Lesson 9

Dilations

9.1: Notice and Wonder: Concentric Circles (5 minutes)

The goal of this warm-up is to introduce the circular grid which students will examine in greater detail throughout this unit. The circles in the grid all have the same center and the distance between consecutive circles is the same. The circular grid is particularly useful for showing dilations where the center of dilation is the center of the grid.

Students engage in MP7 as they look for structure and relationships between the circles and lines in the picture.

Teacher Notes for IM 6–8 Math Accelerated

Add the following to the end of the activity synthesis:

Display this image for all to see, and ask students to picture the small circle \(c\) expanding away from \(P\) until it is the larger circle \(d\). Tell students that they can think of circle \(d\) as a dilation of circle \(c\). A dilation is defined by a center and a scale factor. In this case, the center is \(P\) and the scale factor is 3.

Two circles c and d on a circular grid with center point P. Circle c has radius 2 and circle d has radius 6.

Tell students that in the next activity, they are going to dilate points. Plot a point \(B\) on the circle \(c\) at one of the radial lines. If time allows, invite students to discuss with a partner how they could measure distance on the circular grid, such as the distance from point \(P\) to point \(B\). Otherwise, demonstrate that distance on the circular grid is measured by counting units along one of the rays that start at the center, \(P\). Demonstrate how to dilate point \(B\) using \(P\) as the center of dilation and a scale factor of 3 (so the image is on circle \(d\)). Highlight that the dilated point is on the ray \(PB\) three times as far away from \(P\) as the original point.

If using the digital activity, demonstrate the mechanics of dilating using the applet. You can also use the measurement tool to confirm.

Arrange students in groups of 2. Tell students that they will look at an image, and their job is to think of at least one thing they notice and at least one thing they wonder. Display the image for all to see. Ask students to give a signal when they have noticed or wondered about something. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

What do you notice? What do you wonder?

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Ask students to share their responses, highlighting these features of the picture:

The circles share the same center
The center of the circles is the point where the lines meet
The distance from one circle to the next is always the same (the radius of each successive circle is one unit more than its predecessor)

Students may also notice that the angle made by successive rays from the center is always 30 degrees. Some things students may wonder include

When is this grid useful?
Why are the circles equally spaced?
Why are the lines there?

9.2: Quadrilateral on a Circular Grid (15 minutes)

This activity continues studying dilations on a circular grid, this time focusing on what happens to points lying on a polygon. Students first dilate the vertices of a polygon as in the previous activity. Then they examine what happens to points on the sides of the polygon. They discover that when these points are dilated, they all lie on a side of another polygon. Just as the image of a grid circle is another circle, so the dilation of a polygon is another polygon. Moreover the dilated polygon is a scaled copy of the original polygon. These important properties of dilations are not apparent in the definition.

Monitor for students who notice that the sides of the scaled polygon \(A'B'C'D'\) are parallel to the sides of \(ABCD\) and that \(A'B'C'D'\) is a scaled copy of \(ABCD\) with scale factor 2. Also monitor for students who notice the same structure for \(EFGH\) except this time the scale factor is \(\frac{1}{2}\). Invite these students to share during the discussion.

Digital

Provide access to geometry toolkits. Tell students that they are going to dilate some points. Before they begin, demonstrate the mechanics of dilating a point using a center of dilation and a scale factor. Tell students, “In the previous activity, each point was dilated to its image using a scale factor of 3. The dilated point was three times as far from the center as the original point. When we dilate point \(D\) using \(P\) as the center of dilation and a scale factor of 2, that means we’re going to take the distance from \(P\) to \(D\) and place a new point on the ray \(PD\) twice as far away from \(P\).” Display for all to see:

If using the digital activity, demonstrate the mechanics of dilating using the applet. You can also use the measurement tool to confirm.

Representation: Internalize Comprehension. Begin with a physical demonstration of the process of dilating a point using a center of dilation and a scale factor to support connections between new situations and prior understandings. Consider using these prompts: “How does this build on the previous activity in which the main task was to find distances and scale factor?” or “How does the point D’ correspond to the points D and P?”
Supports accessibility for: Conceptual processing; Visual-spatial processing

Conversing, Reading: MLR2 Collect and Display. Circulate and listen to students as they make observations about the polygon with a scale factor of 2 and the polygon with a scale factor of \(\frac12\). Write down the words and phrases students use to compare features of the new polygons to the original polygon. As students review the language collected in the visual display, encourage students to clarify the meaning of a word or phrase. For example, a phrase such as “the new polygon is the same as the original polygon but bigger” can be clarified with the phrase “the new polygon is a scaled copy with scale factor 2 of the original polygon.” A phrase such as “the polygons have the same angles” can be clarified with the phrase “each angle in the original polygon is the same as the corresponding angle in the new polygon.” This routine will provide feedback to students in a way that supports sense-making while simultaneously increasing meta-awareness of language.
Design Principle(s): Support sense-making; Maximize meta-awareness

Here is a polygon \(ABCD\)

Dilate each vertex of polygon \(ABCD\) using \(P\) as the center of dilation and a scale factor of 2.
Draw segments between the dilated points to create a new polygon.
What are some things you notice about the new polygon?
Choose a few more points on the sides of the original polygon and transform them using the same dilation. What do you notice?
Dilate each vertex of polygon \(ABCD\) using \(P\) as the center of dilation and a scale factor of \(\frac{1}{2}\).
What do you notice about this new polygon?

