# Lesson 3

Representing Proportional Relationships

## 3.1: Number Talk: Multiplication (5 minutes)

### Warm-up

This Number Talk encourages students to think about the numbers in a computation problem and rely on what they know about structure, patterns, multiplication, fractions, and decimals to mentally solve a problem. Only one problem is presented to allow students to share a variety of strategies for multiplication. Notice how students handle the multiplication by a decimal. Some students may use fraction equivalencies while others may use the decimal given in the problem. For each of those choices, ask students why they made that decision.

### Launch

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. Be sure to elicit as many strategies as possible.

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

### Student Facing

Find the value of each product mentally.

$$15 \boldcdot 2$$

$$15 \boldcdot 0.5$$

$$15 \boldcdot 0.25$$

$$15 \boldcdot (2.25)$$

### Activity Synthesis

Invite students to share their strategies. Use MLR 2 (Collect and Display) to record and display student explanations for all to see. Ask students to explain their choice of either using a decimal or fraction for $$2.25$$ in their solution path. 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?”

At the end of discussion, if time permits, ask a few students to share a story problem or context that $$15\boldcdot (2.25)$$ would represent.

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)

## 3.2: Representations of Proportional Relationships (15 minutes)

### Activity

The purpose of this activity is for students to graph a proportional relationship when given a blank pair of axes. They will need to label and scale the axes appropriately before adding the line representing the given relationship. In each problem, students are given two representations and asked to create two more representations so that each relationship has a description, graph, table, and equation. Then, they explain how they know these are different representations of the same situation (MP3). In the next lesson, students will use these skills to compare two proportional relationships represented in different ways.

Identify students making particularly clear graphs and using situation-appropriate scales for their axes. For example, since the second problem is about a car wash, the scale for the axis showing the number of cars does not need to extend into the thousands.

### Launch

Arrange students in groups of 2. Provide access to straightedges. Give 3 minutes of quiet work time for students to begin the first problem and then tell students to check in with their partners to compare tables and how they are labeling and scaling the axes. Ask students to pause their work and select a few students to share what scale they are using for the axes and why they chose it. It is important to note that the scale chosen should be reasonable based on the context. For example, using a very small scale for steps taken does not make sense.

Give partners time to finish the remaining problems and follow with a whole-class discussion.

Action and Expression: Internalize Executive Functions. Provide students with grid or graph paper to organize their work with creating a table and graph for the situation.
Supports accessibility for: Language; Organization

### Student Facing

1. Here are two ways to represent a situation.

Description:

Jada and Noah counted the number of steps they took to walk a set distance. To walk the same distance, Jada took 8 steps while Noah took 10 steps. Then they found that when Noah took 15 steps, Jada took 12 steps.

Equation:

Let $$x$$ represent the number of steps Jada takes and let $$y$$ represent the number of steps Noah takes. $$\displaystyle y=\frac54x$$

1. Create a table that represents this situation with at least 3 pairs of values.

2. Graph this relationship and label the axes.

3. How can you see or calculate the constant of proportionality in each representation? What does it mean?

4. Explain how you can tell that the equation, description, graph, and table all represent the same situation.

2. Here are two ways to represent a situation.

Description:

The Origami Club is doing a car wash fundraiser to raise money for a trip. They charge the same price for every car. After 11 cars, they raised a total of $93.50. After 23 cars, they raised a total of$195.50.

Table:

number of
cars
amount raised
in dollars
11 93.50
23 195.50
1. Write an equation that represents this situation. (Use $$c$$ to represent number of cars and use $$m$$ to represent amount raised in dollars.)
2. Create a graph that represents this situation.
3. How can you see or calculate the constant of proportionality in each representation? What does it mean?
4. Explain how you can tell that the equation, description, graph, and table all represent the same situation.

### Activity Synthesis

Ask previously identified students to share their graphs and how they chose the scales for their axes. If possible, display several graphs from each question for all to see as students share.

Ask students “Which representation makes it more difficult (and less difficult) to calculate the constant of proportionality? Why?” and give 1 minute of quiet think time. Invite several students to share their responses.

Tell students that the constant of proportionality can be thought of as the rate of change of one variable with respect to the other. In the case of Jada and Noah, the rate of change of $$y$$, the number of steps Noah takes, with respect to $$x$$, the number of steps Jada takes, is $$\frac54$$ Noah steps per Jada steps. In the case of the Origami Club’s car wash, the rate of change of $$m$$, the amount they raise in dollars, with respect to $$c$$, the number of cars they clean, is 8.50 dollars per car.

