2014
Common Core
Mathematics Teacher Resource Book 7
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 ReadyA8
The Standards for Mathematical Practice
A9
Depth of Knowledge Level 3 Items in Ready Common CoreA10
Cognitive Rigor Matrix
A11
Using Ready Common CoreA12
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: The Number System1
Lesson 1
Understand Addition of Positive and Negative Integers
3
M
11
M
19
M
29
M
39
M
49
M
CCSS Focus - 7.NS.A.1a, 7.NS.A.1b Embedded SMPs - 2–4
Lesson 2
Understand Subtraction of Positive and Negative Integers
CCSS Focus - 7.NS.A.1c Embedded SMPs - 4, 8
Lesson 3
Add and Subtract Positive and Negative Integers
CCSS Focus - 7.NS.A.1d Embedded SMPs - 2–4
Lesson 4
Multiply and Divide Positive and Negative Integers
CCSS Focus - 7.NS.A.2a, 7.NS.A.2b, 7.NS.A.2c Embedded SMPs - 2, 4, 7
Lesson 5
Terminating and Repeating Decimals
CCSS Focus - 7.NS.A.2d Embedded SMPs - 1–3, 7
Lesson 6
Multiply and Divide Rational Numbers
CCSS Focus - 7.NS.A.2a, 7.NS.A.2b, 7.NS.A.2c Embedded SMPs - 1, 2, 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 1: The Number System (continued)
Lesson 7
Add and Subtract Rational Numbers
CCSS Emphasis
59
M
69
M
CCSS Focus - 7.NS.A.1a, 7.NS.A.1b, 7.NS.A.1c, 7.NS.A.1d Embedded SMPs - 2, 4, 7
Lesson 8
Solve Problems with Rational Numbers
CCSS Focus - 7.NS.A.3, 7.EE.B.3 Embedded SMPs - 1
Unit 1 Interim Assessment
Unit 2: Ratios and Proportional Relationships
Lesson 9
Ratios Involving Complex Fractions
79
82
84
M
94
M
102
M
110
M
120
M
CCSS Focus - 7.RP.A.1 Embedded SMPs - 1, 6, 7
Lesson 10 Understand Proportional Relationships
CCSS Focus - 7.RP.A.2a, 7.RP.A.2b Embedded SMPs - 3, 4
Lesson 11 Equations for Proportional Relationships
CCSS Focus - 7.RP.A.2c, 7.RP.A.2d Embedded SMPs - 1, 2, 4, 6, 8
Lesson 12 Problem Solving with Proportional Relationships
CCSS Focus - 7.RP.A.3 Embedded SMPs - 1–4, 6
Lesson 13 Proportional Relationships
CCSS Focus - 7.RP.A.3 Embedded SMPs - 1–4, 6
Unit 2 Interim Assessment
Unit 3: Expressions and Equations
Lesson 14 Equivalent Linear Expressions
131
134
137
M
147
M
157
M
167
M
CCSS Focus - 7.EE.A.1 Embedded SMPs - 2, 6–8
Lesson 15 Writing Linear Expressions
CCSS Focus - 7.EE.A.2 Embedded SMPs - 2, 4, 6–8
Lesson 16 Solve Problems with Equations
CCSS Focus - 7.EE.B.3, 7.EE.B.4a Embedded SMPs - 1–7
Lesson 17 Solve Problems with Inequalities
CCSS Focus - 7.EE.B.3, 7.EE.B.4b Embedded SMPs - 1, 2, 4, 6, 7
Unit 3 Interim Assessment
Unit 4: Geometry
Lesson 18 Problem Solving with Angles
177
180
182
S/A
192
S/A
CCSS Focus - 7.G.B.5 Embedded SMPs - 2–7
Lesson 19 Understand Conditions for Drawing Triangles
CCSS Focus - 7.G.A.2 Embedded SMPs - 1, 2, 4–6
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 4: Geometry (continued)
Lesson 20 Area of Composed Figures
CCSS Emphasis
200
S/A
210
S/A
222
S/A
232
S/A
242
S/A
252
S/A
CCSS Focus - 7.G.B.6 Embedded SMPs - 1–8
Lesson 21 Area and Circumference of a Circle
CCSS Focus - 7.G.B.4 Embedded SMPs - 1–8
Lesson 22 Scale Drawings
CCSS Focus - 7.G.A.1, 7.RP.A.1 Embedded SMPs - 1–8
Lesson 23 Volume of Solids
CCSS Focus - 7.G.B.6 Embedded SMPs - 1–8
Lesson 24 Surface Area of Solids
CCSS Focus - 7.G.B.6 Embedded SMPs - 1–8
Lesson 25 Understand Plane Sections of Prisms and Pyramids
CCSS Focus - 7.G.A.3 Embedded SMPs - 2, 4, 5, 7
Unit 4 Interim Assessment
Unit 5: Statistics and Probability
Lesson 26 Understand Random Samples
261
264
267
S/A
275
S/A
285
S/A
293
S/A
301
S/A
309
S/A
319
S/A
331
S/A
CCSS Focus - 7.SP.A.1 Embedded SMPs - 3–5
Lesson 27 Making Statistical Inferences
CCSS Focus - 7.SP.A.2 Embedded SMPs - 1–3, 5–7
Lesson 28 Using Mean and Mean Absolute Deviation to Compare Data
CCSS Focus - 7.SP.B.3 Embedded SMPs - 1–7
Lesson 29 Using Measures of Center and Variability to Compare Data
CCSS Focus - 7.SP.B.4 Embedded SMPs - 1–7
Lesson 30 Understand Probability Concepts
CCSS Focus - 7.SP.C.5 Embedded SMPs - 3–7
Lesson 31 Experimental Probability
CCSS Focus - 7.SP.C.6 Embedded SMPs - 1–5
Lesson 32 Probability Models
CCSS Focus - 7.SP.C.7a, 7.SP.C.7b Embedded SMPs - 1–8
Lesson 33 Probability of Compound Events
CCSS Focus - 7.SP.C.8a, 7.SP.C.8b, 7.SP.C.8c Embedded SMPs - 1, 2, 4, 5, 7, 8
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
343
Develop Skills and Strategies
Lesson 9
(Student Book pages 78–87)
Ratios Involving Complex Fractions
Lesson Objectives
The Learning Progression
•Compute unit rates involving ratios with a fraction
in the denominator.
Ratios (including rates, ratios, proportions, and
percents) are commonplace in everyday life and critical
for further study in math and science. In Grade 7,
students extend the concepts of unit rate developed in
Grade 6 to applications involving complex fractions.
They transition from solving problems primarily with
visual models to applying familiar algorithms. This
lesson focuses on solving unit-rate problems that
involve complex fractions. Students model real-world
situations that involve ratios with fractions in the
numerator and/or denominator. They learn to connect
the process of simplifying complex fractions with the
algorithm for the division of fractions. They learn how
to interpret simplified ratios as unit rates to solve
real-world problems.
•Compute unit rates involving ratios with a fraction
in the numerator.
•Compute unit rates involving ratios with fractions in
both the numerator and denominator.
Prerequisite Skills
•Compute unit rates involving ratios with whole
numbers.
•Find equivalent fractions.
•Divide fractions.
•Write whole numbers as fractions.
Teacher Toolbox
Vocabulary
Teacher-Toolbox.com
Prerequisite
Skills
unit rate: a rate in which the first quantity is compared
to 1 unit of the second quantity
Ready Lessons
complex fraction: a fraction where either the
numerator is a fraction, the denominator is a fraction,
or both the numerator and the denominator are
fractions
Tools for Instruction
Interactive Tutorials
7.RP.A.1
✓
✓✓
✓
✓
✓
CCSS Focus
7.RP.A.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like
​  1 ​
or different units. For example, if a person walks ​ 1 ​mile in each ​ 1 ​hour, compute the unit rate as the complex fraction __
​ ··12 ​miles per
2
··
4
··
hour, equivalently 2 miles per hour.
