Ready Common Core Mathematics Teacher Resource Book 6
2014
Common Core Mathematics Teacher Resource Book 6
Table of Contents Ready® Common Core Program Overview
A6
Supporting the Implementation of the Common Core
A7 Answering the Demands of the Common Core with Ready A8 The Standards for Mathematical Practice A9 Depth of Knowledge Level 3 Items in Ready Common Core A10 Cognitive Rigor Matrix A11
Using Ready Common Core A12 Teaching with Ready Common Core Instruction Content Emphasis in the Common Core Standards Connecting with the Ready® Teacher Toolbox Using i-Ready® Diagnostic with Ready Common Core Features of Ready Common Core Instruction Supporting Research
A14 A16 A18 A20 A22 A38
Correlation Charts Common Core State Standards Coverage by Ready Instruction Interim Assessment Correlations
A42 A46
Lesson Plans (with Answers) CCSS Emphasis Unit 1: Ratios and Proportional Relationships Lesson 1
Ratios
1 3 M
CCSS Focus - 6.RP.A.1 Embedded SMPs - 4, 6
Lesson 2
Understand Unit Rate
11
M
19
M
29
M
41
M
CCSS Focus - 6.RP.A.2 Embedded SMPs - 2, 6, 7
Lesson 3
Equivalent Ratios
CCSS Focus - 6.RP.A.3a Embedded SMPs - 1, 2, 4, 5, 7, 8
Lesson 4
Solve Problems with Unit Rate
CCSS Focus - 6.RP.A.3b, 6.RP.A.3d Embedded SMPs - 2–4
Lesson 5
Solve Problems with Percent
CCSS Focus - 6.RP.A.3c Embedded SMPs - 1, 2, 4, 5, 7
Unit 1 Interim Assessment
M = Lessons that have a major emphasis in the Common Core Standards S/A = Lessons that have supporting/additional emphasis in the Common Core Standards
51
Unit 2: The Number System Lesson 6
Understand Division with Fractions
CCSS Emphasis 54 58
M
66
M
78
S/A
88
S/A
98
S/A
110
S/A
120
M
128
M
138
M
CCSS Focus - 6.NS.A.1 Embedded SMPs - 1–4, 7, 8
Lesson 7
Divide with Fractions
CCSS Focus - 6.NS.A.1 Embedded SMPs - 1–4, 7, 8
Lesson 8
Divide Multi-Digit Numbers
CCSS Focus - 6.NS.B.2 Embedded SMPs - 1, 2, 7, 8
Lesson 9
Add and Subtract Decimals
CCSS Focus - 6.NS.B.3 Embedded SMPs - 3, 6, 7
Lesson 10 Multiply and Divide Decimals CCSS Focus - 6.NS.B.3 Embedded SMPs - 2, 3, 6–8
Lesson 11 Common Factors and Multiples CCSS Focus - 6.NS.B.4 Embedded SMPs - 2, 3, 7
Lesson 12 Understand Positive and Negative Numbers CCSS Focus - 6.NS.C.5, 6.NS.C.6a, 6.NS.C.6c Embedded SMPs - 1, 2, 4
Lesson 13 Absolute Value and Ordering Numbers CCSS Focus - 6.NS.C.5, 6.NS.C.7a, 6.NS.C.7b, 6.NS.C.7c, 6.NS.C.7d Embedded SMPs - 1–4, 6
Lesson 14 The Coordinate Plane CCSS Focus - 6.NS.C.6b, 6.NS.C.6c, 6.NS.C.8 Embedded SMPs - 1, 2, 4–7
Unit 2 Interim Assessment
Unit 3: Expressions and Equations Lesson 15 Numerical Expressions with Exponents
151 154 157
M
167
M
179
M
191
M
199
M
CCSS Focus - 6.EE.A.1 Embedded SMPs - 1–8
Lesson 16 Algebraic Expressions CCSS Focus - 6.EE.A.2a, 6.EE.A.2b, 6.EE.A.2c Embedded SMPs - 1–4, 6
Lesson 17 Equivalent Expressions CCSS Focus - 6.EE.A.3, 6.EE.A.4 Embedded SMPs - 2, 3
Lesson 18 Understand Solutions to Equations CCSS Focus - 6.EE.B.5 Embedded SMPs - 1–5, 7
Lesson 19 Solve Equations CCSS Focus - 6.EE.B.6, 6.EE.B.7 Embedded SMPs - 1–4, 7
M = Lessons that have a major emphasis in the Common Core Standards S/A = Lessons that have supporting/additional emphasis in the Common Core Standards
Unit 3: Expressions and Equations (continued) Lesson 20 Solve Inequalities
CCSS Emphasis 211
M
221
M
CCSS Focus - 6.EE.B.5, 6.EE.B.8 Embedded SMPs - 1–4, 6, 7
Lesson 21 Dependent and Independent Variables CCSS Focus - 6.EE.C.9 Embedded SMPs - 1–4, 7, 8
Unit 3 Interim Assessment
Unit 4: Geometry Lesson 22 Area of Polygons
231 234 236
S/A
246
S/A
256
S/A
CCSS Focus - 6.G.A.1 Embedded SMPs - 1–7
Lesson 23 Polygons in the Coordinate Plane CCSS Focus - 6.G.A.3 Embedded SMPs - 1, 2, 4, 5, 7
Lesson 24 Nets and Surface Area CCSS Focus - 6.G.A.4 Embedded SMPs - 7, 8
Lesson 25 Volume
268 S/A
CCSS Focus - 6.G.A.2 Embedded SMPs - 1, 4
Unit 4 Interim Assessment
Unit 5: Statistics and Probability Lesson 26 Understand Statistical Questions
279 282 285
S/A
293
S/A
305
S/A
317
S/A
CCSS Focus - 6.SP.A.1 Embedded SMPs - 1, 3, 6
Lesson 27 Measures of Center and Variability CCSS Focus - 6.SP.A.2, 6.SP.A.3 Embedded SMPs - 2, 4–7
Lesson 28 Display Data on Dot Plots, Histograms, and Box Plots CCSS Focus - 6.SP.B.4 Embedded SMPs - 2–7
Lesson 29 Analyze Numerical Data CCSS Focus - 6.SP.B.5a, 6.SP.B.5b, 6.SP.B.5c, 6.SP.B.5d Embedded SMPs - 2–5
Unit 5 Interim Assessment
M = Lessons that have a major emphasis in the Common Core Standards S/A = Lessons that have supporting/additional emphasis in the Common Core Standards
328
Focus on Math Concepts
Lesson 6
(Student Book pages 52–57)
Understand Division with Fractions Lesson Objectives
The Learning Progression
• Understand the meanings of division.
In Grade 5, students divided whole numbers by unit fractions.
• Use a model to show division of fractions. • Use an understanding of multiplication of fractions to explain division of fractions.
Prerequisite Skills • Know that multiplication and division are inverse operations. • Know that division is either fair sharing (partitive) or repeated subtraction (quotative).
Students continue this understanding in Grade 6 by using visual models and equations to divide whole numbers by fractions and fractions by fractions to solve word problems. In Grade 7, students will continue their work with fractions to include all rational number operations (positive and negative). Students will build on understanding of number lines developed in Grade 6.
• Divide with whole numbers.
Teacher Toolbox
• Divide a whole number by a fraction. • Model division with manipulatives, diagrams and story contexts.
Prerequisite Skills
6.NS.A.1
✓ ✓
Ready Lessons Tools for Instruction
Vocabulary
Teacher-Toolbox.com
✓ ✓ ✓
Interactive Tutorials
There is no new vocabulary.
CCSS Focus 6.NS.A.1 Interpret . . . quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for 1 2 24 1 3 2and use a visual 3 ··
4 ··
fraction model to show the quotient; use the relationship between multiplication and division to explain that 1 2 24 1 3 25 8 because 3 3 ··
4 ··
9 ··
of 8 is 2 . (In general, 1 a 24 1 c 25 ad .) How much chocolate will each person get if 3 people share 1 lb of chocolate equally? How 9 ··
3 ··
b ·
d ·
bc ··
4 ··
2 ··
many 3 -cup servings are in 2 of a cup of yogurt? How wide is a rectangular strip of land with length 3 mi and area 1 square mi? 4 ··
3 ··
4 ··
2 ··
STANDARDS FOR MATHEMATICAL PRACTICE: SMP 1–4, 7, 8 (see page A9 for full text)
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L6: Understand Division with Fractions
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Part 1: Introduction
Lesson 6
At a Glance
Focus on Math concepts
Students explore dividing a fraction and a whole number by a fraction.
Lesson 6
Part 1: introduction
ccss 6.ns.a.1
Understand Division with Fractions
Step By Step
What does it mean to divide a fraction by a fraction?
• Introduce the Question at the top of the page.
You know how to divide a whole number by a unit fraction. For example, you can think
• Read the description of how to divide a whole number by a unit fraction.
divide 6 into fourths and count to see there are 24 fourths in 6.
of 6 divided by 1 as “how many one-fourths are there in 6?” Using a number line, you can 4 ··
6 4 1 5 24 4 ··
0
• Ask students to explain in their own words why
1
2
3
4
5
6 4 1 is the same as 6 3 4. Ask them to use the 4 ··
6 4 1 is the same as 6 3 4, or 24 4 ··
number line illustration to make the explanation
think What does dividing a whole number by a fraction mean? circle the multiplication expression that is the same as the division expression.
Madison cuts a 6-yard length of ribbon into 3 yard pieces.
clear.
4 ··
To figure out how many pieces Madison cut, think, “How many three-fourths are in 6?”
• Have students explain a reciprocal. Remind them that the reciprocal is the same as the multiplicative inverse.
You can draw the same number line to represent the 6 yards of ribbon and divide it into fourths. 0
to explain in his or her own words how the model shows 6 4 3 .
0
4 ··
2
3
4
5
6
1
2
3
4
5
6
63458 3 ··
ELL Support
52
Discuss the difference between the phrases “dividing by” and “dividing into.” Dividing by 1 , for 4 ··
number, but dividing into fourths means to separate a number into four equal groups—i.e., multiplying by a unit fraction.
1
You can circle three 1 sections to represent 3 yard pieces. You can see there are 4 4 ·· ·· eight 3 yard pieces in 6 yards. 4 ·· 64358 4 ··
• Discuss the number line in Think and ask a student
example, asks how many groups of 1 are in a 4 ··
6
You also learned that dividing a number by a fraction is the same as multiplying the number by the reciprocal of the fraction.
Understand L6: Understand Division Division withwith Fractions Fractions
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Mathematical Discourse • When you divide a whole number by a fraction less than 1, the quotient is larger than the dividend. Can you explain why? Students might explain that since the fraction is less than 1, you would be able to take out more than one group from each whole in the dividend. Ask other students to add to the explanation and to use the same reasoning to solve 10 4 3 and explain why the answer 5 ··
makes sense.
L6: Understand Division with Fractions
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Part 1: Introduction
Lesson 6
At a Glance Students explore dividing a fraction by a whole number.
Part 1: introduction
Lesson 6
think What does dividing a fraction by a whole number mean?
Step By Step
Cory wants to pour 3 of a quart of juice equally into 6 glasses. This means he needs to 4 ··
divide 3 into 6 equal parts. You can represent the problem with an area model. First,
• Read Think as a class. Emphasize that in the
4 ··
you can show the 3 quart of juice. Then, you can draw vertical lines to divide the model 4 ··
into 6 equal parts.
previous problem about Madison and the ribbon, students were shown an example of dividing a whole number by a fraction: 6 4 3 . This question is about 4 ··
dividing a fraction by a whole number: 3 4 6. 4 ··
• Discuss the area model with students and have them explain how the first model represents 3 . Then have 4 ··
a student explain how the second model represents
1 2 3 4 5 6 glasses
3465 3 51 24 ·· 8 ··
4 ··
3 quart of juice divided equally into 6 glasses means Cory will pour 3 or 1 quart of juice 4 24 ·· 8 ·· ··
into each glass. 3 4 6 is the same as 3 3 1 . 4 ·· 6 ·· Cory pours 1 of 3 quart of juice into each glass.
4 ··
6 ··
4 ··
reflect
dividing the model into 6 parts. • Ask students how they know how much is in each group. Be sure students understand that 3 refers to 24 ··
the whole.
• Ask students to use the model to show why 3 5 1 . 24 ··
1 Use the number line to show and explain why 4 4 2 and 4 3 1 both equal 2 . 10 10 ·· 2 10 ·· ·· ·· 0
2 10
4 10
1
Possible explanation: see number line above. you can think of 4 divided by 10 ··· 2 as dividing 4 into 2 equal parts. this is the same as finding 1 of 4 because 10 2 10 ··· ·· ··· each half is one of the equal parts.
8 ··
• Ask students to explain to their partner why 3 4 6 4 ··
is the same as 3 3 1 . 4 ··
53
L6: Understand Division with Fractions
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6 ··
• Have students read and reply to the Reflect directive.
Real-World Connection
Visual Model Folding paper to divide a fraction by a whole number. Materials: rectangular sheet of paper • Paul uses 3 of his back yard for gardening. He 4 ··
will divide this gardening section into 3 equal parts for vegetables. Model 3 4 3 using paper. 4 ··
• Start by folding the paper into fourths and shading 3 . 4 ··
• Now fold the paper into thirds to show 3 sections for vegetables. • Unfold. Ask, How is the paper divided now? [twelfths]
Ask students to describe real world situations in which they would have to divide a fraction by a whole number. Ask students to think of a problem or situation that represents 3 4 6. Share a situation in which 3 of 4 ··
4 ··
something is divided into 6 equal groups or shared equally by 6 people. Example: Susan wants to create a picture frame in the shape of a regular hexagon. She has a strip of trim that is 3 of a meter long. If she uses the entire 4 ··
strip, how long would each side be?
• Ask, So how much of the gardening section of Paul’s back yard is used for each vegetable? [The shaded portion of each column represents 3 5 1 of the whole.]
60
12 ··
4 ··
L6: Understand Division with Fractions
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Part 2: Guided Instruction
Lesson 6
At a Glance Part 2: guided instruction
Students use a number line to divide a fraction by a fraction.
Lesson 6
explore it explore dividing a fraction by a fraction with the problem below.
Step By Step
Kate has 2 yards of fabric to make small flags. Each flag requires 1 yard of fabric. How 3 ··
6 ··
many flags can Kate make?
• Tell students that they will have time to work individually on the Explore It problems on this page and then share their responses in groups. You may choose to work through problem 2 together as a class.
2 You need to find out how many
1s 6 ··
are in
2 3 ··
.
3 The number lines below are divided into thirds. Label 2 on the top number line to 3 ·· represent 2 yards of fabric. 3 ·· 2 3
0
1
0
• As students work individually, circulate among them. This is an opportunity to assess student understanding and address student misconceptions. Use the Mathematical Discourse questions to engage student thinking.
