few_shots: Example:
Problem/activity:
Narrative
The purpose of this activity is for students to create composite shapes and compose new shapes from composite shapes. For example, students may find that a rectangular prism can be made from four triangular prisms or from eight smaller cubes. Students then discover that they can combine two of the composed rectangular prisms to make a new rectangular prism.
Required Materials
Materials to Gather
Geoblocks
Launch
Groups of 4
Give each group geoblocks.
Display a large cube.
“This cube can be built from other shapes. See how many ways you can build a cube using other shapes.”
3 minutes: partner work time
Share responses.
Activity
“Choose a different shape. Try to find smaller shapes you can use to build that shape.”
5 minutes: partner work time
Student Response
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for free access to Student Response.
Activity Synthesis
Display a cube and a composite cube—for example, a cube composed of four small cubes and one rectangular prism.
Demonstrate putting the shapes together.
“What new shape did I make with the cube and the cube I made?”
“Put your shapes together to make a rectangular prism.”
Display a rectangular prism and two triangular prisms.
“Find these shapes. Put them together. What new shape can you make?” (a cube, a rectangular block)


Mathematical concept/skill options: 
A. Reason with shapes and their attributes.
B. Understand similarity in terms of similarity transformations.
C. Use coordinates to prove simple geometric theorems algebraically.
D. Translate between the geometric description and the equation for a conic section.
E. Understand congruence and similarity using physical models, transparencies, or geometry software.
F. Analyze, compare, create, and compose shapes.
G. Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
H. Understand congruence in terms of rigid motions.
I. Classify two-dimensional figures into categories based on their properties.
J. Draw, construct, and describe geometrical figures and describe the relationships between them.
K. Explain volume formulas and use them to solve problems.
L. Prove theorems involving similarity.
M. Prove geometric theorems.
N. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
O. Define trigonometric ratios and solve problems involving right triangles.
P. Apply trigonometry to general triangles.
Q. Visualize relationships between two-dimensional and three-dimensional objects.
R. Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
S. Make geometric constructions.
T. Apply geometric concepts in modeling situations.
U. Solve real-world and mathematical problems involving area, surface area, and volume.
V. Graph points on the coordinate plane to solve real-world and mathematical problems.
W. Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
X. Understand and apply theorems about circles.
Y. Understand and apply the Pythagorean Theorem.
Z. Experiment with transformations in the plane.
AA. Find arc lengths and areas of sectors of circles.


Your response:
A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_lesson_009431

1
options: Analyze proportional relationships and use them to solve real-world and mathematical problems.

Understand ratio concepts and use ratio reasoning to solve problems.
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.
| water (cups) | oats (cups) |
|--------------|-------------|
|      15      |      6      |
|      1       |             |
|              |      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
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
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


correct_option_index: 1