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Continuity of Learning Assignments

Grade: 7th grade Subject: Math

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Essential Activity

Iready Teacher assigned Lesson on Ratios involving complex fractions through Clever

Complete Lesson 9 in packet pg. 87-90.

Essential Activity

Iready Teacher assigned Lesson on Ratios involving complex fractions through Clever

Finish Lesson 9 in packet pg. 91-94

Essential Activity

Iready Teacher assigned Lesson on Proportional Relationships

Complete Lesson 10 97-102

Essential Activity

Iready Teacher assigned Lesson on wrting equations from proportional relationships.

Complete Lesson 11 in packet pg. 105-108

Essential Activity

Iready Teacher assigned Lesson on wrting equations from proportional relationships.

Finish Lesson 11 in packet pg. 109-110

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©Curriculum Associates, LLC Copying is not permitted. 87Lesson 9 Ratios Involving Complex Fractions

Ratios Involving Complex Fractions

Lesson 9

Name:

Vocabularyrate the ratio that compares the first quantity to one unit of the second quantity. For a ratio of 2 dollars for every 5 apples, the rate is 2

··

5 dollars per apple.

unit rate the numerical part of the rate, without the units. For a rate of 2 ··

5

dollar per apple, the unit rate is 2

··

5 .

Prerequisite: Find Unit Rates

Study the example problem showing how to find a unit rate. Then solve problems 1–6.

1 In the example, how much would it cost to buy 3, 4, and 5 pounds of grapefruit? Explain how you found your answers.

2 Leila uses 5 cups of grated zucchini to make 2 loaves of zucchini bread. How many cups of zucchini does she need to make 1 loaf?

3 Jenna walks 4 miles in 1 hour. At this rate, how many miles can she walk in 3.5 hours?

Example

Six pounds of grapefruit cost $3. What is the cost per pound?

You can use a double number line to help you find rate and unit rate. Start with the ratio you know: $3 for 6 pounds.

Divide both $3 and 6 pounds by 3 to get $1 for 2 pounds. Then divide $1 and 2 pounds by 2 to find the cost per pound, which is $0.50 for 1 pound.

Price ($)

Pounds

0.500 1.00 ? ? ? 3.00

6543210

Notice that you can also divide $3 by 6 to get the cost per pound directly.

©Curriculum Associates, LLC Copying is not permitted.88 Lesson 9 Ratios Involving Complex Fractions

Solve.

4 It takes 30 cups of milk to make 4 sticks of butter. Use this ratio to complete the double number line. Describe the unit rate.

5 Rashid is paid by the hour. He earned $50 for a 4-hour workday. How much does he earn for a

5 1 ·· 2 -hour workday?

Show your work.

Solution:

6 Ace Bike Rentals rents bikes for $28 per day. Renters can keep the bike for 8 hours. Bart’s Bikes rents bikes for $30 per day. Renters can keep the bike for 10 hours. Which company charges a lower hourly rate? How much lower?

Show your work.

Solution:

0

0

Cupsof Milk

Sticksof Butter 4

30


Name: Lesson 9

Find Unit Rates with Fractions

Study the example problem showing how to find a unit rate. Then solve problems 1–8.

1 If Helena uses 1 1 ·· 4 cups of water to make fabric softener, how much vinegar does she need?

2 Use the recipe to find how much hair conditioner Helena needs to make a half batch and a double batch.

3 Explain how to find the unit rate of water to vinegar. What does the unit rate mean in this context?

Example

Helena uses this recipe to make homemade fabric softener. How much water and vinegar does she use for a half batch? How much water and vinegar does she use for a double batch?

You can use a double number line to help you solve this problem.

You can use the recipe to see that you need 5 cups of water for 2 1 ·· 2 cups of vinegar. Label these on the number line. Find half of each quantity for a half batch. Double each quantity for a double batch.

Cups ofWater

Cups ofVinegar

0 2 5 10

50

12

2 121 1

4

Recipe forFabric Softener

Ingredients:

Directions:

Mix all ingredients together.

5 cups hot water

2 cups white vinegar12

1 cups hair conditioner12

Helena needs 2 1 ·· 2 cups water and 1 1 ·· 4 cup vinegar for a half batch. She needs 10 cups of water and 5 cups of vinegar for a double batch.


Solve.

Use the following recipe for granola bars to solve problems 4–8.

