Lesson 9
Part 1: Introduction
CCLS
7.RP.1
Ratios Involving Complex Fractions
In Grade 6, you learned about unit rates. Take a look at this problem.
Jana is training for a triathlon that includes a 112-mile bike ride. Today, she rode her
bike 12 miles in 45 minutes. What is Jana’s rate in miles per hour?
12 miles
45 minutes
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?
At the same rate, how many miles did she bike in the next 15 minutes? At the same rate, how many miles did she bike in the last 15 minutes? How many more minutes would Jana need to bike to total one hour? At the same rate, how many miles would she bike in that amount of time? Explain how you could find the number of miles Jana bikes in one hour.
78
L9: Ratios Involving Complex Fractions
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Part 1: Introduction
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
in 45 minutes. That’s the same thing as 12 miles in ​ 3 ​hour.
4
··
of  
miles  ​ 
​ number  
5 ​ 12  ​
number of hours ··
·············
​ 3 ​
4
··
​ 12  ​
The fraction ··
​ 3 ​ is a complex fraction. A complex fraction is a fraction where either the
4
··
numerator is a fraction, the denominator is a fraction, or both the numerator and the
denominator are fractions. You can simplify a complex fraction by dividing, just as you
would do with whole numbers.
The fraction bar represents division, so you can think of ​ 6 miles 
 ​ 
as 6 4 2 5 3 miles per hour.
​ 12 miles   
​ 
2 hours
······
You can think about ·······
in the same way.
3
​   ​hour
4
··
12
12
3
​    ​5 ​    ​4 ​   ​
1
4
··
··
··
​ 3 ​
4
··
5 ​ 12  ​3 ​ 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.
L9: Ratios Involving Complex Fractions
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79
Part 2: Modeled Instruction
Lesson 9
Read the problem below. Then explore different ways to understand how to find a
unit rate.
Max’s favorite recipe for oatmeal raisin cookies makes 48 servings. He wants to make
some cookies but only has one egg. Max has to adjust the amounts of the other
ingredients. How much flour will he need?
Oatmeal Raisin Cookies
​ 3 ​cup butter
1 ​ 1 ​cups brown sugar
​ 3 ​teaspoon cinnamon
2 eggs
1 teaspoon vanilla
2 ​ 3 ​cups oats
1 ​ 1 ​cups flour
2
··
1 teaspoon baking soda
1 cup raisins
4
··
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.
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.
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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Part 2: Guided Instruction
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
··
3 How could you find the number that is between 0 and 1 ​ 1 ​ ?
2
··
4 How many cups of flour does Max need to use if he has just 1 egg? Show your work.
5 Write the ratio that compares 1 ​ 1 ​cups of flour to 2 eggs.
2
··
   
​ 
5 ​ 
​ cups of flour
eggs
··········
  
​ 
······
6 Write and simplify a division expression to find the number of cups of flour Max needs to
use if he has just 1 egg.
7 The unit rate is . The number of cups of flour is times the number of eggs.
8 Explain how to find a unit rate.
Try It
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? 10 If Max has only one cup of flour, how much vanilla will he need? L9: Ratios Involving Complex Fractions
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81
Part 3: Modeled Instruction
Lesson 9
Read the problem below. Then explore different ways to understand how to find and
compare unit rates.
José’s mother is trying to decide whether or not she should buy a 12-ounce package of
coffee on sale for $7.50. She knows that she can buy the same coffee for $9.00 per
pound. Which is the better buy?
Model It
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.
There are 16 ounces in 1 pound, so 12 ounces is ​ 12 ​  or ​ 3 ​pound.
16
··
4
··
You can write $7.50 using fractions. $7.50 is the same as $7 ​ 1 ​.
2
··
Start both
number lines
at 0.
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 .
1
1
4
1
2
Find x, the cost
for 1 pound of
coffee.
72
x
3
4
1
Find the cost for each quarter-pound of coffee. Then find the unit cost.
