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18.2

Activity

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

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 situation that can be represented with a ratio and an associated unit rate , there is another unit rate that also has meaning and purpose within the situation.
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 wheat per 1 cup of water, or cups of water per 1 cup of wheat) accurately convey a given recipe. After students conclude that both rates are valid, they use the rates to determine unknown amounts of wheat or water.

As students work and discuss, monitor for those who use different representations (a table or a double number line diagram) and those who construct logical arguments in different ways.

Launch

Tell students to close their books or devices (or to keep them closed). Display the instructions for making kinche (an Ethiopian breakfast dish made with cracked wheat and pronounced KEEN-jay) for all to see.

KINCHE

To make a large batch, bring 15 cups of water to a boil. Then add 6 cups of crushed wheat.

Give students 1 minute of quiet think time, and ask them to be prepared to share at least one thing that they notice and one that thing they wonder about.

Students are likely to notice that this is a recipe involving water and crushed wheat in a ratio of 15 to 6. If the amount or ratio of water and crushed wheat does not come up during the conversation, ask students to discuss this idea.

Students may wonder about what kinche is. Ask students if they have had oatmeal, porridge, or another kind of dish that involves mixing boiling water or milk with a kind of grain or a ground ingredient. Invite a few students to briefly share some information about the dish. (What grain does it use? Does it come in large bags or small packets? Is it served with seasonings or toppings?)

Tell students that kinche is a traditional Ethiopian breakfast dish. It is popular because it is easy to make, healthy, and can be flavored in different ways. Like oatmeal or porridge, kinche can be cooked to different consistencies using different ratios of wheat or water. Sometimes milk is also added into the mixture.

Arrange students in groups of 2. Give students 2 minutes of quiet think time for the first question. Ask them to pause and share their response with their partner afterward. Encourage partners to reach a consensus and to be prepared to justify their thinking.

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.

Engagement: Develop Effort and Persistence. Provide tools to facilitate information processing or computation, enabling students to focus on key mathematical ideas. For example, allow students to use calculators to support their reasoning.
Supports accessibility for: Memory, Conceptual Processing

Activity

None

Activity Synthesis

Focus the discussion on how students found the amount of water for 5 cups of wheat and the amount of wheat for 10 cups of water. Ask questions such as:

“To find the amount of water for 5 cups of wheat, which “per cup” rate did you use: Priya’s (0.4 cup of wheat per cup of water) or Han’s (2.5 cups of water per cup of wheat)? Why?”
“To find the amount of wheat for 10 cups of water, which “per cup” rate did you use? Why?”

If not uncovered in students’ explanations, highlight that when the amount of wheat is known but the amount of water is not, it helps to use the “per 1 cup of wheat” rate. In that situation, 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. Consider using additional rows in the given table to illustrate this line of reasoning.

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

18.3

Activity

Standards Alignment

Building On

Addressing

Instructional Routines

MLR6: Three Reads

Materials

None

Activity Narrative

In this activity, students are introduced to the term "unit rate" as a general name for describing the amount of one quantity in a ratio per 1 of the other quantity. Given a ratio , they calculate and and interpret both unit rates in context. Students also practice selecting the better unit rate to use ( or ) based on the question posed.

Though the activity prompts students to think in terms of unit rate, some students may still reason in other ways that are more intuitive to them. For example, to find out how many weeks it would take to use 3 gallons of detergent, students may:

Triple the ratio of gallons to weeks to arrive at , and then divide each 30 and 18 by 10 to get 3 gallons and 1.8 weeks.
Divide both 10 gallons and 6 weeks by 5 to get 2 gallons in 1.2 weeks, and then add half of each quantity (1 gallon and 0.6 week) to get 3 gallons in 1.8 weeks.

Such lines of reasoning show depth of understanding and should be celebrated. Students should also see, however, that some problems (such as in the context of detergent use and ticket price) may be more efficiently solved using unit rates.

As they decide which unit rates to use, students have an opportunity to interpret numbers in context and practice reasoning quantitatively and abstractly (MP2). For example, they may comment that of a ticket costing a dollar doesn’t make sense since it is not possible to purchase a fraction of a ticket. Encourage such observations and use them as entry points for discussing when or how each unit rate could make sense.

