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True Color Murals: A Unit on Ratios © 2015 SRI International This material is provided through Department of Education grant U411B130019. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of the Department of Education.

Writing and Editing: Jennifer Knudsen, Ken Rafanan, Teresa Lara-Meloy, Natasha Arora, Elizabeth Christiano Editing: Phil Vahey, Natasha Arora, Mimi Campbell, Soren Rosier Design: Ken Rafanan, Natasha Arora, Teresa Lara-Meloy, Jennifer Knudsen Editorial Advice: Phil Vahey, Nicholas Jackiw, Jorge Lara y Góngora

Photo and Art Credits: Ken Rafanan, Natasha Arora Cover: True Colors Murals by Ken Rafanan No. 14, 1960, Mark Rothko. Source: https://www.flickr.com/photos/dennisredfield/8191030174/in/photostream/ Urban Mural in Germany. Source: https://www.flickr.com/photos/steffireichert/5716416993 Clouds and Water, Arthur Dove. Source: commons.wikimedia.org A Sunday on La Grande Jatte, Georges Seurat. Source: commons.wikimedia.org Starry Night Over the Rhone, Vincent Van Gogh. Source: commons.wikimedia.org Water Lilies, Evening Effect, Claude Monet. Source: www.wikiart.org Wheatfield with Cypresses, Vincent Van Gogh. Source: wikipedia.org Woman with a Flower, Paul Gauguin. Source: commons.wikimedia.org Sunflowers, Vincent Van Gogh. Source: commons.wikimedia.org

Videos: Sasha Fera-Schanes

Table of Contents

Investigation 1: The Perfect Orange ................................................................................ 1 Warm Up ............................................................................................................................................................ 1 Main .................................................................................................................................................................... 1 Wrap Up ............................................................................................................................................................. 3 Problem Solving ................................................................................................................................................. 5

Investigation 2: Grayscale Murals .................................................................................... 7 Warm Up ............................................................................................................................................................ 7 Main .................................................................................................................................................................... 7 Wrap Up ........................................................................................................................................................... 10 Problem Solving ............................................................................................................................................... 11

Investigation 3: It’s Great to Be Green ........................................................................... 13 Warm Up .......................................................................................................................................................... 13 Main .................................................................................................................................................................. 13 Wrap Up ........................................................................................................................................................... 15 Problem Solving ............................................................................................................................................... 17

Investigation 4: Helping Out .......................................................................................... 19 Warm Up .......................................................................................................................................................... 19 Main .................................................................................................................................................................. 19 Wrap Up ........................................................................................................................................................... 22 Problem Solving ............................................................................................................................................... 23

Investigation 5: Starry Night .......................................................................................... 25 Warm Up .......................................................................................................................................................... 25 Main .................................................................................................................................................................. 25 Wrap Up ........................................................................................................................................................... 27 Problem Solving ............................................................................................................................................... 29

Investigation 6: Water Lilies .......................................................................................... 31 Warm Up .......................................................................................................................................................... 31 Main .................................................................................................................................................................. 32 Wrap Up ........................................................................................................................................................... 34 Problem Solving ............................................................................................................................................... 35

Investigation 7: Reddish Orange .................................................................................... 37 Warm Up .......................................................................................................................................................... 37 Main .................................................................................................................................................................. 38 Wrap Up ........................................................................................................................................................... 39 Problem Solving ............................................................................................................................................... 41

Investigation 8: Just Tell Me How Many ........................................................................ 43 Warm Up .......................................................................................................................................................... 43 Main .................................................................................................................................................................. 43 Wrap Up ........................................................................................................................................................... 46 Problem Solving ............................................................................................................................................... 47

Optional. Investigation 9: Walls of Metal and Wood .................................................... 49

Optional. Investigation 10: Sunflowers ......................................................................... 53

Investigation 1: The Perfect Orange 1

Investigation 1: The Perfect Orange

Warm Up

Write 20 as the sum of two numbers. (In other words, write two numbers that add up 1.to 20.) How many different ways can you do this?

11 different ways if you don’t count 9+2 and 2+9 as different ways, and if you do count 20+0 as one of the ways.

Main

Introduction

Welcome to the True Colors Mural Company!

We make copies of beautiful paintings and reproduce them as murals on buildings. In our company, we buy only the primary colors: blue, yellow, and red. But by using the right mix of these primary colors, we can create almost any color we need.

You probably know that if you mix blue and yellow paint, you get green paint. But exactly which color of green depends on how much blue is used in relationship to how much yellow is used.

To make our murals, we have to match the colors in the original paintings. To help us make our colors before we start painting, we have Ratio Visualizer, a special online tool. This online tool uses ratios to make particular colors.

Your job will be to determine colors, compare colors, record how to make various colors, and figure out how much paint to make. Also, because we are a business, there are other matters for you to work on. The mathematical idea of ratio will be important in your work.

No. 14, 1960, by Mark Rothko.

2 Investigation 1: The Perfect Orange

Rothko

We have been hired to reproduce Mark Rothko’s painting No. 14, 1960 on the side of a large building. We must make the same orange that is in the painting. We will use Ratio Visualizer, a set of online tools to make this color of orange using what we call “pips” of paint. Using Activity 1.2, click on the Show Artwork button to see a picture of the painting.

Using 20 total paint pips or fewer, create a color of orange. Notice how for each 2.tap on the color bar, a pip of red or yellow paint is added. On the screen, re-label the color “Blend 1” with a name that you choose. My color is____________

Write down the recipe for the orange that you created:

Yellow pips ____ Red pips____ Answers will vary. Accept any two amounts of red and yellow pips as long as there are 20 or fewer total.

Now let your partner(s) use the online tool. Create two more blends: one a color 3.of orange that is more yellow than your first blend, and one that is more red. Click “Add new” to create more blends. Label your two new blends “more yellow” and “more red.”

Copy the table into your notebook and write all the recipes for the different colors of orange that you created:

Name Yellow pips Red pips

Blend 1:

More Yellow

More Red

Answers will vary based on the student answers to #3 above.


Wrap Up

Compare the three blends of orange and their recipes. 4.

Look at the numbers in the recipes. How do the numbers in the recipes show A.which blend is the most yellow? The most red?

Students may notice that reddish orange has more red pips than yellow pips and that yellowish orange has more yellow pips than red pips without paying attention to the ratio. This is ok for now.