82
1
1
Cost, in dollars
0
22
5
72
10
Pounds of coffee
0
1
4
1
2
3
4
1
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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
··
3
line is divided into 3 equal parts from 0 to ​   ​ . How can you use this to find the cost of
4
··
1 pound of coffee?
dollars
12 Write the ratio that compares $7 ​ 1 ​dollars to ​ 3 ​pound of coffee. ​ 
      
​5 ​ 
  
​ 
2
4
pounds of coffee ······
··
··
·············
13 Write and simplify a division expression to find the cost of 1 pound of coffee.
14 Which is the better buy, 12 ounces for $7.50 or 1 pound for $9.00? Explain your reasoning.
15 If you started the problem by converting 1 pound to 16 ounces, would you get the same
result? Justify your conclusion.
16 Can you compare any two unit rates? Explain.
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?
L9: Ratios Involving Complex Fractions
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83
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.
Oliver is training for a marathon. In practice, he runs 15 kilometers
in 72 minutes. What is his speed in kilometers per hour?
Convert the time in minutes to hours to find kilometers per hour.
72 minutes 5 1 hour 12 minutes
5 1 ​ 12  ​hours or 1 ​ 1 ​hours
60
···
15
​ km  ​5 ​   ​ 
hr
···
···
1 ​ 1 ​
5
··
5
··
Pair/Share
How did you decide how
to write the ratio?
How do you evaluate a
complex fraction?
5 15 4 1 ​ 1 ​
5 15 4 ​ 6 ​
5 15 3 ​ 5 ​
5 ​ 75   ​ or 12 ​ 1 ​
5
··
5
··
6
··
6
···
Solution: 2
··
Oliver runs 12 ​ 1 ​kilometers per hour.
2
··
18 Alexis washes 10 ​ 1 ​windows in ​ 3 ​hour. At this rate, how many windows
2
4
··
··
can she wash in one hour?
Pair/Share
How can you tell if your
answer is reasonable?
84
Solution: L9: Ratios Involving Complex Fractions
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Part 4: Guided Practice
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?
Lesson 9
What is the ratio of
pounds of carrots
to cakes?
Show your work.
Pair/Share
Solution: 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 steps did you take
to find the unit rate?
What unit rate do you
need to find?
letter of the correct answer.
A ​ 1 ​ block
5
··
B ​ 1 ​block
2
··
C 2 blocks
D 5 blocks
Dee chose C as the correct answer. What was her error?
L9: Ratios Involving Complex Fractions
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Pair/Share
Does Dee’s answer make
sense?
85
Part 5: Common Core Practice
Lesson 9
Answer Form
Solve the problems. Mark your answers to problems 1–4 on
the Answer Form to the right. Be sure to show your work.
1 1 
2 
3 
4 
B
B
B
B
C
C
C
C
D
D
D
D
Number
Correct
4
1  ​ teaspoons of
A 14-ounce energy drink contains 10 ​ }
2
sugar. How much sugar is in one ounce of the drink?
3 ​  teaspoon
A​ }
4
2 B
1  ​ teaspoons
1 ​ }
3
C
1 ​  teaspoons
1 ​ }
2
D
1  ​ teaspoons
3 ​ }
2
1  ​ minutes. What is this rate in minutes per lap?
Bennie swims 6 laps in 4 ​ }
2
2 ​  minute per lap
A​ }
3
3  ​ minute per lap
B​ }
4
3 C
1 ​  minutes per lap
1 ​ }
3
D
1  ​ minutes per lap
1 ​ }
2
One of the highest snowfall rates ever recorded was in Silver Lake, Colorado, in April 1921,
1  ​ hours. What was that rate in inches per hour?
when just over 7 feet of snow fell in 27 ​ }
2
14 ​  inch per hour
A​ }}
55
55  ​ inch per hour
B​ }}
158
86
C
3  ​ inches per hour
3 ​ }}
55
D
13 ​  inches per hour
3 ​ }}
14
L9: Ratios Involving Complex Fractions
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Part 5: Common Core Practice
4 Lesson 9
A grocery store sells different types of Trail Mix. The cost and weight of each type is shown in
the table.
