1.1: Printing Portraits (10 minutes)
CCSS Standards
Addressing
- 7.G.A.1
Routines and Materials
Instructional Routines
- Think Pair Share
1.2: Scaling F (10 minutes)
CCSS Standards
Addressing
- 7.G.A.1
Routines and Materials
Instructional Routines
- MLR1: Stronger and Clearer Each Time
- Think Pair Share
Activity
This task enables students to describe more precisely the characteristics of scaled copies 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.
Students engage inMP7 in multiple ways in this task. Identifying distinguishing features of the scaled copies means finding similarities and differences in the shapes. In addition, the fact that corresponding parts increase by the same scale factor is a vital structural property of scaled copies.
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, suggestthey do so.
As students work, monitor for studentswho 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 copieshave equivalentwidth-to-height ratios.
- We can use ascale factor (or a multiplier)to compare the lengths ofdifferent 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.
Launch
Keep students in the same groups. Give them 3–4 minutes of quiet work time, and then 1–2 minutes to share their responses with their partner. Tell students that how they decide whether each of the seven drawings is a scaled copy may be very different than how their partner decides. Encourage students to listen carefully to each other’s approach and to be prepared to share their strategies. Use gestures to elicit from students the words “horizontal” and “vertical” and ask groups to agree internally on common terms to refer to the parts of the F (e.g., “horizontal stems”).
Engagement: Internalize Self Regulation. Display sentence frames to support small group discussion. For example, “That could/couldn’t be true because…,” “We can agree that…,” and “Is there another way to say/do...?”
Supports accessibility for: Social-emotional skills; Organization; Language
Speaking: Math Language Routine 1 Stronger and Clearer Each Time.This is the first time Math Language Routine 1 is suggested as a support in this course. In this routine, students are given a thought-provoking question or prompt and asked to create a first draft response in writing. Students meet with 2–3 partners to share and refine their response through conversation. While meeting, listeners ask questions such as, “What did you mean by . . .?” and “Can you say that another way?” Finally, students write a second draft of their response reflecting ideas from partners, and improvements on their initial ideas. The purpose of this routine is to provide a structured and interactive opportunity for students to revise and refine their ideas through verbal and written means.
Design Principle(s): Optimize output (for explanation)
How It Happens:
Use this routine to provide students a structured opportunity to refine their explanations for the first question: “Identify all the drawings that are scaled copies of the original letter F drawing. Explain how you know.” Allow students 2–3 minutes to individually create first draft responses in writing.
Invite students to meet with 2–3 other partners for feedback.
Instruct the speaker to begin by sharing their ideas without looking at their written draft, if possible. Provide the listener with these prompts for feedback that will help their partner strengthen their ideas and clarify their language: “What do you mean when you say….?”, “Can you describe that another way?”, “How do you know that _ is a scaled copy?”, “Could you justify that differently?” Be sure to have the partners switch roles. Allow 1–2 minutes to discuss.
Signal for students to move on to their next partner and repeat this structured meeting.
Close the partner conversations and invite students to revise and refine their writing in a second draft.
Provide these sentence frames to help students organize their thoughts in a clear, precise way: “Drawing _ is a scaled copy of the original, and I know this because.…”, “When I look at the lengths, I notice that.…”, and “When I look at the angles, I notice that.…”
Here is an example of a second draft:
“Drawing 7 is a scaled copy of the original, and I know this because it is enlarged evenly in both the horizontal and vertical directions. It does not seem lopsided or stretched differently in one direction. When I look at the length of the top segment, it is 3 times as large as the original one, and the other segments do the same thing. Also, when I look at the angles, I notice that they are all right angles in both the original and scaled copy.”
If time allows, have students compare their first and second drafts. If not, have the students move on by working on the following problems.
Student Facing
Here is an original drawing of the letter F and some other drawings.
- Identify all the drawings that are scaled copies of the original letter F drawing. Explain how you know.
- Examine all the scaled copies more closely, specifically, the lengths of each part of the letter F. How do they compare to the original? What do you notice?
- On the grid, draw a different scaled copy of the original letter F.
Student Response
For access, consult one of our IM Certified Partners.