2 cups of oats 1 ·· 4 cup of flaxseed

1 ·· 2 cup of quinoa 2 ·· 3 cup of soy protein

4 Write ratios for the amount of oats to the amount of each of the other grains.

5 Show how to find the unit rate of oats to quinoa. Tell what the unit rate means.

6 Find unit rates for the other two ratios that you wrote in problem 4.

7 Suppose you are making granola bars with 3 cups of oats. How much quinoa, flaxseed, and soy protein would you need to use? Explain.

8 In addition to the grains listed above, the recipe also

includes 1 1 ·· 4 cups of dried fruit. Show how to find the

amount of dried fruit needed if you use 3 cups of oats.


Name: Lesson 9

Compare Unit Rates

Study the example problem showing how to compare unit rates. Then solve problems 1–6.

1 Use division to find the price per ounce for the bulk cranberries and the package of cranberries. Round to the nearest cent. (1 pound 5 16 ounces)

2 Compare the prices per ounce for packaged and bulk cranberries. Which price is less? How much less?

3 The Produce Place sells a 12-ounce package of cranberries for the same unit price as the 8-ounce package. What is the price of the 12-ounce package? Explain.

Example

The Produce Place sells cranberries in bulk for $2.40 per pound. The store also sells an 8-ounce package of cranberries for $1.50. Which is the better buy?

Use a double number line to find the cost per pound for the package of cranberries. Remember that

8 ounces 5 1 ·· 2 pound.

Multiply both $1.50 and 1 ·· 2 pound by

2 to find the price for 1 pound.

The bulk price is $2.40 per pound, and the package price is $3.00 per pound. Because $2.40 , $3.00, the bulk cranberries are a better buy.

Cost,in Dollars

Pounds ofCranberries

0 1.50 3.00

10 12


Solve.

4 Brandi jogged 2 1 ·· 2 miles in 30 minutes. Sharna jogged

3 1 ·· 2 miles in 3 ·· 4 hour. Who jogged at a slower rate?

Show your work.

Solution:

5 Amhed pays $2.10 for 5 1 ·· 4 pounds of bananas. Janel

pays $3.25 for 6 1 ·· 2 pounds of bananas. Which is the

better buy? Explain why.

6 The directions for Clean-All say to mix 1 1 ·· 2 cups of cleaner

with 2 quarts of water. The directions for Mega-Clean say

to mix 3 1 ·· 2 cups of cleaner with 1 gallon of water. Which

solution has more cleaner per quart of water?

(1 gallon 5 4 quarts)

Show your work.

Solution:


Name:

1 A road sign gives drivers information about tra!c on busy highways. One sign shows “15 miles in 20 minutes.” What is this speed in miles per hour?

Show your work.

Solution:

Ratios Involving Complex Fractions

Solve the problems.

3 Jeremy swims 5 3 ·· 5 kilometers in a 7-day period. He swims

the same distance each day. What distance does he swim

in a day?

A 39 1 ·· 5 km C 4 ·· 5 km

B 1 1 ·· 5 km D 4 ·· 35 km

Eli chose A as the correct answer. How did he get that answer?

2 During a heavy rainstorm, a city in Florida received

12 1 ·· 4 inches of rain in 25 1 ·· 2 hours. What is the approximate

rainfall rate in inches per hour?

A about 2 inches per hour

B about 1 ·· 4 inch per hour

C about 1 inch per hour

D about 1 ·· 2 inch per hour

What fraction of an hour is 20 minutes?

You can write a ratio and estimate the unit rate.

What rate do you need to find?

Lesson 9


6 Rida and Elisa are each paid by the hour. Rida worked

5 1 ·· 2 hours and earned $77. Elisa worked 3 3 ·· 4 hours and

earned $60. Whose hourly rate is greater?

Show your work.

Solution:

Solve.

4 Mrs. Cain’s coleslaw recipe calls for 1 ·· 3 cup of oil, 1 ·· 2 cup of

vinegar, and 1 ·· 4 cup of sugar. Tell whether each statement

is True or False.

a. The recipe uses 1 1 ·· 2 cups of

oil for each cup of vinegar. u True u False

b. The recipe uses 2 cups of sugar for each cup of vinegar. u True u False

c. The recipe uses 2 cups of vinegar for each cup of sugar. u True u False

d. The recipe uses 2 ·· 3 cup of

oil for each cup of vinegar. u True u False

5 Fill in the table to show the amount of each ingredient needed to make di"erent-size batches of coleslaw. Use the information from problem 4.

