Lesson 6

This warm-up activates students’ prior knowledge around “something per something” language. It gives them a chance to both recall and hear examples and contexts in which such language was used, either in past lessons or outside of the classroom, in preparation for the work ahead.

Arrange students in groups of 3–4. Give students a minute of quiet think time to complete the first question, and then 2 minutes to share their ideas in groups and compile a list. Consider asking one or two volunteers to share an example or sharing one of your own. Challenge students to come up with something that is not an example of either unit price or speed, since these have already been studied.

If students are stuck, encourage them to think back to past lessons and see if they could remember any class activities in which the language of “per” was used or could be used.

Ask each group to share 1–2 of their examples and record unique responses for all to see.

After each group has shared, select one response (or more than one if time allows) that is familiar to students. For example, if one of the groups proposed 30 miles per hour, ask “What are some things we know for sure about an object moving 30 miles per hour?” (The object is traveling a distance of 30 miles every 1 hour.)

It is not necessary to emphasize “per 1” language at this point. The following activities in the lesson focus on the usefulness of “per 1” in the contexts of comparing multiple ratios.

In this activity, students explore two unit rates associated with the ratio, think about their meanings, and use both to solve problems. The goals are to:

• Help students see that for every context that can be represented with a ratio \(a:b\) and an associated unit rate \(\frac{b}{a}\), there is another unit rate \(\frac{a}{b}\) that also has meaning and purpose within the context.
• Encourage students to choose a unit rate flexibly depending on the question at hand. Students begin by reasoning whether the two rates per 1 (cups of oats per 1 cup of water, or cups of water per 1 cup of oats) accurately convey a given oatmeal recipe. As students work and discuss, notice those who use different representations (a table or a double number line diagram) or different arguments to make their case (MP3). Once students conclude that both Priya and Han's rates are valid, they use the rates to determine unknown amounts of oats or water.

Some students may not be familiar with oatmeal; others may only have experience making instant oatmeal, which comes in pre-measured packets. Explain that oatmeal is made by mixing a specific ratio of oats to boiling water.

Arrange students in groups of 2. Give students 2–3 minutes of quiet think time for the first question. Ask them to pause and share their response with their partner afterwards. Encourage partners to reach a consensus and to be prepared to justify their thinking. See MLR 3 (Clarify, Critique, Correct).

After partners have conferred, select several students to explain their reasoning and display their work for all to see. When the class is convinced that both Priya and Han are correct, ask students to complete the rest of the activity.

Some students may think that Priya and Han cannot both be right because they came up with different numbers. Ask them to explain what each number means, so that they have a chance to notice that the numbers mean different things. Point out that the positioning of the number 1 appears in different columns within the table. 

Focus the discussion on students’ responses to the last question and how they knew which rate to use to solve for unknown amounts of oats and water. If not uncovered in students’ explanations, highlight that when the amount of oats is known but the amount of water is not, it helps to use the “per 1 cup of oats” rate; a simple multiplication will tell us the missing quantity. Conversely, if the amount of water is known, it helps to use the “per 1 cup of water” rate. Since tables of equivalent ratios are familiar, use the completed table to support reasoning about how to use particular numbers to solve particular problems.

Consider connecting this idea to students’ previous work. For example, when finding out how much time it would take to wash all the windows on the Burj Khalifa, it was simpler to use the “minutes per window” rate than the other way around, since the number of windows is known.

Leave the table for this activity displayed and to serve as a reference in the next activity.

In this task, students calculate and interpret both \(\frac{a}{b}\) and \(\frac{b}{a}\) from a ratio \(a:b\) presented in a context. They work with less-familiar units. The term unit rate is introduced so that students have a general name for a “how many per 1” quantity.

In the first half of the task, students practice computing unit rates from ratios. In the second half they practice selecting the better unit rate to use (\(\frac{a}{b}\) or \(\frac{b}{a}\)) based on the question posed.

As students work on the second half of the task, identify 1–2 students per question to share their choice of unit rate and how it was used to answer the question.

Recap that in the previous activity the ratio of 15 cups water for every 6 cups oats can be expressed as two rates “per 1”. These rates are 0.4 cups of oats per cup of water or 2.5 cups of water per cup of oats. Emphasize that, in a table, each of these rates reflects a value paired with a “1” in a row, and that both can be useful depending on the problem at hand. Tell students that we call 0.4 and 2.5 “unit rates” and that a unit rate means “the amount of one quantity for 1 of another quantity.”

Arrange students in groups of 2. Tell students that they will now solve some problems using unit rates. Give students 3–4 minutes to complete the first half of the task (the first three problems). Ask them to share their responses with their partner and come to an agreement before moving on to the second half. Clarify that “oz” is an abbreviation for “ounce.”

If students are not sure how to use the unit rates they found for each situation to answer the second half of the task, remind them of how the oatmeal problem was solved. Suggest that this problem is similar because they can scale up from a unit rate to answer the questions.

Invite previously identified students to share their work on the second half (the last three questions) of the task.

Though the task prompts students to think in terms of unit rate, some students may still reason in ways that feel safer. For example, to find out how much cream cheese Lin would mix with 35 oz of sugar, they may double the 12 oz of cream cheese to 15 oz of sugar ratio to obtain 24 oz of cream cheese for 30 oz of sugar, and then add 4 oz more of cream cheese for the additional 5 oz of sugar. Such lines of reasoning show depth of understanding and should be celebrated. Guide students to also see, however, that some problems (such as the milk problem) can be more efficiently solved using unit rates.

For the ticket problem, students may comment that \(\frac14\) of a ticket costing a dollar does not make sense, since it is not possible to purchase \(\frac14\) of a ticket. Take this opportunity to applaud the student(s) for reasoning about the interpretation of the number in the context, which is an example of engaging in MP2. If students do not raise this concern, ask: “How can \(\frac14\) of a ticket costing a dollar make sense?” Students may argue that the quantity, on its own, does not make sense. Challenge them to figure out how the rate could be used in the context of the problem. For example, ask, “If I had $80, how many tickets could I buy? What if I had $75? Can the ‘\(\frac14\) of a ticket per dollar’ rate help answer these questions?”