1
4 Each flag requires 1 yard of fabric. Divide the bottom number line into sixths to show 6 ·· how many sixths are in 2. 3 ·· 4 5 Look at the bottom number line. How many sixths are there in 2 ? 3 ·· 6 How many flags can Kate make? 7 2415 3 ·· 6 ·· 6 8 23 3 ··
4
4 54
• Check to see that students label the number line correctly before they use it to answer questions. • Take note of students who are still having difficulty and wait to see if their understanding progresses as they work in their groups during the next part of the lesson. 54
STUDENT MISCONCEPTION ALERT: Students may believe that dividing by 1 is the same as 2 ··
dividing into half. Dividing by one half means how many 1 s are in a quantity, 7 divided by 1 5 14. 2 ··
2 ··
Dividing in half means “to take a quantity and split into two equal parts.” 7 divided in half equals 3 1 . 2 ··
L6: Understand Division with Fractions
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Mathematical Discourse • How would you explain dividing a fraction by a fraction using the number line model to a student who was absent from class? Students should explain that you first draw and label the dividend on a number line. Then, draw another number line beneath it and divide it equally into the number of parts that are in the divisor. They should then count and circle how many groups of the divisor there are up to the dividend. Students should model the number lines on the board for the class to see. • What is another way you could model this problem? Students might describe an area model or paper folding. • Which method for solving fraction division problems do you prefer and why? Students may prefer multiplying by the reciprocal or modeling. Listen for and encourage correct usage of math vocabulary.
L6: Understand Division with Fractions
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Part 2: Guided Instruction
Lesson 6
At a Glance Students read fraction division word problems and solve them in pairs or groups.
Step By Step • Organize students into pairs or groups. You may choose to work through the first Talk About It problem together as a class. • Walk around to each group, assessing how the groups are solving the problem. Use the Mathematical Discourse questions to help support or extend students’ thinking. • Students may need to be reminded what direction horizontal is. Review horizontal and vertical as needed. • Direct the group’s attention to Try It Another Way. Have a volunteer from each group come to the board to explain the group’s solutions to problems 15 and 16.
Part 2: guided instruction
Lesson 6
talk about it solve the problem below as a group. Kevin has 6 cups of flour. It takes 3 cup of flour to make one cake. How many cakes can 8 ·· Kevin make? 9 You need to find out how many
3s
8 ··
are in
verbal or written explanations accompanied by models. Provide students an opportunity to refine their mathematical communication skills through discussions in which they evaluate their own thinking and the thinking of other students (SMP 3).
.
11 Represent 6 cups with 6 rectangles. 4 rectangles are shown below. Draw 2 more
rectangles. Draw horizontal lines to divide each rectangle into eighths.
12 Circle and count groups of 3 in the model. How many did you circle? 8 ·· 13 How many 3 -cups of flour are in 6 cups of flour? 8 ·· 14 6 4 3 5 16 8 ··
16
16
try it another Way explore dividing by a unit fraction using a common denominator. To solve 5 4 1, write 5 as a fraction with a denominator of 2 and think, “How many halves 2 ··
are in ten halves?” 10 4 1 5 10. Use the same reasoning to find 8 4 2. 2 ··
2 ··
15 Write 8 as a fraction with a denominator of 3. 6 ··
many two-thirds are in four-thirds”?
many four-sixths are in eight-sixths”?
4
6 ··
3 ··
3 ··
To solve 4 4 2, think, “How
4 6 ··
To solve 8 4 4, think, “How
2
16 Write 2 as a fraction with a denominator of 6. 3 ··
SMP Tip: Students construct arguments using
6
10 Do you think the number of cakes Kevin can make is greater than or less than 6? Why? greater than 6. it takes less than 1 cup of flour to make one cake. 2 ··
2
3 ··
6 ··
3 ··
6 ··
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L6: Understand Division with Fractions
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Mathematical Discourse • Why would you use rectangles to model the problem? Using an area model is a strategy for solving fraction division problems. • The problem is asking you to divide by 3 s. Why didn’t you divide it into thirds?
8 ··
To count by 3 s, you would need to start by 8 ··
dividing it into eighths. • How would you explain dividing a fraction by a fraction using a common denominator? Students should realize that once they have a common denominator they know the parts are the same size, so they can simply divide numerators.
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L6: Understand Division with Fractions
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Part 3: Guided Practice
Lesson 6
At a Glance Part 3: guided Practice
Students demonstrate their understanding of dividing a fraction by a fraction.
connect it talk through these problems as a class, then write your answers below.
Step By Step
17 explain: Look at the model below. Write the division equation that the model
represents. Explain how to find the quotient using the model.
• Discuss each Connect It problem as a class using the discussion points outlined below.
5 6
0
1
0
1
5 4 1 5 10 ; the bottom number line is divided into twelfths and shows 6 ··· 12 ·· there are 10 twelfths in 5. 6 ··
Explain: • Ask, What does the top number line represent? 3 5 4
18 analyze: Sam said that 3 4 1 equals 3 . Draw a model and use words to explain why 2 ·· 4 8 ·· ··
6 ··
Sam’s statement is not reasonable.
• Ask, What does the bottom number line represent? [twelfths]
1 2
0
0
• Ask, What do the circled portions represent?
3 2
1
1
2 4
6 4
3 1 Possible explanation: 4 asks how many fourths are in three halves. there
[Groups of 1 in 5 . There are 10.] 12 ··
Lesson 6
2 ··
4 ··
is more than 1 group of fourths in three-halves so the quotient must be
6 ··
3
. the greater than 1. the quotient could not be a fraction less than 1, like ·· 8
Analyze:
3 4 1 5 6. model shows that ·· 2 ·· 4
• Students may draw an area model or a number line.
19 justify: Show that 2 4 4 5 3 by using a model. Explain why the answer is greater 6 ··
than the number you started with.
• Remind students that the problem is asking how many
0
groups of 1 s are in 3 . The quotient must be greater 4 ··
2 ··
than one because there is more than one 1 in 3 . 4 ··
1
2
Possible explanation: When you divide a given number by a fraction less than 1, the quotient is always greater than the given number.
2 ··
Justify:
56
L6: Understand Division with Fractions
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• Read the problem as a class. Remind students that the problem is asking how many groups of 4 are in 2. 6 ··
• Students should draw a number line or area model to model the problem.
SMP Tip: Students need many opportunities to connect and explain the connections between different representations. Encourage students to compare the number line and area models and explain how they help to solve the problem (SMP 4).
L6: Understand Division with Fractions
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63
Part 4: Common Core Performance Task
Lesson 6
At a Glance Students choose a fraction division word problem and solve by writing an expression and drawing a model.
Part 4: common core Performance task
Lesson 6
Put it together 20 Use what you have learned to complete this task.
Step By Step • Direct students to complete Put It Together. • As students work, walk around to assess their progress and understanding, answer their questions, and give additional support, if needed. • If time permits, have students share their problem with the class and discuss how they solved it.
Scoring Rubrics See student facsimile page for possible student answers.
Choose one of the following problems to solve. Circle the problem you choose. Greg made 2 gallon of lemonade and plans to share it equally among 4 friends. 3 ··
How much lemonade will each friend get? Keisha plans to run 4 miles this week. If she runs 2 of a mile each day, how many 3 ··
days will it take her to run 4 miles? Will she be able to run 4 miles in a week? a Write a division expression and draw a model to represent the problem. Division expression for the first problem: 2 4 4 3 ··
Division expression for the second problem: 4 4 2
3 ··
Models will vary, but should represent the problem. b Estimate what you think the quotient will be. Will the quotient will greater than or less than the dividend? How do you know? For the first problem: the quotient will be less than 2 because 2 divided 3 ··
3 ··
by 4 means dividing 2 into 4 parts; each part will be less than 2. the 3 ··
3 ··
quotient will be a fraction less than 2. 3 ··
For the second problem: the quotient will be greater than 4 because 4 divided by 2 means finding how many groups of 2 are in 4. there are 4
A
3 ··
Points Expectations 2 1 0
The student wrote a division expression and drew a model to represent the problem. The student only draws the model or the expression, does not include both.
3 ··
groups of 1 in 4. 2 is less than 1 so there are more than 4 groups of 2 in 4. 3 ··
3 ··
c Use your model to explain how to find the quotient and what the quotient means. For the first problem: 2 4 4 5 2 or 1. this means each of greg’s 4 friends 3 ··
For the second problem: 4 4 2 5 6. this means it will take keisha 6 days to 3 ··
run 4 miles. yes. she will be able to run 4 miles in a week.
The model or expression is incorrect. ©Curriculum Associates, LLC Copying is not permitted.
C
64
6 ··
6 ··
L6: Understand Division with Fractions
B
12 ···
will get 1 gallon of lemonade.
57
Points Expectations 2
Student uses the model to explain how to find the quotient and what the quotient means.
1
Student does not explain how to find the quotient or what the quotient means.
0
Student provides an incorrect explanation or response.
Points Expectations 2
The student estimates what they think the quotient should be and proves if it will be greater than or less than the dividend.
1
The student does not explain why he/she thinks the quotient would be greater than or less than the dividend.
0
The student estimates incorrectly or gives an incorrect explanation.
L6: Understand Division with Fractions
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Differentiated Instruction
Lesson 6
Intervention Activity
On-Level Activity
Explore fraction division on a number line.
Make a fraction division handbook or poster.
Materials: painter’s tape, index cards, markers, string
Guide students to create a three-page “handbook” or three-part poster highlighting the steps to solving a fraction division problem.
Use painter’s tape to create a number line on the floor across the classroom. Label the number line 1 through 5 with index cards spaced evenly apart on the number line. Write the problem 5 4 1 on 3 ··
the board.
Examples should include dividing a fraction by a whole number, dividing a whole number by a fraction, and dividing a fraction by a fraction. Each example should include an expression, a model, and an explanation for how to solve the problem.
Ask students to explain how the number line can be used to solve the problem. Ask students to use painter’s tape to divide the line into 1 s. Ask them to 3 ··
explain why 5 4 1 is 15. 3 ··
Next, ask students to model 5 4 2 . Students should 3 ··
use string to circle groups of 2 . There will be 3 ··
7 groups of 2 and 1 left over. The quotient would be 3 ··
3 ··
7 1 . Help students to realize that 1 is half of 2 , so 2 ··
they have 7 and 1 groups.
3 ··
3 ··
2 ··
Continue to model using the number line for problems such as 10 4 1 , then 10 4 2 . 3 ··
3 ··
Challenge Activity Think about dividing whole numbers by fractions and dividing fractions by whole numbers. Tell students, Yolanda wants to cut 1 -foot pieces from a 4-foot rope. Ask, How would you record this number 2 ··
sentence? [4 4 1 5 8] What do you know about the quotient? [It will be larger than the dividend.] How many 2 ··
pieces will she cut? [8] Tell students, Now Yolanda wants to cut 4-foot pieces of rope from a 1 -foot piece of rope. How would you record this
2 ·· 1 1 number sentence? 1 4 4 5 2 What do you know about this quotient? [It is smaller than dividend.] Is it possible to 2 8 ·· ·· cut 4 feet pieces of rope from a 1 foot piece of rope? [No, so the real world answer is 0.] 2 ··
Ask students to think of other real-world situations in which they would use division with fractions. Work with students to determine that their situations make sense.
L6: Understand Division with Fractions
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65
Develop Skills and Strategies
Lesson 7
(Student Book pages 58–69)
Divide with Fractions Lesson Objectives
The Learning Progression
• Solve word problems using division of fractions.
In Grade 5, students learn to understand fractions as division and to divide whole numbers by unit fractions.
• Write an equation to solve a problem using division of fractions. • Write a story problem that will use division of fractions.
Prerequisite Skills • Know that multiplication and division are inverse operations. • Know that division is either fair sharing (partitive) or repeated subtraction (quotative).
In Lesson 6, students built upon the understanding from Grade 5 using models to show division of fractions. In this lesson, students continue to build upon their knowledge by using visual models and equations to divide whole numbers by fractions, fractions by fractions, and mixed numbers by fractions to solve word problems. In Grade 7, students will continue their work with fractions to include all rational number operations.
• Divide with whole numbers.
Teacher Toolbox
• Divide a whole number by a fraction. • Model division with manipulatives, diagrams, and story contexts.
Teacher-Toolbox.com
Prerequisite Skills
Ready Lessons
Vocabulary
Tools for Instruction
multiplicative inverse: a number which when multiplied by x yields the multiplicative identity, 1
Interactive Tutorials
6.NS.A.1
✓ ✓
✓ ✓ ✓✓
reciprocal: the multiplicative inverse of a number; with fractions, the numerator and denominator are switched
CCSS Focus 6.NS.A.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for 2 4 3 and use a 3 ··
4 ··
visual fraction model to show the quotient; use the relationship between multiplication and division to explain that 2 4 3 5 8 3 ··
4 ··
9 ··
because 3 of 8 is 2 . 1 In general, a 4 c 5 ad . 2How much chocolate will each person get if 3 people share 1 lb of chocolate equally? 4 ··
9 ··
3 ··
b ·
d ·
bc ··
2 ··
How many 3 -cup servings are in 2 of a cup of yogurt? How wide is a rectangular strip of land with length 3 mi and area 1 square mi? 4 ··
3 ··
4 ··
2 ··
STANDARDS FOR MATHEMATICAL PRACTICE: SMP 1–4, 7, 8 (see page A9 for full text)
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L7: Divide with Fractions
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Part 1: Introduction
Lesson 7
At a Glance
Develop skills and strategies
Students read a word problem and explore dividing a whole number by a fraction using a model.
Lesson 7
Part 1: introduction
ccss 6.ns.a.1
Divide with Fractions
Step By Step
in the previous lesson, you learned what dividing by fractions means. in this lesson you will divide with fractions to solve problems. take a look at this problem.
• Tell students that this page models dividing whole numbers by fractions using a visual model.
Charlie is growing vegetables in planters. He has 4 bags of soil and uses 2 of a bag 3 ··
of soil to fill each planter. How many planters can he fill?
• Have students read the problem at the top of the page.
explore it use the math you already know to solve the problem.
• Work through Explore It as a class.
Think of the number of planters that Charlie can fill as how many 2s are in 4. Will that 3 ··
number be greater than or less than 4? Explain your reasoning.
• Ask students how they determined whether the number of planters would be greater or less than 4.
the number of planters will be greater than 4. When you divide a given number by a number less than 1, the answer will be greater than the given number. The model below represents the 4 bags of soil. Draw horizontal lines to divide each bag into thirds.
ELL Support Use the diagram on the page to review the words divisor, dividend, and quotient. Throughout the lesson, use models or manipulatives to demonstrate concepts and processes. Allow students to use the models to demonstrate their learning.
Circle and count groups of 2 in the model. How many did you circle? 3 ··
3 ··
you are trying to find how many 2s are in 4. 3 ··
How many planters can Charlie fill?
Visual Model Model dividing by a number less than 1. • Draw 2 identical circles on the board. Write 2 4 1 . 4 ··
• Ask, Is the divisor, 1 , greater than or less than 1? 4 ·· [less] • How many 1 s are in 2 circles? Let a volunteer draw 4 ··
lines in the circles to show fourths. Ask, Which is greater: the number of 1 parts or the number of 4 ··
whole circles? [The number of 1 parts is more.] 4 ··
• Write 1 4 1 5 . Ask, Will the quotient be greater 2 ··
4 ··
than or less than 1? [greater than 1] Then let a volunteer draw a model to illustrate.