Batch 1 Batch 2 Batch 3

Vinegar (cups) 1

Oil (cups) 1

Sugar (cups) 3 ·· 2

How can you use unit rates to solve the problem?

In this problem, hourly rate means how much the person is paid per hour.

Make sure the terms in each ratio are in the correct order.

©Curriculum Associates, LLC Copying is not permitted. 97Lesson 10 Understand Proportional Relationships

Name: Understand Proportional Relationships

Lesson 10

Study the example problem showing how to find equivalent ratios. Then solve problems 1–7.

1 Explain how to use the unit rate to make the table in the example.

2 One ordered pair from the table is plotted on the coordinate plane. Finish plotting the ordered pairs.

3 Suppose you had 9 counselors available. How many campers could you have?

Example

There are 3 counselors assigned to a group of 24 campers. At this rate, how many counselors are needed for 40 campers?

The given ratio is 24 campers to 3 counselors. That’s a rate of 8 campers to 1 counselor. You can use this information to make a table of equivalent ratios.

Number of Campers 8 16 24 32 40

Number of Counselors 1 2 3 4 5

From the table you can see that 5 counselors are needed for 40 campers.

Prerequisite: How do you find equivalent ratios?

Vocabularyequivalent ratios two or more ratios that are equal to one another.

x

y

8O

1

2

3

4

5

16 24 32 40Number of Campers

Num

ber o

f Cou

nsel

ors

©Curriculum Associates, LLC Copying is not permitted.98 Lesson 10 Understand Proportional Relationships

Solve.

4 In 10 seconds, Jamal travels 550 feet on his bicycle. At this speed, how many feet can he travel in 1 minute? Explain.

5 Cala uses 2 pounds of Feed-All fertilizer for a 100-foot row of vegetables at her farm. At this rate, how many pounds of fertilizer would she use for a 450-foot row of vegetables? Explain.

6 The directions for Grow Better fertilizer say that the 25-pound bag covers a 1,000-foot row of vegetables. Which brand would Cala need more of to fertilize her vegetables? Explain.

7 Giselle has a catering business. There is a proportional relationship between the number of people and the amount of meat she uses for an event. The graph shows the amount of meat Giselle uses for 10 people. Find the amount of meat needed for 20, 30, 40, and 50 people. Finish labeling the graph and plotting ordered pairs. Finally, explain how to calculate the amount of meat needed for a party of 75 people.

Am

ount

of M

eat

(pou

nds)

Number of People20 4010 30 50O

x

y

4


Name: Lesson 10

Identify Proportional Relationships

Study the example showing one way to test whether a relationship is proportional. Then solve problems 1–7.

1 Explain how you can tell whether a group of ratios represents a proportional relationship.

2 Look at the ratios for Company A in the example. What is the constant of proportionality and what does it mean?

3 Write an equation to represent the relationship between the number of mugs and cost for Company A. Use c for cost and m for the number of mugs.

4 Complete the table to show a proportional relationship. Write the constant of proportionality.

Number of Yoga Classes 2 4 6 8

Cost ($) 60

constant of proportionality:

Example

The tables show the prices for ordering photo mugs from two different companies. You can use the information to write ratios showing the relationship between the cost and the corresponding number of mugs.

Company A Company B

Number of Mugs 5 10 25 50 Number of Mugs 5 10 25 50

Cost ($) 15 30 75 150 Cost ($) 20 35 80 155

15 ·· 5 5 3 30 ·· 10 5 3 75 ·· 25 5 3 150 ·· 50 5 3 20 ·· 5 5 4 35 ·· 10 5 3.5 80 ·· 25 5 3.2 155 ·· 50 5 3.1

These ratios are all equivalent. These ratios are not all equivalent.This relationship is proportional. This relationship is not proportional.

Vocabularyconstant of proportionality another term for the unit rate in a proportional relationship.


Solve.

5 Plot a point for some of the ordered pairs from the example problem shown at the right. Model the relationships by drawing a line from the y-axis through each point. Explain how the graphs show which relationship is proportional and which is not proportional.

6 Determine whether each equation below does or does not represent a proportional relationship. Support your answer using either a table or a graph.

