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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click here to register or sign in
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)


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, in json format:
{
  "explanation": "This activity asks students to compose three-dimensional shapes, using geoblocks, to create composite shapes.",
  "answer": "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: Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
problem_activity: Activity
The purpose of this task is for students to use the coordinates of a unit circle to determine information about coordinates on circles of different sizes. Returning to the context of clock hands, this activity focuses on the location of the end of a minute hand relative to the center of the clock.
While there are 3 different clock sizes in this activity, students taking advantage of the similarity between the circles (MP7) can use their solutions for the first clock to identify the solutions for the second clock that is 3 times the size of the first.
Monitor for students who express their answers using:
approximate values based on the unit circle display
exact values using
\(\cos(\theta)\)
and
\(\sin(\theta)\)
Launch
If needed, provide students with copies of separate clock faces.
Student Facing
Here is a clock face.
[IMG]
The length of the minute hand on a clock is 5 inches and the center of the clock is at
\((0,0)\)
on a coordinate plane. Determine the coordinates of the end of the minute hand at the following times. Explain or show your reasoning.
45 minutes after the hour
10 minutes after the hour
40 minutes after the hour
The minute hand on another clock, also centered at
\((0,0)\)
, has a length of 15 inches. Determine the coordinates of the end of the minute hand at the following times. Explain or show your reasoning.
45 minutes after the hour
10 minutes after the hour
40 minutes after the hour
At a certain time, the end of the minute hand of a third clock centered at
\((0,0)\)
has coordinates approximately
\((7.5,7.5)\)
. How long is the minute hand of the clock if each grid square is one inch by one inch? Explain or show your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
The center of a clock is
\((0,0)\)
on a coordinate grid. Its hour hand is half the length of its minute hand. The coordinates of the end of the hour hand are about
\((3,0.5)\)
. What are the approximate coordinates of the end of the minute hand? Explain how you know.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Select students to share their strategies for identifying the coordinates on the clocks with a minute hand of length 5 inches and 15 inches in the order listed in the Activity Narrative. If no students wrote their answers using exact values, such as
\(\left(5\cos\left(\frac \pi 6\right), 5\sin\left(\frac \pi 6\right)\right)\)
for the 5 inch minute hand 10 minutes after the hour, ask students to consider how they could write the exact coordinates. After a brief quiet think time, invite students to share their thinking, highlighting any callbacks to their previous work identifying exact values by using cosine and sine without approximating. If needed, here are some questions for discussion:
“What angle does the minute hand make, with the positive
\(x\)
-axis, at 10 minutes after the hour?” (
\(\frac \pi 6\)
radians.)
“If the minute hand was 1 foot long, could you use sine and cosine to express the coordinates of its tip?” (The coordinates, in feet, are
\(\left(\cos\left(\frac \pi 6\right), \sin\left(\frac \pi 6\right)\right)\)
.)
“What about when the minute hand is 5 inches long? 15 inches long?” (The coordinates, in inches, are
\(\left(5\cos\left(\frac \pi 6\right), 5\sin\left(\frac \pi 6\right)\right)\)
and
\(\left(15\cos\left(\frac \pi 6\right),1 5\sin\left(\frac \pi 6\right)\right)\)
.)
Conclude the discussion by selecting students to share how they calculated the length of the minute hand when the coordinates of the end of the minute hand are known. Make sure to highlight these strategies by inviting students to consider each and share a reason why they prefer one over the other:
use the Pythagorean Theorem
recognize the angle as
\(\frac{\pi}{4}\)
radians due to the
\(x\)
- and
\(y\)
-coordinates matching
Representation: Internalize Comprehension.
Use color and annotations to illustrate student thinking. As students share their strategies for identifying the coordinates on the clocks, scribe their thinking on a visual display. Highlight the exact coordinates (expressions using cosine and sine) and the approximate coordinates in the same color. Use annotation to draw attention to the scale factors of 5 and 15 in the expressions.
Supports accessibility for: Visual-spatial processing; Conceptual processing


correct_option_index: 0

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