Relate fractions as division to fraction 

of a set. 









lesson. 



• What pattern did you notice in Problem 1(a)? 

(Students may say it skip-counts by threes or that 

all the answers are multiples of 3.) Based on this 

pattern, what do you think the answer for of 9 

is? Why is this more than 9? (Because 4 thirds is 

more than a whole, and 9 is the whole.) 

• How did you solve for the last question in 1(c)? 

Explain to your partner. 

• In Problem 1(d), what did you notice about the 

two fractions and ? Can you name them using 

a larger unit (simplify them)? What connections 

did you make about of 24 and of 24, of 24 

and of 24? 

• When solving these problems (fraction of a set), 

how important is it to first find out how many are 


Date: 11/10/13 4.C.8 





in each group (unit)? Explain your thinking to a partner. 

• Is this a true statement? (Write of 18 2 ) Two-thirds of 18 is the same as 18 divided by 3, 

times 2. Why or why not? 






Date: 11/10/13 4.C.9 




NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 Sprint 5•4 


Date: 11/10/13 4.C.10 






Date: 11/10/13 4.C.11 





Name 

Date 

1. Find the value of each of the following. 

a. b. 

of 9 = 

of 9 = 

of 9 = 

of 15 = 

of 15 = 

of 15 = 

c. 

of 20= 

of 20 = 

of 20 = 20 

d. 

of 24 = of 24 = 

of 24 = of 24 = 

of 24 = 


Date: 11/10/13 4.C.12 





2. Find of 14. Draw a set and shade to show your thinking. 

3. How does knowing of 24 help you find three-eighths of 24? Draw a picture to explain your thinking. 

4. There are 32 students in a class. Of the class, bring their own lunch. How many students bring their 

lunch? 

5. Jack collected 18 ten dollar bills while selling tickets for a show. He gave of the bills to the theater and 

kept the rest. How much money did he keep? 


Date: 11/10/13 4.C.13 





Name 

Date 


 

 

 

 

 

 

 

 

a. of 16 = 

b. of 16 = 

2. Out of 18 cookies, are chocolate chip. How many of the cookies are chocolate chip? 


Date: 11/10/13 4.C.14 





Name 

Date 


a. 

 

 

 

 

 

 

of 12 = 

of 12 = 

of 12 = 

b. 

 

 

 

 

 

 

 

 

 

 

 

 

of 20 = of 20 = 

of 20 = of 20 = 

c. 

 

 

 

 

 

of 35 = of 35 = of 35 = 

of 35 = of 35 = of 35 = 


Date: 11/10/13 4.C.15 





2. Find of 18. Draw a set and shade to show your thinking. 

3. How does knowing of 10 help you find of 10? Draw a picture to explain your thinking. 

4. Sara just turned 18 years old. She spent of her life living in Rochester, NY. For how many years did Sara 

live in Rochester? 

5. A farmer collects 12 dozen eggs from her chickens. She sells of the eggs at the farmers’ market and 

gives the rest to friends and neighbors. 

a. How many eggs does she give away? 

b. If she sells each dozen for $4.50, how much will she earn from the eggs she sells? 


Date: 11/10/13 4.C.16 





Lesson 7 

Objective: Multiply any whole number by a fraction using tape diagrams. 






Total Time 

(12 minutes) 

(5 minutes) 

(33 minutes) 

(10 minutes) 

(60 minutes) 


• Read Tape Diagrams 5.NF.4 

• Half of Whole Numbers 5.NF.4 

• Fractions as Whole Numbers 5.NF.3 

(4 minutes) 

(4 minutes) 

(4 minutes) 

Read Tape Diagrams (4 minutes) 


Note: This fluency prepares students to multiply fractions by whole numbers during the Concept 

Development. 

T: (Project a tape diagram with 10 partitioned into 2 equal units.) Say the whole. 

S: 10. 

T: On your boards, write the division sentence. 

S: (Write 10 ÷ 2 = 5.) 

Continue with the following possible sequence: 6 ÷ 2, 9 ÷ 3, 12 ÷ 3, 8 ÷ 4, 12 ÷ 4, 25 ÷ 5, 40 ÷ 5, 42 ÷ 6, 63 ÷ 7, 

64 ÷ 8, and 54 ÷ 9. 

Half of Whole Numbers (4 minutes) 


Note: This fluency reviews G5–M4–Lesson 6 content and prepares students to multiply fractions by whole 

numbers during the Concept Development using tape diagrams. 

T: Draw 4 counters. What’s half of 4? 

S: 2. 

Lesson 7: Multiply any whole number by a fraction using tape diagrams. 

Date: 11/10/13 4.C.17 





T: (Write of 4 = 2.) Say a division sentence that helps you find the answer. 

S: 4 ÷ 2 = 2. 

Continue with the following possible sequence: half of 10, half of 8, 1 half of 30, 1 half of 54, 1 fourth of 20, 1 

fourth of 16, 1 third of 9, and 1 third of 18. 

Fractions as Whole Numbers (4 minutes) 


Note: This fluency reviews G5–M4–Lesson 5 and reviews denominators that are equivalent to hundredths. 

Direct students to use their personal white boards for calculations that they cannot do mentally. 

T: I’ll say a fraction. You say it as a division problem. 4 halves. 

S: 4 ÷ 2 = 2. 


and 


Mr. Peterson bought a case (24 boxes) of fruit juice. 

One-third of the drinks were grape and two-thirds were 

cranberry. How many boxes of each flavor did Mr. 

Peterson buy? Show your work using a tape diagram or 

an array. 

Note: This Application Problem requires students to use 

skills explored in G5–M4–Lesson 6. Students are finding 

fractions of a set and showing their thinking with 

models. 



Problem 1 

What is of 35? 

T: (Write of 35 = ___ on the board.) We used two 

different models (counters and arrays) yesterday to 

find fractions of sets. We will use tape diagrams to 

help us today. 

T: We have to find 3 fifths of 35. Draw a bar to represent 

NOTES ON 



Please note throughout the lesson that 

division sentences are written as 

fractions in order to reinforce the 

interpretation of a fraction as division. 

When reading the fraction notation, 

the language of division should be 

used. For example, in Problem 1, 

1 unit = should be read as 1 unit 

equals 35 divided by 5. 


Date: 11/10/13 4.C.18 





S: 35. 

our whole. What’s our whole? 

T: (Draw a bar and label 35.) How many units 

should we cut the whole into? 

S: 5. 

T: How do you know? 

S: The denominator tells us we want fifths. 

That is the unit being named by the fraction. 

We are asked about fifths so we know we need 

5 equal parts. 

T: (Cut the bar into 5 equal units.) We know 5 units are equal to 35. How do we find the value of 1 

unit? Say the division sentence. 

S: 35 ÷ 5 = 7. 

T: (Write 5 units = 35, 1 unit = 35 ÷ 5 = 7.) Have we 

answered our question? 

S: No, we found 1 unit is equal to 7, but the question is to 

find 3 units. We need 3 fifths. When we divide by 

5, that’s just 1 fifth of 35. 

T: How will we find 3 units? 

S: Multiply 3 and 7 to get 21. We could add 7 + 7 + 7. 

We could put 3 of the 1 fifths together. That would 

be 21. 

T: What is of 35? 

S: 21. 

Problem 2 

NOTES ON 


ACTION AND 

EXPRESSION: 

Students with fine motor deficits may 

find drawing tape diagrams difficult. 

Graph paper may provide some 

support, or online sources like the 

Thinking Blocks website may also be 

helpful. 

Aurelia buys 2 dozen roses. Of these roses, are red and the rest are white. How many white roses did she 

buy? 

T: What do you know about this problem? Turn and share with your partner. 

S: I know the whole is 2 dozen, which is 24. 

are red roses, and are white roses. The total is 

24 roses. The information in the problem is 

about red roses, but the question is about the 

other part, the white roses. 

T: Discuss with your partner how you’ll solve this 

problem. 

S: We can first find the total red roses, then 

subtract from 24 to get the white roses. 


Date: 11/10/13 4.C.19 





Since I know of the whole is white roses, I can find of 24 to find the white roses. And that’s 

faster. 

T: Work with a partner to draw a tape diagram and solve. 

T: Answer the question for this problem. 

S: She bought 6 white roses. 

Problem 3 

Rosie had 17 yards of fabric. She used one-third of it to make a 

quilt. How many yards of fabric did Rosie use for the quilt? 

T: What can you draw? Turn and share with your 

partner. 

T: Compare this problem with the others we’ve done 

today. 

S: The answer is not a whole number. The quotient is 

not a whole number. We were still looking for 

fractional parts, but the answer isn’t a whole number. 

T: We can draw a bar that shows 17 and divide it into 

thirds. How do we find the value of one unit? 

S: Divide 17 by 3. 

T: How much fabric is one-third of 17 yards? 

S: yards. 5 yards. 

T How would you find 2 thirds of 17? 

S: Double 5 . Multiply 5 times 2. 

Subtract 5 from 17. 

Repeat this sequence with of 11, if necessary. 

Problem 4 

of a number is 8. What is the number? 

T: How is this problem different from the ones we 

just solved? 

S: In the first problem, we knew the total and 

wanted to find a part of it. In this one, we know 

how much 2 thirds is, but not the whole. They 

told us the whole and asked us about a part last 

time. This time they told us about a part and 

asked us to find the whole. 

NOTES ON 



The added complexity of finding a 

fraction of a quantity that is not a 

multiple of the denominator may 

require a return to concrete materials 

for some students. Allow them access 

to materials that can be folded and cut 

to model Problem 3 physically. Five 

whole squares can be distributed into 

each unit of 1 third. Then the 

remaining whole squares can be cut 

into thirds and distributed among the 

units of thirds. Be sure to make the 

connection to the fraction form of the 

division sentence and the written 

recording of the division algorithm. 

T: Draw a bar to represent the whole. What kind of units will we need to divide the whole into? 


Date: 11/10/13 4.C.20 





S: Thirds. 

T: What else do we know? Turn and tell your partner. 

S: We know that 2 thirds is the same as 8 so it means we can label 2 of the units with a bracket and 8. 

The units are thirds. We know about 2 of them. They are equal to 8 together. We don’t know 

what the whole bar is worth so we have to put a question mark there. 

T: How can knowing what 2 units are worth help us find the whole? 

S: Since we know that 2 units = 8, then we can divide to find 1 unit is equal to 4. 

T: (Write 2 units = 8 ÷ 2 = 4.) Let’s record 4 inside each unit. Can we find the whole now? 

S: Yes. We can add 4 + 4 + 4 =12. We can multiply 3 times 4, which is equal to 12. 

T: (Write 3 units = 3 × 4 = 12.) Answer the question for this problem. 

S: The number is 12. 

T: Let’s think about it and check to see if it makes sense. (Write of 12 = 8.) Work independently on 

Problem 5 

your personal board and solve to find what 2 thirds of 12 is. 

Tiffany spent of her money on a teddy bear. If the teddy bear cost $24, how much money did she have at 

first? 

T: Which problem that we’ve worked today is 

most like this one? 

S: This one is just like Problem 4. We have 

information about a part, and we have to 

find the whole. 

T: What can you draw? Turn and share with 

your partner. 

S: We can draw a bar for all the money. We can show what the teddy bear costs. It costs $24, and it’s 

of her total money. We can put a question mark over the whole bar. 

T: Do we have enough information to find the value of 1 unit? 

S: Yes. 

T: How much is one unit? How do you know? 

S: 4 units = $24, so 1 unit = $6. 

T: How will we find the amount of money she had at first? 

S: Multiply $6 by 7. 

T: Say the multiplication sentence starting with 7. 

S: 7 × $6 = $42. 

T: Answer the question in this problem. 

S: Tiffany had $42 at first. 


Date: 11/10/13 4.C.21 














Lesson Objective: Multiply any whole number by a 

fraction using tape diagrams. 









lesson. 



• What pattern relationships did you notice 

between Problems 1(a) and 1(b)? (The whole of 

36 is double of 18. That’s why the answer is 12, 

which is also double of 6.) 

• What pattern did you notice between Problems 

1(c) and 1(d)? (The fraction of 3 eighths is half of 

3 fourths. That is why the answer is 9, which is 

also half of 18.) 

• Look at Problems 1(e) and 1(f). We know that 4 

fifths and 1 seventh aren’t equal, so how did we 

get the same answer? 

• Compare Problems 1(c) and 1(k). How are they 

similar, and how are they different? (The 

questions involve the same numbers, but in 

Problem 1(c), 3 fourths is the unknown quantity, 

and in Problem 1(k) it is the known quantity. In 

Problem 1(c) the whole is known, but in Problem 


Date: 11/10/13 4.C.22 





1(k) the whole is unknown.) 

• How did you solve for Problem 2(b)? Explain your strategy or solution to a partner. 

• There are a couple of different methods to solve Problem 2(c). Find someone who used a different 

approach from yours and explain your thinking. 






Date: 11/10/13 4.C.23 





Name 

Date 

1. Solve using a tape diagram. 

a. of 18 b. of 36 

c. × 24 d. × 24 

e. × 25 f. × 140 

g. × 9 h. × 12 

i. of a number is 10. What’s the number? j. of a number is 24. What’s the number? 


Date: 11/10/13 4.C.24 





2. Solve using tape diagrams. 

a. There are 48 students going on a field trip. One-fourth are girls. How many boys are going on the 

trip? 

b. Three angles are labeled below with arcs. The smallest angle is as large as the 1 0 angle. Find the 

value of angle a. 

 

c. Abbie spent of her money and saved the rest. If she spent $45, how much money did she have at 

first? 

d. Mrs. Harrison used 16 ounces of dark chocolate while baking. She used of the chocolate to make 

some frosting and used the rest to make brownies. How much more chocolate did Mrs. Harrison use 

in the brownies than in the frosting? 


Date: 11/10/13 4.C.25 




NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 Exit Ticket 5 

Name 

Date 

Solve using a tape diagram. 

a. of 30 b. of a number is 30. What’s the number? 

c. Mrs. Johnson baked 2 dozen cookies. Two-thirds of them were oatmeal. How many oatmeal cookies did 

Mrs. Johnson bake? 


Date: 11/10/13 4.C.26 





Name 

Date 

1. Solve using a tape diagram. 

a. of 24 b. of 48 

c. × 18 d. × 18 

e. × 49 f. × 120 

g. × 31 h. × 20 

i. × 25 j. × 25 

k. of a number is 27. What’s the number? l. of a number is 14. What’s the number? 


Date: 11/10/13 4.C.27 





2. Solve using tape diagrams. 

a. A skating rink sold 66 tickets. Of these, were children’s tickets, and the rest were adult tickets. How 

many adult tickets were sold? 

b. A straight angle is split into two smaller angles as shown. The smaller angle’s measure is that of a 

straight angle. What is the value of angle a? 

c. Annabel and Eric made 17 ounces of pizza dough. They used of the dough to make a pizza and used 

the rest to make calzones. What is the difference between the amount of dough they used to make 

pizza and the amount of dough they used to make calzones? 

d. The New York Rangers hockey team won of their games last season. If they lost 21 games, how 

many games did they play in the entire season? 


Date: 11/10/13 4.C.28 





Lesson 8 

Objective: Relate fraction of a set to the repeated addition interpretation 

of fraction multiplication. 






Total Time 

(12 minutes) 

(8 minutes) 

(30 minutes) 

(10 minutes) 

(60 minutes) 


• Convert Measures 4.MD.1 


• Multiply a Fraction Times a Whole Number 5.NF.4 

(5 minutes) 

(3 minutes) 

(4 minutes) 

Convert Measures (5 minutes) 

Materials: (S) Personal white boards, Grade 5 Mathematics Reference Sheet 

Note: This fluency prepares students for G5–M4–Lessons 9─12 content. Allow students to use the Grade 5 

Mathematics Reference Sheet if they are confused, but encourage them to answer questions without looking 

at it. 

T: (Write 1 ft = ____ in.) How many inches are in 1 foot? 

S: 12 inches. 

T: (Write 1 ft = 12 in. Below it, write 2 ft = ____ in.) 2 feet? 

S: 24 inches. 


S: 36 inches. 


S: 48 inches. 

T: (Write 4 ft = 48 in. Below it, write 10 ft = ____ in.) On your boards, write the equation. 

S: (Write 10 ft = 120 in.) 

T: (Write 10 ft × ____ = ____ in.) Write the multiplication equation you used to solve it. 

S: (Write 10 ft × 12 = 120 in.) 

Lesson 8: Relate fraction of a set to the repeated addition interpretation of 

fraction multiplication. 

Date: 11/10/13 

4.C.29 





Continue with the following possible sequence: 1 pint = 2 cups, 2 pints = 4 cups, 3 pints = 6 cups, 9 pints = 18 

cups, 1 yd = 3 ft, 2 yd = 6 ft, 3 yd = 9 ft, 7 yd = 21 ft, 1 gal = 4 qt, 2 gal = 8 qt, 3 gal = 12 qt, and 8 gal = 32 qt. 



Note: This fluency reviews G5–M4–Lesson 5 and reviews denominators that are equivalent to hundredths. 

Direct students to use their personal boards for calculations that they cannot do mentally. 

T: I’ll say a fraction. You say it as a division problem. 4 halves. 

S: 4 ÷ 2 = 2. 


Multiply a Fraction Times a Whole Number (4 minutes) 


and 


T: (Project a tape diagram of 12 partitioned into 3 equal 

units. Shade in 1 unit.) What fraction of 12 is shaded? 

S: 1 third. 

T: Read the tape diagram as a division equation. 

S: 12 ÷ 3 = 4. 

T: (Write 12 × ____ = 4.) On your boards, write the 

equation, filling in the missing fraction. 

S: (Write 12 × = 4.) 


12 × = 4 

and 

! 


Sasha organizes the art gallery in her town’s 

community center. This month she has 24 new pieces 

to add to the gallery. 

Of the new pieces, of them are photographs and of 

them are paintings. How many more paintings are 

there than photos? 

Note: This Application Problem requires students to find two fractions of the 

same set—a recall of the concepts from G5–M4–Lessons 6–7 in preparation for today’s lesson. 



Date: 11/10/13 

4.C.30 







Problem 1 

× 6 = ____ 

T: (Write 2 × 6 on the board.) Read this expression out loud. 

S: 2 times 6. 

T: In what different ways can we interpret the meaning of this expression? Discuss with your partner. 

S: We can think of it as 6 times as much as 2. 6 + 6. We could think of 6 copies of 2. 2 + 2 + 2 

+ 2 + 2 + 2. 

T: True, we can find 2 copies of 6, but we could also think about 2 added 6 times. What is the property 

that allows us to multiply the factors in any order? 

S: Commutative property. 

T: (Write × 6 on the board.) How can we interpret this expression? Turn and talk. 

S: 2 thirds of 6. 6 copies of 2 thirds. 2 thirds added together 6 times. 

T: This expression can be interpreted in different ways, just as the whole number expression. We can 

say it’s of 6 or 6 groups of . (Write and on the board as shown below.) 

T: Use a tape diagram to find 2 thirds of 6. (Point to .) 

S: (Solve.) 

of 

T: Let me record our thinking. We see in the 

diagram that 3 units is 6. (Write 3 units = 6.) 

We divide 6 by 3 find 1 unit. (Write .) So, 2 

units is 2 times 6 divided by 3. (Write 2 × 

and the rest of the thinking in the table as 

shown above.) 

T: Now, let’s think of it as 6 groups (or copies) 

of like you did in Grade 4. Solve it using 

repeated addition on your board. 

S: (Solve.) 

T: (Write on the board.) 

T: What multiplication expression gave us 12? 



Date: 11/10/13 

4.C.31 





S: 6 × 2. 

T: (Write on board.) What unit are we counting? 

S: Thirds. 

T: Let me write what I hear you saying. (Write (6 × 2) 

thirds on the board.) Now let me write it another way. 

(Write = 

.) 6 times 2 thirds. 

T: In both ways of thinking what is the product? Why is it 

the same? 

S: It’s thirds because × 6 thirds is the same as 6 × 2 

thirds. It’s the commutative property again. It 

doesn’t matter what order we multiply, it’s the same 

product. 

T: How many wholes is 12 thirds? How much is 12 

divided by 3? 

S: 4. 

T: Let’s use something else we learned in Grade to 

rename this fraction using larger units before we 

multiply. (Point to 

.) Look for a factor that is 

shared by the numerator and the denominator. Turn 

and talk. 

NOTES ON 



If students have difficulty remembering 

that dividing by a common factor 

allows a fraction to be renamed, 

consider a return to the Grade 4 

notation for finding equivalent 

fractions as follows: 

The decomposition in the numerator 

makes the common factor of 3 

apparent. Students may also be 

reminded that multiplying by is the 

same as multiplying by 1. 

S: Two and 3 only have a common factor of 1, but 3 and 6 have a common factor of 3. I know the 

numerator of 6 can be divided by 3 to get 2, and the denominator of 3 can be divided by 3 to get 1. 

T: We can rename this fraction just like in Grade 4 by dividing both the numerator and the denominator 

by 3. Watch me. 6 divided by 3 is 2. (Cross out 6, write 2 above 6.) 3 divided by 3 is 1. (Cross out 3, 

write 1 below 3.) 

T: What does the numerator show now? 

S: 2 × 2. 

T: What’s the denominator? 

S: 1. 

T: (Write = .) This fraction was 12 thirds, now 

it is 4 wholes. Did we change the amount of the 

fraction by naming it using larger units? How do 

you know? 

S: It is the same amount. Thirds are smaller than 

wholes, so it takes 12 thirds to show the same amount as 4 wholes. It is the same. The unit got 

larger, so the number we needed to show the amount got smaller. There are 3 thirds in 1 whole 

so 12 thirds makes 4 wholes. It is the same. When we divide the numerator and the denominator 

by the same the number, it’s like dividing by and dividing by doesn’t change the value of the 

number. 



Date: 11/10/13 

4.C.32 





Problem 2 

× 10 = ____ 

T: Finding of 10 is the same as finding the product of 10 

copies of . I can rewrite this expression in unit form as 

(10 × 3) fifths or as a fraction. (Write .) 10 times 3 

fifths. Multiply in your head and say the product. 

S: 30 fifths. 

T: is equivalent to how many wholes? 

S: 6 wholes. 

T: So, if 10 × is equal to 6, is it also true that 3 fifths of 10 is 6? How do you know? 

S: Yes, it is true. 1 fifth of 10 is 2, so 3 fifths would be 6. 

The commutative property says we can multiply in 

any order. This is true of fractional numbers too, so 

the product would be the same. 3 fifths is a little 

more than half, so 3 fifths of 10 should be a little more 

than 5. 6 is a little more than 5. 

T: Now, let’s work this problem again, but this time let’s 

find a common factor and rename before we multiply. 

(Follow the sequence from Problem 1.) 

S: (Work.) 

T: Did dividing the numerator and the denominator by 

the same common factor change the quantity? Why or 

why not? 

S: (Share.) 

Problem 3 

× 24= ____ 

× 27= ____ 

T: Before we solve, what do you notice that is different this time? 

NOTES ON 


ACTION AND 

EXPRESSION: 

While the focus of today’s lesson is the 

transition to a more abstract 

understanding of fraction of a set, do 

not be too quick to drop pictorial 

representations. Tape diagrams are 

powerful tools in helping students 

make connections to the abstract. 

Throughout the lesson, continue to 

ask, “Can you draw something?” 

These drawings also provide formative 

assessment opportunities for teachers, 

and allow a glimpse into the thinking of 

students in real time. 

S: The fraction of the set that we are finding is more than a whole this time. All the others were 

fractions less than 1. 

T: Let’s estimate the size of our product. Turn and talk. 

S: This is like the one from the Problem Set yesterday. We need more than a whole set, so the answer 

will be more than 24. We need 1 sixth more than a whole set of 24, so the answer will be a little 

more than 24. 



Date: 11/10/13 

4.C.33 





T: (Write on the board.) 24 times 7 sixths. Can you multiply 24 times 7 in your head? 

S: You could, but it’s a lot to think about to do it mentally. 

T: Because this one is harder to calculate mentally, let’s use the renaming strategies we’ve seen to 

solve this problem. Turn and share how we can get started. 

S: We can divide the numerator and denominator by the same common factor. 

Continue with the sequence from Problem 2 having students name the common factor and rename as shown 

above. Then proceed to × 27= ____. 

T: Compare this problem to the last one. 

S: The whole is a little more than last time. The fraction we are looking for is the same, but the 

whole is bigger. We probably need to rename this one before we multiply like the last one 

because 7 × 27 is harder to do mentally. 

T: Let’s rename first. Name a factor that 27 and 6 

share. 

S: 3. 

T: Let’s divide the numerator and denominator by this 

common factor. 27 divided by 3 is 9. (Cross out 27, 

and write 9 above 27.) 6 divided by 3 is 2. (Cross out 

6, and write 2 below 6.) We’ve renamed this fraction. 

What’s the new name? 

S: . (9 times 7 divided by 2.) 

T: Has this made it easier for us to solve this mentally? 

Why? 

S: Yes, the numbers are easier to multiply now. The numerator is a basic fact now and I know 

9 × 7! 

T: Have we changed the amount that is 

represented by this fraction? Turn and talk. 

S: No, it’s the same amount. We just renamed it using a bigger unit. We renamed it just like any 

other fraction by looking for a common factor. This doesn’t change the amount. 

T: Say the product as a fraction greater than one. 

S: 63 halves. (Write = .) 

T: We could express as a mixed number, but we don’t have to. 

T: (Point to .) To compare, let’s multiply without renaming and see if we get the same product. 

T: What’s the fraction? 

S: . 

T: (Write = .) Rewrite that as a fraction greater than 1, using the largest units that you can. What do 

you notice? 



Date: 11/10/13 

4.C.34 





S: (Work to find .) We get the same answer, but it was harder 

to get to it. 9 is a large number, so it’s harder for me to 

find the common factor with 6. I can’t do it in my head. I 

needed to use paper and pencil to simplify. 

T: So, sometimes, it makes our work easier and more efficient to 

rename with larger units, or simplify, first and then multiply. 

Repeat this sequence with 

Problem 4 

hour = ____ minutes 

× 28 = ____. 

T: We are looking for part of an hour. Which part? 

S: 2 thirds of an hour. 

T: Will 2 thirds of an hour be more than 60 minutes or less? 

Why? 

S: It should be less because it isn’t a whole hour. A whole 

hour, 60 minutes, would be 3 thirds, we only want 2 thirds 

so it should be less than 60 minutes. 

T: Turn and talk with your partner about how you might find 

2 thirds of an hour. 

S: I know the whole is 60 minutes, and the fraction I want is 

. We have to find what’s of 60. 

T: (Write × 60 min = ____ min.) Solve this problem independently. You may use any method you 

like. 

S: (Solve.) 

T: (Select students to share their solutions with the class.) 

Repeat this sequence with of a foot. 






Today’s Problem Set is lengthy. Students may benefit from additional guidance. Consider working one 

problem from each section as a class before directing students to solve the remainder of the problems 

independently. 



Date: 11/10/13 

4.C.35 






Lesson Objective: Relate fraction of a set to the repeated 

addition interpretation of fraction multiplication. 









lesson. 



• Share and explain your solution for Problem 1 

with a partner. 

• What do you notice about Problems 2(a) and 

2(c)? (Problem 2(a) is 3 groups of , which is 

equal to , and 2(c) is 3 groups of , 

which is equal to .) 

• What do you notice about the solutions in 

Problems 3 and 4? (All the products are whole 

numbers.) 

• We learned to solve fraction of a set problems 

using the repeated addition strategy and 

multiplication and simplifying strategies today. 

Which one do you think is the most efficient way 

to solve a problem? Does it depend on the 

problems? 

• Why is it important to learn more than one 

strategy to solve a problem? 






future lessons. You may read the questions aloud to the students. 



Date: 11/10/13 

4.C.36 




NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 Reference Sheet 5•4 



Date: 11/10/13 

4.C.37 





Name 

Date 

1. Laura and Sean find the product of using different methods. 

Laura: It’s thirds of . Sean: It’s groups of thirds. 

Use words, pictures, or numbers to compare their methods in the space below. 

2. Rewrite the following addition expressions as fractions as shown in the example. 

Example: 

a. b. c. 

3. Solve and model each problem as a fraction of a set and as repeated addition. 

Example: . 6 

a. 

b. 



Date: 11/10/13 

4.C.38 





4. Solve each problem in two different ways as modeled in the example. 

2 

Example: 

a. 

1 

b. 

c. 

d. 

5. Solve each problem any way you choose. 

a. minute = __________ seconds 

b. hour = __________ minutes 

c. kilogram = __________ grams 

d. meter = __________ centimeters 



Date: 11/10/13 

4.C.39 





Name 

Date 


a. Example: b. 

1 

2 

a. 

b. 



Date: 11/10/13 

4.C.40 





Name 

Date 

1. Rewrite the following expressions as shown in the example. 

Example: 

a. b. c. 


Example: b. 

1 

2 

a. 

b. 

c. 

d. 

e. 



Date: 11/10/13 

4.C.41 





f. 

g. 

3. Solve each problem any way you choose. 

a. minute = _________ seconds 

b. hour = _________ minutes 

c. kilogram = _________ grams 

d. meter = _________ centimeters 



Date: 11/10/13 

4.C.42 





Lesson 9 

Objective: Find a fraction of a measurement, and solve word problems. 






Total Time 

(12 minutes) 

(8 minutes) 

(30 minutes) 

(10 minutes) 

(60 minutes) 


• Multiply Whole Numbers by Fractions with Tape Diagrams 5.NF.4 


• Multiply a Fraction and a Whole Number 5.NF.4 

(4 minutes) 

(4 minutes) 

(4 minutes) 

Multiply Whole Numbers by Fractions with Tape Diagrams (4 minutes) 



T: (Project a tape diagram of 8 partitioned into 2 equal units. Shade in 1 unit.) What fraction of 8 is 

shaded? 

S: 1 half. 

T: Read the tape diagram as a division equation. 

S: 8 ÷ 2 = 4. 

T: (Write 8 __ = 4.) On your boards, write the equation, filling in the missing fraction. 

S: (Write 8 = 4.) 

Continue with the following possible suggestions: . 


Materials: (S) Personal white boards, Grade 5 Mathematics Reference Sheet (G5–M4–Lesson 8) 

Note: This fluency prepares students for G5–M4–Lessons 9–12. Allow students to use the conversion 

reference sheet if they are confused, but encourage them to answer questions without looking at it. 

Lesson 9: Find a fraction of a measurement, and solve word problems. 

Date: 11/9/13 4.C.43 





T: (Write 1 pt = __ c.) How many cups are in one pint? 

S: 2 cups. 

T: (Write 1 pt = 2 c. Below it, write 2 pt = __ c.) 2 pints? 

S: 4 cups. 

T: (Write 2 pt = 4 c. Below it, write 3 pt = __ c.) 3 pints? 

S: 6 cups. 

T: (Write 3 pt = 6 c. Below it, write 7 pt = __ c.) On your boards, write the equation. 

S: (Write 7 pt = 14 c.) 

T: Write the multiplication equation you used to solve it. 

S: (Write 7 pt × 2 = 14 c.) 

Continue with the following possible sequence: 1 ft = 12 in, 2 ft = 24 in, 4 ft = 48 in, 1 yd = 3 ft, 

2 yd = 6 ft, 3 yd = 9 ft, 9 yd = 27 ft, 1 gal = 4 qt, 2 gal = 8 qt, 3 gal = 12 qt, and 6 gal = 24 qt. 

Multiply a Fraction and a Whole Number (4 minutes) 



T: (Write 4 = .) On your boards, write the equation as a repeated addition sentence and solve. 

S: (Write .) 

T: (Write .) On your boards, fill in the multiplication expression for the numerator. 

S: (Write 4 = .) 

T: (Write 4 = = .) Fill in the missing numbers. 

S: (Write 4 = = 2.) 

T: (Write 4 = = .) Find a common factor to simplify, then multiply. 

S: (Write 4 = = = 2.) 

1 

2 

Continue with the following possible suggestions: 6 , 6 , 8, and 9 . 


Date: 11/9/13 4.C.44 






There are 42 people at a museum. Two-thirds of them are children. How many children are at the museum? 

Extension: If 13 of the children are girls, how 

many more boys than girls are at the museum? 

Note: To y’s Application Problem is a multi-step 

problem. Students must find a fraction of a set 

and then use that information to answer the 

question. The numbers are large enough to 

encourage simplifying strategies as taught in G5– 

M4–Lesson 8 without being overly burdensome 

for students who prefer to multiply and then 

simplify or still prefer to draw their solution using 

a tape diagram. 


Materials: (T) Grade 5 Mathematics Reference Sheet (posted) (S) Personal white board, Grade 5 

Mathematics Reference Sheet (G5–M4–Lesson 8) 

Problem 1 

lb = _____ oz 

T: (Post Problem 1 on the board.) Which is a larger unit, pounds or ounces? 

S: Pounds. 

T: So, we are expressing a fraction of a 

larger unit as the smaller unit. We 

want to find of 1 pound. (Write × 

1 lb.) We know that 1 pound is the 

same as how many ounces? 

S: 16 ounces. 

T: Let’s re me the pound in our 

expression as ounces. Write it on 

your personal board. 

S: (Write × 16 ounces.) 

T: (Write × 1 lb = × 16 ounces.) How do you know this is true? 

S: It’s true bec use we just re me the pound as the same amount in ounces. One pound is the 

same amount as 16 ounces. 

T: How will we find how many ounces are in a fourth of a pound? Turn and talk. 


Date: 11/9/13 4.C.45 





S: We can find of 16. We can multiply × 16. It’s fr ctio of set. We’ll just multiply by 

fourth. We can draw a tape diagram and find one-fourth of 16. 

T: Choose one with your partner and solve. 

S: (Work.) 

T: How many ounces are equal to one-fourth of a pound? 

S: 4 ounces. (Write lb = 4 oz.) 

T: So, each fourth of a pound in our tape diagram is equal to 4 ounces. How many ounces in twofourths 

of a pound? 

S: 8 ounces. 

T: Three-fourths of a pound? 

S: 12 ounces. 

Problem 2 

ft = _____ in 

T: Compare this problem to the first one. Turn and talk. 

S: We’re still re mi g fr ctio of l rger u it 

as a smaller unit. This time we’re 

changing feet to inches, so we need to think 

about 12 instead of 16. We were only 

finding 1 unit last time; this time we have to 

find 3 units. 

T: (Write × 1 foot.) We know that 1 foot is 

the same as how many inches? 

S: 12 inches. 

T: Let’s re me the foot in our expression as inches. Write it on your white 

board. 

S: (Write × 12 inches.) 

T: (Write × 1 ft = × 12 inches.) Is this true? How do you know? 

S: This is just like last time. We i ’t ch ge the mou t th t we have in the expression. We just 

renamed the 1 foot using 12 inches. Twelve inches and one foot are exactly the same length. 

T: Before we solve this let’s estim te our swer. We re fi i g p rt of foot. Will our swer be 

more than 6 inches or less than 6 inches? How do you know? Turn and talk. 

S: Six inches is half a foot. We are looking for 3 fourths of a foot. Three-fourths is greater than onehalf 

so our answer will be more than 6. It will be more than 6 inches. Six is only half and 3 

fourths is almost a whole foot. 


Date: 11/9/13 4.C.46 





T: Work with a neighbor to solve this problem. One of 

you can use multiplication to solve and the other can 

use a tape diagram to solve. Check your eighbor’s 

work whe you’re fi ishe . 

S: (Work and share.) 

T: Reread the problem and fill in the blank. 

S: feet = 9 inches. 

T: How can 3 fourths be equal to 9? Turn and talk. 

NOTES ON 


ENGAGEMENT: 

Challenge students to make 

conversions between fractions of 

gallons to pints or cups, or fractions of 

a day to minutes or even seconds. 

S: Because the units are different, the numbers will be different, but show the same amount. Feet 

are larger than inches, so it takes more inches than feet to show the same amount. If you 

measured 3 fourths of a foot with a ruler and then measured 9 inches with a ruler, they would be 

exactly the same length. If you measure the same length using feet and then using inches, you 

will always have more inches than feet because inches are smaller. 

Problem 3 

Mr. Corsetti spends of every year in Florida. How many months does he spend in Florida each year? 

T: Work independently. You may use either a 

tape diagram or a multiplication sentence to 

solve. 

T: Use your work to answer the question. 

S: Mr. Corsetti spends 8 months in Florida each 

year. 

Repeat this sequence with yard = ______ft and 

hour = ________minutes. 


Students should do their personal best to complete the Problem Set within the allotted 10 minutes. For some 

classes, it may be appropriate to modify the assignment by specifying which problems they work on first. 

Some problems do not specify a method for solving. Students solve these problems using the RDW approach 



Lesson Objective: Find a fraction of a measurement, and solve word problems. 





lesson. 


Date: 11/9/13 4.C.47 








with your partner. 

• In Problem 3, could you tell, without calculating, 

whether Mr. Paul bought more cashews or 

walnuts? How did you know? 

• How did you solve Problem 3(c)? Is there more 

than one way to solve this problem? (Yes, there 

is more than one way to solve this problem, i.e., 

finding of 16 and of 16, and then subtracting, 

versus subtracting 

, and then finding the 

fraction of 16.) Share your strategy with a 

partner. 

• How did you solve Problem 3(d)? Share and 

explain your strategy with a partner. 




the stu e ts’ u erst i g of the co cepts th t were 



the students. 


Date: 11/9/13 4.C.48 





Name 

Date 

1. Convert. Show your work using a tape diagram or an equation. The first one is done for you. 

a. yard = ________ feet 

yd = × 1 yard 

= × 3 feet 

= feet 

= feet 

b. foot = ________ inches 

foot = × 1 foot 

= × 12 inches 

= 

? 

12 

c. year = ________ months d. meter = ________ centimeters 

e. hour = ________ minutes f. yard = ________ inches 


Date: 11/9/13 4.C.49 





2. Mrs. Lang told her class that the class’s pet hamster is ft in length. How long is the hamster in inches? 

3. At the market, Mr. Paul bought lb of cashews and lb of walnuts. 

a. How many ounces of cashews did Mr. Paul buy? 

b. How many ounces of walnuts did Mr. Paul buy? 

c. How many more ounces of cashews than walnuts did Mr. Paul buy? 

d. If Mrs. Toombs bought pounds of pistachios, who bought more nuts, Mr. Paul or Mrs. Toombs? 

How many ounces more? 

4. A jewelry maker purchased 20 inches of gold chain. She used of the chain for a bracelet. How many 

inches of gold chain did she have left? 


Date: 11/9/13 4.C.50 





Name 

Date 

1. Express 36 minutes as a fraction of an hour: 36 minutes = _________hour 

2. Solve. 

a. ft = _______inches b. meter = _______ cm c. year = ______ months 


Date: 11/9/13 4.C.51 





Name 

Date 

1. Convert. Show your work using a tape diagram or an equation. The first one is done for you. 

a. yard = ________ inches 

b. foot = ________ inches 

yd = × 1 yard 


= inches 

= 9 inches 

foot = × 1 foot 


= 

? 

12 

c. year = ________ months d. meter = ________ centimeters 

e. hour = ________ minutes f. yard = ________ inches 

2. Michelle measured the length of her forearm. It was of a foot. How long is her forearm in inches? 


Date: 11/9/13 4.C.52 





3. At the market, Ms. Winn bought lb of grapes and lb of cherries. 

a. How many ounces of grapes did Ms. Winn buy? 

b. How many ounces of cherries did Ms. Winn buy? 

c. How many more ounces of grapes than cherries did Ms. Winn buy? 

d. If Mr. Phillips bought pounds of raspberries, who bought more fruit, Ms. Winn or Mr. Phillips? 

How many ounces more? 

4. A gardener has 10 pounds of soil. He used of the soil for his garden. How many pounds of soil did he 

use in the garden? How many pounds did he have left? 


Date: 11/9/13 4.C.53 






G R A D E 


Topic D 

Fraction Expressions and Word 

Problems 

5.OA.1, 5.OA.2, 5.NF.4a, 5.NF.6 

Focus Standard: 5.OA.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions 

with these symbols. 


5.OA.2 

5.NF.4a 

5.NF.6 

Write simple expressions that record calculations with numbers, and interpret 

numerical expressions without evaluating them. For example, express the calculation 

“add 8 and 7, then multiply by 2” as 2 × (8 +7). Recognize that 3 × (18932 + 921) is three 

times as large as 18932 + 921, without having to calculate the indicated sum or product. 

Apply and extend previous understandings of multiplication to multiply a fraction or 



equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a 

visual fraction model to show (2/3 × 4 = 8/3, and create a story context for this 


Solve real world problems involving multiplication of fractions and mixed numbers, e.g., 

by using visual fraction models or equations to represent the problem. 

Coherence -Links from: G4–M2 Unit Conversions and Problem Solving with Metric Measurement 


Interpreting numerical expressions opens Topic D as students learn to evaluate expressions with parentheses, 

such as 3 × (2/3 – 1/5) or 2/3 × (7 +9) (5.OA.1). They then learn to interpret numerical expressions such as 3 

times the difference between 2/3 and 1/5 or two thirds the sum of 7 and 9 (5.OA.2). Students generate word 

problems that lead to the same calculation (5.NF.4a), such as, “Kelly combined 7 ounces of carrot juice and 5 

ounces of orange juice in a glass. Jack drank 2/3 of the mixture. How much did Jack drink?” Solving word 

problems (5.NF.6) allows students to apply new knowledge of fraction multiplication in context, and tape 

diagrams are used to model multi-step problems requiring the use of addition, subtraction, and multiplication 

of fractions. 

Topic D: Fraction Expressions and Word Problems 

Date: 11/10/13 4.D.1 




NYS COMMON CORE MATHEMATICS CURRICULUM Topic D 5•4 

A Teaching Sequence Towards Mastery of Fraction Expressions and Word Problems 

Objective 1: Compare and evaluate expressions with parentheses. 

(Lesson 10) 

Objective 2: Solve and create fraction word problems involving addition, subtraction, and 



Topic D: Fraction Expressions and Word Problems 

Date: 11/10/13 4.D.2 





Lesson 10 

Objective: Compare and evaluate expressions with parentheses. 






Total Time 

(12 minutes) 

(5 minutes) 

(33 minutes) 

(10 minutes) 

(60 minutes) 


• Convert Measures from Small to Large Units 4.MD.1 


• Find the Unit Conversion 5.MD.2 

(5 minutes) 

(3 minutes) 

(4 minutes) 

Convert Measures from Small to Large Units (5 minutes) 


Note: This fluency reviews G5–M4–Lesson 9 and prepares students for G5–M4–Lessons 10–12 content. 

Allow students to use the conversion reference sheet if they are confused, but encourage them to answer 

questions without looking at it. 

T: (Write 12 in = __ ft.) How many feet are in 12 inches? 

S: 1 foot. 

T: (Write 12 in = 1 ft. Below it, write 24 in = __ ft.) 24 inches? 

S: 2 feet. 


S: 3 feet. 


S: 4 feet. 

T: (Write 48 in = 4 ft. Below it, 120 in = __ ft.) On your boards, write the equation. 

S: (Write 120 in = 10 ft.) 

T: (Write 120 in ÷ __ = __ ft.) Write the division equation you used to solve it. 

S: (Write 120 in ÷ 12 = 10 ft.) 

Continue with the following possible sequence: 2 c = 1 pt, 4 c = 2 pt, 6 c = 3 pt, 16 c = 8 pt, 3 ft = 1 yd, 

Lesson 10: Compare and evaluate expressions with parentheses. 

Date: 11/10/13 4.D.3 





6 ft = 2 yd, 9 ft = 3 yd, 27 ft = 9 yd, 4 qt = 1 gal, 8 qt = 2 gal, 12 qt = 3 gal, and 24 qt = 6 gal. 




T: ( .) On your boards, fill in the multiplication expression for the numerator. 

S: (Write × 6 = .) 

T: (Write × 6 = = .) Fill in the missing numbers. 

S: (Write × 6 = = 3.) 

T: (Write × 6 = = .) Find a common factor to simplify. Then multiply. 

S: (Write × 6 = = = 3.) 

Continue with the following possible suggestions: 6 × , 9 × , × 12, and 12 × . 

Find the Unit Conversion (4 minutes) 



T: 1 foot is equal to how many inches? 

S: 12 inches. 

T: (Write 1 ft = 12 inches to label the tape diagram. 

Below it, write ft = 

× 1 ft.) Rewrite the 

expression on the right of the equation on your 

board substituting 12 inches for 1 foot. 

S: (Write × 12 inches.) 

T: How many inches in each third? Represent the 

division using a fraction. 

S: . 

1 

T: How many inches in 2 thirds of a foot? Keep the division as a fraction. 

S: (Write × 2 = 8 inches.) 

T: Let’s read it. 12 divided by 3 times 2. 

3 


Date: 11/10/13 4.D.4 





T: How many inches are equal to foot? 

S: 8 inches. 

Continue with the following possible sequence: 

lb = __ oz, yr = __ months, lb = __ oz, and hr = __ min. 


Bridget has $240. She spent of her money and saved the rest. How much more money did she spend than 

save? 

Note: This Application Problem provides a quick review of fraction of a set, which students have been 

working on in Topic C, and provides a bridge to the return to this work in G5–M4–Lesson 11. It is also a 

multi-step problem. 



Problem 1: Write an expression to match a tape diagram. Then 

evaluate. 

a. 

T: (Post the first tape diagram.) Read the expression that 

names the whole. 

S: 9 + 11. 

T: What do we call the answer to an addition sentence? 


Date: 11/10/13 4.D.5 





S: A sum. 

T: So our whole is the sum of 9 and 11. (Write the sum of 

9 and 11 next to tape diagram.) How many units is the 

sum being divided into? 

S: Four. 

T: What is the name of that fractional unit? 

S: Fourths. 

T: How many fourths are we trying to find? 

S: 3 fourths. 

T: So, this tape diagram is showing 3 fourths of the sum 

of 9 and 11. (Write 3 fourths next to the sum of 9 and 

11.) Work with a partner to write a numerical 

expression to match these words. 

S: ( ) . . 3. 

T: I noticed that many of you put parentheses around 9 + 

11. Explain to a neighbor why that is necessary. 

S: The parentheses tell us to add 9 and 11 first, and then 

multiply. If the parentheses weren’t there, we 

would have to multiply first. We want to find the sum 

first and then multiply. We can find the sum of 9 

and 11 first, and then divide the sum by 4. 

T: Work with a partner to evaluate or simplify this expression. 

S: (Work to find 15.) 

NOTES ON 



If students have difficulty 

understanding the value of a unit as a 

subtraction sentence, write a whole 

number on one side of a small piece of 

construction paper and the equivalent 

subtraction expression on the other. 

Place the paper on the model with the 

whole number facing up and ask what 

needs to be done to find the whole. 

(Most will understand the process of 

multiplying to find the whole.) Write 

the multiplication expression using the 

paper, whole number side up. Then 

flip the paper over and show the 

parallel expression using the 

subtraction sentence. 

b. 

T: (Post the second tape diagram.) Look at this model. How is it 

different from the previous example? 

S: This time we don’t know the whole. In this diagram, the 

whole is being divided into fifths, not fourths. Here we 

know what 1 fifth is. We know it is the difference of and 

We have to multiply the difference of and by 5 to find 

the whole. 

T: Read the subtraction expression that tells the value of one 

unit (or 1 fifth) in the model. (Point to .) 


Date: 11/10/13 4.D.6 





S: One-third minus one-fourth. 

T: What is the name for the answer to a subtraction problem? 

S: A difference. 

T: This unit is the difference of one-third and one-fourth. (Write the difference of and next to the 

tape diagram.) How many of these ( 

S: 5 units of . 

) units does our model show? 

T: Work with a partner to write a numerical expression to match these words. 

S: ( ) or ( ) . 

T: Do we need parentheses for this expression? 

S: Yes, we need to subtract first before multiplying. 

T: Evaluate this expression independently. Then compare your work with a neighbor. 

Problem 2: Write and evaluate an expression from word form. 

T: (Write the product of 4 and 2, divided by 3 on the board.) Read the expression. 

S: The product of 4 and 2, divided by 3. 

T: Work with a partner to write a matching numerical 

expression. 

S: (4 × 2) ÷ 3. . 4 × 2 3. 

T: Were the parentheses necessary here? Why or why 

not? 

S: No. Because the product came first and we can do 

multiplication and division left to right. We didn’t need 

them. I wrote it as a fraction. I didn’t use 

parentheses because I knew before I could divide by 3. 

I needed to find the product in the numerator. 

T: Work independently to evaluate your expression. 

Express your answer as both a fraction greater than 

one and as a mixed number. Check your work with a 

neighbor when you’re finished. 

S: (Work to find and . Then check.) 

Problem 3: Evaluate and compare equivalent expressions. 

NOTES ON 



It may be necessary to prompt 

students to use fraction notation for 

the division portion of the expression. 

Pointing out the format of the division 

sign—dot over dot—may serve as a 

good reminder. Reminding students of 

problems from the beginning of G5– 

Module 4 may also be helpful, (e.g., 

2 3 = ). 

a. 2 3 × 4 b. 4 thirds doubled 

c. 2 (3 × 4) d. × 4 

e. 4 copies of the sum of one-third and one-third f. (2 3) × 4 

T: Evaluate these expressions with your partner. Keep working until I call time. Be prepared to share. 


