In this activity, students analyze the workouts of several people on a treadmill given time-distance ratios, and they work toward more efficient ways to compare speeds, namely, by computing rates per 1. Students see that when such ratios can be expressed with the same number of meters per minute (or per hour), the ratios are equivalent and the moving objects (such as people or cars) have the same speed.

Speed is typically expressed as a distance per 1 unit of time, so the context lends itself for comparing rates per 1. The numbers have been chosen such that any two workouts being compared have the same time, same distance, or same speed.

Encourage students to use “per 1” and “for each” language throughout, as this language supports the development of the concept of unit rate.

As students discuss the problems, listen closely for those who use these terms as well as descriptions of speed (such as “same speed,” “faster,” or “slower”). Also monitor for students who first make the connections between the rates per 1 that they calculated in the first half of the task and then use them to answer questions in the second half. Invite some of these students or groups to share later.

If students are not sure how to begin, suggest that they try using a table or a double number line diagram that associates meters and minutes.

Focus the conversation on the speed of each runner and the idea of “same speed,” including clues that two objects are moving equally fast or slow. Ask questions such as:

• “How can you tell when runners are going at the same speed?” (They keep up with one another while moving. They run the same distance in the same amount of time.) 
• “How can you tell when two cars are moving at the same speed?” (They move at the same miles per hour or kilometers per hour.)

Invite a few students to share their analyses of how the runners compare, starting with how Tyler's workout compares to Kiran's, and how Kiran's compares to Mai's. Descriptions such as “slower,” “faster,” or “higher or lower speed” should begin to emerge. 

After students share their analyses of Mai's and Tyler's workouts, emphasize that even though they ran different distances in different amounts of time, they each ran 140 meters per minute (or 8,400 meters per hour), so we can say “they ran at the same speed.” This also means that Mai and Tyler's original ratios— and —are equivalent ratios. 

In the last problem, students need to understand that since Mai and Tyler ran at the same speed they traveled the same distance for the first 30 minutes on the treadmill. This may be difficult for students to articulate with precision, so allowing multiple students to share their thinking may be beneficial.

In this activity, students compare rates per 1 in a shopping context as they look for “the best deal.” The work reminds students how unit price works and encourages them to look for efficient ways to compare unit prices.

The phrase “the best deal” may have different meanings to students and should be discussed. For instance, students may account for various factors besides price, such as distance to store, transportation costs, store preference, loyalty points, and range of item selections and sizes in a store. Discussing these real-life considerations, and choosing which to prioritize and which to disregard, is an important part of modeling with mathematics (MP4). For the purposes of this activity, however, it is also appropriate to clarify that we are looking for “the best deal” in the sense of the lowest cost per can.

As students work, monitor for students who use representations like double number line diagram or tables of equivalent ratios. These are useful for making sense of a strategy that divides the price by the number of cans to find the price per 1. Also monitor for students using more efficient strategies, such as dividing price by the number of cans or the number of ounces.

The purpose of this discussion is to help students see that computing and comparing the price per 1 is an efficient way to compare rates in a price context.

Select students who supported their reasoning with a double number line diagram or table to share their responses first. Keep these representations visible and make connections with subsequent explanations from students who used more efficient strategies. Highlight the use of division to compute the price per can and the use of “per 1” language.