L7: Divide with Fractions
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6
Explain how the model helped you solve the problem. you can count the groups of 2 that are in 4.
SMP Tip: Students map important quantities in the problem to the diagram as a way of understanding dividing with fractions (SMP 4). Students need many opportunities to explain the connections between different representations. Have students explain how the model helps them solve the problem.
6
Why do you circle groups of 2 to represent this problem? 3 ·· each planter holds 2 of a bag of soil, so each circled group fills one planter.
3 ··
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L7: Divide with Fractions
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Mathematical Discourse • How does using a model help you solve the problem? Students may answer that drawing the model helps them to see and count the groups. • Explain, in your own words, dividing fractions with a model. Responses should discuss drawing the dividend and dividing it into groups of the divisor. • Why does it make sense that the quotient is greater than the dividend when you divide with a fraction less than 1? If the divisor is a fraction greater than 1, will the quotient be greater or less than the dividend? Responses should show understanding that taking out groups that are less than one whole will mean that there are more groups than the dividend. Dividing by a fraction greater than 1 will result in fewer groups than the dividend.
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Part 1: Introduction
Lesson 7
At a Glance Part 1: introduction
Students explore solving a division problem using multiplication.
Lesson 7
Find out More When you found the number of 2 s that are in 4, you were dividing. You are solving the
Step By Step
3 ··
problem 4 4 2. You can solve this problem by multiplying. 3 ··
You know that multiplication and division are related. 4 divided by 2 is the same as 1 of 4,
• Read Find Out More as a class.
2 ··
or multiplying 4 by 1 . 2 ··
• Remind students that a fraction and its reciprocal must have a product of 1. • Point out that when you multiply fractions you multiply the numerators, then multiply the denominators, and then simplify if possible.
44252
Think of 2 as 2. Dividing by 2 is the
43152
same as multiplying by 1.
1 ··
1 ··
2 ··
2 ··
When dividing with unit fractions, you learned that dividing 4 by 1 is the same as 3 ·· multiplying 4 by 3. Dividing by 1 is the same as
4 4 1 5 12
3 ··
3 ··
multiplying by 3 or 3.
4 3 3 5 12
1 ··
Dividing with any fraction works the same way. Dividing 4 by 2 is the same as 3 ··
multiplying 4 by 3. 2 ··
• Discuss Reflect. Guide students to think about dividing as being the same as multiplying by the reciprocal (multiplicative inverse).
Dividing by 2 is the same as
44256 3 ·· 4 3 3 5 12 2 ·· 2 ··
3 ··
multiplying by 3. 2 ··
56
You can solve any division problem using multiplication. To divide by any number, you can multiply by its multiplicative inverse, which is also known as the reciprocal.
Visual Model
reflect 1 Explain how you can solve this division problem by using multiplication.
Use a model to understand using reciprocals in division.
642 3 ·· Dividing by 2 is the same as multiplying by 3. so, you can solve 6 divided by 2 3 ··
2 ··
by multiplying 6 by 3 . 6 4 2 5 6 3 3 5 18 = 9 2 ··
3 ··
2 ··
3 ··
2 ···
• Help students understand why they can solve any division problem by multiplying the dividend by
L7: Divide with Fractions
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59
the reciprocal of the divisor. • Draw this model on the board for 5 4 1 : 3 ··
1 3
Real-World Connection Ask students to think of everyday places or situations where people might need to divide by fractions. Encourage them to share their ideas with the class.
1
1
1
1
1
• Explain that the expression can be read as “how
Examples: cooking, making crafts, measuring, building structures
many groups of 1 are in 5?” Show on the model 3 ··
that 1 contains 3 groups of 1 , and there are 5 3 ··
groups of 1 in 5. The total number of groups of 1 in 5 is simply 5 3 3 5 15.
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3 ··
L7: Divide with Fractions
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Part 2: Modeled Instruction
Lesson 7
At a Glance Part 2: Modeled instruction
Students read a word problem and explore how to divide a whole number by a fraction using a bar picture and by modeling the problem using words and an equation.
Lesson 7
read the problem below. then explore how to divide a whole number by a fraction. Kelly drank 2 of the water in her bottle. She drank 3 cups of water. How many 5 ··
total cups of water were in her bottle?
Step By Step • Read the problem at the top of the page as a class.
Picture it
• Read and discuss Picture It. Ask, What does the shaded part of the bar show? [how much of the whole bottle Kelly drank, or two fifths]
The bar represents Kelly’s water bottle. You can divide the bar into fifths and shade 2 to 5 ·· represent the amount of water Kelly drank, 3 cups.
you can draw a picture to understand the problem.
? cups
SMP Tip: Students reason abstractly when they
3 cups
analyze a problem and represent it as an equation with a missing factor in order to find a solution (SMP 2). Ask students to explain how their equations or models represent the context of the problem.
112 cups
Model it you can use words and equations to understand the problem. 2 of the total amount of water equals 3.
5 ··
2
of
the total amount of water
equals
3
2 5 ··
3
?
5
3
5 ··
• Read and discuss Model It. Walk through each step to be sure students understand how to use the inverse operation.
Hands-On Activity
112 cups
To solve a missing factor problem like 2 3 ? 5 3, you can divide. 5 ··
?5342
5 ··
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L7: Divide with Fractions
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Make a model to show how many 2 s are in 4. 3 ·· Materials: sheet of paper for each student, pencils
• Tell students they will make a model to show how many 2 s are in 4. 3 ·· • Give each student a sheet of paper. Have them fold it into 4 equal parts and draw lines on the folds to show 4 equal sections. Tell them this will be a model for 4. • Ask, What is the first thing you could do to the model to start showing how many 2 s are in 4? [Draw lines to 3 ·
divide each of the 4 whole sections into thirds.] What might you do next? [Circle each group of 2 of the thirds and count them.]
Mathematical Discourse • What is another way you could solve the problem regarding Kelly’s water bottle on page 60? Responses may include dividing 3 cups in half to find 1 of the amount in the bottle, and then 5 ··
multiplying by 5 to find the total amount in the bottle. • How is dividing fractions similar to dividing whole numbers? Listen for responses that indicate dividing a quantity into groups.
• Let students finish their models. Ask them what the solution is. Suggest they label their model with the equation solved: 4 4 2 5 6. 3 ··
L7: Divide with Fractions
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Part 2: Guided Instruction
Lesson 7
At a Glance Students revisit the problem on page 60 to learn how to solve it using the bar picture and the equation model.
Part 2: guided instruction
Lesson 7
connect it now you will solve the problem from the previous page using the picture and model.
Step By Step • Read Connect It as a class. Be sure to point out that the problems refer to the problem on page 60. • For problem 4, remind students to change 1 1 to an 2 ··
improper fraction before multiplying by 5. Multiply the denominator times the whole number, and then add the numerator. The result is the numerator, and the denominator stays the same. • In problem 5, review how to change an improper fraction to a mixed number: Divide the numerator by the denominator to get a whole number, the remainder is the new numerator, and the denominator stays the same.
2 Look at Picture It on the previous page. Why do you divide the bar into fifths? the problem says kelly drank 2 of the bottle, so you need to show fifths. 5 ·· 3 How can you use Picture It to find out how many cups of water are in the bottle? since 3 cups 5 2, you know that 1 is 1 1 cups. you can multiply 1 1 by 5 to find 5 5 2 2 ·· ·· ·· ··
the total number of cups of water that were in the bottle. 4 How many total cups of water were in Kelly’s bottle? 5 3 1 1 5 7 1 cups. 2 2 ·· ·· 5 Look at Model It on the previous page. Find 3 4 2 . Show your work. 5 ·· 3 4 2 5 3 3 5 5 15 cups or 7 1 cups. 5 2 ··· 2 2 ·· ·· ·· 6 Explain how to use multiplication to divide a whole number by a fraction.
Dividing by a fraction is the same as multiplying by its inverse. Multiply the whole number by the reciprocal of the fraction.
try it use what you just learned about dividing with fractions to solve these problems. show your work on a separate sheet of paper. 8 servings 7 How many 1 1 -cup servings are there in 12 cups of juice? 2 ·· 8 It takes Emily 9 minutes to bicycle 3 of the way to school. How many minutes does it 10 ·· take Emily to bicycle all the way to school?
30 minutes
• Have students work through Try It on their own. Then discuss with them how they solved the problems. 61
L7: Divide with Fractions
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Try It Solutions 7 Solution: 8 servings; Students solve the problem by using the equation 12 4 3 5 ? or a drawing such 2 ··
as 12 cups with lines dividing each cup into halves and circled groups of 1 1 . 2 ··
ERROR ALERT: Students who wrote 18 servings forgot to multiply by the reciprocal of the divisor. Remind them that they need to find the reciprocal of the divisor before multiplying.
8 Solution: 30 minutes; Students may use the equation 9 4 3 5 ? They may use a drawing such as a bar 10 ··
divided into tenths with 3 shaded to model the 10 ··
distance she went in 9 minutes. This would show that 1 is equal to 3 minutes; 3 3 10 5 30 minutes. 10 ··
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L7: Divide with Fractions
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Part 3: Modeled Instruction
Lesson 7
At a Glance Part 3: Modeled instruction
Students read a word problem and explore how to divide a fraction by a fraction using a double number line and by modeling the problem using words and an equation.
Lesson 7
read the problem below. then explore how to divide a fraction by a fraction. Eli ran 3 of a mile. Every 1 of a mile, he jumped over a hurdle. There was a final 4 ··
8 ··
hurdle at the 3 mile mark. How many hurdles did Eli jump over? 4 ··
Picture it
Step by Step
you can draw a picture to understand the problem. The top number line shows the distance Eli ran, 3 mile.
• Read the problem at the top of the page as a class.
4 ··
The bottom number line shows the number of 1 s that are in 3. 8 ··
• Discuss Picture It:
1 4
0
Ask, What does the top number line represent? [the distance divided into fourths]
2 4
4 ··
3 4
1
0
1
Model it
Ask, What does the bottom number line represent? [the distance divided into eighths]
you can use words and equations to understand the problem. Think: How many 1 s are in 3? 8 ··
4 ··
Use division to find how many 1 s are in 3. 8 ··
• Discuss Model It. Ask, What does the question mark in the equation represent? [the number of hurdles Eli jumped over during his 3 -mile run] 4 ··
4 ··
3 4 ··
divided into
1s 8 ··
equals
the number of hurdles
3 4 ··
4
1 8 ··
5
?
3415? 4 ·· 8 ··
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L7: Divide with Fractions
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Mathematical Discourse • Which method for dividing fractions do you prefer? Why? Are there situations when one method may be easier to use than another? Encourage students to support their opinions and to listen to the opinions of others. Point out that there is no correct answer, and that different students may have different preferences. • Can you think of another way to describe dividing fractions? Explain. Encourage students to suggest ideas or knowledge on other ways to divide fractions.
L7: Divide with Fractions
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Part 3: Guided Instruction
Lesson 7
At a Glance Part 3: guided instruction
Students revisit the problem on page 62 and solve it using the double number line and the equation model.
Lesson 7
connect it now you will solve the problem from the previous page using the picture and model.
Step by Step
9 Look at Picture It. Why is the top number line divided into fourths? Why is the bottom
number line divided into eighths? the top number line is divided into fourths to mark 3, the total distance eli
• Read and discuss Connect It as a class. Refer to the problem on page 62.
4 ··
ran. the bottom number line is divided into eighths because eli jumped over a hurdle every eighth of a mile.
• For problem 13, remind students that they should find the inverse (reciprocal) only of the divisor and not the dividend. Also remind them they should simplify improper fractions to a mixed- or wholenumber answer.
10 Explain how Picture It helps you figure out how many hurdles Eli jumped over. i could count how many 1 s are in 3. 8 4 ·· ·· 11 How many hurdles did Eli jump over?
6
12 Look at Model It. Explain how to use multiplication to find 3 4 1 . 4 8 ·· 8 ·· 1
Dividing by is the same as multiplying by its inverse, or 8, so 3 4 1 is the 8 ··
1 ··
4 ··
8 ··
same as 3 3 8. 4 ··
1 ··
3 3 8 5 24 or 6 4 ·· 1 ··· 4 13 Evaluate 3 4 1 . Show your work. ·· 4 ·· 8 ··
• Have students work through Try It on their own. Then discuss their answers and solutions.
14 Explain how to divide a fraction by a fraction.
to divide a fraction by a fraction, multiply the first fraction by the reciprocal of the second fraction.
SMP Tip: Give students multiple opportunities to
try it
solve and model problems. Students use repeated reasoning to understand algorithms and make generalizations about patterns (SMP 8).
use what you just learned to solve these problems. show your work on a separate sheet of paper. 15 Keisha cuts a 2 -foot rope into 1 -foot pieces. How many pieces of rope did 3 12 ·· ··
8
she cut?
16 Jade makes half a liter of lemonade. She pours 1 liter of lemonade into each glass. 10 ··
How many glasses is Jade able to fill?
5
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L7: Divide with Fractions
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Try It Solutions 15 Solution: 8; Students may draw a double number line or may use the standard algorithm to solve the problem. 2 4 1 5 2 3 12 5 24 5 8 3 ··
12 ··
3 ··
1 ··
3 ··
ERROR ALERT: Students who wrote 18 pieces of rope may have multiplied the inverse of both the dividend and the divisor. Remind students that they should only multiply by the inverse (reciprocal) of the divisor. Review Find Out More on page 59 of this lesson to help students understand why multiplying by the reciprocal is mathematically valid. 16 Solution: 5; Students may draw a bar picture and shade half of the figure, then draw lines to divide the figure into tenths. 5 parts would be shaded. Or, they may use the standard algorithm. 1 4 1 5 1 3 10 5 5 2 ··
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10 ··
2 ··
1 ··
L7: Divide with Fractions
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Part 4: Modeled Instruction
Lesson 7
At a Glance Part 4: Modeled instruction
Students explore how to divide a mixed number by a fraction using bar pictures and by modeling the problem using words and an equation.
Lesson 7
read the problem below. then explore how to divide a mixed number by a fraction. Mari divides 1 4 pounds of granola into 2 -pound bags for a bake sale. How many 5 ··
5 ··
bags of granola can she sell?
Step by Step
Picture it
• Read the problem at the top of the page as a class.
you can draw a picture to understand the problem. The shaded bars represent 1 4 pounds of granola.
• Discuss Picture It:
5 ··
Each circle shows a 2-pound bag of granola.
Ask, How do the shaded bars in the first picture
5 ··
represent 1 4 pounds of granola? [One whole and 4 5 ··
The remainder is half of 25 .
1 bag of granola
more parts are shaded.] Model it
you can use words and equations to understand the problem.