Date: 11/10/13 4.D.7 





S: (Work.) 

T: Share your work with someone else’s partner. What do you notice? 

S: The answer is 8 thirds every time except in (c). All of the expressions are equivalent except (c). 

These are just different ways of expressing . 

T: What was different about (c)? 

S: Since the expression had parentheses, we had to multiply first, then divide. It was equal to 2 

twelfths. It’s tricky because all of the digits and operations are the same as all the others, but the 

order of them and the parentheses resulted in a different value. 

T: Work with a partner to find another way to express . 

S: (Work. Possible expressions include: 3 × (6 ÷ 5). 3 × 6 ÷ 5. 3 . Six times the value of 3 

divided by 5, etc.) 

Invite students to share their expressions on the board and to discuss. 

Problem 5: Compare expressions in word form and numerical form. 

a. the sum of 6 and 14 (6 + 14) 8 

b. 4 × 4 times the quotient of 3 and 8 

c. Subtract 2 from of 9 (11 2) – 2 

T: Let’s use , or to compare expressions. Write the sum of 6 and 14 and (6 + 14) 8 on the 

board.) Draw a tape diagram for each expression and compare them. 

S: (Write the sum of 6 and 14 = (6 + 14) 

8.) 

T: What do you notice about the 

diagrams? 

S: They are drawn exactly the same way. 

We don’t even need to evaluate the 

expressions in order to compare them. 

You can see that they will simplify to 

the same quantity. I knew it would 

be the same before I drew it because 

finding 1 eighth of something and 

dividing by 8 are the same thing. 

T: Look at the next pair of expressions. Work with your partner to compare them without calculating. 

S: (Work and write 4 × > 4 times the quotient of 3 and 8.) 

T: How did you compare these expressions without calculating? 


Date: 11/10/13 4.D.8 





S: They both multiply something by 4. Since 8 thirds 

is greater than 3 eighths, the expression on the 

left is larger. Since both expressions multiply 

with a factor of 4, the fraction that shows the 

smaller amount results in a product that is also 

less. 

T: Compare the final pair of expressions 

independently without calculating. Be prepared 

to share your thoughts. 

S: (Work and write subtract 2 from of 9 < (11 2) 

– 2.) 

T: How did you know which expression was greater? 


S: Eleven divided by 2 is 11 halves and 11 halves is 

greater than 9 halves. Half of 9 is less than 

half of 11, and since we’re subtracting 2 from 

both of them, the expression on the right is 

greater. 




classes, it may be appropriate to modify the assignment 

by specifying which problems they work on first. Some 





Lesson Objective: Compare and evaluate expressions 

with parentheses. 




Set. They should check work by comparing answers 

with a partner before going over answers as a class. 

Look for misconceptions or misunderstandings that can 

be addressed in the Debrief. Guide students in a 

conversation to debrief the Problem Set and process 

the lesson. 


Date: 11/10/13 4.D.9 






• What relationships did you notice between the 

two tape diagrams in Problem 1? 



• What were your strategies of comparing 

Problem 4? Explain it to your partner. 

• How does the use of parentheses affect the 

answer in Problems 4(b) and 4(c)? 

• Were you able to compare the expressions in 

Problem 4(c) without calculating? What made 

it more difficult than (a) and (b)? 

• Explain to your partner how you created the 

line plot in Problem 5(d)? Compare your line 

plot to your partner’s. 







the students. 


Date: 11/10/13 4.D.10 





Name 

Date 

1. Write expressions to match the diagrams. Then evaluate. 

2. Write an expression to match, then evaluate. 

a. the sum of 16 and 20. b. Subtract 5 from of 23. 

c. 3 times as much as the sum of and . d. of the product of and 42. 

e. 8 copies of the sum of 4 thirds and 2 more. f. 4 times as much as 1 third of 8. 


Date: 11/10/13 4.D.11 





3. Circle the expression(s) that give the same product as . Explain how you know. 

( ) ( ) ( ) 

4. Use , or = to make true number sentences without calculating. Explain your thinking. 

a. 

b. ( ) ( ) 

c. ( ) ( ) 


Date: 11/10/13 4.D.12 





5. Collette bought milk for herself each month and recorded the amount in the table below. For (a–c) write 

an expression that records the calculation described. Then solve to find the missing data in the table. 

a. She bought of July’s total in June. 

Month 

Amount (in gallons) 

January 3 

February 2 

March 

b. She bought as much in September as she did in January 

and July combined. 

April 

May 

June 

July 2 

August 1 

c. In April she bought gallon less than twice as much as 

she bought in August. 

September 

October 

d. Display the data from the table in a line plot. 

e. How many gallons of milk did Collette buy from January to October? 


Date: 11/10/13 4.D.13 





Name 

Date 

1. Rewrite these expressions using words. 

a. ( ) b. 

2. Write an equation, then solve. 

a. Three less than one-fourth of the product of eight thirds and nine. 


Date: 11/10/13 4.D.14 





Name 

Date 

1. Write expressions to match the diagrams. Then evaluate. 

2. Circle the expression(s) that give the same product as 6 ×1 . Explain how you know. 

( ) ( ) ( ) × 6 

3. Write an expression to match, then evaluate. 

a. the sum of 23 and 17. b. Subtract 4 from of 42. 

c. 7 times as much as the sum of and . d. of the product of and 16. 

e. 7 copies of the sum of 8 fifths and 4. f. 15 times as much as 1 fifth of 12. 


Date: 11/10/13 4.D.15 





4. Use , or = to make true number sentences without calculating. Explain your thinking. 

a. (9 + 12) 15 

b. ( ) ( ) 

c. ( ) ( ) 

5. Fantine bought flour for her bakery each month and 

recorded the amount in the table to the right. For (a–c) 

write an expression that records the calculation 

described. Then solve to find the missing data in the 

table. 

a. She bought of January’s total in August. 

Month Amount (in pounds) 

January 3 

February 2 

March 

April 

May 

June 

b. She bought as much in April as she did in October 

and July combined. 

July 

August 

September 

October 


Date: 11/10/13 4.D.16 





c. In June she bought pound less than six times as much as she bought in May. 

d. Display the data from the table in a line plot. 

e. How many pounds of flour did Fantine buy from January to October? 


Date: 11/10/13 4.D.17 





Lesson 11 

Objective: Solve and create fraction word problems involving addition, 






Total Time 

(12 minutes) 

(38 minutes) 

(10 minutes) 

(60 minutes) 



• Multiply Whole Numbers by Fractions Using Two Methods 5.NF.4 

• Write the Expression to Match the Diagram 5.NF.4 

(5 minutes) 

(3 minutes) 

(4 minutes) 



Note: This fluency reviews G5–M4–Lessons 9–10 and prepares students for G5–M4–Lessons 11─12 content. 



T: (Write 2 c = __ pt.) How many pints are in 2 cups? 

S: 1 pint. 

T: (Write 2 c = 1 pt. Below it, write 4 c = __ pt.) 4 cups? 

S: 2 pints. 


S: 3 pints. 

T: (Write 6 c = 3 pt. Below it, write 20 c = __ pt.) On your boards, write the equation. 

S: (Write 20 c = 10 pt.) 

T: (Write 20 c ÷ __ = __ pt.) Write the division equation you used to solve it. 

S: (Write 20 c ÷ 2 = 10 pt.) 

Continue with the following possible sequence: 12 in = 1 ft, 24 in = 2 ft, 48 in = 4 ft, 3 ft = 1 yd, 6 ft = 2 yd, 9 ft 

= 3 yd, 24 ft = 8 yd, 4 qt = 1 gal, 8 qt = 2 gal, 12 qt = 3 gal, and 36 qt = 9 gal. 

Lesson 11: Solve and create fraction word problems involving addition, 


Date: 11/10/13 

4.D.18 





Multiply Whole Numbers by Fractions Using Two Methods (3 minutes) 



T: (Write 8 = .) On your boards, write the equation and fill in the multiplication 

expression for the numerator. 

S: (Write × 8 = .) 

T: (Write 8 = = .) Fill in the missing numbers. 

S: (Write 8 = = 4.) 

T: (Write × 8 = .) Divide by a common factor and solve. 

S: (Write 8 = = 4.) 

T: Did you get the same answer using both methods? 

S: Yes. 

Continue with the following possible suggestions: 12 × , 12 × , × 15, and 18 × . 

Write the Expression to Match the Diagram (4 minutes) 



T: (Project a tape diagram partitioned into 3 equal parts with 8 + 3 as 

the whole.) Say the value of the whole. 

S: 11. 

T: On your boards, write an expression to match the diagram. 

S: (Write (8 + 3) ÷ 3.) 

T: Solve the expression. 

S: (Beneath (8 + 3) ÷ 3, write . Beneath , write .) 

Repeat this sequence for the following suggestion: (5 + 6) ÷ 3. 

T: (Project a tape diagram partitioned into 3 equal parts. Beneath one 

of the units, write 

match the diagram. 

S: (Write ( ) 

1 

T: Solve the expression. 

4 

.) On your boards, write an expression to 



Date: 11/10/13 

4.D.19 





S: (Beneath write Beneath it, write . And, beneath that, write .) 

Continue with the following possible suggestion: ( ) 



Note: Because today’s lesson involves solving word problems time allocated to the Application Problem has 

been allotted to the Concept Development. 



Have two pairs of student who can successfully model the problem work at the board while the others work 

independently or in pairs at their seats. Review the following questions before beginning the first problem: 










Give everyone two minutes to finish work on that question, 

sharing their work and thinking with a peer. All should then 

write their equations and statements of the answer. 


Give students one to two minutes to assess and explain the 

reasonableness of their solution. 

NOTES ON 


ENGAGEMENT: 

When a task offers varied approaches 

for solving, an efficient way to have all 

students’ work shared is to hold a 

“museum walk.” 

This method for sharing works best 

when a purpose for the looking is 

given. For example, students might be 

asked to note similarities and 

differences in the drawing of a model 

or approach to calculation. Students 

can indicate the similarities or 

differences by using sticky-notes to 

color code the displays, or they may 

write about what they notice in a 

journal. 

A general instructional note on today’s problems: The problem 

solving in today’s lesson requires that students combine their 

previous knowledge of adding and subtracting fractions with new knowledge of multiplying to find fractions 

of a set. The problems have been designed to encourage flexibility in thinking by offering many avenues for 

solving each one. Be sure to conclude the work with plenty of time for students to present and compare 

approaches. 



Date: 11/10/13 

4.D.20 





Problem 1 

Kim and Courtney share a 16-ounce box of cereal. By the end of the week, Kim has eaten of the box, and 

Courtney has eaten of the box of cereal. What fraction of the box is left? 

To complete Problem 1, students must find fractions of a set and use skills learned in Module 3 to add or 

subtract fractions. 

As exemplified, students may solve this multi-step word problem using different methods. Consider 

demonstrating these two methods of solving Problem 1 if both methods are not mentioned by students. 

Point out that the rest of today’s Problem Set can be solved using multiple strategies as well. 

If desired this problem’s complexity may be increased by changing the amount of the cereal in the box to 0 

ounces and Courtney’s fraction to . This will produce a mixed number for both girls Kim’s portion becomes 

ounces and Courtney’s becomes . 

Problem 2 

Mathilde has 20 pints of green paint. She uses of it to paint a landscape and 

of it while painting a clover. 

She decides that for her next painting she will need 14 pints of green paint. How much more paint will she 

need to buy? 



Date: 11/10/13 

4.D.21 





Complexity is increased here as students are called on to maintain a high level of organization as they keep 

track of the attribute of paint used and not used. Multiple approaches should be encouraged. For Methods 1 

and 2, the used paint is the focus of the solution. Students may choose to find the fractions of the whole 

(fraction of a set) Mathilde has used on each painting or may first add the separate fractions before finding 

the fraction of the whole. Subtracting that portion from the 14 pints she’ll need for her next project yields 

the answer to the question. Method 3 finds the left over paint and simply subtracts it from the 14 pints 

needed for the next painting. 

Problem 3 

Jack, Jill, and Bill each carried a 48-ounce bucket full of water down the hill. By the time they reached the 

bottom Jack’s bucket was only full Jill’s was full and Bill’s was full. How much water did they spill 

altogether on their way down the hill? 



Date: 11/10/13 

4.D.22 





This problem is much like Problem 2 in that students keep track of one attribute—spilled water or un-spilled 

water. However, the inclusion of a third person, Bill, requires that students keep track of even more 

information. In Method 1, a student may opt to find the fraction of water still remaining in each bucket. This 

process requires the students to then subtract those portions from the 48 ounces that each bucket held 

originally. In Method 2 students may decide to find what fraction of the water has been spilled by counting 

on to a whole (e.g., if 3 fourths remain in Jack’s bucket then only 1 fourth has been spilled.) This is a more 

direct approach to the solution as subtraction from 48 is not necessary. 

Problem 4 

Mrs. Diaz makes 5 dozen cookies for her class. One-ninth of her 27 students are absent the day she brings the 

cookies. If she shares the cookies equally among the students who are present, how many cookies will each 

student get? 



Date: 11/10/13 

4.D.23 





This problem is straightforward, yet the division of the cookies at the end provides an opportunity to call out 

the division interpretation of a fraction. With quantities like 60 and 24, students will likely lean toward the 

long division algorithm, so using fraction notation to show the division may need to be discussed as an 

alternative. Using the fraction and renaming using larger units may be the more efficient approach given the 

quantities The similarities and differences of these approaches certainly bear a moment’s discussion In 

addition, the practicality of sharing cookies in twenty-fourths can lead to a discussion of renaming as . 

Problem 5 

Create a story problem about a fish tank for the tape diagram below. Your story must include a fraction. 

84 

? 

In this problem, students are shown a tape diagram marking 84 

as the whole and partitioned into 6 equal units (or sixths). The 

question mark should signal students to find of the whole. 

Students are asked to create a word problem about a fish tank. 

Students should be encouraged to use their creativity while 

generating a word problem, but remain mathematically sound. 

Two sample stories are included here, but this is a good 

opportunity to have students share aloud their own word 

problems. 

NOTES ON 


ACTION AND 

EXPRESSION: 

In general, early finishers for this 

lesson will be students who use an 

abstract, more procedural approach to 

solving. These students might be 

asked to work the problems again 

using a well-drawn tape diagram using 

color that would explain why their 

calculation is valid. These models 

could be displayed in hallways or 

placed in a class book. 



Date: 11/10/13 

4.D.24 






Lesson Objective: Solve and create fraction word 

problems involving addition, subtraction, and 










the lesson. 



• How are the problems alike? How are they 

different? 

• How many strategies can you use to solve the 

problems? 

• How was your solution the same and different 

from those that were demonstrated? 

• Did you see other solutions that surprised you or 

made you see the problem differently? 

• How many different story problems can you 

create for Problem 5? 







students. 



Date: 11/10/13 

4.D.25 





Name 

Date 

1. Kim and Courtney share a 16-ounce box of cereal. By the end of the week, Kim has eaten of the box, 

and Courtney has eaten of the box of cereal. What fraction of the box is left? 

2. Mathilde has 20 pints of green paint. She uses of it to paint a landscape and of it while painting a 

clover. She decides that for her next painting she will need 14 pints of green paint. How much more 

paint will she need to buy? 



Date: 11/10/13 

4.D.26 





3. Jack, Jill, and Bill each carried a 48-ounce bucket full of water down the hill. By the time they reached the 

bottom Jack’s bucket was only full Jill’s was full and Bill’s was full. How much water did they spill 

altogether on their way down the hill? 

4. Mrs. Diaz makes 5 dozen cookies for her class. One-ninth of her 27 students are absent the day she 

brings the cookies. If she shares the cookies equally among the students who are present, how many 

cookies will each student get? 

5. Create a story problem about a fish tank for the tape diagram below. Your story must include a fraction. 

84 

? 



Date: 11/10/13 

4.D.27 





Name 

Date 

1. Use a tape diagram to solve. 

a. of 5 



Date: 11/10/13 

4.D.28 





Name 

Date 

1. Jenny’s mom says she has an hour before it’s bedtime Jenny spends of the hour texting a friend and 

of the remaining time brushing her teeth and putting on her pajamas. She spends the rest of the time 

reading her book. How long did Jenny read? 

2. A-Plus Auto Body is painting flames on a customer’s car They need pints of red, 3 pints of orange, 

pint of yellow, and 7 pints of blue paint. They use of the blue paint to make the flames. They need 

pints to paint the next car blue. How much more blue paint will they need to buy? 

3. Giovanna, Frances, and their dad each carried a 10-pound bag of soil into the backyard. After putting soil 

in the first flower bed, Giovanna’s bag was full Frances’ bag was full and their dad’s was full. How 

many ounces of soil did they put in the first flower bed altogether? 



Date: 11/10/13 

4.D.29 





4. Mr. Chan made 252 cookies for the Annual Fifth Grade Class Bake Sale. They sold of them and of the 

remaining cookies were given to P.T.A. members. Mr. Chan allowed the 12 student-helpers to divide the 

cookies that were left equally. How many cookies will each student get? 

5. Create a story problem about a farm for the tape diagram below. Your story must include a fraction. 

105 

? 



Date: 11/10/13 

4.D.30 





Lesson 12 

Objective: Solve and create fraction word problems involving addition, 







Total Time 

(12 minutes) 

(6 minutes) 

(32 minutes) 

(10 minutes) 

(60 minutes) 




• Write the Expression to Match the Diagram 5.NF.4 

(4 minutes) 

(4 minutes) 

(4 minutes) 



Note: This fluency reviews G5–M4–Lessons 9–11 and prepares students for G5–M4–Lesson 12 content. 



T: (Write 1 ft = __ in.) How many inches are in 1 foot? 

S: 12 inches. 

T: (Write 1 ft = 12 in. Below it, write 2 ft = __ in.) 2 feet? 

S: 24 inches. 


S: 48 inches. 

T: Write the multiplication equation you used to solve it. 

S: (Write 4 ft × 12 = 48 in.) 

Continue with the following possible sequence: 1 pint = 2 cups, 2 pints = 4 cups, 7 pints = 14 cups, 1 yard = 3 

ft, 2 yards = 6 ft, 6 yards = 18 ft, 1 gal = 4 qt, 2 gal = 8 qt, 9 gal = 36 qt. 

T: (Write 2 c = __ pt.) How many pints are in 2 cups? 



Date: 11/10/13 

4.D.31 





S: 1 pint. 


S: 2 pints. 

T: (Write 4 c = 2 pt. Below it, write 10 c = __ pt.) 10 

cups? 

S: 5 pints 

T: Write the division equation you used to solve it. 

S: (Write 10 c ÷ 2 = 5 pt.) 

Continue with the following possible sequence: 12 in = 1 ft, 36 

in = 3 ft, 3 ft = 1 yd, 12 ft = 4 yd, 4 qt = 1 gal, and 28 qt = 7 gal. 

NOTES ON 


ENGAGEMENT: 

For English language learners and 

struggling students, provide the 

conversion reference sheet included in 

G5–M4–Lesson 9. 



Note: This fluency reviews G5–M4–Lessons 9–11. 

T: (Write 9 ÷ 3 =__.) Say the division sentence. 

S: 9 ÷ 3 = 3. 

T: (Write 9 =__.) Say the multiplication sentence. 

S: 9 = 3. 

T: (Write 9 =__.) On your boards, write the multiplication sentence. 

S: (Write 9 = 6.) 

T: (Write 9 =__.) On your boards, write the multiplication sentence. 

S: (Write 9 = 6.) 

Continue with the following possible sequence: 12 ÷ 6, 12, 12, 12 , 24, 24 , 24 , 12, and 

12 . 

Write the Expression to Match the Diagram (4 minutes) 


Note: This fluency reviews G5–M4–Lessons 10─11. 

T: (Project a tape diagram partitioned into 3 equal units with 15 as the 

whole and 2 units shaded.) Say the value of the whole. 

S: 15. 

T: On your boards, write an expression to match the diagram using a fraction. 



Date: 11/10/13 

4.D.32 





S: (Write 

T: To solve we can write 15 divided by 3 to find the value of one unit, times 2. (Write as you say 

the words.) 

T: Find the value of the expression. 

S: (Write 

Continue this process for the following possible suggestion: , , and . 


Complete the table. 

yds 

_______ feet 

4 pounds _______ ounces 

8 tons _______ pounds 

gallon 

year 

hour 

_______ quarts 

_______months 

_______minutes 

Note: The chart requires students to work within many customary systems reviewing the work of G5–M4– 

Lesson 9. Students may need a conversion chart (see G5–M4–Lesson 9) to scaffold this problem. 



Date: 11/10/13 

4.D.33 









Have two pairs of student who can successfully model the problem work at the board while the others 

work independently or in pairs at their seats. Review the following questions before beginning the first 

problem: 




As students work, circulate. Reiterate the questions above. After two minutes, have the two pairs of 

students share only their labeled diagrams. For about one minute, have the demonstrating students receive 

and respond to feedback and questions from their peers. 



sharing their work and thinking with a peer. All should then 

write their equations and statements of the answer. 


Give students one to two minutes to assess and explain the 

reasonableness of their solution. 

A general instructional note on today’s problems: T day’s 

problems are more complex than those found in G5–M4–Lesson 

11. All are multi-step. Students should be strongly encouraged 

to draw before attempting to solve. As in G5–M4–Lesson 11, 

multiple approaches to solving all the problems are possible. 

Students should be given time to share and compare thinking 

during the Debrief. 

NOTES ON 


ENGAGEMENT: 

The complexity of the language 

inv lved in t day’s p blems may p se 

significant challenges to English 

language learners or those students 

with learning differences that affect 

language processing. Consider pairing 

these students with those in the class 

who are adept at drawing clear 

models. These visuals and the peer 

interaction they generate can be 

invaluable bridges to making sense of 

the written word. 



Date: 11/10/13 

4.D.34 





Problem 1 

A baseball team played 32 games and lost 8. Katy was the catcher in of the winning games and of the 

losing games. 

a. What fraction of the games did the team win? 

b. In how many games did Katy play catcher? 

While Part A is relatively straightforward, there are still varied approaches for solving. Students may find the 

difference between the number of games played and lost to find the number of games won (24) expressing 

this difference as a fraction ( 

). Alternately they may conclude that the losing games are of the total and 

deduce that winning games must constitute . Watch for students distracted by the fractions and written 

in the stem and somehow try to involve them in the solution to Part (a). Complexity increases as students 

must employ the fraction of a set strategy twice, carefully matching each fraction with the appropriate 

number of games and finally combining the number of games that Katy played to find the total. 



Date: 11/10/13 

4.D.35 





Problem 2 

In M s Elli tt’s ga den of the flowers are red, of them are purple, and of the remaining flowers are pink. 

If there are 128 flowers, how many flowers are pink? 

The increase in complexity for this problem comes as students are asked to find the number of pink flowers in 

the garden. This portion of the flowers refers to 1 fifth of the remaining flowers (i.e., 1 fifth of those that are 

not red or purple), not 1 fifth of the total. Some students may realize (as in Method 4) that 1 fifth of the 

remainder is simply equal to 1 unit or 16 flowers. Multiple methods of drawing and solving are possible. 

Some of the possibilities are pictured above. 



Date: 11/10/13 

4.D.36 





Problem 3 

Lillian and Darlene plan to get their homework finished within one hour. Darlene completes her math 

homework in hour. Lillian completes her math homework with hour remaining. Who completes her 

homework faster and by how many minutes? 

Bonus: Give the answer as a fraction of an hour. 

The way in which Lillian’s time is exp essed makes for a bit of complexity in this problem. Students must 

recognize that she only took hour to complete the assignment. Many students may quickly recognize that 

Lillian worked faster as 

. However, students must go further to find exactly how many minutes faster. 

The bonus requires them to give the fraction of an hour. Simplification of this fraction should not be required 

but may be discussed. 



Date: 11/10/13 

4.D.37 





Problem 4 

Create and solve a story problem about a baker and some flour whose solution is given by the 

expression . 

Working backwards from expression to story may be challenging for some students. Since the expression 

given contains parentheses, the story created must first involve the addition or combining of 3 and 5. For 

students in need of assistance, drawing a tape diagram first may be of help. Then, asking the simple prompt, 

“What would a baker add together or combine?” may be enough to get the students started. Evaluating 

should pose no significant challenge to students. Note that the story of the chef interprets the 

expression as repeated addition of a fourth where the story of the baker interprets the expression as a 

fraction of a set. 

Problem 5 

Create and solve a story problem about a baker and 36 kilograms of an ingredient that is modeled by the 

following tape diagram. Include at least one fraction in your story. 

36 

? 

Again, students are asked to both create and then solve a story problem, this time using a given tape diagram. 

The challenge here is that this tape diagram implies a two-step word problem. The whole, 36, is first 

partitioned into thirds, and then one of those thirds is divided in half. The story students create should reflect 

this two-part drawing. Students should be encouraged to share aloud and discuss their stories and thought 

process for solving. 



Date: 11/10/13 

4.D.38 





Problem 6 

Of the students in M Smith’s fifth grade class, 

class, 

we e absent n M nday Of the students in M s Jac bs’ 

were absent on Monday. If there were 4 students absent in each class on Monday, how many 

students are in each class? 

For this problem, students need to find the 

whole. An interesting aspect of this 

problem is that fractional amounts of 

different wholes can be the same amount. 

In this case, two-fifths of 10 is the same as 

one-third of 12. This should be discussed 

with students. 


Lesson Objective: Solve and create fraction word 

problems involving addition, subtraction, and 










the lesson. 

You may choose to use any combination of the 

questions below to lead the discussion. 

• How are the problems alike? How are they 

different? 

• How many strategies can you use to solve the 

problems? 

• How was your solution the same and different 

from those that were demonstrated? 

• Did you see other solutions that surprised you or made you see the problem differently? 

• How many different story problems can you create for Problem 5 and Problem 6? 



Date: 11/10/13 

4.D.39 








the students’ unde standing f the c ncepts that we e 



the students. 



Date: 11/10/13 

4.D.40 





Name 

Date 

1. A baseball team played 32 games, and lost 8. Katy was the catcher in of the winning games and of the 

losing games. 

a. What fraction of the games did the team win? 

b. In how many games did Katy play catcher? 

2. In M s Elli tt’s ga den of the flowers are red, of them are purple, and of the remaining flowers are 

pink. If there are 128 flowers, how many flowers are pink? 



Date: 11/10/13 

4.D.41 





3. Lillian and Darlene plan to get their homework finished within one hour. Darlene completes her math 

homework in hour. Lillian completes her math homework with hour remaining. Who completes her 

homework faster and by how many minutes? 

Bonus: Give the answer as a fraction of an hour. 

4. Create and solve a story problem about a baker and some flour whose solution is given by the expression 

. 



Date: 11/10/13 

4.D.42 





5. Create and solve a story problem about a baker and 36 kilograms of an ingredient that is modeled by the 

following tape diagram. Include at least one fraction in your story. 

36 

kg 

? 

6. Of the students in M Smith’s fifth grade class, were absent on Monday. Of the students in Mrs. 

Jac bs’ class, were absent on Monday. If there were 4 students absent in each class on Monday, how 

many students are in each class? 



Date: 11/10/13 

4.D.43 





Name 

Date 

In a classroom, of the students are wearing blue shirts and are wearing white shirts. There are 36 students 

in the class. How many students are wearing a shirt other than blue or white? 



Date: 11/10/13 

4.D.44 





Name 

Date 

1. Terrence finished a word search in the time it took Frank. Charlotte finished the word search in the 

time it took Terrence. Frank finished the word search in 32 minutes. How long did it take Charlotte to 

finish the word search? 

2. Ms. Phillips ordered 56 pizzas for a school fundraiser. Of the pizzas ordered, of them were pepperoni, 

19 were cheese, and the rest were veggie pizzas. What fraction of the pizzas was veggie? 



Date: 11/10/13 

4.D.45 





3. In an auditorium, of the students are fifth graders, are fourth graders, and of the remaining students 

are second graders. If there are 96 students in the auditorium, how many second graders are there? 

4. At a track meet, Jacob and Daniel compete in the 220 m hurdles. Daniel finishes in of a minute. Jacob 

finishes with 

of a minute remaining. Who ran the race in the faster time? 

Bonus: Give the answer as a fraction of a minute. 



Date: 11/10/13 

4.D.46 





5. Create and solve a story problem about a runner who is training for a race. Include at least one fraction in 

your story. 

48 km 

? 

6. Create and solve a story problem about a two friends and their weekly allowance whose solution is given 

by the expression . 



Date: 11/10/13 

4.D.47 






G R A D E 


Topic E 

Multiplication of a Fraction by a 

Fraction 

5.NBT.7, 5.NF.4a, 5.NF.6, 5.MD.1, 5.NF.4b 

Focus Standard: 5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or 


relationship between addition and subtraction; relate the strategy to a written method 

and explain the reasoning used. 


5.NF.4a 

5.NF.6 

5.MD.1 

Coherence -Links from: G4–M6 Decimal Fractions 

G5–M2 

Apply and extend previous understandings of multiplication to multiply a fraction or 



equivalently, as the result of a sequence of operations a × q ÷ b. For example, use 

a visual fraction model to show (2/3 × 4 = 8/3, and create a story context for this 




Convert among different-sized standard measurement units within a given 

measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in 

solving multi-step, real world problems. 

Multi-Digit Whole Number and Decimal Fraction Operations 

-Links to: G6–M2 Arithmetic Operations Including Division by a Fraction 

G6–M4 

Expressions and Equations 

Topic E introduces students to multiplication of fractions by fractions—both in fraction and decimal form 

(5.NF.4a, 5.NBT.7). The topic starts with multiplying a unit fraction by a unit fraction, and progresses to 

multiplying two non-unit fractions. Students use area models, rectangular arrays, and tape diagrams to 

model the multiplication. These familiar models help students draw parallels between whole number and 

fraction multiplication and solve word problems. This intensive work with fractions positions students to 

extend their previous work with decimal-by-whole number multiplication to decimal-by-decimal 

multiplication. Just as students used unit form to multiply fractional units by wholes in Module 2 (e.g., 3.5 × 2 

= 35 tenths × 2 ones = 70 tenths), they will connect fraction-by-fraction multiplication to multiply fractional 

units‐by-fractional units. (3.5 × 0.2 = 35 tenths × 2 tenths = 70 hundredths). 

Topic E: Multiplication of a Fraction by a Fraction 

Date: 11/10/13 4.E.1 




NYS COMMON CORE MATHEMATICS CURRICULUM Topic E 5 

3 

of a foot = 3 × 12 inches 

Express 5 3 ft as inches. 

4 4 4 

1 foot = 12 inches 

5 3 ft = (5 × 12) inches + 4 (3 × 12) inches 

4 

= 60 + 9 inches 

= 69 inches 

3 

× 12 4 

Reasoning about decimal placement is an integral part of these lessons. Finding fractional parts of customary 

measurements and measurement conversion (5.MD.1) concludes Topic E. Students convert smaller units to 

fractions of a larger unit (e.g., 6 inches = 1/2 ft). The inclusion of customary units provides a meaningful 

context for many common fractions (1/2pint = 1 cup, 1/3 yard = 1 foot, 1/4 gallon = 1 quart, etc.). This topic, 

together with the fraction concepts and skills learned in Module 3, opens the door to a wide variety of 

application word problems (5.NF.6). 

A Teaching Sequence Towards Mastery of Multiplication of a Fraction by a Fraction 

Objective 1: Multiply unit fractions by unit fractions. 

(Lesson 13) 

Objective 2: Multiply unit fractions by non-unit fractions. 

(Lesson 14) 

Objective 3: Multiply non-unit fractions by non-unit fractions. 

(Lesson 15) 

Objective 4: Solve word problems using tape diagrams and fraction-by-fraction multiplication. 

(Lesson 16) 

Objective 5: Relate decimal and fraction multiplication. 


Objective 6: Convert measures involving whole numbers, and solve multi-step word problems. 

(Lesson 19) 

Objective 7: Convert mixed unit measurements, and solve multi-step word problems. 

(Lesson 20) 

Topic E: Multiplication of a Fraction by a Fraction 

Date: 11/10/13 4.E.2 





Lesson 13 

Objective: Multiply unit fractions by unit fractions. 





Total Time 

(8 minutes) 

(42 minutes) 

(10 minutes) 

(60 minutes) 




(4 minutes) 

(4 minutes) 



Note: This fluency reviews G5─M4─Lessons 9─11. 

T: (Write .) Say the division sentence. 

S: 8 ÷ 4 = 2. 

T: (Write × 8 = .) Say the multiplication sentence. 

S: × 8 = 2. 

T: (Write × 8 = .) On your boards, write the multiplication sentence. 

S: (Write × 8 = 6.) 

T: (Write 8 × = .) On your boards, write the multiplication sentence. 

S: (Write 8 × = 6.) 

Continue with the following possible sequence: 

20 × . 

, × 18, × 18, 18 × , × 16, 16 × , 16 × , × 15, and 

Lesson 13: Multiply unit fractions by unit fractions. 

Date: 11/10/13 4.E.3 







Note: This fluency reviews G5─M4─Lesson 12 and prepares students for this lesson. Allow students to use 

the conversion reference sheet if they are confused, but encourage them to answer questions without 

looking at it. 

Convert the following. Draw a tape diagram if it helps you. 

a. yd = ________ft = _______inches 

b. yd = ________ft = _______inches 

c. hour = ________minutes 

d. hour = ________minutes 

e. year = ________months 

f. year = ________months 


Materials: (S) Personal white boards, 4″ × 2″ rectangular paper 

(several pieces per student), scissors 

Note: Today’s lesson is lengthy, so the time normally allotted 

for an Application Problem has been allocated to the Concept 

Development. The last problem in the sequence can be 

considered the Application Problem for today. 

Problem 1 

Jan has 4 pans of crispy rice treats. She sends of the pans to 

school with her children. How many pans of crispy rice treats 

does Jan send to school? 

NOTES ON 


ENGAGEMENT: 

While the lesson moves to the pictorial 

level of representation fairly quickly, 

be aware that many students may 

need the scaffold of the concrete 

model (paper folding and shading) to 

fully comprehend the concepts. Make 

these materials available and model 

their use throughout the remainder of 

the module. 

Note: To progress from finding a fraction of a whole number to a fraction of a fraction, the following 

sequence is then used: 2 pans, 1 pan, pan. 

T: (Post Problem 1 on the board and read it aloud with the students.) Work with your partner to write 

a multiplication sentence that explains your thinking. Be prepared to share. (Allow students time to 

work.) 

T: What fraction of the pans does Jan send to school? 

S: One-half of them. 


Date: 11/10/13 4.E.4 





T: How many pans did Jan have? 

S: 4 pans. 

T: What is one-half of 4 pans? 

S: 2 pans. 

T: Show the multiplication sentence that you wrote to explain your thinking. 

S: (Show 4 ans pans or 4 pans.) 

T: Say the answer in a complete sentence. 

S: Jan sent 2 pans of crispy rice treats to school. 

T: (Erase the 4 in the text of the problem and replace it with a 2.) Imagine that Jan has 2 pans of treats. 

If she still sends half of the pans to school, how many pans will she send? Write a multiplication 

sentence to show how you know. 

S: (Write ans pan.) 

T: (Replace the 2 in the problem with a 1.) Now, imagine that she only has 1 pan. If she still sends half 

to school, how many pans will she send? Write the multiplication sentence. 

S: (Write an pan.) 

MP.4 

T: (Erase the 1 in problem and replace it with . Read the problem aloud with students.) What if Jan 

only has half a pan and wants to send half of it to school? What is different about this problem? 

S: There’s only of a pan instead of a whole pan. Jan is still sending half the treats to school but 

now we’ll find half of a half, not half of 1. The amount we have is less than a whole. 

T: Let’s say that your iece of a er re resents the an of treats. Turn and talk to your partner about 

how you can use your rectangular paper to find out what fraction of the whole pan of treats Jan sent 

to school. 

S: (May fold or shade the paper to show the problem.) 

T: Many of you shaded half of your paper, then partitioned that 

half into 2 equal parts and shaded one of them, like this. 

(Model as seen at right.) 

T: We now have two different size units shaded in our model. I 

can see the part that Jan sent to school, but I need to name 

this unit. In order to name the part she sent (point to the 

double shaded unit), all of the units in the whole must be the 

same size as this one. Turn and talk to your partner about 

how we can split the rest of the pan so that all the units are 

the same as our double-shaded one. Use your paper to show 


S: We could cut the other half in half too. That would make 4 units the same size. We could keep 

cutting across the rest of the whole. That would make the whole pan cut into 4 equal parts. Half 

of a half is a fourth. 

T: Let me record that. (Partition the un-shaded half using a dotted line.) Look at our model. What’s 


Date: 11/10/13 4.E.5 





the name for the smallest units we have drawn now? 

S: Fourths. 

T: She sent half of the treats she had, but what fraction of the whole pan of treats did Jan send to 

school? 

S: One-fourth of the whole pan. 

T: Write a multiplication sentence that shows your thinking. 

S: (Write .) 

Problem 2 

Jan has pan of crispy rice treats. She sends of the treats to school with her children. How many pans of 

crispy rice treats does Jan send to school? 

T: (Erase in the text of Problem 1 and replace it with .) Imagine that Jan only has a third of a pan, 

and she still wants to send half of the treats to school. Will she be sending a greater amount or a 

smaller amount of treats to school than she sent in our last problem? How do you know? Turn and 

discuss with your partner. 

S: It will be a smaller part of a whole pan because she had half a pan before. Now she only has 1 third 

of a pan. 1 third is less than 1 half, so half of a third is less than half of a half. 1 half is larger 

than 1 third, so she sent more in the last problem than this one. 

T: We need to find of pan. (Write of = on the board.) I’ll draw a 

model to represent this problem while you use your paper to model it. 

(Draw a rectangle on the board.) This rectangle shows 1 whole pan. 

(Label 1 above the rectangle.) Fold your paper then shade it to show 

how much of this one pan Jan has at first. 

S: (Fold in thirds and shade 1 third of the whole.) 

T: (On the board, partition the rectangle vertically into 3 parts, shade in 1 of 

them, and label below it.) What fraction of the treats did Jan send to 

school? 

S: One-half. 

T: Jan sends of this part to school. (Point to 1 shaded 

portion.) How can I show of this part? Turn and talk 

to your partner, and show your thinking with your 

paper. 

S: We can draw a line to cut it in half. We need to split 

it into 2 equal parts and shade only 1 of them. 

T: I hear you saying that I should partition the one-third 

into 2 equal parts and then shade only 1. (Draw a 

horizontal line through the shaded third and shade the 

bottom half. Label the double shaded area as ) 

NOTES ON 



There will be students who notice the 

patterns within the algorithm quickly 

and want to use it to find the product. 

Be sure those students are questioned 

deeply and can articulate the reasoning 

and meaning of the product in 

relationship to the whole. 


Date: 11/10/13 4.E.6 





Again, now I have two different size shaded units. What do I need to do with this horizontal line in 

order to be able to name the units? Turn and talk. 

S: We could cut the other thirds in half too. That would make 6 units the same size. We could keep 

cutting across the rest of the whole. That would make the whole pan cut into 6 equal parts. 1 

third is the same as 2 sixths. Half of 2 sixths is 1 sixth. 

T: Let me record that. (Partition the un-shaded thirds using a dotted line.) What’s the name for the 

units we have drawn now? 

S: Sixths. 

T: What fraction of the pan of treats did Jan send to school? 

S: One-sixth of the whole pan. 

T: One-half of one-third is one-sixth. (Write .) 

Repeat a similar sequence with Problem 3, but have students 

draw a matchbook-size model on their paper rather than folding 

their paper. Be sure that students articulate clearly the finding of 

a common unit in order to name the product. 

Problem 3 

Jan has a pan of crispy rice treats. She sends of the treats to school 

with her children. How many pans of crispy rice treats does Jan send to 

school? 

T: (Write of and of on the board.) Let’s com are 

finding 1 fourth of 1 third with finding 1 third of 1 fourth. 

What do you notice about these problems? Turn and talk. 

S: They both have 1 fourth and 1 third in them, but 

they’re fli -flopped. They have the same 

factors, but they are in a different order. 

T: Will the order of the factors affect the size of the 

product? Talk to your partner. 

S: It doesn’t when we multi ly whole numbers. But 

is that true for fractions too? That means 1 

fourth of 1 third is the same as a third of a fourth. 

T: We just drew the model for of . Let’s draw an 

area model for of to find out if we will have the 

same answer. In of , the amount we start with is 

1 fourth pan. Draw a whole, shade , and label it. 

(Draw a rectangular box and cut it vertically into 4 equal parts and label . Point to the 1 shaded 

part.) How do I take a third of this fourth? 

S: Cut the fourth into 3 parts. 


Date: 11/10/13 4.E.7 





T: How will we name this new unit? 

S: Cut the other fourths into 3 equal parts, too. 

T: (Partition each unit into thirds and label .) How many of these units make our whole? 

S: Twelve. 

T: What is their name? 

S: Twelfths. 

T: What’s of ? 

S: 1 twelfth. 

T: (Write .) These multiplication sentences have the same 

answer. But the shape of the twelfth is different. How do you know 

that 12 equal parts can be different shapes but the same fraction? 

S: What matters is that they are 12 equal parts of the same whole. 

It’s like if we have a square, there are lots of ways to show a half, 

or 2 equal parts. The area has to be the same, not the shape. 

T: True. What matters is the parts have the same area. We can prove 

with another drawing. Start with the same brownie pan. 

Draw fourths horizontally, and shade 1 fourth. Now let’s double 

shade 1 third of that fourth (extend the units with dotted lines). Is 

the exact same amount shaded in the two pans? 

S: Yes! 

T: So, we see in another way that of = of . Review how to prove that with our rectangles. Turn 

and talk. 

S: We shade a fourth of a third, drawing the thirds vertically first, then we shaded a third of a fourth, 

drawing the fourths horizontally first. They were exactly the same part of the whole. I can shade 

a fourth and then take a third of it, or I can shade a third and then take a fourth of it, and I get the 

same answer either way. 

T: What do we know about multiplication that supports the truth of the number sentence ? 

S: The commutative property works with fractions the same as whole numbers. The order of the 

factors doesn’t change the roduct. Taking a fourth of a third is like taking a smaller part of a 

bigger unit, while taking a third of a fourth is like taking a bigger part of a smaller unit. Either way, 

you’re getting the same size share. 

T: We can express of as or (Write .) They 

are equivalent expressions. 


Date: 11/10/13 4.E.8 





Problem 4 

A sales lot is filled with vehicles for sale. of the vehicles are pickup trucks. of the trucks are white. What 

fraction of all the vehicles are white pickup trucks? 

T: (Post Problem 4 on the board and read it aloud with 

students.) Work with your partner to draw an area model 

and solve. Write a multiplication sentence to show your 

thinking. (Allow students time to work.) 

T: What is a third of one-third? 

S: . 

T: Say the answer to the question in a complete sentence. 

S: One-ninth of the vehicles in the lot are white pickup trucks. 










Lesson Objective: Multiply unit fractions by unit 

fractions. 









the lesson. 


• In Problem 1, what is the relationship between Parts (a) and (d)? (Part (a) is double Part (d)). 

Between Parts (b) and (c)? (Part (b) is double (c).) Between Parts (b) and (e)? (Part (b) is double 

(e).) 


Date: 11/10/13 4.E.9 





• Why is the product for Problem 1(d) smaller 

than 1(c)? Explain your reasoning to your 

partner. 

• Share and compare your solution with a partner 

for Problem 2. 

• Compare and contrast Problem 3 and Problem 

1(b). Discuss with your partner. 

• How is solving for the product of fraction and a 

whole number the same as or different from 

solving fraction of a fraction? Can you use some 

of the similar strategies? Explain your thinking to 

a partner. 




the students’ understanding of the conce ts that were 



the students. 


Date: 11/10/13 4.E.10 





Name 

Date 

1. Solve. Draw an area model to show your thinking. Then write a multiplication sentence. The first one 

has been done for you. 

1 

a. Half of pan of brownies = _____ pan of brownies 

4 

b. Half of pan of brownies = _____ pan 

of brownies 

c. A fourth of pan of brownies = _____ 

pan of brownies 

d. of e. of 


Date: 11/10/13 4.E.11 





2. Draw models of 3 

and . Compare multiplying a number by 3 and by 1 third. 

4 4 

3. of Ila’s workspace is covered in paper. of the paper is yellow sticky notes. What fraction of Ila’s 

workspace is covered in yellow sticky notes? Draw a picture to support your answer. 

4. A marching band is rehearsing in rectangular formation. of the marching band members play 

percussion instruments. of the percussionists play the snare drum. What fraction of all the band 

members play the snare drum? 

5. Marie is designing a beds read for her grandson’s new bedroom. of the bedspread is covered in race 

cars and the rest is striped. 

stripes? 

of the stripes are red. What fraction of the bedspread is covered in red 


Date: 11/10/13 4.E.12 





Name 

Date 

1. Solve. Draw an area model and write a number sentence to show your thinking. 

a. = 

2. Ms. Sheppard cuts of a piece of construction paper. She uses of the piece to make a flower. What 

fraction of the sheet of paper does she use to make the flower? 


Date: 11/10/13 4.E.13 





Name 

Date 

1. Solve. Draw an area model to show your thinking. 

a. Half of cake = _____ cake b. One-third of cake = _____ cake 

c. of d. 

e. f. 

2. Noah mows of his property and leaves the rest wild. He decides to use of the wild area for a vegetable 

garden. What fraction of the property is used for the garden? Draw a picture to support your answer. 


Date: 11/10/13 4.E.14 





3. Fawn plants of the garden with vegetables. Her son plants the remainder of the garden. He decides to 

use of his space to plant flowers, and in the rest he plants herbs. What fraction of the entire garden is 

planted in flowers? Draw a picture to support your answer. 

4. Diego eats of a loaf of bread each day. On Tuesday, Diego eats of the day’s ortion before lunch. 

What fraction of the whole loaf does Diego eat before lunch on Tuesday? Draw a model to support your 

thinking. 


Date: 11/10/13 4.E.15 





Lesson 14 

Objective: Multiply unit fractions by non-unit fractions. 






Total Time 

(12 minutes) 

(6 minutes) 

(32 minutes) 

(10 minutes) 

(60 minutes) 


• Sprint: Multiply a Fraction and a Whole Number 5.NF.3 


(8 minutes) 

(4 minutes) 

Sprint: Multiply a Fraction and Whole Number (8 minutes) 

Materials: (S) Multiply a Fraction and Whole Number Sprint 

Note: This Sprint reviews G5─M4─Lessons 9─12 content. 



Note: This fluency reviews G5─M4─Lesson 5 and reviews denominators that are easily converted to 

hundredths. Direct students to use their personal boards for calculations that they cannot do mentally. 

T: I’ll say a fraction. You say it as a division problem, and give the quotient. 4 halves. 

S: 4 ÷ 2 = 2. 


and 

Lesson 14: Multiply unit fractions by non-unit fractions. 

Date: 11/10/13 4.E.16 






Solve by drawing an area model and writing a multiplication sentence. 

Beth had box of candy. She ate of the candy. What 

fraction of the whole box does she have left? 

Extension: If Beth decides to refill the box, what 

fraction of the box would need to be refilled? 

Note: This Application Problem activates prior 

knowledge of the multiplication of unit fractions by unit 

fractions in preparation for today’s lesson. 



Problem 1 

Jan had pan of crispy rice treats. She sent of the treats to school. What fraction of the whole pan did she 

send to school? 

T: (Write Problem 1 on the board.) How is this problem different 

than the ones we solved yesterday? Turn and talk. 

S: Yesterday, Jan always had 1 fraction unit of treats. She had 

1 half or 1 third or 1 fourth. Today she has 3 fifths. This one 

has a 3 in one of the numerators. We only multiplied 

unit fractions yesterday. 

T: In this problem, what are we finding of? 

S: 3 fifths of a pan of treats. 

T: Before we find of Jan’s visualize this. If there are 3 

bananas, how many would of the bananas be? Turn 

and talk. 

S: Well, if you have 3 bananas, one-third of that is just 1 

banana. One-third of 3 of any unit is just one of 

those units. 1 third of 3 is always 1. It doesn’t 

matter what the unit is. 

T: What is of 3 pens? 

S: 1 pen. 

NOTES ON 


ENGAGEMENT: 

Consider allowing learners who grasp 

these multiplication concepts quickly 

to draw models and create story 

problems to accompany them. If there 

are technology resources available, 

allow these students to produce 

screencasts explaining fraction by 

fraction multiplication for absent or 

struggling classmates. 


Date: 11/10/13 4.E.17 





MP.2 

T: What is of 3 books? 

S: 1 book. 

T: (Write = of 3 fifths.) So, then, what is of 3 fifths? 

S: 1 fifth. 

T: (Write = 1 fifth.) of 3 fifths equals 1 fifth. Let’s draw a model to prove your thinking. Draw an area 

model showing 

S/T: (Draw, shade, and label the area model.) 

T: If we want to show of , what must we do to each of these 3 units? (Point to each of the shaded 

fifths.) 

S: Split each one into thirds. 

T: Yes, partition each of these units, these fifths, into 3 equal parts. 

S/T: (Partition, shade, and label the area model.) 

