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Unit 1, Lesson 1

What Are Scaled Copies?

Grade 7

1.1

Warm-up

Instructional Routines

None

Materials

To Copy (from Blackline Masters)

6–12 Blank Math Community Chart

To Gather

Chart paper Sticky notes

Activity Narrative

This opening task introduces the term scaled copy. It prompts students to observe several copies of a picture, visually distinguish scaled and unscaled copies, and articulate the differences in their own words. Besides allowing students to have a mathematical conversation about properties of figures, it provides an accessible entry into the concept and gives an opportunity to hear the language and ideas that students associate with scaled figures.

Students are likely to have some intuition about the term “to scale,” either from previous work in grade 6 (e.g., scaling a recipe, or scaling a quantity up or down on a double number line) or from outside the classroom. This intuition can help them identify scaled copies. As students apply their previous experience with scaling to analyze the images, they are making sense of problems (MP1).

Expect them to use adjectives such as “stretched,” “squished,” “skewed,” “reduced,” and so on, in imprecise ways. This is fine because students’ intuitive definitions of scaled copies will be refined over the course of the lesson. As students discuss the pictures, note the range of descriptions used. Monitor for students whose descriptions are particularly supportive of the idea that lengths in a scaled copy are found by multiplying the original lengths by the same value. Invite them to share their responses later.

In the digital version of the activity, students use an applet to manipulate copies of the original image. The applet allows students to explore different transformations of the original portrait. This activity works best when each student has access to the applet because students will benefit from seeing the transformations in a dynamic way. If students don't have individual access, displaying the applet for all to see would be helpful during the launch and synthesis.

Activity Synthesis

Select a few students to share their observations. Record and display students’ explanations for the second question. Consider organizing the observations in terms of how certain pictures are or are not distorted. For example, students may say that C and D are scaled copies because each is a larger or smaller version of the picture, but the face (or the sleeve, or the outline of the picture) has not changed in shape. They may say that A, B, and E are not scaled copies because something other than size has changed. If not already mentioned in the discussion, guide students in seeing features of C and D that distinguish them from A, B, and E.

Invite a couple of students to share their working definition of scaled copies. Some of the students’ descriptions may not be completely accurate. That is appropriate for this lesson because the goal is to build on and refine this language over the course of the next few lessons until students have a more precise notion of what it means for a picture or figure to be a scaled copy.

Math Community
After the Warm-up, tell students that today is the start of planning the type of mathematical community they want to be a part of for this school year. The start of this work will take several weeks as the class gets to know one another, reflects on past classroom experiences, and shares their hopes for the year.

Display and read aloud the question “What do you think it should look like and sound like to do math together as a mathematical community?” Give students 2 minutes of quiet think time and then 1–2 minutes to share with a partner. Ask students to record their thoughts on sticky notes and then place the notes on the sheet of chart paper. Thank students for sharing their thoughts and tell them that the sticky notes will be collected into a class chart and used at the start of the next discussion.

After the lesson is complete, review the sticky notes to identify themes. Make a Math Community Chart to display in the classroom. See the blackline master Blank Math Community Chart for one way to set up this chart. Depending on resources and wall space, this may look like a chart paper hung on the wall, a regular sheet of paper to display using a document camera, or a digital version that can be projected. Add the identified themes from the students’ sticky notes to the student section of the “Doing Math” column of the chart.

1.2

Activity

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

The goal of this activity is for students to describe the characteristics of scaled copies more precisely and to refine the meaning of the term. Students observe copies of a line drawing on a grid and notice how the lengths of line segments and the angles formed by them compare to those in the original drawing.

As students identify distinguishing features of the scaled copies and recognize that corresponding parts increase by the same scale factor, they are making use of structure (MP7).

For the first question, expect students to explain their choices of scaled copies in intuitive, qualitative terms. For the second question, students should begin to distinguish scaled and unscaled copies in more specific and quantifiable ways. If it does not occur to students to look at lengths of segments, suggest they do so.

As students work, monitor for students who notice the following aspects of the figures. Students are not expected to use these mathematical terms at this point, however.

The original drawing of the letter F and its scaled copies have equivalent width-to-height ratios.
We can use a scale factor (or a multiplier) to compare the lengths of different figures and see if they are scaled copies of the original.
The original figure and scaled copies have corresponding angles that have the same measure.

In the digital version of the activity, students use an applet to create a scaled copy of the original letter F. The applet allows students to draw segments on a grid. The digital version may help students draw quickly and accurately so they can focus more on the mathematical analysis of corresponding lengths.

