options: Use coordinates to prove simple geometric theorems algebraically.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Understand and apply theorems about circles.

Find arc lengths and areas of sectors of circles.

Prove geometric theorems.

Make geometric constructions.

Solve real-world and mathematical problems involving area, surface area, and volume.

Visualize relationships between two-dimensional and three-dimensional objects.

Translate between the geometric description and the equation for a conic section.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Apply trigonometry to general triangles.

Classify two-dimensional figures into categories based on their properties.

Explain volume formulas and use them to solve problems.

Understand similarity in terms of similarity transformations.

Apply geometric concepts in modeling situations.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Reason with shapes and their attributes.

Understand congruence in terms of rigid motions.

Experiment with transformations in the plane.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Analyze, compare, create, and compose shapes.

Prove theorems involving similarity.

Understand and apply the Pythagorean Theorem.

Define trigonometric ratios and solve problems involving right triangles.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
problem_activity: Activity
This activity addresses the common misconception that if a figure is dilated by a factor of
\(k\)
, the image’s area also changes by a factor of
\(k\)
. As students decide which response (if either) they agree with, they are critiquing the reasoning of others (MP3).
Launch
Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their written responses for the first question. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “How is the area of the triangle affected when it is dilated by a factor of
\(k=0.25\)
?” or “How do you know that the area of the dilated triangle is 6.25 square inches?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify their reasoning for why the area of the triangle is multiplied by
\(k^2\)
when dilated by scale factor
\(k\)
.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Representation: Access for Perception.
Read the task statement aloud. Students who both listen to and read the information will benefit from extra processing time. Check for understanding by inviting students to rephrase the problem in their own words.
Supports accessibility for: Language
Student Facing
A triangle has area 100 square inches. It’s dilated by a factor of
\(k=0.25\)
.
Mai says, “The dilated triangle’s area is 25 square inches.”
Lin says, “The dilated triangle’s area is 6.25 square inches.”
For each student, decide whether you agree with their statement. If you agree, explain why. If you disagree, explain what the student may have done to arrive at their answer.
Calculate the area of the image if the original triangle is dilated by each of these scale factors:
\(k=9\)
\(k=\frac34\)
Student Response
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Anticipated Misconceptions
Students may believe they can’t calculate the area of the dilated triangle if they don’t have its dimensions. Remind them of the two approaches highlighted in the warm-up and ask if either of those applies here.
Activity Synthesis
Here are some questions for discussion:
“Why didn’t we need the dimensions of the original triangle to calculate the area of the dilated triangle?” (We could multiply
\(k^2\)
by the original triangle’s area.)
“What are some different ways to deal with the scale factor
\(\frac34\)
?” (We can square the fraction to get
\(\left(\frac34 \right)^2 = \frac{9}{16}\)
, then multiply that fraction by the area. Or, we can divide 3 by 4 to get 0.75, square that to get 0.5625, then multiply that by 100.)


correct_option_index: 9

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