In this activity, students work with pairs of ratios involving distance run and elapsed time to determine which of two runners ran faster. In making comparisons, students interpret objects with equivalent distance-time ratios as having the same speed or moving “at the same rate.” 

In the first question, students compare two distance-time ratios in which one quantity (distance) has the same value and the other quantity (time) has different values. In the second question, students compare distance-time ratios in which neither quantities have the same value but the values in one ratio are multiples of those in the other. 

There are several ways to reason about each situation, with or without using diagrams, as shown in the sample student responses. 
To reason about the second questions, students may:

• Draw number line diagrams to see if Han and Tyler’s distance-to-time ratios are equivalent.
• Multiply the values in one ratio by the same number to see if they obtain the values in the other ratio. 
• Find the distance for 1 unit of time in each person’s run and compare them. To make this approach less attractive, the numbers have been deliberately chosen so Han and Tyler’s running time per kilometer ( minutes) is not a whole number.

In the digital version of the activity, students can choose to use an applet to create double number lines to represent given quantities, explore the problems, and solidify their thinking. The applet is similar to those in earlier lessons.

Invite students to share how they determined if Han and Tyler ran at the same rate. Draw students’ attention to the ways in which equivalent ratios are used to make one of the corresponding quantities—either the distance or the elapsed time—the same in both situations. For instance, we can find how far Han would run in 30 minutes and compare it with Tyler’s 30-minute run, or how far Tyler would run in 15 minutes and compare it with Han’s 15-minute run. Emphasize that doing so can help us see if the situations involve the same rate.

In this activity, students encounter the idea of “same rate” in a cost context. Students have previously reasoned about prices of different numbers of items. They have interpreted the phrase “at this rate” in terms of equivalent ratios, building their intuition for what it means to “pay at the same rate.” This activity reinforces the idea that paying at the same rate means paying the same unit price. 

As in the previous activity, students compare ratios in which neither quantity in one ratio has the same value as in the other, but the values in one ratio are multiples of those in the other. 

Monitor for different ways in which students reason about whether the ratios are equivalent. Here are some strategies they may use, from less efficient to more efficient:

• Find the dollar value for 1 ticket in each purchase and compare them. To make this approach less attractive, the numbers have been deliberately chosen so that the cost of a ticket (\$) is not a whole number.
• Draw number line diagrams.
• Multiply the values in one ratio by the same number to see if they obtain the values in the other ratio. 

In the digital version of the activity, students can choose to use an applet to create double number lines to represent given quantities, explore the problems, and solidify their thinking. The applet is similar to those in earlier lessons.

Invite previously selected students to share their reasoning. Sequence the discussion of strategies in the order listed in the Activity Narrative.

Connect the different responses to the learning goals by asking questions such as:

• “What does it mean for Andre and Diego to be paying at the same rate?” (The ratios of dollars paid to tickets bought are equivalent.)
• “What are some ways of using equivalent ratios to see if Diego and Andre are paying at the same rate?” (One way is to find the price for 9 tickets that is equivalent to Diego’s 49 to 3 dollars-to-tickets ratio, and then compare it with Andre’s ratio. Another way is to find the price for 3 tickets that is equivalent to Andre’s 147 to 9 dollars-to-tickets ratio, and then compare it with Diego’s ratio.) 
• “When comparing two ratios, why might it be helpful to make one quantity in the two situations the same?” (It would make it easier to see if the ratios are equivalent. If the two situations were happening at the same rate, when one quantity in both situations is the same, the other quantity also has to be the same.)

Emphasize that making one of the corresponding quantities the same—either the number of tickets or the cost in this case—allows us to more easily compare the other quantity and tell whether the situations involve the same rate.

In this activity, students compare the tastes of two sparkling orange juice mixtures, which involves reasoning about whether the two situations involve equivalent ratios. The problem is more challenging because no values of the quantities match or are multiples of one another. Instead of finding an equivalent ratio for one recipe so that it matches the other, students need to do so for both recipes.

To answer the question, students can either make the values of the soda water match and then compare the orange juice amounts, or make the orange juice amounts match and compare the values for soda water. Again, they may use a double number line, multiplication, and possibly finding how much of one quantity per 1 unit of the other quantity. 

Display two double number line diagrams for all to see: one that represents batches of Lin’s recipe and another that represents batches of Noah’s recipe. Extend the number lines to show enough batches of each recipe to be able to make comparisons.

Ask students to explain how they can tell that the recipe has a stronger orange flavor. Elicit both explanations: comparing the amount of soda water for the same amount of orange juice, and the other way around. Ensure that students can articulate why each way of comparing means that the second recipe has a stronger orange flavor.

If any students calculated a unit rate for each recipe, consider inviting them to share, but support them with their choice of words. It is important to say, for example, “In Lin’s recipe, there is or 0.75 cup of orange juice per cup of soda water, but in Noah’s recipe, there is or 0.8 cup of orange juice per cup of soda water.” A student with this response would be comparing the number of cups of orange juice for every 1 cup of soda water in each mixture.

Ask students: “Did Lin and Noah mix orange juice and soda water at the same rate? Why or why not?” Highlight that the ingredients were not mixed at the same rate because the ratios of orange juice to soda water in the two recipes are not equivalent. As a result, the mixtures don’t taste the same.