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If using the digital activity, demonstrate the mechanics of dilating using the applet. You can also use the measurement tool to confirm.

Here is a polygon \(ABCD\).

Dilate each vertex of polygon \(ABCD\) using \(P\) as the center of dilation and a scale factor of 2. Label the image of \(A\) as \(A’\), and label the images of the remaining three vertices as \(B’\), \(C’\), and \(D’\).
Draw segments between the dilated points to create polygon \(A’B’C’D’\).
What are some things you notice about the new polygon?

Quadrilateral A B C D on a circular grid with center point P. The coordinates of A B C D are A(negative 4 comma negative 1), B(negative 2 comma 2), C(5 comma 1) and D(3 comma negative 1).

4. Choose a few more points on the sides of the original polygon and transform them using the same dilation. What do you notice?

5. Dilate each vertex of polygon \(ABCD\) using \(P\) as the center of dilation and a scale factor of \(\frac{1}{2}\). Label the image of \(A\) as \(E\), the image of \(B\) as \(F\), the image of \(C\) as \(G\) and the image of \(D\) as \(H\).

6. What do you notice about polygon \(EFGH\)?

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Are you ready for more?

Suppose \(P\) is a point not on line segment \(\overline{WX}\). Let \(\overline{YZ}\) be the dilation of line segment \(\overline{WX}\) using \(P\) as the center with scale factor 2. Experiment using a circular grid to make predictions about whether each of the following statements must be true, might be true, or must be false.

\(\overline{YZ}\) is twice as long \(\overline{WX}\).
\(\overline{YZ}\) is five units longer than \(\overline{WX}\).
The point \(P\) is on \(\overline{YZ}\).
\(\overline{YZ}\) and \(\overline{WX}\) intersect.

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Students may think only grid points can be dilated. In fact, any point can, but they may have to measure or estimate the distances from the center. Grid points are convenient because you can measure by counting.

Display the original figure and its image under dilation with scale factor 2 and center \(P\).

Ask selected students to share what they notice about the new polygon. Ensure that the following observations are made. Encourage students to verify each assertion using geometry tools like tracing paper, a ruler, or a protractor.

The new figure is a scaled copy of the original figure.
The sides of the new figure are twice the length of the sides of the original figure.
The corresponding segments are parallel.
The corresponding angles are congruent.

Ask students what happened to the additional points they dilated on polygon \(ABCD\). Note that a good strategic choice for these points are points where \(ABCD\) meets one of the circles: in these cases, it is possible to double the distance from that point to the center without measuring. The additional points should have landed on a side of the dilated polygon (because of measurement error, this might not always occur exactly). The important takeaway from this observation is that dilating the polygon’s vertices, and then connecting them, gives the image of the entire polygon under the dilation.

9.3: Getting Perspective (15 minutes)

In this activity, students apply dilations without a grid. Students will need to find an appropriate way to take measurements (MP5), most likely with the aid of a ruler or the edge of an index card. Different students will work with different scale factors and will produce perspective drawings of a box.

Watch for students who pick a point close to one vertex of the given rectangle. If the point is too close, it will be more difficult to visualize the box. Suggest that they move the point further away. Monitor for students who produce accurate drawings with different scale factors and invite them to share during the discussion.

Digital

Provide access to geometry toolkits.

First, a demonstration about dilating a point on a plane with no grid.

We want to dilate point \(B\) using \(A\) as the center of dilation and a scale factor of 3.

Use a straightedge to draw ray \(AB\).

Measure the distance from \(B\) to \(A\). Multiply the distance by 3. Draw \(B’\) so that it is 3 times as far away from \(A\). For scale factors that are integers, an unmarked edge of an index card or a compass can also be used to transfer the distance along the ray.

If we wanted a scale factor that is not an integer the procedure is the same. Measure the distance from \(A\) to \(B\), multiply by the scale factor, and place \(B’\) at that new distance from \(A\).

Let’s say the distance from \(A\) to \(B\) is 4.8 cm.

If we wanted to use a scale factor of 2.5, the distance from \(A\) to the dilated point \(B’\) would be 12 cm, because \((4.8) \boldcdot (2.5) = 12\).

