In previous lessons, students learned that if two situations involve equivalent ratios, we can say that the situations are described by the same rate. In this lesson, students compare ratios to see if two situations in familiar contexts involve the same rate. The contexts and questions are:
 Two people run different distances in the same amount of time. Do they run at the same speed?
 Two people pay different amounts for different numbers of concert tickets. Do they pay the same cost per ticket?
 Two recipes for a drink are given. Do they taste the same?
In each case, the numbers are purposely chosen so that reasoning directly with equivalent ratios is a more appealing method than calculating howmanyperone and then scaling. The reason for this is to reinforce the concept that equivalent ratios describe the same rate, before formally introducing the notion of unit rate and methods for calculating it. However, students can use any method. Regardless of their chosen approach, students need to be able to explain their reasoning (MP3) in the context of the problem.
Lesson overview
 10.1 Warmup: Treadmills (10 minutes)

10.2 Activity: Concert Tickets (10 minutes)
 There is a digital applet in this activity.

10.3 Activity: Sparkling Orange Juice (15 minutes)
 Includes "Are you Ready for More?" extension problem
 There is a digital applet in this activity.
 Lesson Synthesis
 10.4 Cooldown: Comparing Runs (5 minutes)
Learning goals:
 Choose and create diagrams to help compare two situations and explain whether they happen at the same rate.
 Justify that two situations do not happen at the same rate by finding a ratio to describe each situation where the two ratios share one value but not the other, i.e., \(a:b\) and \(a:c\), or \(x:z\) and \(y:z\).
 Recognize that a question asking whether two situations happen “at the same rate” is asking whether the ratios are equivalent.
Learning goals (student facing):
 Let’s use ratios to compare situations.
Learning targets (student facing):
 I can decide whether or not two situations are happening at the same rate.
 I can explain what it means when two situations happen at the same rate.
 I know some examples of situations where things can happen at the same rate.
Glossary:
 same rate  We use the words same rate to describe two situations that have equivalent ratios. For example, a sink is filling with water at a rate of 2 gallons per minute. If a tub is also filling with water at a rate of 2 gallons per minute, then the sink and the tub are filling at the same rate.
 Access the complete Grade 6 glossary.
IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 20172019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/mathcurriculum/.
Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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