Trail Mix A
Trail Mix B
Trail Mix C
Cost ($)
6
8.50
2.25
Weight
​    ​ lb
}
4
1 lb
4 oz
3
1 lb 5 16 oz
Which statement is correct?
5 A
Trail Mix A is the best buy.
B
Trail Mix B is the best buy.
C
Trail Mix C is the best buy.
D
They are all the same price.
Two friends worked out on treadmills at the gym.
• Alden walked 2 miles in }
​ 3  ​ hour.
4
3  ​ miles in 30 minutes.
• Kira walked 1 ​ }
4
Who walked at a faster rate? Explain your reasoning.
Show your work.
Answer Self Check Go back and see what you can check off on the Self Check on page 77.
L9: Ratios Involving Complex Fractions
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87
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.
Toolbox
Vocabulary
Teacher-Toolbox.com
Prerequisite
Skills
unit rate: the part of the rate that is being compared
to 1
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.1
✓
✓✓
✓
✓
✓
CCLS Focus
7.RP.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 ​ __··21 ​miles per hour,
2
··
4
··
equivalently 2 miles per hour.
​   ​
4
··
STANDARDS FOR MATHEMATICAL PRACTICE: SMP 1, 6, 7 (see page A9 for full text)
80
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.
Lesson 9
Part 1: introduction
CCLs
7.rP.1
ratios involving Complex Fractions
in grade 6, you learned about unit rates. take a look at this problem.
Step By Step
Jana is training for a triathlon that includes a 112-mile bike ride. Today, she rode her
bike 12 miles in 45 minutes. What is Jana’s rate in miles per hour?
•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.
•Work through Explore It as a class.
If Jana biked at a constant rate, how many miles did she bike in the first 15 minutes?
4
•Have students look at the diagram and explain how
to figure out how many rectangles it takes to
represent 15 minutes.
At the same rate, how many miles did she bike in the next 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.
78
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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Copying is not permitted.
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.
81
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
12
 
 
5  
number of hours ··
·············
3 
4
··
•Read Find Out More as a class.
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
•Have students look at the ratio __
​ 12
  ​ . 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.
​   ​
2 hours
······
12 miles
4
··
 
You can think about ·······
in the same way.
3
 hour
4
··
not a unit rate? [The number of hours must be one.]
12
12
3
 5  4  
1
4
··
··
··
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.
5 12  3 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
•Reinforce the idea that the fraction bar can mean
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.
79
L9: Ratios Involving Complex Fractions
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ELL Support
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.
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.
82
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
··
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’s favorite recipe for oatmeal raisin cookies makes 48 servings. He wants to make
some cookies but only has one egg. Max has to adjust the amounts of the other
ingredients. How much flour will he need?
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.
3
1 1 cups brown sugar
2 eggs
1 teaspoon vanilla
2 3 cups oats
3  teaspoon cinnamon
1 1 cups flour
1 teaspoon baking soda
1 cup raisins
2
··
2
··
4
··
4
··
Model it
•Have students use their own words to explain what
they are trying to find in order to solve this problem.
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 relates 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
numbers on the top number line.
 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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Hands-On Activity
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 line across the entire
length of the paper then divide it into ​ 1 ​-inch
4
··
segments.
•Have students fold the paper in half 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
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Mathematical Discourse
•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.
83
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
··
Divide 1 1  by 2 or multiply 1 1 by 1. 
2
2
2
··
··
··
•Read Connect It as a class. Be sure to point out that
the questions refer to the problem on page 80.
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 1 
cups of flour  
2  
5 ··
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
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.