A perspective drawing is an optical illusion that makes an image printed on paper have a three-dimensional look. Display at least one example of a perspective drawing:

Students will practice some simple dilations of points, and then they will create a perspective drawing. Tell students to complete the first part of the activity dilating points \(P\) and \(Q\). After you review their work, assign each student a scale factor to use for the second part. Appropriate scale factors include \(\frac{1}{3}\), \(\frac{1}{2}\), \(1\frac{1}{2}\), and 2. It will work best if the center of the dilation is not too close to the rectangle the students are dilating.

Representation: Provide Access for Perception. Display or provide students with a physical copy of the Launch demonstration about dilating a point on a plane with no grid. Check for understanding by inviting students to rephrase directions for creating a dilation of points in their own words. Consider keeping the display of directions visible throughout the activity.
Supports accessibility for: Language; Memory

Dilate \(P\) using \(C\) as the center and a scale factor of 4. Follow the directions to perform the dilations in the applet.
1. Select the Dilate From Point tool.
2. Click on the object to dilate, and then click on the center of dilation.
3. When the dialog box opens, enter the scale factor. Fractions can be written with plain text, ex. 1/2.
4. Click
5. Use the Ray tool and the Distance tool to verify.
Dilate \(Q\) using \(C\) as the center and a scale factor of \(\frac12\).
Draw a simple polygon.
1. Choose a point outside the polygon to use as the center of dilation. Label it \(C.\)
2. Using your center \(C\) and the scale factor you were given, draw the image under the dilation of each vertex of the polygon, one at a time. Connect the dilated vertices to create the dilated polygon.
3. Draw a segment that connects each of the original vertices with its image. This will make your diagram look like a cool three-dimensional drawing of a box! If there's time, you can shade the sides of the box to make it look more realistic.
4. Compare your drawing to other people’s drawings. What is the same and what is different? How do the choices you made affect the final drawing? Was your dilated polygon closer to \(C\) than to the original polygon, or farther away? How is that decided?

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Provide access to geometry toolkits. Display two points, \(A\) and \(B\), like the ones shown here and ask students to copy the image onto their own paper with \(A\) and \(B\) about a finger width apart.

Ask, “To dilate point \(B\) using point \(A\) as the center of dilation and a scale factor of 3, how could we do it?” (Draw ray \(AB\), and then use a ruler to identify the point \(B’\) on ray \(AB\) that is 3 times further from point \(A\) than point \(B\) is.) After some quiet work time, select students to share their thinking starting with students who estimated the distance and ending with students who drew in the ray, measured precisely, and then plotted the dilated point 3 times as far from point \(A\). Make sure students understand why \(B’\) has to be on the ray \(AB\).

Tell students that if we use a scale factor that is not an integer, the procedure is the same. Measure the distance from \(A\) to \(B\), multiply by the scale factor, and place \(B’\) at that new distance from \(A\). For example, if the scale factor is 2.5 and the distance from \(A\) to \(B\) is 4.8, here is what the dilation would look like:

For the last question, a perspective drawing is an optical illusion that makes an image printed on paper have a three-dimensional look. Display at least one example of a perspective drawing:

Students practice some simple dilations of points, and then they create a perspective drawing. Tell students to complete the first part of the activity dilating points \(P\) and \(Q\). After you review their work, assign each student a scale factor to use for the second part. Appropriate scale factors include \(\frac{1}{3}\), \(\frac{1}{2}\), \(1\frac{1}{2}\), and 2. It will work best if the center of the dilation is not too close to the rectangle the students are dilating.

Using one colored pencil, draw the images of points \(P\) and \(Q\) using \(C\) as the center of dilation and a scale factor of 4. Label the new points \(P’\) and \(Q’\).
Using a different color, draw the images of points \(P\) and \(Q\) using \(C\) as the center of dilation and a scale factor of \(\frac12\). Label the new points \(P’’\) and \(Q’’\).

Pause here so your teacher can review your diagram. Your teacher will then give you a scale factor to use in the next part.
Now you’ll make a perspective drawing. Here is a rectangle.
1. Choose a point inside the shaded circular region but outside the rectangle to use as the center of dilation. Label it \(C\).
2. Using your center \(C\) and the scale factor you were given, draw the image under the dilation of each vertex of the rectangle, one at a time. Connect the dilated vertices to create the dilated rectangle.
3. Draw a segment that connects each of the original vertices with its image. This will make your diagram look like a cool three-dimensional drawing of a box! If there’s time, you can shade the sides of the box to make it look more realistic.
4. Compare your drawing to other people’s drawings. What is the same and what is different? How do the choices you made affect the final drawing? Was your dilated rectangle closer to \(C\) than to the original rectangle, or farther away? How is that decided?

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Are you ready for more?