Look at the marker on the spectrum. How do the blend markers show which B.blend is the most red? The most yellow?

Encourage students to describe the relative locations of their markers in the spectrum. The reddish orange blend’s marker will be further towards the pure red end of the spectrum. The yellowish orange’s marker will be further towards the pure yellow end of the spectrum.


Problem Solving

LaToya created a blend of green using 8 blue pips and 8 yellow pips. Complete 5.the table to make two more blends of green: one that is more blue than LaToya’s blend and one that is more yellow.

Name Blue pips Yellow pips

LaToya’s Blend 8 8

More Blue 6

More Yellow Answers will vary.

From the table above, compare the three blends of green and their recipes. Look 6.at the numbers in the recipes. How do the numbers in the recipes show which blend is the most blue? The most yellow?

LaToya’s green has the same amount of blue and yellow pips. The blend that is more blue should have more blue pips than yellow pips. The blend that is more yellow should have more yellow pips than blue pips.

Tony created a blend of purple with 1 blue pip and 1 red pip. 7.

Complete the table and make two more blends of purple: one that is more A.blue than Tony’s blend and one that is more red.

Name Blue pips Red pips

Tony’s Blend

More Blue

More Yellow

Answers will vary.

Compare the three blends of purple and their recipes. Look at the numbers in B.the recipes. How do the numbers in the recipes show which blend is the most blue? The most red?

LaToya’s green has the same amount of blue and red pips. The blend that is more blue should have more blue pips than red pips. The blend that is more red should have more red pips than blue pips.

Investigation 2: Grayscale Murals 7

Investigation 2: Grayscale Murals

Warm Up

Evaluate. 1.

A. B. C.

14 doubled 10 doubled half of 40

28 20 20

D. E. F.

40÷2 28•2 20•2

20 56 40

Main We have another job where we need to reproduce the grayscale mural you see here. To create a shade of gray we will blend black paint and white paint. Your job is to use the computer to create the same gray with different numbers of paint pips.

Imagine a certain shade of gray that has 2 white pips and 9 black pips. What would that look like? Circle where in the mural you find that shade of gray.

Urban mural in Germany.

Students should predict that the shade of gray would be a very dark color of gray. The dark gray is used in the darker parts of the mural, for example the iris, or the edges of the eyelids.

8 Investigation 2: Grayscale Murals

Xenia’s Gray

One of our co-workers, Xenia, makes a blend of gray paint using 10 pips of black and 14 pips of white. We can say—

The ratio of black pips to white pips is 10 to 14. We write it as “10:14.”

Use Activity 2.1 to make Xenia’s gray using a black pip to white pip ratio of 2.10:14. Click “Container.” Make a container so that the black and white pips form two rows of equal size, with the same number of black pips and white pips in each. It should look like this.

We want to create a second blend that is the same color as Xenia’s gray, but uses 3.only 5 pips of black.

Predict. How many white pips do you think we will need? A.

Accept any predictions.

Check. Create the blend that makes the same color. Use the spectrum to B.check that the two blends are the same color. Copy the following into your notebook and fill in—

The ratio of black pips to white pips is 5 : 7

Click “Container” and form a single column with the black and white pips C.like this.

Explain. Examine this table with the drawings of both ratio’s containers. D.How do the containers for the two blends show us that both ratios give the same color?

5:7 10:14

In the first blend, there are two columns of 5 black pips to 7 white pips. In the second blend, there is one identical column (5


black to 7 white). So, both blends have a ratio of 5 black pips to 7 white pips.

Make a third blend of paint that makes the same shade as Xenia’s gray using 20 4.pips of black. Copy the following into your notebook.

In this blend—


Let’s use a table to compare all of the blends. 5.

Copy the table in your notebook and complete the third blend: A.

Xenia’s gray Black pips White pips

1st Blend 10 14

2nd Blend 5 7

3rd Blend 20 28

4th Blend 40 56

Predict. Fill in the table. Try to create a fourth blend that will make Xenia’s B.gray using different amounts of pips.

In the fourth blend—


Answers may vary. The ratio 40:56 represents doubling again but students could use other strategies.

Check. Use the spectrum and blend bars to check whether you are right. C.

Explain. How do you know that the fourth blend is the same color of gray as D.the other blends?

Students may decide on the ratio by doubling or halving both numbers from another row of the table or by multiplying by another number. It works because when both numbers in a row are multiplied by the same number, the ratio is the same. Students can see this by using Container in the online tool. If the ratio is correct, they will always get five rows of black and seven rows of white.


Wrap Up

Explain to a muralist joining the group how to use your new strategies for 6.making blends that are the same shade of gray.

When you want to make blends that are the same shade of gray, you should take the original amounts and double or halve the number of black pips and white pips.


Problem Solving

Another co-worker, Liam, made a blend of a different shade of gray paint by 7.mixing 4 black pips with 10 white pips. We want to make a few blends of Liam’s gray, but using different amounts of white pips.

Complete the table: A.

Black pips White pips Black : White

Liam’s Gray 4 10 4 : 10

Half 2 5 2 : 5

Double 8 20 8 : 20

Double Again 16 40 16 : 40

Use a drawing of two containers and numbers to explain how you know you B.made the same shade for Liam’s gray and the Double Again blend.

Sample answers: I know the same shade is made because you can get columns with the same numbers of rows of black pips and white pips when they are contained, or because the numbers of each term of the second ratio are 4 times the terms of the first ratio.

A recipe for noodle soup calls for 2 cups of noodles and 4 cups of chicken broth. 8.

To make half the recipe, how much of each ingredient should you use? A.

1 cups noodles 2 cups chicken broth

To make double the recipe, how much of each ingredient should you use? B.

4 cups noodles 8 cups chicken broth


Write the ratios for cups of noodles to cups of chicken broth for all of the C.recipes above.

noodles (cups)

chicken broth (cups)

2 : 4

1 : 2

4 : 8

Gina made a bird feed mix of 4 cups sunflower seeds to 12 cups of corn. We want 9.to make a few blends of Gina’s bird feed mix but using different amounts of sunflower seeds.

Complete the table: A.

Sunflower seeds (cups)

Corn (cups)

Sunflower : Corn

Gina’s Mix 4 12 4 : 12

Half 2 6 2 : 6

Double 8 24 8 : 24

Double Again 16 48 16 : 48

Explain how you know you made the same mix of bird feed as Gina. B.

