Comparing and Scaling: Ratios, Rates, Percents...

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Comparing and Scaling: Ratios, Rates, Percents & Proportions Name: _______________________ Per: _____

Investigation 1: Ratios and Proportions and Investigation 2: Comparing and Scaling Rates

Standards:

7.RP.1: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.

7.RP.2a: 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 (observing whether the graph is a straight line through the origin).

7.RP.2c: Represent proportional relationships by equations.

7.RP.2d: Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

7.RP.3: Use proportional relationships to solve multistep ratio and percent problems.

Date Learning Target/s Classwork (Check Off Completed/ Corrected Items)

Homework (Check Off Completed/ Corrected Items)

Self-Assess Your Understanding of the Learning Target/s

Thursday, Dec. 3

I can write part-to-part and part-to-whole ratios to compare mixtures.

Pg. 2-3 – CS 1.2: Comparing Ratios

Pg. 4 – CS 1.2 Zaption

Friday, Dec. 4 (Half Day)

I can review my mistakes on my Stretching and Shrinking test.

Coversheet for Stretching and Shrinking Test (separate sheet)


Monday, Dec. 7

I can write part-to-part and part-to-whole ratios to compare mixtures.

Pg. 6-7 – CS 1.3: Scaling Ratios


Tuesday, Dec. 8

I can solve proportions by scaling ratios.

Pg. 9-10 – CS 1.4: Scaling to Solve Proportions

Exit Ticket

Pg. 11-12 – Inv. 1 Additional Practice Zaption

Wednesday, Dec. 9


Pg. 13-14 – Inv. 1 Additional Practice Puzzle

Make sure Pg. 2-14 are completed and corrected

Get packet signed

Thursday, Dec. 10


Check Up

Retake for S&S (if needed)

MathXL

MathXL I completed 30 minutes of MathXL between Thursday, Dec. 3 and Thursday, Dec. 10.

Exit Ticket

#1


Score:

Parent/Guardian Signature: _______________________________________ Due: ______________________________

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CS 1.2: Mixing Juice – Comparing Ratios

Every year, the Grade 7 students at Langston Hughes School go on an

outdoor education camping trip. During the week-long trip everyone pitches

in to help with cooking and cleanup. This year, Arvin and Mariah were in

charge of making orange juice for the campers. They planned to make the

juice by mixing water and frozen orange juice concentrate. To find the mix

that would taste best, they decided to test some mixes.

Think and Ink: Explain your thinking.

Which mix will make juice that is the most “orangey”?

Which mix will make juice that is the least “orangey”?

Pair and Share: Discuss your ideas with your group.

Work with your group to create a poster that includes the following:

How did you compare the mixes?

Which mix will make juice that is the most orangey?

Which mix will make the juice that is the least orange?

Be prepared to share your poster in a class gallery walk.

After gallery walk: Did you change your answer? Why or why not?

Which mix will make juice that is the most “orangey”? Which mix will make juice that is the least “orangey”?

Mix A

2 cups concentrate 3 cups water

Mix B


Mix C

1 cup concentrate 2 cups water

Mix D


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A. For each recipe, find the information and then compare the mixes:

How much concentrate is needed to make 1 cup of juice?

How much water is needed to make 1 cup of juice?

Rate the mixes in order from the most ‘orangey’ (1) to the least ‘orangey’ (4).

Mix A

Mix B

Mix C

Mix D

B. Isabelle and Doug used fractions to express their reasoning.

Isabelle: 5/9 of Mix B is concentrate

Doug: 5/14 of Mix B is concentrate

Do you agree with either of them? Explain.

C. Max thinks that Mix A and Mix C are the same. Max says, “They are

both the most ‘orangey’ since the difference between the number of

cups of water and the number of cups of concentrate is 1.” Is Max’s

thinking correct? Explain.

D. Assume that each camper will get ½ cup of juice. Answer the questions below for each recipe:

How many batches are needed to make juice for 240 campers?

How much concentrate is needed to make juice for 240 campers?

How much water is needed to make juice for 240 campers?

Mix A

Mix B

Mix C

Mix A


Mix B


Mix C


Mix D


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Homework CS 1.2 Zaption: “Appley” Juice

Complete the problems and correct them with

the Zaption.

A. Compare these four mixes for apple juice.

1. Which mix would make the most

appley juice? Explain your reasoning.