Activity Synthesis

Display the seven copies of the letter F for all to see. For each copy, ask students to indicate whether they think it is a scaled copy of the original F. Record and display the results for all to see. For any drawing where there is disagreement, ask 1–2 students to share why they think that drawing is not a scaled copy.

To help students generalize about the characteristics of scaled copies, consider asking questions such as:

“What features do the scaled copies have in common?” (The sides all change by the same amount. The angles stay the same.)
“How do the other copies fail to show these features?" (Some lengths in the copy get bigger or smaller by different amounts. Sometimes the angles in the copy do not match the angles in the original.)

If there is a misconception that scaled copies must have vertices on intersections of grid lines, use Drawing 1 (or a relevant drawing by a student) to discuss how that is not the case.

Some students may not be familiar with words such as “twice,” “double,” or “triple.” Clarify the meanings of these words by also saying “two times as long” or “three times as long.”

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer collaboration. When students share their work with a partner, display sentence frames to support conversation, such as “It looks like . . . .” “That could/couldn’t be true because . . . .” “One thing that is the same is . . . .” or “One thing that is different is . . . .”
Supports accessibility for: Language, Social-Emotional Functioning

1.3

Activity

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

MLR8: Discussion Supports Take Turns

Materials

To Copy (from Blackline Masters)

Pairs of Scaled Polygons Cards

Activity Narrative

In this partner activity, students take turns matching pairs of polygons that are scaled copies. The polygons appear comparable to one another, so students need to look very closely at all side lengths of the polygons to tell if they are scaled copies.

As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3). Monitor for students who use precise language (MP6) to articulate their reasoning (for example, “The top side of A is half the length of the top side of G, but the vertical sides of A are a third of the lengths of those in G.”).

Launch

Choose a student to be your partner and demonstrate how to set up and do the matching activity:

Mix up the cards and place them face-up.
Select two cards and then explain to your partner why you think the cards do or do not match (in other words, whether the polygons are or are not scaled copies).
The partner's job is to listen and make sure they agree. If they don't agree, the partners discuss until they come to an agreement. Consider demonstrating productive ways to agree or disagree, such as explaining your mathematical thinking, asking clarifying questions, etc.
When both partners agree on the match, they switch roles.

Arrange students in groups of 2. Give each group a set of 10 slips cut from the blackline master. Tell students that each polygon has one and only one match. Ask students to take turns finding scaled copies and explaining their thinking.

Representation: Internalize Comprehension. Use multiple examples and non-examples to emphasize relationships between scaled copies.
Supports accessibility for: Conceptual Processing, Attention

MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “ matches because. . . . “ and “I noticed , so I matched . . . .”
Advances: Speaking, Conversing

Activity

None

Your teacher will give you a set of cards that have polygons drawn on a grid. Mix up the cards and place them all face up.

Take turns with your partner to match a polygon with another polygon that is a scaled copy.
1. For each match you find, explain to your partner how you know it’s a match.
2. For each match your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
When you agree on all of the matches, check your answers with the answer key. If there are any errors, discuss why and revise your matches.
Select one pair of polygons to examine further. Explain or show how you know that one polygon is a scaled copy of the other. You can draw both polygons on the grid if it helps.

Student Response

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Building on Student Thinking

Some students may think a figure has more than one match. Remind them that there is only one scaled copy for each polygon and ask them to recheck all the side lengths.

Some students may think that vertices must land at intersections of grid lines and conclude that, for example, G cannot be a copy of F because not all vertices on F are on such intersections. Ask them to consider how a 1-unit-long segment would change if scaled to be half its original size. Where must one or both of its vertices land?

Activity Synthesis

The purpose of this discussion is to draw out concrete methods for deciding whether or not two polygons are scaled copies of one another, and in particular, to understand that just "eyeballing" to see whether they look roughly the same is not enough to determine that they are scaled copies.

Display the image of all the polygons. Ask students to share their pairings and guide a discussion about how students went about finding the scaled copies. Ask questions such as:

“When you looked at another polygon, what exactly did you check or look for?” (General shape, side lengths)
“How many sides did you compare before you decided that the polygon was or was not a scaled copy?” (One pair of sides can be enough to tell that polygons are not scaled copies; all pairs of sides need to be checked to make sure a polygon is a scaled copy.)
“Did anyone check the angles of the polygons? Why or why not?” (No; the sides of the polygons all follow grid lines.)

If students do not agree about some pairings after the discussion, ask the groups to explain their case and discuss which of the pairings is correct. Highlight the use of quantitative descriptors such as “half as long” or “three times as long” in the discussion. Ensure that students see that when a figure is a scaled copy of another, all of its segments are the same number of times as long as the corresponding segments in the other.