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

Prove theorems involving similarity.

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

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

Understand and apply theorems about circles.

Explain volume formulas and use them to solve problems.

Make geometric constructions.

Analyze, compare, create, and compose shapes.

Apply trigonometry to general triangles.

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

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

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

Define trigonometric ratios and solve problems involving right triangles.

Prove geometric theorems.

Understand and apply the Pythagorean Theorem.

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

Find arc lengths and areas of sectors of circles.

Understand congruence in terms of rigid motions.

Understand similarity in terms of similarity transformations.

Apply geometric concepts in modeling situations.

Experiment with transformations in the plane.

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

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

Use coordinates to prove simple geometric theorems algebraically.

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

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

Reason with shapes and their attributes.
problem_activity: Optional activity
In this activity, students are given pictures and descriptions of planes cutting prisms and pyramids. Students are asked to draw cross sections freehand but this is not a skill that is required in order for students to be able to describe two-dimensional shapes created from cross sections, which is why this is an optional activity. Some pictures are of a moving plane. Students describe how the cross section changes as the plane moves.
If students have access to the digital activity there are applets to explore the cross sections.
Adapted from applets created in GeoGebra by
Anthony C.M. OR
.
Digital
Launch
Arrange students in groups of 2. Give students 3–5 minutes of quiet work time followed by time to discuss shapes of cross sections with a partner. Follow with a whole-class discussion.
Student Facing
Use the applet to draw each cross section and describe it in words. Start each one by dragging the gray bar on the left side across the screen until you see the whole 3D image on the left and the controls on the right.
Here is an applet with a rectangular prism, 4 units by 2 units by 3 units.
A plane cuts the prism parallel to the bottom and top faces.
The plane moves up and cuts the prism at a different height.
A vertical plane cuts the prism diagonally.
A square pyramid has a base that is 4 units by 4 units. Its height is also 4 units.
A plane cuts the pyramid parallel to the base.
A vertical plane cuts the prism.
A cube has an edge of length 4.
A plane cuts off the corner of the cube.
The plane moves farther from the corner and makes a cut through the middle of the cube.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Arrange students in groups of 2. Give students 3–5 minutes of quiet work time followed by time to discuss shapes of cross sections with a partner. Follow with a whole-class discussion.
Student Facing
Draw and describe each cross section.
Here is a picture of a rectangular prism, 4 units by 2 units by 3 units.
[IMG]
A plane cuts the prism parallel to the bottom and top faces.
[IMG]
The plane moves up and cuts the prism at a different height.
[IMG]
A vertical plane cuts the prism diagonally.
[IMG]
A square pyramid has a base that is 4 units by 4 units. Its height is also 4 units.
A plane cuts the pyramid parallel to the base.
[IMG]
A vertical plane cuts the prism.
[IMG]
A cube has an edge of length 4.
A plane cuts off the corner of the cube.
[IMG]
The plane moves farther from the corner and makes a cut through the middle of the cube.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Select students to share their drawings and descriptions. Consider asking some of the following questions:
“How did you figure out the shape of the cross section?”
“What helped you visualize the shape?”
“Were any of the shapes you drew here similar to the shapes you described in the previous activity?”
Conversing: MLR8: Discussion Supports.
To support a rich discussion while students are describing two-dimensional shapes created from cross sections, invite students to include details in their descriptions and drawings of each cross section. Provide sentence frames such as: “The cross section is ____ because . . .”, “A plane cuts _____ as shown by . . .”, or “The dimensions of a cross section are ______ since . . . .” This will help students make sense of complex language as they draw and describe cross sections shown in each figure.
Design Principle(s): Cultivate conversation


correct_option_index: 3