In this activity, students see that when two ratios are equivalent, they have the same unit rate (if is equivalent to , then ).

Students explore this idea by analyzing the values in a table that represent two quantities in a ratio. They notice that in addition to the values in the rows being equivalent ratios, the values in the columns have a multiplicative relationship. Students learn that a unit rate is a factor that relates the values in one column to those in another column. They also recognize that this unit rate can be used to reason about one quantity of the ratio when the other is known.

In investigating the relationships in the table, students practice looking for and making use of structure (MP7). As they use variables to generalize the pattern seen in numerical examples, they also practice expressing regularity in repeated reasoning (MP8).

As students work and discuss, identify those who observed and can describe the structure in the table. Select them to share during discussion later.

Focus the discussion on the relationships that students noticed in the tables.

Display a completed version of the first table for all to see. Select previously identified students to share their observations. As they explain, illustrate their comments on the table. Students may bring up that:

• The ratios shown in the first two columns are equivalent across all rows.
• The ratios in all rows have the same unit price.
• When the number of binders doubles from 2 to 4 (or from 5 to 10), the cost also doubles, but the unit price is still 7 dollars.
• Multiplying the number of binders by the unit price gives the cost.
• Dividing the cost in dollars by the number of binders gives the unit price.

Highlight the first three observations, or bring them up if students do not mention them. (The last two observations can be emphasized when discussing the second table about notebooks.)

Next, invite selected students to share how they completed the second table. To highlight that the unit price relates the values in the columns by multiplication, ask questions such as:

• “How are the values in the first two columns related to each other?” (The values in the second column are 3 times those in the first column. The value in the first column is of that in the second column.)
• “How is the 3 in the last column related to the values in the first two columns?” (It’s the number that relates the two columns. Multiplying the number of notebooks by 3 gives the cost in dollars. Dividing the cost in dollars by 3 gives the number of notebooks.)
• “What is the cost for buying notebooks?” ( dollars) “Why does it make sense to express that cost as ?” (No matter how many notebooks are bought, the cost is 3 times that number.)

This activity further develops students' understanding that equivalent ratios have the same unit rate. Students practice finding an unknown value in a ratio given the other value and a unit rate, reinforcing their understanding of the relationships among the values. Then, they work to generalize this relationship using variables, by reasoning about whether the ratios such and , or and , have the same unit rate.

As students work, monitor for different approaches taken. Some students may be inclined to create a different table—such as the one shown here—as an intermediate step for completing the given table. 

Though appropriate, this intermediate strategy is less efficient than directly dividing or multiplying the value of one column by the unit rate to get the value of the other column. 

Also monitor for students who can explain how they know the per-item cost is the same given two ratios with one or more variables.

Select a few students to share with the class their strategies for completing the first table. If any students created an additional row to show 1 hour and another row to show 1 bracelet, invite them to share first. Progress toward strategies that use the given unit rate to navigate from column to column in efficient ways, such as multiplying the time in hours by 6 to find the number of bracelets, and multiplying the number of bracelets by (or dividing it by 6) to find time in hours.

Then focus the discussion on how students know whether Lin and Noah paid the same unit price. Display the table for all to see and annotate it with students’ responses and reasoning. Highlight explanations that make clear that and are equivalent fractions or have the same value.

To help students further generalize this relationship, direct students’ attention to the last problem. Tell students: “Mai bought bracelets and paid dollars. Diego bought 3 times as many bracelets as Mai did and paid 3 times what Mai paid.” Give students 1–2 minutes to complete the table. Then discuss:

• “What is the unit price Mai paid? How do you know?” ( dollars. Dividing the amount paid by the number of items gives the unit price.)
• “How many bracelets did Diego buy and for how much?” ( bracelets for )
• “Did Mai and Diego pay the same unit price? Why or why not?” (Yes. Diego paid dollars, which is equivalent to .)

This activity is a chance to apply newly-learned techniques to answer questions in another situation involving equivalent ratios. None of the given numbers are multiples of each other from row to row (for example, 7 isn't a multiple of 4), which encourages students to reason about unit rates. The activity also prompts students to generalize how to use unit rates to find an unknown value in equivalent ratios and to describe their understanding in words.

Focus the discussion on the questions that prompted students to generalize their reasoning. Invite students to share how they could compute the number of cups of applesauce given the number of pounds of apples, and vice versa. Highlight explanations that incorporate unit rates and multiplicative reasoning across columns.