options: Apply geometric concepts in modeling situations.

Prove geometric theorems.

Make geometric constructions.

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

Understand and apply the Pythagorean Theorem.

Prove theorems involving similarity.

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

Explain volume formulas and use them to solve problems.

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

Define trigonometric ratios and solve problems involving right triangles.

Understand and apply theorems about circles.

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

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

Apply trigonometry to general triangles.

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

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

Use coordinates to prove simple geometric theorems algebraically.

Experiment with transformations in the plane.

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

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

Understand congruence in terms of rigid motions.

Find arc lengths and areas of sectors of circles.

Understand similarity in terms of similarity transformations.

Reason with shapes and their attributes.

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

Analyze, compare, create, and compose shapes.

Graph points on the coordinate plane to solve real-world and mathematical problems.
problem_activity: Activity
In this activity, students get another opportunity to practice drawing dilations precisely. They also get to see the effects of different scale factors on an image. By assigning students scale factors less than and greater than one, students get a chance to see how the image is taken closer to or farther away from the center of dilation. By assigning equivalent scale factors, such as
\(\frac32\)
and 1.5, students get a chance to explore how images dilated with the same scale factor are congruent. They also get a chance to remind themselves of decimal representations of fractions, which will be useful when they calculate scale factors from ratios of side lengths in later lessons.
Launch
Arrange students in groups of 2. Assign each student a scale factor from the list:
\(\frac14\)
,
\(\frac12\)
,
\(\frac34\)
,
\(\frac76\)
,
\(\frac32\)
,
\(\frac52\)
, 0.25, 0.5, 1.5, 2.5
Representation: Internalize Comprehension.
Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Assign
\(\frac12\)
and
\(\frac32\)
to a pair of students who would benefit from additional support.
Supports accessibility for: Conceptual processing; Organization; Attention
Student Facing
Here is a center of dilation and a triangle.
[IMG]
Measure the sides of triangle
\(EFG\)
(to the nearest mm).
Your teacher will assign you a scale factor. Predict the relative lengths of the original figure and the image after you dilate by your scale factor.
Dilate triangle
\(EFG\)
using center
\(C\)
and your scale factor.
How does your prediction compare to the image you drew?
Use tracing paper to copy point
\(C\)
, triangle
\(EFG\)
, and your dilation. Label your tracing paper with your scale factor.
Align your tracing paper with your partner’s. What do you notice?
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
[IMG]
Dilate triangle
\(FEG\)
using center
\(C\)
and scale factors:
\(\frac12\)
2
What scale factors would cause some part of triangle
\(E'F'G'\)
’ to intersect some part of triangle
\(EFG\)
?
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
If students are unclear about relative lengths, tell them to make a prediction and write a description such as “a little smaller” or “a lot bigger” before they calculate.
Activity Synthesis
The goal of this discussion is to identify the effects of various scale factors. Collect and align several tracing papers on the original image. Invite students to make generalizations about different scenarios:
compare scale factors which are greater and less than one
compare equivalent scale factors with different representations (such as 0.5 and
\(\frac12\)
)
Speaking: MLR8 Discussion Supports.
Use this routine to help students produce statements about the scale factor of a dilation. Provide sentence frames for students to use when they generalize the effects of various scale factors, such as: “If _____ , then _____ because….” and “_____ will always _____ because….”
Design Principle(s): Support sense-making; Optimize output (for generalization)


correct_option_index: 17