# Lesson 3

Scaled Relationships

## 3.1: Three Quadrilaterals (Part 1) (5 minutes)

### Warm-up

This warm-up gives students a chance to practice identifying corresponding angles of scaled copies, measure angles using a protractor, and test their earlier conjecture that corresponding angles have the same measure.

### Launch

Have students look at the figures in the activity, and ask “What do you notice? What do you wonder?” Call out in particular questions about the angles in the figures (e.g., whether corresponding ones have the same measure). Tell students that they will test their previous observation about the angles of scaled figures, this time by using protractors instead of tracing paper.

Provide access to protractors. Clear protractors with no holes and with radial lines printed on them are recommended here. Some angles may be challenging to measure because of the size of the polygons. If students find the sides of a polygon not long enough to accommodate angle measurements, suggest that they extend the lines, or demonstrate how to do so (especially if available protractors are opaque with holes in the middle).

### Student Facing

Each of these polygons is a scaled copy of the others.

1. Name two pairs of corresponding angles. What can you say about the sizes of these angles?
2. Check your prediction by measuring at least one pair of corresponding angles using a protractor. Record your measurements to the nearest $$5^\circ$$.

### Anticipated Misconceptions

Some students may read the wrong number on the protractor, moving down from the $$180^\circ$$ mark instead of up from the $$0^\circ$$ mark, or reading the measurement outside of one of the lines forming the angle instead of between the two lines. Clarify the angle being measured, how to line up the protractor, or how to read the markings correctly.

### Activity Synthesis

Select a few students to share their angle measurements and poll the class briefly for agreement and disagreement. Discuss major discrepancies, if any. Students should be able to confirm that all corresponding angles in the scaled polygons are equal.

If desired, ask students whether recording the angles to the nearest 1 degree would be appropriate: in general, the thickness of the line segments and the markings on the protractor limit accuracy, so reporting to the nearest 5 degrees is appropriate (as long as none of the angles are too close to halfway between two increments).

## 3.2: Three Quadrilaterals (Part 2) (10 minutes)

### Activity

Students have seen that the lengths of corresponding segments in a figure and its scaled copy vary by the same scale factor. Here, they learn that in such a pair of figures, any corresponding distances—not limited to lengths of sides or segments—are related by the same scale factor. The side lengths of the polygons in this task cannot be easily determined, so students must look to other distances to compare.

Students must take care when they identify corresponding vertices and distances. As students work, urge them to attend to the order in which points or segments are listed.

If students are not sure what to make out of the values in the table (for the second question), encourage them to consider the corresponding distances of two figures at a time. For example, ask: What do you notice about the corresponding vertical distances in $$IJKL$$ and $$EFGH$$? What about the corresponding horizontal distances in those two figures?

Teacher Notes for IM 6–8 Math Accelerated

Adjust this activity to 15 minutes. Conclude the activity synthesis by displaying this image:

Ask, “Here are two quadrilaterals. Kiran says that Polygon $$EFGH$$ is a scaled copy of $$ABCD$$, but Lin disagrees. Do you agree with either of them?” (Lin is correct, because the corresponding distances are not multiplied by the same number. Kiran may have noticed that the corresponding angles are equal and thought this meant the polygons are scaled copies. I noticed that the scale factors for the corresponding sides are not the same. $$AB$$ and $$EF$$ are related by a scale factor of $$\frac23$$, but $$DC$$ and $$HG$$ are related by a scale factor of $$\frac12$$.)

After a brief quiet think time, select students to share their responses. The goal of this discussion is to make clear that angle measurements and distances are both important when deciding whether two polygons are scaled copies.

### Launch

Arrange students in groups of 2. Ask if they can tell the lengths of segments $$GF$$ or $$DC$$ from the grid (without using rulers). Explain that they will explore another way to compare length measurements in scaled copies.

Give students 2–3 minutes of quiet work time for the first two questions, and 1 minute to discuss their responses with a partner before continuing on to the last question.

### Student Facing

Each of these polygons is a scaled copy of the others. You already checked their corresponding angles.

1. The side lengths of the polygons are hard to tell from the grid, but there are other corresponding distances that are easier to compare. Identify the distances in the other two polygons that correspond to $$DB$$ and $$AC$$, and record them in the table.
corresponds to $$DB$$
distance that
corresponds to $$AC$$
$$ABCD$$ $$DB = 4$$ $$AC = 6$$
$$EFGH$$
$$IJKL$$
2. Look at the values in the table. What do you notice?