81
L9: Ratios Involving Complex Fractions
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Try It Solutions
9 Solution: ​ 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
··
simplify the ratio ___
​  11  ​  . They may also draw a double
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
··
84
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’s mother is trying to decide whether or not she should buy a 12-ounce package of
coffee on sale for $7.50. She knows that she can buy the same coffee for $9.00 per
pound. Which is the better buy?
•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
··
Start both
number lines
at 0.
Make sure students understand why 12 ounces is
equivalent to ​ 3 ​pound.
4
··
•Have students study the first double number line.
Go over the steps used to draw the number line
accurately.
•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.
Ask, How do we know that 2 ​ 1 ​dollars lines up
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
©Curriculum Associates, LLC
Copying is not permitted.
Mathematical Discourse
•What are some equivalent ratios shown by the
number line?
2 ​ 1 ​
___
Students may list ​  1··2  ​ ,
​   ​
4
··
__
​ 51 ​ ,
​   ​
2
··
7 ​ 1 ​
___
​  3··2  ​ , and ​ 10  ​ .
1
··
​   ​
4
··
•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.
85
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.
7 1 
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.
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.
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
©Curriculum Associates, LLC Copying is not permitted.
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
  ​ 5 1 ​ 1 ​However, that
Rina: ___
​  2··2  ​5 1 ​ 1 ​ ; Jonah: ​ ___
1
4
··
2 ​   ​
4
··
3
··
means Jonah’s recipe has more cookies per cup of
sugar, not more sugar per dozen cookies.
86
L9: Ratios Involving Complex Fractions
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Part 4: Guided Practice
Lesson 9
Part 4: guided Practice
Lesson 9
study the model below. then answer questions 18–20.
StudentModel
The student knew that
60 minutes 5 1 hour, so
72 minutes 5 60 minutes
1 12 minutes, or
1 hour 12 minutes.
Oliver is training for a marathon. In practice, he runs 15 kilometers
in 72 minutes. What is his speed in kilometers per hour?
Convert the time in minutes to hours to find kilometers per hour.
5 1 12 hours or 1 1  hours
60
···
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
··
5
··
5 15 4 1 1  
Show your work.
8 1 
pounds of carrots  
5 6··4  
···
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
km  5 15  
hr
···
···
1 1 
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
··
20 It takes Zach 15 minutes to walk 7 1  blocks to the swimming pool.
2
··
oliver runs 12 1  kilometers per hour.
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  
number of  
windows  
5 ··2  
number of hours
················
····
3 
4
··
10 1 4 3  5 21 4 3 
2
··
4
··
2
···
4
··
1
  block
5
··
1
 block
2
··
C
2 blocks
D
5 blocks
Dee chose C as the correct answer. What was her error?
5 21 3 4  
2
···
a
b
3
··
she found the number of minutes per block instead of the
5 84 
number of blocks per minute.
6
···
Pair/share
How can you tell if your
answer is reasonable?
84
Solution:
Pair/share
Does Dee’s answer make
sense?
5 14
alexis can wash 14 windows in one hour.
L9: Ratios Involving Complex Fractions
85
L9: Ratios Involving Complex Fractions
©Curriculum Associates, LLC
Copying is not permitted.
©Curriculum Associates, LLC Copying is not permitted.
At a Glance
Solutions
Students write and simplify ratios to solve word
problems involving unit rate. They may also use double
number lines to find the solution.
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.
Step By Step
18 Solution: 14; Students could solve the problem by
•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.
10 ​ 1 ​
simplifying ____
​  3 ··2 ​   or use a double number line.
​   ​
4
··
19 Solution: 1 ​ 3 ​ ; Students could solve the problem by
8 1
··
8 ​   ​
4
··
  ​ or use a double number line.
simplifying ​ ___
6
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.
2
··
L9: Ratios Involving Complex Fractions
©Curriculum Associates, LLC Copying is not permitted.
