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

• To uncover a limitation of a double number line (namely, 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)
• To introduce a table as a way to represent equivalent ratios

Monitor for the different ways students find equivalent ratios involving large values. Here are some strategies they may use, from less practical or less flexible to more accommodating:

• Extend the given double number line and then try to squeeze numbers on the extreme right end, ignoring the previously equal intervals.
• Draw a new double number line diagram that is longer or where the intervals represent multiples of 4 and 5 (rather than 4 and 5).
• Use multiplication (or division) and write expressions or equations to represent the given scenarios.

Select students who use different strategies to share later, including those who find the given double number line inadequate—not long enough to accommodate large numbers, requiring more marking or writing, and so on—and are consequently motivated to find a more efficient strategy.

After students have a chance to share with a partner, ask previously selected students to share their reasoning for the last few questions. Sequence the discussion of students’ strategies in the order listed in the Activity Narrative. Discuss any challenges of using the double number line and merits of alternative methods students might have used.

Explain that there is a more appropriate tool—a table—that can be used to represent equivalent ratios. Display for all to see a double number line representing the sparkling juice recipe 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. Complete the table using the values from the orange-soda ratios. Along the way, connect the two representations and compare and contrast how they show the quantities in the recipe.

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.
• 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, that is, pairs of values can be placed in any order that is convenient. When complete, the display should look something like this:

This activity invites 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 (for instance, for every row, they think “7 times what . . . ” or “5 times what . . .”). They may also reason with values from another row (for instance, they may see 250 as rather than as ). As students work, monitor for different approaches.

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.