​   ​
4
··
STANDARDS FOR MATHEMATICAL PRACTICE: SMP 1, 6, 7 (see page A9 for full text)
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L9: Ratios Involving Complex Fractions
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Part 1: Introduction
Lesson 9
At a Glance
Students read a word problem and answer a series of
questions designed to help them find a unit rate when
one of the given quantities is a fraction.
Develop skills and strategies
Lesson 9
Part 1: Introduction
CCss
7.RP.a.1
Ratios Involving Complex Fractions
In grade 6, you learned about unit rates. take a look at this problem.
Step By Step
Janaistrainingforatriathlonthatincludesa112-milebikeride.Today,sherodeher
bike12milesin45minutes.WhatisJana’srateinmilesperhour?
•Tell students that this page models how to use a
diagram to find a rate in miles per hour when the
time is given as a number of minutes less than
an hour.
12 miles
45 minutes
•Have students read the problem at the top of
the page.
explore It
use the math you already know to solve the problem.
If Jana biked at a constant rate, how many miles did she bike in the first 15 minutes?
4
•Work through Explore It as a class.
At the same rate, how many miles did she bike in the next 15 minutes?
•Have students look at the diagram and explain how
to figure out how many rectangles are needed to
represent 15 minutes.
4
At the same rate, how many miles did she bike in the last 15 minutes?
4
How many more minutes would Jana need to bike to total one hour?
15
At the same rate, how many miles would she bike in that amount of time?
4
Explain how you could find the number of miles Jana bikes in one hour.
Possible answers: I could add 4 1 4 1 4 1 4 to get 16;
•Help students understand that in the diagram,
4 rectangles represent the ratio 4 miles : 15 minutes.
I could multiply 4 3 4 to get 16.
•Ask student pairs or groups to explain their answers
for the remaining questions.
SMP Tip: Help students make sense of problems
and persevere in solving them (SMP 1) by asking
them to explain what they are asked to find and to
identify the needed information. Allow plenty of
wait time.
Visual Model
•Tell students that you will extend the diagram to
show the number of miles per hour.
•Sketch the diagram on the board. Ask a volunteer
to explain how many more rectangles you would
need to draw to show 60 minutes instead of 45.
[4] Add them to the diagram.
•Ask another volunteer to explain how to use the
extended diagram to solve the problem.
L9: Ratios Involving Complex Fractions
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L9: Ratios Involving Complex Fractions
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Mathematical Discourse
•Why is it important that the first question says, “If
Jana biked at a constant rate”?
Listen for responses that indicate that a
constant rate means the distance traveled is the
same during each minute, so the problem can
be solved with multiplication or division.
•The information is given in miles and minutes. Why
might Jana want to know her rate in miles per hour
instead of miles per minute?
Listen for responses that note that she only
rides a small part of a mile in one minute.
85
Part 1: Introduction
Lesson 9
At a Glance
Part 1: Introduction
Students revisit the problem on page 78 to learn how to
model it using a ratio written as a complex fraction.
Then students simplify the complex fraction by
dividing.
Lesson 9
Find out More
The number of miles Jana bikes in one hour is a unit rate. A unit rate compares two
quantities where one of the quantities is 1. A unit rate tells you how many units of the first
quantity correspond to one unit of the second quantity.
The units in this problem are miles and hours. The problem tells us that Jana bikes 12 miles
Step By Step
in 45 minutes. That’s the same thing as 12 miles in 3 hour.
4
··
number of miles 5 12
number of hours
·············
•Read Find Out More as a class.
3
··
4
··
12
3 is a complex fraction. A complex fraction is a fraction where either the
The fraction ··
4
··
•Review the meaning of unit rate.
numerator is a fraction, the denominator is a fraction, or both the numerator and the
12
•Have students look at the ratio ​ __
  ​ . Ask, Why is it
3
would do with whole numbers.
denominator are fractions. You can simplify a complex fraction by dividing, just as you
The fraction bar represents division, so you can think of 6 miles as 6 4 2 5 3 miles per hour.
​   ​
You can think about ·······
3 hour in the same way.
not a unit rate? [The number of hours must be one.]
4
··
12
3
··
4
··
•Have students describe how the ratio looks different
from other fractions they have seen. Discuss the
definition of a complex fraction. Ask students to give
examples of complex fractions.
1
··
4
··
1
··
3
··
48
5 ··
3 or 16 miles per hour
The unit rate is 16. The number of miles Jana bikes is 16 times the number of hours.
Reflect
1 On another training ride, Jana bikes 15 miles in 50 minutes. Explain how you could find
the number of miles she bikes in 1 hour.
division. Give other examples such as ​ 15  ​ and ​ 20  ​ .
Possible answers: jana bikes 3 miles every 10 minutes, so she would bike
3
··
5
··
•Work through the steps used to divide 12 4 ​ 3 ​ .
4
··
15
18 miles in 60 minutes or 1 hour; Write the ratio ···
and then divide to get
5
6
··
18 miles per hour.
•Have students assess the reasonableness of the
answer. Note that 1 hour is slightly more than
45 minutes and 16 miles is slightly more than
12 miles.
Write the word per on the board. Next to it, write for
each and in each. Give examples such as “5 crayons
for each student” means “5 crayons per student” and
“driving 50 miles in each hour” means “50 miles per
hour.” Give other examples and such as “$1.50 for
each pound of peaches” or “3 cups of flour in each
loaf of bread.” Have students restate each using the
word per.
5 12 4 3
5 12 3 4
•Reinforce the idea that the fraction bar can mean
ELL Support
2 hours
······
12 miles
4
··
79
L9: Ratios Involving Complex Fractions
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Real-World Connection
Encourage students to think of everyday situations
in which measurements are given as fractions. Have
volunteers share their ideas.
Examples: Cooking ​1 ​ 3 ​ cup, ​ 1 ​ dozen 2​; sewing ​1 ​ 5 ​ yard,
4
··
2
··
8
··
2 ​ 1 ​ feet 2​; traveling ​1 12 ​ 1 ​ miles in ​ 1 ​ hour, 3 ​ 1 ​ blocks 2
··
in 7 ​ 1 ​ minutes 2​
2
··
2
··
4
··
2
··
Then write unit rate on the board. Circle the word
unit and write a 1 above it. Say that in 50 miles per
hour, the unit rate is 50 because it tells the number
of miles in 1 hour. The word per can mean in one or for one. Give more examples. Have students
restate the ratio using the word per and then give the unit rate.
86
L9: Ratios Involving Complex Fractions
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Part 2: Modeled Instruction
Lesson 9
At a Glance
Part 2: Modeled Instruction
Students use number lines to solve problems that
require them to find unit rates by simplifying ratios
involving complex fractions.
Lesson 9
Read the problem below. then explore different ways to understand how to find a
unit rate.
Max’sfavoriterecipeforoatmealraisincookiesmakes48servings.Hewantstomake
somecookiesbutonlyhasoneegg.Maxhastoadjusttheamountsoftheother
ingredients.Howmuchflourwillheneed?
Step By Step
oatmeal Raisin Cookies
•Read the problem at the top of the page as a class.
•Ask students to look at the recipe to find the number
of eggs and cups of flour needed.
1 teaspoon vanilla
2 3 cups oats
3 teaspoon cinnamon
1 1 cups flour
1 teaspoon baking soda
1 cup raisins
2
··
4
··
4
··
Model It
you can draw a double number line to show the relationship described in the
problem.
The units you need to compare are cups of flour and eggs.
•Have students read Model It. Call students’ attention
to the first double number line. Have them read the
information and note the labels. Ask how the number
line is related to the problem.
•Read the information above the second double
1
Start both
number lines
at 0.
Line up 1 2
cups of flour
with 2 eggs.
Cups of flour
0
12
Eggs
0
2
1
You need to find the unit rate, the number of cups of flour needed for 1 egg.
The point for one egg is halfway
between 0 and 2. Draw a line
halfway between 0 and 2.
number line. Discuss how to find the number that is
halfway between 0 and 1 ​ 1 ​ . Guide students to see
2
··
how they can use the unit rate to find the other
Hands-On Activity
1 1 cups brown sugar
2 eggs
2
··
•Have students use their own words to explain what
they are trying to find in order to solve this problem.
numbers on the top number line.
