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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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)


Standard options: 
A. Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.
B. Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.
C. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. (Students do not need to learn formal names such as “right rectangular prism.”)


Your response:
C

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: im_lesson_009431

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options: Prove that all circles are similar.

Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Construct a tangent line from a point outside a given circle to the circle.

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
problem_activity: In Class Launch
Use after Unit 3, Lesson 11
In this task, students will experiment with cups in order to see what shape they seem to trace out when they are rolled on a flat surface, and then refine their thinking in order to prove what the shape is. If possible, provide cups with a variety of sizes and shapes. Note that cylindrical cups will not curve as they are rolled, and so will not make the same kind of shape as cups with slanted sides.
Ask students to roll some cups around. Then ask them to guess what shape the cup traces out on the table or floor. Sketch the suggested shapes for all to see. Tell students their task is to try to prove what the shape is.
Teacher Instructions
Consider the types of cups to make available to students and how they will be able to manipulate the cups. For example, paper cups can be cut apart and unrolled, but plastic cups can’t. If materials are limited and need to be reused in multiple class periods, consider providing an intact cup and a cup that has already been cut apart so that students can examine both of them.
After students have completed the task, either invite them to share how they knew what shape the cup would trace out, or have a gallery walk so that students can see and comment on each other’s arguments.
Then invite them to share their responses to the last question: how did they make a cup that would trace out the largest possible area? If no student mentions it, ask what would happen if the sides of the cup weren’t slanted at all. (In theory, the cup will roll forever and trace out an infinite area, but in reality, it will run into something eventually and stop.)
Technology can be helpful when students are experimenting with different possible cup shapes. If technology is available, here is an applet students can use:
Student-Facing Statement
What shape does a cup trace out when it is rolled on a flat surface? Use words and pictures to explain how you know.
How can you locate the center and calculate the area of the shape that is traced out?
Using one sheet of 8.5 inch by 11 inch paper, how could you make a cup that will trace out the largest possible area?
Teacher Instructions
Consider the types of cups to make available to students and how they will be able to manipulate the cups. For example, paper cups can be cut apart and unrolled, but plastic cups can’t. If materials are limited and need to be reused in multiple class periods, consider providing an intact cup and a cup that has already been cut apart so that students can examine both of them.
If students see the questions in Part 2 before they have finished Part 1, it will give away the shape that the cup traces out, so hand out Part 2 after the discussion of Part 1. Part 1 can be discussed before all students have finished their explanations, as long as they all have some developed ideas about what the shape is. For the discussion, ask students to share what shape they think the cup traces out and why they think so. Students do not need to have fully developed proofs, but take the opportunity to press them to clarify their terminology and justify their assumptions.
After students have completed the task, invite them to share their responses to the last question: how did they make a cup that would trace out a larger area? (The bottom radius should be closer to the top radius so that the radius of the ring will be larger, or the cup should be taller so that the ring is thicker.) Then ask how they would make a shape that traces out the largest possible area. If no student mentions it, ask what would happen if the sides of the cup weren’t slanted at all. (In theory, the cup will roll forever and trace out an infinite area, but in reality, it will run into something eventually and stop.)
Technology can be helpful when students are experimenting with different possible cup shapes. If technology is available, here is an applet students can use:
Student-Facing Statement
Part 1:
What shape does a cup trace out when it is rolled on a flat surface? Explain or show your reasoning.
Pause for discussion.
Part 2:
The outside of this ring has a radius of 55 cm. The inside of the ring has a radius of 38 cm. Find the area of the ring.
[IMG]
Triangle
\(ABC\)
is isosceles. Find the length of side
\(AC\)
using the information given. The two vertical lines are parallel.
[IMG]
Here is the cross-section of a cup. It shows the width of the cup at the top and at the bottom, and the slant height of the cup. If we extend the sides of the cup down until they meet at a point, we will make a triangle. What would the slant height of that triangle be?
[IMG]
Measure the widths of the top and bottom of your cup, and measure its slant height. If you drew the cross-section of the cup and then extended the sides to make a triangle, what would the slant height of the triangle be?
If you rolled your cup in a circle to make a ring, what would the area of the ring be?
Give an example of a cup that would trace out a larger area than your cup would.


correct_option_index: 0

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