Equation A: y 5 x

Equation B: y 5 x 1 2

7 Zahra has paper rectangles of di!erent sizes. Every rectangle is 5 centimeters longer than it is wide. Is there a proportional relationship between the lengths and widths of these rectangles? Explain.

Company A: (5, 15), (10, 30), (25, 75)Company B: (5, 20), (10, 35), (25, 80)

Cost

($)

Number of Mugs10 20 30O

x

y

20

10

40

60

70

80

30

50


Name:

Example

Describe a relationship involving some product or service and its cost that is NOT proportional. Explain how you know that it is not a proportional relationship.

Show your work. Use tables, graphs, words, and numbers to explain your answer.

Best Bike Rentals rents bikes by the day. The longer you rent the bike, the better their rate is. The table shows the cost of renting a bike for up to 7 days. The ratios of cost to days in this table are not equivalent because the relationship is not proportional.

Number of Days 1 2 3 4 5 6 7

Cost ($) 60 100 130 150 170 180 190

For example, 60 ··· 1 5 60, 100 ···· 2 5 50, and 130 ···· 3 5 43 1 ·· 3 . These

three ratios are not equivalent. The rate of dollars per day is less the longer you keep the bike.

I can also plot the ordered pairs from the table on a coordinate grid.

The points cannot be connected with a straight line that goes through the origin. This is another way to show that the relationship is not proportional.

Lesson 10

Reason and Write

Study the example. Underline two parts that you think make it a particularly good answer and a helpful example.

Where does the example . . . • answer both parts

of the problem?• use a table or graph

to explain?• use numbers to

explain?• use words to

explain?• give details?

Cost

($)

Number of Days2 4 61 3 5 7O

x

y

40

20

80

120

140

160

180

60

100


Solve the problem. Use what you learned from the model.

Describe a relationship involving some product or service and its cost that IS proportional. Explain why it is a proportional relationship, and identify the constant of proportionality.

Show your work. Use tables, graphs, words, and numbers to explain your answer. Did you . . .

• answer both parts of the problem?

• use a table or graph to explain?

• use numbers to explain?

• use words to explain?

• give details?

©Curriculum Associates, LLC Copying is not permitted. 105Lesson 11 Equations for Proportional Relationships

Name: Equations for Proportional Relationships

Lesson 11

Prerequisite: Understand Proportional Relationships

Study the example showing proportional relationships in a table and a graph. Then solve problems 1–5.

1 What is the cost of 12 bottles? Explain.

2 Find the ratio of the cost of 12 bottles to the number of bottles. Is the ratio equivalent to the unit rate?

3 How much will the total cost increase for each additional bottle of water purchased? Compare this value to the constant of proportionality.

Example

Bottles of water cost $2 each. Is the relationship of the cost to the number of bottles a proportional relationship?

You can use a table or a graph to see whether the relationship is proportional.

All of the ratios in the table are equal to 2 ·· 1 .

This is the unit rate and means that it costs $2 for one bottle

of water. The graph is a straight line through the origin (0, 0).

So this is a proportional relationship.

Vocabularyproportional relationship the relationship among a group of ratios that are equivalent.

constant of proportionality the unit rate in a proportional relationship.

Number of Bottles 1 2 3 4

Total Cost ($) 2 4 6 8

Total Cost ················ Number of Bottles 2 ·· 1 4 ·· 2 5 2 ·· 1 6 ·· 3 5 2 ·· 1 8 ·· 4 5 2 ·· 1 To

tal C

ost (

$)

Number of Bottles2 4 6 81 3 5 7O

x

y

2

1

4

6

7

8

9

3

5

©Curriculum Associates, LLC Copying is not permitted.106 Lesson 11 Equations for Proportional Relationships

Solve.

4 The table shows the number of grams of fiber in di!erent amounts of winter squash.

Number of Cups 2 4 5 8

Grams of Fiber 20 40 50 80

a. What is the ratio of the number of grams of fiber to the number of cups?

For 2 cups For 4 cups

For 5 cups For 8 cups

b. Are the data in the table in a proportional relationship? If so, what is the constant of proportionality? Explain.

c. How could you use a graph to show whether the data are in a proportional relationship?

5 There are currently 2 teachers and 22 students signed up for a field trip. For every additional 10 students, the school will assign 1 more teacher. Mel says that this is a proportional relationship. Do you agree? Complete the graph and the table to explain.

