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# Ratio Tables and Graphs

# Relate Ratio Tables to Graphs

## Overview

Students focus on interpreting, creating, and using ratio tables to solve problems. They also relate ratio tables to graphs as two ways of representing a relationship between quantities.

# Key Concepts

Ratio tables and graphs are two ways of representing relationships between variable quantities. The values shown in a ratio table give possible pairs of values for the quantities represented and define ordered pairs of coordinates of points on the graph representing the relationship. The additive and multiplicative structure of each representation can be connected, as shown:

# Goals and Learning Objectives

- Complete ratio tables.
- Use ratio tables to compare ratios and solve problems.
- Plot values from a ratio table on a graph.
- Understand the connection between the structure of ratio tables and graphs.

# Ratio Tables and Graphs

# Lesson Guide

Have students read the prompt, examine the ratio table, and graph the points.

Ask:

- Why do you think the table is called a ratio table?
- How can you complete the table?
- How is the information in the ratio table represented in the graph?
- How can you plot the points on the graph?

Use any disagreement to prompt a discussion.

# Mathematics

The start of the lesson focuses students’ attention on the connection between the pairs of values in the table and the points on the graph and asks them to add information to each representation. Incorrect attempts at adding values or points to the representations are especially productive at this point in the unit; use them to surface misconceptions about the connections between tables and graphs and to provide opportunities for revising student thinking.

Note: You may need to point out that the packages are all the same size if students raise the question.

## Opening

# Ratio Tables and Graphs

Ms. Lopez sells muffins at her bakery. She sells them in packages.

She uses a ratio table to determine the number of packages she will need for different numbers of muffins. The table she uses shows equivalent ratios of number of packages to number of muffins.

She can also use a graph to show this information.

Complete the ratio table with your class.

Using the graph on the handout, graph paper, or a graphing tool, plot all the points in the ratio table on a graph.

Examine the graph. How is the information in the ratio table represented in the graph?

HANDOUT: Ratio Tables and Graphs

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will explore the relationship between ratio tables and graphs and use them to solve problems.

## Opening

Explore the relationship between ratio tables and graphs and use them to solve problems.

# Biscuits

# Lesson Guide

Have students work in pairs on the problem.

SWD: Students with disabilities may have difficulty graphing information accurately. Provide direct instruction on graphing skills and then allow students to complete the graphs during guided practice.

# Mathematical Practices

**Mathematical Practice 7: Look for and make use of structure.**

In making connections between the tables and graphs, students will be working with the additive and multiplicative structure of linear relationships.

**Mathematical Practice 8: Look for and express regularity in repeated reasoning.**

The regularity in the relationships between values often becomes clearer when students move between tables and graphs.

# Interventions

**Student looks for a relationship between values within a row.**

- What do you notice about the relationship between values in each column?
- Focus on the ratio between values in each column, and then compare this ratio across the columns.
- Are the ratios between the two values in each column equivalent?

**Student thinks ratios in a table are not equivalent.**

- [Other pair of students] said they are equivalent. How can we figure out who is right?
- What strategies have you used in other lessons to compare two ratios?
- Does it make sense that the ratios would be different in this situation?

**Student doesn’t adequately relate the ratio table to the graph.**

- How can you see the ratio on the graph? How can you see the ratio in the table?
- Look at the coordinates of the points on the graph. Where did those coordinates come from?

# Answers

- In 39 packages, there will be 936 biscuits. 720 + 144 + 72 = 936

## Work Time

# Biscuits

Ms. Lopez also sells biscuits in packages.

She uses this ratio table to determine the number of packages she will need for different numbers of biscuits.

- Complete the ratio table.
- Make a graph that shows the information in the ratio table.
- Use the ratio table to find the number of biscuits that will fill 39 packages.

Ask yourself:

- The second column of the ratio table (with a 1 in the top cell) tells you the number of biscuits in 1 package.
- Each pair of numbers in the columns of the ratio table represents an
*ordered pair*that defines the coordinates of a point on the graph. - How could you add to or subtract from numbers in the table to find the number of biscuits in 39 packages?

