Lesson 4

Color Mixtures

4.1: Number Talk: Adjusting a Factor (10 minutes)

This number talk encourages students to use the structure of base ten numbers and the properties of operations to find the product of two whole numbers (MP7).

While many strategies may emerge, the focus of this string of problems is for students to see how adjusting a factor impacts the product, and how this insight can be used to reason about other problems. Four problems are given, however, it may not be possible to share every possible strategy. Consider gathering only two or three different strategies per problem. Each problem was chosen to elicit a slightly different reasoning, so as students explain their strategies, ask how the factors impacted how they approached the problem.

Display one problem at a time. Give students 1 minute of quiet think time per problem and ask them to give a signal when they have an answer and a strategy. Follow with a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Find the value of each product mentally.

\(6\boldcdot 15\)

\(12\boldcdot 15\)

\(6\boldcdot 45\)

\(13\boldcdot 45\)

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Ask students to share their strategies for each problem. Record and display their explanations for all to see. Ask students if or how the factors in the problem impacted the strategy choice. To involve more students in the conversation, consider asking:

“Who can restate ___’s reasoning in a different way?”
“Did anyone solve the problem the same way but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to _____’s strategy?”
“Do you agree or disagree? Why?”

Speaking: MLR8 Discussion Supports.: Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . . ." Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)

4.2: Turning Green (35 minutes)

In this activity, students mix different numbers of batches of a color recipe to obtain a certain shade of green. They observe how multiple batches of the same recipe produce the same shade of green as a single batch, which suggests that the ratios of blue to yellow for the two situations are equivalent. They also come up with a ratio that is not equivalent to produce a mixture that is a different shade of green.

As students make the mixtures, ensure that they measure accurately so they will get accurate outcomes. As students work, note the different diagrams students use to represent their recipes. Select a few examples that could be highlighted in discussion later.

Digital

Arrange students in groups of 2–4. (Smaller groups are better, but group size might depend on available equipment.) Each group needs a beaker of blue water and one of yellow water, one graduated cylinder, a permanent marker, a craft stick, and 3 opaque white cups (either styrofoam, white paper, or with a white plastic interior).

For classes using the print-only version: Show students the blue and yellow water. Demonstrate how to pour from the beakers to the graduated cylinder to measure and mix 5 ml of blue water with 15 ml of yellow water. Demonstrate how to get an accurate reading on the graduated cylinder by working on a level surface and by reading the measurement at eye level. Tell students they will experiment with different mixtures of green water and observe the resulting shades.

For classes using the digital version: Display the dynamic color mixing cylinders for all to see. Tell students, “The computer mixes colors when you add colored water to each cylinder. You can add increments of 1 or 5. You can’t remove water (once it’s mixed, it’s mixed), but you can start over. The computer mixes yellow and blue.” Ask students a few familiarization questions before they start working on the activity:

What happens when you mix yellow and blue? (A shade of green is formed.)
What happens if you add more blue than yellow? (Darker green, blue green, etc.)

If necessary, demonstrate how it works by adding some yellows and blues to both the left and the right cylinder. Show how the “Reset” button lets you start over.

Representation: Internalize Comprehension. Provide appropriate reading accommodations and supports to ensure students access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing

In the left cylinder, mix 5 ml of blue and 15 ml of yellow. This is a single batch of green.
Suppose you have one batch of green but want to make more. Which of the following would produce the same shade of green?

If you're unsure, try creating the mixture in the right cylinder. Start with the amounts in a single batch (5 ml of blue and 15 ml of yellow) and . . .
1. add 20 ml of blue and 20 ml of yellow
2. double the amount of blue and the amount of yellow
3. triple the amount of blue and the amount of yellow
4. mix a single batch with a double batch
5. double the amount of blue and triple the amount of yellow
For one of the mixtures that produces the same shade, write down the number of ml of blue and yellow used in the mixture.
For the same mixture that produces the same shade, draw a diagram of the mixture. Make sure your diagram shows the number of milliliters of blue, yellow, and the number of batches.
Someone was trying to make the same shade as the original single batch, but started by adding 20 ml of blue and 20 ml of yellow. How can they add more but still create the same shade of green?
Invent a recipe for a bluer shade of green. Write down the amounts of yellow and blue that you used, and draw a diagram. Explain how you know it will be bluer than the original single batch of green before testing it out.

