standard_description: Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
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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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
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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)


aligns: no