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Unit 2, Lesson 4

Dilating Lines and Angles

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Integrated Math 2

4.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

To Gather

Geometry toolkits (HS)Math Community Chart Protractors

Activity Narrative

In this activity, students use an example to confirm that dilations preserve angle measure. They examine a dilation of a triangle, make a conjecture, and verify it for this example. Listen for students who make a conjecture about the angles in the triangle.

Launch

None

Activity

None

Triangle is a dilation of triangle using center and a scale factor of 2.

<p>Smaller Triangle A B C and larger triangle A prime B prime C prime with Point P. Point P at lower left corner.</p>

What is the same about the two triangles? What is different? Make a conjecture about what stays the same after dilation.
Use the tools available to figure out if what you thought was true is definitely true for these triangles.
Do you think your conjecture will be true for any figure after dilation?

Activity Synthesis

Invite students to share their conjectures. Students may have many conjectures, but this discussion should focus on the idea that angles are preserved after dilation. If that idea does not come up, ask students to consider what happened to these angles in this dilation.

Once students have shared several conjectures, ask students to generalize a claim about angles under dilation. (Dilations keep the measures of angles constant.) They may recall from middle school that dilations preserve angle measure. They may also connect the preservation of angle to the idea that a dilation doesn’t distort the shape of a figure, it just changes its size.

Tell students that we aren’t going to prove that dilations preserve angle measure in this class, we’re going to assert that it is true.

Add the following assertion to the class reference chart, and ask students to add it to their reference charts:

If a figure is dilated, then corresponding angles are congruent. (Assertion)

<p>Triangles A B C and A prime B prime C prime after dilation with center P.</p> — is a dilation of so

Math Community

After the Warm-up, display the Math Community Chart. Remind students that norms are agreements that everyone in the class shares responsibility for, so it is important that everyone understands the intent of each norm and can agree with it. Tell students that today’s Cool-down includes a question asking for feedback on the drafted norms. This feedback will help identify which norms the class currently agrees with and which norms need revising or removing.

4.2

Activity

Instructional Routines

None

Materials

To Gather

Geometry toolkits (HS)Rulers

Activity Narrative

The purpose of this activity is for students to verify experimentally that dilations take lines through the center of dilation to the same line, even though specific points on the line get farther from or closer to the center of dilation according to the same ratio given by the scale factor. Students first dilate points on given lines, and then they are asked to describe what happens to the lines when they are dilated.

Launch

For the sake of time, invite students to estimate rather than measure precisely.

Engagement: Internalize Self Regulation. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Invite students to choose and respond to one scale factor greater than 1 and one scale factor less than 1.
Supports accessibility for: Organization, Attention

4.3

Activity

Instructional Routines

MLR1: Stronger and Clearer Each Time

Materials

None

Activity Narrative

In this activity, students figure out that because dilations preserve angle measures, we can prove that dilations take lines to parallel lines. They draw on the many proofs they did in previous units that use congruent angles to prove that lines are parallel. For students who struggled with proofs in prior units, make sure that they have their reference charts and proof-writing sentence frames available.

Launch

Arrange students in groups of 3–4.

In an earlier lesson, students were given different scale factors to draw dilations, beginning with the same triangle. Display several of these examples for all to see. Superimpose the examples so that the center of dilation and original figure are lined up.

Ask students what they notice about the angles in the problem. (Corresponding angles in the image and original figure are congruent.) Ask students what they notice about the line segments in the triangles. (They are longer or shorter according to the scale factor. They are parallel.)

If students don’t mention that the lines are parallel, use a highlighter or colored pencil to extend a pair of corresponding segments into lines such as and . Ask students what they notice about these lines in particular. Listen for students to suggest that the line segments are parallel in all of the triangles, regardless of the scale factor of the dilation. If it does not come up, tell students, “Jada claims that all of the segments in triangle are parallel to the corresponding segments in .”

Ask students to write their claim as a conjecture. (Dilations take lines to parallel lines.)

Tell students that we can prove this conjecture by showing that is parallel to . Give students 5 minutes to discuss in their small groups how they might go about proving this statement. After this time, pause the class and ask for ideas from the class.

If the ideas do not arise, ask students,

“What do you know about the images of points in a dilation?” (They are on a ray through the center of the dilation and the original point.)
“What information in your reference chart will help you prove that two lines are parallel?” (If two lines are cut by a transversal and corresponding angles are congruent, then the lines have to be parallel.)

Tell students to think about these ideas and how they would draw them on their diagram when they prove the conjecture about parallel lines.

As students work, monitor for groups that draw ray for the dilation as well as those who identify angle and as being congruent, corresponding angles. Ask them to share their ideas during the whole-class discussion.

MLR1 Stronger and Clearer Each Time. Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their first draft response to the proof they have written. Invite listeners to ask questions and give feedback that will help clarify and strengthen their partner's ideas and writing. Give students 3–5 minutes to revise their first draft based on the feedback they receive.
Advances: Writing, Speaking, Listening

Lesson Synthesis

Invite students to sketch each diagram.

“Point is dilated, using center , by a scale factor of . What must be true about and ?” ( and are on the same line through .)

If students don’t mention collinearity, sketch for all to see a picture where is closer to than but clearly not collinear, and ask “Could this be an accurate picture of , and ?”
If students don’t mention length, sketch for all to see a picture where is farther from than , and ask “Could this be an accurate picture of and ?”

“Segment is dilated, using center , by a scale factor of . What must be true about segments and ?” ( and are parallel or on the same line, the ratio of the length of to the length of is ).

Ask students to share diagrams of , and with a partner and then with the class.

“Angle was dilated, using center , by a scale factor of 1.5. What must be true about angle compared to angle ?” (They are congruent.)

If students say that angle is “bigger” than angle , ask students to say more about what they mean. Clarify that the segments are longer and the points are farther apart, but that the angles have the same measure. Extend segments and , and then ask if that changes the measure of angle .

to access Cool-down.

Student Lesson Summary

When one figure is a dilation of the other, we know that corresponding side lengths of the original figure and the dilated image are in the same proportion, and are all related by the same scale factor, . What is the relationship of corresponding angles in the original figure and the dilated image?

For example, if triangle is dilated, using center , with a scale factor of 2, we can verify experimentally that each angle in triangle is congruent to its corresponding angle in triangle . is congruent to . is congruent to . is congruent to .

<p>Figure showing triangle A B C dilated to triangle A’ B’ C’ using center P and scale factor 2.</p>

What is the image of a line not passing through the center of dilation? For example, what will be the image of line when it is dilated with center and a scale factor of 2? We can use congruent corresponding angles to show that line is taken to parallel line .

<p>Figure showing triangle A B C dilated to triangle A’ B’ C’ using center P.</p>

What is the image of a line passing through the center of dilation?

<p>Line G H with point C on the right.</p>

For example, what will be the image of line when it is dilated with center and a scale factor of ? When line is dilated with center and a scale factor of , line is unchanged, because dilations take points on a line through the center of a dilation to points on the same line, by definition.

<p>Line G H with point C on the right. Points G prime and H prime between points H and C, closer together than points G and H.<br>
</p>

So a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

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Lesson 4

Dilating Lines and Angles

Warm-up

Standards Alignment

Building On

HSG-CO.A.2

Addressing

HSG-SRT.A.1

Building Toward

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Activity Synthesis

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

HSG-CO.A.2

Addressing

HSG-SRT.A.1.a

Building Toward

HSG-SRT.A.2

Standards Alignment

Building On

HSG-CO.C.9

Addressing

HSG-SRT.A.1.a

Building Toward

HSG-SRT.A.2

HSG-SRT.A.3