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What happens when you mix yellow and blue? (A shade of green is formed.)
What happens if you add more blue than yellow? (Darker green, blue green, etc.)

If necessary, demonstrate how it works by adding some yellows and blues to both the left and the right cylinder. Show how the “Reset” button lets you start over.

Your teacher mixed milliliters of blue water and milliliters of yellow water in the ratio \(5:15\).

Doubling the original recipe:
1. Draw a diagram to represent the amount of each color that you will combine to double your teacher’s recipe.
2. Use a marker to label an empty cup with the ratio of blue water to yellow water in this double batch.
3. Predict whether these amounts of blue and yellow will make the same shade of green as your teacher’s mixture. Next, check your prediction by measuring those amounts and mixing them in the cup.
4. Is the ratio in your mixture equivalent to the ratio in your teacher’s mixture? Explain your reasoning.
Tripling the original recipe:
1. Draw a diagram to represent triple your teacher’s recipe.
2. Label an empty cup with the ratio of blue water to yellow water.
3. Predict whether these amounts will make the same shade of green. Next, check your prediction by mixing those amounts.
4. Is the ratio in your new mixture equivalent to the ratio in your teacher’s mixture? Explain your reasoning.
Next, invent your own recipe for a bluer shade of green water.
1. Draw a diagram to represent the amount of each color you will combine.
2. Label the final empty cup with the ratio of blue water to yellow water in this recipe.
3. Test your recipe by mixing a batch in the cup. Does the mixture yield a bluer shade of green?
4. Is the ratio you used in this recipe equivalent to the ratio in your teacher’s mixture? Explain your reasoning.

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Are you ready for more?

Someone has made a shade of green by using 17 ml of blue and 13 ml of yellow. They are sure it cannot be turned into the original shade of green by adding more blue or yellow. Either explain how more can be added to create the original green shade, or explain why this is impossible.

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If any students come up with an incorrect recipe, consider letting this play out. A different shade of green shows that the ratio of blue to yellow in their mixture is not equivalent to the ratio of blue to yellow in the other mixtures. Rescuing the incorrect mixture to display during discussion may lead to meaningful conversations about what equivalent ratios mean.

After each group has completed the task, have the students rotate through each group’s workspace to observe the mixtures and diagrams. As they circulate, pose some guiding questions. (For students using the digital version, these questions refer to the mixtures on their computers.)

Are each group’s results for the first two mixtures the same shade of green?
Are the ratios representing the double batch, the triple batch, and your new mixture all equivalent to each other? How do you know?
What are some different ways groups drew diagrams to represent the ratios?

Highlight the idea that a ratio is equivalent to another if the two ratios describe different numbers of batches of the same recipe.

Conversing: MLR8 Discussion Supports. Assign one member from each group to stay behind to answer questions from students visiting from other groups. Provide visitors with question prompts such as, "These look like the same shade, how can we be sure the ratios are equivalent?", "If you want a smaller amount with the same shade, what can you do?" or "What is the ratio for your new mixture?"
Design Principle(s): Cultivate conversation

4.3: Perfect Purple Water (10 minutes)

Students revisit color mixing—this time by producing purple-colored water—to further understand equivalent ratios. They recall that doubling, tripling, or halving a recipe for colored water yields the same resulting color, and that equivalent ratios can represent different numbers of batches of the same recipe.

As students work, monitor for students who use different representations to answer both questions, as well as students who come up with different ratios for the second question.

Arrange students in groups of 2. Remind students of the previous “Turning Green” activity. Ask students to discuss the following questions with a partner. Then, discuss responses together as a whole class:

Why did the different mixtures of blue and yellow water result in the same shade of green? (If mixed correctly, the amount of the ingredients were all doubled or all tripled. The ratio of blue water to yellow water was equivalent within each recipe.)
How were you able to get a darker shade of green? (We changed the ratio of ingredients, so there was more blue for the same amount of yellow.)

Explain to students that the task involves producing purple-colored water, but they won’t actually be mixing colored water. Ask students to use the ideas just discussed from the previous activity to predict the outcomes of mixing blue and red water.

Ensure students understand the abbreviation for milliliters is ml.

Representation: Develop Language and Symbols. Use virtual or concrete manipulatives to connect symbols to concrete objects or values. Provide students with snap cubes, blocks or printed representations.
Supports accessibility for: Conceptual processing

The recipe for Perfect Purple Water says, “Mix 8 ml of blue water with 3 ml of red water.”