Sample answer: The mixes are the same because I always doubled or halved the amount of corn and the amount of seeds. Or the corn is always 3 times as much as the seeds.

Investigation 3: It’s Great to Be Green 13

Investigation 3: It’s Great to Be Green

Warm Up

Copy the table into your notebook. Based on the ratio given in the first row, 1.complete the table so that each row makes the same color.

Yellow pips Red pips

2 6

4 12

8 24

16 48

32 96

64 192

1 3

Main In the previous Investigations, you used doubling and halving to make different blends of the same color paint. There are more ways to use ratio to get the same color paint, and you will learn about them in the Investigations to come.

A Sunday on La Grande Jatte, Georges Seurat.

14 Investigation 3: It’s Great to Be Green

Palm Green

Copy the table into your notebook. The blends of blue and yellow in this table are 2.all supposed to make Palm Green. Each row represents a ratio. (Don’t fill in the blank row until later.)

Blue pips Yellow pips

1 3

2 6

4 12

6 18

7 21

Predict. Look at the table. Do you think that all of the rows in this table make A.the same color?

Accept any predictions.

Check. Use Activity 3.1. Make blends for each row of the table. B.

Explain. How do the spectrum, the blend bars, or the pips show that the C.blends all make Palm Green?

Sample answers: The pointers on the spectrum are in the same location, the blend bars for each blend have the same amount of blue and the same amount of yellow, or because pips placed in a container have the same numbers of rows of blue and yellow.

Explain. When the ratios all make the exact same color paint, we can say D.they are equivalent ratios. How can you tell from the numbers in the table that the ratio for each row will make Palm Green, or that the ratios are equivalent?

Students can reason “down the rows,” multiplying each number in a row by another number to get the other number in the row. For the row representing 7:21, students need to see the multiplicative relationship across the rows, to say something like “2 times 3 is 6, so 7 times 3 is 21 .”


Copy the sentence into your notebook. Write a rule so that anyone can create E.a blend that results in Palm Green paint.

For every 1 blue pip, there are 3 yellow pips.

In your notebook, rewrite your rule as a blue to yellow ratio in the top blank F.row of the table.

Use your rule and create another ratio that makes Palm Green. G.

Answers will vary. Possibil ities include 1 blue and 3 yellow; 10 blue and 30 yellow; 4 blue and 12 yellow.

Ways to Describe Ratios For example, we can describe the Palm Green ratio like this:

• The ratio of blue pips to yellow pips is 1 to 3. • The ratio of blue pips to yellow pips is 1 : 3. • Blue pips and yellow pips are in the ratio 1 to 3. • The blend is 1 blue pip to 3 yellow pips. • The ratio 1:3 is equivalent to the ratio 2:6.

Wrap Up

Joey creates a blend of orange with a ratio of 4 red pips to 20 yellow pips. Write a 3.rule using ratio language so that anyone using more or fewer than 4 red pips can create a blend of paint that results in Joey’s orange.

Answers may vary. Student answers should reflect that Joey’s orange can be made with any ratio that is equivalent to 1 red pip to 5 yellow pips.


Problem Solving

People’s Purple

Grace, Jake, and Nina have each mixed one batch of purple paint using the 4.numbers of blue pips and red pips shown in the table below. You may use Activity 3.2 to check your work.

Red pips Blue pips

Grace 3 15

Jake 12 60

Nina 15 75

How can you tell that Jake’s and Nina’s purples are the same as Grace’s, using A.the numbers?

Sample answers: Both Jake’s, Nina’s, and Grace’s purples have 5 blue pips for every 1 red pip. Also, Jake uses 4 times as many red pips and 4 times as many blue pips as Grace.

How can you tell from the numbers that Nina’s paint amount is 5 times as B.much as Grace’s?

Sample answer: Nina uses 5 times as many red pips and 5 times as many blue pips as Grace. Using Activity 3.2 you can use the container. Grace's ratio has one row of 3 red pips and five rows of blue pips. There are 3 pips in each row. For Nina, keeping the number of rows the same, there are 15 pips in each row, which is 5 times more.

At True Colors, we play games to keep our creativity flowing. One game is called 5.Tip, Tap, Top. One rule is that for every 6 tips there are supposed to be 24 tops. Fill in the table for this relationship to show equivalent ratios. Some numbers are given to you.

Tips Tops

1 4

6 24

4 16

2 8


Represent the following situations with a ratio in different ways (using a colon, 6.“for every” language, or “__times as many” language):

Situation Colon notation

___:___

For every _____ there are _____

There are _____ times as many

The color Blue Raspberry is made with 5 red pips for every 20 blue pips of paint.

The ratio of red to blue pips is 5:20

For every 1 red pip, there are 4 blue pips.

There are 4 times as many blue pips as red pips.

The color Sky Purple is made with 4 red pips for every 28 blue pips of paint.

The ratio of red to blue pips is 4:28.

For every 4 red pips, there are

28 blue pips.

There are 7 times as many

blue pips as red pips.

Fred’s fizzy punch is made by mixing 12 quarts of lemonade for every 20 quarts of sparkling water.

The ratio of quarts of lemonade to quarts of sparkling water is 12:20

For every 12 quarts of lemonade, there are 20 quarts of sparkling water.

Challenge: There is 20/12 times as much sparkling water as lemonade.

Anu’s pancake recipes calls for 7 cups of pancake mix for every 3 cups of milk.

The ratio of cups of pancake mix to cups of milk is 7:3

For every 7 cups of pancake mix there are 3 cups of milk.

Challenge: There are 7/3 times as much pancake mix as milk.

Challenge: There are 9 red marbles and some yellow marbles. There are a total of 7.15 marbles. Write the ratio of the number of red marbles to the number of yellow marbles. Use a complete sentence with colon notation.

The ratio of red marbles to yellow marbles is 9:6 (or 3:2, not required yet). Until now, students have been given two parts to write a part:part ratio. In this problem, students are given one part and one whole and asked to write a part:part ratio. Students may write 9:15 or 15:9. Point out that these ratios (with 15 marbles) are NOT the ratio of red to yellow marbles, but of red to total marbles.

Investigation 4: Helping Out 19

Investigation 4: Helping Out

Warm Up

Evaluate. 1.

