# Lesson 11

Representing Ratios with Tables

## 11.1: How Is It Growing? (10 minutes)

### Warm-up

This warm-up encourages students to look for regularity in how the tiles in the image are growing. Students may use each color to reason about the total, while others may reason about the way the total tiles increase each time. Emphasize both insights as students share their strategies.

### Launch

Arrange students in groups of 2. Display the image for all to see and tell students that the collection of images of green and blue tiles is growing. Ask how many total tiles will be in the 4th, 5th and 10th image if it keeps growing in the same way. Tell students to give a signal when they have an answer and strategy. Give students 3 minutes of quiet think time, and then time to discuss their responses and reasoning with their partner.

### Student Facing

Look for a pattern in the figures.

1. How many total tiles will be in:

1. the 4th figure?
2. the 5th figure?
3. the 10th figure?
2. How do you see it growing?

### Activity Synthesis

Invite students to share their responses and reasoning. Record and display the different ways of thinking for all to see. If possible, record the relevant reasoning on or near the images themselves. After each explanation, ask the class if they agree or disagree and to explain alternative ways of thinking, referring back to what is happening in the images each time.

## 11.2: A Huge Amount of Sparkling Orange Juice (15 minutes)

### Activity

Here, students are asked to find missing values for significantly scaled-up ratios. The activity serves several purposes:

• To uncover a limitation of a double number line (i.e., that it is not always practical to extend it to find significantly scaled-up equivalent ratios),
• To reinforce the multiplicative reasoning needed to find equivalent ratios (especially in cases when drawing diagrams or skip counting is inefficient), and
• To introduce a table as a way to represent equivalent ratios.

To find equivalent ratios involving large values, some students may simply try to squeeze numbers on the extreme right side of the paper, ignoring the previously equal intervals. Others may use multiplication (or division) and write expressions or equations to capture the given scenarios. Notice students’ reasoning processes, especially any struggles with the double number line (e.g., the lines not being long enough, requiring much marking and writing, the numbers being too large, etc.), as these can motivate a need for a more efficient strategy.

### Launch

Give students 2–3 minutes to work on the first two questions and then ask them to pause. As a class, discuss the two approaches students are likely to take: counting multiples of 4 and 5 up to 36 and 45; and multiplicative reasoning (asking “What number times 4 equals 36?”). Also discuss how a double number line like the one below might be used to support reasoning.

Reiterate the multiplicative relationship between equivalent ratios before students move on.

Representation: Internalize Comprehension. Activate or supply background knowledge about equivalent ratios describing situations that happen at the same rate. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing

### Student Facing

Noah’s recipe for one batch of sparkling orange juice uses 4 liters of orange juice and 5 liters of soda water.

1. Use the double number line to show how many liters of each ingredient to use for different-sized batches of sparkling orange juice.

2. If someone mixes 36 liters of orange juice and 45 liters of soda water, how many batches would they make?
3. If someone uses 400 liters of orange juice, how much soda water would they need?
4. If someone uses 455 liters of soda water, how much orange juice would they need?
5. Explain the trouble with using a double number line diagram to answer the last two questions.

### Launch

Give students 2–3 minutes to work on the first two questions and then ask them to pause. As a class, discuss the two approaches students are likely to take: counting multiples of 4 and 5 up to 36 and 45; and multiplicative reasoning (asking “What number times 4 equals 36?”). Also discuss how a double number line like the one below might be used to support reasoning.

Reiterate the multiplicative relationship between equivalent ratios before students move on.

Representation: Internalize Comprehension. Activate or supply background knowledge about equivalent ratios describing situations that happen at the same rate. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing

### Student Facing

Noah’s recipe for one batch of sparkling orange juice uses 4 liters of orange juice and 5 liters of soda water.

1. Use the double number line to show how many liters of each ingredient to use for different-sized batches of sparkling orange juice. 2. If someone mixes 36 liters of orange juice and 45 liters of soda water, how many batches would they make?
3. If someone uses 400 liters of orange juice, how much soda water would they need?
4. If someone uses 455 liters of soda water, how much orange juice would they need?
5. Explain the trouble with using a double number line diagram to answer the last two questions.

### Anticipated Misconceptions

Students may become frustrated when they “run out of number line,” but remind them of what they know about how to find ratios equivalent to $$4:5$$ (they need to multiply both 4 and 5 by the same number). Consider directing their attention to a definition of equivalent ratios displayed in your room or in a previous lesson, or suggesting they reexamine some of the simpler cases (e.g., the relationship between $$4:5$$ and $$36:45$$). Be on the lookout for students trying to tape on more paper to extend their number lines.

### Activity Synthesis

After students have a chance to share with a partner, select a few to share their reasoning with the class for the last few questions. Start with students who tried to extend the double number line (if anyone did so). Discuss any challenges of using the double number line and merits of alternative methods students might have come up with.