Ask, What does each circle in the second picture
Think: How many 2 s are in 1 4? 5 ··
represent? [Each circle represents one 2 -pound bag 5 ··
5 ··
Use division to find how many 2 s are in 1 4. 5 ··
14
of granola.]
5 ··
divided into
14 5 ·· 4 1 425? 5 ·· 5 ·· 9425? 5 ·· 5 ··
• In Picture It, some students might be confused by the circle that contains part of the whole bar and part of
4
5 ··
2s 5 ··
equals
the number of bags of granola
2 5 ··
5
?
the partial bar. Suggest to students that they think of the entire 1 bar plus 4 bar as one entity. 5 ··
• Discuss Model It. Remind students that they must write the mixed number in the dividend as an improper fraction before dividing.
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L7: Divide with Fractions
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Mathematical Discourse • Mari is selling bags of granola to raise money, so she wants to have as many bags as possible to sell. What else might she think about when dividing up the granola? Listen for responses that show students making connections to personal experiences to make sense of the problem. They might discuss how the size of each bag might make a difference to buyers: Customers might not buy if they think there is not enough granola in a bag. Students might also mention cost to customers or the amount left over after filling the bags.
L7: Divide with Fractions
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Part 4: Guided Instruction
Lesson 7
At a Glance Part 4: guided instruction
Students revisit the problem on page 64 and solve it using the bar picture and the equation model.
Lesson 7
connect it now you will solve the problem from the previous page using the picture and model.
Step by Step
17 Look at Picture It. Why do you circle groups of 2 to solve this problem? 5 ·· Mari divided the granola into 2 -pound bags. you are trying to find how 5 ·· 2 4 many s there are in 1 . 5 5 ·· ··
• Discuss Connect It as a class. Point out that Connect It refers to the problem on the previous page.
18 Count the circles. How many 2 -pound bags of granola can Mari sell? 5 ··
4
19 What fraction of a bag would the remaining 1 pound of granola be? Explain your 5 ··
• When discussing problem 20, be sure students make the connection between the bar model and the mathematical process for changing a mixed number to an improper fraction. For students having trouble understanding why writing a mixed number as an improper fraction is mathematically valid, ask students to count the total number of shaded squares (9) in the first bar picture and point out that the bars are divided into fifths, so there are 9 fifths altogether.
answer. each bag is 2 pounds. the remaining 1 pound of granola is half of 2, so the 5 ··
5 ··
5 ··
remainder is 1 of a bag. 2 ··
20 Look at the Model It. Explain how you know 1 4 is equal to 9 . 5 5 ·· ·· 1 4 is 5 1 4, which equals 9. 5 ·· 5 ·· 5 5 ·· ·· 21 Explain how to use multiplication to evaluate 9 4 2 . 5 ·· 5 ·· Dividing by 2 is the same as multiplying by its inverse, 5, so 9 4 2 is the same 5 2 5 ·· 5 ·· ·· ·· as 9 3 5. 5 ·· 2 ·· 9 3 5 5 45 5 4 5 or 4 1 5 ·· 2 ··· 10 10 2 ··· ·· 22 Evaluate 9 4 2 . Show your work. ·· 5 ·· 5 ·· 23 Explain how to divide with mixed numbers.
Dividing with mixed numbers is just like dividing with fractions. you can write mixed numbers as fractions, then multiply by the reciprocal of the divisor.
• Have students work through Try It on their own. Let volunteers share their solutions and answers with the class. Clear up misconceptions and discuss any questions students may have.
try it use what you just learned to solve these problems. show your work on a separate sheet of paper. 24 A recipe requires 3 of a cup of water. Kyle has a 1 1 -cup measuring cup. How much of 4 2 ·· ·· 1 of the cup the measuring cup is filled with water? 2 ·· 2 1 servings 1 5 2 ·· 25 How many -cup servings are in cup? 3 6 ·· ··
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L7: Divide with Fractions
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Try It Solutions 24 Solution: 1 of the cup; Students may draw a model 2 ··
that shows 1 1 and divide it into fourths. 3 of a cup 2 ··
4 ··
would be half of the model.
25 Solution: 2 1 ; Students may write an equation. 2 ··
5 4 1 5 5 3 3 5 15 5 2 1 6 ··
3 ··
6 ··
1 ··
6 ··
2 ··
ERROR ALERT: Students who wrote 2 transposed 5 ··
the fractions and found 1 4 5 . Encourage students 3 ··
6 ··
who made this error to draw a model to help them visualize the problem.
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L7: Divide with Fractions
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Part 5: Guided Practice
Lesson 7
Part 5: guided Practice
Lesson 7
study the student model below. then solve problems 26–28. Student Model
Lydia bought 2 1 gallons of paint and used 1 1 gallons of paint.
The student divided the number of gallons of paint used, 1 1, by the 2 ·· gallons of paint she bought, 2 1.
2 ··
2 ··
Part 5: guided Practice
27 A marathon is 131 miles long. If 4 people divide up the distance 5 ···
Dividing by 4 is the same as multiplying by what number?
equally, how many miles does each person need to run? Show your work.
What fraction of the paint did she use? 1314 4 5 1314 4 5 ···· 5 1313 1 5 4·· ···· 5 131 20 ···· 5 6 11 20 ···
Look at how you can show your work using a model.
5 ····
think: What fraction of 2 1 is 1 1? 2 ··
2 ··
2 ··
some fraction of 21 equals 1 1. 2 ··
2 ··
3 21 5 11 2 2 ·· ·· 1 to solve ? 3 2 5 11 , divide. 2 2 ·· ·· ? 5 11 4 21 2 2 ·· ·· 5345 2 ·· 2 ·· 3 4 5 5 3 3 2 ; 3 3 2 5 6 or 3 2 ·· 2 ·· 2 ·· 5 ·· 2 ·· 5 10 5 ·· ··· ·· Lydia used 3 of the paint she bought. 5 ·· Solution: ?
Pair/share How could you justify your answer with a picture?
Will the answer be less than 1 or greater than 1? Why?
Lesson 7
26 Lexi has planted seeds in 3 of the garden. She used 1 pound of 5 2 ·· ··
seeds. How many pounds will she use for the entire garden? Show your work. 1 4 3 5 1 3 5; 1 3 5 5 5. 2 ·· 5 ·· 2 ·· 3 ·· 2 ·· 3 ·· 6 ··
Pair/share
Solution:
How is this problem different from the others you’ve seen in this lesson?
each person will need to run 6 11 miles. 20 ···
28 Which of the following problems can be solved by finding 4 4 2 ? 3 ·· a 4 people equally share 2 of a pizza. How much of the pizza does 3 ··
What kind of picture could represent the expression?
each person eat?
b
How many 2 -cup servings of soup are in 4 cups of soup?
c
A pie recipe requires 2 pounds of apples. How many apples are 3 ·· needed for 4 pies?
3 ··
D A family ate 2 of a 4-foot sandwich. How much did they eat? 3 ··
Arthur chose a as the correct answer. How did he get that answer? arthur confused the dividend and divisor. the situation he
Pair/share How did you and your partner decide which fraction is the dividend and which is the divisor?
66
chose could be solved by dividing 2 by 4. 3 ··
Pair/share Does Arthur’s answer make sense?
Lexi will use 5 pound of seeds for the entire garden. 6 ·· Solution:
L7: Divide with Fractions
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L7: Divide with Fractions
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At a Glance
Solutions
Students practice solving problems that require dividing a whole number by a fraction, dividing a fraction by a fraction, and dividing a mixed number by a fraction.
Ex Using a model of words and an equation is one way for students to show their solution to the problem.
Step by Step • Ask students to solve the problems individually on pages 66 and 67. • In the student model at the top of the page, call attention to writing each mixed number as an improper fraction before multiplying. • When students have completed each problem, have them Pair/Share to discuss their solutions with a partner or in a group.
26 Solution: Lexi will use 5 pound of seeds for the 6 ··
entire garden; Students could solve the problem by using an equation. 1 4 3 5 1 3 5 5 5 (DOK 1) 2 ··
5 ··
2 ··
3 ··
6 ··
27 Solution: Each person will need to run 6 11 miles; 20 ··
Students could solve the problem by using an equation. 131 4 4 5 131 1 5 131 3 5 6 11 (DOK 1)
5 ···
5 ···
4 ··
20 ···
20 ··
28 Solution: B; Students must recognize the language as a division problem “how many are in ?” Explain to students why the other two answer choices are not correct: C is not correct because the problem asks “how many are needed for,” which is a multiplication problem. 2 3 4 3 ·· D is not correct because the problem states 2 “of” 4, 3 ··
which is a multiplication problem. 2 3 4 (DOK 3) 3 ··
L7: Divide with Fractions
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Part 6: Common Core Practice
Part 6: common core Practice
Lesson 7
2
3
Part 6: common core Practice 4
Solve the problems.
1
Lesson 7
3 411? What is the value of the expression } } 8 2 9 A }} 16 6 B } 8
C
4
D
} 4
Write each expression in the correct column to show whether the quotient is less than, greater than, or equal to 1. 3
1
1
4} } 4 2 2
1
4
3
4} } 5 3
quotient is less than 1
3
4} } 4 2
4 }} } 9 27
5
20
4 }} } 3 6 19
3
4 2} }} 8 8
quotient is equal to 1
143 } 4 2 } 5 20 4 } 6 3 }}
quotient is greater than 1
3 19 4 2} }} 8 8
3
1
4 } 4 } 2 24 1 } 9 }} 27
3
4
4 } 5 3 }
1
1 ?” Find the expression that does NOT answer the question: “What fraction of 8 is 2 } 2 1 48 A 2} 2 5 31 B } 8 ·· 2 1 C 8 4 2} 2 1 D ? 3 8 5 2} 2
The area and one dimension of a piece of land are given. From the list, write the fraction inside each box that represents the second dimension of the piece of land described.
3 } 7
7 } 8
4 } 7
4 } 9
5 } 7
} 9
5 } 8
Lesson 7
4 } 7 7 } 8
5
7
4 } 9
} 9
1 square mile. The area of a rectangular piece of land is } 2 7 mile. One dimension of this piece of land is } 8 The area of a piece of land that is in the shape of a 1 square mile. One dimension of this piece of triangle is }} 12 4 mile. land is }} 21 2 square mile. The area of a rectangular piece of land is } 3 1 miles. One dimension is 1} 2
5
Explain the difference between dividing in half and dividing by half using pictures, models, or numbers.
Possible answer: Dividing in half means dividing into 2 parts or multiplying by 1. 2 ··
Dividing by half means finding how many 1 s there are in the number. if you divide 4 2 ··
in half, you get 2. if you divide 4 by 1 , you get 8. there are eight 1 s in 4. 2 ··
6
2 ··
3 . Draw a model and use multiplication to show Write a story to represent the expression 6 4 } 4 the solution. Explain how the dividend, divisor, and quotient relate to the story.
stories will vary. Possible answer: a recipe calls for 6 cups of flour. if the only
Possible student model: 3 4
3 4
3 4
3 4
3 4
3 4
3 4
3 4
measuring cup you have is 3 cup, how many 4 ··
times will you have to fill the measuring cup
1
1
1
1
1
1
to get 6 cups of flour? 6 4 3 5 6 3 4 4 3 ·· ·· 5 24
3 ···
58
self check Go back and see what you can check off on the Self Check on page 51. 68
L7: Divide with Fractions
L7: Divide with Fractions
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At a Glance Students divide by fractions to solve word problems that might appear on a mathematics test.
Solutions 1 Solution: D; 3 4 3 5 3 3 2 5 6 5 1 (DOK 1) 8 ··
2 ··
8 ··
3 ··
24 ··
4 ··
2 Solution: C; transposed the dividend and divisor.
©Curriculum Associates, LLC Copying is not permitted.
69
4 Solution: See student book page above for solution; To divide fractions, multiply the first fraction by the reciprocal of the second fraction. (DOK 2) 5 See student book page above for possible student explanation. (DOK 3) 6 See student book page above for possible student model and explanation. (DOK 3)
Correct reasoning should be ? of 8 5 2 1 , or ? 5 2 1 4 8. (DOK 2)
2 ··
2 ··
3 Solution: See student book page above for solution; Use the area formula for either a rectangle or triangle to find the unknown dimension. (DOK 2)
76
L7: Divide with Fractions
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Differentiated Instruction
Lesson 7
Assessment and Remediation • Ask students to evaluate the expression 1 1 4 2 3 2 1 4. 2 ··
3 ··
4 ··
• For students who are struggling, use the chart below to guide remediation. • After providing remediation, check students’ understanding. Ask students to explain their thinking while evaluating 2 1 4 3 3 3 1 4. 3 ··
4 ··
9 ··
• If a student is still having difficulty, use Ready Instruction, Level 6, Lesson 6. If the error is . . . 1
Students may . . .
To remediate . . .
have failed to find the multiplicative inverse (reciprocal) of the divisor before multiplying.
Remind students that they must multiply by the multiplicative
3 3 2 5 6 5 1
by 1 fourth. Ask how many fourths altogether. Point out that
2 ··
3 ··
6 ··
inverse of the divisor. For students not understanding why this works, write 6 4 1 . Draw 6 circles. Ask how to divide each circle 4 ··
6 wholes are each split into 4 parts, so 6 wholes are multiplied by 4 parts to get 24. 6 3 4 is the same as 6 4 1 . 1 ··
3 4 ··
have forgotten to include the whole number. 1 4 2 5 1 3 3 5 3
2 ··
3 ··
2 ··
2 ··
4 ··
4 ··
Encourage students to write all mixed numbers as improper fractions as the first step in setting up their computation so they won’t forget.
Hands-On Activity
Challenge Activity
Make a number line to model division.
Write problems involving division of fractions.
Materials: half or whole sheets of paper, pencils
Materials: index cards or sheets of paper, pencils
Tell students they will draw a model to solve a problem. Display this problem: “Mari has 1 1 hours 2 ·· left to prepare for the bake sale. It takes her 1 hour to 4 ·· prepare each item. How many items can she prepare?” Tell students to draw a number line on their paper and use tic marks to divide the line into halves from 0 to 2. They should label each whole and half number mark (0, 1 , 1, 1 1 , 2). Point out that the 2 2 ·· ·· problem asks how many fourths are in one and a half. Tell students to divide each half on their number line with a tic mark to create a fourth. Then tell them to circle each fourth between 0 and 1 1 . Ask students 2 ·· what each circled part represents 3 1 hour 4 and how 4 ·· many circled parts there are [6]. Ask how many fourths are in one and a half [6]. Ask how many items Mari can prepare in the time she has left. [6 items]
Give each student 1–3 index cards or sheets of paper. Tell them to make up a word problem involving division of fractions and mixed numbers for each card. Have students write the word problem on one side of the card and the solution on the other side. Tell them their solution needs to be complete enough so that someone who doesn’t know how to solve the problem can figure it out and why it works. The solution can include such things as a drawing, words explaining the process, or an equation. Have students exchange cards with another student. The other student is to solve the problem and then look at the solution and offer suggestions for changes if the student sees any. Alternatively, let students verbally show and tell how to solve the problems they’ve made up.