3 cup butter
4
··
80
The number that lines up
with 1 is halfway
1
between 0 and 1 2 .
Cups of flour
0
3
4
12
1
24
1
3
Eggs
0
1
2
3
4
L9: Ratios Involving Complex Fractions
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Mathematical Discourse
Fold paper strips to model unit rate.
Materials: strips of paper, scissors, markers, rulers
•Have students cut a strip of paper so that it is
1 ​ 1 ​ inches long.
2
··
•Direct students to draw a horizontal line across
the entire length of the paper then divide it into ​ 1 ​-inch segments.
4
··
•Why is it helpful to know a unit rate when shopping?
Student responses may include that unit rates
allow shoppers to compare similar products of
different sizes.
•Does the problem ask you to find a unit rate? Explain
why or why not.
Students should explain that it does ask for a
unit rate because it asks for the amount of flour
needed for 1 egg.
•Have students fold the paper in half vertically and
then determine the length of each half.
•On the board write 1 ​ 1 ​ 4 2 5 ​ 3 ​ and ​ 3 ​ 3 2 5 1 ​ 1 ​ .
4
··
2
··
4
··
2
··
•Have students relate the result to the number line
used to model the problem.
L9: Ratios Involving Complex Fractions
©Curriculum Associates, LLC Copying is not permitted.
87
Part 2: Guided Instruction
Lesson 9
At a Glance
Part 2: guided Instruction
Students revisit the problem on page 80 to learn how to
solve the problem by simplifying a ratio involving a
complex fraction.
Lesson 9
Connect It
now you will see how to solve the problem from the previous page by writing a ratio.
2 Why do you need to find the number that is halfway between 0 and 1 1 ?
2
··
Step By Step
that is the amount of flour to use if you have just one egg.
3 How could you find the number that is between 0 and 1 1 ?
2
··
1
1
1
•Read Connect It as a class. Be sure to point out that
the questions refer to the problem on page 80.
Divide 1 by 2 or multiply 1 by .
2
··
2
··
2
··
4 How many cups of flour does Max need to use if he has just 1 egg? Show your work.
3 cup of flour; 1 1 3 1 5 3 3 1 5 3
4
2 ··
2 ··
2 ··
2 ··
4
··
··
•Emphasize the idea that since 1 egg is halfway
5 Write the ratio that compares 1 1 cups of flour to 2 eggs.
2
··
1
cups of flour 5 1 ··
2
eggs
··········
······
2
between 0 and 2 eggs, the amount of flour must be
halfway between 0 and 1 ​ 1 ​ cups.
6 Write and simplify a division expression to find the number of cups of flour Max needs to
2
··
use if he has just 1 egg.
1 1 4 2 5 3 4 2; 3 4 2 5 3 3 1 5 3 ; 3 cup of flour
2
··
•Once students have written the ratio, have them
explain why it is a complex fraction.
2
··
7 The unit rate is
1
··
2
··
3
4
··
1
··
2
··
2
··
4
··
4
··
. The number of cups of flour is
3
4
··
times the number of eggs.
8 Explain how to find a unit rate.
Possible answer: Write a ratio that compares the quantities described in the
•Reinforce the idea that students can divide to
simplify a complex fraction because the fraction bar
indicates division.
1
problem. then divide the first quantity by the second quantity.
try It
1 ​   ​
2
··
•Have students simplify ​ ___
  ​ . Have them compare the
2
steps they use to the steps used to find ​ 1 ​ of 1​ 1 ​ on the
2
2
··
··
use what you just learned about finding a unit rate to solve these problems. show
your work on a separate sheet of paper.
use the information in the recipe on the previous page.
9 If Max has only one egg, how much butter will he need?
3 cup
8
··
2 teaspoon
3
10 If Max has only one cup of flour, how much vanilla will he need? ··
number line.
SMP Tip: Students look for and make use of
81
L9: Ratios Involving Complex Fractions
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structure (SMP 7) when they explain how dividing
1 ​ 1 ​ by 2 is the same as multiplying 1 ​ 1 ​ by ​ 1 ​ . Remind
2
··
2
··
2
··
students that division by a number and multiplication
by its reciprocal are equivalent operations.
Try It Solutions
9Solution: ​ 3 ​ cup; Students may draw a number line to
8
··
show that ​ 3 ​ is halfway between 0 and ​ 3 ​ . They may
8
··
4
··
​ 3 ​
also write and simplify the ratio ​ __··24 ​ .
Concept Extension
10 Solution: ​ 2 ​ teaspoon; Students may write and
Help students see how the unit rate helps
them find equivalent ratios.
•Draw a ratio table on the board. Label the first
row Cups of flour and the second row Eggs.
•Fill in the first two columns with information
from Connect It.
•Have students fill in two more columns by
multiplying the number of eggs by ​ 3 ​ .
4
··
•Compare the results with the number line on
page 80.
3
··
1
simplify the ratio ​ ___
  ​  . They may also draw a double
1
1 ​   ​
2
··
number line with 1 teaspoon on the top line and 1 ​ 1 ​ cups on the bottom line. They would show that
2
··
1 cup is ​ 2 ​ of 1​ 1 ​ cups and then show ​ 2 ​ of
3
··
1 teaspoon.
2
··
3
··
ERROR ALERT: Students who wrote ​ 1 ​ found the
2
··
amount of vanilla needed for 1 egg instead of for
1 cup of flour.
•Ask students to explain how to show that each
ratio of flour to eggs is equal to ​ 3 ​ .
4
··
88
L9: Ratios Involving Complex Fractions
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Part 3: Modeled Instruction
Lesson 9
At a Glance
Part 3: Modeled Instruction
Students use double number lines to find a unit rate.
Then students solve a problem by comparing unit rates.
Read the problem below. then explore different ways to understand how to find and
compare unit rates.
Step By Step
José’smotheristryingtodecidewhetherornotsheshouldbuya12-ouncepackageof
coffeeonsalefor$7.50.Sheknowsthatshecanbuythesamecoffeefor$9.00per
pound.Whichisthebetterbuy?
•Read the problem at the top of the page as a class.
Model It
•Ask, Why can’t you just say that $7.50 is less than $9.00
so it is a better buy? [The packages are different
weights so they do not contain the same amount.]
you can draw a double number line to show the relationship described in the
problem.
To find the better buy, compare the unit rate of each option.
The problem gives you one unit rate: $9.00 per pound. To compare unit rates, the units you
use must be the same. So, find the weight of the other coffee in pounds.
•Read Model It as a class. Reinforce that when
There are 16 ounces in 1 pound, so 12 ounces is 12 or 3 pound.
16
··
2
··
Make sure students understand why 12 ounces is
Start both
number lines
at 0.
equivalent to ​ 3 ​ pound.
4
··
•Have students study the first double number line.
Go over the steps used to draw the number line
accurately.
Ask, How do we know that 2 ​ 1 ​dollars lines up
•Write 1 pound on the board. Ask students to
describe what we measure with pounds. Pounds
are a unit of measure to find weight or how heavy
an object is.
•Ask students how we would measure the weight
of something less than a pound. Accept the idea
that we could use a fraction of a pound. If no one
mentions the term ounce, introduce it as a unit of
measure less than 1 pound.
•Write 1 pound 5 16 ounces on the board. Discuss
the equivalency in concrete terms. Dora has
16 ounces of grapes. That is the same as 1 pound of
grapes.
L9: Ratios Involving Complex Fractions
©Curriculum Associates, LLC Copying is not permitted.
Cost, in dollars
0
Pounds of coffee
0
Divide the bottom
number line into
fourths.
3
Line up 4 pound
of coffee with
1
the cost, $7 2 .
Find x, the cost
for 1 pound of
coffee.
1
1
4
1
2
72
x
3
4
1
Find the cost for each quarter-pound of coffee. Then find the unit cost.
•Direct students’ attention to the second number line.
ELL Support
4
··
You can write $7.50 using fractions. $7.50 is the same as $7 1.
comparing unit rates, the units must be the same.
