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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

Create a graph that represents this situation.

quadrant 1 grid. horizontal axis, c. 25 units. vertical axis, m. 10 units.

3.2

Activity

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

MLR4: Information Gap Cards

Materials

To Copy (from Blackline Masters)

Graphing Proportional Relationships Cards

Activity Narrative

In this activity, students graph a proportional relationship but do not initially have enough information to do so. To bridge the gap, they need to exchange questions and ideas.

The Info Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask 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).

This activity uses the Information Gap math language routine, which facilitates meaningful interactions by positioning some students as holders of information that is needed by other students, creating a need to communicate.

Launch

Tell students they will be graphing some proportional relationships. Display the Info Gap graphic that illustrates a framework for the routine for all to see.

Remind students of the structure of the Info Gap routine, and consider demonstrating the protocol if students are unfamiliar with it.

Arrange students in groups of 2. In each group, give a problem card to 1 student and a data card to the other student. After reviewing their work on the first problem, give students the cards for a second problem and instruct them to switch roles.

Action and Expression: Internalize Executive Functions. Check for understanding by inviting students to rephrase directions in their own words. Keep a display of the Info Gap graphic visible throughout the activity or provide students with a physical copy.
Supports accessibility for: Memory, Organization

Activity

None

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:

Silently read your card and think about what information you need to answer the question.
Ask your partner for the specific information that you need. “Can you tell me ?”
Explain to your partner how you are using the information to solve the problem. “I need to know because . . . .”

Continue to ask questions until you have enough information to solve the problem.
Once you have enough information, share the problem card with your partner, and solve the problem independently.
Read the data card, and discuss your reasoning.

If your teacher gives you the data card:

Silently read your card. Wait for your partner to ask for information.
Before telling your partner any information, ask, “Why do you need to know ?”
Listen to your partner’s reasoning and ask clarifying questions. Only give information that is on your card. Do not figure out anything for your partner!

These steps may be repeated.
Once your partner says they have enough information to solve the problem, read the problem card, and solve the problem independently.
Share the data card, and discuss your reasoning.

Student Response

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Building on Student Thinking

If students are unsure how to scale the axes on their graphs, consider asking:

“What are the largest values that need to be shown on the graph?”
“How many grid lines are there?”

Activity Synthesis

After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Here are some questions for discussion:

“How did you decide what to label the two axes?”
“How did you decide to scale the horizontal axis? The vertical axis?”
“Where can you see the rate of change of grams of honey per cups of flour on the graph?”
“Where can you see the rate of change of grams of salt per cups of flour on the graph?”

Student Lesson Summary

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 potatoes and carrots, then .

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: carrots.

Now suppose the recipe is used in a restaurant that makes the stew in large batches of different sizes depending on how busy of a day it is, using up to 300 potatoes at a 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 . Then we can read how many carrots are needed for any number of potatoes up to 300.

graph, horizontal axis, number of potatoes, scale 0 to 300, by 50's. vertical axis, number of carrots, scale 0 to 450, by 50's.

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

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

No matter the representation or the scale used, the constant of proportionality, , is evident in each. In the equation, it is the number we multiply by. In the graph, it is the slope. In the table, it is the number we multiply values in the left column to get values in the right column. We can think of the constant of proportionality as a rate of change: the amount one variable changes by when the other variable increases by 1. In this case, the rate of change of with respect to is carrots per potato.