A. B. C. D.

27÷ 9 2 •___ = 6 ___•7 = 28 4•4 =

3 3 4 16

E. F. G.

1÷ 3 3•___ = 1 ___•3 = 10.5

1/3 1/3 3.5

Main

Running True Colors Murals as a company means solving math problems. Please help out. Make tables and use ratios as needed.

Clouds and Water, Arthur Dove.

20 Investigation 4: Helping Out

True Colors is hosting an art event for elementary school students. Amira is 2.making clay from salt and flour. She knows that the ratio of the amount of flour to the amount of salt should be 3:1. Use Activity 4.1 to model these ratios when possible.

Copy the table into your notebook. Complete the table. (Leave the blank rows A.blank, for now.)

Flour (cups) Salt (cups)

12 4

3 1

1 1/3

Use Graph to show the graph representation of the clay blends. 3.

Explain what each point means. A.

Each point shows one of the clay blends ratios from the table above.

Use Lines to show the graph lines. Why are the points all on the same line? B.

Because each ratio point lies on the line that represents a 3 flour to 1 salt ratio. The line is another way to represent blends with 3 times as many flour pips as salt pips.

Using the graph, determine the amounts of flour and water needed to create C.two new blends of the same clay. In your notebook, write the ratios of the two new blends in the table above.

Answers may vary but the blends should reflect a 3 to 1 flour to salt ratio.


When True Colors muralists go to paint a mural, they bring 7 brushes for every 2 4.palettes so that they have enough brushes for mixing different colors.

Use Activity 4.3 to show the 7:2 brush to palette ratio in a blend bar. A.

Estrella wants to take 6 palettes to her mural site. How many brushes should B.she take? Create another blend bar to show how many brushes she should take. Explain your answer in words and numbers.

Estrella should take 21 brushes because 2 times 3 is 6 and 7 times 3 is 21.

Copy the table into your notebook. Sometimes Estrella needs to bring as C.many as 12 palettes to a mural site. Fill in the table so that she can easily know how many brushes to bring to each job.

Number of brushes Number of palettes

7 2

14 4

21 6

28 8

42 12

Challenge: Jacob wants to carry 3 palettes. Estrella isn’t sure how many D.brushes to send along with him. Advise Jacob as best you can and explain your advice.

If we tried to make a ratio in the table for 3, the number of brushes is halfway between 7 and 14, which is 10.5. We can round up to 11 OR round up to 14 brushes, the number needed for 4 palettes. Either answer could be right. It depends on how the brushes are to be used with the palettes.


Wrap Up

A recipe for a gazpacho (a cold soup) uses 3 cups of cucumbers and 4 cups of 5.tomatoes. Describe how to use a ratio to determine how many cups of cucumbers to use with 1 cup of tomatoes.

Answer should reflect that the ratio of cups of cucumber to cups of tomato is 3:4. To find out the amount of cucumbers to use with 1 cup of tomato, divide each term by 4. And 3 divided by 4 is 3/4.


Problem Solving

True Colors donates murals to communities. For every 9 murals sold, True 6.Colors donates 2 murals for free.

If we have 27 sold murals, how many donated murals do we need to keep the A.same 9:2 ratio?

To keep the same 9:2 ratio, True Colors should donate 6 murals.

Explain in words the relationship between the 9:2 ratio and the ratio you B.created using the 27 sold murals.

27 sold murals is 3 times as many as 9. Three times as many as 2 is 6. So True Colors would donate 6 murals.

True Colors is working with the parks department, which has some problems to 7.solve.

The parks department created a graph using ratios of park benches to basketball hoops in 3 different parks. They are now building a new park and want to keep the same bench to hoop ratio.

If there are 16 park benches in the new park, how many hoops should the A.park have to keep the same bench to hoop ratio?

The existing parks have a 4:1 bench to hoop ratio. The new park should have an equivalent ratio of 16:4. So there should be 4 hoops in the new park.


Explain in words the relationship between the existing parks’ bench to hoop B.ratio and the ratio you found for the new park.

The existing parks have 4 times as many benches as hoops. Four times as many as 4 is 16. The new park has 4 hoops and 16 benches.

The parks department found in their existing park that 5 maple trees can shade 8.7 picnic tables. They are now building a new park and have bought 45 maple trees. Assume that the new trees will shade the picnic tables in the same ratio as in the old park.

How many picnic tables can be shaded by these maple trees? You can make a ratio table to help you find out.

63 picnic tables

The new park will also have a lake with boats. Each boat can fit 2 adults and 9.3 children. There are 20 children going on a boating field trip. How many adults can go on the boats with them? (Hint: Remember that you might need to round as you did in problem 3. You can use a table to organize your thinking.)

14 adults

The park wants to paint their picnic tables Ecological Green, which is made in 10.the ratio of 5 buckets of blue paint to 2 buckets of yellow paint. If you only have 1 bucket of blue paint, how many buckets of yellow paint do you need to make Ecological Green?

2/5 buckets of yellow paint

Investigation 5: Starry Night 25

Investigation 5: Starry Night

Warm Up

Copy the sentences into your notebook. Then, fill in the blanks. 1.

A. 3 multiplied by 7 is 21

B. 5 multiplied by 7 is 35

C. 1 multiplied by 7 is 7

D. 21 divided by 0.6 is 35

E. 35 divided by 3 is 11.67

F. 7 divided by 1 is 7

Determine the greatest common factor of 2.

A. B.

28 and 12 35 and 15

4 5

Main Starry Night Over the Rhone is a famous painting by Vincent Van Gogh. If you look at the dark blue water, you’ll notice the color has small amounts of a bluish purple. We call this Plum Blue. True Colors is trying to make Plum Blue for this painting, as well as other colors. In this investigation we will learn what it means to find the simplest form of a ratio.

Starry Night Over the Rhone, by Vincent Van Gogh

26 Investigation 5: Starry Night

Plum Blue

Our newest paint color, Plum Blue, can be made with 21 blue pips for every 3 red pips in the mix. Terrance is trying to make the same color, but he accidentally put 5 red pips in his mixer.

How can Terrance make Plum Blue using equivalent ratios? 3.

Predict. Since he already has 5 red pips, how many blue pips do you think A.Terrance needs to make Plum Blue?

Predictions will vary; the correct answer is 35 blue pips.

Check. Use Activity 5.1 to make Terrance’s blend of Plum Blue. B.

Using Terrance’s blend, students may simply click on blue until the blend bars’ dividing lines match up.