2
··
with ​ 1 ​ pounds? ​3 ​ 1 ​ is one third of the way from 0 to ​ 3 ​ ,
4
4
4
··
··
··
and 2 ​ 1 ​ is one third of the way from 0 to 7 ​ 1 ​ . 4​ 2
2
··
··
Lesson 9
82
1
1
Cost, in dollars
0
22
5
72
10
Pounds of coffee
0
1
4
1
2
3
4
1
L9: Ratios Involving Complex Fractions
©CurriculumAssociates,LLC Copyingisnotpermitted.
Mathematical Discourse
•What are some equivalent ratios shown by the
number line?
2 ​ 1 ​
7 ​ 1 ​
5
2 __
2
··
··
  ​ , ​   ​ , ​ ___
  ​ , and ​ 10  ​ .
Students may list ​ ___
1 1 3
​   ​ ​   ​ ​   ​
4
··
2
··
4
··
1
··
•Using the number line, how can you tell the ratios
are equivalent? Can you explain it another way?
Students may note that they are the same
distances apart on the number line. They may
also explain that when you double 2 ​ 1 ​ and ​ 1 ​ 2
··
4
··
you get 5 and ​ 1 ​, and when you triple them you
2
··
get 7 ​ 1 ​ and ​ 3 ​ . They may also explain that when
2
··
4
··
you simplify each ratio, the result is 10 to 1.
89
Part 3: Guided Instruction
Lesson 9
At a Glance
Students revisit the problem on page 82. They learn to
solve it by simplifying ratios to find unit rates. Then
students compare the unit rates to solve the problem.
Step By Step
•Read page 83 as a class. Be sure to point out that
Connect It refers to the problem on page 82.
•Ask, Once you know that each ​ 1 ​pound costs $2.50, how
4
··
can you figure out how much a full pound costs? [There
are 4 fourths in a whole, so you would multiply $2.50
Part 3: guided Instruction
Lesson 9
Connect It
now you will see how to use a ratio to solve the problem.
11 The top number line is divided into 3 equal parts from 0 to 7 1 , and the bottom number
2
··
line is divided into 3 equal parts from 0 to 3 . How can you use this to find the cost of
4
··
1 pound of coffee?
Divide 7 1 by 3 to find the length of each part. 7 1 4 3 5 2 1 , so each part is 2 1 .
2
··
2
··
2
··
2
··
to get to the number that lines up with 1, you need 4 of these parts.
71
dollars
2
12 Write the ratio that compares $7 1 dollars to 3 pound of coffee.
5 ··
2
4
pounds of coffee ······
··
··
·············
3
4
··
13 Write and simplify a division expression to find the cost of 1 pound of coffee.
7 1 4 3 5 15 4 3 5 15 3 4 5 60 5 10; $10
2 ··
4 ···
2
4 ···
2
3 ···
6
··
··
··
14 Which is the better buy, 12 ounces for $7.50 or 1 pound for $9.00? Explain your reasoning.
the 12-ounce package of coffee is $10.00 per pound and the 16-ounce package
is $9.00 per pound. the better buy is 1 pound for $9.00.
by 4.]
15 If you started the problem by converting 1 pound to 16 ounces, would you get the same
•Have students explain why you would divide to
simplify a ratio involving a complex fraction. Have
students complete the division process individually
and then review it as a class.
•Have volunteers present the reasoning they used to
find the cost per ounce. Ask whether finding cost per ounce or the cost per pound is easier in this
situation.
result? Justify your conclusion.
yes. students may draw a double number line or write and solve a proportion.
16 Can you compare any two unit rates? Explain.
no, the rates must use the same units to be able to compare them.
try It
use what you just learned about unit rates to solve this problem. show your work on a
separate sheet of paper.
17 Rina’s recipe uses 2 cups of sugar to make 2 1 dozen cookies. Jonah’s recipe uses 2 1 cups
2
4
··
··
of sugar to make 3 dozen cookies. Which recipe uses more sugar for a dozen cookies?
Why?
Rina’s recipe; Rina’s ratio is 4, which is greater than jonah’s ratio, 3.
5
··
4
··
83
L9: Ratios Involving Complex Fractions
SMP Tip: Students should realize that it is
important to specify cost as per pound or per ounce
when writing and simplifying ratios. Be sure to
model this language as you attend to precision
(SMP 6) when working through this problem with
students.
©CurriculumAssociates,LLC Copyingisnotpermitted.
Try It Solution
17 Solution: Rina’s recipe.; Students may simplify ratios
2
to find the unit rate. Rina: ​ ___
  ​  5 2 4 2 ​ 1 ​ 5 ​ 4 ​ ;
1
2
··
2 ​   ​
2 ​ 1 ​
2
··
4
··
Jonah: ​ ___
  ​ 5 2 ​ 1 ​ 4 3 5 ​ 3 ​ 3
4
4
··
··
5
··
Each dozen of Rina’s cookies contains ​ 4 ​ cup sugar.
5
··
Each dozen of Jonah’s contains ​ 3 ​ cup of sugar.
4
··
Rina’s cookies use more sugar per dozen. ​ 4 ​ is
5
··
greater than ​ 3 ​ .
4
··
ERROR ALERT: Students who wrote Jonah may
have found the rate of dozens of cookies per cup of
sugar.
2 ​ 1 ​
3
2
··
  ​ 5 1 ​ 1 ​ ; Jonah: ​ ___  ​  5 1 ​ 1 ​ However, that
Rina: ​ ___
1
2
4
··
2 ​   ​
4
··
3
··
means Jonah’s recipe has more cookies per cup of
sugar, not more sugar per dozen cookies.
90
L9: Ratios Involving Complex Fractions
©Curriculum Associates, LLC Copying is not permitted.
Part 4: Guided Practice
Lesson 9
Part 4: guided Practice
Lesson 9
study the model below. then answer questions 18–20.
Student Model
The student knew that
60 minutes 5 1 hour, so
72 minutes 5 60 minutes
1 12 minutes, or
1 hour 12 minutes.
Oliveristrainingforamarathon.Inpractice,heruns15kilometers
in72minutes.Whatishisspeedinkilometersperhour?
Convert the time in minutes to hours to find kilometers per hour.
Lesson 9
19 A restaurant uses 8 1 pounds of carrots to make 6 carrot cakes. Frank
4
··
wants to use the same recipe. How many pounds of carrots does Frank
need to make one carrot cake?
5 1 12 hours or 1 1 hours
5
··
km 5 15
hr
···
···
11
5
··
5 15 4 1 1
Show your work.
81
4
pounds of carrots 5 ··
···
6
8 1 4 6 5 33 4 6
4
4
··
···
5 33 3 1
4
6
···
··
5 33
24
···
5 1 9 or 1 3
24
8
···
··
5
··
5 15 4 6
Pair/share
5
··
5 15 3 5
6
··
Pair/share
How did you decide how
to write the ratio?
Solution:
5 75 or 12 1
6
···
Solution:
What is the ratio of
pounds of carrots
to cakes?
cakes
···············
72 minutes 5 1 hour 12 minutes
60
···
Part 4: guided Practice
Frank will need 13 pounds of carrots for each cake.
8
··
What steps did you take
to find the unit rate?
2
··
oliver runs 12 1 kilometers per hour.
2
··
20 It takes Zach 15 minutes to walk 7 1 blocks to the swimming pool.
2
··
At this rate, how many blocks can he walk in one minute? Circle the
What unit rate do you
need to find?
letter of the correct answer.
How do you evaluate a
complex fraction?
18 Alexis washes 10 1 windows in 3 hour. At this rate, how many windows
2
4
··
··
can she wash in one hour?
10 1 4 3 5 21 4 3
2
··
4
··
a
b
10 1
2
number of windows 5 ··
number of hours
3
·················
····
4
··
2
···
4
··
5 21 3 4
2
···
3
··
5 84
6
···
1 block
5
··
1 block
2
··
C
2 blocks
D
5 blocks
Dee chose C as the correct answer. What was her error?
she found the number of minutes per block instead of the
number of blocks per minute.