Explain that there is a more appropriate tool—a table—that can be used to represent equivalent ratios. Display for all to see the double number line from the activity above and a table of equivalent ratios. Explain that even though the table is oriented vertically and the double number line is oriented horizontally, the two representations represent the same ratios. Explain what we mean by row and column and demonstrate the use of these words. Fill in the table using the values from the orange-soda ratios and, along the way, compare and contrast how the two representations work. A few other key insights to convey:

• Just as it was important to label the double number line, it is important to label the columns of the table to indicate what the values represent (MP6).
• Each row of a table shows a pair of values from a collection of equivalent ratios. Unlike a number line, distances between values do not matter.
• On each line of a double number line, numbers are shown in order. In each column of a table, order is not important, i.e., pairs of values can be placed in any order that is convenient. When complete, the display should look something like this:
Representing: MLR7 Compare and Connect. Use this routine to help students make connections between specific features of tables and double number lines. Ask students to describe to a partner how multiplication appears in each representation, and then invite listeners to restate or revoice what they heard, back to their partner, using mathematical language (e.g., product, row, column, table, equivalent ratio, etc.). After students have a chance to share with a partner, select a few to share their reasoning with the class.
Design Principle(s): Maximize meta-awareness

## 11.3: Batches of Trail Mix (10 minutes)

### Activity

This task gets students to interact with a table in a way that discourages skip counting. Numbers within each column are deliberately out of order. This is intended to encourage students to multiply the pairs of values from a given ratio by the same number and to emphasize that the order in which pairs of values appear is not a necessary part of the structure of a table. (Order within rows, however, is necessary.) The last question reinforces the definition of equivalent ratios.

Students may use the given values (7 and 5) as the basis for every calculation (e.g., for every row, they think “7 times what . . . ” or “5 times what . . .”). They may also reason with values from another row (e.g., they may see 250 as $$10\boldcdot 25$$ rather than as $$5\boldcdot 50$$). As students work, notice different approaches.

### Launch

Explain that a table is just a list of equivalent ratios. In this case, one column contains amounts of almonds, and the other column contains corresponding amounts of raisins. Each row shows the amount of each ingredients in a particular batch.

Reiterate that multiplying both parts of a ratio by the same non-zero number always creates a ratio that is equivalent to the original ratio.

### Student Facing

A recipe for trail mix says: “Mix 7 ounces of almonds with 5 ounces of raisins.” Here is a table that has been started to show how many ounces of almonds and raisins would be in different-sized batches of this trail mix.

almonds (oz) raisins (oz)
7 5
28
10
3.5
250
56
1. Complete the table so that ratios represented by each row are equivalent.

2. What methods did you use to fill in the table?
3. How do you know that each row shows a ratio that is equivalent
to $$7:5$$? Explain your reasoning.

### Student Facing

#### Are you ready for more?

You have created a best-selling recipe for chocolate chip cookies. The ratio of sugar to flour is $$2:5$$.

Create a table in which each entry represents amounts of sugar and flour that might be used at the same time in your recipe.

• One entry should have amounts where you have fewer than 25 cups of flour.
• One entry should have amounts where you have between 20–30 cups of sugar.
• One entry can have any amounts using more than 500 units of flour.

### Anticipated Misconceptions

Students may make patterns that do not yield equivalent ratios. For example, they may think “7 minus 2 is 5, so for the next row, 28 minus 2 is 26.” Or they may think “7 plus 21 is 28, so then 5 plus 21 is 26.” If so, consider:

• Appealing to what students know about batches of recipes. “The second row represents how many batches of trail mix?” (4, because 28 is $$7\boldcdot 4$$.) “Okay, so to make 4 batches of trail mix, how will we figure out how many raisins?” (Also multiply the 5 by 4.)
• Refreshing what students learned about equivalent ratios. “We need a ratio that is equivalent to the ratio represented in row 1. So what do we need to do to the 7 and the 5?” (Multiply them by the same number.)

Students may be unsure about how to find the missing value in the row with 3.5. Encourage them to reason about it the same way they reasoned about the other rows. “We need a ratio that is equivalent to the ratio represented in row 1. So what do we need to do to the 7 and the 5?” They may have to get there by way of division. 7 divided by 2 is 3.5, so 7 times $$\frac12$$ is 3.5; this means multiplying 5 by $$\frac12$$ as well.

### Activity Synthesis

Invite one or more students who used multiplicative approaches to share their reasoning with the class. Consider displaying the table and using it to facilitate gesturing and arrow-drawing while students explain. Highlight the strategy of multiplying the 7 and 5 values by the same number.

## Lesson Synthesis

### Lesson Synthesis

Sometimes it is easier to use a table rather than a double number line to represent equivalent ratios. Each row contains a ratio that is equivalent to all the other ratios, so if we know one row, we can multiply both of its values by the same number to find another row’s values.
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