# Lesson 2

Scale Factors and Making Scaled Copies

## 2.1: Number Talk: Multiplying by a Unit Fraction (5 minutes)

### Warm-up

This number talk allows students to review multiplication strategies, refreshing the idea that multiplying by a unit fraction is the same as dividing by its whole number reciprocal. It encourages students to use the structure of base ten numbers and the properties of operations to find the product of two whole numbers (MP7). For example, a student might find $$72 \boldcdot \frac{1}{9}$$ (or $$72 \div 9$$) and then shift the decimal one place to the right in order to evaluate $$(7.2) \boldcdot \frac{1}{9}$$. Each problem was chosen to elicit different approaches, so as students share theirs, ask how the factors in each problem impacted their strategies.

Before students begin, consider establishing a small, discreet hand signal (such as a thumbs-up) students can display to indicate they have an answer that they can support by reasoning. Discreet signaling is a quick way for teachers to gather feedback about timing. It also keeps students from being distracted or rushed by raised hands around the class.

### Launch

Display one problem at a time. Give students up to 1 minute of quiet think time per problem and ask them to give a signal when they have an answer and a strategy. Follow with a brief whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

### Student Facing

Find each product mentally.

$$\frac14 \boldcdot 32$$

$$(7.2) \boldcdot \frac19$$

$$\frac14 \boldcdot (5.6)$$

### Activity Synthesis

Ask students to share their strategies for each problem. Record and display their explanations for all to see. If students express strategies in terms of division, ask if that strategy would work for any multiplication problem involving fractions. Highlight that these problems only involve unit fractions and division by the denominator is a strategy that works when multiplying by a unit fraction.

To involve more students in the conversation, consider asking:

• Who can restate ___’s reasoning in a different way?
• Did anyone solve the problem the same way but would explain it differently?
• Did anyone solve the problem in a different way?
• Does anyone want to add on to _____’s strategy?
• Do you agree or disagree? Why?
Speaking: MLR8 Discussion Supports.: Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . . ." Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)

## 2.2: Scaled Triangles (15 minutes)

### Activity

In this activity, students continue to practice identifying corresponding parts of scaled copies. By organizing corresponding lengths in a table, students see that there is a single factor that relates each length in the original triangle to its corresponding length in a copy (MP8). They learn that this number is called a scale factor.

As students work on the first question, listen to how they reason about which triangles are scaled copies. Identify groups who use side lengths and angles as the basis for deciding. (Students are not expected to reason formally yet, but should begin to look to lengths and angles for clues.)

As students identify corresponding sides and their measures in the second and third questions, look out for confusion about corresponding parts. Notice how students decide which sides of the right triangles correspond.

If students still have access to tracing paper, monitor for students who use this tool strategically (MP5).

Teacher Notes for IM 6–8 Math Accelerated
Due to students’ prior work with percent increase and decrease, they may want to describe the scale factor as a percent change. For example, the side lengths of Triangle D are $$50\%$$ the side lengths of Triangle O. Make sure that these students understand that a $$50\%$$ decrease in length corresponds to a scale factor of $$\frac{1}{2}$$. In a later lesson, students will identify the effect a scale factor has on the area of a figure.

### Launch

Arrange students into groups of 4. Assign each student one of the following pairs of triangles in the first question.

• A and E
• B and F
• C and G
• D and H

Give students 2 minutes of quiet think time to determine if their assigned triangles are scaled copies of the original triangle. Give another 2–3 minutes to discuss their responses and complete the first question in groups.

Discuss briefly as a class which triangles are scaled copies and select a couple of groups who reasoned in terms of lengths and angles to explain their reasoning. Some guiding questions:

• What information did you use to tell scaled copies from those that are not?
• How were you able to tell right away that some figures are not scaled copies?

Give students quiet work time to complete the rest of the task after the class recognizes that A, C, F, and H are not scaled copies.

### Student Facing

Here is Triangle O, followed by a number of other triangles.

Your teacher will assign you two of the triangles to look at.

1. For each of your assigned triangles, is it a scaled copy of Triangle O? Be prepared to explain your reasoning.
2. As a group, identify all the scaled copies of Triangle O in the collection. Discuss your thinking. If you disagree, work to reach an agreement.
3. List all the triangles that are scaled copies in the table. Record the side lengths that correspond to the side lengths of Triangle O listed in each column.
Triangle O 3 4 5
4. Explain or show how each copy has been scaled from the original (Triangle O).

### Student Facing

#### Are you ready for more?

Choose one of the triangles that is not a scaled copy of Triangle O. Describe how you could change at least one side to make a scaled copy, while leaving at least one side unchanged.

