Lesson 8

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.

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.

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.

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.

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.

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:

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.

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.

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.

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.