87
Part 5: Common Core Practice
Part 5: Common Core Practice
Lesson 9
Lesson 9
Part 5: Common Core Practice
Lesson 9
answer Form
Solve the problems. Mark your answers to problems 1–4 on
the Answer Form to the right. Be sure to show your work.
1
2
3
1
2
3
4
A
A
A
A
B
B
B
B
C
C
C
C
D
D
D
D
4
number
Correct
A grocery store sells different types of Trail Mix. The cost and weight of each type is shown in
the table.
4
1 teaspoons of
A 14-ounce energy drink contains 10 }
2
sugar. How much sugar is in one ounce of the drink?
trail Mix b
trail Mix C
6
8.50
2.25
Weight
lb
}
4
1 lb
4 oz
3
3
A
teaspoon
}
4
B
1 teaspoons
1}
3
C
1 teaspoons
1}
2
D
1 teaspoons
3}
2
1 lb 5 16 oz
Which statement is correct?
1 minutes. What is this rate in minutes per lap?
Bennie swims 6 laps in 4 }
2
A
Trail Mix A is the best buy.
B
Trail Mix B is the best buy.
C
Trail Mix C is the best buy.
D
They are all the same price.
2
A
minute per lap
}
3
B
3 minute per lap
}
4
C
1 minutes per lap
1}
3
D
1 minutes per lap
1}
2
5
3 miles in 30 minutes.
• Kira walked 1 }
4
Who walked at a faster rate? Explain your reasoning.
1 hours. What was that rate in inches per hour?
when just over 7 feet of snow fell in 27 }
2
A
14 inch per hour
}}
55
B
inch per hour
}}
158
C
3 inches per hour
3 }}
55
Two friends worked out on treadmills at the gym.
3 hour.
• Alden walked 2 miles in }
4
One of the highest snowfall rates ever recorded was in Silver Lake, Colorado, in April 1921,
D
trail Mix a
Cost ($)
Show your work.
1  
kira’s rate: ··4  
2 4 3 5 2 3 4 
1 3 4 1 5 7 3 2 
··
3 
4
··
4
··
55
3
alden’s rate: 2  
1
··
···
1 
2
··
3
··
5 8 or 2 2 miles per hour
3
··
3
··
4
··
2
··
4
··
1
··
5 14 or 3 1 miles per hour
4
···
2
··
kira walks at a faster rate.
Answer
13 inches per hour
3 }}
14
self Check Go back and see what you can check off on the Self Check on page 77.
86
L9: Ratios Involving Complex Fractions
87
L9: Ratios Involving Complex Fractions
©Curriculum Associates, LLC
Copying is not permitted.
©Curriculum Associates, LLC Copying is not permitted.
At a Glance
Solutions
Students find unit rates to solve word problems that
might appear on a mathematics test.
1 Solution: A; Write and simplify the ratio of ounces
Step By Step
2 Solution: B; Write and simplify the ratio of minutes
•First, tell students that they will find unit rates to
solve word problems. Then have students read the
directions and answer the questions independently.
Remind students to fill in the correct answer choices
on the Answer Form.
to laps, ___
​  6··2  ​ .
•After students have completed the Common Core
Practice problems, review and discuss correct
answers. Have students record the number of correct
answers in the box provided.
10 ​ 1 ​
of sugar to ounces of energy drink, ____
​  14··2  ​  .
4 ​ 1 ​
3 Solution: C; Rewrite 7 feet as 84 inches and then
84
write and simplify the ratio of inches to hours, ​ ____
  ​  .
1
27 ​   ​
2
··
4 Solution: 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.
2   ​or 2 ​ 2 ​miles per hour.
5 Solution: Alden’s rate is ​ __
3
3
··
​   ​
1 ​ 3 ​
4
··
4
··
  ​or 3 ​ 1 ​miles per hour. Kira’s rate
Kira’s rate is ​ ___
2
··
​ 1 ​
is faster.
88
2
··
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
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 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.
89