L7: Divide with Fractions
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77
Common Core Mathematics Teacher Resource Book 6
Table of Contents Ready® Common Core Program Overview
A6
Supporting the Implementation of the Common Core
A7 Answering the Demands of the Common Core with Ready A8 The Standards for Mathematical Practice A9 Depth of Knowledge Level 3 Items in Ready Common Core A10 Cognitive Rigor Matrix A11
Using Ready Common Core A12 Teaching with Ready Common Core Instruction Content Emphasis in the Common Core Standards Connecting with the Ready® Teacher Toolbox Using i-Ready® Diagnostic with Ready Common Core Features of Ready Common Core Instruction Supporting Research
A14 A16 A18 A20 A22 A38
Correlation Charts Common Core State Standards Coverage by Ready Instruction Interim Assessment Correlations
A42 A46
Lesson Plans (with Answers) CCSS Emphasis Unit 1: Ratios and Proportional Relationships Lesson 1
Ratios
1 3 M
CCSS Focus - 6.RP.A.1 Embedded SMPs - 4, 6
Lesson 2
Understand Unit Rate
11
M
19
M
29
M
41
M
CCSS Focus - 6.RP.A.2 Embedded SMPs - 2, 6, 7
Lesson 3
Equivalent Ratios
CCSS Focus - 6.RP.A.3a Embedded SMPs - 1, 2, 4, 5, 7, 8
Lesson 4
Solve Problems with Unit Rate
CCSS Focus - 6.RP.A.3b, 6.RP.A.3d Embedded SMPs - 2–4
Lesson 5
Solve Problems with Percent
CCSS Focus - 6.RP.A.3c Embedded SMPs - 1, 2, 4, 5, 7
Unit 1 Interim Assessment
M = Lessons that have a major emphasis in the Common Core Standards S/A = Lessons that have supporting/additional emphasis in the Common Core Standards
51
Unit 2: The Number System Lesson 6
Understand Division with Fractions
CCSS Emphasis 54 58
M
66
M
78
S/A
88
S/A
98
S/A
110
S/A
120
M
128
M
138
M
CCSS Focus - 6.NS.A.1 Embedded SMPs - 1–4, 7, 8
Lesson 7
Divide with Fractions
CCSS Focus - 6.NS.A.1 Embedded SMPs - 1–4, 7, 8
Lesson 8
Divide Multi-Digit Numbers
CCSS Focus - 6.NS.B.2 Embedded SMPs - 1, 2, 7, 8
Lesson 9
Add and Subtract Decimals
CCSS Focus - 6.NS.B.3 Embedded SMPs - 3, 6, 7
Lesson 10 Multiply and Divide Decimals CCSS Focus - 6.NS.B.3 Embedded SMPs - 2, 3, 6–8
Lesson 11 Common Factors and Multiples CCSS Focus - 6.NS.B.4 Embedded SMPs - 2, 3, 7
Lesson 12 Understand Positive and Negative Numbers CCSS Focus - 6.NS.C.5, 6.NS.C.6a, 6.NS.C.6c Embedded SMPs - 1, 2, 4
Lesson 13 Absolute Value and Ordering Numbers CCSS Focus - 6.NS.C.5, 6.NS.C.7a, 6.NS.C.7b, 6.NS.C.7c, 6.NS.C.7d Embedded SMPs - 1–4, 6
Lesson 14 The Coordinate Plane CCSS Focus - 6.NS.C.6b, 6.NS.C.6c, 6.NS.C.8 Embedded SMPs - 1, 2, 4–7
Unit 2 Interim Assessment
Unit 3: Expressions and Equations Lesson 15 Numerical Expressions with Exponents
151 154 157
M
167
M
179
M
191
M
199
M
CCSS Focus - 6.EE.A.1 Embedded SMPs - 1–8
Lesson 16 Algebraic Expressions CCSS Focus - 6.EE.A.2a, 6.EE.A.2b, 6.EE.A.2c Embedded SMPs - 1–4, 6
Lesson 17 Equivalent Expressions CCSS Focus - 6.EE.A.3, 6.EE.A.4 Embedded SMPs - 2, 3
Lesson 18 Understand Solutions to Equations CCSS Focus - 6.EE.B.5 Embedded SMPs - 1–5, 7
Lesson 19 Solve Equations CCSS Focus - 6.EE.B.6, 6.EE.B.7 Embedded SMPs - 1–4, 7
M = Lessons that have a major emphasis in the Common Core Standards S/A = Lessons that have supporting/additional emphasis in the Common Core Standards
Unit 3: Expressions and Equations (continued) Lesson 20 Solve Inequalities
CCSS Emphasis 211
M
221
M
CCSS Focus - 6.EE.B.5, 6.EE.B.8 Embedded SMPs - 1–4, 6, 7
Lesson 21 Dependent and Independent Variables CCSS Focus - 6.EE.C.9 Embedded SMPs - 1–4, 7, 8
Unit 3 Interim Assessment
Unit 4: Geometry Lesson 22 Area of Polygons
231 234 236
S/A
246
S/A
256
S/A
CCSS Focus - 6.G.A.1 Embedded SMPs - 1–7
Lesson 23 Polygons in the Coordinate Plane CCSS Focus - 6.G.A.3 Embedded SMPs - 1, 2, 4, 5, 7
Lesson 24 Nets and Surface Area CCSS Focus - 6.G.A.4 Embedded SMPs - 7, 8
Lesson 25 Volume
268 S/A
CCSS Focus - 6.G.A.2 Embedded SMPs - 1, 4
Unit 4 Interim Assessment
Unit 5: Statistics and Probability Lesson 26 Understand Statistical Questions
279 282 285
S/A
293
S/A
305
S/A
317
S/A
CCSS Focus - 6.SP.A.1 Embedded SMPs - 1, 3, 6
Lesson 27 Measures of Center and Variability CCSS Focus - 6.SP.A.2, 6.SP.A.3 Embedded SMPs - 2, 4–7
Lesson 28 Display Data on Dot Plots, Histograms, and Box Plots CCSS Focus - 6.SP.B.4 Embedded SMPs - 2–7
Lesson 29 Analyze Numerical Data CCSS Focus - 6.SP.B.5a, 6.SP.B.5b, 6.SP.B.5c, 6.SP.B.5d Embedded SMPs - 2–5
Unit 5 Interim Assessment
M = Lessons that have a major emphasis in the Common Core Standards S/A = Lessons that have supporting/additional emphasis in the Common Core Standards
328
Focus on Math Concepts
Lesson 6
(Student Book pages 52–57)
Understand Division with Fractions Lesson Objectives
The Learning Progression
• Understand the meanings of division.
In Grade 5, students divided whole numbers by unit fractions.
• Use a model to show division of fractions. • Use an understanding of multiplication of fractions to explain division of fractions.
Prerequisite Skills • Know that multiplication and division are inverse operations. • Know that division is either fair sharing (partitive) or repeated subtraction (quotative).
Students continue this understanding in Grade 6 by using visual models and equations to divide whole numbers by fractions and fractions by fractions to solve word problems. In Grade 7, students will continue their work with fractions to include all rational number operations (positive and negative). Students will build on understanding of number lines developed in Grade 6.
• Divide with whole numbers.
Teacher Toolbox
• Divide a whole number by a fraction. • Model division with manipulatives, diagrams and story contexts.
Prerequisite Skills
6.NS.A.1
✓ ✓
Ready Lessons Tools for Instruction
Vocabulary
Teacher-Toolbox.com
✓ ✓ ✓
Interactive Tutorials
There is no new vocabulary.
CCSS Focus 6.NS.A.1 Interpret . . . quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for 1 2 24 1 3 2and use a visual 3 ··
4 ··
fraction model to show the quotient; use the relationship between multiplication and division to explain that 1 2 24 1 3 25 8 because 3 3 ··
4 ··
9 ··
of 8 is 2 . (In general, 1 a 24 1 c 25 ad .) How much chocolate will each person get if 3 people share 1 lb of chocolate equally? How 9 ··
3 ··
b ·
d ·
bc ··
4 ··
2 ··
many 3 -cup servings are in 2 of a cup of yogurt? How wide is a rectangular strip of land with length 3 mi and area 1 square mi? 4 ··
3 ··
4 ··
2 ··
STANDARDS FOR MATHEMATICAL PRACTICE: SMP 1–4, 7, 8 (see page A9 for full text)
58
L6: Understand Division with Fractions
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Part 1: Introduction
Lesson 6
At a Glance
Focus on Math concepts
Students explore dividing a fraction and a whole number by a fraction.
Lesson 6
Part 1: introduction
ccss 6.ns.a.1
Understand Division with Fractions
Step By Step
What does it mean to divide a fraction by a fraction?
• Introduce the Question at the top of the page.
You know how to divide a whole number by a unit fraction. For example, you can think
• Read the description of how to divide a whole number by a unit fraction.
divide 6 into fourths and count to see there are 24 fourths in 6.
of 6 divided by 1 as “how many one-fourths are there in 6?” Using a number line, you can 4 ··
6 4 1 5 24 4 ··
0
• Ask students to explain in their own words why
1
2
3
4
5
6 4 1 is the same as 6 3 4. Ask them to use the 4 ··
6 4 1 is the same as 6 3 4, or 24 4 ··
number line illustration to make the explanation
think What does dividing a whole number by a fraction mean? circle the multiplication expression that is the same as the division expression.
Madison cuts a 6-yard length of ribbon into 3 yard pieces.
clear.
4 ··
To figure out how many pieces Madison cut, think, “How many three-fourths are in 6?”
• Have students explain a reciprocal. Remind them that the reciprocal is the same as the multiplicative inverse.
You can draw the same number line to represent the 6 yards of ribbon and divide it into fourths. 0
to explain in his or her own words how the model shows 6 4 3 .
0
4 ··
2
3
4
5
6
1
2
3
4
5
6
63458 3 ··
ELL Support
52
Discuss the difference between the phrases “dividing by” and “dividing into.” Dividing by 1 , for 4 ··
number, but dividing into fourths means to separate a number into four equal groups—i.e., multiplying by a unit fraction.
1
You can circle three 1 sections to represent 3 yard pieces. You can see there are 4 4 ·· ·· eight 3 yard pieces in 6 yards. 4 ·· 64358 4 ··
• Discuss the number line in Think and ask a student
example, asks how many groups of 1 are in a 4 ··
6
You also learned that dividing a number by a fraction is the same as multiplying the number by the reciprocal of the fraction.
Understand L6: Understand Division Division withwith Fractions Fractions
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Mathematical Discourse • When you divide a whole number by a fraction less than 1, the quotient is larger than the dividend. Can you explain why? Students might explain that since the fraction is less than 1, you would be able to take out more than one group from each whole in the dividend. Ask other students to add to the explanation and to use the same reasoning to solve 10 4 3 and explain why the answer 5 ··
makes sense.
L6: Understand Division with Fractions
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59
Part 1: Introduction
Lesson 6
At a Glance Students explore dividing a fraction by a whole number.
Part 1: introduction
Lesson 6
think What does dividing a fraction by a whole number mean?
Step By Step
Cory wants to pour 3 of a quart of juice equally into 6 glasses. This means he needs to 4 ··
divide 3 into 6 equal parts. You can represent the problem with an area model. First,
• Read Think as a class. Emphasize that in the
4 ··
you can show the 3 quart of juice. Then, you can draw vertical lines to divide the model 4 ··
into 6 equal parts.
previous problem about Madison and the ribbon, students were shown an example of dividing a whole number by a fraction: 6 4 3 . This question is about 4 ··
dividing a fraction by a whole number: 3 4 6. 4 ··
• Discuss the area model with students and have them explain how the first model represents 3 . Then have 4 ··
a student explain how the second model represents
1 2 3 4 5 6 glasses
3465 3 51 24 ·· 8 ··
4 ··
3 quart of juice divided equally into 6 glasses means Cory will pour 3 or 1 quart of juice 4 24 ·· 8 ·· ··
into each glass. 3 4 6 is the same as 3 3 1 . 4 ·· 6 ·· Cory pours 1 of 3 quart of juice into each glass.
4 ··
6 ··
4 ··
reflect
dividing the model into 6 parts. • Ask students how they know how much is in each group. Be sure students understand that 3 refers to 24 ··
the whole.
• Ask students to use the model to show why 3 5 1 . 24 ··
1 Use the number line to show and explain why 4 4 2 and 4 3 1 both equal 2 . 10 10 ·· 2 10 ·· ·· ·· 0
2 10
4 10
1
Possible explanation: see number line above. you can think of 4 divided by 10 ··· 2 as dividing 4 into 2 equal parts. this is the same as finding 1 of 4 because 10 2 10 ··· ·· ··· each half is one of the equal parts.
8 ··
• Ask students to explain to their partner why 3 4 6 4 ··
is the same as 3 3 1 . 4 ··
53
L6: Understand Division with Fractions
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6 ··
• Have students read and reply to the Reflect directive.
Real-World Connection
Visual Model Folding paper to divide a fraction by a whole number. Materials: rectangular sheet of paper • Paul uses 3 of his back yard for gardening. He 4 ··
will divide this gardening section into 3 equal parts for vegetables. Model 3 4 3 using paper. 4 ··
• Start by folding the paper into fourths and shading 3 . 4 ··
• Now fold the paper into thirds to show 3 sections for vegetables. • Unfold. Ask, How is the paper divided now? [twelfths]
Ask students to describe real world situations in which they would have to divide a fraction by a whole number. Ask students to think of a problem or situation that represents 3 4 6. Share a situation in which 3 of 4 ··
4 ··
something is divided into 6 equal groups or shared equally by 6 people. Example: Susan wants to create a picture frame in the shape of a regular hexagon. She has a strip of trim that is 3 of a meter long. If she uses the entire 4 ··
strip, how long would each side be?
• Ask, So how much of the gardening section of Paul’s back yard is used for each vegetable? [The shaded portion of each column represents 3 5 1 of the whole.]
60
12 ··
4 ··
L6: Understand Division with Fractions
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Part 2: Guided Instruction
Lesson 6
At a Glance Part 2: guided instruction
Students use a number line to divide a fraction by a fraction.
Lesson 6
explore it explore dividing a fraction by a fraction with the problem below.
Step By Step
Kate has 2 yards of fabric to make small flags. Each flag requires 1 yard of fabric. How 3 ··
6 ··
many flags can Kate make?
• Tell students that they will have time to work individually on the Explore It problems on this page and then share their responses in groups. You may choose to work through problem 2 together as a class.
2 You need to find out how many
1s 6 ··
are in
2 3 ··
.
3 The number lines below are divided into thirds. Label 2 on the top number line to 3 ·· represent 2 yards of fabric. 3 ·· 2 3
0
1
0
• As students work individually, circulate among them. This is an opportunity to assess student understanding and address student misconceptions. Use the Mathematical Discourse questions to engage student thinking.