Pair/share
5 14
Does Dee’s answer make
sense?
Pair/share
How can you tell if your
answer is reasonable?
84
Solution: alexis can wash 14 windows in one hour.
L9: Ratios Involving Complex Fractions
85
L9: Ratios Involving Complex Fractions
©CurriculumAssociates,LLC Copyingisnotpermitted.
At a Glance
Students write and simplify ratios to solve word
problems involving unit rate. They may also use double
number lines to find the solution.
Step By Step
•Ask students to solve the problems individually and interpret their answers in the context of the
problems.
•When students have completed each problem, have
them Pair/Share to discuss their solutions with a
partner or in a group.
Solutions
Ex The example shows how to write and simplify a
ratio as one way to solve the problem. Students
could also use a double number line.
©CurriculumAssociates,LLC Copyingisnotpermitted.
18 Solution: 14; Students could solve the problem by
10 ​ 1 ​
____
  use a double number line.
simplifying ​  3 ··2 ​ or
​   ​
(DOK 1)
4
··
19 Solution: 1 ​ 3 ​ ; Students could solve the problem by
8 1
··
8 ​   ​
4
··
  ​ or use a double number line.
simplifying ​ ___
6
(DOK 1)
20 Solution: B; Divide 7 ​ 1 ​ by 15 to find the number of
2
··
blocks per minute.
Explain to students why the other two answer
choices are not correct:
A is not correct because 7 ​ 1 ​ 4 15 5 0.5, which is ​ 1 ​,
2
2
··
··
not ​ 1 ​ .
5
··
D is not correct because it does not make sense for
him to walk 5 blocks in one minute if it takes him
15 minutes to walk 7 ​ 1 ​ blocks. (DOK 3)
2
··
L9: Ratios Involving Complex Fractions
©Curriculum Associates, LLC Copying is not permitted.
91
Part 5: Common Core Practice
Part 5: Common Core Practice
Lesson 9
Lesson 9
Part 5: Common Core Practice
4
Solve the problems.
1
One of the highest snowfall rates ever recorded was in Silver Lake, Colorado, in April 1921,
2
inch per hour
}}
55
14
C
3 inches per hour
3 }}
55
B
55 inch per hour
}}
158
D
13 inches per hour
3 }}
14
A restaurant makes a special citrus dressing for its salads. Here is how the ingredients are mixed:
1
of the mixture is orange juice
}
4
1
of the mixture is lemon juice
}
4
Cost ($)
6
Weight
lb
}
4
3
trail Mix b
1
When the ingredients are mixed in the same ratio as shown above, every batch of dressing
tastes the same. Study the measurements for each batch in the table. Fill in the blanks so that
every batch will taste the same.
A grocery store sells different types of Trail Mix, as shown in the table below.
trail Mix a
1
of the mixture is oil
}
3
of the mixture is vinegar
}
6
1 hours. What was that rate in inches per hour?
when just over 7 feet of snow fell in 27 }
2
A
Lesson 9
batch 1
batch 2
Oil (cups)
1
2
Vinegar (cups)
}
2
1
1
3
1
1}
2
1
1}
2
trail Mix C
Orange juice (cups)
}
4
8.50
2.25
Lemon juice (cups)
}
4
1 lb
4 oz
1 lb 5 16 oz
5
Which statement is correct?
A
Trail Mix A is the best buy.
C
Trail Mix C is the best buy.
B
Trail Mix B is the best buy.
D
They are all the same price.
3
batch 3
1
1}
3
2
}
3
1
1
Two friends worked out on treadmills at the gym.
3 hour.
• Alden walked 2 miles in }
4
3 miles in 30 minutes.
• Kira walked 1 }
4
Who walked at a faster rate? Explain your reasoning.
3
Show your work.
1 mile. The display of the
A treadmill counts one lap as }
5
treadmill indicates the number of laps already completed
Lapscompleted:
completed:
Laps
and highlights the completed portion of the current lap.
13 miles run.
Create a display that shows a total of }}
10
• Write one number in the box to indicate
the number of laps already completed.
• Shade in one or more sections of the display to indicate
how much of the current lap has been completed.
Total distance run:
1
kira’s rate: ··4
2435234
13 4 1 5 7 3 2
3
··
4
··
Total
distance
run:
Start
Start
13
10
3
alden’s rate: 2
6
4
··
13
10
1
··
1
···
2
··
3
··
4
··
5 8 or 2 2 miles per hour
miles
3
··
3
··
2
··
4
··
1
··
5 14 or 3 1 miles per hour
4
···
2
··
Answer kira walks at a faster rate.
miles
self Check Go back and see what you can check off on the Self Check on page 77.
86
L9: Ratios Involving Complex Fractions
L9: Ratios Involving Complex Fractions
©CurriculumAssociates,LLC Copyingisnotpermitted.
©CurriculumAssociates,LLC Copyingisnotpermitted.
At a Glance
87
4Solution: See student book page above for solutions.
Students may choose to use proportions to fill in
the missing values in the table. (DOK 2)
Students find unit rates to solve word problems that
might appear on a mathematics test.
2 5Solution: Alden’s rate is ​ __
  ​ or 2 ​ 2 ​ miles per hour.
3
Solutions
1Solution: C; Rewrite 7 feet as 84 inches and then
84
write and simplify the ratio of inches to hours, ​ ____
  ​  .
1
27 ​   ​
3
··
​   ​
4
··
1 ​ 3 ​
4
··
Kira’s rate is ​ ___
  ​ or 3 ​ 1 ​ miles per hour. Kira’s rate 2
··
​ 1 ​
2
··
is faster. (DOK 2)
2
··
(DOK 1)
2Solution: A; Find the cost per pound for each brand.
(Trail Mix A: $8/pound, B: $8.50/pound, C: $9/pound.) Then find the lowest unit rate.
(DOK 2)
3Solution: 6 laps completed, and students shade in 5
sections of the display; Divide ​ 13 ​  by ​ 1 ​ to get 6​ 1 ​ 10
··
5
··
2
··
laps, which is 6 full laps and 5 of 10 sections of the
display shaded. (DOK 2)
92
L9: Ratios Involving Complex Fractions
©Curriculum Associates, LLC Copying is not permitted.
Differentiated Instruction
Lesson 9
Assessment and Remediation
•A recipe calls for 2 ​ 1 ​ cups of sugar for 1 ​ 1 ​ dozen cookies. Have students find the amount of sugar per dozen
4
··
2
··
cookies. ​3 1 ​ 1 ​ cups 4​
2
··
•For students who are struggling, use the chart below to guide remediation.
•After providing remediation, check students’ understanding. Have students find Carlos’ rate in laps per
minute if he runs 6 ​ 1 ​ laps in 10 minutes. ​3 ​ 5 ​ 4​
4
··
8
··
•If a student is still having difficulty, use Ready Instruction, Level 7, Lesson 6.
If the error is . . .
Students may . . .
To remediate . . .
have found the amount
of sugar per cookie, not
per dozen.
Have students reread the problem and state what they need to
find. Have them explain why they do not need to convert
1 ​ 1 ​ dozen to individual cookies.
​  2 ​
3
··
have found the number
of dozens per cup of
sugar.
Write the ratio using words, ​ sugar   
​ . Have students substitute
dozen
·····
numbers for words.
3 ​  3 ​
have multiplied instead
of divided.
Remind students that the fraction bar indicates division. Review
the steps used to divide two fractions.
any other answer
have divided incorrectly.
Go over the student’s work to make sure each step was done
correctly.
​  1 ​
8
··
8
··
2
··
Hands-On Activity
Challenge Activity
Use a paper model to find a unit rate.
Extend the concept of unit rate to solve
problems.