3. The larger figure is a scaled copy of the smaller figure.

1. If $$AE = 4$$, how long is the corresponding distance in the second figure? Explain or show your reasoning.
2. If $$IK = 5$$, how long is the corresponding distance in the first figure? Explain or show your reasoning.

### Anticipated Misconceptions

Students may list the corresponding vertices for distances in the wrong order. For example, instead of writing $$IK$$ as the distance corresponding to $$DB$$, they may write $$KI$$. Remind students of the corresponding points by asking, “Which vertex in $$IJKL$$ corresponds to $$D$$? Which corresponds to $$B$$?” and have them match the order of the vertices accordingly.

### Activity Synthesis

Display the completed tables for all to see. To highlight how all distances in a scaled copy (not just the side lengths of the figure) are related by the same scale factor, discuss:

• How does the vertical distance in $$ABCD$$ compare to that in $$EFGH$$? How do the horizontal distances in the two polygons compare? Do the pairs of vertical and horizontal distances share the same scale factor?
• How do the vertical distances in $$EFGH$$ and $$IJKL$$ compare? What about the horizontal distances? Is there a common scale factor? What is that scale factor?
• What scale factor relates the corresponding lengths and distances in the two drawings of the letter W?
Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between figures. For example, mark corresponding line segments using the same color.
Supports accessibility for: Visual-spatial processing
Speaking: MLR7 Compare and Connect. Use this routine to call attention to the different ways students may identify scale factors. Display the following statements: “The scale factor from $$EFGH$$ to $$IJKL$$ is 3,” and “The scale factor from $$EFGH$$ to $$IJKL$$ is $$\frac13$$.” Give students 2 minutes of quiet think time to read and consider whether either or both of the statements are correct. Invite students to share their initial thinking with a partner before selecting 2–3 students to share with the class. In this discussion, listen for and amplify any comments that refer to the order of the original figure and its scaled copy, as well as those who identify corresponding vertices and distances. Draw students’ attention to the different ways to describe the relationships between scaled copies and the original figure.
Design Principle(s): Maximize meta-awareness

## 3.3: Card Sort: Scaled Copies (15 minutes)

### Activity

Students have studied many examples of scaled copies and know that corresponding lengths in a figure and its scaled copy are related by the same scale factor. The purpose of this activity is for students to examine how the size of the scale factor is related to the original figure and the scaled copy. The activity serves several purposes:

1. To reinforce students’ awareness of scale factors
2. To draw attention to how scaled copies behave when the scale factor is 1, less than 1, and greater than 1; and
3. To help students notice that reciprocal scale factors reverse the scaling.

You will need the Scaled Copies Card Sort blackline master for this activity. Here is an image of the cards for your reference and planning.

Monitor for students who group the cards in terms of:

• Specific scale factors (e.g., 2, 3, $$\frac12$$, etc.)
• Ranges of scale factors producing certain effects (e.g., factors producing larger, unchanged, or smaller copies)
• Reciprocal scale factors (e.g., one factor scales Figure A to B, and its reciprocal reverses the scaling)

Select groups who use each of these approaches (and any others) and ask them to share during the discussion.

### Launch

Arrange students in groups of 3–4. Distribute one set of slips to each group. Give students 7–8 minutes of group work time, followed by whole-class discussion.

Action and Expression: Internalize Executive Functions. Begin with a small-group or whole-class demonstration and think aloud using Card 1 to remind students how to determine the scale factor between Figure A and Figure B.  Keep the worked-out calculations on display for students to reference as they work.
Supports accessibility for: Memory; Conceptual processing

### Student Facing

Your teacher will give you a set of cards. On each card, Figure A is the original and Figure B is a scaled copy.

1. Sort the cards based on their scale factors. Be prepared to explain your reasoning.

2. Examine cards 10 and 13 more closely. What do you notice about the shapes and sizes of the figures? What do you notice about the scale factors?

3. Examine cards 8 and 12 more closely. What do you notice about the figures? What do you notice about the scale factors?

### Student Facing

#### Are you ready for more?

Triangle B is a scaled copy of Triangle A with scale factor $$\frac12$$

1. How many times bigger are the side lengths of Triangle B when compared with Triangle A?
2. Imagine you scale Triangle B by a scale factor of $$\frac12$$ to get Triangle C. How many times bigger will the side lengths of Triangle C be when compared with Triangle A?
3. Triangle B has been scaled once. Triangle C has been scaled twice. Imagine you scale triangle A $$n$$ times to get Triangle N, always using a scale factor of $$\frac12$$. How many times bigger will the side lengths of Triangle N be when compared with Triangle A?