1
4 Each flag requires 1 yard of fabric. Divide the bottom number line into sixths to show 6 ·· how many sixths are in 2. 3 ·· 4 5 Look at the bottom number line. How many sixths are there in 2 ? 3 ·· 6 How many flags can Kate make? 7 2415 3 ·· 6 ·· 6 8 23 3 ··
4
4 54
• Check to see that students label the number line correctly before they use it to answer questions. • Take note of students who are still having difficulty and wait to see if their understanding progresses as they work in their groups during the next part of the lesson. 54
STUDENT MISCONCEPTION ALERT: Students may believe that dividing by 1 is the same as 2 ··
dividing into half. Dividing by one half means how many 1 s are in a quantity, 7 divided by 1 5 14. 2 ··
2 ··
Dividing in half means “to take a quantity and split into two equal parts.” 7 divided in half equals 3 1 . 2 ··
L6: Understand Division with Fractions
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Mathematical Discourse • How would you explain dividing a fraction by a fraction using the number line model to a student who was absent from class? Students should explain that you first draw and label the dividend on a number line. Then, draw another number line beneath it and divide it equally into the number of parts that are in the divisor. They should then count and circle how many groups of the divisor there are up to the dividend. Students should model the number lines on the board for the class to see. • What is another way you could model this problem? Students might describe an area model or paper folding. • Which method for solving fraction division problems do you prefer and why? Students may prefer multiplying by the reciprocal or modeling. Listen for and encourage correct usage of math vocabulary.
L6: Understand Division with Fractions
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61
Part 2: Guided Instruction
Lesson 6
At a Glance Students read fraction division word problems and solve them in pairs or groups.
Step By Step • Organize students into pairs or groups. You may choose to work through the first Talk About It problem together as a class. • Walk around to each group, assessing how the groups are solving the problem. Use the Mathematical Discourse questions to help support or extend students’ thinking. • Students may need to be reminded what direction horizontal is. Review horizontal and vertical as needed. • Direct the group’s attention to Try It Another Way. Have a volunteer from each group come to the board to explain the group’s solutions to problems 15 and 16.
Part 2: guided instruction
Lesson 6
talk about it solve the problem below as a group. Kevin has 6 cups of flour. It takes 3 cup of flour to make one cake. How many cakes can 8 ·· Kevin make? 9 You need to find out how many
3s
8 ··
are in
verbal or written explanations accompanied by models. Provide students an opportunity to refine their mathematical communication skills through discussions in which they evaluate their own thinking and the thinking of other students (SMP 3).
.
11 Represent 6 cups with 6 rectangles. 4 rectangles are shown below. Draw 2 more
rectangles. Draw horizontal lines to divide each rectangle into eighths.
12 Circle and count groups of 3 in the model. How many did you circle? 8 ·· 13 How many 3 -cups of flour are in 6 cups of flour? 8 ·· 14 6 4 3 5 16 8 ··
16
16
try it another Way explore dividing by a unit fraction using a common denominator. To solve 5 4 1, write 5 as a fraction with a denominator of 2 and think, “How many halves 2 ··
are in ten halves?” 10 4 1 5 10. Use the same reasoning to find 8 4 2. 2 ··
2 ··
15 Write 8 as a fraction with a denominator of 3. 6 ··
many two-thirds are in four-thirds”?
many four-sixths are in eight-sixths”?
4
6 ··
3 ··
3 ··
To solve 4 4 2, think, “How
4 6 ··
To solve 8 4 4, think, “How
2
16 Write 2 as a fraction with a denominator of 6. 3 ··
SMP Tip: Students construct arguments using
6
10 Do you think the number of cakes Kevin can make is greater than or less than 6? Why? greater than 6. it takes less than 1 cup of flour to make one cake. 2 ··
2
3 ··
6 ··
3 ··
6 ··
55
L6: Understand Division with Fractions
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Mathematical Discourse • Why would you use rectangles to model the problem? Using an area model is a strategy for solving fraction division problems. • The problem is asking you to divide by 3 s. Why didn’t you divide it into thirds?
8 ··
To count by 3 s, you would need to start by 8 ··
dividing it into eighths. • How would you explain dividing a fraction by a fraction using a common denominator? Students should realize that once they have a common denominator they know the parts are the same size, so they can simply divide numerators.
62
L6: Understand Division with Fractions
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Part 3: Guided Practice
Lesson 6
At a Glance Part 3: guided Practice
Students demonstrate their understanding of dividing a fraction by a fraction.
connect it talk through these problems as a class, then write your answers below.
Step By Step
17 explain: Look at the model below. Write the division equation that the model
represents. Explain how to find the quotient using the model.
• Discuss each Connect It problem as a class using the discussion points outlined below.
5 6
0
1
0
1
5 4 1 5 10 ; the bottom number line is divided into twelfths and shows 6 ··· 12 ·· there are 10 twelfths in 5. 6 ··
Explain: • Ask, What does the top number line represent? 3 5 4
18 analyze: Sam said that 3 4 1 equals 3 . Draw a model and use words to explain why 2 ·· 4 8 ·· ··
6 ··
Sam’s statement is not reasonable.
• Ask, What does the bottom number line represent? [twelfths]
1 2
0
0
• Ask, What do the circled portions represent?
3 2
1
1
2 4
6 4
3 1 Possible explanation: 4 asks how many fourths are in three halves. there
[Groups of 1 in 5 . There are 10.] 12 ··
Lesson 6
2 ··
4 ··
is more than 1 group of fourths in three-halves so the quotient must be
6 ··
3
. the greater than 1. the quotient could not be a fraction less than 1, like ·· 8
Analyze:
3 4 1 5 6. model shows that ·· 2 ·· 4
• Students may draw an area model or a number line.
19 justify: Show that 2 4 4 5 3 by using a model. Explain why the answer is greater 6 ··
than the number you started with.
• Remind students that the problem is asking how many
0
groups of 1 s are in 3 . The quotient must be greater 4 ··
2 ··
than one because there is more than one 1 in 3 . 4 ··
1
2
Possible explanation: When you divide a given number by a fraction less than 1, the quotient is always greater than the given number.
2 ··
Justify:
56
L6: Understand Division with Fractions
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• Read the problem as a class. Remind students that the problem is asking how many groups of 4 are in 2. 6 ··
• Students should draw a number line or area model to model the problem.
SMP Tip: Students need many opportunities to connect and explain the connections between different representations. Encourage students to compare the number line and area models and explain how they help to solve the problem (SMP 4).
L6: Understand Division with Fractions
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63
Part 4: Common Core Performance Task
Lesson 6
At a Glance Students choose a fraction division word problem and solve by writing an expression and drawing a model.
Part 4: common core Performance task
Lesson 6
Put it together 20 Use what you have learned to complete this task.
Step By Step • Direct students to complete Put It Together. • As students work, walk around to assess their progress and understanding, answer their questions, and give additional support, if needed. • If time permits, have students share their problem with the class and discuss how they solved it.
Scoring Rubrics See student facsimile page for possible student answers.
Choose one of the following problems to solve. Circle the problem you choose. Greg made 2 gallon of lemonade and plans to share it equally among 4 friends. 3 ··
How much lemonade will each friend get? Keisha plans to run 4 miles this week. If she runs 2 of a mile each day, how many 3 ··
days will it take her to run 4 miles? Will she be able to run 4 miles in a week? a Write a division expression and draw a model to represent the problem. Division expression for the first problem: 2 4 4 3 ··
Division expression for the second problem: 4 4 2
3 ··
Models will vary, but should represent the problem. b Estimate what you think the quotient will be. Will the quotient will greater than or less than the dividend? How do you know? For the first problem: the quotient will be less than 2 because 2 divided 3 ··
3 ··
by 4 means dividing 2 into 4 parts; each part will be less than 2. the 3 ··
3 ··
quotient will be a fraction less than 2. 3 ··
For the second problem: the quotient will be greater than 4 because 4 divided by 2 means finding how many groups of 2 are in 4. there are 4
A
3 ··
Points Expectations 2 1 0
The student wrote a division expression and drew a model to represent the problem. The student only draws the model or the expression, does not include both.
3 ··
groups of 1 in 4. 2 is less than 1 so there are more than 4 groups of 2 in 4. 3 ··
3 ··
c Use your model to explain how to find the quotient and what the quotient means. For the first problem: 2 4 4 5 2 or 1. this means each of greg’s 4 friends 3 ··
For the second problem: 4 4 2 5 6. this means it will take keisha 6 days to 3 ··
run 4 miles. yes. she will be able to run 4 miles in a week.
The model or expression is incorrect. ©Curriculum Associates, LLC Copying is not permitted.
C
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6 ··
6 ··
L6: Understand Division with Fractions
B
12 ···
will get 1 gallon of lemonade.
57
Points Expectations 2
Student uses the model to explain how to find the quotient and what the quotient means.
1
Student does not explain how to find the quotient or what the quotient means.
0
Student provides an incorrect explanation or response.
Points Expectations 2
The student estimates what they think the quotient should be and proves if it will be greater than or less than the dividend.
1
The student does not explain why he/she thinks the quotient would be greater than or less than the dividend.
0
The student estimates incorrectly or gives an incorrect explanation.
L6: Understand Division with Fractions
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Differentiated Instruction
Lesson 6
Intervention Activity
On-Level Activity
Explore fraction division on a number line.
Make a fraction division handbook or poster.
Materials: painter’s tape, index cards, markers, string
Guide students to create a three-page “handbook” or three-part poster highlighting the steps to solving a fraction division problem.
Use painter’s tape to create a number line on the floor across the classroom. Label the number line 1 through 5 with index cards spaced evenly apart on the number line. Write the problem 5 4 1 on 3 ··
the board.
Examples should include dividing a fraction by a whole number, dividing a whole number by a fraction, and dividing a fraction by a fraction. Each example should include an expression, a model, and an explanation for how to solve the problem.
Ask students to explain how the number line can be used to solve the problem. Ask students to use painter’s tape to divide the line into 1 s. Ask them to 3 ··
explain why 5 4 1 is 15. 3 ··
Next, ask students to model 5 4 2 . Students should 3 ··
use string to circle groups of 2 . There will be 3 ··
7 groups of 2 and 1 left over. The quotient would be 3 ··
3 ··
7 1 . Help students to realize that 1 is half of 2 , so 2 ··
they have 7 and 1 groups.
3 ··
3 ··
2 ··
Continue to model using the number line for problems such as 10 4 1 , then 10 4 2 . 3 ··
3 ··
Challenge Activity Think about dividing whole numbers by fractions and dividing fractions by whole numbers. Tell students, Yolanda wants to cut 1 -foot pieces from a 4-foot rope. Ask, How would you record this number 2 ··
sentence? [4 4 1 5 8] What do you know about the quotient? [It will be larger than the dividend.] How many 2 ··
pieces will she cut? [8] Tell students, Now Yolanda wants to cut 4-foot pieces of rope from a 1 -foot piece of rope. How would you record this
2 ·· 1 1 number sentence? 1 4 4 5 2 What do you know about this quotient? [It is smaller than dividend.] Is it possible to 2 8 ·· ·· cut 4 feet pieces of rope from a 1 foot piece of rope? [No, so the real world answer is 0.] 2 ··
Ask students to think of other real-world situations in which they would use division with fractions. Work with students to determine that their situations make sense.
L6: Understand Division with Fractions
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Develop Skills and Strategies
Lesson 7
(Student Book pages 58–69)
Divide with Fractions Lesson Objectives
The Learning Progression
• Solve word problems using division of fractions.
In Grade 5, students learn to understand fractions as division and to divide whole numbers by unit fractions.
• Write an equation to solve a problem using division of fractions. • Write a story problem that will use division of fractions.
Prerequisite Skills • Know that multiplication and division are inverse operations. • Know that division is either fair sharing (partitive) or repeated subtraction (quotative).
In Lesson 6, students built upon the understanding from Grade 5 using models to show division of fractions. In this lesson, students continue to build upon their knowledge by using visual models and equations to divide whole numbers by fractions, fractions by fractions, and mixed numbers by fractions to solve word problems. In Grade 7, students will continue their work with fractions to include all rational number operations.
• Divide with whole numbers.
Teacher Toolbox
• Divide a whole number by a fraction. • Model division with manipulatives, diagrams, and story contexts.
Teacher-Toolbox.com
Prerequisite Skills
Ready Lessons
Vocabulary
Tools for Instruction
multiplicative inverse: a number which when multiplied by x yields the multiplicative identity, 1
Interactive Tutorials
6.NS.A.1
✓ ✓
✓ ✓ ✓✓
reciprocal: the multiplicative inverse of a number; with fractions, the numerator and denominator are switched
CCSS Focus 6.NS.A.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for 2 4 3 and use a 3 ··
4 ··
visual fraction model to show the quotient; use the relationship between multiplication and division to explain that 2 4 3 5 8 3 ··
4 ··
9 ··
because 3 of 8 is 2 . 1 In general, a 4 c 5 ad . 2How much chocolate will each person get if 3 people share 1 lb of chocolate equally? 4 ··
9 ··
3 ··
b ·
d ·
bc ··
2 ··
How many 3 -cup servings are in 2 of a cup of yogurt? How wide is a rectangular strip of land with length 3 mi and area 1 square mi? 4 ··
3 ··
4 ··
2 ··
STANDARDS FOR MATHEMATICAL PRACTICE: SMP 1–4, 7, 8 (see page A9 for full text)
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L7: Divide with Fractions
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Part 1: Introduction
Lesson 7
At a Glance
Develop skills and strategies
Students read a word problem and explore dividing a whole number by a fraction using a model.
Lesson 7
Part 1: introduction
ccss 6.ns.a.1
Divide with Fractions
Step By Step
in the previous lesson, you learned what dividing by fractions means. in this lesson you will divide with fractions to solve problems. take a look at this problem.
• Tell students that this page models dividing whole numbers by fractions using a visual model.
Charlie is growing vegetables in planters. He has 4 bags of soil and uses 2 of a bag 3 ··
of soil to fill each planter. How many planters can he fill?
• Have students read the problem at the top of the page.
explore it use the math you already know to solve the problem.
• Work through Explore It as a class.
Think of the number of planters that Charlie can fill as how many 2s are in 4. Will that 3 ··
number be greater than or less than 4? Explain your reasoning.
• Ask students how they determined whether the number of planters would be greater or less than 4.
the number of planters will be greater than 4. When you divide a given number by a number less than 1, the answer will be greater than the given number. The model below represents the 4 bags of soil. Draw horizontal lines to divide each bag into thirds.
ELL Support Use the diagram on the page to review the words divisor, dividend, and quotient. Throughout the lesson, use models or manipulatives to demonstrate concepts and processes. Allow students to use the models to demonstrate their learning.
Circle and count groups of 2 in the model. How many did you circle? 3 ··
3 ··
you are trying to find how many 2s are in 4. 3 ··
How many planters can Charlie fill?
Visual Model Model dividing by a number less than 1. • Draw 2 identical circles on the board. Write 2 4 1 . 4 ··
• Ask, Is the divisor, 1 , greater than or less than 1? 4 ·· [less] • How many 1 s are in 2 circles? Let a volunteer draw 4 ··
lines in the circles to show fourths. Ask, Which is greater: the number of 1 parts or the number of 4 ··
whole circles? [The number of 1 parts is more.] 4 ··
• Write 1 4 1 5 . Ask, Will the quotient be greater 2 ··
4 ··
than or less than 1? [greater than 1] Then let a volunteer draw a model to illustrate.