Materials: small pieces of paper that are the same
shape and size
Tell students that when Ginger made applesauce
On the board, write “Sheila buys 9 ​ 1 ​ pounds of nuts
3
··
using 2 ​ 1 ​ pounds of apples, she used 1 ​ 1 ​ tablespoons
for 4 gift baskets. How many pounds of nuts does
of sugar. She now has 8 pounds of apples and
Sheila buy per gift basket?” Distribute 10 pieces of
wonders how much sugar she should use. Ask
paper to each student. Tell students that each piece
students how they could find and use the unit rate to
represents a pound of nuts. Ask, How can you
solve the problem. [Possible answer: Find the unit
represent 9 ​ 1 ​ pounds using the paper? [Tear one sheet
3
··
in thirds and discard two of the thirds.] Direct
students to distribute the paper into 4 piles so that
there is the same amount of paper in each pile. It is
acceptable to tear the paper into pieces that are the
4
··
2
··
1 ​ 1 ​
rate by simplifying ​ ___··12  ​, which is ​ 2 ​ . Then either
2 ​   ​
4
··
3
··
create a ratio table or multiply 8 3 ​ 2 ​ to show that
3
··
Ginger should use 5 ​ 1 ​ tablespoons of sugar for
8 pounds of apples.]
3
··
same size. When students have completed the task,
9 ​ 1 ​
3
··
write ​ ___
  ​ 5 2 ​ 1 ​ . Ask students what 2 ​ 1 ​ represents.
4
3
··
3
··
L9: Ratios Involving Complex Fractions
©Curriculum Associates, LLC Copying is not permitted.
93
Focus on Math Concepts
Lesson 10
(Student Book pages 88–93)
Understand Proportional Relationships
Lesson Objectives
The Learning Progression
•Determine whether two quantities are in a
proportional relationship by looking at values in a
table, a line in the coordinate plane, and an
equation. (Use equivalent fraction relationships and
multiplication/division to find proportional ratios.)
The ability to represent a relationship in multiple
ways—through words, equations, tables of values, or
graphs—and to move smoothly among them gives
students a range of tools to identify the relationships
and solve problems involving them.
•Identify the constant of proportionality (unit rate) in
a table and represented by an equation.
Students have worked with proportional relationships
using tables and equivalent ratios. In this lesson, they
learn that the graph of a proportional relationship is a
straight line that passes through the origin. They learn
that another name for the unit rate is the constant of
proportionality. They use these concepts to analyze
relationships that may or may not be proportional.
They write equations to describe proportional
relationships in the form of y 5 mx, in which m is the
constant of proportionality. Working with different
methods aids in flexible thinking. Students can apply
their understanding to solve a range of problems in
school and everyday life. In later lessons and grades,
they will connect proportional relationships to linear
and non-linear functions.
Prerequisite Skills
In order to be proficient with the concepts in this
lesson, students should:
•Understand ratio, unit rate, and proportions.
•Use ratio and rate reasoning to solve real-world and
mathematical problems, e.g., by reasoning about
tables of equivalent ratios or equations.
•Graph ordered pairs from a table on a coordinate
grid.
•Recognize and generate simple equivalent fractions,
including writing whole numbers as fractions.
Teacher Toolbox
Vocabulary
proportional relationship: the relationship among a
group of ratios that are equivalent
constant of proportionality: what the unit rate is
called in a proportional relationship
Ready Lessons
Tools for Instruction
Interactive Tutorials
Teacher-Toolbox.com
Prerequisite
Skills
7.RP.A.2a
7.RP.A.2b
✓
✓
✓
✓
✓✓
CCSS Focus
7.RP.A.2 Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing
on a coordinate plane and observing whether the graph is a straight line through the origin.
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of
proportional relationships.
STANDARDS FOR MATHEMATICAL PRACTICE: SMP 3, 4 (see page A9 for full text)
94
L10: Understand Proportional Relationships
©Curriculum Associates, LLC Copying is not permitted.
Part 1: Introduction
Lesson 10
At a Glance
Students review the idea that data displayed in a table
show a proportional relationship if all the ratios formed
are equivalent. They learn that the ratio expressed as
the unit rate is called the constant of proportionality.
Focus on Math concepts
Lesson 10
Part 1: introduction
ccss
7.rP.a.2a
7.rP.a.2b
Understand Proportional Relationships
What is a proportional relationship?
Suppose you and some friends plan to go to a movie and the tickets cost $8 each.
Step By Step
You will pay $8 for 1 ticket, $16 for 2 tickets, $24 for 3 tickets, $32 for 4 tickets, and so on.
The ratios of the total cost of the tickets to the number of tickets are all equivalent.
•Introduce the Question at the top of the page.
A group of ratios that are equivalent are in a proportional relationship. When ratios are
equivalent, they all have the same unit rate. In a proportional relationship, the unit rate is
called the constant of proportionality.
•Reinforce the definitions of proportional relationship
and constant of proportionality. Have a volunteer
explain what a unit rate is and relate it to the
constant of proportionality.
think How can you use a table to tell if a relationship is proportional?
The table below shows the total cost of movie tickets based on
the number of tickets you buy.
total cost of tickets ($)
number of tickets
8
1
16
2
24
3
32
4
circle the ratio in the
table that shows the
constant of
proportionality.
The ratios of the total cost of tickets to the number of tickets are equivalent. The ratios all
simplify to 8 or 8, so the ratios are in a proportional relationship.
•Read the first part of Think with students. Make sure
students can connect the data in the first table with
the ratios and the equations. Relate all the
representations to the context of movie tickets.
1
··
858
16 5 8
1
··
24 5 8
2
··
3
··
32 5 8
4
··
The unit rate is 8, so the constant of proportionality is 8. The equation c 5 8t, where c is the
total cost and t is the number of tickets, represents this relationship. The total cost is always
8 times the number of tickets.
The table below shows the cost to play in the town soccer tournament.
•Read the second part of Think with students. Ask
how the simplified ratios formed by the data in the
second table are different from those formed by the
data in the first. Emphasize that when the ratios are
not equivalent, the data do not show a proportional
relationship.
Concept Extension
Reinforce the connection between constant
of variation and unit rate.
Materials: dictionary
•Write constant of variation on the board. Say that
variation means change.
•Have students look up the word constant in the
dictionary. Have them read the various definitions
and decide which definition best applies to the
term constant of variation.
•Have volunteers describe the meaning of constant
of variation in their own words.
•Have students explain why a unit rate expresses a
constant rate of change and therefore can be
called the constant of variation.
L10: Understand Proportional Relationships
©Curriculum Associates, LLC Copying is not permitted.
total cost ($)
number of Family Members
7
1
8
2
9
3
10
4
You can find and simplify the ratios of the total cost to the number of family members.
75 7
1
··
85 4
2
··
95 3
3
··
10 5 2 1
2
··
4
··
The ratios are not equivalent, so the quantities are not in a proportional relationship.
88
L10: Understand Proportional Relationships
©CurriculumAssociates,LLC Copyingisnotpermitted.
Mathematical Discourse
•Relationships can be described in equations and in
words. The relationship of total cost to tickets is
shown in the equation c 5 8t. How could you
describe the relationship in a “word equation”?
Responses should convey the idea that the total cost of the tickets is 8 times the number of tickets.
•How would your word equation be different if the
situation were about teams and players?
The total number of players is 8 times the
number of teams.
•Think of something in our class or school that
c 5 8t could describe and use it in a word equation.
Students might suggest desks in a group,
students at lunch tables, or weeks in a semester.
95
Part 1: Introduction
Lesson 10
At a Glance
•Read Think with the students. Ask students how
they can represent the data in a table using a graph.
•Have students compare and contrast the two graphs.
Discuss why the first graph shows a proportional
relationship but the second graph does not.
•After students have read the information in the table,
have them restate each statement in their own words.
•Have students read and reply to the Reflect directive.
SMP Tip: Using graphs to determine whether or
not a relationship is proportional helps students see
how they can model real-world situations with
mathematics. (SMP 4)
Lesson 10
think How can you use a graph to tell if a relationship is proportional?
You can use a graph to determine if a relationship is proportional.
The data for the cost of movie tickets and the cost to participate in
the soccer tournament can be modeled by the graphs below.
40
36
32
28
24
20
16
12
8
4
0 1 2 3 4 5 6 7 8 9 10
Number of Tickets
Total Cost ($)
Step By Step
Part 1: introduction
Total Cost ($)
Students explore how to use graphs to determine
whether or not relationships are proportional.
