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

Define trigonometric ratios and solve problems involving right triangles.

Explain volume formulas and use them to solve problems.

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

Analyze, compare, create, and compose shapes.

Apply trigonometry to general triangles.

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

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

Reason with shapes and their attributes.

Find arc lengths and areas of sectors of circles.

Understand congruence in terms of rigid motions.

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

Experiment with transformations in the plane.

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

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

Prove theorems involving similarity.

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

Prove geometric theorems.

Use coordinates to prove simple geometric theorems algebraically.

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

Understand similarity in terms of similarity transformations.

Understand and apply the Pythagorean Theorem.

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

Apply geometric concepts in modeling situations.

Make geometric constructions.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Understand and apply theorems about circles.
problem_activity: Warm-up
Students calculate a scale factor given the areas of the circular base of a cone and the base of its dilation. This connects the concept of surface area dilation to cross sections, and gives practice with non-integer roots.
Monitor for pairs of students who initially consider an answer of 20.25 but come to a consensus that the scale factor is 4.5.
Launch
Arrange students in groups of 2. Provide access to scientific calculators. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.
Student Facing
The image shows a cone that has a base with area
\(36\pi\)
square centimeters. The cone has been dilated using the top vertex as a center. The area of the dilated cone’s base is
\(729\pi\)
square centimeters.
[IMG]
What was the scale factor of the dilation?
Student Response
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Anticipated Misconceptions
Some students may struggle to find the square root of 20.25. Remind students that their calculators can find square roots, and prompt them to use an estimate to check the reasonableness of the calculator output.
Activity Synthesis
Select a pair of students to explain their reasoning. If the pair considered 20.25 but moved to an answer of 4.5, ask how they knew 20.25 wasn’t correct.
Discuss with students how they can decide if 4.5 is a reasonable value for the square root of 20.25, considering the fact that 4
2
= 16 and 5
2
= 25. If time allows, ask students to calculate the scale factor for the solid’s volume (4.5
3
= 91.125).


correct_option_index: 2

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