### Anticipated Misconceptions

Students may think that Triangle F is a scaled copy because just like the 3-4-5 triangle, the sides are also three consecutive whole numbers. Point out that corresponding angles are not equal.

### Activity Synthesis

Display the image of all triangles and invite a couple of students to share how they knew which sides of the triangles correspond. Then, display a completed table in the third question for all to see. Ask each group to present its observations about one triangle and how the triangle has been scaled from the original. Encourage the use of “corresponding” in their explanations. As students present, record or illustrate their reasoning on the table, e.g., by drawing arrows between rows and annotating with the operation students are describing, as shown here.

Use the language that students use to describe the side lengths and the numerical relationships in the table to guide students toward scale factor. For example: “You explained that the lengths in Triangle F are all twice those in the original triangle, so we can write those as “2 times” the original numbers. Lengths in Triangle A are half of those in the original; we can write “$$\frac12$$ times” the original numbers. We call those multipliers—the 2 and the $$\frac12$$-scale factors. We say that scaling Triangle O by a scale factor of 2 produces Triangle F, and that scaling Triangle O by $$\frac12$$ produces Triangle A.”

## 2.3: Which Operations? (Part 1) (10 minutes)

### Activity

The purpose of this activity is to contrast the effects of multiplying side lengths versus adding to side lengths when creating copies of a polygon. To find the corresponding side lengths on a scaled copy, the side lengths of a figure are all multiplied (or divided) by the same number. However, students often mistakenly think that adding or subtracting the same number to all the side lengths will also create a scaled copy. When students recognize that there is a multiplicative relationship between the side lengths rather than an additive one, they are looking for and making use of structure (MP7).

Monitor for students who:

• notice that Diego's copy is no longer a polygon while Jada's still is
• notice that the relationships between side lengths in Diego's copy have changed (e.g., Side 1 is twice as long as Side 2 in the original but is not twice as long as Side 2 in the copy.) while in Jada's copy they have not
• notice that all the corresponding angles have equal measures (i.e., 90 or 270 degrees)
• describe Jada's copy as having all side lengths divided by 3
• describe Jada's copy as having all side lengths a third as long as their original lengths
• describe Jada's copy as having a scale factor of $$\frac13$$

### Launch

Give students 2–3 minutes of quiet think time, and then 2 minutes to share their thinking with a partner.  See MLR 3 (Clarify, Critique, Correct) and use the strategy "Critique a Partial or Flawed Explanation".

Engagement: Internalize Self Regulation. Demonstrate giving and receiving constructive feedback. Use a structured process and display sentence frames to support productive feedback. For example, “How did you get…?,” “How do you know…?,” and “That could/couldn’t be true because…”
Supports accessibility for: Social-emotional skills; Organization; Language

### Student Facing

Diego and Jada want to scale this polygon so the side that corresponds to 15 units in the original is 5 units in the scaled copy.

Diego and Jada each use a different operation to find the new side lengths. Here are their finished drawings.

1. What operation do you think Diego used to calculate the lengths for his drawing?
2. What operation do you think Jada used to calculate the lengths for her drawing?
3. Did each method produce a scaled copy of the polygon? Explain your reasoning.

### Activity Synthesis

Invite previously-selected students to share their answers and reasoning. Sequence their explanations from most general to most technical.

Before moving to the next activity, consider asking questions like these:

• What is the scale factor used to create Jada’s drawing? What about for Diego’s drawing? ($$\frac13$$ for Jada's; there isn't one for Diego's, because it is not a scaled copy.)
• What can you say about the corresponding angles in Jada and Diego’s drawings? (They are all equal, even though one is a scaled copy and one is not.)
• Subtraction of side lengths does not (usually) produce scaled copies. Do you think addition would work? (Answers vary.)

Note: There are rare cases when adding or subtracting the same length from each side of a polygon (and keeping the angles the same) will produce a scaled copy, namely if all side lengths are the same. If not mentioned by students, it is not important to discuss this at this point.

Representing, writing, and speaking: Math Language Routine 3 Clarify, Critique, Correct. This is the first time Math Language Routine 3 is suggested as a support in this course. In this routine, students are given an incorrect or incomplete piece of mathematical work. This may be in the form of a written statement, drawing, problem-solving steps, or another mathematical representation. Students analyze, reflect on, and improve the written work by correcting errors and clarifying meaning. Typical prompts are: “Is anything unclear?” and/or “Are there any reasoning errors?” The purpose of this routine is to engage students in analyzing mathematical thinking that is not their own, and to solidify their knowledge through communicating about conceptual errors and ambiguities in language.
Design Principle(s): Support sense-making; Optimize output (for reasoning)

How It Happens:

1. Play the role of Diego and present the following statement along with his flawed drawing to the class. “I used a scale factor of minus 10, and Jada used a scale factor of one third. So my drawing is a different kind of scaled copy from Jada’s.”