Teachers 2

Students 22

Show your work.

Solution:

Num

ber o

f Stu

dent

s

Number of Teachers2 4 6 8 91 3 5 7O

x

y

20

10

40

60

70

80

90

30

50


Name: Lesson 11

Write Equations for Proportional Relationships

Study the example showing how to identify a proportional relationship. Then solve problems 1–9.

1 Graph the relationship between the money earned and the number of lawns and connect the points. How does the graph tell you that the relationship is proportional?

2 What does the ratio 5 ·· 1 represent in terms of the example?

3 How can you use the graph to find the constant of proportionality? What is the constant of proportionality?

4 Use the constant of proportionality to write an equation that represents the amount of money earned, y, for mowing x lawns.

5 If you know the constant of proportionality, m, for two proportional quantities, x and y, what equation can you write to describe the relationship?

Example

The table shows the relationship between the money that Leo earns and the number of lawns that he mows. Is the relationship proportional?

Number of Lawns 2 4 5 8

Money Earned ($) 10 20 25 40

The ratios of the money earned to the number of lawns all

simplify to 5 ·· 1 , or 5, so the relationship is proportional.

Mon

ey E

arne

d ($

)

Number of Lawns2 4 6 8 91 3 5 7O

x

y

10

5

20

30

35

40

45

15

25


Solve.

6 Nila uses the equation c 5 8h to figure out the total amount c she should charge a customer if she babysits for h hours. Find the constant of proportionality and explain what it means.

7 Use the information in problem 6 to solve this problem. Nila decides to increase the rate she charges customers by $2 per hour. What equation should she now use to determine how much to charge her customers? Explain.

8 The table shows the cost of several bunches of bananas. What equation can be used to represent the cost c of a bunch that weighs p pounds?

Show your work.

Solution:

9 The graph shows the relationship between the distance that Dustin can drive his car and the amount of gas needed for that distance. Explain how Dustin can use the graph to predict the number of gallons of gas he will need for a trip of 120 miles. Then find the amount of gas he will need.

Show your work.

Solution:

Number of Pounds 2.5 3.5 4 4.5

Cost ($) 1.05 1.47 1.68 1.89

Dis

tanc

e (m

i)

Gas (gal)1 20.5 1.5O

g

d

10

5

20

30

35

40

45

15

25


Name:

1 Micah paints birdhouses to sell at a fair. The table shows the amount of paint he uses. Is this a proportional relationship? If so, find the constant of proportionality and write an equation for the relationship.

Cans of Paint (p) 1 ·· 4 0.75 1 1 ·· 2 2.5

Number of Birdhouses (b) 3 9 18 30

Show your work.

Solution:

Equations for Proportional Relationships

Solve the problems.

3 Cayley says that the equations p 5 1.5q and 2 ·· 3 p 5 q both

represent the same proportional relationship. Mariah says

that can’t be true because the constants of proportionality

are di!erent. With which student do you agree? Explain.

2 Consider the table, equation, and graph. Which of them represents a proportional relationship?

x 3 5 8

y 3.6 6 7.2

How can you simplify the ratios?

How can you identify a proportional relationship?

How can you identify the equation for a proportional relationship?

Lesson 11

2 41 3Ox

y

2

1

4

3

y 5 2x 1 5


6 When Chef Alice makes rice pilaf for 30 people, she uses 15 cups of chicken broth and 10 cups of rice. Dan wants to make the same recipe for 9 people. Write and use equations to find how much broth and how much rice Dan should use.

Show your work.

Solution:

Solve.

4 Jason runs the same distance each day. In one 7-day

period he ran 40 1 ·· 4 miles. He knows that there is a

proportional relationship between n, the number of

days, and t, the total distance he runs. Tell whether each

statement is True or False.

a. The relationship can be u True u False expressed as n 5 5.25t.

b. The graph of the equation u True u False is a straight line through (0, 0).

c. The unit rate is 5.75. u True u False

5 A farmer charges $6 for 4 pounds of tomatoes. Which equation can the farmer use to find how many dollars d he should charge for p pounds of tomatoes?

A d 5 2 ·· 3 p C d 5 1.5p

B d 5 6p D d 5 4p

Rosa chose A as her answer. Explain her error.

How can finding the unit rate help you?

What is the form of an equation for a proportional relationship?

Finding unit rates could be helpful.

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