Explain. Contain the pips for Terrance’s blend and for Plum Blue. Explain C.how the containers show that Terrance’s blend and Plum Blue have the same red to blue ratio.

The container shows the pips columns each with 1 red and 7 blue pips. Any other batch should have the same ratio of red pips to blue pips.

Explain. How can containing the pips help you figure out how to make a D.batch of the same color, but with different amounts of paint?

After you contain the pips, you can rearrange them to make a rectangular array where the number of reds and blues in each column is, in this case, 1 and 7.

Copy the table into your notebook. Fill in the table to show how many blue pips 4.you need for different amounts of Plum Blue paint.

Number of red pips Number of blue pips

3 21

1 7

5 35


The ratio 1:7 is the called the simplest form of 3:21 and 5:35. It is equivalent to 5.both. Explain why 1:7 would be called "simplest form."

Possible answer: Because I can't divide both 1 and 7 by any other whole number without getting a fraction as one of the terms.

Use Graph to show the graph representation of the two blends. Use Lines to show 6.the graph lines. Why are the points all on the same line?

Because each ratio point lies on the line that represents a 1 red to 7 blue pip ratio. The line is another way to represent blends with 7 times as many blue pips as red pips.

Cherry Tomato

For the color Cherry Tomato, the ratio of red pips to yellow pips is 28 to 12.

Use Activity 5.2 to view Cherry Tomato and contain the pips. What is the 7.simplest form of 28:12?

The ratio 7:3 is the simplest because the pips form rows of 7 red pips and 3 yellow pips. Students will see that simplest form does not have to be a unit ratio.

Connor wants to make Cherry Tomato using 35 red pips. How many yellow pips 8.does he need to use? Why?

He needs 15 because using the simplest form, 7 times 5 is 35 and 3 times 5 is 15.

Wrap Up

Julian makes Sky Green using 30 blue pips and 12 yellow pips. What are two 9.ways to find the simplest form of the Sky Green recipe’s ratio?

The simplest form is 5:2. In the container view, there would be six rows each with 5 blue and 2 yellow pips. Also, because 6 times 5 is 30 and 6 times 2 is 12.


Problem Solving

Rewrite the ratios in simplest form. [Hint: Some may already be in simplest 10.form.]

35:5 is equivalent to 7:1 A.

13:26 is equivalent to 1:2 B.

132:121 is equivalent to 12:11 C.

11:5 is equivalent to 11:5 already in simplest form D.

12:9 is equivalent to 4:3 E.

48:16 is equivalent to 3:1 F.

15:60 is equivalent to 1:4 G.

168:224 is equivalent to 3:4 H.

37:3 is equivalent to 37:3 already in simplest form I.

24:16 is equivalent to 3:2 J.

A garden has 20 rose bushes for every 15 tulip plants. What is the ratio of the 11.number of rose bushes to the number of tulip plants in simplest form?

4:3

A garden has 15 red rose bushes for every 18 white rose bushes. What is the ratio 12.of the number of red rose bushes to the number of white rose bushes in simplest form?

5:6


Ivy used the measuring tape below to measure the wall for a mural. Some of the 13.numbers on the tape have worn off.

Ivy can see that there are 9 feet in 3 yards. She writes this using the ratio 9:3.

Write Ivy’s ratio in simplest form. A.

3:1

Fill in the two missing numbers on the measuring tape in the boxes. B.

7, 9

Investigation 6: Water Lilies 31

Investigation 6: Water Lilies

Warm Up

The blends of paint below have different color ratios. Copy the pips into your 1.notebook. Label the pips to match the ratios given.

1:2 (yellow:blue)

1:5 (red:blue)

2:3 (red:yellow)

2:1 (blue:red)

32 Investigation 6: Water Lilies

Main We want to make a yellowish green for use in our mural based on part of Claude Monet’s Water Lilies, Evening Effect. You will learn to compare the total amount of paint to the amount of each color using ratios.

Yellowish Green

Cecilia needs yellowish green to use in her mural of the Water Lilies. She asks you to make yellowish green for her.

Use Activity 6.1 to make yellowish green using exactly 10 pips total and exactly 6 2.yellow pips. Yellowish green has more yellow in it than blue.

In your notebook, give the color a name and write the ratio. A.

The name of my color is ___________________.

In a blend with 10 total pips there will be _____ blue pips and 6 yellow pips.

In your notebook, write the ratio in simplest form. B.

Answers will vary. Students can add pips so that the color indicator stays to the yellow side of green. Remind them that they must use exactly 10 pips. They should name their blend.

The ratio of blue to yellow is ____:____

Water Lilies, Evening Effect, by Claude Monet.


Copy the table into your notebook. Show how to produce different total C.amounts of your yellowish green paint. Each pip of paint contains 1 fluid ounce.

Ounces of blue 4 6 12 16 20 22

Ounces of yellow 6 9 18 24 30 33

Ounces of blue + yellow = Total ounces 10 15 30 40 50 55

Explain how the total ounces row in your table is related to the ounces of blue D.and yellow rows.

Sample answer: The total row is always the sum of the blue row and the yellow row.

In your notebook, write the ratio of blue pips to total pips in simplest form. E.

The ratio of ounces of blue to total ounces is 2 : 5

In your notebook, write the ratio of yellow pips to total pips in simplest form. F.

The ratio of ounces of yellow to total ounces is 3 : 5

Explain how the first column in your table is related to every other column. G.

Sample answer: I multiplied all the numbers in the first column by 1.5 to get the numbers in the second column. For the other columns, I could multiply all the numbers by 4, 5, and 6. OR: I added the amounts in two of the rows to get the amounts in a third row. OR: (less likely) The amount of blue was 4/6 the amount of yellow.

Make blends in the software for each of the rows in the table above. Show the H.graph. How does the graph show the ratios? How are they related on the graph?

The graph shows the ratios as points whose coordinates are the terms in the ratios. The ratios are related by a straight line on the graph because they all have the same blue to yellow ratio.


Jaxon wants you to give him a table for Pretty Purple, showing how many pips to 3.use for batches of different total amounts. The ratio of blue pips to red pips in Pretty Purple is 4:5.

Copy the table into your notebook. Then, fill in the ratios. A.