2. Suppose you make a single batch of each mix. What fraction of each batch is concentrate?

Mix W _____________ Mix X _____________

Mix Y _____________ Mix Z _____________

3. Rewrite your answers to the previous question as percents.

Mix W _____________ Mix X _____________

Mix Y _____________ Mix Z _____________

4. Suppose you make only 1 cup of Mix W. How much water and how much concentrate do you need?

1 cup of Mix W = _______cup of Water and ______ cup of Concentrate

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Homework CS 1.3 Zaption: Time to Concentrate – Scaling Ratios

In Problem 1.2, you may have used the ratios below to determine which

recipe was the most ‘orangey’. Below are two ratios describing Mix A:

The first ratio is a _______________-to-________________ ratio. It compares

one ____________ (the water) of the whole (the juice) to the other

____________ (the concentrate).

The second ratio is a _____________-to-_______________ ratio. It compares

one ____________ (the concentrate) to the ____________ (the juice).

Write ratios for the mixes:

Part-to-Part Ratio Concentrate to Water

Part-to-Whole Ratio Concentrate to Juice

Part-to-Whole Ratio Water to Juice

Mix A 2:3 2/3 2 to 3

2:5 2/5 2 to 5

3:5 3/5 3 to 5

Mix B

Mix C

Mix D

Scaling ratios was one of the comparison strategies Sam used in Problem 1.2. Practice scaling ratios:



Mix A 2

3=

4

=

6

=

12=

15=

12

Mix C

1

=

2

=

6=

8=

10=

6

Mix B 5

=

18=

27=

20

=

45=

30

Mix D

5=

6

=

9

=

20=

25=

18

How can you use these ratios to compare Mix A and Mix B?

Mix A


Mix B


Mix C


Mix D


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CS 1.3: Time to Concentrate – Scaling Ratios

A. A typical can of orange juice concentrate holds 12 fluid ounces. The standard recipe is: Mix one can of

concentrate with three cans of cold water. How large of a pitcher will you need to hold the juice made from a

typical can? Show or explain how you arrived at your answer.

B. A typical can of lemonade concentrate holds 12 fluid ounces. The standard recipe is: Mix one can of concentrate

with 4 ⅓ cans of cold water. How large of a pitcher will you need to hold the lemonade from a typical can?

Show or explain how you arrived at your answer.

The pitchers hold ½ gallon, 60 ounces,

and 1 gallon. Which container should

you use for the lemonade from one

can? Explain your reasoning. Note: 1

gallon = 128 ounces.

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C. Solve these mixing problems:

1. Cece is making orange juice using one 16-ounce can of concentrate. She is using the standard ratio of one

can of concentrate to three cans of cold water. How large of a pitcher will she need?

2. Olivia has a one-gallon pitcher to fill with orange juice. She uses the standard ratio of one can of

concentrate to three cans of cold water. How much concentrate does she need?

3. August has some leftover cans of lemonade concentrate in his freezer. He uses 1 ½ ten-ounce cans of

concentrate and the standard ratio of one can of concentrate to 4 ⅓ cans of cold water. How large of a

pitcher does he need?

D. Otis likes to use equivalent ratios. For Olivia’s problem in Question C, he wrote ratios in fraction form:

1

4=

𝑥

128

1. What do the numbers 1, 4, and 128 mean in each ratio? What does x mean in this equation?

2. How can Otis find the correct value of x? (think about what you know about scale factors…)

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Homework CS 1.4 Zaption: Keeping Things in Proportion – Scaling to Solve Proportions

Questions Notes

What is a proportion? What are some other ways Otis might have written the proportion? How can you solve the proportion by scaling up? Suppose that among American doctors men outnumber women by a ratio of 12 to 5. If about 600,000 American doctors are men, how can you figure out how many are women? Does one of the proportions seem easier to solve than the others? How many women doctors are there?

In Problem 1.3, you used ratios and scaling to solve problems. When you write two equivalent ratios in fraction form and set them equal to each other, you form a _______________________. Otis’s strategy for solving a problem involving a ratio of orange concentrate to juice was to write this proportion:

1

4=

𝑥

128

=

=

=

Otis solved the proportion by scaling up. He wrote:

1 • 32

4 • 32=

𝑥

128

There are four ways to write this as a proportion:

Using what you know about equivalent ratios, you can find the number of women doctors from any one of these proportions. Finding the missing value in a proportion is called solving the proportion.