### Anticipated Misconceptions

Students may sort by the types of figures rather than by how the second figure in each pair is scaled from the first. Remind students to sort based on how Figure A is scaled to create Figure B.

Students may think of the change in lengths between Figures A and B in terms of addition or subtraction, rather than multiplication or division. Remind students of an earlier lesson in which they explored the effect of subtracting the same length from each side of a polygon in order to scale it. What happened to the copy? (It did not end up being a polygon and was not a scaled copy of the original one.)

Students may be unclear as to how to describe how much larger or smaller a figure is, or may not recall the meaning of scale factor. Have them compare the lengths of each side of the figure. What is the common factor by which each side is multiplied?

### Activity Synthesis

Select groups to explain their sorting decisions following the sequence listed in the Activity Narrative. If no groups sorted in terms of ranges of scale factors (less than 1, exactly 1, and greater than 1) or reciprocal scaling, ask:

• What can we say about the scale factors that produce larger copies? Smaller copies? Same-size copies?
• Some cards had the same pair of figures on them, just in a reversed order (i.e., pairs #1 and 7, #10 and 13). What do you notice about their scale factors?

Highlight the two main ideas of the lesson: 1) the effects of scale factors that are greater than 1, exactly 1, and less than 1; and 2) the reversibility of scaling. Point out that if Figure B is a scaled copy of Figure A, then A is also a scaled copy of B. In other words, A and B are scaled copies of one another, and their scale factors are reciprocals.

Suggest students add these observations to their answer for the last question.

Writing: MLR3 Clarify, Critique, Correct. Present an incorrect statement that reflects a possible misunderstanding from the class for the last prompt. For example, “The scale factor of cards 8 and 12 is 0 because the shapes are the same and there was no change.” Prompt students to identify the error, and then write a correct version. In this discussion, highlight the use of disciplinary language by revoicing student ideas. This helps students evaluate, and improve on, the written mathematical arguments of others.
Design Principle(s): Maximize meta-awareness

## 3.4: Scaling A Puzzle (15 minutes)

### Optional activity

This activity gives students a chance to apply what they know about scale factors, lengths, and angles and create scaled copies without the support of a grid. Students work in groups of 3 to complete a jigsaw puzzle, each group member scaling 2 non-adjacent pieces of a 6-piece puzzle with a scale factor of $$\frac12$$. The group then assembles the scaled pieces and examines the accuracy of their scaled puzzle. Consider having students use a color in place of the cross hatching.

As students work, notice how they measure distances and whether they consider angles. Depending on how students determine scaled distances, they may not need to transfer angles. Look out for students who measure only the lengths of drawn segments rather than distances, e.g., between the corner of a square and where a segment begins. Suggest that they consider other measurements that might help them locate the beginning and end of a segment.

You will need the Scaling a Puzzle blackline master for this activity.

Teacher Notes for IM 6–8 Math Accelerated
If time allows, allow students to start this activity in class and then finish as part of the practice problem set.

### Launch

Arrange students in groups of 3. Give pre-cut puzzle squares 1 and 5 to one student in the group, squares 2 and 6 to a second student, and squares 3 and 4 to the third. After students have answered the first question, give each student 2 blank squares cut from the second section of the blackline master, whose sides are half of the side length of the puzzle squares. Provide access to geometry toolkits.

Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success. Before students begin, ask them to predict which tools, from their geometry toolkits, they anticipate they will need, and to describe how they might use them. During the activity, make sure that students are using their rulers and protractors correctly.
Supports accessibility for: Visual-spatial processing; Conceptual processing; Fine-motor skills

### Student Facing

Your teacher will give you 2 pieces of a 6-piece puzzle.

1. If you drew scaled copies of your puzzle pieces using a scale factor of $$\frac12$$, would they be larger or smaller than the original pieces? How do you know?
2. Create a scaled copy of each puzzle piece on a blank square, with a scale factor of $$\frac12$$.
3. When everyone in your group is finished, put all 6 of the original puzzle pieces together like this: Next, put all 6 of your scaled copies together. Compare your scaled puzzle with the original puzzle. Which parts seem to be scaled correctly and which seem off? What might have caused those parts to be off?
4. Revise any of the scaled copies that may have been drawn incorrectly.
5. If you were to lose one of the pieces of the original puzzle, but still had the scaled copy, how could you recreate the lost piece?