Ounces of blue Ounces of red Ounces of blue + red =

Total ounces

4 5 9

8 10 18

12 15 27

40 50 90

Use Activity 6.2 to create the batches represented in your table. Using the B.graph, explain how you know that all the rows in your table make Pretty Purple.

Sample answer: I know that all the blends make Pretty Purple because their ratios make a straight line in the graph.

Wrap Up

Jaxon notices a pattern in Pretty Purple in the ratios of ounces of blue to total 4.ounces, and wonders if they are all equivalent. In your notebook, write a note to Jaxon explaining the relationship between the ratios of ounces of blue to total ounces for all the batches of Pretty Purple. Use ratio language and what you know about the graph, the spectrum, the numbers, and the blend bars.

Sample answer: For all batches of Pretty Purple the ratios of ounces of blue to total ounces are all equivalent.

No matter how large a batch, there are always 4 parts blue to 9 total parts. The table shows the specific number of ounces that you need to make a batch. The blend bars for all the batches are aligned on the spectrum. On the graph, the points for the ratios are all on the same line.


Problem Solving

Jordan has two containers of Cherry Tomato paint with the batches shown in the 5.table below. Make up your own container of Cherry Tomato and put it in the table.

(answers will vary) Ounces of red Ounces of yellow Total ounces

Large container 28 12 40

Medium container 14 6 20

Small container 7 3 10

Your own container

The ratio of boys to girls in a class is 2:3. Fill in the missing numbers in the table. 6.

Number of boys 2 4 8 6

Number of girls 3 6 12 9

Total number of students 5 10 20 15

Sonya made some lemonade with the original recipe below. She doesn’t have 7.enough lemonade for the party, so she adds 4 more ounces of lemon juice and 4 more ounces of water to the jar. Sonya will add sugar later.

Ounces of lemon juice Ounces of water

Original recipe 4 12

New recipe 8 16

Your recipe

Fill in the “new recipe” in the second row of the table. A.

Does the new recipe taste the same? B.

No, it is different. The new juice has twice as much water as lemon juice. The original recipe was more watery with 3 times as much water as lemon juice. Adding the same amount to each amount in the row or ratio does not guarantee the same taste in the new recipe.


Create your own recipe for Sonya’s lemonade that makes more lemonade, but C.tastes the same as her first recipe.

Answers will vary; example: 6:18 Your recipe can be obtained by multiplying each number in the same row by the same number or by finding another ratio whose simplest form is 1:3.

Tina made some berry soda with the original recipe below. She doesn’t have 8.enough, so she adds 1 more ounce of cranberry juice and 1 more ounce of sparkling water to the jar.

Ounces of cranberry juice

Ounces of sparkling water

Original recipe 2 3

New recipe 3 4

Your recipe

Fill in the “new recipe” in second row in the table. A.

Does the new recipe taste the same as the original? B.

No, it is different. The new berry soda has about 1.3 times as much sparkling water as cranberry juice. The original recipe was more watery with 1.5 times as much sparkling water as cranberry juice. Adding the same amount to each amount in the row or ratio does not guarantee the same taste in the new recipe.

Create your own recipe for Tina’s berry soda that makes more berry soda, but C.tastes the same as her original recipe.

Answers will vary; example: 6:9. Your recipe can be obtained by multiplying each number in the same row by the same number or by finding another ratio whose simplest form is 2:3.

Investigation 7: Reddish Orange 37

Investigation 7: Reddish Orange

Warm Up

Evaluate. 1.

A. B. C. D. E. F.

2•4 5•4 4•5 5•5 8•3 5•3

8 20 20 25 24 15

Choose the blend that is the reddest (closest to pure red) purple. 2.

50 blue pips to 40 red pips

2 blue pips to 20 red pips (reddest)

20 blue pips to 30 red pips

Choose the blend that is the reddest (closest to pure red) orange. 3.

17 yellow pips to 10 red pips

30 yellow pips to 30 red pips

7 yellow pips to 20 red pips (reddest)

38 Investigation 7: Reddish Orange

Main

Carlos makes a green color that he 4.calls Glowing Green with 8 blue pips for every 18 yellow pips.

Vanessa makes a green color with 2 blue pips for every 5 yellow pips. Vanessa says that her color is more yellow than Glowing Green.

Predict. Do you think that A.Vanessa’s green is more yellow than Glowing Green?

This is a prediction. Accept all reasonable predictions.

Check. Use Activity 7.1 to create Vanessa’s blend. B.

Explain. Using both the blend bars and the spectrum, describe how you C.know which green is more yellow.

Vanessa’s green is closer to the yellow side of the spectrum.

Copy the table in your notebook. Rewrite the ratio for Vanessa’s green. D.

Blue pips Yellow pips Ratio

Glowing Green 8 18 8:18

Vanessa’s 2 5 2 :5

Vanessa’s 8 20 8 :20

Explain: Compare the two greens using the numbers of their ratios. Explain E.how you know whether Vanessa’s green was more yellow.

Vanessa’s ratio can be written as 8 blue for every 20 yellow. That is more yellow per 8 than Glowing Green’s paint ratio. So Vanessa’s green was more yellow, and she was correct.

Wheatfield with Cypresses, by Vincent Van Gogh.


Sara makes an orange color called Sunglow using 4 red pips for every 5 yellow 5.pips. Larry makes a color called Tangerine using 20 red pips for every 26 yellow pips. Larry says that Tangerine is more reddish than Sunglow because he used more red pips than Sara.

Use Activity 7.2 to create Sunglow and Tangerine. Compare them, using the A.container and the spectrum.

Tangerine is further to the yellow side of the spectrum. By containing, Sunglow can be lined up in columns of 9, with 4 red and 5 yellow. You can’t do the same with Tangerine. There is an extra yellow sticking out so that you can’t make a “nice” rectangle.

In your notebook, explain why Larry is correct or incorrect. Provide a B.diagram and calculation in your explanation.

Larry is incorrect. Sara’s ratio, rewritten as red to yellow, is 20:25. Then we can see that Tangerine has one more yellow for every 20 red than Sunglow. Tangerine is more yellow than Sunglow , which means that Larry’s blend is less red.

Wrap Up

True Colors buys snacks for the muralists to take on the road. Monet’s Mix has 6.5 chocolate bits for every 8 nuts. Van Gogh’s Mix has 8 chocolate bits for every 16 nuts. We want to know which mix has a greater chocolate to nut ratio. Elli says to compare mixes we should always double one of the recipes. Explain how Elli’s strategy might or might not always work.