# Granola

# Lesson Guide

Have students work in pairs on the problem.

ELL: Allow ELLs to write up parts of their answers. It can be hard for ELLs to explain the whole problem, but they can either draw what they mean or show a graph or a table. This will help them prepare for the Ways of Thinking section of the lesson.

# Mathematical Practices

**Mathematical Practice 7: Look for and make use of structure.**

In making connections between the tables and graphs, students will be working with the additive and multiplicative structure of linear relationships.

**Mathematical Practice 8: Look for and express regularity in repeated reasoning.**

The regularity in the relationships between values often becomes clearer when students move between tables and graphs.

# Interventions

**Student looks for a relationship between values within a row.**

- What do you notice about the relationship between values in each column?
- Focus on the ratio between values in each column, and then compare this ratio across the columns.
- Are the ratios between the two values in each column equivalent?

**Student thinks ratios in a table are not equivalent.**

- [Other pair of students] said they are equivalent. How can we figure out who is right?
- What strategies have you used in other lessons to compare two ratios?
- Does it make sense that the ratios would be different in this situation?

**Student doesn’t adequately relate the ratio table to the graph.**

- How can you see the ratio on the graph? How can you see the ratio in the table?
- Look at the coordinates of the points on the graph. Where did those coordinates come from?

# Answers

- Mr. Lee’s granola has more almonds. Explanations will vary, but should include a comparison of the ratios shown in each table.
- Explanations will vary. Possible answer: The graph of Mr. Lee’s ratio is steeper than Ms. Lopez’s ratio. The greater the ratio, the steeper the graph will be.

## Work Time

# Granola

Ms. Lopez makes granola in her bakery. Her friend, Mr. Lee, has a bakery on the other side of town, and he makes granola, too.

Both Ms. Lopez and Mr. Lee use ratio tables to show equivalent ratios of almonds to oats in their granola.

- Use the ratio tables to determine whose granola has a higher ratio of almonds to oats. Explain your answer.
- Using a piece of graph paper or a graphing tool, make a graph of the information from each granola ratio table using the same coordinate grid.
- Use the two graphs to justify your answer as to whose granola has the higher ratio of almonds to oats.

Hint:

- To compare the amounts of oats and almonds in the two types of granola, find a row from each table that has either the same number for oats or the same number for almonds.
- The steepness of each graph—called the
*slope*—gives you a visual way to compare the ratios of almonds to oats for the two types of granola.

# Prepare a Presentation

# Preparing for Ways of Thinking

Listen and look for the following student work to highlight during the Ways of Thinking discussion:

- Students who understand the connection between the ratio table and the graph
- Students who disagree—and discuss to resolve it—about which granola has more almonds and about how to interpret the tables
- Students who attend to the role of the ratio in determining the points of the graph

ELL: When eliciting answers, be cognizant of the difficulties some ELLs encounter when they have to express themselves in a foreign language. Allow them to use graphs or drawings to show what they mean at all times.

# Challenge Problem

## Possible Answers

- Yes, Anna is correct. Explanations will vary. In comparing the graphs in this context, the graph that is “lower” (i.e., has greater horizontal change per unit of vertical change) shows a greater ratio of oats to almonds.

## Work Time

# Prepare a Presentation

Be prepared to show:

How you completed the ratio table of packages to biscuits

How you used the information in the table to make a graph

How you found the number of biscuits in 39 packages

Also be prepared to demonstrate and explain:

How you used the granola ratio tables to determine whose granola has a higher ratio of almonds to oats

How you graphed the information from the two tables

How the graphs provide support for your answer as to whose granola has the higher ratio of almonds to oats

# Challenge Problem

Anna graphed the ratios of cups of almonds to cups of oats, using the *x*-axis to represent cups of oats and the *y*-axis to represent cups of almonds. She says that if you make this type of graph for any two types of granola mixtures, the graph of the granola that has the higher ratio of almonds to oats will always be steeper.