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CS 1.4: Keeping Things in Proportion – Scaling to Solve Proportions

For each question, set up a proportion that shows the relationship between known and unknown quantities. Then use

equivalent fractions, ratios, and scaling to solve each proportion.

A. Imani gives vitamins to her dogs. The recommended dosage is 1 teaspoon per day for adult dogs weighing 10

pounds. She needs to give vitamins to Bruiser, who weighs 80 pounds, and Dust Ball, who weighs 7 pounds.

What is the correct dosage for each dog?

Proportion:

B. Jogging 5 miles burns about 500 calories. How many miles does Tyler need to jog to burn off the 1,200 calorie

lunch he ate?

Proportion:

C. Tyler jogs about 8 miles in 2 hours. How long will it take him to jog 12 miles?

Proportion:

D. The triangles in this picture are similar. Find

the height of the tree.

Proportion:

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E. Solve these proportions for the variable x. Use the reasoning you applied in the earlier problems.

1. 8

5=

32

𝑥

2. 7

12=

𝑥

9

3. 25: 𝑥 = 5: 7

4. 𝑥

3=

8

9

5. 𝑥

5=

120

3

6. x : 6 = 10 : 150

7. 𝑥

15=

20

30

8. 0.1

48=

𝑥

960

9. 𝑥

900=

3.5

15

10. 18

𝑥=

4.5

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F. Nic was working on the proportion at right:

1. He could not see a way to scale 10 to make 6. Instead, he scaled both sides of the

proportion. His work is shown at right. How could Nic complete his solution?

2. Kevin thinks Nic’s idea is great, but he used 30 as a common denominator. Show what Kevin’s version of the

proportion would look like.

Does Kevin’s scaled up proportion give the same answer as Nic’s? Explain your reasoning.

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Homework CS Inv. 1 Zaption: Additional Practice

Complete and then correct with the Zaption.

For each problem: Set up and solve with a proportion.

A. Jared and Pedro walk 1 mile in about 15 minutes. They can keep up this pace for several hours. Set up and

solve each problem with a proportion.

a. About how far can they walk in 90 minutes?

b. About how far can they walk in 65 minutes?

B. Swimming ¼ of a mile uses about the same number of calories as running 1 mile. Set up and solve each problem

with a proportion.

a. Gilda ran a 26-mile marathon. About how far would her sister have to swim to use the same number of

calories Gilda used during the marathon?

b. Juan swims 5 miles a day. About how many miles would he have to run to use the same number of

calories used during his swim?

C. After testing many samples, an electric company determined that approximately 2 of every 1,000 light bulbs on

the market are defective. Americans buy more than 1 billion light bulbs every year. Estimate how many of

these bulbs are defective. Set up and solve the problem with a proportion.

Assignment continues on next page…

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D. The organizers of an environmental conference order buttons for the participants. They pay $18 for 12 dozen

buttons. Set up and solve each problem with a proportion.

a. How much do 4 dozen buttons cost?

b. How much do 50 dozen buttons cost?

c. How many dozens of buttons can the organizers buy for $27?

d. How many dozens of buttons can the organizers buy for $63?

E. Denzel makes 10 of his first 15 shots in a

basketball free-throw contest. His success rate

stays about the same for his next 100 free

throws. Write and solve a proportion for each

part. Round your answer to the nearest whole

number.

a. About how many baskets do you expect

Denzel to make in his next 60 attempts?

b. About how many free throws do you expect him to make in his next 80 attempts?

c. About how many attempts do you expect Denzel to take to make 30 free throws?

d. About how many attempts do you expect him to take to make 45 free throws?

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Investigation 1 Additional Practice Puzzle: Setting Up and Solving Proportions with Scaling

Set up and solve a proportion for each problem:

Puzzle continues on the next page…

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Challenge Problem:

A fruit bar is 5 inches long. The bar will be split into two pieces. For each situation, find the lengths of the two pieces.

Hint: Draw a diagram of the bar each time.

a. One piece is 3/10 of the whole bar.

b. One piece is 60% of the whole bar.

c. One piece is 1 inch longer than the other.

Set up and solve a proportion for each problem:

Comparing and Scaling: Ratios, Rates, Percents...

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