options: Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
problem_activity: Activity
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.
Launch
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.
Action and Expression: Develop Expression and Communication.
Invite students to talk about their ideas with their partner before writing them down. Display sentence frames to support students when they explain their ideas. For example, “I noticed _____ so I . . .”, “Both _____ and _____ are alike because . . .”, and “How do you know . . . ?”
Supports accessibility for: Language; Organization
Student Facing
Priya, Han, Lin, and Diego are all on a camping trip with their families. The first morning, Priya and Han make oatmeal for the group. The instructions for a large batch say, “Bring 15 cups of water to a boil, and then add 6 cups of oats.”
Priya says, “The ratio of the cups of oats to the cups of water is
\(6:15\)
. That’s 0.4 cups of oats per cup of water.”
Han says, “The ratio of the cups of water to the cups of oats is
\(15:6\)
. That’s 2.5 cups of water per cup of oats.”
Who is correct? Explain your reasoning. If you get stuck, consider using the table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "water (cups)"}, {"_attributes": {"scope": "col"}, "_value": "oats (cups)"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "15"}, {"_value": "6"}]}, {"td": [{"_value": "1"}, {}]}, {"td": [{}, {"_value": "1"}]}]}]}]}
The next weekend after the camping trip, Lin and Diego each decide to cook a large batch of oatmeal to have breakfasts ready for the whole week.
Lin decides to cook 5 cups of oats. How many cups of water should she boil?
Diego boils 10 cups of water. How many cups of oats should he add into the water?
Did you use Priya’s rate (0.4 cups of oats per cup of water) or Han’s rate (2.5 cups of water per cup of oats) to help you answer each of the previous two questions? Why?
Student Response
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Anticipated Misconceptions
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.
Activity Synthesis
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.
Writing, Listening, Conversing: MLR3 Clarify, Critique, and Correct.
Before discussing students’ approaches to the final question. Present the following explanation: “I did not use either Priya’s rate (0.4 cups of oats per cup of water) or Han’s rate (2.5 cups of water per cup of oats) because they are not equal.” Ask students to critique the reasoning and identify the error(s), as they work in pairs to propose an improved response that details how either rate can be used to answer the question. If students can’t make the connections between these two rates, consider asking them what each number means. This will help students reflect on efficiently using a unit rate associated with the ratio, cups of water to the cups of oats.
Design Principle(s): Optimize output (for explanation); Cultivate conversation; Maximize meta-awareness


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