Jada mixes 24 ml of blue water with 9 ml of red water. Andre mixes 16 ml of blue water with 9 ml of red water.

Which person will get a color mixture that is the same shade as Perfect Purple Water? Explain or show your reasoning.
Find another combination of blue water and red water that will also result in the same shade as Perfect Purple Water. Explain or show your reasoning.

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At a quick glance, students may think that since Andre is mixing a multiple of 8 with a multiple of 3, it will also result in Perfect Purple Water. If this happens, ask them to take a closer look at how the values are related or draw a diagram showing batches.

Select students to share their answers to the questions.

For the first question, emphasize that not only did Jada triple each amount of red and blue, but this means that amount of each color is being multiplied by the same value, in this case, 3.
For the second question, list all the different ratios students brought up for all to see. Discuss how each ratio differed from that for the original mixture. Point out that as long as both terms are multiplied by the same quantity, the resulting ratio will be equivalent and will yield the same shade of purple.

Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their combination of blue water and red water that will make the same shade as Perfect Purple Water, present a a flawed response. For example, “Mixing 18 ml of blue water with 13 ml of red water would result in the same shade of purple because I added 10 ml of each color.” Ask students to identify the error, critique the reasoning, and write a correct explanation. Invite students to share their critiques and corrected combinations and explanations with the class. Listen for and amplify the language students use to justify the ratios are equivalent. This may include such language as multiply by the same quantity or represent the same shade. This helps students evaluate, and improve upon, the written mathematical arguments of others, as they clarify their understanding of equivalent ratios.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

Lesson Synthesis

The important take-aways from this lesson are:

To create more batches of a color recipe that will come out to be the same shade of the color, multiply each ingredient by the same number.
We can think of equivalent ratios as representing different numbers of batches of the same recipe.

Remind students of the work done and observations made in this lesson. Some questions to guide the discussion might include:

How did you decide that 10 ml blue and 30 ml yellow would make 2 batches of 5 ml blue and 15 ml yellow? (Multiply each part by 2.)
How did you decide that 15 ml blue and 45 ml yellow would make 3 batches? (Multiply each part by 3.)
How did we know that \(5:15\), \(10:30\), and \(15:45\) were equivalent? (They created the same shade of green. Also, \(10:30\) has both parts of the original recipe multiplied by 2, and \(15:45\) has both parts of the original recipe multiplied by 3.)

4.4: Cool-down - Orange Water (5 minutes)

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Student Lesson Summary

When mixing colors, doubling or tripling the amount of each color will create the same shade of the mixed color. In fact, you can always multiply the amount of each color by the same number to create a different amount of the same mixed color.

For example, a batch of dark orange paint uses 4 ml of red paint and 2 ml of yellow paint.

To make two batches of dark orange paint, we can mix 8 ml of red paint with 4 ml of yellow paint.
To make three batches of dark orange paint, we can mix 12 ml of red paint with 6 ml of yellow paint.

Here is a diagram that represents 1, 2, and 3 batches of this recipe.

A discrete diagram for two quantities labeled “red paint (ml)” and “yellow paint (ml)”.

The data are as follows: 1 batch orange, 4 red squares and 2 yellow squares. 2 batches orange, 8 red squares and 4 yellow squares. Three batches orange, 12 red squares and 6 yellow squares.

We say that the ratios \(4:2\), \(8:4\), and \(12:6\) are equivalent because they describe the same color mixture in different numbers of batches, and they make the same shade of orange.

Lesson 4

4.1: Number Talk: Adjusting a Factor (10 minutes)

Warm-up

Launch

Student Facing

Student Response

Activity Synthesis

4.2: Turning Green (35 minutes)

Activity

Launch

Student Facing

Student Response

Launch

Student Facing

Student Response

Student Facing

Are you ready for more?

Student Response

Anticipated Misconceptions

Activity Synthesis

4.3: Perfect Purple Water (10 minutes)

Optional activity

Launch

Student Facing

Student Response

Anticipated Misconceptions

Activity Synthesis

Lesson Synthesis

Lesson Synthesis

4.4: Cool-down - Orange Water (5 minutes)

Cool-Down

Student Lesson Summary

Student Facing