Unit 2, Lesson 18
Interpreting Rates
Accelerated 6
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This Math Talk focuses on the connections between division, fractions, and decimals. It encourages students to think about the relationship between a fraction and division (an idea from grade 5) and to rely on what they know about equivalent fractions and decimals to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students interpret or find unit rates in a situation given a ratio of two quantities.
To mentally determine if the statements involving division, fractions, and decimals are true, students need to look for and make use of structure (MP7).
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Keep all previous problems and work displayed throughout the talk.
Decide mentally whether each statement is true.
To involve more students in the conversation, consider asking:
Highlight that a fraction can be seen as division of the numerator by the denominator:
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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:
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.
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.
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KINCHE To make a large batch, bring 15 cups of water to a boil. Then add 6 cups of crushed wheat. |
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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.
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:
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.
Help us improve by sharing suggestions or reporting issues.
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.”
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.
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.
For each situation, find the unit rates. Then choose a unit rate to solve the problem in part c. Show your reasoning.
A laundry service uses 10 gallons of detergent every 6 weeks.
Tyler paid $16 for 4 raffle tickets.
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.
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