Launch
Keep students in the same groups. Give them 3–4 minutes of quiet work time, and then 1–2 minutes to share their responses with their partner. Tell students that how they decide whether each of the seven drawings is a scaled copy may be very different than how their partner decides. Encourage students to listen carefully to each other’s approach and to be prepared to share their strategies. Use gestures to elicit from students the words “horizontal” and “vertical” and ask groups to agree internally on common terms to refer to the parts of the F (e.g., “horizontal stems”).
Engagement: Internalize Self Regulation. Display sentence frames to support small group discussion. For example, “That could/couldn’t be true because…,” “We can agree that…,” and “Is there another way to say/do...?”
Supports accessibility for: Social-emotional skills; Organization; Language
Speaking: Math Language Routine 1 Stronger and Clearer Each Time. This is the first time Math Language Routine 1 is suggested as a support in this course. In this routine, students are given a thought-provoking question or prompt and asked to create a first draft response in writing. Students meet with 2–3 partners to share and refine their response through conversation. While meeting, listeners ask questions such as, “What did you mean by . . .?” and “Can you say that another way?” Finally, students write a second draft of their response reflecting ideas from partners, and improvements on their initial ideas. The purpose of this routine is to provide a structured and interactive opportunity for students to revise and refine their ideas through verbal and written means.
Design Principle(s): Optimize output (for explanation)
How It Happens:
Use this routine to provide students a structured opportunity to refine their explanations for the first question: “Identify all the drawings that are scaled copies of the original letter F drawing. Explain how you know.” Allow students 2–3 minutes to individually create first draft responses in writing.
Invite students to meet with 2–3 other partners for feedback.
Instruct the speaker to begin by sharing their ideas without looking at their written draft, if possible. Provide the listener with these prompts for feedback that will help their partner strengthen their ideas and clarify their language: “What do you mean when you say….?”, “Can you describe that another way?”, “How do you know that _ is a scaled copy?”, “Could you justify that differently?” Be sure to have the partners switch roles. Allow 1–2 minutes to discuss.
Signal for students to move on to their next partner and repeat this structured meeting.
Close the partner conversations and invite students to revise and refine their writing in a second draft.
Provide these sentence frames to help students organize their thoughts in a clear, precise way: “Drawing _ is a scaled copy of the original, and I know this because.…”, “When I look at the lengths, I notice that.…”, and “When I look at the angles, I notice that.…”
Here is an example of a second draft:
“Drawing 7 is a scaled copy of the original, and I know this because it is enlarged evenly in both the horizontal and vertical directions. It does not seem lopsided or stretched differently in one direction. When I look at the length of the top segment, it is 3 times as large as the original one, and the other segments do the same thing. Also, when I look at the angles, I notice that they are all right angles in both the original and scaled copy.”
If time allows, have students compare their first and second drafts. If not, have the students move on by working on the following problems.
Student Facing
Here is an originaldrawing of the letter F andsomeother drawings.
- Identify all the drawings that are scaled copies of the original letter F. Explain how you know.
- Examine all the scaled copies more closely, specifically the lengths of each part of the letter F. How do they compare to the original? What do you notice?
- On the grid, draw a different scaled copy of the original letter F.
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
Students may make decisions by “eyeballing” rather than observing side lengths and angles. Encourage them to look for quantifiable evidence and notice lengths and angles.
Some may think vertices must land at intersections of grid lines (e.g., they may say Drawing 4 is not a scaled copy because the endpoints of the shorter horizontal segment are not on grid crossings). Address this during the whole-class discussion, after students have a chance to share their observations about segment lengths.
Activity Synthesis
Display the seven copies of the letter F for all to see. For each copy, ask students to indicatewhether they think each one is a scaled copy of the original F. Record and display the results for all to see. For contested drawings, ask 1–2 students to briefly say why they ruled these out.
Discuss the identified scaled and unscaled copies.
- What features do the scaled copies have in common? (Be sure to invite students who were thinking along the lines of scale factors and angle measures to share.)
- How do the other copies fail to show these features? (Sometimes lengths of sides in the copy use different multipliers for different sides. 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 by saying “two times as long” or “three times as long.”
