options: Explain volume formulas and use them to solve problems.

Understand congruence in terms of rigid motions.

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

Prove geometric theorems.

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

Understand similarity in terms of similarity transformations.

Prove theorems involving similarity.

Understand and apply theorems about circles.

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

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

Use coordinates to prove simple geometric theorems algebraically.

Analyze, compare, create, and compose shapes.

Define trigonometric ratios and solve problems involving right triangles.

Apply trigonometry to general triangles.

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

Experiment with transformations in the plane.

Apply geometric concepts in modeling situations.

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

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

Reason with shapes and their attributes.

Understand and apply the Pythagorean Theorem.

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

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

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

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

Make geometric constructions.

Find arc lengths and areas of sectors of circles.
problem_activity: Warm-up
The purpose of this warm-up is to ensure students understand that
they can infer the coordinates of a point based on knowing the coordinates of points on the same horizontal and vertical lines;
the length of a horizontal or vertical line segment can be determined based on the coordinates of its endpoints. Each of these is important background knowledge for this lesson.
Launch
Give students 2 minutes of quiet think time. Ask them to share their reasoning with a partner followed by a whole-class discussion.
Student Facing
[IMG]
Find each of the following and explain your reasoning:
The length of segment
\(BE\)
.
The coordinates of
\(E\)
.
Student Response
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Anticipated Misconceptions
In order to find the length of segment
\(BE\)
, it is first necessary to find the lengths of
\(CD\)
,
\(AC\)
, and
\(AB\)
, and reason about side lengths in similar triangles. If students have trouble getting started, scaffold the work by suggesting they first find any
other
segment lengths in the diagram that they can find.
Activity Synthesis
There are three important things students should understand or recall as a result of working on the warm-up:
Points on the same vertical line have the same
\(x\)
-coordinate and points on the same horizontal line have the same
\(y\)
-coordinate.
In order to find the length of a vertical segment, you can subtract the
\(y\)
-coordinates of its endpoints.
The side lengths of triangle
\(ACD\)
are the same as the side lengths of the similar triangle
\(ABE\)
multiplied by the scale factor
\(AC \div AB\)
.
Display the image from the task. For each question, invite a student to share their reasoning. The displayed image should be used as a tool for gesturing. For example, if you start at the origin and want to navigate to point
\(D\)
, you would move 4 units to the right. You move the same distance to the right to navigate from the origin to
\(C\)
. Therefore,
\(C\)
must have the same
\(x\)
-coordinate as
\(D\)
.


correct_option_index: 22