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

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

Prove theorems involving similarity.

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

Make geometric constructions.

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

Reason with shapes and their attributes.

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

Define trigonometric ratios and solve problems involving right triangles.

Experiment with transformations in the plane.

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

Apply geometric concepts in modeling situations.

Understand and apply theorems about circles.

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

Prove geometric theorems.

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

Apply trigonometry to general triangles.

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

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

Find arc lengths and areas of sectors of circles.

Understand and apply the Pythagorean Theorem.

Use coordinates to prove simple geometric theorems algebraically.

Analyze, compare, create, and compose shapes.

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

Explain volume formulas and use them to solve problems.

Understand congruence in terms of rigid motions.

Understand similarity in terms of similarity transformations.
problem_activity: Warm-up
This task helps students think strategically about what kinds of transformations they might use to show two figures are congruent. Being able to recognize when two figures have either a mirror orientation or rotational orientation is useful for planning out a sequence of transformations.
Launch
Provide access to geometry toolkits. Allow for 2 minutes of quiet work time followed by a whole-class discussion.
Student Facing
All of these triangles are congruent. Sometimes we can take one figure to another with a translation. Shade the triangles that are images of triangle
\(ABC\)
under a translation.
[IMG]
Student Response
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Anticipated Misconceptions
If any students assert that a triangle is a translation when it isn’t really, ask them to use tracing paper to demonstrate how to translate the original triangle to land on it. Inevitably, they need to rotate or flip the paper. Remind them that a translation consists only of sliding the tracing paper around without turning it or flipping it.
Activity Synthesis
Point out to students that if we just translate a figure, the image will end up pointed in the same direction. (More formally, the figure and its image have the same mirror and rotational orientation.) Rotations and reflections usually (but not always) change the orientation of a figure.
For a couple of the triangles that are not translations of the given figure, ask what sequence of transformations would show that they are congruent, and demonstrate any rotations or reflections required.


correct_option_index: 1