Elli ’s strategy does not always work. Elli ’s strategy works in this case because doubling Monet’s Mix allows us to see the amount of chocolate bits for 16 nuts. A better general strategy is to use equivalent ratios to compare when a term is equal.


Problem Solving

The bagel shops in town are competing to see who can make the cheesier bagel. 7.Better Batch Bakery makes 7 cheese bagels for every 14 ounces of cheese. The Cheese House makes 21 cheese bagels for every 36 ounces of cheese. The bagels are exactly the same, except for the amount of cheese used. Who makes cheesier bagels?

Better Batch. If they made 21 bagels, they would use 42 ounces of cheese, compared with The Cheese House’s 36.

True Colors muralists are playing Tip, Tap, Top again. Taps and tops can be used 8.in different ratios. Cassat’s team played 9 taps for every 14 tops. Moriset’s team played 8 taps for 7 tops. Which team played a “tappier” game of Tip, Tap, Top?

Cassat’s team played taps to tops in the ratio 9:14 as compared with the Moriset’s team’s 16:14 ratio. So Moriset’s team was “tappier.”

Each day, a deli buys 27 pounds of sandwich meat and 21 pounds of cheese. 9.

What is the ratio of pounds of meat to pounds of cheese in simplest form? A.

9:7

The deli makes sandwiches that contain pounds of meat to pounds cheese in B.a ratio of 10:7. Assume that the deli sells only sandwiches. Will the deli run out of meat or cheese first? Use a picture to help you solve.

The deli will run out of meat first, because 21 pounds of cheese will require 30 pounds of meat in the sandwiches. The deli bought only 27 pounds of meat.


Vanessa makes a green color with 2 blue pips and 5 yellow pips. Her classmates 10.claim their greens are more yellow than Vanessa’s green. Are their claims true or false?

Jenny’s mixture is made of 6 blue pips and 15 yellow pips. A.

False. Same as Vanessa’s.

Teddy’s mixture is made of 3 blue pips and 6 yellow pips. B.

False.

Raj’s mixture is made of 5 blue pips and 15 yellow pips C.

True.

Ali’s mixture is made of !! blue pips and 1 yellow pip. D.

False.

Josh’s mixture is made of 1 blue pips and !! yellow pips. E.

True.

Torus’s mixture is made of 7 blue pips and 18 yellow pips. F.

False.

Investigation 8: Just Tell Me How Many 43

Investigation 8: Just Tell Me How Many

Warm Up

Evaluate. 1.

A. B. C. D.

12÷ 3 6÷ 4 150÷ 30 1÷ 4

4 1½ or 1.5 5 1/4 or 0.25

E. F. G.

4÷ 16 40÷ 160 2÷ 3

1/4 or 0.25 1/4 or 0.25 2/3 or 0.66

Main The recipe for Eggplant Purple is 12 ounces of red paint for every 3 ounces of blue paint. Our artist, Ying, is tired of working with ratios such as 12:3. “Just tell me how many ounces of red you need for every ounce of blue, please,” she requests.

Ying is asking for a unit rate—the number of ounces of red paint per one ounce of blue. For the ratio of red to blue, 12:3, the unit rate is 4 ounces of red per 1 ounce of blue.

Woman with a Flower, by Paul Gauguin.

44 Investigation 8: Just Tell Me How Many

Use Activity 8.1 to make one blend of Eggplant Purple with 12 red and 3 blue. 2.

Make a second blend showing the unit rate. A.

Use the Container tool to compare the ratio and the unit rate. B.

In your notebook, draw diagrams of the Container tool that shows the C.relationship between the ratio and unit rate.

Ratio container Unit rate container

Explain how your diagrams show this relationship. D.

Diagrams of both the ratio and the unit rate should show that for every 4 red pips there is 1 blue pip. Both Container tools show that each column has 4 red pips and 1 blue pip.

Ying makes another purple called Iris using a ratio of 6 red to 4 blue. She rewrites 3.the ratio in simplest form as 3 red to 2 blue. She thinks Iris has a unit rate of 1.5 ounces red per 1 ounce blue. She draws the diagram below. Explain how the diagram shows the relationship of the ratio and the unit rate forms of Iris.

6:4

The diagram of 6:4 and the diagram of 3:2 each use rows of 3 red to 2 blue. The bottom diagram shows that the 3 red and 2 blue could be divided in half so that for every 1 blue there are 1.5 red. So the diagrams show that a unit rate of 1.5 red per 1 blue is equivalent to a red to blue ratio of 3:2, which is equivalent to a red to blue ratio of 6:4. The video in Activity 8.2 helps show how a ratio and a unit rate are related.

3:2

3:2


If Ying is using 3 ounces of blue paint to make Iris, how many ounces of red A.paint does she need?

4.5 ounces

If Ying is using 15 ounces of blue paint to make Iris, how many ounces of red B.paint does she need?

22.5 ounces

Unit rates can help us to compare. The following table shows several different 4.colors of purple that Ying created. She needs to figure out which are more red and which are more blue (less red) than Eggplant Purple.

Copy the table into your notebook. Determine the red per blue ounce (oz.) A.unit rate for each color. You can round to the nearest 100th.

Color name Ratio of red oz. to blue oz. Unit rate

red per 1 blue oz.

Eggplant Purple 10:5 2

Iris 6:4 1.5

Lavender 32:16 2

Lyla 40:160 0.25

Orchid 150:30 5

Violet 20:30 0.67 or 2/3

Which colors, if any, are more red than Eggplant Purple? B.

Orchid

How do you know that these colors are more red? C.

Orchid is more red because it has a greater number of red ounces for each blue ounce, or a higher unit rate of red to blue.

Which colors, if any, are more blue than Eggplant Purple? D.

Iris, Lyla, and Violet


How do you know that these colors are more blue? E.

They have a smaller number of red ounces for each blue ounce, or a lower unit rate of red to blue.

Wrap Up

Explain how unit rates can help us to compare ratios. 5.

The unit rate shows us, for example, how many red pips there are for every blue. If we compare two unit rates, the fact that they both compare red to ONE blue means we can just compare two numbers instead of four.


Problem Solving

Our last mural measured 56 square feet. We used 14 containers of paint to 6.complete this mural. How many square feet per container were painted?

