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

Apply geometric concepts in modeling situations.

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

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

Make geometric constructions.

Prove theorems involving similarity.

Find arc lengths and areas of sectors of circles.

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

Prove geometric theorems.

Apply trigonometry to general triangles.

Define trigonometric ratios and solve problems involving right triangles.

Use coordinates to prove simple geometric theorems algebraically.

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

Analyze, compare, create, and compose shapes.

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

Understand and apply the Pythagorean Theorem.

Explain volume formulas and use them to solve problems.

Understand similarity in terms of similarity transformations.

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.

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

Understand congruence in terms of rigid motions.

Reason with shapes and their attributes.

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

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

Experiment with transformations in the plane.

Understand and apply theorems about circles.
problem_activity: Optional activity
In both subsequent lessons and units, students will work extensively with similar right triangles. This activity gives students opportunities to practice the Pythagorean Theorem, and apply similar triangles in a contextual problem.
Launch
If students struggled to use the Pythagorean Theorem in a previous lesson:
Begin by asking students what they notice and wonder about the image from this activity.
Draw students’ attention to the right triangles, and ask them what must be true about the side lengths.
Provide some possible lengths for sides and have students practice finding an unknown side length in an isosceles right triangle to allow students to practice working with the square root of 2.
Speaking, Reading: MLR5 Co-Craft Questions.
Use this routine to increase awareness of the language used to talk about similar right triangles. Before revealing the questions in this activity, display the image of the quilt design made of right isosceles triangles. Ask students to write down possible mathematical questions that could be asked about the image. Invite students to compare their questions before revealing the actual questions. Listen for and amplify any questions about the features of the right triangles. For example, “Are all the right triangles similar?”, “What is the scale factor between the small and large right triangle?”, and “What are the side lengths of the right triangles?”
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
Here is a quilt design made of right isosceles triangles. The smallest squares in the center have an area of 1 square unit. Find the dimensions of the triangles.
[IMG]
Are the triangles similar? If so, what are the scale factors?
This quilt is meant for a baby (1 unit = 6 inches). To make a quilt for a queen-size bed, it needs to be 90 inches wide. What dimensions should the center squares of the big quilt have to reach that width?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Here is a quilt design made of right triangles. Find as many different size triangles as you can.
[IMG]
Write similarity statements for 2 pairs of triangles.
The smallest triangles have legs 2 units and 3 units long. Write some equivalent ratios or equations that will help you determine the dimensions of triangle
\(LCQ\)
. Use your equivalent ratios or equations to find the dimensions of triangle
\(LCQ\)
.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students who struggle with the number of similar triangles in the image can trace each size triangle separately onto tracing paper and label the corresponding sides using colored pencils.
Activity Synthesis
A key point to emphasize in this discussion is that ratios in which each ratio compares side lengths within each figure are also equivalent across similar figures. That is, for a triangle with side lengths
\(a, b, c\)
, and a similar triangle with corresponding side lengths
\(x,y,z\)
, we have equivalent ratios such as
\(\frac{a}{b} = \frac{x}{y}\)
.
Invite students to share strategies for scaling the figure up from a baby blanket to a quilt for a queen-size bed.


correct_option_index: 5