T: In order to name these parts, what must we do to the 

rest of the whole? 

S: Partition the other fifths into 3 equal parts also. 

T: Show that using dotted lines. What new unit have we 

created? 

S: Fifteenths. 

T: How many fifteenths are in the whole? 

S: 15. 

T: How many fifteenths are double-shaded? 

S: 3. 

T: (Write next to the area model.) I thought we said that our answer was 1 fifth. So, how is it 

that our model shows 3 fifteenths? Turn and talk. 

S: 3 fifteenths is another way to show 1 fifth. I can see 5 equal groups in this model. They each 

have 3 fifteenths in them. Only 1 of those 5 is double shaded, so it’s really only 1 fifth shaded here 

too. The answer is 1 fifth. It’s just chopped into fifteenths in the model. 

T: Let’s explore that a bit. Looking at your model, how many groups of 3 fifteenths do you see? Turn 

and talk. 

S: There are 5 groups of 3 fifteenths in the whole. I see 1 group that’s double-shaded. I see 2 more 

groups that are single-shaded, and then there are 2 groups that aren’t shaded at all. That makes 5 

groups of 3 fifteenths. 

T: Out of the 5 groups that we see, how many are double-shaded? 

S: 1 group. 

T: 1 out of 5 groups is what fraction? 

S: 1 fifth. 


Date: 11/10/13 4.E.18 





MP.2 

T: Does our area model support our thinking from before? 

S: Yes, of 3 fifths equals 1 fifth. 

Problem 2 

Jan had pan of crispy rice treats. She sent of the treats to school. What fraction of the whole pan did she 

send to school? 

T: What are we finding of this time? 

S: of 3 fourths. 

T: (Write = of fourths.) Based on what we learned in the previous problem, what do you think the 

answer will be for of 3 fourths? Whisper and tell a partner. 

S: Just like of 3 apples is equal to 1 apple, and of 3 fifths is equal to 1 fifth. We know that of 3 

fourths is equal to 1 fourth. We are taking 1 third of 3 units again. The units are fourths this time, 

so the answer is 1 fourth. 

T: Work with a neighbor to solve one-third of 3 fourths. One of you can draw the area model while the 

other writes a matching number sentence. 


T: In your model, when you partitioned each of the fourths into 3 equal parts, what new unit did you 

create? 

S: Twelfths. 

T: How many twelfths represent 1 third of 3 fourths? 

S: 3 twelfths. 

T: Say 3 twelfths in its simplest form. 

S: 1 fourth. 

T: So, of 3 fourths is equal to what? 

S: 1 fourth. 

T: Look back at the two problems we just solved. If of 3 fifths is 1 fifth and of 3 fourths is 1 fourth, 

what then is of 3 eighths? 

S: 1 eighth. 

T: of 3 tenths? 

S: 1 tenth. 

T: of 3 hundredths? 


T: Based on what you’ve just learned, what is of 4 fifths? 

S: 1 fifth. 


Date: 11/10/13 4.E.19 





T: of 2 fifths? 

S: 1 fifth. 

T: of 4 sevenths? 

S: 1 seventh. 

Problem 3 

T: We need to find 1 half of 4 fifths. If this were 1 half of 4 bananas, how many bananas would we 

have? 

S: 2 bananas. 

T: How can you use this thinking to help you find 1 

half of 4 fifths? Turn and talk. 

S: It’s half of 4, so it must be 2. This time it’s 4 

fifths, so half would be 2 fifths. Half of 4 is 

always 2. It doesn’t matter that it is fifths. The 

answer is 2 fifths. 

T: It sounds like we agree that 1 half of 4 fifths is 2 fifths. Let’s draw a model to confirm our thinking. 

Work with your partner and draw an area model. 

S: (Draw.) 

T: I notice that our model shows that the product is 4 tenths, but we said a moment ago that our 

product was 2 fifths. Did we make a mistake? Why or why not? 

S: No, 4 tenths is just another name for 2 fifths. I 

can see 5 groups of 4 tenths, but only 2 of them are 

double-shaded. Two out of 5 groups is another way 

to say 2 fifths. 

Repeat this sequence with 

T: What patterns do you notice in our multiplication 

sentences? Turn and talk. 

S: I notice that the denominator in the product is the product of the two denominators in the factors 

until we simplified. I notice that you can just multiply the numerators and then multiply the 

denominators to get the numerator and denominator in the final answer. When you split the 

amount in the second factor into thirds, it’s like tripling the units, so it’s just like multiplying the first 

unit by 3. But the units get smaller so you have the same amount that you started with. 

T: As we are modeling the rest of our problems, let’s notice if this pattern continues. 


Date: 11/10/13 4.E.20 





Problem 4 

of Benjamin’s garden is planted in vegetables. Carrots are planted in of his vegetable section of the 

garden. How much of Benjamin’s garden is planted in carrots? 

T: Write a multiplication expression to represent the amount of his garden planted in carrots. 

S: . of . 

T: I’ll write this in unit form. (Write of 3 fourths on the board.) Compare this problem with the last 

ones. Turn and talk. 

S: This one seems trickier because all the others were easy to halve. They were all even numbers of 

units. This is half of . I know that’s 1 and 1 half, but the unit is fourths and I don’t know how to 

say 

fourths. 

T: Could we name 3 fourths of Benjamin’s garden using another unit that makes it easier to halve? 

Turn and talk with your partner, and then write the amount in unit form. 

S: We need a unit that lets us name 3 fourths with an even 

number of units. We could use 6 eighths. 6 eighths is 

the same amount as 3 fourths and 6 is a multiple of 2. 

T: What is 1 half of 6? 

S: 3. 

T: So, what is 1 half of 6 eighths? 

S: 3 eighths. 

T: Let’s draw our model to confirm our thinking. (Allow 

students time to draw.) 

T: Looking at our model, what was the new unit that we used to 

name the parts of the garden? 

S: Eighths. 

T: How much of Benjamin’s garden is planted in carrots? 

S: 3 eighths. 

Problem 5 

of 

T: (Post Problem 5 on the board.) Solve this by drawing a model 

and writing a multiplication sentence. (Allow students time to 

work.) 

T: Compare this model to the one we drew for Benjamin’s 

garden. Turn and talk. 

S: It’s similar. The fractions are the same, but when you draw this 

one you have to start with 1 half and then chop that into 

fourths. The model for this problem looks like what we drew 


Date: 11/10/13 4.E.21 





for Benjamin’s garden, except it’s been turned on its side. When we wrote the multiplication 

sentence, the factors are switched around. This time we’re finding 3 fourths of a half, not a half 

of 3 fourths. If this were another garden, less of the garden is planted in vegetables overall. Last 

time it was 3 fourths of the garden, this time it would be only half. The fraction of the whole garden 

that is carrots is the same, but now there is only 1 eighth of the garden planted in other vegetables. 

Last time, 3 eighths of the garden would have had other vegetables. 

T: I hear you saying that of and of are equivalent expressions. (Write .) Can you give 

an equivalent expression for ? 

S: of . of . . 

T: Show me another pair of equivalent expressions that involve fraction multiplication. 


Problem 6 

Mr. Becker, the gym teacher, uses of his kickballs in class. Half of the remaining balls are given to students 

for recess. What fraction of all the kickballs is given to students for recess? 

T: (Post Problem 6 and read it aloud 

with students.) This time, let’s solve 

using a tape diagram. 

S/T: (Draw a tape diagram.) 

T: What fraction of the balls does Mr. 

Becker use in class? 

S: 3 fifths. (Partition the diagram into 

fifths and label used in class.) 

T: What fraction of the balls is remaining? 

S: 2 fifths. 

T: How many of those are given to students for recess? 

S: One half of them. 

T: What is one-half of 2? 

S: 1. 

T: What’s one half of 2 fifths? 

S: 1 fifth. 

T: Write a number sentence and make a statement to answer the question. 

S: of 2 fifths = 1 fifth. One-fifth of Mr. Becker’s kickballs are given to students to use at recess. 

Repeat this sequence using . 


Date: 11/10/13 4.E.22 














Lesson Objective: Multiply unit fractions by non-unit 

fractions. 









the lesson. 



• In Problem 1, what is the relationship between 

Parts (a) and (b)? (Part (b) is double (a).) 

• Share and explain your solution for Problem 

1(c) to your partner. Why is taking 1 half of 2 

halves equal to 1 half? Is it true for all 

numbers? 1 half of ? 1 half of ? 1 half of 8 

wholes? 

• How did you solve Problem 3? Explain your 

strategy to a partner. 

• What kind of picture did you draw to solve 

Problem 4? Share and explain your solution to 

a partner. 

• We noticed some patterns when we wrote our 

multiplication sentences. Did you notice the 

same patterns in your Problem Set? (Students 

should note the multiplication of the numerators and denominators to produce the product.) 


Date: 11/10/13 4.E.23 





• Explore with students the commutative property in real life situations. While the numeric product 

(fraction of the whole) is the same, are the situations also the same? (For example, 

.) Is a 

class of fifth-graders in which half are girls (a third of which wear glasses) the same as a class of fifthgraders 

in which 1 third are girls (half of which wear glasses)? 






Date: 11/10/13 4.E.24 






Date: 11/10/13 4.E.25 






Date: 11/10/13 4.E.26 





Name 

Date 

1. Solve. Draw a model to explain your thinking. Then write a number sentence. An example has been 

done for you. 

Example: 

of 

of 2 fifths = 1 fifth 

a. of = of ____ fourths = ____ fourth b. of = of ____ fifths = ____fifths 

c. of = d. of = 

e. f. = 


Date: 11/10/13 4.E.27 





2. of the songs on Harrison’s iPod are hip-hop. of the remaining songs are rhythm and blues. What 

fraction of all the songs are rhythm and blues? Use a tape diagram to solve. 

3. Three-fifths of the students in a room are girls. One-third of the girls have blond hair. One-half of the 

boys have brown hair. 

a. What fraction of all the students are girls with blond hair? 

b. What fraction of all the students are boys without brown hair? 

4. Cody and Sam mowed the yard on Saturday. Dad told Cody to mow of the yard. He told Sam to mow 

of the remainder of the yard. Dad paid each of the boys an equal amount. Sam said, “Dad, that’s not fair! 

I had to mow one-third and Cody only mowed one-fourth!” Explain to Sam the error in his thinking. 

Draw a picture to support your reasoning. 


Date: 11/10/13 4.E.28 





Name 

Date 

1. Solve. Draw a model to explain your thinking. Then write a number sentence. 

a. of = 

2. In a cookie jar, of the cookies are chocolate chip, and of the rest are peanut butter. What fraction of 

all the cookies are peanut butter? 


Date: 11/10/13 4.E.29 





Name 

Date 

1. Solve. Draw a model to explain your thinking. 

a. of = of ____ thirds = ____ thirds b. of = of ____ thirds = ____ thirds 

c. of = d. = 

e. = f. = 

2. Sarah has a photography blog. of her photos are of nature. of the rest are of her friends. What 

fraction of all Sarah’s photos is of her friends? Support your answer with a model. 


Date: 11/10/13 4.E.30 





3. At Laurita’s Bakery, of the baked goods are pies, and the rest are cakes. of the pies are coconut. of 

the cakes are angel-food. 

a. What fraction of all of the baked goods at Laurita’s Bakery are coconut pies? 

b. What fraction of all of the baked goods at Laurita’s Bakery are angel-food cakes? 

4. Grandpa Mick opened a pint of ice cream. He gave his youngest grandchild of the ice cream and his 

middle grandchild of the remaining ice cream. Then he gave his oldest grandchild of the ice cream 

that was left after serving the others. 

a. Who got the most ice cream? How do you know? Draw a picture to support your reasoning. 

b. What fraction of the pint of ice cream will be left if Grandpa Mick serves himself the same amount as 

the second grandchild? 


Date: 11/10/13 4.E.31 





Lesson 15 

Objective: Multiply non-unit fractions by non-unit fractions. 






Total Time 

(12 minutes) 

(7 minutes) 

(31 minutes) 

(10 minutes) 

(60 minutes) 


• Multiply Fractions 5.NF.4 



(4 minutes) 

(4 minutes) 

(4 minutes) 

Multiply Fractions (4 minutes) 


Note: This fluency reviews G5─M4─Lesson 13. 

T: (Write of .) Say the fraction of a set as a multiplication sentence. 

S: . 

T: Draw a rectangle and shade in 1 third. 

S: (Draw a rectangle, partition it into 3 equal units, and shade 1 of the units.) 

T: To show of , how many parts do you need to break the 1 third into? 

S: 2. 

T: Shade 1 half of 1 third. 

S: (Shade 1 of the 2 parts.) 

T: How can we name this new unit? 

S: Partition the other 2 thirds in half. 

T: Show the new units. 

S: (Partition the other thirds into 2 equal parts.) 

T: How many new units do you have? 

Lesson 15: Multiply non-unit fractions by non-unit fractions. 

Date: 11/10/13 4.E.32 





S: 6 units. 

T: Write the multiplication sentence. 

S: (Write × = .) 

Continue the process with the following possible sequence: of , of , and of . 


Note: This fluency prepares students for G5─M4─Lessons 17─18. 


S: 1 tenth. 


S: Zero point one. 


T: (Write = .) Say the fraction. 



S: Zero point zero one. 

Continue with the following possible suggestions: , , , , , and . 

T: (Write 0.01 = .) Say it as a fraction. 


T: (Write 0.01 = .) 




Note: This fluency prepares students for G5─M4─Lessons 17─18. 

T: (Write = .) Write the equivalent fraction. 


T: (Write = = .) Write 1 fifth as a decimal. 

S: (Write = = 0.2.) 

Continue with the following possible suggestions: , , , , , , , , , , and . 


Date: 11/10/13 4.E.33 






Kendra spent of her allowance on a book and 

on a snack. If she had four dollars remaining after 

purchasing a book and snack, what was the total 

amount of her allowance? 

Note: This problem reaches back to addition and 

subtraction of fractions as well as fraction of a set. 

Keeping these skills fresh is an important goal of 




Problem 1 

of 

T: (Post Problem 1 on the board.) How is this problem 

different from the problems we did yesterday? Turn 

and talk. 

S: In every problem we did yesterday, one factor had a 

numerator of 1. There are no numerators that are 

ones today. Every problem multiplied a unit 

fraction by a non-unit fraction, or a non-unit fraction 

by a unit fraction. This is two non-unit fractions. 

T: (Write of 3 fourths.) What is 1 third of 3 fourths? 

S: 1 fourth. 

T: If 1 third of 3 fourths is 1 fourth, what is 2 thirds of 3 

fourths? Discuss with your partner. 

NOTES ON 



Notice the dotted lines in the area 

model shown below. 

If this were an actual pan partially full 

of brownies, the empty part of the pan 

would obviously not be cut! However, 

to name the unit represented by the 

double-shaded parts, the whole pan 

must show the same size or type of 

unit. Therefore, the empty part of the 

pan must also be partitioned as 

illustrated by the dotted lines. 

S: 2 thirds would just be double 1 third, so it would be 2 fourths. 3 fourths is 3 equal parts so of 

that would be 1 part or 1 fourth. We want this time, so 

that is 2 parts, or 2 fourths. 

T: Name 2 fourths using halves. 

S: 1 half. 

T: So, 2 thirds of 3 fourths is 1 half. Let’s draw an area model 

to show the product and check our thinking. 

T: I’ll draw it on the board, and you’ll draw it on your personal board. Let’s draw 3 fourths and label it 

on the bottom. (Draw a rectangle and cut it vertically into 4 units, and shade in 3 units.) 


Date: 11/10/13 4.E.34 





T: (Point to the 3 shaded units.) We now have to take 2 thirds of these 3 shaded units. What do I have 

to do? Turn and talk. 

S: Cut each unit into thirds. Cut it across into 3 equal parts, and shade in 2 parts. 

T: Let’s do that now. (Partition horizontally into thirds, shade in 2 thirds and label.) 

T: (Point to the whole rectangle.) What unit have we used to rename our whole? 

S: Twelfths. 

T: (Point to the 6 double-shaded units.) How many twelfths are double-shaded when we took of ? 

S: 6 twelfths. 

T: Compare our model to the product we thought about. Do they represent the same product or have 

we made a mistake? Turn and talk. 

S: The units are different, but the answer is the same. 2 fourths and 6 twelfths are both names for 1 

half. When we thought about it, we knew it would be 2 fourths. In the area model, there are 12 

parts and we shaded 6 of them. That’s half. 

T: Both of our approaches show that 2 thirds of 3 fourths is what simplified fraction? 

S: . 

T: Let’s write this problem as a multiplication sentence. (Write on the board.) Turn and talk 

to your partner about the patterns you notice. 

S: If you multiply the numerators you get 6 and the denominators you get 12. That’s 6 twelfths just 

like the area model. It’s easy to get a fraction of a fraction, just multiply the top numbers to get 

the numerator and the bottom to get the denominator. Sometimes you can simplify. 

T: So, the product of the denominators tells us the total number of units, 12 (point to the model). The 

product of the numerators tells us the total number of units selected, 6. 

Problem 2 

T: (Post Problem 2 on the board.) We need 2 thirds of 2 thirds this 

time. Draw an area model to solve and then write a multiplication 

sentence. Talk to your partner about whether the patterns are the 

same as before. 

S: It’s the same as before. When you multiply the numerators, you 

get the numerator of the double-shaded part. When you multiply 

the denominators, you get the denominator of the double-shaded 

part. It’s pretty cool! The denominator of the product gives the area of the whole rectangle (3 by 

3) and the numerator of the product gives the area of the double-shaded part (2 by 2)! 

T: Yes, we see from the model that the product of the denominators tells us the total number of units, 

9. The product of the numerator tells us the total number of units selected, 4. 


Date: 11/10/13 4.E.35 





Problem 3 

a. of 

b. 

c. 

T: (Post Problem 3(a) on the board.) How would this problem look if we drew an area model for it? 

Discuss with your partner. 

S: We’d have to draw 3 sevenths first, and then split each 

seventh into ninths. We’d end up with a model showing 

sixty-thirds. It would be really hard to draw. 

T: You are right. It’s not really practical to draw an area 

model for a problem like this because the units are so 

small. Could the pattern that we’ve noticed in the 

multiplication sentences help us? Turn and talk. 

S: of is the same as . Our pattern lets us just 

multiply the numerators and the denominators. We can 

multiply and get 21 as the numerator and 63 as the 

denominator. Then we can simplify and get 1 third. 

T: Let me write what I hear you saying. (Write = 

on the board.) 

T: What’s the simplest form for ? Solve it on your board. 

S: . 

T: Let’s use another strategy we learned recently and rename this 

fraction using larger units before we multiply. (Point to .) 

Look for factors that are shared by the numerator and the 

denominator. Turn and talk. 

S: There’s a 7 in both the numerator and the denominator. The numerator and denominator have a 

common factor of 7. I know the 3 in the numerator can be divided by 3 to get 1 and the 9 in the 

denominator can be divided by 3 to get 3. Seven divided by 7 is 1, so both sevens change to ones. 

The factors of 3 and 9 can both be divided by 3 and changed to 1 and 3. 

T: We can rename this fraction by dividing both the numerators and denominators by common factors. 

Seven divided by 7 is 1, in both the numerator and denominator. (Cross out both sevens and write 

ones next to them.) Three divided by 3 is 1 in the numerator, and 9 divided by 3 is 3 in the 

denominator. (Cross out the 3 and 9 and write 1 and 3 respectively, next to them.) 

T: What does the numerator show now? 

S: 1 1. 

T: What’s the denominator? 

S: 3 1. 


Date: 11/10/13 4.E.36 





T: Now multiply. What is of equal to? 

S: . 

T: Look at the two strategies, which one do you think is the easier and more efficient to use? Turn and 

talk. 

S: The first strategy of simplifying after I multiply is a little bit harder because I have to find the 

common factors between 21 and 63. Simplifying first is a little easier. Before I multiply, the 

numbers are a little smaller so it’s easier to see common factors. Also, when I simplify first, the 

numbers I have to multiply are smaller, and my product is already expressed using the largest unit. 

T: (Post Problem 3(b) on the board.) Let’s practice using the strategy of simplifying first before we 

multiply. Work with a partner and solve. Remember, we are looking for common factors before we 

multiply. (Allow students time to work and share their answers.) 

T: What is of ? 

S: . 

T: Let’s confirm that by multiplying first and then 

simplifying. 

S: (Rework the problem to find .) 

T: (Post Problem 3(c) on the board.) Solve 

independently. (Allow students time to solve the 

problem.) 

T: What is of ? 

S: . 

Problem 4 

Nigel completes of his homework immediately after school 

and of the remaining homework before supper. He finishes 

the rest after dessert. What fraction of his work did he finish 

after dessert? 

T: (Post the problem on the board, and read it aloud 

with students.) Let’s solve using a tape diagram. 

S/T: (Draw diagram.) 

T: What fraction of his homework does Nigel finish 

immediately after school? 

S: . 

T: (Partition diagram into sevenths and label 3 of them 

after school.) What fraction of the homework does Nigel have remaining? 

S: . 


Date: 11/10/13 4.E.37 





T: What fraction of the remaining homework does Nigel finish before supper? 

S: One-fourth of the remaining homework. 

T: Nigel completes of 4 sevenths before supper. (Point to the remaining 4 units on the tape diagram.) 

What’s of these 4 units? 

S: 1 unit. 

T: Then what’s of 4 sevenths? (Write of 4 sevenths = 

_________ sevenths on the board.) 

S: 1 seventh. (Label 1 seventh of the diagram before 

supper.) 

T: When does Nigel finish the rest? (Point to the 

remaining units.) 

S: After dessert. (Label the remaining after dessert.) 

T: Answer the question with a complete sentence. 

S: Nigel completes of his homework after dessert. 

T: Let’s imagine that Nigel spent 70 minutes to complete 

all of his homework. Where would I place that 

information in the model? 

S: Put 70 minutes above the diagram. We just found 

out the whole, so we can label it above the tape 

diagram. 

T: How could I find the number of minutes he worked on 

homework after dessert? Discuss with your partner, 

then solve. 

NOTES ON 



In these examples, students are 

simplifying the fractional factors before 

they multiply. This step may eliminate 

the need to simplify the product, or 

make simplifying the product easier. 

In order to help struggling students 

understand this procedure, it may help 

to use the Commutative Property to 

reverse the order of the factors. For 

example: 

3 × 4 = 4 × 3 

4 × 7 4 × 7 

In this example, students may now 

more readily see that is equivalent to 

, and can be simplified before 

multiplying. 

S: He finished already, so we can find of 70 minutes and then just subtract that from 70 to find how 

long he spent after dessert. It’s fraction of a set. He does of his homework after dessert. We 

can multiply to find of 70. That’ll be how long he worked after dessert. We can first find the 

total minutes he spent after school by solving of 70. Then we know each unit is 10 minutes. 

We find what one unit is equal to, which is 10 minutes. Then we know the time he spent after 

dessert is 3 units. 10 times 3 = 30 minutes. 

T: Use your work to answer the question. 

S: Nigel spends 30 minutes working after dessert. 







Date: 11/10/13 4.E.38 






Lesson Objective: Multiply non-unit fractions by nonunit 

fractions. 









the lesson. 



• What is the relationship between Parts (c) and 

(d) of Problem 1? (Part(d) is double (c).) 

• In Problem 2, how are Parts (b) and (d) 

different from Parts (a) and (c)? (Parts (b) and 

(d) have two common factors each.) 

• Compare the picture you drew for Problem 3 

with a partner. Explain your solution. 

• In Problem 5, how is the information in the 

answer to Part (a) different from the information 

in the answer to Part (b)? What are the different 

approaches to solving, and is there one strategy 

that is more efficient than the others? (Using 

fraction of a set might be more efficient than 

subtraction.) Explain your strategy to a partner. 







students. 


Date: 11/10/13 4.E.39 





Name 

Date 

1. Solve. Draw any model to explain your thinking. Then write a multiplication sentence. The first one is 

done for you. 

a. of 

1 

2 

3 

3 

5 

b. of c. of = 

d. = e. 

2. Multiply. Draw a model if it helps you, or use the method in the example. 

Example: 

3 

4 

a. b. 


Date: 11/10/13 4.E.40 





c. d. 

3. Phillip’s family traveled of the distance to his grandmother’s house on Saturday. They traveled of the 

remaining distance on Sunday. What fraction of the distance to his grandmother’s house was traveled on 

Sunday? 

4. Santino bought a lb bag of chocolate chips. He used of the bag while baking. How many pounds of 

chocolate chips did he use while baking? 

5. Farmer Dave harvested his corn. He stored of his corn in one large silo and of the remaining corn in a 

small silo. The rest was taken to market to be sold. 

a. What fraction of the corn was stored in the small silo? 

b. If he harvested 18 tons of corn, how many tons did he take to market? 


Date: 11/10/13 4.E.41 





Name 

Date 

1. Solve. 

a. of 

b. × 

2. A newspaper’s cover page is text, and photographs fill the rest. If of the text is an article about 

endangered species, what fraction of the cover page is the article about endangered species? 


Date: 11/10/13 4.E.42 





Name 

Date 

1. Solve. Draw a model to explain your thinking. Then write a multiplication sentence. 

a. of b. of 

c. of d. of 

2. Multiply. Draw a model if it helps you. 

a. × b. × 

c. × d. × 

e. × f. × 


Date: 11/10/13 4.E.43 





3. Every morning, Halle goes to school with a 1 liter bottle of water. She drinks of the bottle before school 

starts and of the rest before lunch. 

a. What fraction of the bottle does Halle drink before lunch? 

b. How many milliliters are left in the bottle at lunch? 

4. Moussa delivered of the newspapers on his route in the first hour and of the rest in the second hour. 

What fraction of the newspapers did Moussa deliver in the second hour? 

5. Rose bought some spinach. She used of the spinach on a pan of spinach pie for a party, and of the 

remaining spinach for a pan for her family. She used the rest of the spinach to make a salad. 

a. What fraction of the spinach did she use to make the salad? 

b. If Rose used 3 pounds of spinach to make the pan of spinach pie for the party, how many pounds of 

spinach did Rose use to make the salad? 


Date: 11/10/13 4.E.44 





Lesson 16 

Objective: Solve word problems using tape diagrams and fraction-byfraction 






Total Time 

(8 minutes) 

(42 minutes) 

(10 minutes) 

(60 minutes) 



• Multiply Whole Numbers by Decimals 5.NBT.7 

(3 minutes) 

(5 minutes) 



Note: This fluency reviews G5─M4─Lessons 14─15. 

T: (Write of is _______.) Write the fraction of a set as a multiplication sentence. 

S: × . 

T: Draw a rectangle and shade in . 

S: (Draw a rectangle, partition it into 5 equal units, and 

shade 2 of the units.) 

T: To show of , how many equal parts to we need? 

S: 3. 

T: Show 1 third of 2 fifths. 

S: (Partition the 2 fifths into thirds, and shade 1 third.) 

T: Make the other units the same size as the double 

shaded ones. 

S: (Extend the horizontal thirds across the remaining 

units using dotted lines.) 

T: What unit do we have now? 

S: Fifteenths. 

NOTES ON 


ENGAGEMENT: 

Sixty minutes is allotted for all lessons 

in Grade 5. All 60 minutes, however, 

do not need to be consecutive. For 

example, fluencies can be completed 

as students wait in line, or when they 

are transitioning between subjects. 

Lesson 16: Solve word problems using tape diagrams and fraction-by-fraction 


Date: 11/10/13 

4.E.45 





T: How many fifteenths are double-shaded? 

S: Two. 

T: Write the product and say the sentence. 

S: (Write × = .) of is 2 fifteenths. 

Continue this process with the following possible sequence: × , × , and × . 

Multiply Whole Numbers by Decimals (5 minutes) 


Note: This fluency prepares students for G5─M4─Lessons 17–18. 

T: (Write + + ). Say the repeated addition sentence. 

S: + + = . 

T: (Write 3 × __ = .) On your boards, write the number sentence, filling in the missing number. 

S: (Write 3 × = .) 

T: (Write 3 × = 3 × 0.__.) On your boards, fill in the missing digit. 

S: (Write 3 × = 3 × 0.1.) 

T: (Write 3 × 0.1 = 0.__.) Say the missing digit. 

S: 3. 

Continue with the following expression: + + + + . 

T: (Write 7 × 0.1 = .) On your boards, write the number sentence. 

S: (Write 7 × 0.1 = 0.7.) 


S: (Write 7 × 0.01 = 0.07.) 

Continue this process with the following possible sequence: 9 × 0.1 and 9 × 0.01. 

T: (Write 20 × = .) On your boards, write the number sentence. 

S: (Write 20 × = = 2.) 

T: (Write 20 × 0.1 = .) Write the number sentence on your boards. 

S: (Write 20 × 0.1 = 2.) 

T: (Write 20 × 0.01 = .) Write the number sentence on your boards. 

S: (Write 20 × 0.01 = 0.2.) 

Continue this process with the following possible sequence: 80 × 0.1 and 80 × 0.01. 

T: (Write 15 × = .) On your boards, write the number sentence. 



Date: 11/10/13 

4.E.46 





S: (Write 15 × = .) 

T: (Write 15 × 0.1 = .) On your boards, write the number sentence and answer as a decimal. 

S: (Write 15 × 0.1 = 1.5.) 

T: (Write 15 × 0.01 = .) On your boards, write the number sentence and answer as a decimal. 

S: (Write 15 × 0.01 = 0.15.) 

Continue with the following possible sequence: 37 × 0.1 and 37 × 0.01. 


Materials: (S) Problem Set, personal white boards 

Note: Because today’s lesson involves students in learning a new type of tape diagram, the time normally 

allotted to the Application Problem has been used in the Concept Development to allow students ample time 

to draw and solve the story problems. 

Note: There are multiple approaches to solving these problems. Modeling for a few strategies is included 

here, but teachers should not discourage students from using other mathematically sound procedures for 

solving. The dialogues for the modeled problems are detailed as a scaffold for teachers unfamiliar with 

fraction tape diagrams. 

Problem 2 from the Problem Set opens the lesson and is worked using two 

different fractions (first 1 fifth, then 2 fifths) so that diagramming of two 

different whole–part situations may be modeled. 

Problem 2 

Joakim is icing 30 cupcakes. He spreads mint icing on of the cupcakes and 

chocolate on of the remaining cupcakes. The rest will get vanilla frosting. 

How many cupcakes have vanilla frosting? 

T: (Display Problem 2, and read it aloud with students.) 

Let’s use a tape diagram to model this problem. 

T: This problem is about Joakim’s cupcakes. What does 

the first sentence tell us? 

S: Joakim has 30 cupcakes. 

T: (Draw a diagram and label with a bracket and 30.) Joakim is icing the 

cupcakes. What fraction of the cupcakes get mint icing? 

S: of the cupcakes. 

T: How can I show fifths in my diagram? 

S: Partition the whole into 5 equal units. 

T: How many of those units have mint icing? 

S: 1. 



Date: 11/10/13 

4.E.47 





T: Let’s show that now. (Partition the diagram into fifths and label 1 unit mint.) 

T: Read the next sentence. 

S: (Read.) 

T: Where are the remaining cupcakes in our tape? 

S: The unlabeled units. 

T: Let’s drop that part down and draw a new tape to represent the remaining cupcakes. (Draw a new 

diagram underneath the original whole.) 

T: What do we know about these remaining cupcakes? 

S: Half of them get chocolate icing. 

T: How can we represent that in our new diagram? 

S: Cut it into 2 equal parts and label 1 of them chocolate. 

T: Let’s do that now. (Partition the lower diagram into 2 units and label 1 unit chocolate.) What about 

the rest of the remaining cupcakes? 

S: They are vanilla. 

T: Let’s label the other half vanilla. (Model.) What is the question asking us? 

S: How many are vanilla? 

T: Place a question mark below the portion showing vanilla. (Put a question mark beneath vanilla.) 

T: Let’s look at our diagram to see if we can find how many cupcakes get vanilla icing. How many units 

does the model show? (Point to original tape.) 

S: 5 units. 

T: (Write 5 units.) How many cupcakes does Joakim have in all? 

S: 30 cupcakes. 

T: (Write = 30 cupcakes.) If 5 units equals 30 cupcakes, how can we find the value of 1 unit? Turn and 

talk. 

S: It’s like 5 times what equals 30. 5 × 6 = 30, so 1 unit equals 6 cupcakes. We can divide. 30 

cupcakes ÷ 5 = 6 cupcakes. 

T: What is 1 unit equal to? (Write 1 unit = .) 

S: 6 cupcakes. 

T: Let’s write 6 in each unit to show its value. (Write 6 in each unit of original diagram.) That means 

that 6 cupcakes get mint icing. How many cupcakes remain? (Point to 4 remaining units.) Turn and 

talk. 

S: 30 – 6 = 24. 6 + 6 + 6 + 6 = 24. 4 units of 6 is 24. 4 × 6 = 24. 

T: Let’s label that on the diagram showing the remaining cupcakes. (Label 24 above the second 

diagram.) How can we find the number of cupcakes that get vanilla icing? Turn and talk. 

S: Half of the 24 cupcakes get chocolate and half get vanilla. Half of 24 is 12. 24 ÷ 2 = 12. 

T: What is half of 24? 

S: 12. 

T: (Write = 12 and label 12 in each half of the second diagram.) Write a statement to answer the 

question. 



Date: 11/10/13 

4.E.48 





S: 12 cupcakes have vanilla icing. 

T: Let’s think of this another way. When we labeled the 1 fifth for the mint icing, what fraction of the 

cupcakes were remaining? 

S: . 

S: What does Joakim do with the remaining cupcakes? 

S: of the remaining cupcakes get chocolate icing. 

MP.4 

T: (Write of.) of what fraction? 

S: 1 half of 4 fifths. 

T: (Write 4 fifths.) What is of 4 fifths? 

S: . 

T: So, 2 fifths of all the cupcakes got chocolate, and 2 fifths of all the cupcakes got vanilla. The question 

asked us how many cupcakes got vanilla icing. Let’s find 2 fifths of all the cupcakes—2 fifths of 30. 

Work with your partner to solve. 

S: 1 fifth of 30 is 6, so 2 fifths of 30 is 12. 5 

30 

T: So, using fraction multiplication, we got the same answer,12 cupcakes. 

5 

30 

60 

5 

30 6 . 

T: This time, let’s imagine that Joakim put mint icing on fifths of the cupcakes. Draw another diagram 

to show that situation. 

S: (Draw.) 

T: What fraction of the cupcakes are remaining this time? 

S: 3 fifths. 

T: Let’s draw a second tape that is the same length 

as the remaining part of our whole. (Draw the 

second tape below the first.) Has the value of 

one unit changed in our model? Why or why 

not? 

S: The unit is still 6 because the whole is still 30 and 

we still have fifths. Each unit is still 6 because 

we still divided 30 into 5 equal parts. 

T: So, how many remaining cupcakes are there this 

time? 

S: 18. 

T: Imagine that Joakim still put chocolate icing on half the remaining cupcakes, and the rest were still 

vanilla. How many cupcakes got vanilla icing this time? Work with a partner to model it in your tape 

diagram and answer the question with a complete sentence. 

S: (Work.) 

T: Let’s confirm that there were 9 cupcakes that got vanilla icing by using fraction multiplication. How 

might we do this? Turn and talk. 



Date: 11/10/13 

4.E.49 





MP.4 

S: We could just multiply and get . Then we can find 3 tenths of 30. That’s 9! We can find 1 

half of 3 fifths. That gives us the fraction of all the cupcakes that got vanilla icing. We need the 

number of cupcakes, not just the fraction, so we need to multiply 3 tenths and 30 to get 9 cupcakes. 

Nine cupcakes got vanilla frosting. 

T: Complete Problem 1 and Problem 3 on the Problem Set. Check your work with a neighbor when 

you’re finished. You may use either method to solve. 

Solutions for Problems 1 and Problem 3 

Problem 5 

Milan puts of her lawn-mowing money in savings and uses of the remaining money to pay back her sister. 

If she has $15 left, how much did she have at first? 

T: (Post Problem 5 on board, and read it aloud with students.) How is this problem different from the 

ones we’ve just solved? Turn and discuss with your partner. 

S: In the others, we knew what the whole was, this time we don’t. We know how much money she 

has left, but we have to figure out what she had at the beginning. It seems like we might have to 

work backwards. The other problems were whole-to-part problems. This one is part-to-whole. 

T: Let’s draw a tape diagram. (Draw a blank tape diagram.) What is the whole in this problem? 

S: We don’t know yet; we have to find it. 

T: I’ll put a question mark above our diagram to show that this is unknown. (Label diagram with a 

question mark.) What fraction of her money does Milan put in savings? 



Date: 11/10/13 

4.E.50 





S: . 

T: How can we show that on our diagram? 

S: Cut the whole into 4 equal parts and bracket one of them. Cut it into fourths and label 1 unit 

savings. 

T: (Record on diagram.) What part of our diagram shows the remaining money? 

S: The other parts. 

T: Let’s draw another diagram to represent the remaining money. Notice that I will draw it exactly the 

same length as those last 3 parts. (Model.) What do we know about this remaining part? 

S: Milan gives half of it to her sister. 

T: How can we model that? 

S: Cut the bar into two parts and label one of them. (Partition the second diagram in halves, and label 

one of them sister.) 

T: What about the other half of the 

remaining money? 

S: That’s how much she has left. It’s $ 5. 

T: Let’s label that. (Write $15 in the 

second equal part.) If this half is $15, 

(point to labeled half) what do we 

know about the amount she gave her 

sister, and what does that tell us about 

how much was remaining in all? Turn 

and talk. 

S: If one half is $15, then the other half is $15 too. That makes $30. $15 + $15 = $30. $15 × 2 = 

$30. 

T: If the lower tape is worth $30, what do we know about these 3 units in the whole? (Point to original 

diagram.) Turn and discuss. 

S: The remaining money is the same as 3 units, so 3 units is equal to $30. They represent the same 

money in two different parts of the diagram. 3 units is equal to $30. 

T: (Label 3 units $30.) If 3 units = $30, what is the value of 1 unit? 

S: (Work and show 1 unit = $10.) 

T: Label $10 inside each of the 3 units. (Model on diagram.) If these 3 units are equal to $10 each, 

what is the value of this last unit? (Point to savings unit.) 

S: $10. 

T: (Label $10 inside savings unit.) Look at our diagram. We have 4 units of $10 each. What is the value 

of the whole? 

S: (Work and show 4 units = $40.) 

T: Make a statement to answer the question. 

S: Milan had $40 at first. 

T: Let’s check our work using a fraction of a set. What multiplication sentence tells us what fraction of 



Date: 11/10/13 

4.E.51 





all her money Milan gave to her sister? What fraction did she give to her sister? 

S: . 

T: So, $15 should be 3 eighths of $40. Is that true? Let’s see. Find of $40 with your partner. 

S: (Work and show of $40 = $15.) 

T: Does this confirm our answer of $40 as Milan’s money at first? 

S: Yes! 

T: Complete Problem 4 and Problem 6 on the Problem Set. Check your work with a neighbor when 

you’re finished. You may use either method to solve. 

Solutions for Problem 4 and Problem 6 

Note: Problem 7 may be used as an extension 

for early finishers. 



Date: 11/10/13 

4.E.52 






The Problem Set forms the basis for today’s lesson. Please see 

the Concept Development for modeling suggestions. 


Lesson Objective: Multiply non-unit fractions by non-unit 

fractions. 

The Student Debrief is intended to invite reflection and active 

processing of the total lesson experience. 

Invite students to review their solutions for the Problem Set. 

They should check work by comparing answers with a partner 

before going over answers as a class. Look for misconceptions 

or misunderstandings that can be addressed in the Debrief. 

Guide students in a conversation to debrief the Problem Set and 

process the lesson. 

You may choose to use any combination of the questions below 

to lead the discussion. 

• Did you use the same method for solving Problem 

1 and Problem 3? Why or why not? Did you use 

the same method for solving Problem 4 and 

Problem 6? Why or why not? 

• Were any alternate methods used? If so, explain 

what you did. 

• How was setting up Problem 1 and Problem 3 

different from the process for solving Problem 4 

and Problem 6? What were your thoughts as you 

worked? 

• Talk about how your tape diagrams helped you to 

find the solutions today. Give some examples of 

questions that you could have been able to 

answer, using the information in your tape 

diagram. 

• Questions for further analysis of tape diagrams: 

• Problem 1: Half of the cookies sold were 

oatmeal raisin. How many oatmeal raisin 

cookies were sold? 

• Problem 3: What fraction of the burgers 

had onions? How many burgers had 

onions? 

NOTES ON 


ENGAGEMENT: 

If it is anticipated that the student may 

struggle with a homework assignment, 

there are several ways to provide 

support. 

• Complete one of the problems or 

a portion of a problem as an 

example before the pages are 

duplicated for students. 

• Staple the Problem Set to the 

homework as a reference. 

• Provide a copy of completed 

homework as a reference. 

• Differentiate homework by using 

some of these strategies for 

specific students or specifying that 

only certain problems be 

completed. 



Date: 11/10/13 

4.E.53 





• Problem 4: How many more metamorphic 

rocks does DeSean have than igneous 

rocks? 

• Problem 6: If Parks takes off 2 tie-dye 

bracelets, and puts on 2 more camouflage 

bracelets, what fraction of all the 

bracelets would be camouflage? 



the Exit Ticket. A review of their work will help you 

assess the students’ understanding of the concepts that 

were presented in the lesson today and plan more 

effectively for future lessons. You may read the 

questions aloud to the students. 



Date: 11/10/13 

4.E.54 





Name 

Date 

1. Mrs. Onusko made 60 cookies for a bake sale. She sold of them and gave of the remaining cookies to 

the students working at the sale. How many cookies did she have left? 

2. Joakim is icing 30 cupcakes. He spreads mint icing on of the cupcakes and chocolate on of the 

remaining cupcakes. The rest will get vanilla icing. How many cupcakes have vanilla icing? 

3. The Booster Club sells 240 cheeseburgers. of the cheeseburgers had pickles, of the remaining burgers 

had onions, and the rest had tomato. How many cheeseburgers had tomato? 



Date: 11/10/13 

4.E.55 





4. DeSean is sorting his rock collection. of the rocks are metamorphic and of the remainder are igneous 

rocks. If the 3 rocks left over are sedimentary, how many rocks does DeSean have? 

5. Milan puts of her lawn-mowing money in savings and uses of the remaining money to pay back her 

sister. If she has $15 left, how much did she have at first? 

6. Parks is wearing several rubber bracelets. of the bracelets are tie-dye, are blue, and of the 

remainder are camouflage. If Parks wears 2 camouflage bracelets, how many bracelets does he have on? 

7. Ahmed spent of his money on a burrito and a water. The burrito cost 2 times as much as the water. 

The burrito cost $4, how much money does Ahmed have left? 



Date: 11/10/13 

4.E.56 





Name 

Date 

1. Three-quarters of the boats in the marina are white, of the remaining boats are blue, and the rest are 

red. If there are 9 red boats, how many boats are in the marina? 



Date: 11/10/13 

4.E.57 





Name 

Date 

Solve using tape diagrams. 

1. Anthony bought an 8-foot board. He cut off of the board to build a shelf, and gave of the rest to his 

brother for an art project. How many inches long was the piece Anthony gave to his brother? 

2. Riverside Elementary School is holding a school-wide election to choose a school color. Five-eighths of 

the votes were for blue, of the remaining votes were for green, and the remaining 48 votes were for 

red. 

a. How many votes were for blue? 

b. How many votes were for green? 



Date: 11/10/13 

4.E.58 





c. If every student got one vote, but there were 25 students absent on the day of the vote, how many 

students are there at Riverside Elementary School? 

d. Seven-tenths of the votes for blue were made by girls. Did girls who voted for blue make up more 

than or less than half of all votes? Support your reasoning with a picture. 

e. How many girls voted for blue? 



Date: 11/10/13 

4.E.59 





Lesson 17 

Objective: Relate decimal and fraction multiplication. 






Total Time 

(12 minutes) 

(7 minutes) 

(31 minutes) 

(10 minutes) 

(60 minutes) 




• Multiply Whole Numbers by Decimals 5.NBT.7 

(4 minutes) 

(4 minutes) 

(4 minutes) 



Note: This fluency reviews G5─M4─Lessons 13–16. 

T: (Write × .) Say the number sentence. 

S: × = . 

Continue the process with × and × . 

T: (Write × = .) On your boards, write the number sentence. 

S: (Write × = .) 

T: (Write × = .) Say the number sentence. 

S: = . 

Repeat the process with × , × , and × . 

T: (Write × = .) Say the number sentence. 

S: × = . 

Lesson 17: Relate decimal and fraction multiplication. 

Date: 11/10/13 4.E.60 





Continue the process with × . 

T: (Write × = .) On your boards, write the equation. 

S: (Write × = .) 

T: (Write × = .) On your boards write the equation. 

S: (Write × = = 1.) 

Continue the process with the following possible suggestions: × , × , and × . 



Note: This fluency prepares students for G5─M4─Lessons 17–18. 


S: 1 tenth. 


S: 0.1. 





S: 0.01. 

Continue with the following possible suggestions: , , , , , , and . 

T: (Write 0.01.) Say it as a fraction. 


T: Write it as a fraction. 

S: . 


Multiply Whole Numbers by Decimals (4 minutes) 


Note: This fluency prepares students for G5─M4─Lessons 17─18. In the following dialogue, several possible 

student responses are represented. 


Date: 11/10/13 4.E.61 





T: (Write 2 × 0.1 = .) What is 2 copies of 1 tenth? 

S: 2 tenths. 

T: (Write 0.2 in the number sentence above.) 

T: (Erase the product and replace the 2 with a 3.) What is 3 copies of 1 tenth? 

S: 3 tenths. 

T: Write it as a decimal on your board. 

S: 0.3. 

T: 4 copies of 1 tenth? Write it as a decimal on your board. 

S: 0.4. 

T: 7 × 0.1? 

S: 0.7. 

T: (Write 7 × 0.01 = .) What is 7 copies of 1 hundredth? 

S: 7 hundredths. 


T: What is 5 copies of 1 hundredth? Write it as a decimal. 

T: 5 × 0.01? 


T: (Write 2 × 0.1 = .) Say the answer. 

T: (Write 20 × 0.1 = .) What is 20 copies of 1 tenth? 

S: 20 tenths. 

T: Rename it using ones. 

S: 2 ones. 

T: (Write 20 × 0.01 = .) On your boards, write the number sentence. What are 20 copies of 1 

hundredth? 

S: (Write 20 × 0.01 = 0.20.) 20 hundredths. 

T: Rename the product using tenths. 

S: 2 tenths. 

Continue this process with the following possible suggestions, shifting between choral and board responses: 

30 × 0.1, 30 × 0.01, 80 × 0.01, and 80 × 0.1. If students are successful with the sequence above, continue with 

the following: 83 × 0.1, 83 × 0.01, 53 × 0.01, 53 × 0.1, 64 × 0.01, and 37 × 0.1. 


Ms. Casey grades 4 tests during her lunch. She grades 

of the remainder after school. If she still has 16 tests to 

grade after school, how many tests are there? 

Note: Today’s Application Problem recalls the previous 

lesson’s work with tape diagrams. This is a challenging 


Date: 11/10/13 4.E.62 





problem in that the value of a part is given and then the value of 2 thirds of the remainder. Possibly remind 

students to draw without concern initially for proportionality. They have erasers for a reason and can rework 

the model if they so choose. 



Problem 1: a. 0.1 4 b. 0.1 2 c. 0.01 6 

T: (Post Problem 1(a) on the board.) Read this multiplication expression using unit form and the word 

of. 

S: 1 tenth of 4. 

T: Write this expression as a multiplication sentence using 

a fraction and solve. Do not simplify your product. 

S: (Write 4 .) 

T: Write this as a decimal on your board. 

S: (Write 0.4.) 

T: (Write 0.1 4 = 0.4.) Let’s compare the 4 ones that we started with to the product that we found, 4 

tenths. Place 4 and 0.4 on a place value chart and talk to your partner about what happened to the 

digit 4 when we multiplied by 1 tenth. Why did our answer get smaller? 

S: The answer is 4 tenths because we were taking a part of 4 so the answer got smaller. The digit 4 

will shift one space to the right because the answer is only part of 4. The answer is 4 tenths. This 

is like 4 copies of 1 tenth. There are 40 tenths in 4 wholes. 1 tenth of 40 is 4. The unit is tenths, 

so the answer is 4 tenths. The digit stays the same because we are multiplying by 1 of something, 

but the unit is smaller, so the decimal point is moving one place to the left. 

T: What about of 4? Multiply, then show your thinking on the place value chart. 

S: (Work to show 4 hundredths. 0.04.) 

T: What about of 4? 

S: 4 thousandths. 0.004. 

Repeat the sequence with 0.1 2 and 0.1 6. Ask students to verbalize 

the patterns they notice. 

MP.4 

Problem 2: a. 0.1 0.1 b. 0.2 0.1 c. 1.2 0.1 

T: (Post Problem 2(a) on the board.) Write this as a fraction 

multiplication sentence and solve it with a partner. 


T: Let’s draw an area model to see if this makes sense. What should 

I draw first? Turn and talk. 


