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

Explain volume formulas and use them to solve problems.

Understand similarity in terms of similarity transformations.

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

Prove theorems involving similarity.

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

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

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

Experiment with transformations in the plane.

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

Reason with shapes and their attributes.

Understand congruence in terms of rigid motions.

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

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

Make geometric constructions.

Prove geometric theorems.

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.

Apply geometric concepts in modeling situations.

Find arc lengths and areas of sectors of circles.

Understand and apply the Pythagorean Theorem.

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

Use coordinates to prove simple geometric theorems algebraically.

Apply trigonometry to general triangles.

Define trigonometric ratios and solve problems involving right triangles.

Analyze, compare, create, and compose shapes.
problem_activity: Activity
The purpose of this activity is to have students identify corresponding parts in congruent triangles.
Identify students who:
make general statements such as “The lines look parallel.”
make arguments based on properties of rotations
make arguments based on alternate interior angles being congruent
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, mark congruent angles with the same color.
Supports accessibility
for:
Visual-spatial processing
Student Facing
Draw a triangle.
Find the midpoint of the longest side of your triangle.
Rotate your triangle
\(180^{\circ}\)
using the midpoint of the longest side as the center of the rotation.
Label the
corresponding
parts and mark what must be congruent.
Make a conjecture and justify it.
What type of quadrilateral have you formed?
What is the definition of that quadrilateral type?
Why must the quadrilateral you have fit the definition?
Student Response
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for free access to Student Response.
Anticipated Misconceptions
If students are struggling to perform the rotation, encourage them to use whatever tools they are most comfortable with, including sketching.
Activity Synthesis
The purpose of this discussion is to share examples of convincing arguments.
Select previously identified students to share in this order:
A student who made a general statement such as “The lines look parallel.”
A student who used the fact that rotation by
\(180^{\circ}\)
using a point not on the line as the center takes lines to parallel lines
A student who used corresponding parts to establish that the lines were parallel because alternate interior angles are congruent
If no student used corresponding parts to establish alternate interior angles as congruent, ask students to look for structure by looking at the congruence marks and seeing whether they match any diagrams on the reference chart.
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. As students share their justification for why the opposite sides must be parallel, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students’ attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


correct_option_index: 8