10
9
8
7
6
5
4
3
2
1
0 1 2 3 4 5 6 7 8 9 10
Number of Family Members
The points on the graphs are on a straight line for both sets of data, but only the data for the
cost of movie tickets goes through the origin. This means that only the total cost of the
movie tickets compared to the number of tickets is a proportional relationship.
Proportional relationship
non-Proportional relationship
• The graph can be represented by a
• The graph may or may not be represented by
straight line.
• The line goes through the origin.
a straight line.
• If the graph is a line, it does not go through
the origin.
reflect
1 Look at the graph that compares the total cost to the number of movie tickets you buy.
How can you use the graph to find the cost of 5 movie tickets?
Possible answer: the slope of the line, $8, represents the cost of 1 ticket.
therefore, multiply 5 tickets by $8 to get $40 for 5 tickets.
L10: Understand Proportional Relationships
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ELL Support
•Sketch examples and non-examples of straight line
and through the origin on the board. Model the
correct language such as, This line goes through the
origin but it is not a straight line or This is a straight
line that does not go through the origin.
•Have a volunteer go to the board and draw an
example or non-example on a coordinate plane.
The volunteer will call on classmates to describe
the graph using straight line and through the origin.
Repeat with other volunteers.
•Once students are comfortable with the
vocabulary, tie the terms to the graphs of
proportional and non-proportional relationships.
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Compare the two graphs.
How are they alike? How
are they different?
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Mathematical Discourse
Extend the discussion of the Reflect directive with
these questions.
•Can you repeat that method in your own words?
Responses should paraphrase how the student
found the constant of proportionality from the
graph.
•Is there another way to find the constant of
proportionality?
Responses could include making a table of
ratios from the points on the line, using the
y-coordinate of the point where x 5 1, or
recognizing that each point is 8 units higher on
the y-axis.
L10: Understand Proportional Relationships
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Part 2: Guided Instruction
Lesson 10
At a Glance
Part 2: guided instruction
Students examine data in tables to see if they represent
proportional relationships.
explore it
use the table below to analyze the cost of downloading applications to a phone.
Step By Step
number of Downloads
total cost ($)
4
12
5
15
6
18
10
30
Divide the total cost by the corresponding number of downloads.
3 What is the ratio of the total cost to the number of downloads when you download
12
15
6
4
5
2
2 applications? ··
4 applications? ···
5 applications? ···
18
30
6
10
6 applications? ···
10 applications? ···
•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.
4 Are the data in the table in a proportional relationship? If so, what is the constant
of proportionality?
yes; all the ratios are equivalent so the data are in a proportional relationship.
the constant of proportionality is 3.
now try these problems.
5 The table shows the number of hours needed for different numbers of people to clean
up after a school dance.
•For the second table, suggest to students that they
can use either equivalent ratios or graphs to
determine if the relationships are proportional.
hours needed to clean up
number of People cleaning
12
2
9
3
8
4
6
6
Are the quantities in the table in a proportional relationship? Explain your reasoning.
no. the ratios are not equivalent, so the quantities are not proportional.
6 The students in the Service Club are mixing paint to make a mural. The table below
•Help students understand what they are being asked
to find in the last problem. Help them connect their
answer to the idea of equivalent ratios.
STUDENT MISCONCEPTION ALERT: Some
students may find the ratios but not remember that
all the ratios must be the same for the data to be
proportional and have a constant of proportionality.
Have students find and simplify the ratios for each
problem. Then note that there can be only one
constant of proportionality. If the simplified ratios
are not equivalent, ask students why they cannot
pick one of them to be the constant of
proportionality. Then reinforce the idea that the
relationship is not proportional.
2
6
2 How can you find the ratio of the total cost to the number of downloads?
•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.
•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.
Lesson 10
shows the different parts of paint that the students mix together.
Parts of red Paint
Parts of White Paint
a
1
3
b
2
4
c
4
8
D
2
6
e
3
9
Two mixtures of paint will be the same shade if the red paint and the white paint are in
the same ratio. How many different shades of paint did the students make? Explain.
2; the ratio of white paint to red paint is 3 in a, D, and e and is 2 for b and c.
1
··
90
1
··
L10: Understand Proportional Relationships
©CurriculumAssociates,LLC Copyingisnotpermitted.
Mathematical Discourse
•How can you tell if the data in the table form
equivalent ratios?
Responses might indicate that it they all
simplify to the same ratio, then they are
equivalent.
•Do you think you should check every ratio before you
decide if the relationship is proportional or not? Why
or why not?
Responses might include that you can
recognize a non-proportional relationship with
the first non-equivalent ratio.
•If the relationship is proportional, how do you find
the constant of proportionality? Could you do it
another way?
Responses might use the term “unit rate”
or indicate that it is the ratio with the
denominator of 1.
L10: Understand Proportional Relationships
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97
Part 2: Guided Instruction
Lesson 10
At a Glance
Students graph data from a table to see if there is a
proportional relationship.
Step By Step
Part 2: guided instruction
Lesson 10
talk about it
solve the problems below as a group.
7 Refer to the situation in Problem 6. Which shades of paint are the most red? Why?
Mixtures b and c; Possible explanation: For Mixtures b and c, the ratio of white
•Organize students into pairs or groups. You may
choose to work through the first Talk About It
problem together as a class.
paint to red paint is 2 to 1. For Mixtures a, D, and e, the ratio of white paint to red
paint is 3 to 1, so there is more red paint in Mixtures b and c.
8 Use the table in Problem 6. Plot a point for each ordered pair. After you plot each point,
draw a line connecting the point to (0, 0).
•Walk around to each group, listen to, and join in on
discussions at different points. Use the Mathematical
Discourse questions to help support or extend
students’ thinking.
•If students need more support, have them use the
Hands-On-Activity to help them visualize the
common ratios.
Parts of
of White
White Paint
Paint
Parts
12
12
11
11
10
10
99
88
77
66
55
44
33
22
11
00 11 22 33 44 55 66 77 88 9910
10
Parts
Parts of
of Red
Red Paint
Paint
9 Based on the graph, what do the mixtures that are the same shade have in common?
What does this tell you about their relationship?
the points lie on a line that goes through the point (0, 0) so they are in a
•Direct the group’s attention to Try It Another Way.
Have a volunteer from each group come to the board
to present a table or graph that illustrates the group’s
solutions to problems 10 and 11.
proportional relationship.
try it another Way
Work with your group to determine whether the equation represents a proportional
relationship. explain your choice. you may want to make a table similar to those in
Problems 5 and 6 or a graph similar to that in problem 8 on separate paper to support
your reasoning.
10 y 5 2x 1 4 no; the graph is a line but it does not go through the origin.
11 y 5 2x yes; the graph is a line and goes through the origin.
L10: Understand Proportional Relationships
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Hands-On Activity
Use concrete materials to model ratios.
Materials: red paper, white paper, scissors, drawing
paper, glue sticks
•Have students cut 12 small squares from red
paper and 30 from white paper.
•Have them divide a sheet of drawing paper into
5 sections.
•Students should use glue and the small squares to
illustrate the following ratios of red paint to white
paint: 1 to 3, 2 to 4, 4 to 8, 2 to 6, and 3 to 9.
•Direct students to write 2 or 3 sentences that
identify the two sets of equivalent ratios and
explain why they are equivalent.
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91
Mathematical Discourse
•For the Try It Another Way problem, what did your
group do to get started with the questions?
Responses may include making a table of
values and graphing or checking equivalent
ratios.
•Did other groups use a different way to decide which
relationship is proportional?
Listen for responses that show students have
connected the form of the equation to
proportional and non-proportional
relationships and encourage explanation as a
preview to upcoming lessons.
•How can you use your method to decide if
y 5 3x 1 6 is proportional?
Responses should indicate understanding of
the method.
L10: Understand Proportional Relationships
©Curriculum Associates, LLC Copying is not permitted.
Part 3: Guided Practice
Lesson 10
At a Glance
Part 3: guided Practice
Students demonstrate their understanding of
proportional relationships as they examine
relationships represented using both graphs and tables.