Date: 11/10/13 4.E.63 





S: Draw a rectangle and cut it vertically into 10 units and shade one of them. (Draw and label .) 

T: What do I do next? 

S: Cut each unit horizontally into 10 equal parts, and shade in 1 of those units. 

MP.4 

T: (Cut and label .) What units does our model show now? 

S: Hundredths. 

T: Look at the double-shaded parts, what is of ? (Save this 

model for use again in Problem 3(a).) 

S: 1 hundredth. . 

T: Write the answer as a decimal. 

S: 0.01. 

T: Let’s show this multiplication on the place value chart. When 

writing 1 tenth, where do we put the digit 1? 

S: In the tenths place. 

T: Turn and talk to your partner about what happened to the digit 1 

that started in the tenths place, when we took 1 tenth of it. 

S: The digit shifted 1 place to the right. We were taking only part of 1 tenth, so the answer is 

smaller than 1 tenth. It makes sense that the digit shifted to the right one place again because the 

answer got smaller and we are taking 1 tenth again like in the first problems. 

T: (Post Problem 2(b) on the board.) Show me 2 tenths on your place value chart. 

S: (Show the digit 2 in the tenths place.) 

T: Explain to a partner what will happen to the digit 2 when you multiply it by 1 tenth. 

S: Again, it will shift one place to the right. Every time you multiply by a tenth, no matter what the 

digit, the value of the digit gets smaller. The 2 shifts one place over to the hundredths place. 

T: Show this problem using fraction multiplication and solve. 

S: (Work and show = .) 

T: (Post Problem 2(c) on the board.) If we were to show this 

multiplication on the place value chart, visualize what 

would happen. Tell your partner what you see. 

S: The 1 is in the ones place and the 2 is in the tenths place. 

Both digits would shift one place to the right, so the 1 would be in the tenths place and the 2 would 

be in the hundredths place. The answer would be 0.12. Each digit shifts one place each. The 

answer is 12 hundredths. 

T: How can we express 1.2 as a fraction greater than 1? Turn and talk. 

S: 1 and 2 tenths is the same as 12 tenths. 12 tenths as a fraction is just 12 over 10. 

T: Show the solution to this problem using fraction multiplication. 

S: (Work and show = .) 

1 

10 

1 


Date: 11/10/13 4.E.64 





Problem 3: a. 0.1 0.01 b. 0.5 0.01 c. 1.5 0.01 

T: (Post Problem 3(a) on the board.) Work with a partner to show this as fraction multiplication. 

S: (Work and show ) 

T: What is ? 

S: . 

T: (Retrieve the model drawn in Problem 2(a).) Remember 

this model showed 1 tenth of 1 tenth, which is 1 

hundredth. We just solved 1 tenth of 1 hundredth, 

which is 1 thousandth. Turn and talk with your partner 

about how that would look as a model. 

S: If I had to draw it, I’d have to cut the whole into 100 equal parts and just shade 1. Then I’d have to 

cut just one of those tiny parts into 10 equal parts. If I did that to the rest of the parts, I’d end up 

with 1,000 equal parts and only 1 of them would be double shaded! It would be like taking that 1 

tiny hundredth and dividing it into 10 parts to make thousandths. I’d need a really fine pencil point! 

T: (Point to the tenths on place value chart.) Put 1 tenth on the place value chart. I’m here in the 

tenths place, and I have to find 1 

of this number. The digit 1 will shift in which direction 

and why? 

S: It will shift right, because the product is smaller than what we started with. 

T: How many places will it shift? 

S: Two places. 

T: Why two places? Turn and talk. 

S: We shifted one place when multiplying by a tenth, so it should be two places when multiplying by a 

hundredth. Like when we multiply by 10, that shifts one place to the left, and two places to the left 

when we multiply by 100. Our model showed us that finding a hundredth of something is like 

finding a tenth of a tenth, so we have to shift one place two times. 

T: Yes. (Move finger two places to the right to the thousandths place.) So, is equal to . 

T: (Post Problem 3(b) on the board.) Visualize a place value chart. When writing 0.5, where will the 

digit 5 be? 

S: In the tenths place. 

T: What will happen as we multiply by 1 hundredth? 

S: The 5 will shift two places to the right to the thousandths place. 

T: Say the answer. 

S: 5 thousandths. 

T: Show the solution to this problem using fraction multiplication. 

S: (Write and solve = .) 

T: Show the answer as a decimal. 

S: 0.005. 


Date: 11/10/13 4.E.65 





T: (Post Problem 3(c) on the board.) Express 1.5 as a 

fraction greater than 1. 

S: . 

T: Show the solution to this problem using fraction 


S: (Write and show = .) 

T: Write the answer as a decimal. 

S: 0.015. 

NOTES ON 



It may be too taxing to ask some 

students to visualize a place value 

chart. As in previous problems, a place 

value chart can be displayed or 

provided. To provide further support 

for specific students, teachers can also 

provide place value disks. 

Problem 4: a. 7 × 0.2 b. 0.7 × 0.2 c. 0.07 × 0.2 

T: (Post Problem 4(a) on the board.) I’m going to rewrite this problem expressing the decimal as a 

fraction. (Write 7 × 

.) Are these equivalent expressions? Turn and talk. 

S: Yes, 0.2 = . So they show the same thing. This 

is like multiplying fractions like we’ve been doing. 

T: When we multiply, what will the numerator show? 

S: 7 × 2. 

T: The denominator? 

S: 10. 

T: (Write .) Write the answer as a fraction. 

S: (Write ) 

T: Write 14 tenths as a decimal. 

S: (Write 1.4.) 

T: Think about what we know about the place value chart and multiplying by tenths. Does our product 

make sense? Turn and talk. 

S: Sure! 7 times 2 is 14. So, 7 times 2 tenths is like 7 times 2 

times 1 tenth. The answer should be one-tenth the size of 14. 

It does make sense. It’s like 7 times 2 equals 14, and then 

the digits in 14 both shift one place to the right because we 

took only 1 tenth of it. I know it’s like 2 tenths copied 7 

times. Five copies of 2 tenths is 1 and then 2 more tenths. 

T: (Post Problem 4(b) on the board.) Work with a partner 

and show the solution using fraction multiplication. 

S: (Write and solve = .) 

T: What’s 14 hundredths as a decimal? 

S: 0.14. 

T: (Post Problem 4(c) on the board.) Solve this problem independently. Compare your answer with a 

partner when you’re done. (Allow students time to work and compare answers.) 


Date: 11/10/13 4.E.66 





T: Say the problem using fractions. 

S: 

T: What’s 14 thousandths as a decimal? 

S: 0.014. 


NOTES ON 


ACTION AND 

EXPRESSION: 

Teachers and parents alike may want 

to express multiplying by one-tenth as 

moving the decimal point one place to 

the left. Notice the instruction focuses 

on the movement of the digits in a 

number. Just like the ones place, the 

tens place, and all places on the place 

value chart, the decimal point does not 

move. It is in a fixed location 

separating the ones from the tenths. 









Lesson Objective: Relate decimal and fraction 










lesson. 



Date: 11/10/13 4.E.67 





• In Problem 2, what pattern did you notice 

between (a), (b), and (c); (d), (e), and (f); and (g), 

(h), and (i)? (The product is to the tenths, 

hundredths, and thousandths.) 

• Share and explain your solution to Problem 3 


• Share your strategy for solving Problem 4 with a 

partner. 

• Explain to your partner why and 

. 

• We know that when we take one-tenth of 3, this 

shifts the digit 3 one place to the right on the 

place value chart, because 3 tenths is 1 tenth of 

3. When we compare the standard form of 3 to 

0.3, it appears that the decimal point has moved. 

How could thinking of it this way help us? How 

does the decimal point move when we multiply 

by 1 tenth? By 1 hundredth? 




effectively for future lessons. You may read the questions aloud to the students 


Date: 11/10/13 4.E.68 





Name 

Date 

1. Multiply and model. Rewrite each expression as a multiplication sentence with decimal factors. The first 

one is done for you. 

1 

a. 

b. 

= 

= 

0.1 0.1 = 0.01 

c. 1 

d. 1 


Date: 11/10/13 4.E.69 





2. Multiply. The first few are started for you. 

a. 5 0.7 = _______ b. 0.5 0.7 = _______ c. 0.05 0.7 = _______ 

= 5 = = 

= = = 

= = = 

= 3.5 

d. 6 0.3 = _______ e. 0.6 0.3 = _______ f. 0.06 0.3 = _______ 

g. 1.2 4 = _______ h. 1.2 0.4 = _______ i. 0.12 0.4 = _______ 

3. A boy scout has a length of rope measuring 0.7 meter. He uses 2 tenths of the rope to tie a knot at one 

end. How many meters of rope are in the knot? 

4. After just 4 tenths of a 2.5 mile race was completed, Lenox took the lead and remained there until the 

end of the race. 

a. How many miles did Lenox lead the race? 

b. Reid, the second place finisher, developed a cramp with three-tenths of the race remaining. How 

many miles did Reid run without a cramp? 


Date: 11/10/13 4.E.70 





Name 

Date 

1. Multiply and model. Rewrite each expression as a number sentence with decimal factors. 

a. 1 

2. Multiply. 

a. 1.5 3 = _______ b. 1.5 0.3 = _______ c. 0.15 0.3 = _______ 


Date: 11/10/13 4.E.71 





Name 

Date 

1. Multiply and model. Rewrite each expression as a number sentence with decimal factors. The first one is 

done for you. 

a. b. 

1 

= 

= 

0.1 0.1 = 0.01 

c. 1 

d. 1 


Date: 11/10/13 4.E.72 





2. Multiply. The first few are started for you. 

a. 4 0.6 = _______ b. 0.4 0.6 = _______ c. 0.04 0.6 = _______ 

= 4 = = 

= = = 

= = = 

= 2.4 

d. 7 0.3 = _______ e. 0.7 0.3 = _______ f. 0.07 0.3 = _______ 

g. 1.3 5 = _______ h. 1.3 0.5 = _______ i. 0.13 0.5 = _______ 

3. Jennifer makes 1.7 liters of lemonade. If she pours 3 tenths of the lemonade in the glass, how many liters 

of lemonade are in the glass? 

4. Cassius walked 6 tenths of a 3.6 mile trail. 

a. How many miles did Cassius have left to hike? 

b. Cameron was 1.3 miles ahead of Cassius. How many miles did Cameron hike already? 


Date: 11/10/13 4.E.73 





Lesson 18 

Objective: Relate decimal and fraction multiplication. 






Total Time 

(12 minutes) 

(8 minutes) 

(30 minutes) 

(10 minutes) 

(60 minutes) 


• Sprint: Multiply Fractions 5.NF.4 

• Multiply Whole Numbers and Decimals 5.NBT.7 

(9 minutes) 

(3 minutes) 

Sprint: Multiply Fractions (9 minutes) 

Materials: (S) Multiply Fractions Sprint 


Multiply Whole Numbers and Decimals (3 minutes) 



T: (Write 3 × 2.) Say the number 

sentence. 

S: 3 × 2 = 6. 

T: (Write 3 × 0.2.) On your boards, 

write the number sentence and 

solve. 

S: (Write 3 × 0.2 = 0.6.) 

T: (Write 0.3 × 0.2.) On your boards, write the number sentence. 

S: (Write 0.3 × 0.2 = 0.06.) 

T: (Write 0.03 × 0.2.) On your boards, write the number sentence. 

S: (Write 0.03 × 0.2 = 0.006.) 

3 × 2 = 6 3 × 0.2 = 0.6 3 × 0.02 = 0.06 0.3 × 0.2 = 0.06 

2 × 7 = 14 2 × 0.7 = 1.4 2 × 0.7 = 1.4 0.02 × 0.7 = 0.014 

5 × 3 = 15 0.5 × 3 = 1.5 0.5 × 0.3 = 0.15 0.5 × 0.03 = 0.015 


Date: 11/10/13 4.E.74 





Continue this process with the following possible suggestions: 2 × 7, 2 × 0.7, 0.2 × 0.7, 0.02 × 0.7, 5 × 3, 

0.5 × 3, 0.5 × 0.3, and 0.5 × 0.03. 


An adult female gorilla is 1.4 meters tall when standing upright. Her daughter is 3 tenths as tall. How much 

more will the young female gorilla need to grow before she is as tall as her mother? 

Note: This Application Problem reinforces that multiplying a 

decimal number by tenths can be interpreted in fraction or 

decimal form (as practiced in G5─M4─Lesson 17). Students who 

solve this problem by converting to smaller units (centimeters 

or millimeters) should be encouraged to compare their process 

to solving the problem using 1.4 meters. 


NOTES ON 


ENGAGEMENT: 

With reference to Table 2 of the 

Common Core Learning Standards, this 

Application Problem is considered a 

compare with unknown product 

situation. Table 2 is a matrix that 

organizes story problems or situations 

into specific categories. Consider 

presenting this table in a studentfriendly 

format as a tool to help 

students identify specific types of story 

problems. 


Problem 1: a. 3.2 2.1 b. 3.2 0.44 c. 3.2 4.21 

T: (Post Problem 1(a) on board.) Rewrite this problem as a fraction multiplication expression. 

S: (Write .) 

T: Before we multiply these two decimals, let’s 

estimate what our product will be. Turn and 

talk. 

S: 3.2 is pretty close to 3 and 2.1 is pretty close to 

2. I’d say our answer will be around 6. The 

product will be a little more than 6 because 3.1 

is a little more than 3 and 2.1 is a little more 

than 2. It’s about twice as much as 3. 

T: Now that we’ve estimated, let’s solve. (Write 

= on board.) What do we get when we 

multiply tenths by tenths? 


Date: 11/10/13 4.E.75 






T: Let’s use unit form to multiply 32 tenths and 21 tenths vertically. Solve with your partner. (Allow 

students time to work and solve.) 

T: (Write = .) What is 32 tenths times 21 tenths? 


T: (Write = on the board.) Write this as a decimal. 

S: (Write 6.72.) 

T: Does this answer make sense given what we estimated 

the product to be? 

S: Yes. 

T: (Post Problem 1(b) on the board.) Before we solve this one, turn and talk with your partner to 

estimate the product. 

S: We are still multiplying by 3.2, but this time we want about 3 of almost 1 half. That’s like 3 halves, so 

our answer will be around 1 and a half. This is about 3 times more than 4 tenths, so the answer 

will be around 12 tenths. It will be a little more because it’s a little more than 3 times as much. 

T: Work with a partner and rewrite this problem as a fraction multiplication expression. 

S: (Share and show .) 

T: What is 1 tenth of a hundredth? 

S: 1 thousandth. 

T: (Write = .) Work with a partner to multiply. Express your answer as a fraction and as a 

decimal. 

S: (Work and show = 1.408.) 

T: Does this product make sense given our estimates? 

S: Yes! It’s a little more than 1.2 and a little less than 1.5. 

T: (Post Problem 1(c) on the board.) Estimate this product with your partner. 

S: Three times as much as 4 is 12. This will be a little more than that because it’s a little more than 3 

and a little more than 4. It’s still multiplying by something close to 3. This time it’s close to 4. 3 

fours is 12. 

T: Rewrite this problem as a fraction multiplication expression. 

S: (Write .) 

T: (Write = .) Solve independently. Express your answer as a fraction and as a decimal. 

S: (Write and solve = 13.472.) 

T: Does our answer make sense? Turn and talk. (Allow students time to discuss with their partners.) 


Date: 11/10/13 4.E.76 





Problem 2: 2.6 0.4 

T: (Post Problem 2 on the board.) This time, let’s rewrite this problem vertically in unit form first. 2.6 is 

equal to how many tenths? 

S: 26 tenths. 

T: (Write = 26 tenths.) 0.4 is equal to how many tenths? 

S: 4 tenths. 

T: (Write 4 tenths.) Think, what does tenths times 

tenths result in? 


T: Our product will be named in hundredths. I’ll name those units right now. (Write hundredths at 

bottom of algorithm.) Solve 26 times 4. 

S: (Work and solve to find 104.) 

T: I’ll record 104 as the product. (Write 104 in the algorithm.) 104 what? 

What is our unit? 


T: Write it in standard form. 

S: (Write = 1.04.) 

MP.2 

T: Work with your partner to solve this using fraction multiplication to 

confirm our product. (Allow students time to work.) 

T: Look back at the original problem. What do you notice about the number of decimal places in the 

factors and the number of decimal places in our product? Turn and talk. 

S: There is one decimal place in each factor and two in the answer. I see two total decimal places in 

the factors and two decimal places in the product. They match. 

T: Keep this observation in mind as we continue our work. Let’s see if it’s always true. 

Problem 3: a. 3.1 1.4 b. 0.31 1.4 

T: (Post Problem 3(a) on the board.) Please estimate the product with your partner. 

S: It should be something close to 3, because 3 times 1 is 3. Something between 3 and 6, because 

1.4 is close to the midpoint of 1 and 2. It’s close to 3 times 1 and a half. That’s 4 and a half. 

T: Let’s use unit form again to solve this, but I will record it 

slightly differently. Let’s think of 3.1 as 31 tenths. 

(Record 3.1, and use the arrow to show the movement of 

the decimal and record 31 to the right). If we rename 1.4 

as tenths what will we record? 

S: 14 tenths. 

T: Let me record that. (Record as above showing movement 

with an arrow and writing 14 to the right.) Now multiply 

31 and 14. What is the product? 

S: 434. 

T: 434 what? 


Date: 11/10/13 4.E.77 






T: Name it as a decimal. 

S: 4 and 34 hundredths. 

T: Let me record that using our new method. (Rewrite 434 beneath the decimal multiplication. Show 

movement of the decimal two places to the left using two arrows.) 

T: What do you notice about the decimal places in the factors and the product this time? 

S: This is like before. We have two decimal places in the factors and two decimal places in the answer. 

We had tenths times tenths. That’s one decimal place times one decimal place. We got 

hundredths in our answer that’s two decimal places. It’s just like last time. 

T: Keep observing. Let’s see if this pattern holds true in our next problems. 

T: (Post Problem 3(b) on the board.) Let’s think of 0.31 and 1.4 as whole numbers of units. 0.31 is the 

same as 31 what? 1.4 is the same as 14 what? 

S: 31 hundredths and 14 tenths. 

T: If we were using fractions to multiply these two numbers, what part of the fraction would 31 x 14 

give us? 

S: The numerator. 

T: What does the numerator of a fraction tell us? 

S: The number of units we have. 

T: This whole number multiplication problem is 

the same as our last one. What is 31 times 14? 

S: 434. 

T: While these digits are the same as last time, 

will our product be the same? Why or why not? Turn and talk. 

S: It won’t be the same as last time. We are multiplying hundredths and tenths this time so our unit in 

the answer has to be thousandths. The answer is 434 thousandths. Last time, we had two 

decimal places in our factors, so we had two decimal places in our product. This time, there are 

three decimal places in the factors, so we should have thousandths in the answer. Last time, we 

were multiplying by about 3 times as much as 1 and a half. This time, we want around 3 tenths of 1 

and a half. That’s going to be a lot smaller answer because we only want part of it, so the product 

couldn’t be the same. 

T: What is our product? 

S: 434 thousandths. 

T: Yes, since we remember that 1 hundredth times 1 tenth gives us our unit, the denominator of our 

fraction. Let’s use arrows to show that product. (Write the product and draw corresponding 

arrows.) Did the pattern that we saw earlier concerning the decimal places of factors and product 

hold true here as well? Turn and talk. (Allow students time to discuss with their partners.) 

Problem 4: 4.2 0.12 

T: (Post Problem 4 on the board.) Work independently to solve this problem. You may rename the 

factors as fractions and then multiply, rename the factors in unit form, or show the unit form using 

arrows. When you’re finished, compare your work with a neighbor and explain your thinking. 


Date: 11/10/13 4.E.78 






T: What is the product of 4.2 0.12? 

S: 0.504. 


Students should do their personal best to complete the Problem 

Set within the allotted 10 minutes. For some classes, it may be 

appropriate to modify the assignment by specifying which 

problems they work on first. Some problems do not specify a 

method for solving. Students solve these problems using the 

RDW approach used for Application Problems. 


Lesson Objective: Relate decimal and fraction multiplication. 












the answers for Parts (a) and (b) and the answers 

for Parts (c) and (d)? What pattern did you 

notice between 1(a) and 1(b)? (Part (a) is double 

(b). Part (c) is 4 times as large as (d).) Explain 

why that is. 

• Compare Problems 1(c) and 2(c). Why are the 

products not so different? Use estimation, and 

explain it to your partner. 

• Compare Problems 1(d) and 2(d). Why do they 

have the same digits but a different product? 

Explain it to your partner. 

• What do you notice about the relationship 

between 3(a) and 3(b)? (Part (a) is half of (b).) 

18 

10 

NOTES ON 



Double, twice, and half are words that 

can be confusing to all students, but 

especially English language learners. 

• Pre-teach this vocabulary in ways 

that connect to students’ prior 

knowledge. 

• Display posters with graphic 

representations of these words. 

• Ask questions that specifically 

require students to use this 

vocabulary. 

• Solicit support from physical 

education, art, and music 

teachers. Ask them to carefully 

embed these words into their 

lessons. 


Date: 11/10/13 4.E.79 





• For Problem 5, compare and share your solutions 

with a partner. Explain how you solved. 

• In one sentence, explain to your partner the 

pattern that we discovered today in the number 

of decimal places in our factors compared to the 

number of decimal places in our products. 







the students. 


Date: 11/10/13 4.E.80 






Date: 11/10/13 4.E.81 






Date: 11/10/13 4.E.82 





Name 

Date 

1. Multiply. The first one has been done for you. 

a. 2.3 × 1.8 = 2 3 (tenths) 

b. 2.3 × 0.9 = 

2 3 (tenths) 

= 

= 

1 8 (tenths) 

1 8 4 

+ 2 3 0 

4 1 4 (hundredths) 

9 (tenths) 

= 4.14 

c. 6.6 × 2.8 = d. 3.3 × 1.4 = 


a. 2.38 × 1.8 = 


b. 2.37 × 0.9 = 

= 

= 

× 1 8 (tenths) 

1 9 0 4 

+ 2 3 8 0 

4 2 8 4 (thousandths) 


× 9 (tenths) 

= 4.284 

c. 6.06 × 2.8 = d. 3.3 × 0.14 = 


Date: 11/10/13 4.E.83 





3. Solve using the standard algorithm. Use the thought bubble to show your thinking about the units of 

your product. 

a. 3.2 × 0.6 = __________ b. 3.2 × 1.2 = __________ 

3. 2 

0. 6 

1. 9 2 

3 2 tenths 

6 tenths 

1 9 2 hundredths 

3 2 

1 2 

c. 8.31 × 2.4 = __________ d. 7.50 × 3.5 = __________ 

4. Carolyn buys 1.2 lb of chicken breast. If each pound of chicken costs $3.70, how much will she pay for 

the chicken? 

5. A kitchen measures 3.75 m by 4.2 m. 

a. Find the area of the kitchen. 

b. The area of the living room is one and a half times that of the kitchen. Find the total area of the living 

room and the kitchen. 


Date: 11/10/13 4.E.84 





Name 

Date 

1. Multiply. 

a. 3.2 × 1.4 = b. 1.6 × 0.7 = 

c. 2.02 × 4.2 = d. 2.2 × 0.42 = 


Date: 11/10/13 4.E.85 





Name 

Date 


a. 3.3 × 1.6 = 3 3 

b. 3.3 × 0.8 = 

= 

= 

× 1 6 

1 9 8 

+ 3 3 0 

5 2 8 

3 3 

× 8 

= 5.28 

c. 4.4 × 3.2 = d. 2.2 × 1.6 = 


a. 3.36 × 1.4 = 3 3 6 

b. 3.35 × 0.7 = 

× 1 4 

= 

3 3 5 

× 7 

= 

= 4.704 

c. 4.04 × 3.2 = d. 4.4 × 0.16 = 


Date: 11/10/13 4.E.86 





3. Solve using the standard algorithm. Use the thought bubble to show your thinking about the units of 

your product. 

a. 3.2 × 0.6 = __________ b. 3.2 × 1.2 = __________ 

3. 2 

0. 6 

1. 9 2 

3 2 tenths 

6 tenths 

1 9 2 hundredths 

3 2 

1 2 

c. 7.41 × 3.4 = __________ d. 6.50 × 4.5 = __________ 

32 

10 

6 

10 

192 

100 

4. Erik buys 2.5 lb of cashews. If each pound of cashews costs $7.70, how much will he pay for the cashews? 

5. A swimming pool at a park measures 9.75 m by 7.2 m. 

a. Find the area of the swimming pool. 

b. The area of the playground is one and a half times that of the swimming pool. Find the total area of 

the swimming pool and the playground. 


Date: 11/10/13 4.E.87 





Lesson 19 

Objective: Convert measures involving whole numbers, and solve multistep 







Total Time 

(8 minutes) 

(8 minutes) 

(34 minutes) 

(10 minutes) 

(60 minutes) 


Angle A of a triangle is the size of angle C. Angle B is the size of angle C. If angle C measures 80 degrees, 

what are the measures of angle A and angle B? 

B 

° 

A 

Note: Because today’s fluency activity asks students to recall the content of yesterday’s lesson, this problem 

asks students to recall previous learning to find fraction of a set. The presence of a third angle increases 

complexity. 


• Multiply Decimals 5.NBT.7 


(4 minutes) 

(4 minutes) 

Lesson 19: Convert measures involving whole numbers, and solve multi-step 


Date: 11/10/13 

4.E.88 





Multiply Decimals (4 minutes) 


Note: This fluency reviews G5–M4–Lessons 17−18. 

T: (Write 4 × 2 =____.) Say the number 

sentence. 

S: 4 × 2 = 8. 

T: (Write 4 × 0.2 =____.) On your 

boards, write number sentence. 

S: (Write 4 × 0.2 = 0.8.) 

T: (Write 0.4 × 0.2 =____.) On your boards, write number sentence. 

S: (Write 0.4 × 0.2 = 0.08.) 


0.4 × 3, 0.4 × 0.3, and 0.4 × 0.03. 



Note: This lesson prepares students for G5–M4–Lesson 19. Allow students to use the conversion reference 

sheet if they are confused, but encourage them to answer questions without looking at it. 

T: (Write 1 yd = ____ ft.) How many feet are equal to 1 yard? 

S: 3 feet. 

T: (Write 1 yd = 3 ft. Below it, write 10 yd = ____ ft.) 10 yards? 

S: 30 feet. 

Continue with the following possible sequence: 1 pint = 2 cups, 8 pints = 16 cups, 1 ft = 12 in, 4 ft = 48 in, 1 

gal = 4 qt, and 8 gal = 32 qt. 

T: (Write 2 c = ____ pt.) How many pints are equal to 2 cups? 

S: 1 pint. 


S: 8 pints. 

4 × 2 = 8 4 × 0.2 = 0.8 0.4 × 0.2 = 0.08 0.04 × 0.2 = 0.008 

2 × 9 = 18 2 × 0.9 = 1.8 0.2 × 0.9 = 0.18 0.02 × 0.9 = 0.018 

4 × 3 = 12 0.4 × 3 = 1.2 0.4 × 0.3 = 0.12 0.4 × 0.03 = 0.012 

Continue with the following possible sequence: 12 in = 1 ft, 48 in = 4 ft, 3 ft = 1 yd, 24 ft = 8 yd, 4 qt = 1 gal, 

and 24 qt = 6 gal. 



Problem 1: 30 centimeters = ________meters 



Date: 11/10/13 

4.E.89 





T: (Post Problem 1 on the board.) Which is a larger unit, centimeters or meters? 

S: Meters. 

T: So, we are expressing a smaller unit in terms of a larger unit. Is 30 cm more or less than 1 meter? 

S: Less than 1 meter. 

T: Is it more than or less than half a meter? Talk to your 

partner about how you know. 

S: It’s less than half, because 50 cm is half a meter and this is 

only 30 cm. It’s less than half, because 30 out of a 

hundred is less than half. 

T: Let’s keep that in mind as we work. We want to rename 

these centimeters using meters. 

T: (Write 30 cm = 30 × 1 cm.) We know that 30 cm is the 

same as 30 copies of 1 cm. Let’s rename 1 cm as a fraction 

of a meter. What fraction of a meter is 1 cm? Turn and 

talk. 

S: It takes 100 cm to make a meter, so 1 cm would be 1 

hundredth of a meter. 100 cm = 1 meter so 1 cm = 

meter. 100 out of 100 cm makes 1 whole 

meter. We’re looking at 1 out of 100 cm, so that is 1 

hundredth of a meter. 

T: (Write 30 cm = 30 × 1 cm = 30 × meter.) How do 

you know this is true? 

S: It’s true because we just renamed the centimeter as 

the same amount in meters. One centimeter is the 

same thing as 1 hundredth of a meter. 

T: Now we have 30 copies of meter. How many 

hundredths of a meter is that in all? 

S: 30 hundredths of a meter. 

T: Write it as a fraction on your board, and then work 

with a neighbor to express it in simplest form. 

S: (Work.) 

T: Answer the question in simplest form. 

S: 30 cm = m. 

T: (Write = m.) Think about our estimate. Does this 

answer make sense? 

S: Yes, we thought it would be less than a half meter, and 

meter is less than half a meter. 

NOTES ON 


ACTION AND 

EXPRESSION: 

Teachers can provide students with 

meter sticks and centimeter rulers to 

help answer these important first 

questions of this Concept 

Development. These tools will help 

students see the relationship of 

centimeters to meters, and meters to 

centimeters. 



Date: 11/10/13 

4.E.90 





Problem 2: 9 inches = ________foot 

T: (Write 9 inches = 9 × 1 inch on board.) 9 inches is 9 copies of 1 inch. What fraction of a foot is 1 

inch? Draw a tape diagram if it helps you. 

S: 1 twelfth foot. 

T: Before we rename 1 inch, let’s estimate. Will 9 inches be more than half a foot or less than half a 

foot? Turn and tell your partner how you know. 

S: Half a foot is 6 inches. Nine is more than that so it will be more than half. Half of 12 inches is 6 

inches. Nine inches is more than that. 

T: (Write = 9 × foot.) Let’s rename 1 inch as a 

S: Yes. 

fraction of a foot. Now we have written 9 copies of 

foot. Are these expressions equivalent? 

T: Multiply. How many feet is the same amount as 9 

inches? 

S: 9 twelfths of a foot 3 fourths of a foot. 

T: Does this answer make sense? Turn and talk. 

Repeat sequence for 24 inches = ______ yard. 

Problem 3: Koalas will often sleep for 20 hours a day. For what fraction of a day does a Koala often sleep? 

T: (Post Problem 3 on the board.) What will we need to do to 

solve this problem? Turn and talk. 

S: We’ll need to express hours in days. We’ll need to convert 

20 hours into a fraction of a day. 

T: Work with a partner to solve. Express your answer in its 

simplest form. 

S: (Work and share and show 20 hours = day.) 

Problem 4: 15 inches = ________ feet 

T: (Post Problem 4 on the board.) Compare this conversion to the others we’ve done. Turn and talk. 

S: We’re still converting from a small unit to a larger one. The last one converted something smaller 

than a whole day. This is converting something more than a whole foot. Fifteen inches is more than 

a foot, so our answer will be greater than 1. We still have to think about what fraction of a foot is 

1 inch. 

T: Yes, the process of converting will be the same, but our answer will be greater than 1. Let’s keep 

that in mind as we work. Write an equation showing how many copies of 1 inch we have. 

S: (Work and show 15 inches = 15 1 inch.) 

T: What fraction of a foot is 1 inch? Turn and talk. 



Date: 11/10/13 

4.E.91 





S: It takes 12 inches to make a foot, so 1 inch 

would be 1 twelfth of a foot. 12 inches = 1 

foot so 1 inch = 

foot. 

T: Now we have 15 copies of foot. How many 

twelfths of a foot is that in all? 

S: feet. 

T: Work with a neighbor to express in its 

simplest form. 

S: (Work and show 15 inches = feet.) 

Problem 5: 24 ounces = ________ pound 

T: (Post Problem 5 on the board.) Work 

independently to solve this conversion. 

S: (Work.) 

T: Show the conversion in its simplest form. 

S: (Show 24 ounces = 1 pounds.) 










Lesson Objective: Convert measures involving whole 

numbers, and solve multi-step word problems. 







addressed in the Debrief. Guide students in a conversation to debrief the Problem Set and process the 

lesson. 



Date: 11/10/13 

4.E.92 






• In Problem 1, what did you notice about all of the 

problems in the left-hand column? The right-hand 

column? Did you solve the problems differently as a 

result? 

• Explain your process for solving Problem 4. How did 

you convert from cups to gallons? What is a cup 

expressed as a fraction of a gallon? How did you figure 

that out? 

• In Problem 2, you were asked to find the fraction of a 

yard of craft trim Regina bought. Tell your partner how 

you solved this problem. 

• How did today’s second fluency activity help prepare 

you for this lesson? 

• Look back at Problem 1(e). Five ounces is equal to how 

many pounds? What would 6 ounces be equal to? 7 

ounces? 8 ounces? 9 ounces? Think carefully. 

pound equals how many ounces? 

pound? 

pound? 

pound? Talk about your thinking as you 

answered those questions. 


NOTES ON 


ACTION AND 

EXPRESSION: 

Some students may struggle as they try 

to articulate their ideas. Some 

strategies that may be used to support 

these students are given below. 

• Ask students to repeat in their 

own words the teacher’s thinking. 

• Ask students to add on to either 

the teacher’s thinking or another 

student’s thoughts. 

• Give students time to practice 

with their partners before 

answering in a larger group. 

• Pose a question and ask students 

to use specific vocabulary in their 

answers. 






Date: 11/10/13 

4.E.93 





Name 

Date 

1. Convert. Express your answer as a mixed number if possible. The first one is done for you. 

a. 2 ft = ________ yd 

2 ft = 2 1 ft 

= 2 yd 

= yd 

b. 4 ft = ______ yd 

4 ft = 4 1 ft 

= 4 yd 

= yd 

c. 7 in = ________ ft d. 13 in = _________ft 

e. 5 oz = ________ lb f. 18 oz = _________lb 

2. Regina buys 24 inches of trim for a craft project. 

a. What fraction of a yard does Regina buy? 

b. If a whole yard of trim costs $6, how much did Regina pay? 



Date: 11/10/13 

4.E.94 





3. At Yo-Yo Yogurt, the scale says that Sara has 8 oz of vanilla yogurt in her cup. Her father’s yogurt weighs 

11 oz. How many pounds of frozen yogurt did they buy altogether? Express your answer as a mixed 

number. 

4. Pheng-Xu drinks 1 cup of milk every day for lunch. How many gallons of milk does he drink in 2 weeks? 



Date: 11/10/13 

4.E.95 





Name 

Date 

1. Convert. Express your answer as a mixed number if possible. 

a. 5 in = ___________ft b. 13 in = _____________ft 

c. 9 oz = ___________lb d. 18 oz = _____________lb 



Date: 11/10/13 

4.E.96 





Name 

Date 

1. Convert. Express your answer as a mixed number if possible. 

a. 2 ft = ________ yd 

b. 6 ft = ______ yd 

2 ft = 2 1 ft 

= 2 yd 

= yd 

6 ft = 6 1 ft 

= 6 yd 

= yd 

c. 5 in = ________ ft d. 14 in = _________ft 

e. 7 oz = ________ lb f. 20 oz = _________lb 

g. 1 pt = ________ qt h. 4 pt = __________qt 



Date: 11/10/13 

4.E.97 





2. Marty buys 12 oz of granola. 

a. What fraction of a pound of granola did Marty buy? 

b. If a whole pound of granola costs $4, how much did Marty pay? 

3. Sara and her dad visit Yo-Yo Yogurt again. This time, the scale says that Sara has 14 oz of vanilla yogurt in 

her cup. Her father’s yogurt weighs half as much. How many pounds of frozen yogurt did they buy 

altogether on this visit? Express your answer as a mixed number. 

4. An art teacher uses 1 quart of blue paint each month. In one year, how many gallons of paint will she 

use? 



Date: 11/10/13 

4.E.98 





Lesson 20 

Objective: Convert mixed unit measurements, and solve multi-step word 

problems. 






Total Time 

(12 minutes) 

(6 minutes) 

(32 minutes) 

(10 minutes) 

(60 minutes) 


• Count by Fractions 5.NF.7 


• Multiply Decimals 5.NBT.7 


(3 minutes) 

(3 minutes) 

(3 minutes) 

(3 minutes) 

Count by Fractions (3 minutes) 


T: Count by ones to 10. (Write as students count.) 

S: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 

T: Count by halves to 10 

halves. (Write as 

students count.) 

S: 1 half, 2 halves, 3 

halves, 4 halves, 5 



halves, 10 halves. 

1 2 3 4 5 6 7 8 9 10 

1 2 3 4 5 

1 1 2 2 3 3 4 4 5 

T: Let’s count by halves 

again. This time, when we arrive at a whole number, say the whole number. (Write as students 

count.) 

S: 1 half, 1 whole, 3 halves, 2 wholes, 5 halves, 3 wholes, 7 halves, 4 wholes, 9 halves, 5 wholes. 

Lesson 20: Convert mixed unit measurements, and solve multi-step word 

problems. 

Date: 11/10/13 

4.E.99 





T: Let’s count by halves again. This time, change improper fractions to mixed numbers. (Write as 


S: 1 half, 1, 1 and 1 half, 2, 2 and 1 half, 3, 3 and 1 half, 4, 4 and 1 half, 5. 



Note: This fluency reviews G5–M4–Lessons 19–20. Allow students to use the conversion reference sheet if 

they are confused, but encourage them to answer questions without looking at it. 

T: (Write 1 ft = __ in.) How many inches are equal to 1 foot? 

S: 12 inches. 


S: 24 inches. 


S: 48 inches. 

Continue with the following possible sequence: 1 pint = 2 cups, 7 pints = 14 cups, 1 yard = 3 feet, 6 yd = 18 ft, 

1 gal = 4 qt, and 9 gal = 36 qt. 

T: (Write 2 c = __ pt.) How many pints are equal to 2 cups? 

S: 1 pint. 


S: 2 pints. 


S: 5 pints. 

Continue with the following possible sequence: 12 in = 1 ft, 36 in = 3 ft, 3 ft = 1 yd, 12 ft = 4 yd, 4 qt = 1 gal, 

and 28 qt = 7 gal. 

Multiply Decimals (3 minutes) 



T: (Write 3 × 3 = .) Say the 

multiplication sentence. 

S: 3 × 3 = 9. 

T: (Write 3 × 0.3 = .) On your 

boards, write the number sentence. 

S: (Write 3 × 0.3 = 0.9.) 

T: (Write 0.3 × 0.3 = .) On your boards, write the number sentence. 

S: (Write 0.3 × 0.3 = 0.09.) 

3 × 3 = 9 3 × 0.3 = 0.9 0.3 × 0.3 = 0.09 0.03 × 0.3 = 0.009 

2 × 8 = 16 2 × 0.8 = 1.6 0.2 × 0.8 = 0.16 0.02 × 0.8 = 0.016 

5 × 5 = 25 0.5 × 5 = 2.5 0.5 × 0.5 = 0.25 0.5 × 0.05 = 0.025 


problems. 

Date: 11/10/13 

4.E.100 






0.5 × 5, 0.5 × 0.5, and 0.5 × 0.05. 




T: How many feet are in 1 yard? 

S: 3 feet. 

T: (Write 3 ft = 1 yd. Below it, write 1 ft = __ yd.) 

What fraction of 1 yard is 1 foot? 

S: 1 third. 

T: On your boards, draw a tape diagram to explain your thinking. 

Continue with the following possible sequence: 2 ft = __ yd, 5 in = __ ft, 1 in = __ ft, 1 oz = __ lb, 9 oz = __ lb, 

1 pt = __ qt, 3 pt = __ qt, 4 days = _____week, and 18 hours = _____day. 


A recipe calls for lb of cream 

cheese. A small tub of cream 

cheese at the grocery store 

weighs 12 oz. Is this enough 

cream cheese for the recipe? 

Note: This problem builds on 

previous lessons involving unit 

conversions and multiplication of 

a fraction and a whole number. 

In addition to the method shown, 

students may also simply realize 

that is equal to . 

NOTES ON 



Another approach to this Application 

Problem is to think of it as a 

comparison problem. (See Table 2 of 

the Common Core Learning Standards.) 

Students can draw two bars, one 

showing the amount needed for the 

recipe, and another showing the 

amount sold in the small tub. The tape 

diagram would help students recognize 

the need to convert one of the 

amounts so that like units can be 

compared. 



Problem 1: Conversion of large units to small units. 


problems. 

Date: 11/10/13 

4.E.101 





yd = ________ ft 

T: (Write yd = ________ ft on board.) Which units are larger, yards or feet? 

S: Yards. 

T: Compare this problem with the conversions we worked on yesterday. What do you notice about the 

units? Turn and talk. 

S: This is starting with larger units and converting to smaller ones. Yesterday every conversion we 

did was little to big unit. This is big to little. 

T: Let’s draw a tape diagram to model this problem. We want to name yards using feet. (Draw a 

bar and label yd.) Let’s partition the bar into 4 equal units to represent the 4 whole yards and 1 

smaller unit to represent of a yard. yards is the same as 1 yard. (Write on the board.) 

Think back to our fluency activity. How many feet are in 1 yard? 

S: 3 feet. 

T: On your boards, draw a tape diagram to explain your 

thinking. 

S: (Draw.) 

T: (Show 1 yard equal to 3 feet below tape diagram.) 

Write a new expression to rename the yard in feet. 

S: (Write 3 feet.) 

T: Before we multiply, let’s express as an improper 

fraction. How many thirds are in 1? 

S: 3 thirds. 

T: So how many thirds are in 4? 

S: 12 thirds. 

T: How many thirds in 4 and 1 third? 

S: 13 thirds. 

T: Write a new multiplication expression that uses the 

improper fraction we just found. 

S: (Write 3.) 

T: Work with a partner to find the product of 13 thirds and 3. 

S: (Work.) 

T: (Point to the original problem.) Fill in the blank using a complete sentence. 

S: yd is equal to 13 ft. 

Repeat the same process with the following as necessary. 

NOTES ON 



The lessons in this topic require 

students to know basic conversions, 

such as 16 ounces are combined to 

make 1 pound. Teachers can provide 

this background knowledge in the form 

of posters or other graphic organizers. 

Students may also need support with 

some of the abbreviations used. For 

example, the abbreviation of pound 

(lb) and the abbreviation for ounce (oz) 

may seem confusing to some students. 

Teachers can post this information in a 

poster, or provide reference sheets for 

all students. 


problems. 

Date: 11/10/13 

4.E.102 





ft = _____ in 

gal = _____ qt 

hr = _____ min 

Problem 2: Conversion of small units to large units. 

11 ft = ________ yd 

T: (Write 11 ft = ___ yd on the board.) Which units 

are larger, feet or yards? 

S: Yards. 

T: Compare this problem to the others we’ve solved. 

S: This one gives us the measurement in small units 

and wants the amount of large units. This one 

goes from little units to big units like the ones we 

did yesterday. 

T: What fraction of 1 yard is 1 foot? 

S: 1 third. 

T: On your boards, draw a tape diagram to show the relationship between feet and yards. 

S: (Draw.) 


problems. 

Date: 11/10/13 

4.E.103 





T: What two whole number of yards will 11 feet fall between? Turn and talk. 

S: 9 feet is 3 yards, and 12 ft is 4 yards, so 11 feet must be somewhere between 3 and 4 yards. 

T: We know that 11 ft = 11 1 ft. (Write on the board.) Write a multiplication sentence that is 

equivalent to this one using yards. 

S: (Work.) 

T: Let’s record the equivalent expression beneath our first one. (Record as shown.) What is 11 ? 

S: 11 thirds. 

T: Express your answer as yards. 

S: 3 and 2 thirds yards. 

Repeat the same process with the following as necessary. 

ft = ______ yd 

qts = ______ gal 

If time permits, the following may be explored with students. 

Problem 3: A container can hold 

pints of water. How many cups of water can 2 containers hold? 

T: (Post the problem on the board, and read it out loud with the students.) How do you solve this 

problem? Turn and share your idea with a partner. 

S: It’s a two-step problem. I first have to convert pints to cups, and then I’ll have to double it. 

MP.2 

T: Let’s draw a tape diagram for pt. I’ll do it on the board and you draw it on your personal board. 

S: (Draw and label.) 

T: Say the multiplication expression to 

convert 

S: × 2 cups. 

pints to cups. 

T: Express as an improper fraction 

and restate the expression. 

S: × 2 cups. 

T: What’s the answer? 

S: cups. 


problems. 

Date: 11/10/13 

4.E.104 





MP.2 

T: How many whole cups is that? 

S: 9 cups. 

T: Finish by finding the amount of water in two 

containers. Turn and talk. 

S: We have to find the water in 2 containers. 

Since 1 container holds 9 cups, then we’ll have to 

double it. 9 cups + 9 cups = 18 cups. To find 

the amount 2 containers hold, we have to 

multiply. 2 × 9 cups = 18 cups. 










Lesson Objective: Convert mixed unit measurements, and 

solve multi-step word problems. 









lesson. 



• Share and compare your solutions for Problem 1 


• Explain to your partner how to solve Problem 3. 

Did you have a different strategy than your 

partner? 

• How did you solve for Problem 4? Explain your strategy to a partner. 


problems. 

Date: 11/10/13 

4.E.105 










problems. 

Date: 11/10/13 

4.E.106 





Name 

Date 

1. Convert. Show your work. Express your answer as a mixed number. (Draw a tape diagram if it helps 

you.) The first one is done for you. 

a. 2 yd = 8 ft 

b. qt = _____ gal 

2 yd = 2 1 yd 

qt = 

1 qt 

= 2 3 ft 

= 3 ft 

= ft 

= gal 

= gal 

= 

= 8 ft 

c. ft = _____ in d. pt = _____ qt 

e. 3 hr = _____ min f. 3 ft = _____ yd 


problems. 

Date: 11/10/13 

4.E.107 





2. Three dump trucks are carrying topsoil to a construction site. Truck A carries 3,545 lb, Truck B carries 

1,758 lb, and Truck C carries 3,697 lb. How many tons of topsoil are the 3 trucks carrying all together? 

3. Melissa buys gallons of iced tea. Denita buys 7 quarts more than Melissa. How much tea do they buy 

all together? Express your answer in quarts. 

4. Marvin buys a hose that is feet long. He already owns a hose at home that is the length of the new 

hose. How many total yards of hose does Marvin have now? 


problems. 

Date: 11/10/13 

4.E.108 





Name 

Date 

1. Convert. Express your answer as a whole number. 

a. ft = ___________ in b. ft = _____________yd 

c. c = ___________pt d. years = _____________ months 


problems. 

Date: 11/10/13 

4.E.109 





Name 

Date 

1. Convert. Show your work. Express your answer as a mixed number. The first one is done for you. 

a. yd = 8 ft 

yd = yd 

= ft 

= 3 ft 

= ft 

= 8 ft 

b. ft = _____ yd 

ft = 1 ft 

= yd 

= yd 

= 

c. ft = _____ in d. pt = ________ qt 

e. hr = ________ min f. months = ________ years 


problems. 

Date: 11/10/13 

4.E.110 





2. Four members of a track team run a relay race in 165 seconds. How many minutes did it take them to run 

the race? 

3. Horace buys lb of blueberries for a pie. He needs 48 oz of blueberries for the pie. How many more 

pounds of blueberries does he need to buy? 

4. Tiffany is sending a package that may not exceed 16 lb. The package contains books that weigh a total of 

lb. The other items to be sent weigh the weight of the books. Will Tiffany be able to send the 

package? 


problems. 

Date: 11/10/13 

4.E.111 






G R A D E 


Topic F 

Multiplication with Fractions and 

Decimals as Scaling and Word 

Problems 

5.NF.5, 5.NF.6 

Focus Standard: 5.NF.5 Interpret multiplication as scaling (resizing), by: 


5.NF.6 

Coherence -Links from: G4–M3 Multi-Digit Multiplication and Division 

G5–M2 

a. Comparing the size of a product to the size of one factor on the basis of the size of 

the other factor, without performing the indicated multiplication. 


product greater than the given number (recognizing multiplication by whole 

numbers greater than 1 as a familiar case); explaining why multiplying a given 

number by a fraction less than 1 results in a product smaller than the given number; 

and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of 

multiplying a/b by 1. 





G6–M4 


Students interpret multiplication in Grade 3 as equal groups, and in Grade 4 students begin to understand 

multiplication as comparison. Here, in Topic F, students once again extend their understanding of 

multiplication to include scaling (5.NF.5). Students compare the product to the size of one factor, given the 

size of the other factor (5.NF.5a) without calculation (e.g., 486 × 1327.45 is twice as large as 243 × 1327.45, 

because 486 = 2 × 243). This reasoning, along with the other work of this module, sets the stage for students 

to reason about the size of products when quantities are multiplied by 1, by numbers larger than 1, and 

smaller than 1. Students relate their previous work with equivalent fractions to interpreting multiplication by 

n/n as multiplication by 1 (5.NF.5b). 

Topic F: 

Date: 11/10/13 


Multiplication with Fractions and Decimals as Scaling and 

Word Problems 


Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported.License. 

