options: Apply trigonometry to general triangles.

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

Prove geometric theorems.

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

Understand and apply theorems about circles.

Understand similarity in terms of similarity transformations.

Understand congruence in terms of rigid motions.

Understand and apply the Pythagorean Theorem.

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

Find arc lengths and areas of sectors of circles.

Make geometric constructions.

Reason with shapes and their attributes.

Apply geometric concepts in modeling situations.

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

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

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

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

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

Analyze, compare, create, and compose shapes.

Explain volume formulas and use them to solve problems.

Experiment with transformations in the plane.

Define trigonometric ratios and solve problems involving right triangles.

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

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

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

Use coordinates to prove simple geometric theorems algebraically.

Prove theorems involving similarity.
problem_activity: Activity
The purpose of this activity is to give students practice finding missing side lengths in a right triangle using the Pythagorean Theorem.
Teacher Notes for IM 6–8 Math Accelerated
Adjust time to 15 minutes. After 2–3 minutes of work time, select 1–2 groups to share their work for the first question before continuing with the remaining questions. If time is limited, tell students that the final question is optional.
Launch
Arrange students in groups of 2. Give students 10 minutes of quiet work time and then have them compare with a partner. If partners disagree about any of their answers, ask them to explain their reasoning to one another until they reach agreement. Follow with a whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge by displaying the Pythagorean Theorem with a labeled diagram. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Find
\(c\)
.
[IMG]
Find
\(b\)
.
[IMG]
A right triangle has sides of length 2.4 cm and 6.5 cm. What is the length of the hypotenuse?
A right triangle has a side of length
\(\frac14\)
and a hypotenuse of length
\(\frac13\)
. What is the length of the other side?
Find the value of
\(x\)
in the figure.
[IMG]
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
The spiral in the figure is made by starting with a right triangle with both legs measuring one unit each. Then a second right triangle is built with one leg measuring one unit, and the other leg being the hypotenuse of the first triangle. A third right triangle is built on the second triangle’s hypotenuse, again with the other leg measuring one unit, and so on.
[IMG]
Find the length,
\(x\)
, of the hypotenuse of the last triangle constructed in the figure.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Ask students to share how they found the missing side lengths. If students drew triangles for the two questions that did not have an image, display a few of these for all to see, noting any differences between them. For example, students may have drawn triangles with different orientations or labeled different sides as
\(a\)
and
\(b\)
.
For the last question, ask students to say what they did first to try and solve for
\(x\)
. For example, while many students may have found the length of the unknown altitude first and then used that value to find
\(x\)
, others may have set up the equation
\(34-5^2=18-x^2\)
.
Point out that when you know two sides of a right triangle, you can always find the third by using the Pythagorean identity
\(a^2 + b^2 = c^2\)
. Remind them that it is important to keep track of which side is the hypotenuse.
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct.
Before students share their method for the questions that did not have an image, present an incorrect solution based on a common error related to labeling the sides of a right triangle. For the right triangle with a side of length
\(\frac{1}{4}\)
and a hypotenuse of length
\(\frac{1}{3}\)
, draw a right triangle with the legs labeled
\(\frac{1}{4}\)
and
\(\frac{1}{3}\)
. Provide an incorrect explanation such as: “I know that
\(a = \frac{1}{4}\)
and
\(b = \frac{1}{3}\)
, so when I use the Pythagorean Theorem, I get the equation
\((\frac{1}{4})^2 + (\frac{1}{3})^2 = c^2\)
.” Ask students to identify the error, critique the reasoning, and revise the original statement. As students discuss in partners, listen for students who clarify the meaning of the hypotenuse and identify
\(c\)
as the length of the hypotenuse in the Pythagorean Theorem. This routine will engage students in meta-awareness as they critique and correct a common error when labeling the sides of a right triangle.
Design Principles(s): Cultivate conversation; Maximize meta-awareness


correct_option_index: 7