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Unit 2, Lesson 3

Measuring Dilations

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Integrated Math 2

3.1

Warm-up

5 mins

Dilating Out

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

To Gather

Rulers marked with centimeters

Activity Narrative

In this Warm-up, students dilate a figure with the center of dilation in the interior of the figure. This slight difference should allow students to practice using precision with their understanding of dilations.

Launch

None

Activity

None

Student Task Statement

Dilate triangle using center and a scale factor of 3.

<p>Triangle F G H with point C interior.</p>

Activity Synthesis

Display the image, and ask students why it is incorrect. (The image of should be on the same side of as . You can tell it's wrong because the triangles don't look like scaled copies.)

<p>Center of dilation, C. 2 triangles, F G H and its image, F prime G prime H prime. C is located within triangle F G H.</p>

Make sure that all students understand how to draw rays from the center point through the points to be dilated, and find the image of the points by measuring along the rays. Students will have more opportunities to dilate figures in other activities in this lesson, so there is no need to have them correct any errors they made.

3.2

Activity

15 mins

All the Scale Factors

Instructional Routines

MLR8: Discussion Supports

Materials

To Gather

Geometry toolkits (HS)Rulers marked with centimeters

Activity Narrative

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 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.

The triangles that students draw in this activity will be used later in this unit.

3.3

Activity

15 mins

What Stays the Same?

Instructional Routines

MLR1: Stronger and Clearer Each Time

Materials

To Gather

Geometry toolkits (HS)Rulers marked with centimeters

Activity Narrative

In this activity, students dilate quadrilateral using center and different scale factors, . They notice that not only are , and equal to because that is how dilations are defined, but for example, and are also equal to . This assertion is not proved in this course. is a consequence of the definition of dilation, but the fact that dilation results in figures that are scaled copies of one another by a scale factor of is taken as an assertion after students make a conjecture based on their measurements. Students should be familiar with this assertion from middle school. In this activity, students extend that understanding by making a distinction between the distances from points in the original figure and image to the center of dilation (determined by the definition of dilation), and the lengths in the image and original figure (an unproven property of dilations).

Lesson Synthesis

Display this image:

<p>Dotted rays O C and O E that share vertex O. Point F is between the rays. Triangle E C F is drawn in. O E has length of 8.</p>

Ask students where they would place points along ray in order to make a scaled copy of the triangle with different scale factors. Record their thinking for all to see.

“If you wanted to make a scaled copy with side lengths twice the size of the original, where should you place ? Explain how you know.” (16 units from along ray , or 8 units past along ray . In order for the scaled copy to be twice as big, it has to be twice as far from the center. So because is 8 units from , make 16 units from .)
“If you wanted to make a scaled copy with side lengths half the size of the original, where should you place ? Explain how you know.” (4 units from along ray . In order for the scaled copy to be half as big, it has to be half as far from the center. So because is 8 units from , make 4 units from .)
“If you wanted to make a scaled copy with side lengths the size of the original, where should you place ?” (2 units from along ray .)
“If you wanted to make a scaled copy with side lengths the size of the original, where should you place ?” (6 units from along ray .)

Invite students to explain what they learned about dilations and specifically what can be determined from knowing the distance to . (The ratio of the distances from the original to the center and the scaled copy to the center is the same as the scale factor from the original to the scaled copy.)

to access Cool-down.

Student Lesson Summary

We know that a dilation with center and positive scale factor, , takes a point along the ray to another point whose distance is times farther away from than is.

The triangle is a dilation of the triangle with center and with a scale factor of 2. So is 2 times farther away from than is, is 2 times farther away from than is, and is 2 times farther away from than is.

Because of the way dilations are defined, all of these quotients give the scale factor: .

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

If triangle is dilated from point with scale factor , then each vertex in is on the ray from P through the corresponding vertex of , and the distance from to each vertex in is one-third as far as the distance from to the corresponding vertex in .

<p>Figure showing triangle A B C dilated to triangle A” B” C” using center P and scale factor fraction 1 over 3.</p>

The dilation of a line segment is longer or shorter according to the same ratio given by the scale factor. In other words, if segment is dilated from point with a scale factor of , then the length of segment is multiplied by to get the corresponding length of .

Corresponding side lengths of the original figure and dilated image are all in the same proportion, and are related by the same scale factor, .

Launch

Arrange students in groups of 2–4. Assign each group a scale factor: 2, 3, , . Encourage students to split up the measuring tasks to save time.

Explain to students that although they may be used to a ratio referring to an association between two or more quantities, people often also use the word to refer to a quotient of two quantities in a ratio relationship. For example, we might say the ratio of juice to sparkling water in a punch is 3 to 2. We could also say that the ratio of juice to sparkling water is to 1, because this is equivalent. As a shorthand, people sometimes say the ratio of juice to sparkling water is three halves. In this activity, ratio is used to refer to the quotient of two side lengths, and in this unit we frequently use ratio as a shorthand for quotient.

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 their proof of the conjecture. 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

Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, color code the segments in the shapes that correspond to the ratios in the calculations.
Supports accessibility for: Visual-Spatial Processing

Activity

None

Student Task Statement

Dilate quadrilateral using center and your scale factor.
Complete the table.

Ratio

Value
What do you notice? Can you prove your conjecture?
Complete the table.

Ratio

Value
What do you notice? Does the same reasoning you just used also prove this conjecture?

Ratio
Value

Ratio
Value

Student Response

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Lesson 3

Measuring Dilations

Warm-up

Dilating Out

Standards Alignment

Building On

8.G.A.3

Addressing

HSG-CO.A.2

HSG-SRT.A.1

Building Toward

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Student Task Statement

Activity Synthesis

Activity

All the Scale Factors

Instructional Routines

Materials

To Gather

Activity Narrative

Activity

What Stays the Same?

Instructional Routines

Materials

To Gather

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Are You Ready for More?

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Standards Alignment

Building On

8.G.A.3

Addressing

HSG-CO.A.2

Building Toward

HSG-SRT.A.1.b

Standards Alignment

Building On

HSG-CO.A.2

Addressing

HSG-SRT.A.1.b

Building Toward

HSG-SRT.A.2