Ask students, “What steps did Diego take to make the drawing?” and “Did he create a scaled copy? How do you know?”

2. Give students 1 minute of quiet think time to analyze the statement, and then 3 minutes to work on improving the statement with a partner.

As pairs discuss, provide these sentence frames for scaffolding: “I believe Diego created the drawing by ___ because ___.”, “Diego created/did not create a scaled copy. I know this because ___.”, “You can’t ___ because ___.” Encourage the listener to ask clarifying questions by referring to the statement and the drawings. Allow each partner to take a turn as the speaker and listener.

Listen for students identifying the type of operation used and justification for whether or not a scaled drawing was produced. Have the pairs reach a mutual understanding and agreement on a correct statement about Diego’s drawing.

3. Invite 3 or 4 pairs to present their improved statement to the class, both orally and in writing. . Ask students to listen for order/time transition words (first, next, then, etc.), and any elements of justifications (e.g., First, ___ because ___.).

Here are two sample improved statements:

“I subtracted 10 from each side length and Jada used a scale factor of one third. So my drawing is not a scaled copy and Jada’s is. Jada’s is a scaled copy because I know that multiplying—not subtracting—creates a scaled copy. Her drawing created a polygon with no gaps.”
or
“I minused 10 from each side, but I should have realized that in order to scale 15 units in the original down to 5 units in the copy, you have to divide by 3. Jada used a scale factor of one third, which is the same as dividing by 3. My drawing is not a scaled copy and Jada’s is because hers is not a polygon with no gaps, and minusing 10 is not a scale factor.”

Call attention to statements that generalize that the method for finding the side lengths of a scaled copy is by multiplying or dividing, not adding or subtracting. Revoice student thoughts with an emphasis on knowing whether or not they created a scaled polygon.

4. Close the conversation on Diego’s drawing, discuss the accuracy of Jada’s scaled copy, and then move on to the next lesson activity.

## 2.4: Which Operations? (Part 2) (10 minutes)

### Activity

In the previous activity, students saw that subtracting the same value from all side lengths of a polygon did not produce a (smaller) scaled copy. This activity makes the case that adding the same value to all lengths also does not produce a (larger) scaled copy, reinforcing the idea that scaling involves multiplication.

This activity gives students a chance to draw a scaled copy without a grid and to use paper as a measuring tool. To create a copy using a scale factor of 2, students need to mark the length of each original segment and transfer it twice onto their drawing surface, reinforcing—in a tactile way—the meaning of scale factor. The angles in the polygon are right angles (and a 270 degree angle in one case) and can be made using the corner of an index card.

Some students may struggle to figure out how to use an index card or a sheet of paper to measure lengths. Before demonstrating, encourage them to think about how a length in the given polygon could be copied onto an index card and used as an increment for measuring. If needed, show how to mark the 4-unit length along the edge of a card and to use the mark to determine the needed lengths for the copy.

Teacher Notes for IM 6–8 Math Accelerated
This activity is optional. Use this activity if your students need more practice with the ideas from Which Operations? (Part 1).

### Launch

Have students read the task statement and check that they understand which side of the polygon Andre would like to be 8 units long on his drawing. Provide access to index cards, so that students can use it as a measuring tool. Consider not explicitly directing students as to its use to give them a chance to use tools strategically (MP5). Give students 5–6 minutes of quiet work time, and then 2 minutes to share their work with a partner.

### Student Facing

Andre wants to make a scaled copy of Jada's drawing so the side that corresponds to 4 units in Jada’s polygon is 8 units in his scaled copy.

1. Andre says “I wonder if I should add 4 units to the lengths of all of the segments?” What would you say in response to Andre? Explain or show your reasoning.

2. Create the scaled copy that Andre wants. If you get stuck, consider using the edge of an index card or paper to measure the lengths needed to draw the copy.

### Student Facing

#### Are you ready for more?

The side lengths of Triangle B are all 5 more than the side lengths of Triangle A. Can Triangle B be a scaled copy of Triangle A? Explain your reasoning.

### Anticipated Misconceptions

Some students might not be convinced that making each segment 4 units longer will not work. To show that adding 4 units would work, they might simply redraw the polygon and write side lengths that are 4 units longer, regardless of whether the numbers match the actual lengths. Urge them to check the side lengths by measuring. Tell them (or show, if needed) how the 4-unit length in Jada’s drawing could be used as a measuring unit and added to all sides.

