# Lesson 1

Scale Drawings

## 1.1: Is That the Same Hippo? (10 minutes)

### Warm-up

In middle school, students explored scale drawings including how measurements in a scaled copy of a figure relate to measurements in the original figure. In this activity students remind themselves of these relationships by studying an example and a non-example.

### Student Facing

Diego took a picture of a hippo and then edited it. Which is the distorted image? How can you tell?

Is there anything about the pictures you could measure to test whether there’s been a distortion?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The important concepts in this discussion are that in a scale drawing:

- the ratio of the distance between two points in the original figure to the distance between two corresponding points in the scaled figure is constant
- the corresponding angles are congruent

Remind students of the term *scale drawing* from middle school. Ask students what they could measure to figure out which hippo was a scale drawing of the original hippo. (If the distance between the original hippo’s pupils is twice the distance between the scaled hippo’s pupils, then the length of the original hippo’s tail should also be twice the length of the scaled hippo’s tail.) If no student mentions angles, ask students if the protractor could also help them confirm which hippo is the scale drawing. (The angles defined by points in the scaled figure should have the same measure as the angles defined by points in the original figure.) Students need not do any measuring.

Ask students to add this definition to their reference charts as you add it to the class reference chart:

**Scale factor** is the factor by which every length in an original figure is multiplied when you make a scaled copy.

## 1.2: Sketching Stretching (10 minutes)

### Activity

Students drew dilations in middle school. This activity is the first time students are asked to dilate a figure in this course. Monitor for students with misconceptions, such as making the distance from \(H\) to \(H’\) 120 mm, rather than making the distance from \(C\) to \(H’\) 120 mm so that the distance from \(H\) to \(H’\) is 80 mm

### Launch

*Representation: Internalize Comprehension.*Differentiate the degree of difficulty or complexity by beginning with an example with more accessible values. For instance, demonstrate dilating a 10 mm long segment \(OP\) with a scale factor of 2. Highlight connections between representations by using color to show the 2 equal segments which reflect the scale factor of 2.

*Supports accessibility for: Conceptual processing*

### Student Facing

A **dilation** with center \(O\) and positive **scale factor** \(r\) takes a point \(P\) along the ray \(OP\) to another point whose distance is \(r\) times farther away from \(O\) than \(P\) is. If \(r\) is less than 1 then the new point is really closer to \(O\), not farther away.

- Dilate \(H\) using \(C\) as the center and a scale factor of 3. \(H\) is 40 mm from \(C\).

- Dilate \(K\) using \(O\) as the center and a scale factor of \(\frac{3}{4}\). \(K\) is 40 mm from \(O\).

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

If students aren't sure how to get started, support them in understanding the definition of dilation given in the task. Draw an example using the labels in the definition and invite them to match the labels in the task to the example.

### Activity Synthesis

Invite students to share ideas they had for how to get started with the problem. (Draw a ray from \(C\) through \(H\).)

Display the images for all to see and ask students why they are incorrect. (The total distance \(CH'\) should be 120 mm. They used the wrong center of dilation, \(K'O\) should be \(\frac34\) as long as \(KO\).)

Make sure that all students understand how to draw rays from the center point through the point(s) to be dilated, and find the image of the points by measuring along the rays. Students will dilate more complex figures in subsequent activities.

*Conversing, Representing: MLR8 Discussion Supports.*Use this routine to amplify the language students use to critique the images. After students share a response, invite them to repeat their reasoning using mathematical language relevant to the lesson, such as dilation, center of dilation, and scale factor. For example, ask students, “Can you say that again, using the terms ‘center of dilation’ and ‘scale factor’?” Consider inviting the remaining students to repeat these phrases to provide additional opportunities for all students to produce this language.

*Design Principle(s): Support sense-making*

## 1.3: Mini Me (15 minutes)

### Activity

Students dilated a single point in a previous activity. In this activity, they dilate a figure with multiple points and multiple shapes. The goal is for them to practice dilation and begin to confirm some properties of dilation. Students will have additional opportunities to formalize properties of dilation in subsequent lessons (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. The dilation of a line segment is longer or shorter according to the same ratio given by the scale factor.)