L7: Divide with Fractions
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6
Explain how the model helped you solve the problem. you can count the groups of 2 that are in 4.
SMP Tip: Students map important quantities in the problem to the diagram as a way of understanding dividing with fractions (SMP 4). Students need many opportunities to explain the connections between different representations. Have students explain how the model helps them solve the problem.
6
Why do you circle groups of 2 to represent this problem? 3 ·· each planter holds 2 of a bag of soil, so each circled group fills one planter.
3 ··
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L7: Divide with Fractions
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Mathematical Discourse • How does using a model help you solve the problem? Students may answer that drawing the model helps them to see and count the groups. • Explain, in your own words, dividing fractions with a model. Responses should discuss drawing the dividend and dividing it into groups of the divisor. • Why does it make sense that the quotient is greater than the dividend when you divide with a fraction less than 1? If the divisor is a fraction greater than 1, will the quotient be greater or less than the dividend? Responses should show understanding that taking out groups that are less than one whole will mean that there are more groups than the dividend. Dividing by a fraction greater than 1 will result in fewer groups than the dividend.
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Part 1: Introduction
Lesson 7
At a Glance Part 1: introduction
Students explore solving a division problem using multiplication.
Lesson 7
Find out More When you found the number of 2 s that are in 4, you were dividing. You are solving the
Step By Step
3 ··
problem 4 4 2. You can solve this problem by multiplying. 3 ··
You know that multiplication and division are related. 4 divided by 2 is the same as 1 of 4,
• Read Find Out More as a class.
2 ··
or multiplying 4 by 1 . 2 ··
• Remind students that a fraction and its reciprocal must have a product of 1. • Point out that when you multiply fractions you multiply the numerators, then multiply the denominators, and then simplify if possible.
44252
Think of 2 as 2. Dividing by 2 is the
43152
same as multiplying by 1.
1 ··
1 ··
2 ··
2 ··
When dividing with unit fractions, you learned that dividing 4 by 1 is the same as 3 ·· multiplying 4 by 3. Dividing by 1 is the same as
4 4 1 5 12
3 ··
3 ··
multiplying by 3 or 3.
4 3 3 5 12
1 ··
Dividing with any fraction works the same way. Dividing 4 by 2 is the same as 3 ··
multiplying 4 by 3. 2 ··
• Discuss Reflect. Guide students to think about dividing as being the same as multiplying by the reciprocal (multiplicative inverse).
Dividing by 2 is the same as
44256 3 ·· 4 3 3 5 12 2 ·· 2 ··
3 ··
multiplying by 3. 2 ··
56
You can solve any division problem using multiplication. To divide by any number, you can multiply by its multiplicative inverse, which is also known as the reciprocal.
Visual Model
reflect 1 Explain how you can solve this division problem by using multiplication.
Use a model to understand using reciprocals in division.
642 3 ·· Dividing by 2 is the same as multiplying by 3. so, you can solve 6 divided by 2 3 ··
2 ··
by multiplying 6 by 3 . 6 4 2 5 6 3 3 5 18 = 9 2 ··
3 ··
2 ··
3 ··
2 ···
• Help students understand why they can solve any division problem by multiplying the dividend by
L7: Divide with Fractions
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59
the reciprocal of the divisor. • Draw this model on the board for 5 4 1 : 3 ··
1 3
Real-World Connection Ask students to think of everyday places or situations where people might need to divide by fractions. Encourage them to share their ideas with the class.
1
1
1
1
1
• Explain that the expression can be read as “how
Examples: cooking, making crafts, measuring, building structures
many groups of 1 are in 5?” Show on the model 3 ··
that 1 contains 3 groups of 1 , and there are 5 3 ··
groups of 1 in 5. The total number of groups of 1 in 5 is simply 5 3 3 5 15.
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3 ··
L7: Divide with Fractions
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Part 2: Modeled Instruction
Lesson 7
At a Glance Part 2: Modeled instruction
Students read a word problem and explore how to divide a whole number by a fraction using a bar picture and by modeling the problem using words and an equation.
Lesson 7
read the problem below. then explore how to divide a whole number by a fraction. Kelly drank 2 of the water in her bottle. She drank 3 cups of water. How many 5 ··
total cups of water were in her bottle?
Step By Step • Read the problem at the top of the page as a class.
Picture it
• Read and discuss Picture It. Ask, What does the shaded part of the bar show? [how much of the whole bottle Kelly drank, or two fifths]
The bar represents Kelly’s water bottle. You can divide the bar into fifths and shade 2 to 5 ·· represent the amount of water Kelly drank, 3 cups.
you can draw a picture to understand the problem.
? cups
SMP Tip: Students reason abstractly when they
3 cups
analyze a problem and represent it as an equation with a missing factor in order to find a solution (SMP 2). Ask students to explain how their equations or models represent the context of the problem.
112 cups
Model it you can use words and equations to understand the problem. 2 of the total amount of water equals 3.
5 ··
2
of
the total amount of water
equals
3
2 5 ··
3
?
5
3
5 ··
• Read and discuss Model It. Walk through each step to be sure students understand how to use the inverse operation.
Hands-On Activity
112 cups
To solve a missing factor problem like 2 3 ? 5 3, you can divide. 5 ··
?5342
5 ··
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L7: Divide with Fractions
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Make a model to show how many 2 s are in 4. 3 ·· Materials: sheet of paper for each student, pencils
• Tell students they will make a model to show how many 2 s are in 4. 3 ·· • Give each student a sheet of paper. Have them fold it into 4 equal parts and draw lines on the folds to show 4 equal sections. Tell them this will be a model for 4. • Ask, What is the first thing you could do to the model to start showing how many 2 s are in 4? [Draw lines to 3 ·
divide each of the 4 whole sections into thirds.] What might you do next? [Circle each group of 2 of the thirds and count them.]
Mathematical Discourse • What is another way you could solve the problem regarding Kelly’s water bottle on page 60? Responses may include dividing 3 cups in half to find 1 of the amount in the bottle, and then 5 ··
multiplying by 5 to find the total amount in the bottle. • How is dividing fractions similar to dividing whole numbers? Listen for responses that indicate dividing a quantity into groups.
• Let students finish their models. Ask them what the solution is. Suggest they label their model with the equation solved: 4 4 2 5 6. 3 ··
L7: Divide with Fractions
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Part 2: Guided Instruction
Lesson 7
At a Glance Students revisit the problem on page 60 to learn how to solve it using the bar picture and the equation model.
Part 2: guided instruction
Lesson 7
connect it now you will solve the problem from the previous page using the picture and model.
Step By Step • Read Connect It as a class. Be sure to point out that the problems refer to the problem on page 60. • For problem 4, remind students to change 1 1 to an 2 ··
improper fraction before multiplying by 5. Multiply the denominator times the whole number, and then add the numerator. The result is the numerator, and the denominator stays the same. • In problem 5, review how to change an improper fraction to a mixed number: Divide the numerator by the denominator to get a whole number, the remainder is the new numerator, and the denominator stays the same.
2 Look at Picture It on the previous page. Why do you divide the bar into fifths? the problem says kelly drank 2 of the bottle, so you need to show fifths. 5 ·· 3 How can you use Picture It to find out how many cups of water are in the bottle? since 3 cups 5 2, you know that 1 is 1 1 cups. you can multiply 1 1 by 5 to find 5 5 2 2 ·· ·· ·· ··
the total number of cups of water that were in the bottle. 4 How many total cups of water were in Kelly’s bottle? 5 3 1 1 5 7 1 cups. 2 2 ·· ·· 5 Look at Model It on the previous page. Find 3 4 2 . Show your work. 5 ·· 3 4 2 5 3 3 5 5 15 cups or 7 1 cups. 5 2 ··· 2 2 ·· ·· ·· 6 Explain how to use multiplication to divide a whole number by a fraction.
Dividing by a fraction is the same as multiplying by its inverse. Multiply the whole number by the reciprocal of the fraction.
try it use what you just learned about dividing with fractions to solve these problems. show your work on a separate sheet of paper. 8 servings 7 How many 1 1 -cup servings are there in 12 cups of juice? 2 ·· 8 It takes Emily 9 minutes to bicycle 3 of the way to school. How many minutes does it 10 ·· take Emily to bicycle all the way to school?
30 minutes
• Have students work through Try It on their own. Then discuss with them how they solved the problems. 61
L7: Divide with Fractions
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Try It Solutions 7 Solution: 8 servings; Students solve the problem by using the equation 12 4 3 5 ? or a drawing such 2 ··
as 12 cups with lines dividing each cup into halves and circled groups of 1 1 . 2 ··
ERROR ALERT: Students who wrote 18 servings forgot to multiply by the reciprocal of the divisor. Remind them that they need to find the reciprocal of the divisor before multiplying.
8 Solution: 30 minutes; Students may use the equation 9 4 3 5 ? They may use a drawing such as a bar 10 ··
divided into tenths with 3 shaded to model the 10 ··
distance she went in 9 minutes. This would show that 1 is equal to 3 minutes; 3 3 10 5 30 minutes. 10 ··
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L7: Divide with Fractions
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Part 3: Modeled Instruction
Lesson 7
At a Glance Part 3: Modeled instruction
Students read a word problem and explore how to divide a fraction by a fraction using a double number line and by modeling the problem using words and an equation.
Lesson 7
read the problem below. then explore how to divide a fraction by a fraction. Eli ran 3 of a mile. Every 1 of a mile, he jumped over a hurdle. There was a final 4 ··
8 ··
hurdle at the 3 mile mark. How many hurdles did Eli jump over? 4 ··
Picture it
Step by Step
you can draw a picture to understand the problem. The top number line shows the distance Eli ran, 3 mile.
• Read the problem at the top of the page as a class.
4 ··
The bottom number line shows the number of 1 s that are in 3. 8 ··
• Discuss Picture It:
1 4
0
Ask, What does the top number line represent? [the distance divided into fourths]
2 4
4 ··
3 4
1
0
1
Model it
Ask, What does the bottom number line represent? [the distance divided into eighths]
you can use words and equations to understand the problem. Think: How many 1 s are in 3? 8 ··
4 ··
Use division to find how many 1 s are in 3. 8 ··
• Discuss Model It. Ask, What does the question mark in the equation represent? [the number of hurdles Eli jumped over during his 3 -mile run] 4 ··
4 ··
3 4 ··
divided into
1s 8 ··
equals
the number of hurdles
3 4 ··
4
1 8 ··
5
?
3415? 4 ·· 8 ··
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L7: Divide with Fractions
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Mathematical Discourse • Which method for dividing fractions do you prefer? Why? Are there situations when one method may be easier to use than another? Encourage students to support their opinions and to listen to the opinions of others. Point out that there is no correct answer, and that different students may have different preferences. • Can you think of another way to describe dividing fractions? Explain. Encourage students to suggest ideas or knowledge on other ways to divide fractions.
L7: Divide with Fractions
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Part 3: Guided Instruction
Lesson 7
At a Glance Part 3: guided instruction
Students revisit the problem on page 62 and solve it using the double number line and the equation model.
Lesson 7
connect it now you will solve the problem from the previous page using the picture and model.
Step by Step
9 Look at Picture It. Why is the top number line divided into fourths? Why is the bottom
number line divided into eighths? the top number line is divided into fourths to mark 3, the total distance eli
• Read and discuss Connect It as a class. Refer to the problem on page 62.
4 ··
ran. the bottom number line is divided into eighths because eli jumped over a hurdle every eighth of a mile.
• For problem 13, remind students that they should find the inverse (reciprocal) only of the divisor and not the dividend. Also remind them they should simplify improper fractions to a mixed- or wholenumber answer.
10 Explain how Picture It helps you figure out how many hurdles Eli jumped over. i could count how many 1 s are in 3. 8 4 ·· ·· 11 How many hurdles did Eli jump over?
6
12 Look at Model It. Explain how to use multiplication to find 3 4 1 . 4 8 ·· 8 ·· 1
Dividing by is the same as multiplying by its inverse, or 8, so 3 4 1 is the 8 ··
1 ··
4 ··
8 ··
same as 3 3 8. 4 ··
1 ··
3 3 8 5 24 or 6 4 ·· 1 ··· 4 13 Evaluate 3 4 1 . Show your work. ·· 4 ·· 8 ··
• Have students work through Try It on their own. Then discuss their answers and solutions.
14 Explain how to divide a fraction by a fraction.
to divide a fraction by a fraction, multiply the first fraction by the reciprocal of the second fraction.
SMP Tip: Give students multiple opportunities to
try it
solve and model problems. Students use repeated reasoning to understand algorithms and make generalizations about patterns (SMP 8).
use what you just learned to solve these problems. show your work on a separate sheet of paper. 15 Keisha cuts a 2 -foot rope into 1 -foot pieces. How many pieces of rope did 3 12 ·· ··
8
she cut?
16 Jade makes half a liter of lemonade. She pours 1 liter of lemonade into each glass. 10 ··
How many glasses is Jade able to fill?
5
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L7: Divide with Fractions
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Try It Solutions 15 Solution: 8; Students may draw a double number line or may use the standard algorithm to solve the problem. 2 4 1 5 2 3 12 5 24 5 8 3 ··
12 ··
3 ··
1 ··
3 ··
ERROR ALERT: Students who wrote 18 pieces of rope may have multiplied the inverse of both the dividend and the divisor. Remind students that they should only multiply by the inverse (reciprocal) of the divisor. Review Find Out More on page 59 of this lesson to help students understand why multiplying by the reciprocal is mathematically valid. 16 Solution: 5; Students may draw a bar picture and shade half of the figure, then draw lines to divide the figure into tenths. 5 parts would be shaded. Or, they may use the standard algorithm. 1 4 1 5 1 3 10 5 5 2 ··
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10 ··
2 ··
1 ··
L7: Divide with Fractions
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Part 4: Modeled Instruction
Lesson 7
At a Glance Part 4: Modeled instruction
Students explore how to divide a mixed number by a fraction using bar pictures and by modeling the problem using words and an equation.
Lesson 7
read the problem below. then explore how to divide a mixed number by a fraction. Mari divides 1 4 pounds of granola into 2 -pound bags for a bake sale. How many 5 ··
5 ··
bags of granola can she sell?
Step by Step
Picture it
• Read the problem at the top of the page as a class.
you can draw a picture to understand the problem. The shaded bars represent 1 4 pounds of granola.
• Discuss Picture It:
5 ··
Each circle shows a 2-pound bag of granola.
Ask, How do the shaded bars in the first picture
5 ··
represent 1 4 pounds of granola? [One whole and 4 5 ··
The remainder is half of 25 .
1 bag of granola
more parts are shaded.] Model it
you can use words and equations to understand the problem.
Ask, What does each circle in the second picture
Think: How many 2 s are in 1 4? 5 ··
represent? [Each circle represents one 2 -pound bag 5 ··
5 ··
Use division to find how many 2 s are in 1 4. 5 ··
14
of granola.]