4.F.1

NYS COMMON CORE MATHEMATICS CURRICULUM Topic F 5 

Students build on their new understanding of fraction equivalence as multiplication by n/n to convert 

fractions to decimals and decimals to fractions. For example, 3/25 is easily renamed in hundredths as 12/100 

using multiplication of 4/4. The word form of twelve hundredths will then be used to notate this quantity as a 

decimal. Conversions between fractional forms will be limited to fractions whose denominators are factors of 

10, 100, or 1,000. Students will apply the concepts of the topic to real world, multi‐step problems (5.NF.6). 

A Teaching Sequence Towards Mastery of Multiplication with Fractions and Decimals as Scaling and 

Word Problems 

Objective 1: Explain the size of the product, and relate fraction and decimal equivalence to multiplying a 

fraction by 1. 

(Lesson 21) 

Objective 2: Compare the size of the product to the size of the factors. 


Objective 3: Solve word problems using fraction and decimal multiplication. 

(Lesson 24) 

Topic F: 

Date: 11/10/13 


Multiplication with Fractions and Decimals as Scaling and 

Word Problems 


Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported.License. 

4.F.2


Lesson 21 

Objective: Explain the size of the product, and relate fraction and decimal 







Total Time 

(12 minutes) 

(7 minutes) 

(31 minutes) 

(10 minutes) 

(60 minutes) 


• Sprint: Multiply Decimals 5.NBT.7 


(8 minutes) 

(4 minutes) 

Sprint: Multiply Decimals (8 minutes) 

Materials: (S) Multiply Decimals Sprint 





T: (Write 2 yd = ____ ft.) How many feet are in 1 yard? 

S: 3 feet. 

T: Express yards as an improper fraction. 

S: yards. 

T: Write an expression using the improper fraction and feet. 

Then solve. 

S: (Write feet = 7 feet.) 

2 yd = __ ft 

= 2 1 yd 

= ft 

= ft 

= 7 ft 

T: yards equals how many feet? Answer in a complete sentence. 

Lesson 21: Explain the size of the product, and relate fractions and decimal 


Date: 11/10/13 

4.F.3 





S: yards equals 7 feet. 

Continue with one or more of the following possible suggestions: 2 gal = __ qt, 

and pt = __ c. 

ft = __ in, 


Carol had yard of ribbon. She wanted to use it to decorate two 

picture frames. If she uses half the ribbon on each frame, how 

many feet of ribbon will she use for one frame? Use a tape 

diagram to show your thinking. 

Note: This Application Problem draws on fraction 

multiplication concepts taught in earlier lessons in this 

module. 



Problem 1: 

of 

T: (Post Problem 1 on the board.) Write a multiplication expression for this problem. 

S: (Write .) 

T: Work with a partner to find the product of 2 halves and 3 fourths. 

S: (Work and solve.) 

T: Say the product. 

S: . 

T: (Write = .) Let’s draw an area model to verify our solution. (Draw a 

rectangle and label it 1.) What are we taking 2 halves of? 

S: . 

T: (Partition model into fourths and shade 3 of them.) How do we show 2 halves? 

S: Split each fourth unit into 2 equal parts, and shade both of them. 

T: (Partition fourths horizontally, and shade both halves, or 6 eighths.) What is the product? 

S: 6 eighths. 

T: How does the size of the product, , compare to the size of the original fraction, ? Turn and talk. 

S: They’re exactly same amount. 6 eighths and 3 fourths are equal. They’re the same. 3 fourths is 



Date: 11/10/13 

4.F.4 





just 6 eighths in simplest form. Eighths are a smaller unit than fourths but we have twice as many 

of them, so really the two fractions are equal. 

T: I hear you saying that the product, , is equal to the amount we had at first, . We multiplied. How 

is it possible that our quantity has not changed? Turn and talk. 

S: We multiplied by 2 halves, which is like a whole. So, I’m thinking we showed the whole just using a 

different name. 2 halves is equal to 1, so really we just multiplied 3 fourths by 1. Anything times 

1 is just itself. The fraction two-over-two is equivalent to 1. We just created an equivalent 

fraction by multiplying the numerator and denominator by a common factor. 

T: It sounds like you think that our beginning amount (point to ) didn’t change because we multiplied 

by one. Name some other fractions that are equal to 1. 

S: 3 thirds, 4 fourths, 10 tenths, 1 million millionths! 

T: Let’s test your hypothesis. Work with a partner to find of . One of you can multiply the fractions 

while the other draws an area model. 

S: (Share and work.) 

MP.3 T: What did you find out? 

S: It happened again. The product is 9 twelfths which is still equal to 3 fourths. We were right: 3 

thirds is equal to 1, so we got another product that is equal to 3 fourths. My area model shows it 

very clearly. Even though twelfths are a smaller unit, 9 twelfths is equal to 3 fourths. 

T: Show some other fraction multiplication expressions involving 3 fourths that would give us a product 

that is equal in size to 3 fourths. 

S: (Show .) 

T: Is equal to ? Turn and talk. 

S: Yes, if we multiplied fourths by 6 sixths, we’d get Sure, is in simplest form. I can divide 

18 and 24 by 6. 

T: Is equal to ? Work with a partner to write a multiplication sentence and share your thinking. 

S: Yes. I know 25 cents is 1 fourth of 100 cents. It is equal, because if we 

multiply 1 fourth and 25 twenty-fifths, that renames the same amount 

just using hundredths. It’s like all the others we’ve done today. 

Problem 2: Express fractions as an equivalent decimal. 

T (Write ) Show the product. 

S: . 

T: (Write = ) What are some other ways to express ? Turn and talk. 

S: We could write it in unit form, like 2 tenths. One-fifth. Tenths, that’s a decimal. We could 

write it as 0.2. 



Date: 11/10/13 

4.F.5 





T: Express as a decimal on your board. 

S: (Write 0.2.) 

T: (Write = 0.2.) We multiplied one-fifth by a fraction equal to 1. Did that change the value of onefifth? 

S: No. 

T: So, if is equal to , and is equal to 0.2. Can we 

say that = 0.2? (Write = 0.2.) Turn and talk. 

S: They are the same. We multiplied one-fifth by 1 to get 

to 

, so they must be the same. 

T: Let’s try fifths. How can we change 3 fifths to a 

decimal? 

S: We could multiply by again. Since we know onefifth 

is equal to 0.2, 3 fifths is just 3 times more than 

that, so we could triple 0.2. 

T: Work with a partner to express 3 fifths as a decimal. 


T: Say as a decimal. 

S: 0.6. 

NOTES ON 



Once students are comfortable 

renaming fractions using decimal units, 

make a connection to the powers of 10 

concepts learned back in Module 1. 

Students can be challenged to see that 

tenths can be notated as , 

hundredths as 

as . 

, and thousandths 

T: (Write on the board.) All the fractions we have worked with so far have been related to tenths. 

Let’s think about 1 fourth. We just agreed a moment ago that 1 fourth was equal to 25 hundredths. 

Write 25 hundredths as a decimal. 

S: 0.25. 

T: Fourths were renamed as hundredths in this decimal. Could we have easily renamed fourths as 

tenths? Why or why not? Turn and talk. 

S: We can’t rename fourths as tenths because 4 isn’t a factor of 10. There’s no whole number we 

can use to get from 4 to 10 using multiplication. We could name 1 fourth as tenths, but that 

would be 2 and a half tenths, which is weird. 

T: Since tenths are not possible, what unit did we use and how did we get there? 

S: We used hundredths. We multiplied by 25 twenty-fifths. 

T: Is 25 hundredths the only decimal name for 1 fourth? Is there another unit that would rename 

fourths as a decimal? Turn and talk. 

S: We could multiply 25 hundredths by 10 tenths, that would be 250 thousandths. So, we could do it in 

two steps. If we multiply 1 fourth by 250 over 250, that would get us to 250 thousandths. 

Four 50’s is a thousand. 

T: Work with a neighbor to express as a decimal, showing your work with multiplication sentences. 

One of you multiply by , and the other multiply by . Compare your work when you’re done. 



Date: 11/10/13 

4.F.6 






T: What did you find? Are the products the same? 

S: Some of us got 25 hundredths, and some of us got 250 

thousandths. They look different, but they’re equal. I got 

0.25, which looks like 25 cents, which is a quarter. Wow, that 

must be why we call a quarter! 

T: (Write 0.250 = 0.25.) What about ? How could we express that as a decimal? Tell a neighbor 

what you think, then show as a decimal. 

S: We could multiply by again. 2 fourths is a half. 1 half is 0.5 2 fourths is twice as much as 1 

fourth. We could just double 0.25. (Show = 0.5.) 

T: Think about . Are eighths a unit we can express directly as a decimal, or do we need to multiply by 

a fraction equal to 1 first? 

S: We’ll need to multiply first. 

T: What fraction equal to 1 will help us rename eighths? Discuss with your neighbor. 

S: Eight isn’t a factor of 10 or 100. I’m not sure. I don’t know if 1,000 can be divided by 8 without a 

remainder. I’ll divide. Hey, it works! 

T: Jonah, what did you find out? 

S: 1000 8 = 125. We can multiply by . 

T: Work independently, and try Jonah’s strategy. Show your work when you’re done. 

S: (Work and show 0.125.) 

T: How would you express as a decimal? Tell a 

neighbor. 

S: We could multiply by again. We could just 

double 0.125 and get 0.250. is equal to . We 

already solved that as 0.250 and 0.25. 

T: Work independently to show as a decimal. 

S: (Show = 0.250 or = 0.25.) 

T: It’s a good idea to remember some of these common 

fraction–decimal equivalencies, likes fourths and 

eighths; you will use them often in your future math 

work. 

Follow similar sequence for 1 and . 

NOTES ON 

PROPERTIES OF 

OPERATIONS: 

After completing this lesson, it may be 

interesting to some students to know 

the name of the property they’ve been 

studying: multiplicative identity 

property of 1. Consider asking 

students if they can think of any other 

identity properties. Hopefully they will 

say that zero added to any number 

keeps the same value. This is the 

additive identity property of 0. (See 

Table 3 of the Common Core Learning 

Standards.) 



Date: 11/10/13 

4.F.7 














Lesson Objective: Explain the size of the product and 

relate fractions and decimal equivalence to multiplying a 

fraction by 1. 









lesson. 



• Share your response to Problem 1(d) with a 

partner. 


Parts (a) and (b), Parts(c) and (d), Parts (e) and (f), 

Parts (i) and (k), and Parts (j) and (l)? (They have 

the same denominator.) 

• In Problem 2, what did you notice about Parts (f), 

(g), (h), and (j)? (The fractions are greater than 1, 

thus the answers will be more than one whole.) 

• In Problem 2, what did you notice about Parts (k) 

and (l)? (The fractions are mixed numbers, thus 

the answers will be more than one whole.) 

• Share and explain your thought process for 

answering Problem 3. 



Date: 11/10/13 

4.F.8 





• In Problem 4, did you have the same expressions to represent one on the number line as your 

partner’s? Can you think of more expressions? 

• How did you solve Problem 5? Share your strategy and solution with a partner. 







Date: 11/10/13 

4.F.9 







Date: 11/10/13 

4.F.10 







Date: 11/10/13 

4.F.11 





Name 

Date 

1. Fill in the blanks. The first one has been done for you. 

a. b. = c. = 

d. Use words to compare the size of the product to the size of the first factor. 

2. Express each fraction as an equivalent decimal. 

a. b. 

c. d. 

e. f. 

g. h. 

i. j. 

k. 2 l. 



Date: 11/10/13 

4.F.12 





3. Jack said that if you take a number and multiply it by a fraction, the product will always be smaller than 

what you started with. Is he correct? Why or why not? Explain your answer and give at least two 

examples to support your thinking. 

4. There is an infinite number of ways to represent 1 on the number line. In the space below, write at least 

four expressions multiplying by 1. Represent “one” differently in each expression. 

5. Maria multiplied by one to rename as hundredths. She made factor pairs equal to 10. Use her method 

to change one-eighth to an equivalent decimal. 

Maria’s way: 0.25 

Paulo renamed as a decimal, too. He knows the decimal equal to , and he knows that is half as much 

as . Can you use his ideas to show another way to find the decimal equal to ? 



Date: 11/10/13 

4.F.13 





Name 

Date 

1. Fill in the blanks to make the equation true. 

= 

2. Express the fractions as equivalent decimals: 

a. = b. = 

c. = d. = 



Date: 11/10/13 

4.F.14 





Name 

Date 

1. Fill in the blanks. 

a. 

b. = 

c. = 

d. Compare the first factor to the value of the product. 

2. Express each fraction as an equivalent decimal. 

a. b. = 

c. d. 

e. f. 



Date: 11/10/13 

4.F.15 





g. h. 

i. 3 j. 

3. is equivalent to . How can you use this to help you write as a decimal? Show your thinking to solve. 

4. A number multiplied by a fraction is not always smaller than what you start with. Explain this, and give at 

least two examples to support your thinking. 

5. Elise has dollar. She buys a stamp that costs 44 cents. Change both numbers into decimals, and tell how 

much money Elise has after paying for the stamp. 



Date: 11/10/13 

4.F.16 





Lesson 22 

Objective: Compare the size of the product to the size of the factors. 






Total Time 

(11 minutes) 

(7 minutes) 

(32 minutes) 

(10 minutes) 

(60 minutes) 



• Multiply Fractions by Whole Numbers 5.NF.4 

• Group Count by Multiples of 100 5.NBT.2 

(5 minutes) 

(4 minutes) 

(2 minutes) 




3 gal = __ qt 

T: (Write gal = ____ qt and gal = 1 gallon.) How 

many quarts are in 1 gallon? 

S: 4 quarts. 

T: Write an equivalent multiplication sentence using an improper 

fraction and quarts. 

S: (Write = 4 qts.) 

T: Solve and show. 

S: (Work and hold up board.) 

Continue with one or more of the following possible suggestions: 2 yd = __ ft, 2 ft = __ yd and 

5 pt = __ c, c = __ pt. 

3 gal = 3 1 gallon 

= 4 qts 

= 4 

= 13 qt 

Lesson 22: Compare the size of the product to the size of the factors. 

Date: 11/10/13 4.F.17 





Multiply Fractions by Whole Numbers (4 minutes) 



T: (Write 10 = ____.) Say the multiplication sentence. 

S: 10 = 5. 

T: (Write 10 = ____.) Say the multiplication sentence. 

S: 10 = 5. 

Continue the process with the following possible suggestions: 12, 12 , 15 , and 15. 

T: (Write 6 = ____.) On your boards, write the number sentence. 

S: (Write 6 = 3.) 

T: (Write 6 = ____.) On your boards, write the multiplication sentence. Below it, rewrite the 

multiplication sentence as a whole number times 6. 

S: (Write 6 = ____. Below it, write 1 6 = 6.) 

T: (Write 6 = ____.) On your boards, write the number sentence. 

S: (Write × 6 = 9.) 

Continue with the following possible suggestions: 8 × , 8 × , and 8 × . 

Group Count by Multiples of 100 (2 minutes) 


T: Count by tens to 100. (Extend finger each time a multiple is counted.) 

S: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. 

T: (Show 10 extended fingers.) How many tens are in 100? 

S: 10. 

T: (Write 10 × 10 = 100.) Count by twenties to 100. (Extend finger each time a multiple is counted.) 

S: 20, 40, 60, 80, 100. 

T: (Show 5 extended fingers.) How many twenties are in 100? 

S: 5. 

T: (Write 20 × 5 = 100. Below it, write 5 × __ = 100.) How many fives are in 100? 

S: 20. 

Repeat the process with 4 and 25, 2 and 50. 


Date: 11/10/13 4.F.18 






In order to test her math skills, Isabella’s father told her he would give her of a 

dollar if she could tell him how much money that is and what that amount is in 

decimal form. What should Isabella tell her father? Show your calculations. 

Note: This Application Problem reviews G5–M4–Lesson 21’s Concept Development. 

Among other strategies, students might convert the eighths to fourths, and then 

multiply by 

multiply by 6. 

, or they may remember the decimal equivalent of 1 eighth and 


Materials: (T) 12-inch string (S) Personal white boards 

Problem 1 

a. 12 inches b. 12 inches c. 12 inches 

T: (Post Problem 1(a─c) on the board.) Find the products 

of these expressions. 

S: (Work.) 

T: Let’s compare the size of the products you found to the 

size of this factor. (Point to 12 inches.) Did multiplying 

12 inches by 4 fourths change the length of this string? 

(Hold up the string.) Why or why not? Turn and talk. 

S: The product is equal to 12 inches. We multiplied 

and got 48 twelfths, but that’s just another name for 

12 using a different unit. It’s 4 fourths of the string, 

all of it. Multiplying by 1 means just 1 copy of the 

number, so it stays the same. The other factor just 

named 1 as a fraction, but it is still just multiplying by 

1, so the size of 12 won’t change. 

NOTES ON 



Whenever students are calculating 

problems involving measurements, 

they will benefit if they have 

established mental benchmarks of 

each increment. For example, students 

should be able to think about 12 inches 

not just as a foot, but also as a specific 

length, perhaps as length just a little 

longer than a sheet of paper. Although 

teachers can give benchmarks for 

specific increments, it is probably 

better if students discover benchmarks 

on their own. Establishing mental 

benchmarks may be essential for 

English language learners’ 

understanding. 

T: (Write 12 inches = 12 inches under first expression.) Did multiplying by 3 fourths change the size 

of our other factor, 12 inches? If so, how? Turn and talk. 

S: The string got shorter because we only took 3 of 4 parts of it. We got almost all of 12, but not 

quite. We wanted 3 fourths of it rather than 4 fourths, so the factor got smaller after we multiplied. 

12 got smaller. We got 9 this time. 

T: (Write 12 12 under the second expression.) I hear you saying that 12 inches was shortened, 

resized to 9 inches. How can it be that multiplying made 12 smaller when I thought multiplication 


Date: 11/10/13 4.F.19 





always made numbers get bigger? Turn and talk. 

S: We took only part of 12. When you take just a part of something it is smaller than what you start 

with. We ended up with 3 of the 4 parts, not the whole thing. Adding twelve times is going 

to be smaller than adding one the same number of times. 

T: So, 9 is 3 fourths as much as 12. True or false? 

S: True. 

T: Let’s consider our last expression. How did multiplying by 5 fourths change or not change the size of 

the other factor, 12? How would it change the length of the string? Turn and talk. 

S: The answer to this one was bigger than 12 because it’s more than 4 fourths of it. 12 1 

MP.2 

The product was greater than 12. We copied a number bigger than 1 twelve times. The 

answer had to be greater than copying 1 the same number of times. 5 fourths of the string would 

be 1 fourth longer than the string is now. 

T: (Write 12 12 under the third expression.) So, 15 is 5 fourths as much as 12. True or false? 

S: True. 

T: 15 is 1 and times as much as 12. True or false? 

S: True. 

T: We’ve compared our products to one factor, 12 inches, in each of these expressions. We explained 

the changes we saw by thinking about the other factor. We can call that other factor a scaling 

factor. A scaling factor can change the size of the other factor. Let’s look at the relationships in 

these expressions one more time. (Point to the first expression.) When we multiplied 12 inches by a 

scaling factor equal to 1, what happened to the 12 inches? 

S: 12 didn’t change. The product was the same size as 12 inches, even after we multiplied it. 

T: (Point.) In the second expression, was the scaling factor. Was this scaling factor more than or less 

than 1? How do you know? 

S: Less than 1, because 4 fourths is 1. 

T: What happened to 12 inches? 

S: It got shorter. 

T: And in our last expression, what was the scaling factor? 

S: 5 fourths. 

T: More or less than 1? 

S: More than 1. 

T: What happened to 12 inches? 

S: It got longer. The product was larger than 12 inches. 


Date: 11/10/13 4.F.20 





Problem 2 

a. b. c. 

T: (Post Problem 2 (a–c) on the board.) Keeping in mind the relationships that we’ve just seen between 

our products and factors, evaluate these expressions. 

S: (Work.) 

T: Let’s compare the products that you found to this factor. (Point to .) What is the product of and 

? 

S: . 

T: Did the size of change when we multiplied it by a scaling factor equal to 1? 

S: No. 

T: (Write under the first expression.) Since we are comparing our product to 1 third, what is 

S: . 

the scaling factor in the second expression? 

T: Is this scaling factor more than or less than 1? 

S: Less than 1. 

T: What happened to the size of when we multiplied it by a scaling factor less 

than 1? Why? 

S: The product was 3 twelfths. That is less than 1 third which is 4 twelfths. 

We only wanted part of 1 third this time, so the answer had to be smaller than 

1 third. When you multiply by less than 1, the product is smaller than what 

you started with. 

T: (Write on the board.) In the last expression, was the scaling factor. 

Is the scaling factor more than or less than 1? 


T: Say the product of . 

S: . 

T: Is 5 twelfths more than, less than or equal to 

S: More than 

T: (Write under the third expression.) Explain why product of and is more than . 

S: (Share.) 


Date: 11/10/13 4.F.21 





Problem 3 

a. b. c. d. e. f. g. 

T: I’m going to show you some multiplication expressions where we start with . The expressions will 

have different scaling factors. Think about what will happen to the size of 1 half when it is multiplied 

by the scaling factor. Tell whether the product will be equal to , more than or less than . 

Ready? (Show .) 

S: Equal to . 

T: Tell a neighbor why. 

S: The scaling factor is equal to 1. 

T: (Show .) 

S: Less than . 


S: The scaling factor is less than 1. 

T: (Show .) 

S: More than . 


S: The scaling factor is more than 1. 

Repeat questioning with , , , and . 

Problem 4 

At the book fair, Vlad spends all of his money on new books. Pamela spends as much as Vlad. Eli spends 

as much as Vlad. Who spent the most? The least? 

T: (Post Problem 4 on the board, and read it aloud with students.) Read the first sentence again out 

loud. 

S: (Read.) 

T: Before we begin drawing, to whose money will we 

make the comparisons? 

S: Vlad’s money. 

T: What can we draw from the first sentence? 

S: We can make a tape diagram. We should label a 

tape diagram Vlad’s money. 

T: Vlad spent all of his money at the book fair. I’ll draw a tape diagram and label it Vlad’s money (write 

Vlad’s $). Read the next sentence aloud. 


Date: 11/10/13 4.F.22 





S: (Read.) 

T: What can we draw from this sentence? 

S: We can draw another tape that is shorter than Vlad’s. 

T: Let me record that. (Draw a shorter tape representing Pamela’s money.) How will we know how 

much shorter to draw it? Turn and talk. 

S: We know she spent of the same amount. Since Pamela’s units are thirds, we can split Vlad’s tape 

into 3 equal units, and then draw a tape below it that is 2 units long and label it Pamela’s money. 

I know Pamela’s has 2 units, and those 2 units are 2 out of the that Vlad spent. I’ll draw 2 units for 

Pam, and then make Vlad’s 1 unit longer than hers. 

T: I’ll record that. Thinking of as a scaling factor, did Pamela spend more or less than Vlad? How do 

you know? Does our model bear that out? 

S: Less than Vlad. If you think of as a scaling factor, it’s less than 1, so she spent less than Vlad. That’s 

how we drew it. She spent less than Vlad. She only spent a part of the same amount as Vlad. 

Vlad spent all his money or, of his money. Pamela only spent as much as Vlad. You can see that 

in the diagram. 

T: Read the third sentence and discuss what you can draw from this information. 

S: (Read and discuss.) 

T: Eli spent as much as Vlad. If we think of as a scaling factor, what does that tell us about how 

much money Eli spent? 

S: Eli spent more than Vlad, because is more than 1. Again, Vlad spent all of his money, or of it. 

is more than , so Eli spent more than Vlad. We have to draw a tape that is one-third more than 

Vlad’s. 

T: Since the scaling factor is more than 1, I’ll draw a third tape for Eli that is longer than Vlad’s money. 

What is the question we have to answer? 

S: Who spent the most and least money at the book fair? 

T: Does our tape diagram show enough information to answer this question? 

S: Yes, it’s very easy to see whose tape is longest and shortest in our diagram. Even though we 

don’t know exactly how much Vlad spent, we can still answer the question. Since the scaling factors 

are more than 1 and less than 1, we know who spent the most and least. 

T: Answer the question in a complete sentence. 

S: Eli spent the most money. Pamela spent the least money. 







Date: 11/10/13 4.F.23 






Lesson Objective: Compare the size of the product to the 

size of the factors. 









lesson. 



• In Problem 1, what relationship did you notice 

between Parts (a) and (b)? 

• For Problem 2, compare your tape diagrams with 

a partner. Are your drawings similar to or 

different from your partner’s? 

• Explain to a partner your thought process for solving 

Problem 3. How did you know what to put for the 

missing numerator or denominator? 

• In Problem 4, did you notice a relationship between 

Parts (a) and (b)? How did you solve them? 

• For Problem 5, did you and your partner use the same 

examples to support the solution? Can you also give 

some examples to support the idea that multiplication 

can make numbers bigger? 

• What’s the scaling factor in Problem 6? What is an 

expression to solve this problem? 

• How did you solve Problem 7? Share your solution and 

explain your strategy to a partner. 

NOTES ON 



Some students may find it helpful to 

have a physical representation of the 

tape diagrams as they work to draw 

these models. Students can use square 

tiles or uni-fix cubes. For some 

students, arranging the manipulatives 

first, and then drawing may be easier, 

and it may eliminate the need to 

redraw or erase. 


Date: 11/10/13 4.F.24 











the students. 


Date: 11/10/13 4.F.25 





Name 

Date 

1. Solve for the unknown. Rewrite each phrase as a multiplication sentence. Circle the scaling factor and 

put a box around the number of meters. 

a. as long as 8 meters = ______ meters b. 8 times as long as meter = _______ meters 

2. Draw a tape diagram to model each situation in Problem 1, and describe what happened to the number 

of meters when it was multiplied by the scaling factor. 

a. b. 

3. Fill in the blank with a numerator or denominator to make the number sentence true. 

a. 7 7 b. 15 15 c. 3 3 

4. Look at the inequalities in each box. Choose a single fraction to write in all three blanks that would make 

all three number sentences true. Explain how you know. 

a. 

_____ _____ _____ 

b. 

_____ _____ _____ 


Date: 11/10/13 4.F.26 





5. Johnny says multiplication always makes numbers bigger. Explain to Johnny why this isn’t true. 

Give more than one example to help him understand. 

6. A company uses a sketch to plan an advertisement on the side of a building. The lettering on the sketch is 

in tall. In the actual advertisement, the letters must be 34 times as tall. How tall will the letters be on 

the building? 

7. Jason is drawing the floor plan of his bedroom. He is drawing everything with dimensions that are of 

the actual size. His bed measures 6 ft by 3 ft, and the room measures 14 ft by 16 ft. What are the 

dimensions of his bed and room in his drawing? 


Date: 11/10/13 4.F.27 





Name 

Date 

1. Fill in the blank to make the number sentences true. Explain how you know. 

a. 11 ˃ 11 

b. ˂ 

c. 6 = 6 


Date: 11/10/13 4.F.28 





Name 

Date 

1. Solve for the unknown. Rewrite each phrase as a multiplication sentence. Circle the scaling factor and 

put a box around the number of meters. 

a. as long as 6 meters = ______ meters b. 6 times as long as meter = ______ meters 

2. Draw a tape diagram to model each situation in Problem 1, and describe what happened to the number 

of meters when it was multiplied by the scaling factor. 

a. b. 

3. Fill in the blank with a numerator or denominator to make the number sentence true. 

a. 5 ˃ 9 b. 12 13 c. 4 4 

4. Look at the inequalities in each box. Choose a single fraction to write in all three blanks that would make 

all three number sentences true. Explain how you know. 

a. 

_____ _____ _____ 

b. 

_____ _____ _____ 


Date: 11/10/13 4.F.29 





5. Write a number in the blank that will make the number sentence true. 

3 × _____ ˂ 1 

a. Explain how multiplying by a whole number can result in a product less than 1. 

6. In a sketch, a fountain is drawn yard tall. The actual fountain will be 68 times as tall. How tall will the 

fountain be? 

7. In blueprints, an architect’s firm drew everything of the actual size. The windows will actually 

measure 4 ft by 6 ft and doors measure 12 ft by 8 ft. What are the dimensions of the windows and the 

doors in the drawing? 


Date: 11/10/13 4.F.30 





Lesson 23 

Objective: Compare the size of the product to the size of the factors. 






Total Time 

(12 minutes) 

(7 minutes) 

(31 minutes) 

(10 minutes) 

(60 minutes) 


• Compare the Size of a Product to the Size of One Factor 5.NF.5 

• Compare Decimal Numbers 5.NBT.2 

• Write Fractions as Decimals 5.NBT.2 

(5 minutes) 

(2 minutes) 

(5 minutes) 

Compare the Size of a Product to the Size of One Factor (5 minutes) 



T: (Write 1 = .) On your boards, fill in the missing 

numerator. 

S: (Write 1 = .) 

T: (Write 9 __ = 9.) Say the missing whole number 

factor. 

S: 1. 

T: (Write 9 = 9.) Fill in the missing numerator to make 

a true number sentence. 

S: (Write 9 = 9.) 

T: (Write 9 < 9.) Fill in the missing numerator to make 

a true number sentence. 

NOTES ON 


ACTION AND 

EXPRESSION: 

It is very helpful for most learners to 

know and understand the objective of 

specific lessons. This knowledge helps 

them make connections to what they 

already know and what they need to 

learn. Each lesson in the modules has 

a stated objective. These objectives 

should be posted daily. Today’s goal, 

comparing the size of the factors to the 

size of the product, includes math 

vocabulary that students should know. 

For a quick visual review, write an 

equation below the objective and draw 

lines showing which number is the 

factor and which is the product. 


Date: 11/10/13 4.F.31 





S: (Write 9 < 9.) 

T: (Write 9 > 9.) Fill in a missing numerator to make a true number sentence. 

S: (Write the number sentence filling in a numerator greater than 2.) 

Continue this process with the following possible sequence: 7 = 7, 6 < 6, 6 > 6, 8 < 8, 

9 = 9, and 10 < 10. 

Compare Decimal Numbers (2 minutes) 


Note: This fluency prepares students for today’s lesson. 

T: (Write 1 __ 9.) Say the greater number. 

S: 9. 

T: On your boards, write the symbol to make the number sentence true. 

S: (Write 1 < 9.) 

Continue this process with the following possible sequence: 1 __ 0.9, 0.95 __ 1, and 0.994 __ 1. 




T: (Write = .) How many fifties are in 100? 

S: 2. 

T: (Write = .) is the same as how many 1 hundredths? 

S: 2 one hundredths. 

T: (Write = . Below it, write = __.__.) On your boards, write as a decimal. 

S: (Write = 0.02.) 

Continue this process with the following possible sequence , , , , , , , , , , , , , , 

and . 


Jasmine took as much time to take a math test as Paula. If 

Paula took 2 hours to take the test, how long did it take Jasmine 


Date: 11/10/13 4.F.32 





to take the test? Express your answer in minutes. 

Note: Scaling as well as conversion is required for today’s Application Problem. This both reviews G5–M4– 

Topic E and prepares students to continue a study of scaling with decimals in today’s lesson. 



Problem 1: 2 meters 2 meters 2 meters 

T: (Post Problem 1 on the board.) Let’s compare products to the 2 meters in each expression. Let’s 

notice what happens to 2 meters when we multiply, or scale, 2 meters by the other factors. Read 

the scaling factors out loud in the order they are written. 

S: 97 hundredths, 101 hundredths, 100 hundredths. 

T: Without evaluating them, turn and talk with a neighbor about which expression is greater than, less 

than, and equal to 2 meters. Be sure to explain your thinking. 

S: 2 would be equal to 2 meters, because it’s being scaled by 1. 2 meters would be less 

than 2 meters, because it’s being scaled by a fraction less than 1. 2 meters 

than 2 meters, because it’s being scaled by a fraction more than 1. 

T: Rewrite the expressions using decimals to express the scaling factors. 

S: (Work and show 2 meters 0.97, 2 meters 1.01, and 2 meters 1.0.) 

would be more 

T: (Write decimal expressions below the fractional ones.) Which expression is greater than, less than, 

and equal to 2 meters. Turn and 

talk. 

S: It’s the same as before: 2 times 1 is 

equal to 1, 2 times 0.97 is less than 

2, and 2 times 1.01 is more than 2. 

Nothing has changed; we’ve just 

expressed the scaling factor as a decimal. We haven’t changed the value. 

T: (Write 2 _____ 2 on board.) Write three decimal scaling factors that would make this number 

sentence true. 

S: (Work and show numbers less than 1.0.) 

T: Finish my sentence. To get a product that is less than the number you started with, multiply by a 

scaling factor that is…. 


T: (Write 2 _____ 2 on board.) Show me some more decimal scaling factors that would make this 

number sentence true. 

S: (Work and show numbers more than 1.0.) 

T: Finish this sentence. To get a product that is more than the number you started with, multiply by a 


Date: 11/10/13 4.F.33 





scaling factor that is…. 


Problem 2: 19.4 0.96 19.4 0.02 

T: (Post Problem 2 on the board.) Let’s compare our product to the first factor,19.4. Let’s consider the 

other factors the scaling factors. Read the scaling factors out loud in the order they are written. 

S: 96 hundredths, 2 hundredths. 

T: Look at the first expression. Will the product be more than, less than, or equal to 19.4? Tell a 

neighbor why. 

S: Less than 19.4, because the scaling factor is less than 1. 

T: (Write 19.4 next to the first expression.) Look at the second expression. Will the product be more 

than, less than, or equal to 19.4? Tell a neighbor why. 

S: It’s also less than . , because that scaling factor is also less than . 

T: (Write 19.4 next to the second expression.) So, we know that both scaling factors will lead to a 

product that is less than the 

number we started with. Which 

expression will give a greater 

product? Why? Turn and talk. 

S: 19.4 times 96 hundredths. Even though both scaling factors are less than 1, 96 hundredths is a 

much bigger scaling factor than 2 hundredths. 96 hundredths is close to 1. 2 hundredths is 

almost zero. The first expression will be really close to 19.4 and the second expression will be closer 

to zero. 

T: (Point to first expression.) What is the scaling factor here? 


T: What would the scaling factor need to be in order for the product to be equal to 19.4? 

S: 1. 

T: Isn’t the same as hundredths? 

S: Yes. 

T: So this scaling factor, 96 hundredths is slightly less than 

1. True or false? 

S: True. 

T: If this is true, what can we say about the product of 

19.4 and 0.96? Turn and talk. 

S: If we draw a tape diagram of 19.4 it would be 19.4 

units long. Since 96 hundredths is just slightly less than 

1, that means that 19.4 0.96 is slightly less than 

19.4 1. The tape diagram should be slightly shorter 

than the first one we drew. The expression 19.4 

times 96 hundredths is just a little bit less than 19.4. 

T: Imagine partitioning this tape into 100 equal parts. The tape for 19.4 times 96 hundredths should be 

as long as 96 of those hundredths, or just 4 hundredths less than this whole tape. (Draw a second 


Date: 11/10/13 4.F.34 





tape diagram slightly shorter and label it 19.4 0.96.) 

T: Make a statement about this expression. Is 19.4 times 96 hundredths slightly less than 19.4, or a lot 

less than 19.4? 

S: Slightly less than 19.4. 

T: (Write 19.4 0.96 is slightly less than 19.4.) Let’s look at the other expression now. Is the scaling 

factor, 2 hundredths, slightly less than 1 or a lot less than 1? Turn and talk. 

S: 1 is 100 hundredths this is only 2 hundredths. It’s a lot less than . It’s a lot less than . In fact, 

it’s only slightly more than zero. 

T: This scaling factor is a lot less than 1. Work with a partner to draw two tape diagrams. One should 

show 19.4, like we did before, and the other should show 19.4 times 2 hundredths. 


T: Make a statement about this expression. Is 19.4 times 2 hundredths slightly less than 19.4, or a lot 

less than 19.4? 

S: It is a lot less than 19.4. 

T: (Write 19.4 0.02 is a lot less than 19.4.) 

Problem 3: 1.02 1.73 29.01 1.73 

T: (Post Problem 3 on the board.) Let’s compare our products with the second factor in these 

expressions. (Point to 1.73 in both expressions.) We’ll consider the first factors to be scaling factors. 

Read the scaling factors out loud in the order they are written. 

S: 1 and 2 hundredths, 29 and 1 hundredth. 

T: Think about these expressions. Will the products be more than, less than, or equal to the 1.73? Tell 

your neighbor why. 

S: They’ll both be more than . , because both scaling factors are more than . 

T: Let’s be more specific. Look at the first expression. Will the product be slightly more than 1.73, or a 

lot more than 1.73? Tell a neighbor. 

S: The product will just be slightly more than 1.73. The scaling factor is just 2 hundredths more than 1. 

I can visualize two tape diagrams, and the one showing 1.73 times 1.02 is just a little bit longer, 

like 2 hundredths longer than the tape showing 1.73. The product will be slightly more than what 

we started with because the scaling factor is just slightly more than 1. 

T: (Write 1.02 1.73 is slightly more than 1.73.) Think about the second expression. Will its product be 

slightly more than 1.73, or a lot more than 1.73? Tell a neighbor. 

S: The product will just be a lot more than 1.73. The scaling factor is almost 30 times more than 1, so 

the product will be almost 30 times more too. I can visualize two tape diagrams, and the one 

showing 1.73 times 29.01 is a lot longer, like 29 times longer than the tape showing just 1.73. 

The product will be a lot more than what we started with because the scaling factor is a lot more 

than 1. 


Date: 11/10/13 4.F.35 














Lesson Objective: Compare the size of the product to the 

size of the factors. 









lesson. 



• Share your solutions and explain your thought 

process for solving Problem 1 to a partner. How 

did you decide which number goes into which 

expression? 

• Compare your solutions for Problem 2 with a 

partner. Did you have different answers? If so, 

explain your thinking behind each sorting. 

• What was your strategy for solving Problem 3? 

Share it with a partner. 

• How did you solve Problem 4? Did you make a 

drawing or tape diagram to compare the sprouts? 

Share it with and explain it to a partner. 

• Share your decimal examples for Problem 5 with 

a partner. Did you have the same or different 

examples? 


Date: 11/10/13 4.F.36 










Date: 11/10/13 4.F.37 





Name 

Date 

1. Fill in the blank using one of the following scaling factors to make each number sentence true. 

1.021 0.989 1.00 

a. 3.4 _______ = 3.4 b. _______ 0.21 0.21 c. 8.04 _______ 8.04 

2. 

a. Sort the following expressions by rewriting them in the table. 

The product is less than the 

boxed number: 

The product is greater than the 

boxed number: 

13.89 1.004 602 0.489 102.03 4.015 

0.3 0.069 0.72 1.24 0.2 0.1 

b. Explain your sorting by writing a sentence that tells what the expressions in each column of the table 

have in common. 


Date: 11/10/13 4.F.38 





3. Write a statement using one of the following phrases to compare the value of the expressions. 

Then explain how you know. 

is slightly more than is a lot more than is slightly less than is a lot less than 

a. 4 0.988 _________________________________ 4 

b. 1.05 0.8 _________________________________ 0.8 

c. 1,725 0.013 _______________________________ 1,725 

d. 989.001 1.003 _____________________________ 1.003 

e. 0.002 0.911 _______________________________ 0.002 

4. During science class, Teo, Carson, and Dhakir measure the length of their bean sprouts. Carson’s sprout is 

. times the length of Teo’s, and Dhakir’s is . 8 times the length of Teo’s. Whose bean sprout is the 

longest? The shortest? Explain your reasoning. 

5. Complete the following statements, then use decimals to give an example of each. 

• a b > a will always be true when b is… 

• b < a will always be true when b is… 


Date: 11/10/13 4.F.39 





Name 

Date 

1. Fill in the blank using one of the following scaling factors to make each number sentence true. 

1.009 1.00 0.898 

a. 3.06 _______ 3.06 b. 5.2 _______ = 5.2 c. _______ 0.89 0.89 

2. Will the product of 22.65 0.999 be greater than or less than 22.65? Without calculating, explain how 

you know. 


Date: 11/10/13 4.F.40 





Name 

Date 

1. 

a. Sort the following expressions by rewriting them in the table. 

The product is less than the 

boxed number: 

The product is greater than the 

boxed number: 

12.5 1.989 828 0.921 321.46 1.26 

0.007 1.02 2.16 1.11 0.05 0.1 

b. What do the expressions in each column have in common? 

2. Write a statement using one of the following phrases to compare the value of the expressions. 

Then explain how you know. 

is slightly more than is a lot more than is slightly less than is a lot less than 

a. 14 0.999 _______________________________ 14 

b. 1.01 2.06 _______________________________ 2.06 

c. 1,955 0.019 ______________________________ 1,955 


Date: 11/10/13 4.F.41 





d. Two thousand 1.0001 _______________________________ two thousand 

e. Two-thousandths 0.911 ______________________________ two-thousandths 

3. Rachel is 1.5 times as heavy as her cousin, Kayla. Another cousin, Jonathan, weighs 1.25 times as much as 

Kayla. List the cousins, from lightest to heaviest, and explain your thinking. 

4. Circle your choice. 

a. a b > a 

For this statement to be true, b must be greater than 1 less than 1 

Write two expressions that support your answer. Be sure to include one decimal example. 

b. a b < a 

For this statement to be true, b must be greater than 1 less than 1 

Write two expressions that support your answer. Be sure to include one decimal example. 


Date: 11/10/13 4.F.42 





Lesson 24 

Objective: Solve word problems using fraction and decimal multiplication. 





Total Time 

(12 minutes) 

(38 minutes) 

(10 minutes) 

(60 minutes) 


• Compare the Size of a Product to the Size of One Factor 5.NF.5 


• Write the Scaling Factor 5.NBT.3 

(4 minutes) 

(5 minutes) 

(3 minutes) 

Compare the Size of a Product to the Size of One Factor (4 minutes) 



T: How many halves are in 1? 

S: 2. 

T: How many thirds are in 1? 

S: 3. 

T: How many fourths are in 1? 

S: 4. 

T: (Write 1 = .) On your boards, fill in the missing numerator. 

S: (Write 1 = .) 

T: (Write 6 __ = 6.) Say the missing factor. 

S: 1. 

T: (Write 6 = 6.) On your boards, write the equation, filling in the missing numerator. 

S: (Write 6 = 6.) 

T: (Write 6 < 6.) On your boards, fill in a numerator to make a true number sentence. 

Lesson 24: Solve word problems using fraction and decimal multiplication. 

Date: 11/10/13 4.F.43 





S: (Write 6 < 6 or 6 < 6.) 

T: (Write 9 > 9.) On your boards, fill in a numerator to make a true number sentence. 

S: (Write a number sentence, filling in a numerator greater than 6.) 

Continue this process with the following possible sequence: 5 = 5, 5 < 5, 5 > 5, 9 < 9, 8 = 8, 

and 7 < 7. 




T: (Write = .) How many fifties are in 100? 

S: 2. 

T: (Write = .) is the same as how many 1 hundredths? 

S: 2 one hundredths. 

T: (Write = . Below it, write = __.__.) On your boards, write as a decimal. 

S: (Write = 0.02.) 

Continue this process with the following possible sequence: , , , , , , , , , , , , , , 

and . 

Write the Scaling Factor (3 minutes) 



T: (Write 3 __ = 3.) Say the unknown whole number factor. 

S: 1. 

T: (Write 3.5 __ = 3.5.) Say the unknown whole number factor. 

S: 1. 

T: (Write 4.2 1 =____.) Say the product. 

S: 4.2. 

T: (Write __ 0.58 > 0.58.) Is the unknown factor going to be greater or less than 1? 

S: Greater than 1. 

T: Fill in a factor to make a true number sentence. 


Date: 11/10/13 4.F.44 





S: (Write a number sentence filling in a decimal number greater than 1.) 

T: (Write 7.03 __ < 7.03.) Is the unknown factor greater or less than 1? 


T: Fill in a factor to make a true number sentence. 

Continue this process with the following possible sequence: 6.07 __ < 6.07, __ 6.2 = 6.2, and 0.97 __ > 

0.97. 



Note: The time normally allotted for the Application Problem has been included in the Concept Development 

portion of today’s lesson. 



Have two pairs of student who can successfully model the problem work at the board while the others work 

independently or in pairs at their seats. Review the following questions before beginning the first problem: 




As students work, circulate. Reiterate the questions above. After two minutes, have the two pairs of students 

share only their labeled diagrams. For about one minute, have the demonstrating students receive and 

respond to feedback and questions from their peers. 


Give everyone two minutes to finish work on that question, sharing their work and thinking with a peer. All 

should write their equations and statements of the answer. 



Problem 1 

A vial contains 20 mL of medicine. If each dose is of the vial, how many mL is each dose? Express your 

answer as decimal. 


Date: 11/10/13 4.F.45 





In this fraction of a set problem, students are asked to find one-eighth of 20 mL. Since the final answer needs 

to be expressed as a decimal, students again have some choices in how they solve. As illustrated, some 

students may choose to multiply by 20 to find the fractional mL in each dose. This method requires the 

students to then simply express 

as a decimal. 

Other students may choose to first express as a decimal (0.125), and then multiply that by 20 to find 2.5 mL 

of medicine per dose. This method is perhaps more direct, but it does require that students recall that 8 is a 

factor of 1,000 to express as a decimal. 

Problem 2 

A container holds 0.7 liters of oil and vinegar. 

the container? Express your answer as both a fraction and a decimal. 

of the mixture is vinegar. How many liters of vinegar are in 

In this fraction of a set problem, students are asked to find three-fourths of a set that is expressed using a 

decimal. Since the final answer needs to be expressed as both a fraction and a decimal, students again have 

choice in approach. As illustrated, some students may choose to express 0.7 as a fraction, and then multiply 

by three-fourths to find the fractional liters of vinegar in the container. This method requires the slightly 

complex step of converting a fraction with denominator of 40 to a decimal. This process is not extremely 

challenging, but perhaps unfamiliar to students. 

Other students may choose to first express as a decimal (0.75), and then multiply that by 0.7 to find 0.525 

liters of vinegar are in the container. The decimal 0.525 is easily written 

are not required to simplify this fraction. 

Problem 3 

as a fraction. Students may but 

Andres completed a km race in . minutes. His sister’s time was times longer than his time. How long 

did it take his sister to run the race? 


Date: 11/10/13 4.F.46 





In this problem, Andres’ race time (13.5 minutes) is being 

multiplied by a scaling factor of 

. Students must interpret 

both a decimal and fractional factor, thus giving rise to 

expression of both factors as either decimals or fractions 

( or ). Alternately, students may have chosen 

to draw a tape diagram showing Andres’ sister’s time as and a 

half times more than his. In this manner, students need to 

multiply to find the value of the half-unit that represents the 

additional time that his sister spent running, and then add that 

sum to 13.5 minutes. Student choice of approach provides an 

opportunity to discuss the efficiency of both approaches during 

the Student Debrief. In any case, students should find that it 

took Andres’ sister . (or ) minutes to complete the race. 

NOTES ON 



Problems 3, 4, and 5 require students 

to compare quantities. For example, in 

Problem 3, students are comparing 

Andres’ race time to his sister’s time. 

Typically, when using tape diagrams to 

solve comparison word problems, at 

least two bars are used. 

There is a strong connection between 

tape diagrams used with comparison 

story problems and bar graphs. The 

bars in bar graphs allow readers to 

compare quantities, exactly like the 

bars that are used in comparison word 

problems. Although tape diagrams are 

typically drawn horizontally, they can 

be drawn vertically. Similarly, bar 

graphs can (and should) be drawn 

horizontally and vertically. In Problems 

3, 4, and 5, it is easy to visualize 

additional data that would result in 

additional bars. 


Date: 11/10/13 4.F.47 





Problem 4 

A clothing factory uses 1,275.2 meters of cloth a week to make shirts. How much cloth would they need to 

make times as many shirts? 

In this scaling problem, a length of cloth (1,275.2 m) is being multiplied by a scaling factor of 

. Before 

students solve, ask them to identify the scaling factor and what comparison is being made (that of the initial 

amount of fabric and the resulting amount). Though students do have the option of expressing both factors 

as fractions, the method of converting 

to a decimal is far simpler. The efficiency of this approach can be a 

focus during the Student Debrief. Some students may also have chosen to draw a tape diagram showing 

1,275.2 meters of cloth being scaled to times its original length. In this manner, students could have 

tripled 1,275.2 first, then found three-fifths of it before combining those two totals. In either case, students 

should find that the factory would need 4,590.72 meters of cloth. 

Problem 5 

There are as many boys as girls in a class of fifth-graders. If there are 35 students in the class, how many are 

girls? 

What may seem like a simple problem is 

actually rather challenging, as students 

are required to work backwards as they 

solve. The word problem states that 

there are as many boys as girls in the 

class, yet the number of girls is unknown. 

Students should first reason that since 

the number of boys is a scaled multiple 

of the number of girls, a tape should first 

be drawn to represent the girls. From 

that tape, students can draw a smaller tape (one that is three-fourths the size of the tape representing the 

girls) to represent the boys in the class. In this way, students can see that 3 units are boys and 4 units are 

girls. Since there are 35 students in the class and 7 total units, each unit represents 5 students. Four of those 


Date: 11/10/13 4.F.48 





units are girls, so there are 20 girls in the class. 

