This lesson develops students’ ability to work with a table of equivalent ratios and to compare and contrast different ways of solving equivalent ratio problems.

Students see that a table accommodates different ways of reasoning about equivalent ratios, with some being more direct than others. They notice that to find an unknown quantity, they can:

• Find the multiplier that relates two corresponding values in different rows (for instance, “What times 5 equals 8?”) and use that multiplier to find unknown values. (This follows the multiplicative thinking developed in previous lessons.)
• Find an equivalent ratio with one quantity having a value of 1 and use that ratio to find missing values.

All tasks in the lesson aim to strengthen students’ understanding of the multiplicative relationships between equivalent ratios—that given a ratio , an equivalent ratio may be found by multiplying both and by the same factor. They also aim to build students’ awareness of how a table can facilitate this reasoning to varying degrees of efficiency, depending on one’s approach.

Ultimately, the goal of this unit is to prepare students to make sense of situations involving equivalent ratios and solve problems flexibly and strategically, rather than to rely on a procedure (such as “set up a proportion and cross multiply”) without an understanding of the underlying mathematics.

To reason using ratios in which one of the quantities is 1, students are likely to use division. In the example here, they are likely to divide the 90 by 5 to find the amount earned per hour. Remind students that dividing by a whole number is the same as multiplying by its reciprocal (a unit fraction) and encourage the use of multiplication (as shown in the activity about hourly wages) whenever possible. Doing so will better prepare students to scale down or find equivalent ratios involving values that are smaller than the given ones. 

The optional activity presents a situation where ratios are considerably scaled down, highlighting one limitation of double number lines. 

As they relate the values in a table to the quantities in the situation being represented, students practice reasoning quantitatively and abstractly (MP2).

During the Lesson Synthesis, display the table showing Lin’s earnings for 5, 1, and 8 hours of work, as well as a double number line diagram showing the same quantities.

Invite students to share ways that the representations are alike and different. Highlight the distinctions in terms of distances between the numbers, the order of the numbers, and the vertical or horizontal orientations of the representations. 

Ask students to discuss the benefits and drawbacks of each representation. If not mentioned by students, highlight that tables can be created fairly quickly and offer flexibility in what numbers to use to help solve a problem. Double number line diagrams may be more involved to draw, may need to be pretty long to accommodate the numbers, or may require careful thinking about the scale being used. The numbers need to be listed in order of size—from smaller to larger.