5 ··
divided into
14 5 ·· 4 1 425? 5 ·· 5 ·· 9425? 5 ·· 5 ··
• In Picture It, some students might be confused by the circle that contains part of the whole bar and part of
4
5 ··
2s 5 ··
equals
the number of bags of granola
2 5 ··
5
?
the partial bar. Suggest to students that they think of the entire 1 bar plus 4 bar as one entity. 5 ··
• Discuss Model It. Remind students that they must write the mixed number in the dividend as an improper fraction before dividing.
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L7: Divide with Fractions
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Mathematical Discourse • Mari is selling bags of granola to raise money, so she wants to have as many bags as possible to sell. What else might she think about when dividing up the granola? Listen for responses that show students making connections to personal experiences to make sense of the problem. They might discuss how the size of each bag might make a difference to buyers: Customers might not buy if they think there is not enough granola in a bag. Students might also mention cost to customers or the amount left over after filling the bags.
L7: Divide with Fractions
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Part 4: Guided Instruction
Lesson 7
At a Glance Part 4: guided instruction
Students revisit the problem on page 64 and solve it using the bar picture and the equation model.
Lesson 7
connect it now you will solve the problem from the previous page using the picture and model.
Step by Step
17 Look at Picture It. Why do you circle groups of 2 to solve this problem? 5 ·· Mari divided the granola into 2 -pound bags. you are trying to find how 5 ·· 2 4 many s there are in 1 . 5 5 ·· ··
• Discuss Connect It as a class. Point out that Connect It refers to the problem on the previous page.
18 Count the circles. How many 2 -pound bags of granola can Mari sell? 5 ··
4
19 What fraction of a bag would the remaining 1 pound of granola be? Explain your 5 ··
• When discussing problem 20, be sure students make the connection between the bar model and the mathematical process for changing a mixed number to an improper fraction. For students having trouble understanding why writing a mixed number as an improper fraction is mathematically valid, ask students to count the total number of shaded squares (9) in the first bar picture and point out that the bars are divided into fifths, so there are 9 fifths altogether.
answer. each bag is 2 pounds. the remaining 1 pound of granola is half of 2, so the 5 ··
5 ··
5 ··
remainder is 1 of a bag. 2 ··
20 Look at the Model It. Explain how you know 1 4 is equal to 9 . 5 5 ·· ·· 1 4 is 5 1 4, which equals 9. 5 ·· 5 ·· 5 5 ·· ·· 21 Explain how to use multiplication to evaluate 9 4 2 . 5 ·· 5 ·· Dividing by 2 is the same as multiplying by its inverse, 5, so 9 4 2 is the same 5 2 5 ·· 5 ·· ·· ·· as 9 3 5. 5 ·· 2 ·· 9 3 5 5 45 5 4 5 or 4 1 5 ·· 2 ··· 10 10 2 ··· ·· 22 Evaluate 9 4 2 . Show your work. ·· 5 ·· 5 ·· 23 Explain how to divide with mixed numbers.
Dividing with mixed numbers is just like dividing with fractions. you can write mixed numbers as fractions, then multiply by the reciprocal of the divisor.
• Have students work through Try It on their own. Let volunteers share their solutions and answers with the class. Clear up misconceptions and discuss any questions students may have.
try it use what you just learned to solve these problems. show your work on a separate sheet of paper. 24 A recipe requires 3 of a cup of water. Kyle has a 1 1 -cup measuring cup. How much of 4 2 ·· ·· 1 of the cup the measuring cup is filled with water? 2 ·· 2 1 servings 1 5 2 ·· 25 How many -cup servings are in cup? 3 6 ·· ··
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L7: Divide with Fractions
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Try It Solutions 24 Solution: 1 of the cup; Students may draw a model 2 ··
that shows 1 1 and divide it into fourths. 3 of a cup 2 ··
4 ··
would be half of the model.
25 Solution: 2 1 ; Students may write an equation. 2 ··
5 4 1 5 5 3 3 5 15 5 2 1 6 ··
3 ··
6 ··
1 ··
6 ··
2 ··
ERROR ALERT: Students who wrote 2 transposed 5 ··
the fractions and found 1 4 5 . Encourage students 3 ··
6 ··
who made this error to draw a model to help them visualize the problem.
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L7: Divide with Fractions
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Part 5: Guided Practice
Lesson 7
Part 5: guided Practice
Lesson 7
study the student model below. then solve problems 26–28. Student Model
Lydia bought 2 1 gallons of paint and used 1 1 gallons of paint.
The student divided the number of gallons of paint used, 1 1, by the 2 ·· gallons of paint she bought, 2 1.
2 ··
2 ··
Part 5: guided Practice
27 A marathon is 131 miles long. If 4 people divide up the distance 5 ···
Dividing by 4 is the same as multiplying by what number?
equally, how many miles does each person need to run? Show your work.
What fraction of the paint did she use? 1314 4 5 1314 4 5 ···· 5 1313 1 5 4·· ···· 5 131 20 ···· 5 6 11 20 ···
Look at how you can show your work using a model.
5 ····
think: What fraction of 2 1 is 1 1? 2 ··
2 ··
2 ··
some fraction of 21 equals 1 1. 2 ··
2 ··
3 21 5 11 2 2 ·· ·· 1 to solve ? 3 2 5 11 , divide. 2 2 ·· ·· ? 5 11 4 21 2 2 ·· ·· 5345 2 ·· 2 ·· 3 4 5 5 3 3 2 ; 3 3 2 5 6 or 3 2 ·· 2 ·· 2 ·· 5 ·· 2 ·· 5 10 5 ·· ··· ·· Lydia used 3 of the paint she bought. 5 ·· Solution: ?
Pair/share How could you justify your answer with a picture?
Will the answer be less than 1 or greater than 1? Why?
Lesson 7
26 Lexi has planted seeds in 3 of the garden. She used 1 pound of 5 2 ·· ··
seeds. How many pounds will she use for the entire garden? Show your work. 1 4 3 5 1 3 5; 1 3 5 5 5. 2 ·· 5 ·· 2 ·· 3 ·· 2 ·· 3 ·· 6 ··
Pair/share
Solution:
How is this problem different from the others you’ve seen in this lesson?
each person will need to run 6 11 miles. 20 ···
28 Which of the following problems can be solved by finding 4 4 2 ? 3 ·· a 4 people equally share 2 of a pizza. How much of the pizza does 3 ··
What kind of picture could represent the expression?
each person eat?
b
How many 2 -cup servings of soup are in 4 cups of soup?
c
A pie recipe requires 2 pounds of apples. How many apples are 3 ·· needed for 4 pies?
3 ··
D A family ate 2 of a 4-foot sandwich. How much did they eat? 3 ··
Arthur chose a as the correct answer. How did he get that answer? arthur confused the dividend and divisor. the situation he
Pair/share How did you and your partner decide which fraction is the dividend and which is the divisor?
66
chose could be solved by dividing 2 by 4. 3 ··
Pair/share Does Arthur’s answer make sense?
Lexi will use 5 pound of seeds for the entire garden. 6 ·· Solution:
L7: Divide with Fractions
67
L7: Divide with Fractions
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©Curriculum Associates, LLC Copying is not permitted.
At a Glance
Solutions
Students practice solving problems that require dividing a whole number by a fraction, dividing a fraction by a fraction, and dividing a mixed number by a fraction.
Ex Using a model of words and an equation is one way for students to show their solution to the problem.
Step by Step • Ask students to solve the problems individually on pages 66 and 67. • In the student model at the top of the page, call attention to writing each mixed number as an improper fraction before multiplying. • When students have completed each problem, have them Pair/Share to discuss their solutions with a partner or in a group.
26 Solution: Lexi will use 5 pound of seeds for the 6 ··
entire garden; Students could solve the problem by using an equation. 1 4 3 5 1 3 5 5 5 (DOK 1) 2 ··
5 ··
2 ··
3 ··
6 ··
27 Solution: Each person will need to run 6 11 miles; 20 ··
Students could solve the problem by using an equation. 131 4 4 5 131 1 5 131 3 5 6 11 (DOK 1)
5 ···
5 ···
4 ··
20 ···
20 ··
28 Solution: B; Students must recognize the language as a division problem “how many are in ?” Explain to students why the other two answer choices are not correct: C is not correct because the problem asks “how many are needed for,” which is a multiplication problem. 2 3 4 3 ·· D is not correct because the problem states 2 “of” 4, 3 ··
which is a multiplication problem. 2 3 4 (DOK 3) 3 ··
L7: Divide with Fractions
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Part 6: Common Core Practice
Part 6: common core Practice
Lesson 7
2
3
Part 6: common core Practice 4
Solve the problems.
1
Lesson 7
3 411? What is the value of the expression } } 8 2 9 A }} 16 6 B } 8
C
4
D
} 4
Write each expression in the correct column to show whether the quotient is less than, greater than, or equal to 1. 3
1
1
4} } 4 2 2
1
4
3
4} } 5 3
quotient is less than 1
3
4} } 4 2
4 }} } 9 27
5
20
4 }} } 3 6 19
3
4 2} }} 8 8
quotient is equal to 1
143 } 4 2 } 5 20 4 } 6 3 }}
quotient is greater than 1
3 19 4 2} }} 8 8
3
1
4 } 4 } 2 24 1 } 9 }} 27
3
4
4 } 5 3 }
1
1 ?” Find the expression that does NOT answer the question: “What fraction of 8 is 2 } 2 1 48 A 2} 2 5 31 B } 8 ·· 2 1 C 8 4 2} 2 1 D ? 3 8 5 2} 2
The area and one dimension of a piece of land are given. From the list, write the fraction inside each box that represents the second dimension of the piece of land described.
3 } 7
7 } 8
4 } 7
4 } 9
5 } 7
} 9
5 } 8
Lesson 7
4 } 7 7 } 8
5
7
4 } 9
} 9
1 square mile. The area of a rectangular piece of land is } 2 7 mile. One dimension of this piece of land is } 8 The area of a piece of land that is in the shape of a 1 square mile. One dimension of this piece of triangle is }} 12 4 mile. land is }} 21 2 square mile. The area of a rectangular piece of land is } 3 1 miles. One dimension is 1} 2
5
Explain the difference between dividing in half and dividing by half using pictures, models, or numbers.
Possible answer: Dividing in half means dividing into 2 parts or multiplying by 1. 2 ··
Dividing by half means finding how many 1 s there are in the number. if you divide 4 2 ··
in half, you get 2. if you divide 4 by 1 , you get 8. there are eight 1 s in 4. 2 ··
6
2 ··
3 . Draw a model and use multiplication to show Write a story to represent the expression 6 4 } 4 the solution. Explain how the dividend, divisor, and quotient relate to the story.
stories will vary. Possible answer: a recipe calls for 6 cups of flour. if the only
Possible student model: 3 4
3 4
3 4
3 4
3 4
3 4
3 4
3 4
measuring cup you have is 3 cup, how many 4 ··
times will you have to fill the measuring cup
1
1
1
1
1
1
to get 6 cups of flour? 6 4 3 5 6 3 4 4 3 ·· ·· 5 24
3 ···
58
self check Go back and see what you can check off on the Self Check on page 51. 68
L7: Divide with Fractions
L7: Divide with Fractions
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At a Glance Students divide by fractions to solve word problems that might appear on a mathematics test.
Solutions 1 Solution: D; 3 4 3 5 3 3 2 5 6 5 1 (DOK 1) 8 ··
2 ··
8 ··
3 ··
24 ··
4 ··
2 Solution: C; transposed the dividend and divisor.
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69
4 Solution: See student book page above for solution; To divide fractions, multiply the first fraction by the reciprocal of the second fraction. (DOK 2) 5 See student book page above for possible student explanation. (DOK 3) 6 See student book page above for possible student model and explanation. (DOK 3)
Correct reasoning should be ? of 8 5 2 1 , or ? 5 2 1 4 8. (DOK 2)
2 ··
2 ··
3 Solution: See student book page above for solution; Use the area formula for either a rectangle or triangle to find the unknown dimension. (DOK 2)
76
L7: Divide with Fractions
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Differentiated Instruction
Lesson 7
Assessment and Remediation • Ask students to evaluate the expression 1 1 4 2 3 2 1 4. 2 ··
3 ··
4 ··
• For students who are struggling, use the chart below to guide remediation. • After providing remediation, check students’ understanding. Ask students to explain their thinking while evaluating 2 1 4 3 3 3 1 4. 3 ··
4 ··
9 ··
• If a student is still having difficulty, use Ready Instruction, Level 6, Lesson 6. If the error is . . . 1
Students may . . .
To remediate . . .
have failed to find the multiplicative inverse (reciprocal) of the divisor before multiplying.
Remind students that they must multiply by the multiplicative
3 3 2 5 6 5 1
by 1 fourth. Ask how many fourths altogether. Point out that
2 ··
3 ··
6 ··
inverse of the divisor. For students not understanding why this works, write 6 4 1 . Draw 6 circles. Ask how to divide each circle 4 ··
6 wholes are each split into 4 parts, so 6 wholes are multiplied by 4 parts to get 24. 6 3 4 is the same as 6 4 1 . 1 ··
3 4 ··
have forgotten to include the whole number. 1 4 2 5 1 3 3 5 3
2 ··
3 ··
2 ··
2 ··
4 ··
4 ··
Encourage students to write all mixed numbers as improper fractions as the first step in setting up their computation so they won’t forget.
Hands-On Activity
Challenge Activity
Make a number line to model division.
Write problems involving division of fractions.
Materials: half or whole sheets of paper, pencils
Materials: index cards or sheets of paper, pencils
Tell students they will draw a model to solve a problem. Display this problem: “Mari has 1 1 hours 2 ·· left to prepare for the bake sale. It takes her 1 hour to 4 ·· prepare each item. How many items can she prepare?” Tell students to draw a number line on their paper and use tic marks to divide the line into halves from 0 to 2. They should label each whole and half number mark (0, 1 , 1, 1 1 , 2). Point out that the 2 2 ·· ·· problem asks how many fourths are in one and a half. Tell students to divide each half on their number line with a tic mark to create a fourth. Then tell them to circle each fourth between 0 and 1 1 . Ask students 2 ·· what each circled part represents 3 1 hour 4 and how 4 ·· many circled parts there are [6]. Ask how many fourths are in one and a half [6]. Ask how many items Mari can prepare in the time she has left. [6 items]
Give each student 1–3 index cards or sheets of paper. Tell them to make up a word problem involving division of fractions and mixed numbers for each card. Have students write the word problem on one side of the card and the solution on the other side. Tell them their solution needs to be complete enough so that someone who doesn’t know how to solve the problem can figure it out and why it works. The solution can include such things as a drawing, words explaining the process, or an equation. Have students exchange cards with another student. The other student is to solve the problem and then look at the solution and offer suggestions for changes if the student sees any. Alternatively, let students verbally show and tell how to solve the problems they’ve made up.
L7: Divide with Fractions
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