Is she right? Explain and give examples.

# Make Connections

# Mathematics

Begin the Ways of Thinking discussion with presentations of at least two different ways to use the ratio table to find the number of biscuits that will fill 39 packages. If no student addresses it, invite students to think about what can be added together from the table in order to solve the problem.

Then focus on different strategies for solving the granola problem. Call on students to get a range of approaches. For example:

- Students who compare the ratios where the amount of almonds is the same (3 cups)

1 + 2 = 3 cups almonds

3 + 6 = 9 cups of oats

Ratio of almonds to oats is 3:9.

Ratio of almonds to oats is 3:5.

There is the same amount of almonds. Ms. Lopez has more oats, so Mr. Lee’s granola is “almondier.”

- Students who compare the ratios where the amount of oats is the same (15 cups)

Ratio of almonds to oats is 5:15.

Ratio of almonds to oats is 9:15.

There is the same amount of oats. Mr. Lee has more almonds.

Students who compare the ratios where the total amount of cups is the same (8 cups: the 2:6 ratio in Ms. Lopez’s table has 8 cups total and the 3:5 ratio in Mr. Lee’s table has 8 cups total).

8 cups of granola with a almond to oat ratio of 2:6

8 cups of granola with a almond to oat ratio of 3:5

So, Ms. Lopez’s granola has more oats and Mr. Lee’s granola has more almonds.

Close the discussion with a presentation from at least one pair of students who attempted the Challenge Problem. If students are very clear on interpreting the graphs, ask how their interpretation would change if the quantities on the axes switched positions (almonds on the horizontal axis and oats on the vertical axis). In comparing the graphs in this context, the graph that is “lower” (i.e., has greater horizontal change per unit of vertical change) shows a greater ratio of almonds to oats.

# Mathematical Practices

**Mathematical Practice 8: Look for and express regularity in repeated reasoning.**

As students present their work, you may want to point out that any time they are creating or filling in a ratio table they are comparing ratios to see if they are equivalent. Prompt students to express this regularity in repeated reasoning themselves by asking how they know the values in their table are correct and how they know the points on the graph are correct.

**Mathematical Practice 7: Look for and make use of structure.**

Ask students to highlight the additive structure implicit in the tables, where appropriate, particularly in finding the number of biscuits that will fill 39 packages, and to connect this structure to the graphs as students share their work on the Challenge Problem.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about your classmates' work with ratio tables and graphs.

As your classmates present, ask questions such as:

- Can you say more about how the ratio table relates to the corresponding graph? For example, can you show how an increase in the number of packages would be reflected in the table and on the graph?
- How could you use the graph you made from the ratio table of biscuits to packages to find the number of biscuits in 39 packages? Compare this to how you would use the ratio table.
- How do the graphs of the two granola mixtures tell you which mixture has the higher ratio of almonds to oats?

# Representing Ratios

# Mathematics

Have pairs quietly discuss the relationship between the ratio table, the double number line, and the graph, and explain how each shows the ratio of cups of milk to cups of flour. As student pairs work together, listen for students who may still have misconceptions so you can address them in the class discussion. After a few minutes, discuss the Summary as a class. Talk about how all three models represent the same relationship between the quantities.

## Formative Assessment

# Summary of the Math: Representing Ratios

**Read and Discuss**

- Suppose the ratio of cups of milk to cups of flour in a recipe is 3:2.
- You can show this ratio with a ratio table, a double number line, or a graph.
- For each pair of values in a ratio table, you can place a corresponding lined-up pair of numbers on a double number line and make a graph by plotting a corresponding point (using an ordered pair) on a coordinate grid.

Can you:

- Explain what a ratio table is?
- Describe how to make a graph that shows the information from a ratio table?
- Describe how to make a graph that shows the information from a double number line?

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of class. Review the reflections to find out what about ratios still confuses students.

## Work Time

# Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

**One thing that still confuses me about ratios is …**