Writing, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine for students to respond in writing to the final question, “Explain how you can tell that the equation, description, graph, and table all represent the same situation.” Give students time to meet with 2–3 partners, to share their responses and get feedback. Encourage the listener to ask clarifying questions such as, “How did you identify the same constant of proportionality?” or “Did the scales for the axes cause any confusion?” Have the students write a final draft based on their peer feedback. This will help students to generalize the process for identifying the same constant of proportionality.
Design Principle(s): Optimize output; Cultivate conversation

## 3.3: Info Gap: Proportional Relationships (15 minutes)

### Activity

This info gap activity gives students an opportunity to determine and request the information needed when working with proportional relationships. In order to graph the relationship and the requested information, students need to think carefully about how they can scale the axes.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then asking for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of one of the cards for reference and planning:

Listen for how students request (and supply) information about the relationship between the two ingredients. Identify students using different scales for their graphs that show clearly the requested information to share during the discussion.

### Launch

Tell students that they will continue their work graphing proportional relationships. Explain the Info Gap and consider demonstrating the protocol if students are unfamiliar with it. Arrange students in groups of 2. Provide access to straightedges. In each group, distribute a problem card to one student and a data card to the other student. After you review their work on the first problem, give them the cards for the second problem and instruct them to switch roles.

Representation: Provide Access for Perception. Display or provide students with a physical copy of the written directions and read them aloud. Check for understanding by inviting students to rephrase directions in their own words. Consider keeping the display of directions visible throughout the activity.
Supports accessibility for: Language; Memory
Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to choose a scale and graph a proportional relationship. Display questions or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?”
Design Principle(s): Cultivate Conversation

### Student Facing

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the problem card:

3. Explain how you are using the information to solve the problem.

Continue to ask questions until you have enough information to solve the problem.

4. Share the problem card and solve the problem independently.

If your teacher gives you the data card:

2. Ask your partner “What specific information do you need?” and wait for them to ask for information.

If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.

3. Before sharing the information, ask “Why do you need that information?” Listen to your partner’s reasoning and ask clarifying questions.

4. Read the problem card and solve the problem independently.

5. Share the data card and discuss your reasoning.

### Student Facing

#### Are you ready for more?

Ten people can dig five holes in three hours. If $$n$$ people digging at the same rate dig $$m$$ holes in $$d$$ hours:

1. Is $$n$$ proportional to $$m$$ when $$d=3$$?
2. Is $$n$$ proportional to $$d$$ when $$m=5$$?
3. Is $$m$$ proportional to $$d$$ when $$n=10$$

### Anticipated Misconceptions

Some students may be unsure how large to make their scale before they answer the question on the card. Encourage these students to answer the question on their card and then think about how to scale their graph.

### Activity Synthesis

After students have completed their work, ask previously identified students to share their graphs and explain how they chose their axis. Some guiding questions:

• “Other than the answer, what information would have been nice to have?”
• “How did you decide what to label the two axes?”
• “How did you decide to scale the horizontal axis? The vertical?”
• “What was the rate of change of grams of honey per cups of flour? Where can you see this on the graph you made?” (4.5 grams of honey per cup of flour.)
• “What was the rate of change of grams of salt per cups of flour? Where can you see this on the graph you made?” (2.5 grams of salt per cups of flour)

## Lesson Synthesis

### Lesson Synthesis

Consider asking some of the following questions.

• “The proportional relationship $$y=5.5x$$ includes the point $$(18, 99)$$ on its graph. How could you choose a scale for a pair of axes with a 10 by 10 grid to show this point?” (Have each grid line represent 10 or 20 units.)
• “What are some things you learned about graphing today that you are going to try to remember for later?”

## Student Lesson Summary

### Student Facing

Proportional relationships can be represented in multiple ways. Which representation we choose depends on the purpose. And when we create representations we can choose helpful values by paying attention to the context. For example, a stew recipe calls for 3 carrots for every 2 potatoes. One way to represent this is using an equation. If there are $$p$$ potatoes and $$c$$ carrots, then $$c = \frac32p$$.

Suppose we want to make a large batch of this recipe for a family gathering, using 150 potatoes. To find the number of carrots we could just use the equation: $$\frac32\boldcdot 150= 225$$ carrots.

Now suppose the recipe is used in a restaurant that makes the stew in large batches of different sizes depending on how busy a day it is, using up to 300 potatoes at at time.

Then we might make a graph to show how many carrots are needed for different amounts of potatoes. We set up a pair of coordinate axes with a scale from 0 to 300 along the horizontal axis and 0 to 450 on the vertical axis, because $$450 = \frac32\boldcdot 300$$. Then we can read how many carrots are needed for any number of potatoes up to 300.

Or if the recipe is used in a food factory that produces very large quantities and the potatoes come in bags of 150, we might just make a table of values showing the number of carrots needed for different multiplies of 150.

number of potatoes number of carrots
150 225
300 450
450 675
600 900

No matter the representation or the scale used, the constant of proportionality, $$\frac32$$, is evident in each. In the equation it is the number we multiply $$p$$ by; in the graph, it is the slope; and in the table, it is the number we multiply values in the left column to get numbers in the right column. We can think of the constant of proportionality as a rate of change of $$c$$ with respect to $$p$$. In this case the rate of change is $$\frac32$$ carrots per potato.