Here is line segment \(DE\) and its image \(D’E’\) under a dilation.

Line segment D E and its image D prime E prime under a dilation. D E and D prime E prime appear to be parallel and D E is four times longer than D prime E prime.

Use a ruler to find and draw the center of dilation. Label it \(F\).
What is the scale factor of the dilation?

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Students may all try to make their drawing match any example drawings shown in the launch. For example, if the center of dilation in an example is above and to the right, everyone might place their center of dilation above and to the right of the rectangle. Any point is fine as a dilation point, but the effect on what the picture looks like may vary.

Students may not recall that to dilate a polygon, they can first dilate the vertices and then connect them in the proper order. It may be necessary to show students how to dilate one of the vertices and allow them to perform the dilation on the other three vertices.

Display the work of several students selected based on the different scale factors. Then ask students:

“What are the effects of using a scale factor greater than 1?” (The image is larger than the original and farther away from the center of dilation than the original.)
“What are the effects of using a scale factor less than 1?” (The image is smaller than the original and closer to the center of dilation than the original.)
“What effect does the location of \(C\), the center of dilation, have?” (It impacts the size and location of the dilated rectangle: if the scale factor is less than 1 then the dilated rectangle is closer to \(C\) than the original and if the scale factor is larger than 1 then the dilated rectangle is further away from \(C\) than the original.)

Time permitting, consider showing several student drawings with the same scale factor but a different location for the point \(C\). How are they the same? How are they different? Two faces of these boxes (the original rectangle and the scaled copy) are congruent but the point of view or perspective on them is different.

Speaking, Listening: MLR7 Compare and Connect. As students prepare their perspective drawings, identify the drawings with scale factors greater than 1 or less than 1. As students investigate each other’s work, ask them to share what is similar about the drawings with scale factor greater than 1 (or less than 1). Listen for and amplify statements such as “a scale factor greater than 1 results in an image larger than the original” and “a scale factor less than 1 results in an image smaller than the original.” Then encourage students to make connections between the value of the scale factor and the effect on the image. Listen for and amplify language students use to describe how the scale factor affects the size of the image and its distance from the center of dilation. This will foster students’ meta-awareness and support constructive conversations as they compare perspective drawings and make connections between the value of the scale factor and the image of the original polygon.
Design Principles(s): Cultivate conversation; Maximize meta-awareness

Lesson Synthesis

Begin by asking students how a circular grid helps perform dilations. (The circular grid is mainly useful when the center of dilation is the center of the grid. The radial lines help us plot dilated points when the scale factor is an integer.) Next, ask students to think about how they would explain the steps for dilating a point without a grid, and either write them down or share them with a partner. Ask a few students to share their steps. Ensure that all of the important aspects are mentioned:

“You need to know which point you want to dilate, which point is the center of dilation, and what scale factor to use.”
“Use a straightedge to draw a ray from the center of dilation through the point you want to dilate.”
“Measure the distance from the center of dilation through the point. Multiply this distance by the scale factor. Place the new point at this distance from the center of dilation and also on the ray you drew.”
“If the scale factor is greater than 1, the new point will be farther from the center than the original point. If the scale factor is less than 1, the new point will be closer to the center than the original point.”

9.4: Cool-down - A Single Dilation of a Triangle (5 minutes)

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Student Lesson Summary

Dilations also work without a grid. Consider the dilation shown here:

Ray A B. Point D is between A and B closer to A, and point C is to the left of point B. There is about the same distance between A and B as there is between C and B.

If \(A\) is the center of dilation, how can we find which point is the dilation of \(B\) with scale factor 2? Because the scale factor is larger than 1, the point must be farther away from \(A\) than \(B\) is, which makes \(C\) the point we are looking for. If we measure the distance between \(A\) and \(C\), we would find that it is exactly twice the distance between \(A\) and \(B\).

A dilation with scale factor less than 1 brings points closer. The point \(D\) is the dilation of \(B\) with center \(A\) and scale factor \(\frac{1}{3}\).

Lesson 9

9.1: Notice and Wonder: Concentric Circles (5 minutes)

Warm-up

Launch

Student Facing

Student Response

Activity Synthesis

9.2: Quadrilateral on a Circular Grid (15 minutes)

Activity

Launch

Student Facing

Student Response

Launch

Student Facing

Student Response

Student Facing

Are you ready for more?

Student Response

Anticipated Misconceptions

Activity Synthesis

9.3: Getting Perspective (15 minutes)

Activity

Launch

Student Facing

Student Response

Launch

Student Facing

Student Response

Student Facing

Are you ready for more?

Student Response

Anticipated Misconceptions

Activity Synthesis

Lesson Synthesis

Lesson Synthesis

9.4: Cool-down - A Single Dilation of a Triangle (5 minutes)

Cool-Down

Student Lesson Summary

Student Facing