Problem 6 

Ciro purchased a concert ticket for $56. The cost of the ticket was the cost of his dinner. The cost of his 

hotel was times as much as his ticket. How much did Ciro spend altogether for the concert ticket, hotel, 

and dinner? 

In this problem, students must read and work carefully to identify that the cost of the concert ticket plays 

two roles. In relation to the cost of the dinner, the ticket cost can be considered the scaling factor as it 

represents the cost of dinner. However, in relation to the cost of the hotel, the ticket cost should be 

considered the factor being scaled (as the hotel cost is 

times greater.) This understanding is crucial for 

drawing an accurate model and should be discussed thoroughly as students draw and again in the Student 

Debrief. 

Once the modeling is complete, the steps toward 

solution are relatively simple. Since the ticket cost 

represents the cost of dinner, division shows that each 

unit (or fifth) is equal to $14. Therefore, 5 units (5 

fifths), or the cost of dinner, is equal to $70. The model 

representing the cost of the hotel very clearly shows 2 

units of $56 and a half unit of $56, which in total, equals 

$140. Students must use addition to find the total cost 

of Ciro’s spending. 


Date: 11/10/13 4.F.49 






Lesson Objective: Solve word problems using fraction and 

decimal multiplication. 









lesson. 



• For all the problems in this Problem Set, there are 

a few ways to solve for the solution. Compare 

and share your strategy with a partner. 

• How did you solve Problem 5? Explain your 

strategy to a partner. Can you find how many boys 

there are in the classroom? How many more girls 

than boys are in the classroom? 

• Did you make any drawings or tape diagrams for 

Problem 6? Share and compare with a partner. 

Does drawing a tape diagram help you solve this 

problem? Explain. 


After the Student Debrief, instruct students to complete the 

Exit Ticket. A review of their work will help you assess the 

students’ understanding of the concepts that were 



students. 

NOTES ON 


ENGAGEMENT: 

A daily goal for teachers is to get 

students to talk about their thinking. 

One possible strategy to achieve this 

goal is to partner each student with a 

peer who has a different perspective. 

Ask students to separate themselves 

into groups that solved a specific 

problem in a similar way. Once these 

groups are formed, ask each student to 

partner with a peer in another group. 

Let these partners describe and discuss 

their strategies and solutions with each 

other. It is harder for students to 

explain different approaches than like 

approaches. 


Date: 11/10/13 4.F.50 





Name 

Date 

1. A vial contains 20 mL of medicine. If each dose is of the vial, how many mL is each dose? Express your 

answer as decimal. 

2. A container holds 0.7 liters of oil and vinegar. of the mixture is vinegar. How many liters of vinegar are 

in the container? Express your answer as both a fraction and a decimal. 

3. Andres completed a 5 km race in 13.5 minutes. His sister’s time was times longer than his time. How 

long did it take his sister to run the race? 


Date: 11/10/13 4.F.51 





4. A clothing factory uses 1,275.2 meters of cloth a week to make shirts. How much cloth would they need 

to make 

times as many shirts? 

5. There are as many boys as girls in a class of fifth-graders. If there are 35 students in the class, how many 

are girls? 

6. Ciro purchased a concert ticket for $56. The cost of the ticket was the cost of his dinner. The cost of his 

hotel was times as much as his ticket. How much did Ciro spend altogether for the concert ticket, 

hotel, and dinner? 


Date: 11/10/13 4.F.52 





Name 

Date 

1. An artist builds a sculpture out of metal and wood that weighs 14.9 kilograms. of this weight is metal, 

and the rest is wood. How much does the wood part of the sculpture weigh? 

2. On a boat ride tour, there are half as many children as there are adults. There are 30 people on the tour. 

How many children are there? 


Date: 11/10/13 4.F.53 





Name 

Date 

1. Jesse takes his dog and cat for their annual vet visit. Jesse’s dog weighs 23 pounds. The vet tells him his 

cat’s weight is as much as his dog’s weight. How much does his cat weigh? 

2. An image of a snowflake is 1.8 centimeters wide. If the actual snowflake is the size of the image, what is 

the width of the actual snowflake? Express your answer as a decimal. 

3. A community bike ride offers a short ride for children and families, which is 5.7 miles, followed by a long 

ride for adults, which is times as long. If a woman bikes the short ride with her children, and then the 

long ride with her friends, how many miles does she ride altogether? 


Date: 11/10/13 4.F.54 





4. Sal bought a house for $78,524.60. Twelve years later he sold the house for times as much. What was 

the sale price of the house? 

5. In the fifth grade at Lenape Elementary School, there are as many students who do not wear glasses as 

those who do wear glasses. If there are 60 students who wear glasses, how many students are in the fifth 

grade? 

6. At a factory, a mechanic earns $17.25 an hour. The president of the company earns times as much for 

each hour he works. The janitor at the same company earns as much as the mechanic. How much does 

the company pay for all three people employees’ wages for one hour of work? 


Date: 11/10/13 4.F.55 






G R A D E 


Topic G 

Division of Fractions and Decimal 

Fractions 

5.OA.1, 5.NBT.7, 5.NF.7 




5.NBT.7 

5.NF.7 



relationship between addition and subtraction; relate the strategy to a written method 

and explain the reasoning used. 


numbers and whole numbers by unit fractions. (Students able to multiple fractions in 

general can develop strategies to divide fractions in general, by reasoning about the 

relationship between multiplication and division. But division of a fraction by a fraction 

is not a requirement at this grade level.) 


quotients. For example, create a story context for (1/3) ÷ 4, and use a visual 

fraction model to show the quotient. Use the relationship between multiplication 

and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3. 

b. Interpret division of a whole number by a unit fraction, and compute such 

quotients. For example, create a story context for 4 ÷ (1/5), and use a visual 

fraction model to show the quotient. Use the relationship between multiplication 

and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4. 

c. Solve real world problems involving division of unit fractions by non‐zero whole 

numbers and division of whole numbers by unit fractions, e.g., by using visual 

fraction models and equations to represent the problem. For example, how much 

chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How 

many 1/3-cup servings are in 2 cups of raisins? 


G5–M2 



G6–M4 


Topic G: Division of Fractions and Decimal Fractions 

Date: 11/10/13 4.G.1 




NYS COMMON CORE MATHEMATICS CURRICULUM Topic G 5 

Topic G begins the work of division with fractions, both fractions and decimal fractions. Students use tape 

diagrams and number lines to reason about the division of a whole number by a unit fraction and a unit 

fraction by a whole number (5.NF.7). Using the same thinking developed in Module 2 to divide whole 

numbers, students reason about how many fourths are in 5 when considering such cases as 5 ÷ 1/4. They also 

reason about the size of the unit when 1/4 is partitioned into 5 equal parts: 1/4 ÷ 5. Using this thinking as a 

backdrop, students are introduced to decimal fraction divisors and use equivalent fraction and place value 

thinking to reason about the size of quotients, calculate quotients, and sensibly place the decimal in quotients 

(5.NBT.7). 

A Teaching Sequence Towards Mastery of Division of Fractions and Decimal Fractions 

Objective 1: Divide a whole number by a unit fraction. 

(Lesson 25) 

Objective 2: Divide a unit fraction by a whole number. 

(Lesson 26) 

Objective 3: Solve problems involving fraction division. 

(Lesson 27) 

Objective 4: Write equations and word problems corresponding to tape and number line diagrams. 

(Lesson 28) 

Objective 5: Connect division by a unit fraction to division by 1 tenth and 1 hundredth. 

(Lesson 29) 

Objective 6: Divide decimal dividends by non-unit decimal divisors. 


Topic G: Division of Fractions and Decimal Fractions 

Date: 11/10/13 4.G.2 





Lesson 25 

Objective: Divide a whole number by a unit fraction. 






Total Time 

(12 minutes) 

(7 minutes) 

(31 minutes) 

(10 minutes) 

(60 minutes) 



• Multiply Fractions by Decimals 5.NBT.7 

(7 minutes) 

(5 minutes) 




T: (Write = ) is how many hundredths? 

T: (Write = ) Write as a decimal. 

S: (Write = 0.5 or = 0.50.) 

T: (Write = .) is how many hundredths? 


T: (Write = .) Write as a decimal. 

S: (Write = 0.25.) 

T: (Write = .) is how many hundredths? 


T: (Write = ) Write as a decimal. 

Lesson 25: Divide a whole number by a unit fraction. 

Date: 11/10/13 4.G.3 





S: (Write = 0.75.) 

T: (Write = __.__.) Write as a decimal. 

S: (Write = 1.75.) 

Continue the process for , , , , , , , , , , , , and . 

Multiply Fractions by Decimals (5 minutes) 



T: (Write = ____.) On your boards, write the multiplication sentence. 


T: (Write = . Beneath it, write 0.___ = ) On your boards, fill in the missing digit. 

S: (Write 0.5 = ) 

T: (Write 0.5 = = __.__.) Complete the equation. 

S: (Write 0.5 = = 0.25.) 

T: (Write = __.__.) On your boards, write the multiplication 

sentence. 

S: (Write = = 0.01.) 

T: (Write 0.5 = __.__.) 

S: (Write 0.5 = 0.01.) 

T: (Write 0.7 = __.__.) Rewrite the multiplication sentence as a fraction times a fraction. 

S: (Write = 0.42.) 

T: (Write 0.8 = __.__ × __.__ = = __.__.) Rewrite the multiplication sentence, filling in the blanks. 

S: (Write 0.8 = 0.8 × 0.4 = = 0.32.) 

T: (Write 0.9 = __.__ × __.__ = __.__.) Rewrite the multiplication sentence, filling in the blanks. 

S: (Write 0.9 = 0.8 × 0.9 = 0.72.) 


Date: 11/10/13 4.G.4 






The label on a 0.118-liter bottle of cough syrup recommends a 

dose of 10 milliliters for children aged 6 to 10 years. How many 

10-mL doses are in the bottle? 

Note: This problem requires students to access their knowledge 

of converting among different size measurement units—a look 

back to Modules 1 and 2. Students may disagree on whether 

the final answer should be a whole number or a decimal. There 

are only 11 complete 10-mL doses in the bottle, but many 

students will divide 118 by 10, and give 11.8 doses as their final 

answer. This invites interpretation of the remainder since both 

answers are correct. 


Materials: (S) Personal white boards, 4″ × 2″ rectangular paper (several pieces per student), scissors 

Problem 1 

Jenny buys 2 pounds of pecans. 

a. If Jenny puts 2 pounds in each bag, how many bags can 

she make? 

b. If she puts 1 pound in each bag, how many bags can she 

make? 

c. If she puts pound in each bag, how many bags can she 

make? 

d. If she puts pound in each bag, how many bags can she 

make? 

NOTES ON 



In addition to tape diagrams and area 

models, students can also use region 

models to represent the information in 

these problems. For example, students 

can draw circles to represent the 

apples and divide the circles in half to 

represent halves. 

e. If she puts pound in each bag, how many bags can she 

make? 

Note: Continue this questioning sequence to include thirds, fourths, and fifths. 

T: (Post Problem 1(a) on the board, and read it aloud with students.) Work with your partner to write a 

division sentence that explains your thinking. Be prepared to share. 

S: (Work.) 

T: Say the division sentence to solve this problem. 

S: 2 ÷ 2 = 1. 

T: (Record on board.) How many bags of pecans can she make? 


Date: 11/10/13 4.G.5 





MP.4 

S: 1 bag. 

T: (Post Problem 1(b).) Write a division sentence for this situation and solve. 

S: (Solve.) 

T: Say the division sentence to solve this problem. 

S: 2 ÷ 1 = 2. 

T: (Record directly beneath the first division sentence.) Answer the question in a complete sentence. 

S: She can make 2 bags. 

T: (Post Problem 1(c).) If Jenny puts 1 half-pound in each of the bags, how many bags can she make? 

What would that division sentence look like? Turn and talk. 

S: We still have 2 as the amount that’s divided up, so it should still be 2 . We are sort of putting 

pecans in half-pound groups, so 1 half will be our divisor, the size of the group. It’s like asking 

how many halves are in 2? 

T: (Write 2 directly beneath the other division sentences.) Will the answer be more or less than 2? 

Talk to your partner. 

S: I looked at the other problems and see a pattern. 2 ÷ 2 = 1, 2 ÷ 1 = 2, and now I think 2 ÷ will be 

more than 2. It should be more, because we’re cutting each pound into halves so that will make 

more groups. I can visualize that each whole pound would have 2 halves, so there should be 4 

half-pounds in 2 pounds. 

T: Let’s use a piece of rectangular paper to represent 2 pounds of 

pecans. Cut it into 2 equal pieces, so each piece represents…? 

S: 1 pound of pecans. 

T: Fold each pound into halves, and cut. 

S: (Fold and cut.) 

T: How many halves were in 2 wholes? 

S: 4 halves. 

T: Let me model what you just did using a tape diagram. The tape 

represents 2 wholes. (Label 2 on top.) Each unit (partition the 

tape with one line down the middle) is 1 whole. The dottedlines 

cut each whole into halves. (Partition each whole with a dotted line.) How many halves are in 

1 whole? 

S: 2 halves. 

T: How many halves are in 2 wholes? 

S: 4 halves. 

T: Yes. I’ll draw a number line underneath the tape diagram and label the wholes. (Label 0, 1, and 2 on 

the number line.) Now, I can put a tick mark for each half. Let’s count the halves with me as I label. 

(Label .) 

S: . 


Date: 11/10/13 4.G.6 





MP.4 

T: There are 4 halves in 2 wholes. (Write 2 ÷ = 4.) She can make 4 bags. But how can we be sure 4 

halves is correct? How do we check a division problem? Multiply the quotient and the…? 

S: Divisor. 

T: What is the quotient? 

S: 4. 

T: The divisor? 

S: 1 half. 

T: What would our checking expression be? Write it with your 

partner. 

S: 4 

2 . 

T: Complete the number sentence. (Pause.) Read the 

complete sentence. 

S: 4 

2 

2 or 2 

4 2. 

T: Were we correct? 

S: Yes. 

T: Let’s remember this thinking as we continue. 

Repeat the modeling process with Problem 1(d) and (e), divisors of 1 

third and 1 fourth. 

Extend the dialogue when dividing by 1 fourth to look for patterns: 

T: (Point to all the number sentences in the previous 

problems: 2 ÷ 2 = 1, 2 ÷ 1 = 2, 2 ÷ = 4, 2 ÷ = 6, and 

2 ÷ = 8.) Take a look at these problems, what patterns do 

you notice? Turn and share. 

S: The 2 pounds are the same, but each time it is being divided 

into a smaller and smaller unit. The answer is getting 

bigger and bigger. When the 2 pounds is divided into 

smaller units, then the answer is bigger. 

T: Explain to your partner why the quotient is getting bigger as 

it is divided by smaller units. 

S: When we cut a whole into smaller parts, then we’ll get more 

parts. The more units we split from one whole, then the 

more parts we’ll have. That’s why the quotient is getting bigger. 

T: Based on the patterns, solve how many bags she can make if she puts pound in each bag. Draw a 

tape diagram and a number line on your personal board to explain your thinking. 

S: (Solve.) 

T: Say the division sentence. 


Date: 11/10/13 4.G.7 





S: 2 ÷ = 10. 


S: She can make 10 bags. 

Problem 2 

Jenny buys 2 pounds of pecans. 

a. If this is the number she needs to make pecan pies, 

how many pounds will she need? 

b. If this is the number she needs to make pecan pies, 


c. If this is the number she needs to make pecan pies, 


T: We can also ask different questions about Jenny and 

her two pounds of pecans. (Post Problem 2(a).) Two is 

half of what number? 

S: 4. 

T: Give me the division sentence. 

S: It’s not division! It’s multiplication. It’s 2 twos. 

That’s four. 

T: Give me the multiplication number sentence. 

S: 2 2 = 4. 4 = 2. 

T: Hold on. Stop. Let’s try to write a division expression 

for this whole number situation. (Write 4 ___ = 8.) 

S: What would the division expression be? 

S: 8 ÷ 4. 

T: Tell me the complete number sentence. 

S: 8 ÷ 4 = 2. 

T: Now try the same process with 2 

____= 2. 

Give me the division expression. 

S: 2 ÷ . 

T: Tell me the complete number sentence. 

S: 2 ÷ = 4. 

T: Yes. We are finding how much is in one unit just like we did with 

8 ÷ 2 = 4. In this case, the whole is the unit. 

T: What is the whole unit in this story? 

NOTES ON 

TABLE 2 OF THE 

COMMON CORE 

LEARNING STANDARDS: 

It is important to distinguish between 

interpretations of division when 

working with fractions. When working 

with fractions, it may be easier to 

understand the distinction by using the 

word unit rather than group. 

Number of units unknown (or number 

of groups unknown) is the 

measurement model of division, for 

example, for 12 ÷ 3 and 3 ÷ : 

• 12 cards are put in packs of 3. 

How many packs are there? 

• 3 meters of cloth are cut into 

meter strips. How many strips are 

cut? 

Unknown unit (or group size unknown) 

is the partitive model of division, for 

example, for 12 ÷ 3 and 3 ÷ : 

• 12 cards are dealt to 3 people. 

How many cards does each 

person get? 

• 3 miles is the trip. How far is the 

whole trip? 


Date: 11/10/13 4.G.8 





S: The whole amount she needs for pecan pies. 

T: Let’s go back and answer our question. Jenny buys 2 pounds of pecans. If this is the number she 

needs to make pecan pies, how many pounds will she need? 

S: She will need 4 pounds of pecans. 

T: Yes. 

T: (Post Problem 2(b) on the board.) The answer is…? 

S: 6. 

T: Give me the division sentence. 

S: 2 ÷ 6. 

T: Explain to your partner why that is true. 

S: We are looking for the whole amount of pounds. Two is a third, so we divide it by a third. I still 

think of it as multiplication though, 2 times 3 equals 6. But the problem doesn’t mention 3, it says 

a third, so 2 ÷ = 2 3. So, dividing by a third is the same as multiplying by 3. 

T: We can see in our tape diagram that this is true. (Write 2 ÷ = 2 3.) Explain to your partner why. 

Problem 3 

Use the story of the pecans, if you like. 

Tien wants to cut foot lengths from a board that is 5 feet long. How many boards can he cut? 

T: (Post Problem 3 on the board, and read it together with the class.) What is the length of the board 

Tien has to cut? 

S: 5 feet. 

T: How can we find the number of boards 1 fourth of a foot long? Turn and talk. 

S: We have to divide. The division sentence is 5 ÷ . I can draw 5 wholes, and cut each whole into 

fourths. Then I can count how many fourths are in 5 wholes. 

T: On your personal board, draw and solve this problem independently. 

S: (Work.) 

T: How many quarter feet are in one foot? 


Date: 11/10/13 4.G.9 





S: 4. 

T: How many quarter feet are in 5 feet? 

S: 20. 

T: Say the division sentence. 

S: 5 ÷ = 20. 

T: Check your work, then answer the question in a 


S: Tien can cut 20 boards. 










Date: 11/10/13 4.G.10 






Lesson Objective: Divide a whole number by a unit fraction. 









You may choose to use any combination of the questions below 

to lead the discussion. 

• In Problem 1, what do you notice about (a) and (b), and 

(c) and (d)? What are the whole and the divisor in the 

problems? 

• Share your solution and compare your strategy for 

solving Problem 2 with a partner. 

• Explain your strategy of solving Problem 3 and 4 with a 

partner. 

• Problem 5 on the Problem Set is a partitive division 

problem. Students are not likely to interpret the 

problem as division and will more likely use a missing 

factor strategy to solve (which is certainly appropriate). 

• Problem 5 can be expressed as 3 . This could be 

thought of as “ gallons is 1 out of 4 parts needed to fill 

the pail” or “ is fourth of what number?” Asking 

students to consider this interpretation will be 

beneficial in future encounters with fraction division. 

(See UDL box. The model below puts the two 

interpretations right next to 

each other.) 

NOTES ON 


ENGAGEMENT: 

The second to last bullet in today’s 

Debrief brings out an interpretation of 

fraction division in context that is 

particularly useful for Grade 6’s 

encounters with non-unit fraction 

division. In Grade 6, Problem 5 might 

read: 

gallon of water fills the pail to of 

its capacity. How much water does 

the pail hold? 

This could be expressed as 

is, 

is 3 of the 4 groups needed to 

. That 

completely fill the pail. This type of 

problem can be thought of partitively 

as 2 thirds is 3 fourths of what number 

or 

. This gives rise to 

explaining the invert and multiply 

strategy. Working from a tape 

diagram, this problem would be stated 

as: 

• 3 units = 

• 1 unit = 

We need 4 units to fill the pail: 

• 4 units = 

• = 


Date: 11/10/13 4.G.11 










Date: 11/10/13 4.G.12 





Name 

Date 

1. Draw a tape diagram and a number line to solve. You may draw the model that makes the most sense to 

you. Fill in the blanks that follow. Use the example to help you. 

Example: 2 = 6 

2 

? 

0 1 2 

2 

0 1 2 3 4 5 6 

There are __3__ thirds in 1 whole. 

If 2 is , what is the whole? 6 

There are __6__ thirds in 2 wholes 

a. 4 = _________ There are ____ halves in 1 whole. 

There are ____ halves in 4 wholes. 

If 4 is , what is the whole? ________ 

b. 2 = _________ There are____ fourths in 1 whole. 

There are ____ fourths in 2 wholes. 


c. 5 = _________ There are ____ thirds in 1 whole. 

There are ____ thirds in 5 wholes. 


d. 3 = _________ There are ____ fifths in 1 whole. 

There are ____ fifths in 3 wholes. 

If 3 is , what is the whole? _______ 


Date: 11/10/13 4.G.13 





2. Divide. Then multiply to check. 

a. 5 b. 3 c. 4 d. 1 

e. 2 f. 7 g. 8 h. 9 

3. For an art project, Mrs. Williams is dividing construction paper into fourths. How many fourths can she 

make from 5 pieces of construction paper? 


Date: 11/10/13 4.G.14 





4. Use the chart below to answer the following questions. 

Donnie’s Diner Lunch Menu 

Food 

Serving Size 

Hamburger 

Pickles 

Potato Chips 

Chocolate Milk 

lb 

pickle 

bag 

cup 

a. How many hamburgers can Donnie make with 6 pounds of hamburger meat? 

b. How many pickle servings can be made from a jar of 15 pickles? 

c. How many servings of chocolate milk can he serve from a gallon of milk? 

5. Three gallons of water fills of the elephant’s pail at the zoo. How much water does the pail hold? 


Date: 11/10/13 4.G.15 





Name 

Date 

1. Draw a tape diagram and a number line to solve. Fill in the blanks that follow. 

a. 5 = _________ There are ____ halves in 1 whole. 

There are ____ halves in 5 wholes. 

5 is of what number? _______ 

b. 4 = _________ There are ____ fourths in 1 whole. 

There are ____ fourths in ____ wholes. 

4 is of what number? _______ 

2. Ms. Leverenz is doing an art project with her class. She has a 3-foot piece of ribbon. If she gives each 

student an eighth of a foot of ribbon, will she have enough for her 22-student class? 


Date: 11/10/13 4.G.16 





Name 

Date 

1. Draw a tape diagram and a number line to solve. Fill in the blanks that follow. 

a. 3 = _________ There are ____ thirds in 1 whole. 

There are ____ thirds in __ wholes. 


b. 3 = _________ There are____ fourths in 1 whole. 

There are ____ fourths in __ wholes. 


c. 4 = _________ There are ____ thirds in 1 whole. 

There are ____ thirds in __ wholes. 


d. 5 = _________ There are____ fourths in 1 whole. 

There are ____ fourths in __ wholes. 



Date: 11/10/13 4.G.17 






a. 2 b. 6 c. 5 d. 5 

e. 6 f. 3 g. 6 h. 6 

3. A principal orders 8 sub sandwiches for a teachers’ meeting. She cuts the subs into thirds and puts the 

mini-subs onto a tray. How many mini-subs are on the tray? 

4. Some students prepare 3 different snacks. They make pound bags of nut mix, pound bags of cherries, 

and pound bags of dried fruit. If they buy 3 pounds of nut mix, 5 pounds of cherries, and 4 pounds of 

dried fruit, how many of each type of snack bag will they be able to make? 


Date: 11/10/13 4.G.18 





Lesson 26 

Objective: Divide a unit fraction by a whole number. 






Total Time 

(12 minutes) 

(8 minutes) 

(30 minutes) 

(10 minutes) 

(60 minutes) 



• Divide Whole Numbers by Fractions 5.NF.7 


(5 minutes) 

(4 minutes) 

(3 minutes) 



1 2 3 

1 2 3 

1 2 3 

T: Count by one-fourth to 12 fourths. (Write as students count.) 

S: 1 fourth, 2 fourths, 3 fourths, 4 fourths, 5 fourths, 6 fourths, 7 fourths, 8 fourths, 9 fourths, 10 

fourths, 11 fourths, 12 fourths. 

T: Let’s count by one-fourths again. This time, when we arrive at a whole number, say the whole 

number. (Write as students count.) 

S: 1 fourth, 2 fourths, 3 fourths, 1 whole, 5 fourths, 6 fourths, 7 fourths, 2 wholes, 9 fourths, 10 fourths, 

11 fourths, 3 wholes. 

T: Let’s count by one-fourths again. This time, change improper fractions to mixed numbers. 

Lesson 26: Divide a unit fraction by a whole number. 

Date: 11/10/13 4.G.19 





S: 1 fourth, 2 fourths, 3 fourths, 1 whole. 1 and 1 fourth, 1 and 2 fourths, 1 and 3 fourths, 2 wholes, 2 

and 1 fourth, 2 and 2 fourths, 2 and 3 fourths, 3 wholes. 

T: Let’s count by one-fourths again. This time, simplify 2 fourths to 1 half. (Write as students count.) 

S: 1 fourth, 1 half, 3 fourths, 1 whole, 1 and 1 fourth, 1 and 1 half, 1 and 3 fourths, 2 wholes, 2 and 1 

fourth, 2 and 1 half, 2 and 3 fourths, 3 wholes. 

Continue the process, counting by one-fifths to 15 fifths. 

Divide Whole Numbers by Fractions (4 minutes) 



T: (Write 1 ÷ =____. ) Say the division problem. 

S: 1 ÷ . 

T: How many halves are in 1 whole? 

S: 2. 

T: (Write 1 ÷ = 2. Beneath it, write 2 ÷ .) How many halves are in 2 wholes? 

S: 4. 

T: (Write 2 ÷ = 4. Beneath it, write 3 ÷ .) How many halves are in 3 wholes? 

S: 6. 

T: (Write 3 ÷ = 6. Beneath it, write 8 ÷ .) On your boards, write the complete number sentence. 

S: (Write 8 ÷ = 16.) 

Continue with the following possible suggestions: 1 ÷ , 2 ÷ , 5 ÷ , 1 ÷ , 2 ÷ , 7 ÷ , 3 ÷ , 4 ÷ , and 7 ÷ . 




T: (Write .) Say the multiplication number sentence. 

S: = . 

Continue this process with and . 

T: (Write .) On your boards, write the number sentence. 



Date: 11/10/13 4.G.20 





T: (Write .) Say the multiplication sentence. 

S: = . 

Repeat this process with , , and . 

T: (Write =____.) Say the multiplication sentence. 


Continue this process with . 

T: (Write .) On your boards, write the number sentence. 


T: (Write =____.) On your boards, write the number sentence. 

S: (Write = = 1.) 


A race begins with 

for 

miles through town, continues through the park 

miles, and finishes at the track after the last mile. A volunteer 

is stationed every quarter mile and at the finish line to pass out cups of 

water and cheer on the runners. How many volunteers are needed? 

Note: This multi-step problem requires students to first add three 

fractions, then divide the sum by a fraction, which reinforces yesterday’s 

division of a whole number by a unit fraction. (How many miles are in 5 

miles?) It also reviews adding fractions with different denominators (G5– 

Module 3). 



Problem 1 

Nolan gives some pans of brownies to his 3 friends to share equally. 

a. If he has 3 pans of brownies, how many pans of brownies will each friend get? 

b. If he has 1 pan of brownies, how many pans of brownies will each friend get? 

c. If he has pan of brownies, how many pans of brownies will each friend get? 


Date: 11/10/13 4.G.21 





d. If he has pan of brownies, how many pans of brownies will each friend get? 

T: (Post Problem 1(a) on the board, and read it aloud with students.) Work on your personal board and 

write a division sentence to solve this problem. Be prepared to share. 

S: (Work.) 

T: How many pans of brownies does Nolan have? 

S: 3 pans. 

T: The 3 pans of brownies are divided equally into how many friends? 

S: 3 friends. 

T: Say the division sentence with the answer. 

S: 3 ÷ 3 = 1. 


S: Each friend will get 1 pan of brownies. 

T: (In the problem, erase 3 pans and replace it with 1 

pan.) Imagine that Nolan has 1 pan of brownies. If 

he gave it to his 3 friends to share equally, what 

portion of the brownies will each friend get? Write 

a division sentence to show how you know. 

S: (Write 1 ÷ 3 = pan.) 

T: Nolan starts out with how many pans of brownies? 

S: 1 pan. 

T: The 1 pan of brownie is divided equally by how many 

friends? 

S: 3 friends. 

T: Say the division sentence with the answer. 

S: 1 ÷ 3 = . 

T: Let’s model that thinking with a tape diagram. I’ll draw 

a bar and shade it in representing 1 whole pan of 

brownie. Next, I’ll partition it equally with dotted lines 

into 3 units, and each unit is . (Draw a bar and cut it 

equally into three parts.) How many pans of brownies 

did each friend get this time? Answer the question in a 


S: Each friend will get pan of brownie. (Label underneath one part.) 

T: Let’s rewrite the problem as thirds. How many thirds are in whole? 

S: 3 thirds. 

T: (Write 3 thirds ÷ 3 = ___.) What is 3 thirds divided by 3? 

S: 1 third. (Write = 1 third.) 

NOTES ON 


ENGAGEMENT: 

While the tape diagramming in the 

beginning of this lesson is presented as 

teacher-directed, it is equally 

acceptable to elicit each step of the 

diagram from the students through 

questioning. Many students benefit 

from verbalizing the next step in a 

diagram. 


Date: 11/10/13 4.G.22 





T: Another way to interpret this division expression would be to ask, “ is of what number?” And of 

course, we know that 3 thirds makes 1. 

T: But just to be sure, let’s check our work. How do we check a division problem? 

S: Multiply the answer and the divisor. 

T: Check it now. 

S: (Work and show 3 1.) 

T: (Replace 1 pan in the problem with pan.) Now, 

imagine that he only has pan. Still sharing 

them with 3 friends equally, how many pans of 

brownies will each friend get? 

T: Now that we have half of a pan instead of 1 

whole pan to share, will each friend get more or 

less than pan? Turn and discuss. 

S: Less than pan. We have less to share, but 

we are sharing with the same number of people. They will get less. Since we’re starting out with 

pan which is less than 1 whole pan, the answer should be less than pan. 

T: (Draw a bar and cut it into 2 parts. Shade in 1 part.) How can we show how many people are 

sharing this pan of brownie? Turn and talk. 

S: We can draw dotted lines to show the 3 equal parts that he cuts the half into. We have to show 

the same size units, so I’ll cut the half that’s shaded into 3 parts and the other half into 3 parts, too. 

T: (Partition the whole into 6 parts.) What fraction of the pan will each friend get? 

S: . (Label underneath one part.) 

T: (Write .) Let’s think again, half is equal to how many sixths? Look at the tape diagram to 

help you. 

S: 3 sixths. 

T: So, what is 3 sixths divided by 3? (Write 3 sixths ÷ 3 =____.) 

S: 1 sixth. (Write = 1 sixth.) 

T: What other question could we ask from this division expression? 

S: is 3 of what number? 

T: And 3 of what number makes half? 

S: Three 1 sixths makes half. 

T: Check your work, then answer the question in a complete sentence. 

S: Each friend will get pan of brownie. 

T: (Erase the in the problem, and replace it with .) What if Nolan only has a third of a pan and let 3 

friends share equally? How many pans of brownies will each friend get? Work with a partner to 


Date: 11/10/13 4.G.23 





solve it. 

S: (Share.) 


S: Each friend will get pan of brownie. 

T: (Point to all the previous division sentences: 3 ÷ 3 

= 1, 1 ÷ 3 = , , and .) Compare our 

division sentences. What do you notice about the quotients? Turn and talk. 

S: The answer is getting smaller and smaller because Nolan kept giving his friends a smaller and smaller 

part of a pan to share. The original whole is getting smaller from 3 to 1, to , to , and the 3 

people sharing the brownies stayed the same, that’s why the answer is getting smaller. 

Problem 2 

T: (Post Problem 2 on the board.) Work 

independently to solve this problem on your 

personal board. Draw a tape diagram to show 


S: (Work.) 

T: What’s the answer? 

S: . 

T: How many tenths are in 1 fifth? 

S: 2 tenths. 

T: (Write 2 tenths ÷ ___.) What’s tenths divided by ? 

S: 1 tenth. (Write = 1 tenth.) 

T: Asked another way: (Write = 2 ______.) Fill in the missing factor. 

S: 1 tenth. 

T: Let’s check our work aloud together. What is the quotient? 

S: 1 tenth. 

T: The divisor? 

S: 2. 

T: Let’s multiply the quotient by the divisor. What is 1 tenth times 2? 

S: 2 tenths. 

T: Is 2 tenths the same units as our original whole? 

S: No. 

T: Did we make a mistake? 

S: No, 2 tenths is just another way to say 1 fifth. 


Date: 11/10/13 4.G.24 





T: Say 2 tenths in its simplest form. 

S: 1 fifth. 

Problem 3 

If Melanie pours liter of water into 4 bottles, putting an equal amount in each, how many liters of water will 

be in each bottle? 

T: (Post Problem 3 on the board, and read it together with the class.) How many liters of water does 

Melanie have? 

S: Half a liter. 

T: Half of liter is being poured into how many 

bottles? 

S: 4 bottles. 

T: How do you solve this problem? Turn and discuss. 

S: We have to divide. The division sentence is 

. I need to divide the dividend 1 half by the 

divisor, 4. I can draw 1 half, and cut it into 4 

equal parts. I can think of this as . 

T: On your personal board, draw a tape diagram and solve this problem independently. 

S: (Work.) 

T: Say the division sentence and the answer. 

S: . (Write .) 

T: Now say the division sentence using eighths and unit form. 

S: 4 eighths ÷ 4 = 1 eighth. 

T: Show me your checking solution. 

S: (Work and show 4 = = .) 

T: If you used a multiplication sentence with a missing factor, say it now. 

S: . 

T: No matter your strategy, we all got the same result. Answer the question in a complete sentence. 

S: Each bottle will have liter of water. 







Date: 11/10/13 4.G.25 






Lesson Objective: Divide a unit fraction by a whole 

number. 









lesson. 




(a) and (b), (c) and (d), and (b) and (d)? 

• Why is the quotient of Problem 1(c) greater than 

Problem 1(d)? Is it reasonable? Explain to your 

partner. 

• In Problem 2, what is the relationship between (c) 

and (d) and (b) and (f)? 

• Compare your drawing of Problem 3 with a 

partner. How is it the same as or different from 

your partner’s? 

• How did you solve Problem 5? Share your 

solution and explain your strategy to a partner. 

• While the invert and multiply strategy is not 

explicitly taught (nor should it be while students 

grapple with these abstract concepts of division), 

discussing various ways of thinking about division 

in general can be fruitful. A discussion might 

proceed as follows: 

T: Is dividing something by 2 the same as taking 1 

half of it? For example, is 4 

? (Write 

this on the board and allow some quiet time for 

thinking.) Can you think of some examples? 

S: Yes. If 4 cookies are divided between 2 

people, each person gets half of the cookies. 


Date: 11/10/13 4.G.26 





T: So, if that’s true, would this also be true: 2 = 

? (Write and allow quiet time.) Can you think 

of some examples? 

S: Yes. If there is only 1 fourth of a candy bar 

and 2 people share it, they would each get half of 

the fourth. But that would be 1 eighth of the 

whole candy bar. 

Once this idea is introduced, look for opportunities in 

visual models to point it out. For example, in today’s 

lesson, Problem ’s tape diagram was drawn to show 

divided into 4 equal parts. But, just as clearly as we can 

see that the answer to our question is of that , we can 

see that we get the same answer by multiplying . 







students. 


Date: 11/10/13 4.G.27 





Name 

Date 

1. Draw a model or tape diagram to solve. Use the thought bubble to show your thinking. Write your 

quotient in the blank. Use the example to help you. 

Example: 3 

1 half 3 

= 3 sixths 3 

= 1 sixth 

3 = 

a. 2 = ______ 

b. 4 = ______ 


Date: 11/10/13 4.G.28 





c. 2 = ______ 

d. 3 = ______ 


a. 7 b. 6 c. 5 d. 4 

e. 2 f. 3 g. 2 h. 10 


Date: 11/10/13 4.G.29 





3. Tasha eats half her snack and gives the other half to her two best friends for them to share equally. What 

portion of the whole snack does each friend get? Draw a picture to support your response. 

4. Mrs. Appler used gallon of olive oil to make 8 identical batches of salad dressing. 

a. How many gallons of olive oil did she use in each batch of salad dressing? 

b. How many cups of olive oil did she use in each batch of salad dressing? 

5. Mariano delivers newspapers. He always puts of his weekly earnings in his savings account, then divides 

the rest equally into 3 piggy banks for spending at the snack shop, the arcade, and the subway. 

a. What fraction of his earnings does Mariano put into each piggy bank? 

b. If Mariano adds $2.40 to each piggy bank every week, how much does Mariano earn per week 

delivering papers? 


Date: 11/10/13 4.G.30 





Name 

Date 

1. Solve. Support at least one of your answers with a model or tape diagram. 

a. 4 = ______ 

b. 5 = ______ 

2. Larry spends half of his workday teaching piano lessons. If he sees 6 students, each for the same amount 

of time, what fraction of his workday is spent with each student? 


Date: 11/10/13 4.G.31 





Name 

Date 

1. Solve and support your answer with a model or tape diagram. Write your quotient in the blank. 

a. 4 = ______ b. 6 = ______ 

c. 3 = ______ d. 2 = ______ 


a. 10 b. 10 c. 5 d. 3 

e. 4 f. 3 g. 5 h. 20 


Date: 11/10/13 4.G.32 





3. Teams of four are competing in a quarter-mile relay race. Each runner must run the same exact distance. 

What is the distance each teammate runs? 

4. Solomon has read of his book. He finishes the book by reading the same amount each night for 5 nights. 

a. What fraction of the book does he read each of the 5 nights? 

b. If he reads 14 pages on each of the 5 nights, how long is the book? 


Date: 11/10/13 4.G.33 





Lesson 27 

Objective: Solve problems involving fraction division. 





Total Time 

(12 minutes) 

(38 minutes) 

(10 minutes) 

(60 minutes) 



• Divide Whole Numbers by Unit Fractions 5.NF.7 

• Divide Unit Fractions by Whole Numbers 5.NF.7 

(6 minutes) 

(3 minutes) 

(3 minutes) 



T: Count by sixths to 12 sixths. (Write as students count.) 

S: 1 sixth, 2 sixths, 3 sixths, 4 sixths, 5 sixths, 6 sixths, 7 sixths, 8 sixths, 9 sixths, 10 sixths, 11 sixths, 12 

sixths. 

T: Let’s count by sixths again. This time, when we arrive at a whole number, say the whole number. 

(Write as students count.) 

S: 1 sixth, 2 sixths, 3 sixths, 4 sixths, 5 sixths, 1 whole, 7 sixths, 8 sixths, 9 sixths, 10 sixths, 11 sixths, 2 

wholes. 

T: Let’s count by sixths again. This time, change improper fractions to mixed numbers. (Write as 


S: 1 sixth, 2 sixths, 3 sixths, 4 sixths, 5 sixths, 1 whole, 1 and 1 sixth, 1 and 2 sixths, 1 and 3 sixths, 1 and 

4 sixths, 1 and 5 sixths, 2 wholes. 

T: Let’s count by sixths again. This time, simplify 3 sixths to 1 half. (Write as students count.) 

S: 1 sixth, 2 sixths, 1 half, 4 sixths, 5 sixths, 1 whole, 1 and 1 sixth, 1 and 2 sixths, 1 and 1 half , 1 and 4 

sixths, 1 and 5 sixths, 2 wholes. 

T: Let’s count by 1 sixths again. This time, simplify 2 sixths to 1 third and 4 sixths to 2 thirds. (Write as 


S: 1 sixth, 1 third, 1 half, 2 thirds, 5 sixths, 1 whole, 1 and 1 sixth, 1 and 1 third, 1 and 1 half, 1 and 2 

thirds, 1 and 5 sixths, 2 wholes. 

Lesson 27: Solve problems involving fraction division. 

Date: 11/10/13 4.G.34 





Continue the process counting by 1 eighths to 8 eighths or, if time allows, 16 eighths. 

Divide Whole Numbers by Unit Fractions (3 minutes) 



T: (Write 1 ÷ .) Say the division sentence. 

S: 1 ÷ . 

T: How many halves are in 1 whole? 

S: 2. 

T: (Write 1 ÷ = 2. Beneath it, write 2 ÷ = ____.) How many halves are in 2 wholes? 

S: 4. 

T: (Write 2 ÷ = 4. Beneath it, write 3 ÷ = ____.) How many halves are in 3 wholes? 

S: 6. 

T: (Write 3 ÷ = 6. Beneath it, write 6 ÷ .) On your boards, write the division sentence. 

S: (Write 6 ÷ = 12.) 

Continue with the following possible suggestions: 1 ÷ , 2 ÷ , 7 ÷ , 1 ÷ , 2 ÷ , 9 ÷ , 5 ÷ , 6 ÷ , and 8 ÷ . 

Divide Unit Fractions by Whole Numbers (3 minutes) 



T: (Write ÷ 2 = ____.) Say the division sentence with answer. 

S: ÷ 2 = . 

T: (Write ÷ 2 = . Beneath it, write ÷ 3 = ____.) Say the division sentence with the answer. 

S: ÷ 3 = . 

T: (Write ÷ 3 = . Beneath it, write ÷ 4 = ____.) Say the division sentence with the answer. 

S: ÷ 4 = . 

T: (Write ÷ 7 = ____.) On your boards, complete the number sentence. 

S: (Write ÷ 7 = .) 

Continue with the following possible sequence: ÷ 2, ÷ 3, ÷ 4, ÷ 9, ÷ 3, ÷ 5, ÷ 7, ÷ 4, and ÷ 6. 


Date: 11/10/13 4.G.35 





MP.3 



Note: The time normally allotted for the Application Problem has been reallocated to the Concept 

Development to provide adequate time for solving the word problems. 

Suggested Delivery of Instruction for Solving Lesson 27’s Word Problems. 


Have two pairs of student work at the board while the others work independently or in pairs at their seats. 

Review the following questions before beginning the first problem: 




As students work, circulate. Reiterate the questions above. After two minutes, have the two pairs of 

students share only their labeled diagrams. For about one minute, have the demonstrating students receive 

and respond to feedback and questions from their peers. 


Give everyone two minutes to finish work on that question, sharing his or her work and thinking with a peer. 

All should write their equations and statements of the answer. 



Problem 1 

Mrs. Silverstein bought 3 mini cakes for a 

birthday party. She cut each cake into quarters, 

and plans to serve each guest 1 quarter of a 

cake. How many guests can she serve with all 

her cakes? Draw a model to support your 

response. 

In this problem, students are asked to divide a whole number (3) by a unit fraction ( ), and draw a model. A 

tape diagram or a number line would both be acceptable models to support their responses. The reference 


Date: 11/10/13 4.G.36 





to the unit fraction as a quarter provides a bit of complexity. There are 4 fourths in 1 whole, and 12 fourths in 

3 wholes. 

Problem 2 

Mr. Pham has pan of lasagna left in the refrigerator. He wants to cut the lasagna into equal slices so he can 

have it for dinner for 3 nights. How much lasagna will he eat each night? Draw a picture to support your 

response. 

Problem 2 is intentionally similar to Problem 1. Although the numbers used in the problems are identical, 

careful reading reveals that 3 is now the divisor rather than the dividend. While drawing a supporting tape 

diagram, students should recognize that dividing a fourth into 3 equal parts creates a new unit, twelfths. The 

model shows that the fraction is equal to 

, and therefore a division sentence using unit form (3 twelfths 

3) is easy to solve. Facilitate a quick discussion about the similarities and differences of Problems 1 and 2. 

What do students notice about the division expressions and the solutions? 

Problem 3 

The perimeter of a square is meter. 

a. Find the length of each side in meters. Draw a picture to support your response. 

b. How long is each side in centimeters? 


Date: 11/10/13 4.G.37 





This problem requires students to recall their measurement 

work from Grade 3 and Grade 4 involving perimeter. Students 

must know that all four side lengths of a square are equivalent, 

and therefore the unknown side length can be found by dividing 

the perimeter by 4 ( m 

4). The tape diagram shows clearly 

that dividing a fifth into 4 equal parts creates a new unit, 

twentieths, and that is equal to 

. Students may use a 

division expression using unit form (4 twentieths 4) to solve 

this problem very simply. This problem also gives opportunity 

to point out a partitive division interpretation to students. 

While the model was drawn to depict 1 fifth divided into 5 equal 

parts, the question mark clearly asks “What is of ?” That is, 

. 

NOTES ON 


ENGAGEMENT: 

Perimeter and area are vocabulary 

terms that students often confuse. To 

help students differentiate between 

the terms, teachers can make a poster 

outlining, in sandpaper, the perimeter 

of a polygon. As he uses a finger to 

trace along the sandpaper, the student 

says the word perimeter. This sensory 

method may help some students to 

learn an often confused term. 

Part (b) requires students to rename 

meters as centimeters. This conversion mirrors the work done in 

G5─M4─Lesson 20. Since 1 meter is equal to 100 centimeters, students can multiply to find that 

equivalent to 

Problem 4 

cm, or 5 cm. 

A pallet holding 5 identical crates weighs ton. 

a. How many tons does each crate weigh? Draw a picture to support your response. 

b. How many pounds does each crate weigh? 

m is 

The numbers in this problem are similar to those used in Problem 3, and the resulting quotient is again . 

Engage students in a discussion about why the answer is the same in Problems 3 and 4, but was not the same 

in Problems 1 and 2, despite both sets of problems using similar numbers. Is this just a coincidence? In 

addition, Problem 4 presents another opportunity for students to interpret the division here as 5 . 


Date: 11/10/13 4.G.38 





Problem 5 

Faye has 5 pieces of ribbon each 1 yard long. She cuts each ribbon into sixths. 

a. How many sixths will she have after cutting all the ribbons? 

b. How long will each of the sixths be in inches? 

In Problem 5, since Faye has 5 pieces of ribbon of equal length, students have the choice of drawing a tape 

diagram showing how many sixths are in 1 yard (and then multiplying that number by 5) or drawing a tape 

showing all 5 yards to find 30 sixths in total. 

Problem 6 

A glass pitcher is filled with water. 

equally into 2 glasses. 

of the water is poured 

a. What fraction of the water is in each glass? 

b. If each glass has 3 ounces of water in it, how many 

ounces of water were in the full pitcher? 

c. If of the remaining water is poured out of the pitcher 

to water a plant, how many cups of water are left in the 

pitcher? 

NOTES ON 


ENGAGEMENT: 

Problem 6 in this lesson may be 

especially difficult for English language 

learners. The teacher may wish have 

students act out this problem in order 

to keep track of the different questions 

asked about the water. 


Date: 11/10/13 4.G.39 





In Part (a), to find what fraction of the water is in each glass, students might divide the unit fraction ( ) by 2 or 

multiply . Part (b) requires students to show that since both glasses hold 3 ounces of water each, the 1 

unit (or of the total water) is equal to 6 ounces. Multiplying 6 ounces by 8, provides the total amount of 

water (48 ounces) that was originally in the pitcher. Part (c) is a complex, multi-step problem that may 

require careful discussion. Since of the water (or 6 ounces) has already been poured out, subtraction yields 

42 ounces of water left in the pitcher. After 1 fourth of the remaining water is used for the plant, of the 

water in the pitcher is 31 ounces. Students must then 

rename 31 ounces in cups. 


Lesson Objective: Solve problems involving fraction 

division. 


Date: 11/10/13 4.G.40 













lesson. 



• What did you notice about Problems 1 and 2? 

What are the similarities and differences? What 

did you notice about the division expressions and 

the solutions? 

• What did you notice about the solutions in 

Problems 3(a) and 4(a)? Share your answer and 

explain it to a partner. 

• Why is the answer the same in Problems 3 and 4, 

but not the same in Problems 1 and 2, despite 

using similar numbers in both sets of problems? 

Is this just a coincidence? Can you create similar 

pairs of problems and see if the resulting quotient 

is always equivalent (e.g., 2 and 3)? 

• How did you solve for Problem 6? What strategy 

did you use? Explain it to a partner. 







students. 


Date: 11/10/13 4.G.41 





Name 

Date 

1. Mrs. Silverstein bought 3 mini cakes for a birthday party. She cut each cake into quarters, and plans to 

serve each guest 1 quarter of a cake. How many guests can she serve with all her cakes? Draw a picture 

to support your response. 