Lesson 10
connect it
talk through these problems as a class. then write your answers below.
12 compare: The graphs below show the number of points you earn in each level of a
game. Which games, if any, have a proportional relationship between the number of
points you earn and the level of the game? In which game can you earn the most points
in Level 2? Explain your answer.
Number of Points
Compare:
Game B
350
300
250
200
150
100
50
0 1 2 3 4 5
Level
Number of Points
Game A
•Discuss each Connect It problem as a class using the
discussion points outlined below.
Game C
Number of Points
Step By Step
350
300
250
200
150
100
50
0 1 2 3 4 5
Level
350
300
250
200
150
100
50
0 1 2 3 4 5
Level
games a and c have a proportional relationship. the points are on straight lines
•As students evaluate each graph, have them identify
the two features that show whether or not a graph
shows a proportional relationship.
that go through the origin; you earn more points in game c. (the constant of
proportionality is 100.)
13 apply: Servers at a snack shop use the table below to find the total cost for frozen
yogurt, but some of the numbers have worn off. If the total cost is proportional to the
number of cups of frozen yogurt, find the missing numbers in the table.
•Use the following to lead a class discussion that
relates the idea of a constant ratio to the graphs:
number of cups of Frozen yogurt
total cost ($)
1
2
3
4
4.50
9.00
13.50
18.00
14 analyze: Michael says that the difference between Dani’s and Raj’s ages is always the
W
hat is the number of points possible for Level 1 of
each game? [A: 50; B: 100, C: 100]
same, so Raj’s age is proportional to Dani’s age. Is Michael correct? Explain.
Dani’s age
raj’s age
D
o you think the ratio of points per level will remain
constant for all the levels of each game? [Only for
Games A and C]
2010
5
10
2015
10
15
2020
15
20
2025
20
25
no, Michael is not correct. the difference between their ages is always the same,
but none of the ratios 10 , 15, 20, or 25 are equivalent, so the ages are not in a
5 ···
10 ···
15
···
20
···
proportional relationship.
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L10: Understand Proportional Relationships
©CurriculumAssociates,LLC Copyingisnotpermitted.
Apply:
•The second problem focuses on the idea of the unit
rate or constant of proportionality.
•Once students find the unit rate, have them explain
how they can use it to find the total cost of other
amounts of yogurt.
Analyze:
•This problem requires students to focus on the
multiplicative aspects of proportional relationships.
•Have students suggest different ways to show
whether or not the data is proportional. If they use
ratios, discuss why there is no constant of
proportionality. If students chose to use a graph,
have them explain why it does not display a
proportional relationship.
L10: Understand Proportional Relationships
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•After students have determined that the data are not
proportional, have them examine the table. Ask
questions such as:
How do the boys’ ages compare when you go from one
column to the next? [Raj’s age is always 5 more than
Dani’s.]
Is the ratio of Raj’s age to Dani’s age a constant? [No]
How does this confirm that the data are not
proportional? [They are obtained by adding, not
multiplying.]
SMP Tip: Encourage students to support their
answers by referring to the characteristics of the
graph or the idea of equivalent common ratios. This
helps them practice constructing viable arguments
and critique the reasoning of others (SMP 3) as they
explain whether or not the relationships are
proportional.
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Part 4: Common Core Performance Task
Lesson 10
At a Glance
Part 4: common core Performance task
Students generate one table of data that compares side
length and perimeter and another that compares side
length and area. They analyze the data using both ratios
and graphs to determine if the data are proportional.
Put it together
15 Use what you know to complete this task.
Paigeworksinanartstorethatsellssquarepiecesofcanvas.Thereare5different
squarestochoosefrom.
Step By Step
canvas
Length of side (in feet)
•Direct students to complete the Put It Together task
on their own.
A
1
B
2
C
3
D
4
E
5
a Make a table to show the perimeter for each square piece of canvas. Use the
formula P 5 4s. Then draw a graph to compare the length of a side of each
square to its perimeter. Use your table and graph to explain whether this is a
proportional relationship.
Length of side
(in feet)
1
2
3
4
5
Perimeter
(in feet)
4
8
12
16
20
Perimeter
•As students work on their own, walk around to
assess their progress and understanding, to answer
their questions, and to give additional support, if
needed.
24
20
16
12
8
4
0 1 2 3 4 5
Length of Side
yes; the ratios are all equivalent to 4. the graph is a line through the origin.
1
··
•If time permits, have students share their tables and
graphs and explain why they do or do not show a
proportional relationship.
Length of side
(in feet)
1
2
3
4
5
See student facsimile page for possible student answers.
area
(in square feet)
1
4
9
16
25
Area
b Make a table to show the area for each square piece of canvas. Use the equation
A 5 s2. Then draw a graph to compare the length of a side of each square to its area.
Use your table and graph to explain whether this is a proportional relationship.
Scoring Rubrics
32
28
24
20
16
12
8
4
0
1 2 3 4 5 6
Length of Side
no; Possible explanations: the ratios of the areas to the side lengths are not all
equal; the graph of the ratios of the areas to the side lengths is not a straight line.
A
Points Expectations
2
100
Lesson 10
The response demonstrates the student’s
mathematical understanding of how to
show that a relationship is proportional
using both
• a table of equivalent ratios and
• a graph of a straight line passing through
the origin.
1
The student was able to show that the data
are proportional using either a table of
equivalent ratios or a graph, but not both.
0
There is no response or the response does
not demonstrate that the data are
proportional.
L10: Understand Proportional Relationships
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93
B
Points Expectations
2
The response demonstrates the student’s
mathematical understanding of how to
show that a relationship is not proportional
because
• the ratios formed by the data in the table
are not equivalent and
• the graph formed by the data is not a
straight line.
1
The student was able to show that the data
are not proportional by showing that the
ratios formed are not equivalent or the
graph formed is not a straight line, but not
both.
0
There is no response or the response does
not demonstrate that the data are not
proportional.
L10: Understand Proportional Relationships
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Differentiated Instruction
Lesson 10
Intervention Activity
On-Level Activity
Use graphs to model proportional and
non-proportional relationships.
Analyze real-world situations to see if they are
proportional.
Materials: graph paper
Materials: graph paper
Students will connect graphs, ratios, and
proportional relationships.
Students will generate data from real-world situations
and then analyze the relationships to see if they are
proportional.
Students should label the left half of a sheet of graph
paper “Proportional” and draw and label a coordinate
plane. They should plot 5 points that lie on a line
passing through the origin. Beneath the graph, have
them record the data in a table with rows labeled x
and y. Have them find and simplify the ratios, x:y.
Review the idea of the constant of proportionality
and have them record their constant of
proportionality below the table.
Have students label the right half “Not Proportional”
and repeat the process with 5 points that are not part
of a straight line passing through the origin. After
they find and simplify the ratios, x:y, discuss why the
data do not have a constant of proportionality.
Write the following information on the board.
Video Plan A: $2 for each video you rent
Video Plan B: $1 for each video you rent plus a
$10 monthly fee
Have students make a table of data for each plan to
show the amount it would cost to rent various
numbers of videos in one month. After they have
generated the data, ask students to describe two
methods they can use to tell whether or not either
plan represents a proportional relationship. Then
have them work in pairs to analyze each set of data
using both ratios and a graph. They should then
explain why Plan A is a proportional relationship
and name the constant of proportionality.
Challenge Activity
Develop and interpret a proportional relationship.
Materials: graph paper
Students will develop and interpret a proportional relationship from a point on a coordinate plane.
Have students plot one point such as (3, 6) or (5, 2) on a coordinate plane. They should connect the origin and
their point and extend the line to the edge of the paper. Have them identify several other points on the line and
enter the coordinates in a table with rows labeled x and y.
Have students work individually to find the following:
•the ratio of x to y in simplest form for each point
•the constant of proportionality
•an equation that relates x and y
•a real-world situation that could be modeled by their data
Have students share their work in small groups. They should explain how the graph, the table, the equation,
and the real-world situation are related.
L10: Understand Proportional Relationships
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