56/14 = 4 4 square feet per container

We have to choose which of two customers to work for. The total amount they 7.offer to pay us and the number of hours it will take us to complete the project for each project are shown in the table below. Which customer is offering a better pay rate per hour?

Pay Number of hours

Diego Rivera Middle School $315 15

Ruth Auger Elementary School $276 12

Diego Rivera Middle School offers $21/hour. Ruth Auger Elementary School offers $23/hour. So Ruth Auger Elementary School offers the better rate.

At their practices, the volleyball team ran a total of 10 miles in 14 days. The 8.tennis team ran a total of 8 miles in 7 days. Which team ran more miles per day?

The tennis team ran more miles per day. The tennis team ran 8/7 (or about 1.1) miles per day. The volleyball team ran 10/14 (or about 0.7) miles per day.


Find and use the unit rate. 9.

There are 2 packs of cucumbers are sold for $4. What is the price for each A.pack?

$2 per pack

There are 2 bundles of green onion are sold for $1. What is the price (in B.dollars) per bundle? At that rate, what is the cost of 7 bundles?

$0.50 per bundle; $3.50

Josh typed 280 words in 10 minutes. How many words might Josh type per C.minute?

28 words per minute

Jamal typed 525 words in 15 minutes. How many words might Jamal type D.per minute? At that rate how many words would he type, if he typed for 10 minutes?

35 words per minute; 350 words

In Store A, downloading 5 songs costs $2.50. What is the cost (in dollars) per E.song?

$0.50 per song

In Store B, downloading 15 songs costs $9.75. What is the cost (in dollars) per F.song? At that rate, what is the cost of 18 songs?

$0.65 per song; $8.30

Challenge: How many songs can you buy for 20.00? 30 songs

Optional Investigation 9: Walls of Metal and Wood 49

Optional Investigation 9: Walls of Metal and Wood

Warm Up

Evaluate. 1.

A. B. C. D.

220÷ 10 200÷ 5 450÷ 10 160÷ 8

22 40 45 20

E. F.

125÷ 5 675÷ 15

25 45

Main SunBay Studios, Hollywood’s fastest rising movie studio, has asked us for help in designing a movie set for its new superhero movie. In the movie set, one backdrop is a wall made of 24 panels. In this task, you will be asked to design the wall, calculate ratios, make predictions, and calculate costs.

The following types of panels can be purchased. Panels can be bought individually, but the pricing is given for the whole pack.

Metal Panels

Type A – Solid

$450 for a pack of 10 panels

Type B – Striped


Type C – Checkerboard


50 Optional Investigation 9: Walls of Metal and Wood

Wooden Panels

Type D – Light


Type E – Medium


Type F – Dark


Designing the Wall

The new movie’s villain, Aluminum Joe, kills trees by coating them in metal so that photosynthesis cannot take place.

The director wants to see two wall designs for Aluminum Joe’s lair. Make each 2.with 24 panels using the boxes below. You can use any combination of metal and wood panels. Put the letter for a type of panel in each box.

Design 1 Design 2

Optional Investigation 9: Walls of Metal and Wood 51

Describe each design in terms of ratios for the director. In your notebook, write 3.ratios in their simplest form.

In my first design, A.

the ratio of metal to wood panels is: Answers will vary

the ratio of metal to total panels is: Answers will vary

In my second design, B.

the ratio of metal to wood panels is: Answers will vary

the ratio of metal to total panels is: Answers will vary

The first/second design is more metallic because: Answers will vary

Calculating the Costs

Cost matters too, as well as the ratio of wood to metal. You can use unit rates to help you calculate costs.

Tell the director the total cost of each of your two walls and show how you got 4.those totals, so the director understands.

For your first design: A.

Calculate the unit rate of each type of panel used and write them in order.

Type A = $45/panel Type F = $25/panel

Type B = $45/panel Type D = $22/panel

Type C = $40/panel Type E = $20/panel

Calculate the total cost using the unit rates and the number of panels you used of each type. Count only the panels you actually used.

Total cost: Answers will vary

52 Optional Investigation 9: Walls of Metal and Wood

For your second design: B.

Calculate the unit rate of each type of panel used and write them in order.

Type A = $45/panel Type F = $25/panel

Type B = $45/panel Type D = $22/panel

Type C = $40/panel Type E = $20/panel

Calculate the total cost using the unit rates and the number of panels you used of each type. Count only the panels you actually used.

Total cost: Answers will vary

The director likes both walls but cannot decide which to use. Choose one wall 5.design to recommend for the Aluminum Joe’s lair. Explain your choice by cost and by how the wood to metal ratio works for the Aluminum Joe’s lair. Give other reasons why your design should be chosen. Write your explanation in a paragraph.

Optional Investigation 10: Sunflowers 53

Optional Investigation 10: Sunflowers

One of our muralists, Astrid, needs you to mix a variety of yellowish oranges for a mural based on Van Gogh’s Sunflowers. Use Activity 10.1 and create the following red to yellow blends: 1:1, 1:2, and 1:3.

Predict. Make a sketch to 1.predict where in the spectrum the red to yellow mixtures 1:4, 1:5, and 1:6 will be.

Note that the online tools show yellow to red, but the question asks red to yellow. Check student work to see that they are making blends and predictions that are red:yellow.

Check. Check your 2.prediction by making these red to yellow mixtures.

Explain. Is your prediction right? In your notebook, explain whether your 3.prediction was correct or not and why.

Each ratio’s dividing line is a little bit further toward yellow, but a little less far than the ratio before it: 1:4 is mostly yellow; 1:5 is even more yellow; and 1:6 is really yellow. A good explanation should reflect on how their prediction and what they see in Ratio Visualizer are aligned.

Choose three different colors for Astrid to use in the mural, name them, and 4.write their ratios.

Sunflowers, by Vincent Van Gogh.

Predict Check Explain

SunBayMath

3. Explain• Use the representations to support your reasoning. • Was your prediction correct? Explain how you know. • Was your prediction incorrect? Explain why.• Describe how to get the correct answer.• Were you surprised? What did you learn? • Maybe, make a new prediction!

2. Check• Use the technology or make a calculation to see if your prediction was correct.

1. Predict• Estimate! • Sketch! • Make your best guess.• Be creative.• Work quickly.

• Listen actively to other people's predictions. • Predictions don't have to be correct.• Do mental calculations as needed.

© 2015 SRI International

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