2. Mr. Pham has pan of lasagna left in the refrigerator. He wants to cut the lasagna into equal slices so he 

can have it for dinner for 3 nights. How much lasagna will he eat each night? Draw a picture to support 

your response. 

3. The perimeter of a square is meter. 

a. Find the length of each side in meters. Draw a picture to support your response. 

b. How long is each side in centimeters? 


Date: 11/10/13 4.G.42 





4. A pallet holding 5 identical crates weighs ton. 

a. How many tons does each crate weigh? Draw a picture to support your response. 

b. How many pounds does each crate weigh? 

5. Faye has 5 pieces of ribbon each 1 yard long. She cuts each ribbon into sixths. 

a. How many sixths will she have after cutting all the ribbons? 

b. How long will each of the sixths be in inches? 


Date: 11/10/13 4.G.43 





6. A glass pitcher is filled with water. of the water is poured equally into 2 glasses. 

a. What fraction of the water is in each glass? 

b. If each glass has 3 ounces of water in it, how many ounces of water were in the full pitcher? 

c. If of the remaining water is poured out of the pitcher to water a plant, how many cups of water are 

left in the pitcher? 


Date: 11/10/13 4.G.44 





Name 

Date 

1. Kevin divides 3 pieces paper into fourths. How many fourths does he have? Draw a picture to support 

your response. 

2. Sybil has pizza left over. She wants to share the pizza with 3 of her friends. What fraction of the original 

pizza will Sybil and her 3 friends each receive? Draw a picture to support your response. 


Date: 11/10/13 4.G.45 





Name 

Date 

1. Kelvin ordered four pizzas for a birthday party. The pizzas were cut in eighths. How many slices were 

there? Draw a picture to support your response. 

2. Virgil has of a birthday cake left over. He wants to share the leftover cake with three friends. What 

fraction of the original cake will each of the 4 people receive? Draw a picture to support your response. 

3. A pitcher of water contains L water. The water is poured equally into 5 glasses. 

a. How many liters of water are in each glass? Draw a picture to support your response. 

b. Write the amount of water in each glass in milliliters. 


Date: 11/10/13 4.G.46 





4. Drew has 4 pieces of rope 1 meter long each. He cuts each rope into fifths. 

a. How many fifths will he have after cutting all the ropes? 

b. How long will each of the fifths be in centimeters? 

5. A container is filled with blueberries. of the blueberries are poured equally into two bowls. 

a. What fraction of the blueberries is in each bowl? 

b. If each bowl has 6 ounces of blueberries in it, how many ounces of blueberries were in the full 

container? 

c. If of the remaining blueberries are used to make muffins, how many pounds of blueberries are left 

in the container? 


Date: 11/10/13 4.G.47 





Lesson 28 

Objective: Write equations and word problems corresponding to tape and 






Total Time 

(10 minutes) 

(40 minutes) 

(10 minutes) 

(60 minutes) 



• Divide Whole Numbers by Unit Fractions and Unit Fractions by Whole Numbers 5.NF.7 

(5 minutes) 

(5 minutes) 



Note: This fluency prepares students for G5─M4─Lesson 29. 

T: Count by tenths to 20 tenths. (Write as students count.) 

S: 1 tenth, 2 tenths,… 20 tenths. 

T: Let’s count by tenths again. This time, when we arrive at a whole number, say the whole number. 


S: 1 tenth, 2 tenths,… 1, 11 tenths, 12 tenths,… 2. 

T: Let’s count by tenths again. This time, say the tenths in decimal form. (Write as students count.) 

S: Zero point 1, zero point 2,…. 

T: How many tenths are in 1 whole? 

S: 10 tenths. 

T: (Write 1 = 10 tenths. Beneath it, write 2 = ____ tenths.) How many tenths are in 2 wholes? 

S: 20 tenths. 

T: 3 wholes? 

S: 30 tenths. 

T: (Write 9 = __ tenths.) On your boards, fill in the unknown number. 

S: (Write 9 = 90 tenths.) 

T: (Write 10 = __ tenths.) Fill in the unknown number. 

Lesson 28: Write equations and word problems corresponding to tape and 


Date: 11/10/13 

4.G.48 






Divide Whole Numbers by Unit Fractions and Unit Fractions by Whole Numbers (5 minutes) 


Note: This fluency reviews G5─M4─Lessons 25─26 and prepares students for today’s lesson. 

T: (Write 2 ÷ ) Say the division sentence. 

S: 2 ÷ = 6. 

T: (Write 2 ÷ = 6. Beneath it, write 3 ÷ .) Say the division sentence. 

S: 3 ÷ = 9. 

T: (Write 3 ÷ = 9. Beneath it, write 8 ÷ = ____.) On your boards, write the division sentence. 

S: (Write 8 ÷ = 24.) 

Continue with 2 ÷ , 5 ÷ , and 9 ÷ . 

T: (Write ÷ 2.) Say the division sentence. 

S: ÷ 2 = . 

T: (Write ÷ 2 = . Beneath it, write ÷ 2.) Say the division sentence. 

S: ÷ 2 = . 

T: (Write ÷ 2 = . Erase the board and write ÷ 2.) On your boards, write the sentence. 

S: (Write ÷ 2 = .) 

Continue the process with the following possible sequence: ÷ 2 and ÷ 3. 


Materials: (S) Problem Set, personal white boards 

Note: Today’s lesson involves creating word problems, which can be time intensive. The time for the 

Application Problem has been included in the Concept Development. 

Note: Students create word problems from expressions and visual models in the form of tape diagrams. In 

Problem 1, guide students to identify what the whole and the divisor are in the expressions before they start 

writing the word problems. After about 10 minutes of working time, guide students to analyze the tape 

diagrams in Problems 2, 3, and 4. After the discussion, allow students to work for another 10 minutes. 

Finally, go over the answers, and have students share their answers with the class. 



Date: 11/10/13 

4.G.49 





Problems 1─2 

1. Create and solve a division story problem about 5 meters of rope that is modeled by the tape diagram 

below. 

5 meters 

1 

4 

1 

4 

. . . 

1 

4 

? fourths 

T: Let’s take a look at Problem 1 on our Problem Set and read it out loud together. What’s the whole in 

the tape diagram? 

S: 5. 

T: 5 what? 

S: 5 meters of rope. 

T: What else can you tell me about this tape diagram? Turn and share with a partner. 

S: The 5 meters of rope is being cut into fourths. The 5 meters of rope is being cut into pieces that 

are 1 fourth meter long. The question is, how many pieces can be cut? This is a division drawing, 

because a whole is being partitioned into equal parts. We’re trying to find out how many fourths 

are in 5. 

T: Since we seem to agree that this is a picture of division, what would the division expression look 

like? Turn and talk. 

S: Since 5 is the whole, it is the dividend. The one-fourths are the equal parts, so that is the divisor. 

5 ÷ . 

T: Work with your partner to write a story about this diagram, then solve for the answer. (A possible 

response appears on the student work example of the Problem Set.) 

T: (Allow students time to work.) How can we be sure that 20 fourths is correct? How do we check a 

division problem? 

S: Multiply the quotient and the divisor. 

T: What would our checking equation look like? Write it with your partner and solve. 

S: 20 5. 

T: Were we correct? How do you know? 

S: Yes. Our product matches the dividend that we started with. 



Date: 11/10/13 

4.G.50 





2. Create and solve a story problem about pound of almonds that is modeled by the tape diagram below. 

1 

4 

? 

MP.4 

T: Let’s now look at Problem 2 on the Problem Set, and read it together. 

S: (Read aloud.) 

T: Look at the tape diagram, what’s the whole, or dividend, in this problem? 

S: pound of almonds. 

T: What else can you tell me about this tape diagram? Turn and share with a partner. 

S: The 1 fourth is being cut into 5 parts. I counted 5 boxes. It means the one-fourth is cut into 5 

equal units, and we have to find how much 1 unit is. When you find the value of 1 equal part, that is 

division. I see that we could find of . That would be 

finding 1 part. 

. That’s the same as dividing by 5 and 

T: We must find how much of a whole pound of almonds is in each of the units. Say the division 

expression. 

S: 5. 

T: I noticed some of you were thinking about multiplication here. What multiplication expression 

would also give us the part that has the question mark? 

S: . 

T: Write the expression down on your paper, then work with a partner to write a division story and 

solve. (A possible response appears on the student work example of the Problem Set). 

T: How can we check our division work? 

S: Multiply the answer and the divisor. 

T: Check it now. 

S: (Write 5 

Problem 3 

a. 2 ÷ 

b. ÷ 4 

c. ÷ 3 

d. 3 ÷ 



Date: 11/10/13 

4.G.51 





T: (Write the three expressions on the board.) What do all of these expressions have in common? 

S: They are division expressions. They all have unit fractions and whole numbers. Problems (b) 

and (c) have dividends that are unit fractions. Problems (a) and (d) have divisors that are unit 

fractions. 

T: What does each number in the expression represent? Turn and discuss with a partner. 

S: The first number is the whole, and the second number is the divisor. The first number tells how 

much there is in the beginning. It’s the dividend. The second number tells how many in each group 

or how many equal groups we need to make. In Problem (a), 2 is the whole and is the divisor. 

In Problems (b) and (c), both expressions have a fraction divided by a whole number. 

T: Compare these expressions to the word problems we just wrote. Turn and talk. 

S: Problems (a) and (d) are like Problem 1, and the other two are like Problem 2. Problems (a) and 

(d) have a whole number dividend just like Problem 1. The others have fraction dividends like 

Problem 2. Our tape diagram for (a) should look like the one for Problem 1. The first one is 

asking how many fractional units in the wholes like Problems (a) and (d). The others are asking what 

kind of unit you get when you split a fraction into equal parts. Problems (b) and (c) will look like 

Problem 2. 

T: Work with a partner to draw a tape diagram for each expression, then write a story to match your 

diagram and solve. Be sure to use multiplication to check your work. (Possible responses appear on 

the student work example of the Problem Set. Be sure to include in the class discussion all the 

interpretations of division as some students may write stories that take on a multiplication flavor.) 


The Problem Set forms the basis for today’s lesson. Please 

see the script in the Concept Development for modeling 

suggestions. 


Lesson Objective: Write equations and word problems 

corresponding to tape and number line diagrams. 









lesson. 





Date: 11/10/13 

4.G.52 





• In Problem 3, what do you notice about (a) and (b), (a) 

and (d), and (b) and (c)? 

• Compare your stories and solutions for Problem 3 with 

a partner. 

• Compare and contrast Problems 1 and 2. What is 

similar or different about these two problems? 

• Share your solutions for Problems 1 and 2 and explain 

them to a partner. 


After the Student Debrief, instruct students to complete the Exit 

Ticket. A review of their work will help you assess the students’ 

understanding of the concepts that were presented in the 

lesson today and plan more effectively for future lessons. You 

may read the questions aloud to the students. 

NOTES ON 


EXPRESSION AND 

ACTION: 

Comparing and contrasting is often 

required in English language arts, 

science, and social studies classes. 

Teachers can use the same graphic 

organizers that are successfully used in 

these classes in math class. Although 

Venn Diagrams are often used to help 

students organize their thinking when 

comparing and contrasting, this is not 

the only possible graphic organizer. 

To add variety, charts listing similarities 

in a center column and differences in 

two outer columns can also be used. 



Date: 11/10/13 

4.G.53 





Name 

Date 

1. Create and solve a division story problem about 5 meters of rope that is modeled by the tape diagram 

below. 

5 

1 

4 

1 

4 

. . . 

1 

4 

? fourths 

2. Create and solve a story problem about pound of almonds that is modeled by the tape diagram below. 

1 

4 

? 



Date: 11/10/13 

4.G.54 





3. Draw a tape diagram and create a word problem for the following expressions, and then solve. 

a. 2 ÷ 

b. ÷ 4 

c. ÷ 3 

d. 3 



Date: 11/10/13 

4.G.55 





Name 

Date 

1. Create a word problem for the following expressions, and then solve. 

a. 4 ÷ 

b. ÷ 4 



Date: 11/10/13 

4.G.56 





Name 

Date 

1. Create and solve a division story problem about 7 feet of rope that is modeled by the tape diagram 

below. 

7 

1 1 

. . . 

1 

? halves 

2. Create and solve a story problem about pound of flour that is modeled by the tape diagram below. 

1 

3 

? 



Date: 11/10/13 

4.G.57 





3. Draw a tape diagram and create a word problem for the following expressions. Then solve and check. 

a. 2 ÷ 

b. ÷ 2 

c. ÷ 5 

d. 3 ÷ 



Date: 11/10/13 

4.G.58 





Lesson 29 

Objective: Connect division by a unit fraction to division by 1 tenth and 1 

hundredth. 






Total Time 

(9 minutes) 

(10 minutes) 

(31 minutes) 

(10 minutes) 

(60 minutes) 



• Divide Whole Numbers by Unit Fractions and Fractions by Whole Numbers 5.NF.7 

(5 minutes) 

(4 minutes) 




T: Count by 1 tenths to 20 tenths. When you reach a whole number, say the whole number. 


S: 1 tenth, 2 tenths, 3 tenths, 4 tenths, 5 tenths, 6 tenths, 7 tenths, 8 tenths, 9 tenths, 1 whole, 11 

tenths, 12 tenths, 13 tenths, 14 tenths, 15 tenths, 16 tenths, 17 tenths, 18 tenths, 19 tenths, 2 

wholes. 

T: How many tenths are in 1 whole? 

S: 10. 

T: 2 wholes? 

S: 20. 

T: 3 wholes? 

S: 30. 

T: 9 wholes. 

S: 90. 

T: 10 wholes? 

S: 100. 

Lesson 29: Connect division by a unit fraction to division by 1 tenth and 

1 hundredth. 

Date: 11/10/13 

4.G.59 





T: (Write 10 = 100 tenths. Beneath it, write 20 = ____ tenths.) On your boards, fill in the unknown. 


Continue the process with 30, 50, 70, and 90. 

T: (Write 90 = 900 tenths. Beneath it, write 91 = ____ tenths.) On your boards, fill in the unknown. 


Continue the process with 92, 82, 42, 47, 64, 64.1, 64.2, and 83.5. 

Divide Whole Numbers by Unit Fractions and Fractions by Whole Numbers (4 minutes) 


Note: This fluency reviews G5–M4–Lessons 25─27 and prepares students for today’s lesson. 

T: (Write 2 ÷ .) Say the division sentence. 

S: 2 ÷ = 4. 

T: (Write 2 ÷ = 4. Beneath it, write 3 ÷ .) Say the division sentence. 

S: 3 ÷ = 6. 

T: (Write 3 ÷ = 6. Beneath it, write 8 ÷ .) On your boards, complete the division sentence. 

S: (Write 8 ÷ = 16.) 

Continue the process with 5 ÷ , 7 ÷ , 1 ÷ , 2 ÷ , 7 ÷ , and 10 ÷ . 


S: ÷ 3 = . 

T: (Write ÷ 3 = . Beneath it, write ÷ 4.) Say the division sentence. 

S: ÷ 4 = . 

T: (Write ÷ 4 = . Beneath it, write ÷ 5.) On your boards, write the division sentence. 

S: (Write ÷ 5 = .) 


S: ÷ 5 = . 

Continue the process with 7 ÷ , ÷ 7, 5 ÷ , ÷ 5, ÷ 7, and 9 ÷ . 


1 hundredth. 

Date: 11/10/13 

4.G.60 






Fernando bought a jacket for $185 and sold it for 

times what 

he paid. Marisol spent as much as Fernando on the same 

jacket, but sold it for as much as Fernando sold it for. 

How much money did Marisol make? Explain your thinking 

using a diagram. 

Note: This problem is a multi-step problem requiring a high level 

of organization. Scaling language and fraction multiplication from 

G5–M4–Topic G coupled with fraction of a set and subtraction 

warrant the extra time given to today’s Application Problem. 



Problem 1: 7 0.1 

MP.2 

T: (Post Problem 1 on the board.) Read the division 

expression using unit form. 

S: 7 ones divided by 1 tenth. 

T: Rewrite this expression using a fraction. 


T: (Write = 7 .) What question does this division 

expression ask us? 

S: How many tenths are in 7? 7 is one tenth of what 

number? 

T: Let’s start with just whole. How many tenths are in 1 

whole? 

S: 10 tenths. 

T: (Write 10 in the blank, then below it, write, There are 

_____ tenths in 7 wholes.) So, if there are 10 tenths in 

1 whole, how many are in 7 wholes? 

S: 70 tenths. 

T: (Write 70 in the blank.) Explain how you know. Turn 

and talk. 

NOTES ON 


ACTION AND 

EXPRESSION: 

The same place value mats that were 

used in previous modules can be used 

in this lesson to support students who 

are struggling. Students can start 

Problem 1 by drawing or placing 7 disks 

in the ones column. Teachers can 

follow the same dialogue that is 

written in the lesson. Have the 

students physically decompose the 7 

wholes into 70 tenths, which can then 

be divided by one-tenth. 


1 hundredth. 

Date: 11/10/13 

4.G.61 





MP.2 

S: There are 10 tenths in 1, 20 tenths in 2, and 30 tenths in 3, so there are 70 tenths in 7. Seven is 7 

times greater than 1, and 70 tenths is 7 times more than 10 tenths. Seven times 10 is 70, so there 

are 70 tenths in 7. 

T: Let’s think about it another way. Seven is one-tenth of what number? Explain to your partner how 

you know. 

S: It’s 70, because I think of a tape diagram with 10 parts and 1 part is 7. 7 × 10 is 70. I think of 

place value. Just move each digit one place to left. It’s ten times as much. 

Problem 2: 7.4 0.1 

T: (Post Problem 2 on the board.) Rewrite this division expression using a fraction for the divisor. 

S: (Write 7.4 .) 

T: Compare this problem to the one we just solved. What do you notice? Turn and talk. 

S: There still are 7 wholes, but now there are also 4 more tenths. The whole in this problem is just 4 

tenths more than in problem 1. 

There are 74 tenths instead of 70 tenths. 

We can ask ourselves, 7.4 is 1 tenth 

of what number? 

T: We already know part of this problem. 

(Write, There are _____ tenths in 7 

wholes.) How many tenths are in 7 

wholes? 

S: 70. 

T: (Write 70 in the blank, and below it write, There are _____ tenths in 4 tenths.) How many tenths are 

in 4 tenths? 

S: 4. 

T: (Point to 7 ones.) So, if there are 70 tenths in 7 wholes, and (point to 4 tenths) 4 tenths in 4 tenths, 

how many tenths are in 7 and 4 tenths? 

S: 74. 

T: Work with your partner to rewrite this expression using only tenths to name the whole and divisor. 

S: (Write 74 tenths 1 tenth.) 

T: Look at our new expression. How many tenths are in 74 tenths? 

S: 74 tenths. 

T: (Write 6 0.1.) Read this expression. 

S: 6 divided by 1 tenth. 

T: How many tenths are in 6? Show me on your boards. 

S: (Write and show 60 tenths.) 

T: 6 is 1 tenth of what number? 

S: 60. 

T: (Erase 6 and replace with 6.2.) How many tenths in 6.2? 


1 hundredth. 

Date: 11/10/13 

4.G.62 





S: (Write 62 tenths.) 

T: 6.2 is 1 tenth of what number? 

S: 62. 

Continue the process with 9 and 9.8 and 12 and 12.6. 

Problem 3: a. 7 0.01 b. 7.4 0.01 c. 7.49 0.01 

T: (Post Problem 3(a) on the board.) Read this expression. 

S: 7 divided by 1 hundredth. 

T: Rewrite this division expression using a fraction for the 

divisor. 


T: We can think of this as finding how many hundredths are in 

7. Will your thinking need to change to solve this? Turn and talk. 

S: No, because the question is really the same. How many smaller units in the whole? The units we 

are counting are different, but that doesn’t really change how we find the answer. 

T: Will our quotient be greater or less than our last problem? Again, talk with your partner. 

S: The quotient will be greater because we are counting units that are much smaller, so there’ll be 

more of them in the wholes. Not too much. It’s the same basic idea but since our divisor has 

gotten smaller; the quotient should be larger than before. 

T: Before we think about how many hundredths are in 7 wholes, let’s find how many hundredths are in 

1 whole. (Write on the board: There are _____ hundredths in 1 whole.) Fill in the blank. 

S: 100. 

T: (Write 100 in the blank. Write, There are _____ 

hundredths in 7 wholes.) Knowing this, how many 

hundredths are in 7 wholes? 

S: 700. 

T: (Write 700 in the blank. Then, post Problem 3(b) on 

board.) What is the whole in this division expression? 

S: 7 and 4 tenths. 

T: How will you solve this problem? Turn and talk. 

S: It’s only more tenths than the one we just solved. 

We need to figure out how many hundredths are in 4 

tenths. We know there are 700 hundredths in 7 

wholes, and this is 4 tenths more than that. There are 

10 hundredths in 1 tenth, so there must be 40 

hundredths in 4 tenths. 

T: How many hundredths are in 7 wholes? 

S: 700. 

T: How many hundredths in 4 tenths? 

NOTES ON 



Generally speaking, it is better for 

teachers to use unit form when they 

read decimal numbers. For example, 

seven and four-tenths is generally 

preferable to seven point four. Seven 

point four is appropriate when 

teachers or students are trying to 

express what they need to write. 

Similarly, it is preferable to read 

fractions in unit form, too. For 

example, it’s better to say two-thirds, 

rather than two over three unless 

referring to how the fraction is written. 


1 hundredth. 

Date: 11/10/13 

4.G.63 





MP.2 

S: 40. 

T: How many hundredths in 7.4? 

S: 740. 

T: Asked another way, if 7.4 is 1 hundredth, what is the whole? 

S: 740. 

T: (Post Problem 4(c) on the board.) Work with a partner to solve this problem. Be prepared to explain 


S: (Work and show 7.49 0.1 = 749.) 

T: Explain your thinking as you solved. 

S: 7.49 is just 9 hundredths more in the dividend than 7.4 0.01, so the answer must be 749. There 

are 7 hundredths in 7, and 9 hundredths in 9 hundredths. That’s 7 9 hundredths all together. 

T: Let’s try some more. Think first... how many hundredths are in 6? Show me. 

S: (Show 600.) 

T: Show me how many hundredths are in 6.2? 

S: (Show 620.) 

T: 6.02? 

S: (Show 602.) 

T: 12.6? 

S: (Show 1,260.) 

T: 12.69? 

S: (Show 1,269.) 

T: What patterns are you noticing as we find the number of hundredths in each of these quantities? 

S: The digits stay the same, but they are in a larger place value in the quotient. I’m beginning to 

notice that when we divide by a hundredth each digit shifts two places to the left. It’s like 

multiplying by 100. 

T: That leads us right into thinking of our division expression differently. When we divide by a 

hundredth, we can think, “This number is hundredth of what whole?” or “What number is this 1 

hundredth of?” 

T: (Write 7 ÷ on the board.) What number is 7 one hundredths of? 

S: 700. 

T: Explain to your partner how you know. 

S: It’s like thinking 7 times because 7 is one of a hundred parts. It’s place value again but this 

time we move the decimal point two places to the right.) 

T: You can use that way of thinking about these expressions, too. 





1 hundredth. 

Date: 11/10/13 

4.G.64 








Lesson Objective: Connect division by a unit fraction to 

division by 1 tenth and 1 hundredth. 







addressed in the Debrief. Guide students in a conversation 

to debrief the Problem Set and process the lesson. 



• In Problem 1, did you notice the relationship 

between (a) and (c), (b) and (d), (e) and (g), (f) and 

(h)? 

• What is the relationship between Problems 2(a) 

and 2(b)? (The quotient of (b) is triple that of (a).) 

• What strategy did you use to solve Problem 3? 

Share your strategy and explain to a partner. 

• How did you answer Problem 4? Share your 

thinking with a partner. 

• Compare your answer for Problem 5 to your 

partner’s. 

• Connect the work of Module 1, the movement on 

the place value chart, to the division work of this 

lesson. Back then, the focus was on conversion 

between units. However, it’s important to note 

place value work asks the same question, “How 

many tenths are in whole?” “How many 

hundredths in a tenth?” Further, the partitive 

division interpretation leads naturally to a 

discussion of multiplication by powers of 10, that 

is, if 6 is 1 hundredth, what is the whole? (6 100 

= 600.) This echoes the work students have done 

on the place value chart. 


1 hundredth. 

Date: 11/10/13 

4.G.65 










1 hundredth. 

Date: 11/10/13 

4.G.66 





Name 

Date 

1. Divide. Rewrite each expression as a division sentence with a fraction divisor, and fill in the blanks. 

The first one is done for you. 

Example: 2 0.1 = 2 = 20 

There are 10 tenths in 1 whole. 

There are 20 tenths in 2 wholes. 

a. 5 0.1 = b. 8 0.1 = 

There are 

tenths in 1 whole. 

There are 


There are 

tenths in 5 wholes. 

There are 


c. 5.2 0.1 = 

d. 8.7 0.1 = 

There are 


There are 


There are 

tenths in 2 tenths. 

There are 


There are tenths in 5.2 

There are tenths in 8.7 

e. 5 0.01 = f. 8 0.01 = 

There are 

hundredths in 1 whole. 

There are 


There are 

hundredths in 5 wholes. 

There are 


g. 5.2 0.01 = h. 8.7 0.01 = 

There are 


There are 


There are 


There are 


There are hundredths in 5.2 

There are hundredths in 8.7 


1 hundredth. 

Date: 11/10/13 

4.G.67 





2. Divide. 

a. 6 ÷ 0.1 b. 18 ÷ 0.1 c. 6 ÷ 0.01 

d. 1.7 ÷ 0.1 e. 31 ÷ 0.01 f. 11 ÷ 0.01 

g. 125 ÷ 0.1 h. 3.74 ÷ 0.01 i. 12.5 ÷ 0.01 

3. Yung bought $4.60 worth of bubble gum. Each piece of gum cost $0.10. How many pieces of bubble gum 

did Yung buy? 

4. Cheryl solved a problem: 84 ÷ 0.01 = 8,400. 

Jane said, “Your answer is wrong because when you divide, the quotient is always smaller than the whole 

amount you start with, for example, 6 ÷ 2 = 3, and 100 ÷ = .” Who is correct? Explain your thinking. 

5. The US Mint sells 2 pounds of American Eagle gold coins to a collector. Each coin weighs one-tenth of an 

ounce. How many gold coins were sold to the collector? 


1 hundredth. 

Date: 11/10/13 

4.G.68 





Name 

Date 

1. 8.3 is equal to 2. 28 is equal to 

_______ tenths 

_______ hundredths 

_______ hundredths 

_______ tenths 

3. 15.09 ÷ 0.01 = _______ 4. 267.4 ÷ = _______ 

5. 632.98 ÷ = _______ 


1 hundredth. 

Date: 11/10/13 

4.G.69 





Name 

Date 

1. Divide. Rewrite each expression as a division sentence with a fraction divisor, and fill in the blanks. The 

first one is done for you. 

Example: 4 0.1 = 4 = 40 

There are 10 tenths in 1 whole. 

There are 40 tenths in 4 wholes. 

a. 9 0.1 = There are 40 b. tenths 6 0.1 in = 4 wholes. 

There are 


There are 


There are 


There are 


c. 3.6 0.1 = d. 12.8 0.1 = 

There are _____ tenths in 3 wholes. 

There are 


There are 


There are 


There are _______ tenths in 3.6. 

There are ______ tenths in 12.8. 

e. 3 0.01 = f. 7 0.01 = 

There are 


There are 


There are 


There are 


g. 4.7 0.01 = h. 11.3 0.01 = 

There are _____ hundredths in 4 wholes. 

There are _____ hundredths in 11 wholes. 

There are 


There are 


There are _______ hundredths in 4.7. 

There are _______ hundredths in 11.3. 


1 hundredth. 

Date: 11/10/13 

4.G.70 





2. Divide. 

a. 2 ÷ 0.1 b. 23 ÷ 0.1 c. 5 ÷ 0.01 

d. 7.2 ÷ 0.1 e. 51 ÷ 0.01 f. 31 ÷ 0.1 

g. 231 ÷ 0.1 h. 4.37 ÷ 0.01 i. 24.5 ÷ 0.01 

3. Giovanna is charged $0.01 for each text message she sends. Last month her cell phone bill included a 

$12.60 charge for text messages. How many text messages did Giovanna send? 

4. Geraldine solved a problem: 68.5 ÷ 0.01 = 6,850. 

Ralph said, “This is wrong because a quotient can’t be greater than the whole you start with. For 

example, 8 ÷ 2 = 4, and 250 ÷ = .” Who is correct? Explain your thinking. 

5. The price for an ounce of gold on September 23, 2013, was $1,326.40. A group of 10 friends decide to 

share the cost equally on 1 ounce of gold. How much money will each friend pay? 


1 hundredth. 

Date: 11/10/13 

4.G.71 





Lesson 30 

Objective: Divide decimal dividends by non‐unit decimal divisors. 






Total Time 

(12 minutes) 

(6 minutes) 

(32 minutes) 

(10 minutes) 

(60 minutes) 


• Sprint: Divide Whole Numbers by Fractions and Fractions by Whole Numbers 5.NBT.7 

• Divide Decimals 5.NBT.7 

(9 minutes) 

(3 minutes) 

Sprint: Divide Whole Numbers by Fractions and Fractions by Whole Numbers (9 minutes) 

Materials: (S) Divide Whole Numbers by Fractions and Fractions by Whole Numbers Sprint 


Divide Decimals (3 minutes) 



T: (Write 1 ÷ 0.1 = ____.) How many tenths are in 1? 

S: 10. 

T: 2? 

S: 20. 

T: 3? 

S: 30. 

T: 9? 

S: 90. 

T: (Write 10 ÷ 0.1 = ____.) On your boards, complete the equation, answering how many tenths are in 

10. 

S: (Write 10 ÷ 0.1 = 100.) 

Lesson 30: Divide decimal dividends by non-unit decimal divisors. 

Date: 11/10/13 4.G.72 





T: (Write 20 ÷ 0.1 = ____.) If there are 100 tenths in 10, how many tenths are in 20? 

S: 200. 

T: 30? 

S: 300. 

T: 70? 

S: 700. 

T: (Write 75 ÷ 0.1 = ____.) On your boards, complete the equation. 

S: (Write 75 ÷ 0.1 = 750.) 

T: (Write 75.3 ÷ 0.1 = ____.) Complete the equation. 

S: (Write 75.3 ÷ 0.1 = 753.) 

Continue this process with the following possible sequence: 0.63 ÷ 0.1, 6.3 ÷ 0.01, 63 ÷ 0.1, and 630 ÷ 0.01. 


Alexa claims that 16 4, , and 8 halves are all equivalent expressions. Is Alexa correct? Explain how you 

know. 

Note: This problem reminds students that when you multiply (or divide) both the divisor and the dividend by 

the same factor, the quotient stays the same or, alternatively, we can think of it as the fraction has the same 

value. This concept is critical to the Concept Development in this lesson. 


Date: 11/10/13 4.G.73 







Problem 1: a. 2 0.1 b. 2 0.2 c. 2.4 0.2 d. 2.4 0.4 

T: (Post Problem 1(a) on the board.) We did this yesterday. How many tenths are in 2? 

S: 20. 

T: (Write = 20.) Tell a partner how you know. 

S: I can count by tenths. 1 tenth, 2 tenths, 3 tenths,… all the way up to 20 tenths, which is 2 wholes. 

There are 10 tenths in 1 so there are 20 tenths in 2. Dividing by 1 tenth is the same as 

multiplying by 10, and 2 times 10 is 20. 

T: We also know that any division expression can be 

rewritten as a fraction. Rewrite this expression as a 

fraction. 

S: (Show .) 

T: That fraction looks different from most we’ve seen 

before. What’s different about it? 

S: The denominator has a decimal point; that’s weird. 

T: It is different, but it’s a perfectly acceptable fraction. 

We can rename this fraction so that the denominator is 

a whole number. What have we learned that allows us 

to rename fractions without changing their value? 

S: We can multiply by a fraction equal to 1. 

T: What fraction equal to 1 will rename the denominator 

as a whole number? Turn and talk. 

NOTES ON 



The presence of decimals in the 

denominators in this lesson may pique 

the interest of students performing 

above grade level. These students can 

be encouraged to investigate and 

operate with complex fractions 

(fractions whose numerator, 

denominator, or both contain a 

fraction). 

S: Multiplying by is easy, but that would just make the denominator 0.2. That’s not a whole number. 

I think it is fun to multiply by 

but then we’ll still have 1.3 as the denominator. I’ll multiply 

by . That way I’ll be able to keep the digits the same. If we just want a whole number, would 

work. Any fraction with a numerator and denominator that are multiples of 10 would work, really. 

T: I overheard lots of suggestions for ways to rename this denominator as a whole number. I’d like you 

to try some of your suggestions. Be prepared to share your results about what worked and what 

didn’t. (Allow students time to work and experiment.) 

S: (Work and experiment.) 

T: Let’s share some of the equivalent fractions we’ve created. 

S: (Share while teacher records on board. Possible examples include and .) 

T: Show me these fractions written as division expressions with the quotient. 

S: (Work and show 20 1 = 20, 40 2 = 20, 100 5 = 20, etc.) 


Date: 11/10/13 4.G.74 





T: What do you notice about all of these division sentences? 

S: The quotients are all 20. 

T: Since all of the quotients are equal to each other, can we say then that 

these expressions are equivalent as well? (Write 2 0.1 = 20 1 = 40 2, 

etc.) 

S: Since the answer to them is all the same, then yes, they are equivalent 

expressions. It reminds me of equal fractions, the way they don’t look 

alike but are equal. 

T: These are all equivalent expressions. When we multiply by a fraction 

equal to 1, we create equal fractions and an equivalent division 

expression. 

T: (Post Problem 1(b), 2 0.2, on the board.) Let’s use this thinking as we 

find the value of this expression. Turn and talk about what you think the 

quotient will be. 

S: I can count by 2 tenths. 2 tenths, 4 tenths, 6 tenths,… 20 tenths. That was 

10. The quotient must be 10. Two is like 2.0 or 20 tenths. 20 tenths divided by 2 tenths is going 

to be 10. The divisor in this problem is twice as large as the one we just did so the quotient will 

be half as big. Half of 20 is 10. 

T: Let’s see if our thinking is correct. Rewrite this division expression as a fraction. 

S: (Work and show .) 

T: What do you notice about the denominator? 

S: It’s not a whole number. It’s a decimal. 

T: How will you find an equal fraction with a whole 

number divisor? Share your ideas. 

S: We have to multiply it by a fraction equal to 1. I 

think multiplying by would work. That will make the 

divisor exactly 1. 

make 

would work again. That would 

. This time any numerator and denominator 

that is a multiple of 5 would work. 

T: I heard the fraction 10 tenths being mentioned during 

both discussions. What if our divisor were 0.3? If we 

multiplied by 

be? 

S: 3. 

T: What if the divisor were 0.8? 

S: 8. 

T: What about 1.2? 

S: 12. 

, what would the new denominator 

NOTES ON 


ACTION AND 

EXPRESSION: 

Place value mats can be used here to 

support struggling learners. The same 

concepts that students studied in G5– 

Module 1 apply here. By writing the 

divisor and dividend on a place value 

mat, students can see that 2 ones 

divided by 2 tenths is equal to 10 since 

the digit 2 in the ones place is 10 times 

greater than a 2 in the tenths place. 


Date: 11/10/13 4.G.75 





T: What do you notice about the decimal point and digits when we use tenths to rename? 

S: The digits stay the same, but the decimal point moves to the right. The decimal just moves, so 

that the numerator and the denominator are 10 times as much. 

T: Multiply the fraction by 10 tenths. 

S: (Show .) 

T: What division expression does our renamed fraction represent? 


T: What’s the quotient? 

S: 10. 

T: Let’s be sure. To check our division’s answer (write = 10), we multiply the quotient by the…? 

S: Divisor. 

T: Show me. 

S: (Show 10 0.2 = 2 or 10 2 tenths = 20 tenths.) 

T: (Post Problem 1(c), 2.4 0.2, on the board.) Share your thoughts about what the quotient might be 

for this expression. 

S: I think it is 12. I counted by 2 tenths again and got 12. 2.4 is only 4 tenths more than the last 

problem, and there are two groups of 2 tenths in 4 tenths so that makes 12 altogether. I’m 

thinking 24 tenths divided by 2 tenths is going to be 12. I’m starting to think of it like whole 

number division. It almost looks like 24 divided by 2, which is 12. 

T: Rewrite this division expression as a fraction. 

S: (Write and show .) 

T: This time we have a decimal in both the divisor and the whole. Remind me. What will you do to 

rename the divisor as a whole number? 

S: Multiply by . 

T: What will happen to the numerator when you multiply by ? 

S: It will be renamed as a whole number too. 

T: Show me. 


T: Say the fraction as a division expression with the quotient. 

S: 24 divided by 2 equals 12. 

T: Check your work. 

S: (Check work.) 

T: (Post Problem 1(d) on the board.) Work this one independently. 



Date: 11/10/13 4.G.76 





Problem 2: a. 1.6 0.04 b. 1.68 0.04 c. 1.68 0.12 

T: (Post Problem 2(a) on the board.) Rewrite this expression as a fraction. 

S: (Write .) 

T: How is this expression different from the ones we just evaluated? 

S: This one is dividing by a hundredth. Our divisor is 4 hundredths, rather than 4 tenths. 

T: Our divisor is still not a whole number, and now it’s a hundredth. Will multiplying by 10 tenths 

create a whole number divisor? 

S: No, 4 hundredths times 10 is just 4 tenths. That’s still not a whole number. 

T: Since our divisor is now a hundredth, the most efficient way to rename it as a whole number is to 

multiply by 100 hundredths. Multiply and show me the equivalent fraction. 

S: (Show .) 

T: Say the division expression. 


T: This expression is equivalent to 1.6 divided by 0.04. What is the quotient? 

S: 40. 

T: So, 1.6 divided by 0.04 also equals…? 

S: 40. 

T: Show me the multiplication sentence you can use to check. 

S: (Show 40 0.04 = 1.6, or 40 4 hundredths = 160 hundredths.) 

T: (Post Problem 1(b) on the board.) Work with 

your partner to solve and check. 

S: (Work.) 

T: (Post Problem 1(c) on the board.) Work 

independently to find the quotient. Check your 

work with a partner after each step. 











Date: 11/10/13 4.G.77 






Lesson Objective: Divide decimal dividends by non‐unit 

decimal divisors. 









lesson. 



• In Problem 1, what did you notice about the 

relationship between (a) and (b), (c) and (d), (e) 

and (f), (g) and (h), (i) and (j), and (k) and (l)? 

• Share your explanation of Problem 2 with a 

partner. 

• In Problem 3, what is the connection between (a) 

and (b)? How did you solve (b)? Did you solve it 

mentally or by re-calculating everything? 

• Share and compare your solution for Problem 4 


• How did you solve Problem 5? Did you use 

drawings to help you solve the problem? Share 

and compare your strategy with a partner. 

• Use today’s understanding to help you find the 

quotient of 0.08 0.4. 







students. 


Date: 11/10/13 4.G.78 






Date: 11/10/13 4.G.79 






Date: 11/10/13 4.G.80 





Name 

Date 

1. Rewrite the division expression as a fraction, and divide. The first two have been started for you. 

a. 2.7 ÷ 0.3 = 

= 

= 

= 9 

b. 2.7 ÷ 0.03 = 

= 

= 

= 

c. 3.5 0.5 = d. 3.5 0.05 = 

e. 4.2 ÷ 0.7 = f. 0.42 0.07 = 


Date: 11/10/13 4.G.81 





g. 10.8 0.9 = h. 1.08 0.09 = 

i. 3.6 1.2 = j. 0.36 0.12 = 

k. 17.5 2.5 = l. 1.75 0.25 = 

2. 15 3 = 5. Explain why it is true that 1.5 ÷ 0.3 and 0.15 ÷ 0.03 have the same quotient. 


Date: 11/10/13 4.G.82 





3. Mr. Volok buys 2.4 kg of sugar for his bakery. 

a. If he pours 0.2 kg of sugar into separate bags, how many bags of sugar can he make? 

b. If he pours 0.4 kg of sugar into separate bags, how many bags of sugar can he make? 

4. Two wires, one 17.4 meters long and one 7.5 meters long, were cut into pieces 0.3 meters long. How 

many such pieces can be made from both wires? 

5. Mr. Smith has 15.6 pounds of oranges to pack for shipment. He can ship 2.4 lb of oranges in a large box 

and 1.2 lb in a small box. If he ships 5 large boxes, what is the minimum number of small boxes required 

to ship the rest of the oranges? 


Date: 11/10/13 4.G.83 





Name 

Date 

Rewrite the division expression as a fraction, and divide. 

a. 3.2 ÷ 0.8 = b. 3.2 ÷ 0.08 = 

c. 7.2 0.9 = d. 0.72 0.09 = 


Date: 11/10/13 4.G.84 





Name 

Date 

1. Rewrite the division expression as a fraction, and divide. The first two have been started for you. 

a. 2.4 ÷ 0.8 = 

= 

= 

= 

b. 2.4 ÷ 0.08 = 

= 

= 

= 

c. 4.8 ÷ 0.6 = d. 0.48 0.06 = 

e. 8.4 0.7 = f. 0.84 0.07 = 

g. 4.5 1.5 = h. 0.45 0.15 = 


Date: 11/10/13 4.G.85 





i. 14.4 1.2 = j. 1.44 0.12 = 

2. Leann says 18 6 = 3, so 1.8 ÷ 0.6 = 0.3 and 0.18 ÷ 0.06 = 0.03. Is Leann correct? How would you explain 

how to solve these division problems? 

3. Denise is making bean bags. She has 6.4 pounds of beans. 

a. If she makes each bean bag 0.8 pounds, how many bean bags will she be able to make? 

b. If she decides instead to make mini bean bags that are half as heavy, how many can she make? 

4. A restaurant’s small salt shakers contain 0.6 ounces of salt. Its large shakers hold twice as much. The 

shakers are filled from a container that has 18.6 ounces of salt. If 8 large shakers are filled, how many 

small shakers can be filled with the remaining salt? 


Date: 11/10/13 4.G.86 





Lesson 31 

Objective: Divide decimal dividends by non‐unit decimal divisors. 






Total Time 

(12 minutes) 

(6 minutes) 

(32 minutes) 

(10 minutes) 

(60 minutes) 


• Multiply Decimals by 10 and 100 5.NBT.1 

• Divide Decimals by 1 Tenth and 1 Hundredth 5.NBT.7 


(4 minutes) 

(3 minutes) 

(5 minutes) 

Multiply Decimals by 10 and 100 (4 minutes) 



T: (Write 3 × 10 = ____.) Say the multiplication sentence. 

S: 3 × 10 = 30. 

T: (Write 3 × 10 = 30. Beneath it, write 20 × 10 = ____.) Say the multiplication sentence. 

S: 20 × 10 = 200. 

T: (Write 20 × 10 = 200. Beneath it, write 23 × 10 = ____.) Say the multiplication sentence. 

S: 23 × 10 = 230. 

T: (Write 2.3 × 10 = ____. Point to 2.3.) How many tenths is 2 and 3 tenths? 

S: 23 tenths. 

T: On your boards, write the multiplication sentence. 

S: (Write 2.3 × 10 = 23.) 

T: (Write 2.34 × 100 = ____. Point to 2.34.) How many hundredths is 2 and 34 hundredths? 



S: (Write 2.34 × 100 = 234.) 


Date: 11/9/13 4.G.87 





T: (Write 23.4 × 10 = ____. Point to 23.4.) How many tenths is 23 and 4 tenths? 

S: 234 tenths. 


S: (Write 23.4 × 10 = 234.) 

Continue this process with the following possible suggestions: 47.3 × 10, 4.73 × 100, 8.2 × 10, 38.2 × 10, and 

6.17 × 100. 

Divide Decimals by 1 Tenth and 1 Hundredth (3 minutes) 



T: (Write 1 ÷ 0.1 = ____.) How many tenths are in 1? 

S: 10. 

T: 2? 

S: 20. 

T: 3? 

S: 30. 

T: 7? 

S: 70. 

T: (Write 10 ÷ 0.1.) On your boards, write the complete number sentence, answering how many tenths 

are in 10. 

S: (Write 10 ÷ 0.1 = 100.) 

T: (Write 20 ÷ 0.1.) If there are 100 tenths in 10, how many tenths are in 20? 

S: 200. 

T: 30? 

S: 300. 

T: 90? 

S: 900. 

T: (Write 65 ÷ 0.1.) On your boards, write the complete number sentence. 

S: (Write 65 ÷ 0.1 = 650.) 

T: (Write 65.2 ÷ 0.1.) Write the complete number sentence. 

S: (Write 65.2 ÷ 0.1 = 652.) 

T: (Write 0.08 ÷ 0.1 = ÷ .) On your boards, complete the division sentence. 

S: (Write 0.08 ÷ 0.1 = ÷ .) 



Date: 11/9/13 4.G.88 








T: (Write 15 ÷ 5 = ____.) Say the division sentence. 

S: 15 ÷ 5 = 3. 

T: (Write 15 ÷ 5 = 3. Beneath it, write 1.5 ÷ 0.5 = ____.) Say the division 

sentence in tenths. 

S: 15 tenths ÷ 5 tenths. 

T: Write 15 tenths ÷ 5 tenths as a fraction. 

S: (Write .) 

1.5 ÷ 0.5 = 3 

= 

= 

= 3 

T: (Beneath 1.5 ÷ 0.5, write .) On your boards, rewrite the fraction using whole numbers. 

S: (Write . Beneath it, write .) 

T: (Beneath , write . Beneath it, write = ____. ) Fill in your answer. 

S: (Write = 3.) 

Continue this process with the following possible suggestions: 1.5 ÷ 0.05, 0.12 ÷ 0.3, 1.04 ÷ 4, 4.8 ÷ 1.2, and 

0.48 ÷ 1.2. 


A café makes ten 8-ounce fruit smoothies. Each 

smoothie is made with 4 ounces of soy milk and 

1.3 ounces of banana flavoring. The rest is 

blueberry juice. How much of each ingredient 

will be necessary to make the smoothies? 

Note: This two-step problem requires decimal 

subtraction and multiplication, reviewing concepts from G5–Module 1. Some students will be comfortable 

performing these calculations mentally while others may need to sketch a quick visual model. Developing 

versatility with decimals by reviewing strategies for multiplying decimals serves as a quick warm-up for 

today’s lesson. 


Date: 11/9/13 4.G.89 







Problem 1: a. 34.8 0.6 b. 7.36 0.08 

T: (Post Problem 1 on the board.) Rewrite this 

division expression as a fraction. 


T: (Write = .) How can we express the 

divisor as a whole number? 

S: Multiply by a fraction equal to 1. 

T: Tell a neighbor which fraction equal to 1 

you’ll use. 

S: I could multiply by 5 fifths, which would make the divisor 3, but I’m not sure I want to multiply 34.8 

by 5. That’s not as easy. If we multiply by 10 tenths, that would make both the numerator and 

the denominator whole numbers. There are lots of choices. If I use 10 tenths, the digits will all 

stay the same—they will just move to a larger place value. 

T: As always, we have many fractions equal to 1 that 

would create a whole number divisor. Which fraction 

would be most efficient? 

S: 10 tenths. 

T: (Write .) Multiply, then show me the equivalent 

fraction. 


T: (Write = .) This isn’t mental math like the basic 

facts we saw yesterday, so before we divide, let’s 

estimate to give us an idea of a reasonable quotient. 

Think of a multiple of 6 that is close to 348 and divide. 

(Write _____ 6.) Turn and share your ideas with a 

partner. 

S: I can round 348 to 360. I can use mental math to 

divide 360 by 6 = 60. 

T: (Fill in the blank to get 360 6 = 60.) Now, use the 

division algorithm to find the actual quotient. 

S: (Work.) 

T: What is 34.8 0.6? How many 6 tenths are in 34.8? 

S: 58. 

NOTES ON 


ENGAGEMENT: 

Some students may require a refresher 

on the process of long division. This 

example dialogue might help: 

T: Can we divide 3 hundreds by 6, or 

must we decompose? 

S: We need to decompose. 

T: Let’s work with 34 tens then. What 

is 34 tens divided by 6? 

S: 5 tens. 

T: What is 5 tens times 6? 

S: 30 tens. 

T: How many tens remain? 

S: 4 tens. 

T: Can we divide 4 tens by 6? 

S: Not without decomposing. 

T: 4 tens is equal to 40 ones, plus the 8 

ones in our whole makes 48 ones. 

What is 48 ones divided by 6? 

S: 8 ones. 


Date: 11/9/13 4.G.90 





MP.7 

T: Is our quotient reasonable? 

S: Yes, our estimate was 60. 

T: (Post Problem 1(b), 7.36 0.08, on the board.) Work with a partner to find the quotient. Remember 

to rename your fraction so that the denominator is a whole number. 


T: What is 7.36 0.08? How many 8 

hundredths are in 7.36? 

S: 92. 

T: Is the quotient reasonable considering your 

estimate? 

S: Yes, our estimate was 100. We got an 

estimate of 90, so 92 is reasonable. 