Other students might add 4 units to all sides and manage to make a polygon but changing the angles along the way. If students do so to make the case that the copy will not be scaled, consider sharing their illustrations with the class, as these can help to counter the idea that “scaling involves adding.” If, however, students do this to show that adding 4 units all around does work, address the misconception. Ask them to recall the size of corresponding angles in scaled copies, or remind them that angles in a scaled copy are the same size as their counterparts in the original figure.

### Activity Synthesis

The purpose of the activity is to explicitly call out a potential misunderstanding of how scale factors work, emphasizing that scale factors work by multiplying existing side lengths by a common factor, rather than adding a common length to each.

Invite a couple of students to share their explanations or illustrations that adding 4 units to the length of each segment would not work (e.g. the copy is no longer a polygon, or the copy has angles that are different than in the original figure). Then, select a couple of other students to show their scaled copies and share how they created the copies. Consider asking:

• What scale factor did you use to create your copy? Why?
• How did you use an index card (or a sheet of paper) to measure the lengths for the copy?
• How did you measure the angles for the copy?

Speaking: Math Language Routine 7 Compare and Connect. This is the first time Math Language Routine 7 is suggested as a support in this course. In this routine, students are given a problem that can be approached using multiple strategies or representations, and are asked to prepare a visual display of their method. Students then engage in investigating the strategies (by means of a teacher-led gallery walk, partner exchange, group presentation, etc.), compare approaches, and identify correspondences between different representations. A typical discussion prompt is “What is the same and what is different?”, comparing their own strategy to the others. The purpose of this routine is to allow students to make sense of mathematical strategies by identifying, comparing, contrasting, and connecting other approaches to their own, and to develop students’ awareness of the language used through constructive conversations.
Design Principle(s): Maximize meta-awareness

How It Happens:

1. Use this routine to compare and contrast different methods for creating scaled copies of Jada’s drawing. Before selecting students to share a display of their method with the whole class, first give students an opportunity to do this in a group of 3–4.

Invite students to quietly investigate each other’s work. Ask students to consider what is the same and what is different about each display. Invite students to give a step-by-step explanation of their method using this sentence frame: “In order to create the copy, first I…. Next,…. Then, …. Finally,….”. Allow 1–2 minutes for each display and signal when it is time to switch.

2. Next, give each student the opportunity to add detail to their own display for 1-2 minutes. As students work on their displays, circulate the room to identify at least two different methods or two different ways of representing a method. Also look for methods that were only partially successful.

3. Consider selecting 1–2 students to share methods that were only partially successful in producing scaled copies. Then, select a couple of students to share displays of methods that did produce scaled copies.

Draw students’ attention to the approaches used in each drawing (e.g., adding the same value to each side length, not attending to the angles, multiplying by a common factor, not creating a polygon, etc.). Ask students, “Did this approach create a scaled copy? Why or why not?”

4. After the pre-selected students have finished sharing with the whole class, lead a discussion comparing, contrasting, and connecting the different approaches and representations.

In this discussion, demonstrate using the mathematical language “scale factor”, “corresponding”, and “multiplicative” to amplify student language.

Consider using these prompts:

• “How did the scale factor show up in each method?”,
• “Why did the different approaches lead to the same outcome?”,
• “What worked well in _____’s approach/representation? What did not work well?”, and
• “What role does multiplication play in each approach?”
5. Close the discussion by inviting 3 students to revoice the incorrect method for creating a scaled drawing, and then invite 3 different students to revoice the correct method for creating a scaled drawing. Then, transition back to the Lesson Synthesis and Cool Down.

## Lesson Synthesis

### Lesson Synthesis

Here are some questions for discussion:

• “How do we draw a scaled copy of a figure?” (To draw a scaled copy of a figure, we need to multiply all of the lengths by the scale factor.)
• “Can we create scaled copies by adding or subtracting the same value from all lengths? Why or why not?” (No, because scaling is a multiplicative process. We saw from Diego and Jada's drawings that subtracting the same value to all lengths will not create scaled copies.)

## 2.5: Cool-down - More Scaled Copies (5 minutes)

### Cool-Down

The scale factor between Polygon 1 and Polygon 2 is 2, because all of the lengths in Polygon 2 are 2 times the corresponding lengths in Polygon 1. The angle measures in Polygon 2 are the same as the corresponding angle measures in Polygon 1. For example, the measure of angle $$JKL$$ is the same as the measure of angle $$DEF$$.
For example, to make a scaled copy of triangle $$ABC$$ where the base is 8 units, we would use a scale factor of 4. This means multiplying all the side lengths by 4, so in triangle $$DEF$$, each side is 4 times as long as the corresponding side in triangle $$ABC$$.