### Anticipated Misconceptions

Students may incorporate the scale factor when scaling line segments but neglect to do so when scaling distances between two points not connected by a segment. Remind them that all distances are scaled by the same factor.

Students may not remember to verify that the angles in their copies must remain the same as the original. Ask them to notice the angles and recall what happens to angles when a figure is a scaled copy.

### Activity Synthesis

Much of the conversations about creating accurate scaled copies will have taken place among partners, but consider coming together as a class to reflect on the different ways students worked. Ask questions such as:

• How is this task more challenging than creating scaled copies of polygons on a grid?
• Besides distances or lengths, what helped you create an accurate copy?
• How did you know or decide which distances to measure?
• Before your drawings were assembled, how did you check if they were correct?

Student responses to these questions may differ: for example, for piece 6, the two lines can be drawn by measuring distances on the border of the puzzle piece; the angles work out correctly automatically. For piece 2, however, to get the three lines that meet in a point in the middle of the piece just right, students can either measure angles, or extend those line segments until they meet the border of the piece (and then measure distances).

## Lesson Synthesis

### Lesson Synthesis

When a scaled copy is created from a figure, we know that:

• The distances between any two points in the original figure, even those not connected by drawn segments, are scaled by the same scale factor.
• The corresponding angles in the original figure and scaled copies are congruent.

Polygons are a perfect context in which to apply these two ideas, being made up of line segments meeting at angles. So we can use these observations to check whether a polygon is actually a scaled copy of another. If all the corresponding angles are the same size and all corresponding distances are all scaled by the same factor, then we can conclude that it is a scaled copy of the other.

Here are some questions for discussion:

• “How is showing that a figure is not a scaled copy of another alike or different from showing that two figures are not congruent?” (Like congruence, showing that one figure is not a scaled copy only takes a single pair of distances between two points that are not scaled by the same factor.)
• “What happens to the copy when it is created with a scale factor greater than 1? Less than 1? Exactly 1?” (When the scale factor is greater than 1, the scaled copy is larger than the original. When it is less than 1, the copy is smaller than the original. A scale factor of exactly 1 produces a same-size copy.)
• “How can we reverse the scaling to get back to the original figure when we have a scaled copy?” (Scaling can be reversed by using reciprocal factors. If we scale Figure A by a factor of 4 to obtain Figure B, we can scale B back to A using a factor of $$\frac{1}{4}$$. This means that if B is a scaled copy of A, A is also a scaled copy of B. As such, they are scaled copies of each other.)

## Student Lesson Summary

### Student Facing

When a figure is a scaled copy of another figure, we know that:

• All distances in the copy can be found by multiplying the corresponding distances in the original figure by the same scale factor, whether or not the endpoints are connected by a segment.

For example, Polygon $$STUVWX$$ is a scaled copy of Polygon $$ABCDEF$$. The scale factor is 3. The distance from $$T$$ to $$X$$ is 6, which is three times the distance from $$B$$ to $$F$$.

• All angles in the copy have the same measure as the corresponding angles in the original figure, as in these triangles.

These observations can help explain why one figure is not a scaled copy of another.

For example, even though their corresponding angles have the same measure, the second rectangle is not a scaled copy of the first rectangle, because different pairs of corresponding lengths have different scale factors, $$2 \boldcdot \frac12 = 1$$ but $$3 \boldcdot \frac23 = 2$$.

When one figure is a scaled copy of another, the size of the scale factor affects the size of the copy. When a figure is scaled by a scale factor greater than 1, the copy is larger than the original. When the scale factor is less than 1, the copy is smaller. When the scale factor is exactly 1, the copy is the same size as the original.

Triangle $$DEF$$ is a larger scaled copy of triangle $$ABC$$, because the scale factor from $$ABC$$ to $$DEF$$ is $$\frac32$$. Triangle $$ABC$$ is a smaller scaled copy of triangle $$DEF$$, because the scale factor from $$DEF$$ to $$ABC$$ is $$\frac23$$.

This means that triangles $$ABC$$ and $$DEF$$ are scaled copies of each other. It also shows that scaling can be reversed using reciprocal scale factors, such as $$\frac23$$ and $$\frac32$$.

In other words, if we scale Figure A using a scale factor of 4 to create Figure B, we can scale Figure B using the reciprocal scale factor, $$\frac14$$, to create Figure A.