Monitor for students who use the following (incomplete) strategies:

- use the two given lines to place \(D’\) and \(B’\), and then estimate all of the other shapes and points
- measure distances within the figure and try to replicate them, for example calculating the radii of the two circles and trying to place them without scaling \(A\) and \(C\) along rays through \(P\)

Monitor for students who use the precise strategy of drawing rays from \(P\) through each point in the original figure and placing points in the image halfway between \(P\) and the corresponding point.

### Launch

It's okay if students do not have time to dilate every point, encourage them to record things they notice and wonder as they draw the dilation.

*Engagement: Internalize Self Regulation.*Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Assign students an appropriate number of points to choose and dilate.

*Supports accessibility for: Organization; Attention*

### Student Facing

- Dilate the figure using center \(P\) and scale factor \(\frac12\).

- What do you notice? What do you wonder?

### Student Response

For access, consult one of our IM Certified Partners.

### Student Facing

#### Are you ready for more?

- Dilate segment \(AB\) using center \(P\) by scale factor \(\frac12 \). Label the result \(A'B'\).
- Dilate the segment \(AB\) using center \(Q\) by scale factor \(\frac12\).
- How does the length of \(A''B''\) compare to \(A'B\)? How would the length of \(A''B''\) change if \(Q\) was infinitely far away? Explain or show your answer.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The goal of this discussion is to surface some observations. If most students were comfortable drawing a dilation move directly to that. If not, select multiple students who used different strategies (both precise and incomplete) to share their work. Ask students to explain what was *precise* in each student’s method. The key point is that the most precise way to draw a dilation is to draw rays from the center of the dilation through each point of the original figure, and then measure to figure out where the point goes in the image.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If the relative length of segments, or measure of angles do not come up during the conversation, ask students to discuss these ideas.

*Speaking: MLR8 Discussion Supports.*Use this routine to support whole-class discussion. As students share their strategies for drawing the dilation of the figure, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students’ attention to any words or phrases that helped clarify the original explanation such as, “draw rays from the center of dilation through each point of the original figure.” This provides more students with an opportunity to produce language as they interpret the reasoning of others.

*Design Principle(s): Support sense-making*

## Lesson Synthesis

### Lesson Synthesis

Invite students to examine the figures they have drawn during this lesson.

- Does a point change size when it is dilated? (No, points are always infinitesimally small, only the location changes.)
- What does a segment look like when it is dilated? (A longer or shorter segment depending on the scale factor.)
- Why doesn't it have gaps? (There are infinite points in a segment so even when it is dilated to be longer there are still infinite points so there are no gaps.)

Note that dilations take line segments to line segments and circles to circles. Even though any pair of points in the original figure will be farther apart or closer together in the image, we don’t need to worry that a dilated circle will have tiny holes in it, or a dilated line segment will be dashed if we zoom in far enough. We define dilation to take lines to lines and circles to circles without any gaps.

Ask students to add this definition to their reference charts as you add it to the class reference chart:

A **dilation** with center \(P\) and positive scale factor \(k\) takes a point \(A\) along the ray \(PA\) to another point whose distance is \(k\) times farther away from \(P\) than \(A\) is.

Dilate (object) using center (point) and a scale factor of (number).

## 1.4: Cool-down - Match the Scale Factors (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

A scale drawing of an object is a drawing in which all lengths in the drawing correspond to lengths in the object by the same scale. When we scale a figure we need to be sure to scale all of the parts equally or else the image will become distorted.

Creating a scaled copy involves multiplying the lengths in the original figure by a **scale factor**. The scale factor is the factor by which every length in a original figure is multiplied when you make a scaled copy. A scale factor greater than 1 enlarges an object while a scale factor less than 1 shrinks an object. What would a scale factor equal to 1 do?

For example, segment \(BC\) is a scaled copy of segment \(DE\) with a scale factor of \(\frac14\). So \(BC=\frac14DE\). If \(DE=6\), then \(BC=\frac64\) or 1.5.

Segment \(FG\) is a dilation of segment \(DE\) using center \(A\) and a scale factor of 3. So \(FA=3 \boldcdot DA\). If \(DA=15\), then \(FA=45\).