Problem 2: a. 21.56 0.98 b. 45.5 0.7 c. 4.55 0.7 

T: (Post Problem 2(a) on the board.) Rewrite this division expression as a fraction. 


T: We know that before we divide, we’ll want to rename the divisor as a whole number. Remind me 

how we’ll do that. 

S: Multiply the fraction by . 

T: Then, what would the fraction show after multiplying? 

S: . 

T: In this case, both the divisor and the whole become 

100 times greater. When we write the number that is 

100 times as much, we must write the decimal two 

places to the…? 

S: Right. 

T: Rather than writing the multiplication sentence to show this, I’m going to record that thinking using 

arrows. (Draw a thought bubble around the fraction and use arrows to show the change in value of 

the divisor and whole.) 

T: Is this fraction equivalent to the one we started with? Turn and talk. 

S: It looks a little different, but it shows the fraction we got when we multiplied by 100 hundredths. 

It’s equal. Both the divisor and whole were multiplied by the same amount, so the two fractions 

are still equal. 

T: Because it is an equal fraction, the division will give us the same quotient as dividing 21.56 by 0.98. 

Estimate 98. 

S: 100. 


Date: 11/9/13 4.G.91 





T: (Write _____ 100.) Now estimate the whole, 2,156, 

as a number that we can easily divide by 100. Turn and 

talk. 

S: 100 times 22 is 2,200. 2,156 is between 21 

hundreds and 22 hundreds. It’s closer to 22 hundreds. 

I’ll round to 2,200. 

T: Record your estimated quotient, and then work with a 

partner to divide. 


T: Say the quotient. 

S: 22. 

T: Is that reasonable? 

S: Yes. 

T: (Post Problem 2(b), 45.5 0.7, on the board.) Rewrite 

this expression as a fraction and show a thought 

bubble as you rename the divisor as a whole number. 


T: Work independently to estimate, and then find the 

quotient. Check your work with a neighbor as you go. 


NOTES ON 


ACTION AND 

EXPRESSION: 

Unit form is a powerful means of 

representing these dividends so that 

students can more easily see the 

multiples of the rounded divisor. 

Expressing 2,156 as 21 hundreds + 56 

may allow students to estimate more 

accurately. 

Similarly, students should be using 

easily identifiable multiples to find an 

estimated quotient. Remind students 

about the relationship between 

multiplication and division so they can 

think of the following division 

sentences as multiplication equations: 

2,200 ÷ 100 = → 100 = 2,200 

490 ÷ 7 = → 7 = 490 

T: (Check student work and discuss reasonableness of quotient. Post Problem 2(c), 4.55 0.7, on the 

board.) Use a thought bubble to show this expression as a fraction with a whole number divisor. 


T: How is this problem similar to and different from the previous one? Turn and talk. 

S: The digits are all the same, but the whole is smaller this time. The whole still has a decimal point 

in it. The whole is 1 tenth the size of the previous whole. 

T: We still have a divisor of 7, but this time our whole is 45 and 5 tenths. Is the whole more than or 

less than it was in the previous problem? 

S: Less than. 

T: So, will the quotient be more than 65 or less than 65? Turn and talk. 

S: Our whole is smaller, so we can make fewer groups of 7 from it. The quotient will be less than 65. 

The whole is 1 tenth as large, so the quotient will be too. 

T: Divide. 

S: (Work.) 

T: What is the quotient? 

S: 6 and 5 tenths. 

T: Does that make sense? 

S: Yes. 


Date: 11/9/13 4.G.92 














Lesson Objective: Divide decimal dividends by non‐unit 

decimal divisors. 









lesson. 



• Look at the example in Problem 1. What is 

another way to estimate the quotient? (Students 

could say 78 divided by 1 is equal to 78.) 

Compare the two estimated sentences, 770 ÷ 7 = 

110 and 78 ÷ 7 = 78. Why is the actual quotient 

equal to 112? Does it make sense? 

• In Problems 1(a) and 1(b), is your actual quotient 

close to your estimated quotients? 

• In Problems 2(a) and 2(b), is your actual quotient 

close to your estimated quotients? 

• How did you solve Problem 4? Share and explain 

your strategy to a partner. 

• How did you solve Problem 5? Did you draw a 

tape diagram to help you solve? Share and 

compare your strategy with a partner. 


Date: 11/9/13 4.G.93 










Date: 11/9/13 4.G.94 





Name 

Date 

1. Estimate, then divide. An example has been done for you. 

78.4 0.7 770 7 110 

= 

= 

= 

= 112 

1 1 2 

7 7 8 4 

–7 

8 

–7 

1 4 

–1 4 

0 

a. 53.2 0.4 = b. 1.52 0.8 = 

2. Estimate, then divide. The first one has been done for you. 

7.32 0.06 = 

= 

= 

= 122 

720 6 120 

1 2 2 

6 7 3 2 

–6 

1 3 

–1 2 

1 2 

–1 2 

0 

a. 9.42 0.03 = b. 39.36 0.96 = 


Date: 11/9/13 4.G.95 





3. Solve using the standard algorithm. Use the thought bubble to show your thinking as you rename the 

divisor as a whole number. 

a. 46.2 0.3 = ______ 

b. 3.16 0.04 = ______ 

3 4 6 2 

= = 154 

c. 2.31 0.3 = ______ d. 15.6 0.24 = 

4. The total distance of a race is 18.9 km. 

a. If volunteers set up a water station every 0.7 km, including one at the finish line, how many stations 

will they have? 

b. If volunteers set up a first aid station every 0.9 km, including one at the finish line, how many stations 

will they have? 

5. In a laboratory, a technician combines a salt solution contained in 27 test tubes. Each test tube contains 

0.06 liter of the solution. If he divides the total amount into test tubes that hold 0.3 liter each, how many 

test tubes will he need? 


Date: 11/9/13 4.G.96 





Name 

Date 

Estimate first, and then solve using the standard algorithm. Show how you rename the divisor as a whole 

number. 

1. 6.39 0.09 

2. 82.14 0.6 


Date: 11/9/13 4.G.97 





Name 

Date 


78.4 0.7 

770 7 110 

= 

= 

= 

= 112 

1 1 2 

7 7 8 4 

–7 

8 

–7 

1 4 

–1 4 

0 

a. 61.6 0.8 = b. 5.74 0.7 = 


7.32 0.06 = 

= 

= 

= 122 

720 6 120 

1 2 2 

6 7 3 2 

–6 

1 3 

–1 2 

1 2 

–1 2 

0 

a. 4.74 0.06 = b. 19.44 0.54 = 


Date: 11/9/13 4.G.98 





3. Solve using the standard algorithm. Use the thought bubble to show your thinking as you rename the 

divisor as a whole number. 

a. 38.4 0.6 = ______ 

b. 7.52 0.08 = ______ 

6 

= = 

c. 12.45 0.5 = ______ d. 5.6 0.16 = 

4. Lucia is making a 21.6 centimeter beaded string to hang in the window. She decides to put a green bead 

every 0.4 centimeters and a purple bead every 0.6 centimeters. How many green beads and how many 

purple beads will she need? 

5. A group of 14 friends collects 0.7 pound of blueberries and decides to make blueberry muffins. They put 

0.05 pound of berries in each muffin. How many muffins can they make if they use all the blueberries 

they collected? 


Date: 11/9/13 4.G.99 






G R A D E 


Topic H 

Interpretation of Numerical 

Expressions 

5.OA.1, 5.OA.2 




5.OA.2 

Write simple expressions that record calculations with numbers, and interpret 

numerical expressions without evaluating them. For example, express the calculation 

“add 8 and 7, then multiply by 2” as 2 × (8 +7). Recognize that 3 × (18932 + 921) is three 

times as large as 18932 + 921, without having to calculate the indicated sum or product. 


G5–M2 



G6–M4 

Expressions and Operations. 

The module concludes with Topic H, in which numerical expressions involving fraction-by-fraction 

multiplication are interpreted and evaluated (5.OA.1, 5.OA.2). Students create and solve word problems 

involving both multiplication and division of fractions and decimal fractions. 

A Teaching Sequence Towards Mastery of Interpretation of Numerical Expressions 

Objective 1: Interpret and evaluate numerical expressions including the language of scaling and fraction 

division. 

(Lesson 32) 

Objective 2: Create story contexts for numerical expressions and tape diagrams, and solve word 

problems. 

(Lesson 33) 

Topic H: Interpretation of Numerical Expressions 

Date: 11/10/13 4.H.1 





Lesson 32 

Objective: Interpret and evaluate numerical expressions including the 







Total Time 

(12 minutes) 

(6 minutes) 

(32 minutes) 

(10 minutes) 

(60 minutes) 


• Order of Operations 5.OA.1 

• Divide Decimals by 1 Tenth and 1 Hundredth 5.NBT.7 


(3 minutes) 

(3 minutes) 

(6 minutes) 

Order of Operations (3 minutes) 



T: (Write 12 ÷ 3 + 1.) On your boards, write the complete number sentence. 

S: (Write 12 ÷ 3 + 1 = 5.) 

T: (Write 12 ÷ (3 + 1).) On your boards, copy the expression. 

S: (Write 12 ÷ (3 + 1).) 

T: Write the complete number sentence, performing the operation inside the parentheses. 

S: (Beneath 12 ÷ (3 + 1) = ____, write 12 ÷ 4 = 3.) 

Continue this process with the following possible suggestions: 20 – 4 ÷ 2, (20 – 4) ÷ 2, 24 ÷ 6 – 2, and 24 ÷ (6 – 2). 

Divide Decimals by 1 Tenth and 1 Hundredth (3 minutes) 



T: (Write 1 ÷ 0.1.) How many tenths are in 1? 

Lesson 32: Interpret and evaluate numerical expressions including the 


Date: 11/10/13 

4.H.2 





S: 10. 

T: 2? 

S: 20. 

T: 3? 

S: 30. 

T: 6? 

S: 60. 

T: (Write 10 ÷ 0.1.) On your boards, write the complete number sentence answering how many tenths 

are in 10. 

S: (Write 10 ÷ 0.1 = 100.) 

T: (Write 20 ÷ 0.1 = ____.) If there are 100 tenths in 10, how many tenths are in 20? 

S: 200. 

T: 30? 

S: 300. 

T: 80? 

S: 800. 

T: (Write 43 ÷ 0.1.) On your boards, write the complete number sentence. 

S: (Write 43 ÷ 0.1 = 430.) 

T: (Write 43.5 ÷ 0.1= ____.) Write the complete number sentence. 

S: (Write 43.5 ÷ 0.1 = 435.) 

T: (Write 0.04 ÷ 0.1 = ÷ .) On your boards, complete the division sentence. 

S: (Write 0.04 ÷ 0.1 = ÷ .) 





T: (Write = ____.) Say the division sentence. 

S: 8 ÷ 2 = 4. 

T: (Write = 4. Beneath it, write = ____.) What is 8 tenths ÷ 2 tenths? 

S: 4. 

T: On your boards, complete the division sentence. 

S: (Write 0.8 ÷ 0.02 = 40.) 

Continue this process with 12 ÷ 3, 1.2 ÷ 0.3, 1.2 ÷ 0.03, and 12 ÷ 0.3. 



Date: 11/10/13 

4.H.3 





T: (Write 23.84 ÷ 0.2 = ____.) On your boards, write the division sentence as a fraction. 

S: (Write 

T: (Beneath 23.8 ÷ 0.2 = ____, write = .) What do we need to multiply the divisor by to make it a 

whole number? 

S: 10. 

T: Multiply your numerator and denominator by 10. 

S: (Write .) 

T: (Write . Use the standard algorithm to solve, then write your answer. 

S: (Write 23.8 ÷ 0.2 = 119.2 after solving.) 

Continue this process with 5.76 ÷ 0.4, 9.54 ÷ 0.03, and 98.4 ÷ 0.12. 


Four baby socks can be made from skein of yarn. How 

many baby socks can be made from a whole skein? Draw 

a number line to show your thinking. 

Note: This problem is a partitive fraction division problem 

intended to give students more experience with this 

interpretation of division. 



Problem 1 

Write word form expressions numerically. 

a. Twice the sum of and . 

b. Half the sum of and . 

c. The difference between and divided by 3. 

NOTES ON 



The complexity of language in the word 

form expressions may pose challenges 

for English language learners. Consider 

drawing tape diagrams for each 

expression. Then, have students match 

each expression to its tape diagram 

before writing the numerical 

expression. 

T: (Post Problem 1(a) on the board.) Read the expression aloud with me. 

S: (Read.) 

T: What do you notice about this expression that would help us translate these words into symbols? 


S: It says twice; that means we have to make two copies of something or multiply by 2. It says the 



Date: 11/10/13 

4.H.4 





sum of, so something is being added. The part that says sum of needs to be added before the 

multiplying, so we’ll need parentheses. 

T: Let’s take a look at the first part of the expression. Show me on your personal boards how can we 

write twice numerically? 

S: (Write . 

T: (Write 2 _____ on the board.) Great, what is being multiplied by 2? 

S: The sum of 3 fifths and 1 and one-half. 

T: Show me the sum of 3 fifths and 1 and one-half numerically. 

S: (Write .) 

T: (Write in the blank.) Are we finished? 


S: (Share.) 

T: What else is needed? 

S: Parentheses. 

T: Why? Turn and talk. 

S: You have to have parentheses around the addition expression so that the addition is done before 

the multiplication. If you don’t, the answer will be 

right and multiply first. 

T: (Draw parentheses around the addition expression.) 

Work with a partner to find another way to write this 

expression numerically. 

S: (Work and show ( ) 2.) 

T: (Post Problem 1(b), half the sum of and 1 on the 

board.) Read this expression out loud with me. 

S: (Read.) 

T: Compare this expression to the other one. Turn and 

talk. 

S: (Share.) 

T: Without evaluating this expression, will the value of 

this expression be more than or less than the previous 

one? Turn and talk. 

. Because otherwise, you just go left to 

NOTES ON 



Due to the commutativity of addition 

and multiplication, there are multiple 

ways of writing expressions involving 

these operations. Have students be 

creative and experiment with their 

expressions. Be cautious, though, that 

students do not overgeneralize this 

property and try to apply it to 

subtraction and division. 

S: It’s less. The other one multiplied the sum by 2, and this one is one-half of the same sum. Taking 

half of a number is less than doubling it. 

T: This expression again includes the sum of 3 fifths and 1 and one-half. (Write on the board.) 

We need to show one-half of it. Tell a neighbor how we can show one-half of this expression. 

S: We can multiply it by . We can show times the addition expression, or we can show the 

addition expression times 

Taking a half of something is the same as dividing it by 2, so we could 



Date: 11/10/13 

4.H.5 





divide the addition expression by 2. We could divide the addition expression by 2, but write it like 

a fraction with 2 as the denominator. 

T: Work with a partner to write this expression numerically in at least two different ways. 


T: (Select students to share their work and explain their thought process.) 

Possible student responses: 

T: (Post Problem 1(c), the difference between and divided by 3, on board.) Read this expression 

aloud with me. 

S: (Read.) 

T: Talk with a neighbor about what is happening in this expression. 

S: Difference means subtract, so one-fourth is being subtracted from 10. The last part says divided 

by 3, so we can put parentheses around the subtraction expression and then write 3. Since it 

says divided by 3, we can write that like a fraction with 3 as the denominator. Dividing by 3 is the 

same as taking a third of something, so we can multiply the subtraction expression by . 

T: Work independently to write this expression numerically. Share your work with a neighbor when 

you’re finished. 


Possible student responses: 

T: Look at your numerical expression. Let’s evaluate it. Let’s put this expression in its simplest form. 

What is the first step? 

S: Subtract one-half from 3 fourths. 

T: Why must we do that first? 

S: Because it is in parentheses. Because we need to evaluate the numerator before dividing by 3. 

T: Work with a partner, then show me the difference between 3 fourths and one-half. 


T: Since we wrote our numerical expressions in different ways, tell your partner what your next step 

will be in evaluating your expression. 

S: I need to multiply one-fourth times one-third. I need to divide one-fourth by 3. 

T: Complete the next step and then share your work with a partner. 



Date: 11/10/13 

4.H.6 






T: What is this expression equal to in simplest form? 

S: 1 twelfth. 

T: Everyone got 1 twelfth? 

S: Yes! 

T: What does that show us about the different ways to write these expressions? Turn and talk. 

S: (Share.) 

Problem 2 

Write numerical expressions in word form. 

a. ( 0.4) b. ( + 1.25) c. (2 ) 0.3 

T: (Post Problem 2(a) on the board.) Now we’ll rewrite a numerical expression in word form. What is 

happening in this expression? Turn and talk. 

S: In parentheses, there’s the difference between one-half and 4 tenths. The subtraction expression 

is being multiplied by 2 fifths. 

T: Show me a word form expression for the operation outside the parentheses. 

S: (Show times.) 

T: (Write ______ on the board.) We have 2 fifths times what? 

S: The difference between one-half and 4 tenths. 

T: Exactly! (Write the difference between and 0.4 in the blank, then post Problem 2(b) on the board.) 

Work with a partner to write this expression using words. 

S: (Work and show, the sum of and 1.25 divided by .) 

T: Let’s evaluate the numerical expression. What must we do first? 

S: Add and 1.25. 

T: Work with a partner to find the sum in its simplest form. 

S: (Work and show 2.) 

T: What’s the next step? 

S: Divide 2 by one-third. 

T: How many thirds are in 1 whole? 

S: 3. 

T: How many are in 2 wholes? 

S: 6. 

T: (Post Problem 2(c) on the board.) Work independently to rewrite this expression using words. If you 

finish early, evaluate the expression. Check your work with a partner when you’re both ready. 




Date: 11/10/13 

4.H.7 














Lesson Objective: Interpret and evaluate numerical 

expressions including the language of scaling and fraction 

division. 









lesson. 



• For Problems 1 and 2, explain to a partner how 

you chose the correct equivalent expression(s). 

• Compare your answer for Problem 3 with a 

partner. Is there more than one correct answer? 

• What’s the relationship between Problems 5(a) 

and 5(c)? 

• Share and compare your solutions for Problem 7 

with a partner. Be care with the order of 

operations; calculate the parenthesis first. 

• Share and compare your answers for Problem 8 

with a partner. For the two expressions that did 

not match the story problems, can you think of a 

story problem for them? Share your ideas with a 

partner. 



Date: 11/10/13 

4.H.8 











the students. 



Date: 11/10/13 

4.H.9 





Name 

Date 

1. Circle the expression equivalent to “the sum of 3 and 2 divided by .” 

3 (2 (3 + 2) (3 + 2) 

2. Circle the expression(s) equivalent to “28 divided by the difference between and .” 

(28 – ) 

– 

( – 28 28 ( – ) 

3. Fill in the chart by writing an equivalent numerical expression. 

a. Half as much as the difference between 2 and . 

b. The difference between 2 and divided by 4. 

c. A third of the sum of and 22 tenths. 

d. Add 2.2 and , and then triple the sum. 

4. Compare expressions 3(a) and 3(b). Without evaluating, identify the expression that is greater. Explain 

how you know. 



Date: 11/10/13 

4.H.10 





5. Fill in the chart by writing an equivalent expression in word form. 

a. (1.75 ) 

b. – ( 0.2) 

c. (1.75 ) 

d. 2 ( ) 

6. Compare the expressions in 5(a)and 5(c). Without evaluating, identify the expression that is less. Explain 

how you know. 

7. Evaluate the following expressions. 

a. (9 – 5) b. (2 ) c. (1 ) 

d. e. Half as much as ( 0.2) f. 3 times as much as the 

quotient of 2.4 and 0.6 



Date: 11/10/13 

4.H.11 





8. Choose an expression below that matches the story problem, and write it in the blank. 

(20 – 5) ( 20) – ( 5) 20 – 5 (20 – ) – 5 

a. Farmer Green picked 20 carrots. He cooked of them and then gave 5 to his rabbits. Write the 

expression that tells how many carrots he had left. 

Expression: ____________________________________ 

b. Farmer Green picked 20 carrots. He cooked 5 of them and then gave to his rabbits. Write the 

expression that tells how many carrots the rabbits will get. 

Expression: ____________________________________ 



Date: 11/10/13 

4.H.12 





Name 

Date 

1. Write an equivalent expression in numerical form. 

A fourth as much as the product of two-thirds and 0.8 

2. Write an equivalent expression in word form. 

a. (1 – ) b. (1 – ) 2 

3. Compare the expressions in 2(a) and 2(b). Without evaluating, determine which expression is greater, 

and explain how you know. 



Date: 11/10/13 

4.H.13 





Name 

Date 

1. Circle the expression equivalent to “the difference between 7 and 4, divided by a fifth.” 

7 + (4 (7 - 4) (7 - 4) 

2. Circle the expression(s) equivalent to “42 divided by the sum of and .” 

( 42 (42 ) + 42 ( ) 

3. Fill in the chart by writing the equivalent numerical expression or expression in word form. 

Expression in word form 

Numerical expression 

a. A fourth as much as the sum of 3 and 4.5 

b. (3 + 4.5) 5 

c. Multiply by 5.8, then halve the product 

d. (4.8 – ) 

e. 8 – ( ) 

4. Compare the expressions in 3(a)and 3(b). Without evaluating, identify the expression that is greater. 

Explain how you know. 



Date: 11/10/13 

4.H.14 





5. Evaluate the following expressions. 

a. (11 – 6) b. (4 ) c. (5 ) 

d. e. 50 divided by the difference between and 

6. Lee is sending out 32 birthday party invitations. She gives 5 invitations to her mom to give to family 

members. Lee mails a third of the rest, and then she takes a break to walk her dog. 

a. Write a numerical expression to describe how many invitations Lee has already mailed. 

b. Which expression matches how many invitations still need to be sent out? 

32 – 5 – (32 – 5) 32 – 5 (32 – 5) (32 – 5) 



Date: 11/10/13 

4.H.15 





Lesson 33 

Objective: Create story contexts for numerical expressions and tape 






Total Time 

(12 minutes) 

(38 minutes) 

(10 minutes) 

(60 minutes) 


• Sprint: Divide Decimals 5.NBT.7 

• Write Equivalent Expressions 5.OA.1 

(9 minutes) 

(3 minutes) 

Sprint: Divide Decimals (9 minutes) 

Materials: (S) Divide Decimals Sprint 


Write Equivalent Expressions (3 minutes) 



T: (Write 2 ÷ = ____.) What is 2 ÷ ? 

S: 6. 

T: (Write 2 ÷ + 4.) On your boards, write the complete number sentence. 

S: (Write 2 ÷ + 4. Beneath it, write = 6 + 4. Beneath it, write = 10.) 

Continue this process with the following possible suggestions: , ÷ (2 + 2), (4 + 3) ÷ , and ( – ) ÷ 5. 

Lesson 33: Create story contexts for numerical expressions and tape 


Date: 11/10/13 

4.H.16 







Note: The time normally allotted for the Application Problem has been included in the Concept Development 

portion of today’s lesson in order to give students the time necessary to write story problems. 



Have two pairs of student work at the board while the others 

work independently or in pairs at their seats. Review the 

following questions before beginning the first problem: 











sharing their work and thinking with a peer. All should write 

their equations and statements of the answer. 

NOTES ON 


ENGAGEMENT: 

When selecting students to work at the 

board, teachers typically choose their 

top students to model their thinking. 

Consider asking students who may 

struggle to work at the board, being 

sure to support them as necessary but 

also praising their effort and 

perseverance as they work. This 

approach can help improve the 

classroom climate and reinforce the 

notion that math work is often more 

about determination and persistence 

than it is sheer math skill. 



Problem 1 

Ms. Hayes has liter of juice. She distributes it equally to 6 students in her tutoring group. 

a. How many liters of juice does each student get? 

b. How many more liters of juice will Ms. Hayes need if she wants to give each of the 24 students in her 

class the same amount of juice found in Part (a)? 



Date: 11/10/13 

4.H.17 





In this problem, Ms. Hayes is sharing equally, or dividing, one-half liter of juice among six students. Students 

should recognize this problem as 

6. A tape diagram shows that when halves are partitioned into 6 equal 

parts, twelfths are created. Likewise, the diagram shows that 1 half is equal to 6 twelfths, and when written 

in unit form, 6 twelfths divided by 6 is a simple problem. Each student gets one-twelfth liter of juice. In Part 

(b), students must find how much more juice is necessary to give a total of 24 students 

liter of juice. Some 

students may choose to solve by multiplying 24 by one-twelfth to find that a total of 2 liters of juice is 

necessary. Encourage interpretation as a scaling problem. Help students see that since 24 students is 4 times 

more students than 6, Ms. Hayes will need 4 times more juice as well. Four times one-half is, again, equal to 

2 liters of juice. Either way, Ms. Hayes will need more liters of juice. 

Problem 2 

Lucia has 3.5 hours left in her workday as a car mechanic. Lucia needs of an hour to complete one oil 

change. 

a. How many oil changes can Lucia complete during the rest of her workday? 

b. Lucia can complete two car inspections in the same amount of time it takes her to complete one oil 

change. How long does it take her to complete one car inspection? 

c. How many inspections can she complete in the rest of her workday? 



Date: 11/10/13 

4.H.18 





MP.7 

In Part (a), students are asked to find how many half-hours are in a 3.5 hour period. The presence of both 

decimal and fraction notation in these problems adds a layer of complexity. Students should be comfortable 

choosing which form of fractional number is most efficient for solving. This will vary by problem and, in many 

cases, by student. In this problem, many students may prefer to deal with 3.5 as a mixed number ( ). Then 

a tape diagram clearly shows that 

can be partitioned into 7 units of . Others may prefer to express the 

half hour as 0.5. Still others may begin their thinking with, “How many halves are in 

with similar prompts to find how many halves are in 3 wholes and 1 half. 

In Part (b), students reason that since Lucia can complete 2 

inspections in the time it takes her to complete just one oil 

change, may be divided by 2 to find the fraction of an hour 

that an inspection requires. Students may also reason that 

there are two 15-minute units in one half-hour period, and 

therefore, Lucia can complete an inspection in hour. 

In Part (c), a variety of approaches is also possible. Some may 

argue that since Lucia can work twice as fast completing 

inspections, they need only to double the number of oil changes 

she could complete in 3.5 hours to find the number of 

inspections done. This type of thinking is evidence of a deeper 

understanding of a scaling principle. Other students may solve 

Part (c), just as they did Part (a), but using a divisor of 

either case, Lucia can complete 14 inspections in 3.5 hours. 

Problem 3 

Carlo buys $14.40 worth of grapefruit. Each grapefruit cost $0.80. 

In 

whole?” and continue 

NOTES ON 


ENGAGEMENT: 

Challenge high achieving students (who 

may also be early finishers) to solve the 

problems more than one way. After 

looking at their work, challenge them 

by specifying the operation they must 

use to begin, or change the path of 

their approach by requiring a certain 

operation within their solution. 

Challenge them further by asking them 

to use the same general context but to 

write a different question that results 

in the same quantity as the original 

problem. 

a. How many grapefruit does Carlo buy? 

b. At the same store, Kahri spends one-third as much money on grapefruit as Carlo. How many 

grapefruit does she buy? 

Students divide a decimal dividend by a decimal divisor to solve Problem 3. This problem is made simpler by 

showing the division expression as a fraction. Then, multiplication by a fraction equal to 1 ( or , 

depending on whether 80 cents is expressed as 0.8 or 0.80) results in both a whole number divisor and 



Date: 11/10/13 

4.H.19 





dividend. From here, students must divide 144 by 8 to find a quotient of 18. Carlo buys 18 grapefruit with his 

money. 

In Part (b), since Kahri spends one-third of her money on equally priced grapefruit, students should reason 

that she would be buying one-third the number of fruit. Therefore, 18 3 shows that Kahri buys 6 grapefruit. 

Students may also choose the far less-direct method of solving a third of $14.40 and dividing that number 

($4.80) by $0.80, to find the number of grapefruit purchased by Kahri. 

Problem 4 

Studies show that a typical giant hummingbird can flap its wings once in 0.08 of a second. 

a. While flying for 7.2 seconds, how many times will a typical giant hummingbird flap its wings? 

b. A ruby-throated hummingbird can flap its wings 4 times faster than a giant hummingbird. How many 

times will a ruby-throated hummingbird flap its wings in the same amount of time? 

Problem 4 is another decimal divisor/dividend problem. Similarly, students should express this division as a 

fraction, and then multiply to rename the divisor as a whole number. Ultimately, students should find that 

the giant hummingbird can flap its wings 90 times in 7.2 seconds. Part (b) is another example of the 

usefulness of the scaling principle. Since a ruby-throated hummingbird can flap its wings 4 times faster than 

the giant hummingbird, students need only to multiply 90 by 4 to find that a ruby-throated hummingbird can 

flap its wings a remarkable 360 times in 7.2 seconds. Though not very efficient, students could also divide 

0.08 by 4 to find that it takes a ruby-throated hummingbird just 0.02 seconds to flap its wings once. Then 

division of 7.2 by 0.02 (or 720 by 2, after renaming the divisor as a whole number) yields a quotient of 360. 

Problem 5 

Create a story context for the following expression. 

($20 – $3.20) 



Date: 11/10/13 

4.H.20 





Working backwards from expression to story may be challenging for some students. Since the expression 

given contains parentheses, the story created must first involve the subtraction of $3.20 from $20. For 

students in need of assistance, drawing a tape diagram first may be of help. Note that the story of Jamis 

interprets the multiplication of directly, whereas the story of Wilma interprets the expression as division by 

3. 

Problem 6 

Create a story context about painting a wall for the following tape diagram. 

1 

? 

Again, students are asked to create a story 

problem, this time using a given tape diagram 

and the context of painting a wall. The 

challenge here is that this tape diagram implies 

a two-step word problem. The whole, 1, is first 

partitioned into half, and then one of those 

halves is divided into thirds. The story students create should reflect this two-part drawing. Students should 

be encouraged to share aloud and discuss their stories and thought process for solving. 


Lesson Objective: Create story contexts for numerical 

expressions and tape diagrams, and solve word problems. 

NOTES ON 


ENGAGEMENT: 

Challenge early finishers in this lesson 

by encouraging them to go back to 

each problem and provide an alternate 

means for solution or an additional 

model to represent the problem. 

Students could discuss how their 

interpretation of each problem led 

them to solve it the way they did, and 

how and why alternate interpretations 

could lead to a differing solution 

strategy. 



Date: 11/10/13 

4.H.21 













lesson. 



• For Problems 1 to 5, did you draw a tape diagram 

to help solve the problems? If so, share your 

drawings and explain them to a partner. 

• For Problems 1 to 4, there are different ways to 

solve the problems. Share and compare your 

strategy with a partner. 

• For Problems 5 and 6, share your story problem 

with a partner. Explain how you interpreted the 

expression in Problem 5, and the tape diagram in 

Problem 6. 



Date: 11/10/13 

4.H.22 











Date: 11/10/13 

4.H.23 







Date: 11/10/13 

4.H.24 







Date: 11/10/13 

4.H.25 





Name 

Date 

1. Ms. Hayes has liter of juice. She distributes it equally to 6 students in her tutoring group. 

a. How many liters of juice does each student get? 

b. How many more liters of juice will Ms. Hayes need, if she wants to give each of the 24 students in her 

class the same amount of juice found in Part (a)? 

2. Lucia has 3.5 hours left in her workday as a car mechanic. Lucia needs of an hour to complete one oil 

change. 

a. How many oil changes can Lucia complete during the rest of her workday? 

b. Lucia can complete two car inspections in the same amount of time it takes her to complete one oil 

change. How long does it take her to complete one car inspection? 

c. How many inspections can she complete in the rest of her workday? 



Date: 11/10/13 

4.H.26 





3. Carlo buys $14.40 worth of grapefruit. Each grapefruit cost $0.80. 

a. How many grapefruit does Carlo buy? 

b. At the same store, Kahri spends one-third as much money on grapefruit as Carlo. How many 

grapefruit does she buy? 

4. Studies show that a typical giant hummingbird can flap its wings once in 0.08 of a second. 

a. While flying for 7.2 seconds, how many times will a typical giant hummingbird flap its wings? 

b. A ruby-throated hummingbird can flap its wings 4 times faster than a giant hummingbird. How many 

times will a ruby-throated hummingbird flap its wings in the same amount of time? 



Date: 11/10/13 

4.H.27 





5. Create a story context for the following expression. 

($20 – $3.20) 

6. Create a story context about painting a wall for the following tape diagram. 

1 

? 



Date: 11/10/13 

4.H.28 





Name 

Date 

1. An entire commercial break is 3.6 minutes. 

a. If each commercial takes 0.6 minutes, how many commercials will be played? 

b. A different commercial break of the same length plays commercials half as long. How many 

commercials will play during this break? 



Date: 11/10/13 

4.H.29 





Name 

Date 

1. Chase volunteers at an animal shelter after school, feeding and playing with the cats. 

a. If he can make 5 servings of cat food from a third of a kilogram of food, how much does one serving 

weigh? 

b. If Chase wants to give this same serving size to each of 20 cats, how many kilograms of food will he 

need? 

2. Anouk has 4.75 pounds of meat. She uses a quarter pound of meat to make one hamburger. 

a. How many hamburgers can Anouk make with the meat she has? 

b. Sometimes Anouk makes sliders. Each slider is half as much meat as is used for a regular hamburger. 

How many sliders could Anouk make with the 4.75 pounds? 



Date: 11/10/13 

4.H.30 





3. Ms. Geronimo has a $10 gift certificate to her local bakery. 

a. If she buys a slice of pie for $2.20 and uses the rest of the gift certificate to buy chocolate macaroons 

that cost $0.60 each, how many macaroons can Ms. Geronimo buy? 

b. If she changes her mind and instead buys a loaf of bread for $4.60 and uses the rest to buy cookies 

that cost 

times as much as the macaroons, how many cookies can she buy? 

4. Create a story context for the following expressions. 

a. ( – 2 ) ÷ 4 b. 4 ( ) 

5. Create a story context for the following tape diagram. 

6 

? 



Date: 11/10/13 

4.H.31 




Mid-Module Assessment Task Lesson 

2•3 

NYS COMMON CORE MATHEMATICS CURRICULUM 5•4 

Name 

Date 

1. Multiply or divide. Draw a model to explain your thinking. 

a. b. 

c. d. 

e. f. 

g. ( ) h. 


Date: 11/10/13 4.S.1 




NYS COMMON CORE MATHEMATICS CURRICULUM 



2. If the whole bar is 3 units long, what is the length of the shaded part of the bar? Write a multiplication 

equation for the diagram, and then solve. 

0 

3. Circle the expression(s) that are equal to . Explain why the others are not equal using words, 

pictures, or numbers. 

a. 3 × ( 

b. 3 ÷ (5 × 6) 

c. (3 × 

d. 3 × 


Date: 11/10/13 4.S.2 







4. Write the following as expressions. 

a. One-third the sum of 6 and 3. 

b. Four times the quotient of 3 and 4. 

c. One-fourth the difference between and . 

5. Mr. Schaum used 10 buckets to collect rainfall in various locations on his property. The following line plot 

shows the amount of rain collected in each bucket in gallons. Write an expression that includes 

multiplication to show how to find the total amount of water collected in gallons. Then solve your 

expression. 


Date: 11/10/13 4.S.3 







6. Mrs. Williams uses the following recipe for crispy rice treats. She decides to make of the recipe. 

2 cups melted butter 

24 oz marshmallows 

13 cups rice crispy cereal 

a. How much of each ingredient will she need? Write an expression that includes multiplication. Solve 

by multiplying. 

b. How many fluid ounces of butter will she use? (Use your measurement conversion chart if you wish.) 

c. When the crispy rice treats have cooled, Mrs. Williams cuts them into 30 equal pieces. She gives twofifths 

of the treats to her son and takes the rest to school. How many treats will Mrs. Williams take to 

school? Use any method to solve. 


Date: 11/10/13 4.S.4 







Mid-Module Assessment Task 

Standards Addressed 


5.OA.1 

5.OA.2 

Topics A–D 

Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions 



expressions without evaluating them. For example, express the calculation “add 8 and 7, 

then multiply by 2” as 2 × (8 +7). Recognize that 3 × (18932 + 921) is three times as large 

as 18932 + 921, without having to calculate the indicated sum or product. 

Apply and extend previous understandings of multiplication and division to multiply and divide 

fractions. 

5.NF.3 

5.NF.4 

5.NF.6 

Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve 


fractions or mixed numbers, e.g., by using visual fraction models or equations to represent 

the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 

multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each 

person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by 

weight, how many pounds of rice should each person get? Between what two whole 




a. Interpret the product (a/b) × q as a parts of a partition of q into b parts; equivalently, 

as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction 

model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the 

same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.) 




5.MD.1 





5.MD.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 

1/8). Use operations on fractions for this grade to solve problems involving information 


beakers, find the amount of liquid each beaker would contain if the total amount in all the 

beakers were redistributed equally. 


Date: 11/10/13 4.S.5 







Evaluating Student Learning Outcomes 

A Progression Toward Mastery is provided to describe steps that illuminate the gradually increasing 

understandings that students develop on their way to proficiency. In this chart, this progress is presented 

from left (Step 1) to right (Step 4). The learning goal for each student is to achieve Step 4 mastery. These 

steps are meant to help teachers and students identify and celebrate what the student CAN do now and what 

they need to work on next. 


Date: 11/10/13 4.S.6 







A Progression Toward Mastery 

Assessment 

Task Item 

and 

Standards 

Assessed 

STEP 1 

Little evidence of 

reasoning without 

a correct answer. 

(1 Point) 

STEP 2 

Evidence of some 



(2 Points) 

STEP 3 


reasoning with a 

correct answer or 

evidence of solid 

reasoning with an 

incorrect answer. 

(3 Points) 

STEP 4 

Evidence of solid 


correct answer. 

(4 Points) 

1 

5.NF.4a 

5.MD.1 

The student draws 

valid models and/or 

arrives at the correct 

product for two or 

more items. 




product for at least 

four or more items. 




product for at least six 

or more items. 

The student correctly 

answers all eight items, 

and draws valid 

models: 

a. 3 

b. 3 1/2 

c. 9 

d. 12 

e. 8 inches 

f. 1 1/2 feet 

g. 49 

h. 60 2/3 

2 

5.NF.4a 

5.NF.3 

T u ’ w rk 

shows no evidence of 

being able to express 

the length of the 

shaded area. 

The student 

approximates the 

length of the shaded 

bar, but does not write 

a multiplication 

equation. 

The student is able to 

write the correct 

multiplication equation 

for the diagram, but 

incorrectly states the 

length of the shaded 

part of the bar. 

The student correctly: 

• Writes a 

multiplication 

equation: 3/4 × 3. 

• Finds the length of 

the shaded part of 

the bar as 9/4 or 

2 1/4. 

3 

5.OA.1 

The student is unable 

to identify any equal 

expressions. 


identifies one correct 

expression. 


identifies two equal 

expressions. 


• Identifies (a), (c), 

and (d) as equal to 

3/5 × 6. 

• Explains why (b) is 

not equal. 


Date: 11/10/13 4.S.7 








4 

5.OA.2 

The student is unable 

to write expressions for 

(a), (b), or (c). 


writes one expression. 


writes two expressions. 


writes three 

expressions: 

a. 1/3 × (6 + 3) 

b. 4 × (3 ÷ 4) or 

4 × 3/4 

c. 1/4 × (2/3 – 1/2) 

5 

5.NF.4a 

5.NF.6 

5.MD.2 

The student is neither 

able to produce a 


expression that 

identifies the data from 

the line plot, nor is able 

to find the total gallons 

of water collected. 

The student is either 

able to write a 


expression that 

accounts for all data 

points on the line plot, 

or is able to find the 

total gallons of water 

collected. 

T u ’ 


expression correctly 

accounts for all the 

data points on the line 

plot when finding the 

total gallons of water 

collected, but makes a 

calculation error. 


• Accounts for all data 

points in the line 

plot in the 


expression. 

• Finds the total 

gallons of water 

collected as 15 6/8 

gallons or 15 3/4 

gallons. 

6 

5.NF.4a 

5.NF.6 

5.MD.1 


calculates two correct 

answers. 


correctly calculate 

three correct answers. 


correctly calculate four 

correct answers. 


a. Calculates: 1 1/3 c 

butter; 16 oz of 

marshmallows; 8 

2/3 c of cereal 

b. Converts 1 1/3c 

butter to 10 2/3 

fluid ounces. 

c. Uses an equation or 

model and finds the 

number of treats 

taken to school as 

18 treats. 


Date: 11/10/13 4.S.8 








Date: 11/10/13 4.S.9 








Date: 11/10/13 4.S.10 








Date: 11/10/13 4.S.11 








Date: 11/10/13 4.S.12 




End-of-Module Assessment Task Lesson 

2•3 


Name 

Date 

1. Multiply or divide. Draw a model to explain your thinking. 

a. b. of 

c. d. 

e. f. 

2. Multiply or divide using any method. 

a. b. 

c. d. 

e. f. 


Date: 11/10/13 4.S.13 







3. Fill in the chart by writing an equivalent expression. 

a. One-fifth of the sum of onehalf 

and one-third 

b. Two and a half times the sum 

of nine and twelve 

c. Twenty-four divided by the 

difference between 

and 

4. A castle has to be guarded 24 hours a day. Five knights are ordered to split each day’s guard duty 

equally. How long will each knight spend on guard duty in one day? 

a. Record your answer in hours. 

b. Record it in hours and minutes. 

c. Record your answer in minutes. 


Date: 11/10/13 4.S.14 





2•3 


5. On the blank, write a division expression that matches the situation. 

a. _________________ Mark and Jada share 5 yards of ribbon equally. How 

much ribbon will each get? 

b. __________________ It takes half of a yard of ribbon to make a bow. How many bows 

can be made with 5 yards of ribbon? 

c. Draw a diagram for each problem and solve. 

d. Could either of the problems also be solved by using ? If so, which one(s)? Explain your thinking. 


Date: 11/10/13 4.S.15 







6. Jackson claims that multiplication always makes a number bigger. He gave the following examples: 

• If I take 6, and I multiply it by 4, I get 24, which is bigger than 6. 

• If I take , and I multiply it by 2 (whole number), I get , or which is bigger than . 

Jackson’s reasoning is incorrect Give an example that proves he is wrong, and explain his mistake using 

pictures, words, or numbers. 

7. Jill is collecting honey from 9 different beehives, and recorded the amount collected, in gallons, from each 

hive in the line plot shown: 

a. She wants to write the value of each point marked on the number line above (Points a–d) in terms of 

the largest possible whole number of gallons, quarts, and pints. Use the line plot above to fill in the 

blanks with the correct conversions. (The first one is done for you.) 

0 

3 0 

a. _______ gal ______ qt _______pt 

Gallons 

b. _______ gal ______ qt _______pt 

c. _______ gal ______ qt _______pt 

d. _______ gal ______ qt _______pt 


Date: 11/10/13 4.S.16 







b. Find the total amount of honey collected from the five hives that produced the most honey. 

c. Jill collected a total of 19 gallons of honey. If she distributes all of the honey equally between 9 jars, 

how much honey will be in each jar? 

d. Jill used of a jar for baking. How much honey did she use baking? 


Date: 11/10/13 4.S.17 







e. Jill’s mom used of a gallon of honey to bake 3 loaves of bread. If she used an equal amount of 

honey in each loaf, how much honey did she use for 1 loaf? 

f. Jill’s mom stored some of the honey in a container that held of a gallon. She used half of this 

amount to sweeten tea. How much honey, in cups, was used in the tea? Write an equation and draw 

a tape diagram. 

g. Jill uses some of her honey to make lotion. If each bottle of lotion requires gallon, and she uses a 

total of 3 gallons, how many bottles of lotion does she make? 


Date: 11/10/13 4.S.18 







End-of-Module Assessment Task 

Standards Addressed 


5.OA.1 

5.OA.2 

Topics A–H 

Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions 



expressions without evaluating them. For example, express the calculation “add 8 and 7, 

then multiply by 2” as 2 × (8 +7). Recognize that 3 × (18932 + 921) is three times as large as 

18932 + 921, without having to calculate the indicated sum or product. 

Perform operations with multi-digit whole numbers and with decimals to hundredths. 

5.NBT.7 



relationship between addition and subtraction; relate the strategy to a written method and 

explain the reasoning used. 

Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 

5.NF.3 

5.NF.4 

5.NF.5 

Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve 


fractions or mixed numbers, e.g., by using visual fraction models or equations to represent 

the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 

multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each 

person has a share of size 3/4. If 9 people want to share a 50‐pound sack of rice equally by 

weight, how many pounds of rice should each person get? Between what two whole 





equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a 

visual fraction model to show (2/3 × 4 = 8/3, and create a story context for this 


Interpret multiplication as scaling (resizing) by: 

a. Comparing the size of a product to the size of one factor on the basis of the 

size of the other factor, without performing the indicated multiplication. 


product greater than the given number (recognizing multiplication by whole numbers 

greater than 1 as a familiar case); explaining why multiplying a given number by a 

fraction less than 1 results in a product smaller than the given number; and relating 

the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b 

by 1. 


Date: 11/10/13 4.S.19 







5.NF.6 

5.NF.7 

Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by 

using visual fraction models or equations to represent the problem. 


numbers and whole numbers by unit fractions. (Students able to multiple fractions in 

general can develop strategies to divide fractions in general, by reasoning about the 

relationship between multiplication and division. But division of a fraction by a fraction is 

not a requirement at this grade level.) 


quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction 

model to show the quotient. Use the relationship between multiplication and division 

to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3. 

b. Interpret division of a whole number by a unit fraction, and compute such quotients. 

For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to 

show the quotient. Use the relationship between multiplication and division to explain 

that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4. 

c. Solve real world problems involving division of unit fractions by non‐zero whole 

numbers and division of whole numbers by unit fractions, e.g., by using visual fraction 

models and equations to represent the problem. For example, how much chocolate 

will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup 

servings are in 2 cups of raisins? 


5.MD.1 





5.MD.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). 

Use operations on fractions for this grade to solve problems involving information 


beakers, find the amount of liquid each beaker would contain if the total amount in all the 

beakers were redistributed equally. 

Evaluating Student Learning Outcomes 

A Progression Toward Mastery is provided to describe steps that illuminate the gradually increasing 

understandings that students develop on their way to proficiency. In this chart, this progress is presented 

from left (Step 1) to right (Step 4). The learning goal for each student is to achieve Step 4 mastery. These 

steps are meant to help teachers and students identify and celebrate what the student CAN do now and what 

they need to work on next. 


Date: 11/10/13 4.S.20 








Assessment 

Task Item 

and 

Standards 

Assessed 

STEP 1 

Little evidence of 



(1 Point) 

STEP 2 




(2 Points) 

STEP 3 



correct answer or 

evidence of solid 

reasoning with an 

incorrect answer. 

(3 Points) 

STEP 4 

Evidence of solid 



(4 Points) 

1 

5. NF.4 

5. NF.7 




answer for two or 

more items. 




answer for three or 

more items. 




answer for four or 

more items. 


answers all eight items, 

and draws valid 

models: 

a. 

b. 

c. 

d. 12 

e. 20 

f. 

2 

5.NBT.7 

The student has two or 

fewer correct answers. 

The student has three 


The student has four 



answers all six items: 

a. 48 

b. 0.48 

c. 400 

d. 4 

e. 9.6 or 924/40 or any 

equivalent fraction 

f. 32 

3 

5.OA.2 

The student has no 


The student has one 


The student has two 



answers all three 

items: 

a. 

b. (9 + 12) or 

2 × (9 + 12) 


Date: 11/10/13 4.S.21 








c. 24 ( 

4 

5.NF.3 

5.NF.6 

5.MD.1 

The student has no 


The student has one 


The student has two 



answers all three 

items: 

a. 4.8 hours 

b. 4 hours, 48 minutes 

c. 288 minutes 

5 

5.NF.6 

5.NF.7 

The student gives one 

or fewer correct 

responses among Parts 

(a), (b), (c), and (d). 

The student gives at 

least two correct 


(a), (b), (c), and (d). 

The student gives at 

least three correct 


(a), (b), (c), and (d). 


answers all four items: 

a. 5 ÷ 2 

b. 5 ÷ 1/2 

c. Draws a correct 

diagram for both 

expressions and 

solves. 

d. Correctly identifies 

5 ÷ 2, and offers 

solid reasoning. 

6 

5.NF.5 

The student gives both 

a faulty example and a 

faulty explanation. 

The student gives 

either a faulty example 

or explanation. 

The student gives a 

valid example or a clear 

explanation. 


give a correct example 

and clear explanation. 

7 

5.NF.3 

5.NF.4 

5.NF.6 

5.NF.7 

5.MD.1 

5.MD.2 

The student has two or 

fewer correct answers. 

The student has three 


The student has five 



answers all seven 

items: 

a. 1 gal, 2 qts 

2 gal, 1 pt 

2 gal, 2 qt, 1 pt 

b. 13 gal, 1 pt 

c. 2 1/9 gal 

d. 1 7/12 gal 

e. 1/12 gal 

f. 6 c 

g. 12 bottles 


Date: 11/10/13 4.S.22 








Date: 11/10/13 4.S.23 








Date: 11/10/13 4.S.24 








Date: 11/10/13 4.S.25 








Date: 11/10/13 4.S.26 








Date: 11/10/13 4.S.27

math-g5-m4-full-module

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