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

Launch

Recap that, in the previous activity, the ratio of 15 cups of water for every 6 cups of wheat can be expressed as two rates “per 1":

or 0.4 cup of wheat per cup of water
or 2.5 cups of water per cup of wheat

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 the amounts 0.4 and 2.5 “unit rates.” Explain 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. For each question, ask them to answer the first two parts about unit rates, discuss their responses, and come to an agreement before answering the last part.

MLR6 Three Reads. Keep books or devices closed. For the first situation in the activity, display only the problem stem (“A laundry service uses 10 gallons of detergent every 6 weeks”), without revealing the questions. “We are going to read this statement 3 times.”

After the 1st read: “Tell your partner what this situation is about.”
After the 2nd read: “List the quantities. What can be counted or measured?”
Reveal and read the questions. Ask “What are some ways we might get started on this?”

If necessary, repeat the routine for the second situation.
Advances: Reading, Representing

Representation: Internalize Comprehension. Provide students with a blank or partially labeled double number line diagram to record the ratio of quantities in each situation. Prompt students to identify the amount of each quantity when the other quantity is 1 and to interpret each pair of values in context.
Supports accessibility for: Visual-Spatial Processing, Organization

Activity Synthesis

Invite previously identified students to share their work on the last question about each situation. Highlight observations about if or when one unit rate might be more helpful than the other for answering each question, and why it might be more helpful.

If no students made observations about the ticket-per-dollar unit rate, ask: “How can 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, consider asking:

“If I had \$80, how many tickets could I buy? What if I had \$75?"
“Can the ‘ of a ticket per dollar’ unit rate help answer these questions?”

Lesson Synthesis

The important takeaways from this lesson are:

Any ratio has two associated unit rates.
Unit rates can often be calculated efficiently with a single operation (division or multiplication).
Depending on the problem being solved, one unit rate might be more useful than the other.

Display the table from earlier in the lesson:

water (cups)	wheat (cups)
15	6
1
	1

Ask students the following questions, and complete the table along the way:

“What is a quick way to compute the number of cups of water for 1 cup of wheat?” ()
“What is a quick way to compute the number of cups of wheat for 1 cup of water?” ()
“For what types of problems would it be easier to use ?” (finding how many cups of water when we know the number of cups of wheat)
“For what types of problems would it be easier to use ?” (finding how many cups of wheat when we know the number of cups of water)

Student Lesson Summary

Suppose a farm lets us pick 2 pounds of blueberries for 5 dollars. We can say:

We get pound of blueberries per dollar.
The blueberries cost or dollars per pound.

The “price per pound of blueberries” and the “weight of blueberries per dollar” are the two unit rates describing this situation.

weight of blueberries (pounds)	price (dollars)
2	5
1
	1

A unit rate tells us how much of one quantity for 1 of the other quantity. Each of these numbers is useful in the right situation.

If we want to find out how much 8 pounds of blueberries will cost, it helps to know how much 1 pound of blueberries will cost.

weight of blueberries (pounds)	price (dollars)
1
8

If we want to find out how many pounds we can buy for 10 dollars, it helps to know how many pounds we can buy for 1 dollar.

weight of blueberries (pounds)	price (dollars)
	1
	10

Which unit rate is most useful depends on what question we want to answer, so be ready to find either one!

Priya and Han are making kinche for a breakfast event at school. The instructions for a large batch of kinche say, “Bring 15 cups of water to a boil. Then add 6 cups of crushed wheat.”

Priya says, “The ratio of cups of wheat to cups of water is . That’s 0.4 cup of wheat per cup of water.”
Han says, “The ratio of cups of water to cups of wheat is . That’s 2.5 cups of water per cup of wheat.”

Who is correct? Explain your reasoning. If you get stuck, consider using the table.

water (cups) wheat (cups)

15 6

1

1
Priya and Han are each making a pot of kinche.
1. Priya has 5 cups of wheat. How many cups of water should she boil? Show your reasoning.
2. Han boils 10 cups of water. How many cups of wheat should he add into the water? Show your reasoning.

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Lesson 18

Interpreting Rates

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Addressing

6.RP.A.3.b

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Building on Student Thinking

6.RP.A.3.b

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