1.3: Pairs of Scaled Polygons (15 minutes)
CCSS Standards
Addressing
- 7.G.A.1
Routines and Materials
Instructional Routines
- MLR8: Discussion Supports
- Take Turns
Required Materials
- Pre-printed slips, cut from copies of the blackline master
Activity
In this activity, students hone their understanding of scaled copies by working with more complex figures. Students work with a partner to match 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 confer with one another, notice how they go about looking for a match. Monitor for students who use precise language (MP6) to articulatetheir reasoning (e.g., “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.”).
You will need the Pairs of Scaled Polygons blackline master for this activity.
Launch
Demonstrate how to set up and do the matching activity. Choose a student to be your partner. Mix up the cards and place them face-up. Tell them that each polygon has one and only one match (i.e., for each polygon, there is one and only one scaled copy of the polygon). Select twocards and then explain to your partner why you think the cards do or do not match. Demonstrate productive ways to agree or disagree(e.g., by explaining your mathematical thinking, asking clarifying questions, etc.).
Arrange students in groups of 2. Give each groupa set of 10 slips cut from theblackline master. Encourage students to refer to arunning list of statements and diagrams to refine their language and explanations of how they know one figure is a scaled copy of the other.
Representation: Internalize Comprehension.Provide a range of examples and counterexamples. During the demonstration of how to set up and do the matching activity, select two cards that do not match, and invite students to come up with a shared justification.
Supports accessibility for: Conceptual processing
Speaking: MLR8 Discussion Supports. Use this routine to support small-group discussion. As students take turns finding a match of two polygons that are scaled copies of one another and explaining their reasoning to their partner, display the following sentence frames for all to see: “____ matches ____ because . . .” and “I noticed ___ , so I matched . . . ." Encourage students to challenge each other when they disagree with the sentence frames “I agree because . . .”, and “I disagree because . . . ." This will help students clarify their reasoning about scaled copies of polygons.
Design Principle(s): Support sense-making; Optimize output (for explanation)
Student Facing
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 pair of polygons that are scaled copies of one another.
- For each match you find, explain to your partner how you know it’s a match.
- For each match your partner finds, listen carefully to their explanation, and if you disagree, explain your thinking.
- 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. Use the grid below to produceboth polygons. Explain or show how you know that one polygon is a scaled copy of the other.
Student Response
For access, consult one of our IM Certified Partners.
Launch
Demonstrate how to set up and do the matching activity. Choose a student to be your partner. Mix up the cards and place them face-up. Tell them that each polygon has one and only one match (i.e., for each polygon, there is one and only one scaled copy of the polygon). Select twocards and then explain to your partner why you think the cards do or do not match. Demonstrate productive ways to agree or disagree(e.g., by explaining your mathematical thinking, asking clarifying questions, etc.).
Arrange students in groups of 2. Give each groupa set of 10 slips cut from theblackline master. Encourage students to refer to arunning list of statements and diagrams to refine their language and explanations of how they know one figure is a scaled copy of the other.
Representation: Internalize Comprehension. Provide a range of examples and counterexamples. During the demonstration of how to set up and do the matching activity, select two cards that do not match, and invite students to come up with a shared justification.
Supports accessibility for: Conceptual processing
Speaking: MLR8 Discussion Supports. Use this routine to support small-group discussion. As students take turns finding a match of two polygons that are scaled copies of one another and explaining their reasoning to their partner, display the following sentence frames for all to see: “____ matches ____ because . . .” and “I noticed ___ , so I matched . . . ." Encourage students to challenge each other when they disagree with the sentence frames “I agree because . . .”, and “I disagree because . . . ." This will help students clarify their reasoning about scaled copies of polygons.
Design Principle(s): Support sense-making; Optimize output (for explanation)
Student Facing
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 pair of polygons that are scaled copies of one another.
- For each match you find, explain to your partner how you know it’s a match.
- For each match your partner finds, listen carefully to their explanation, and if you disagree, explain your thinking.
- 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. Draw both polygons on the grid. Explain or show how you know that one polygon is a scaled copy of the other.
Student Response
For access, consult one of our IM Certified Partners.
Student Facing
Are you ready for more?
Is it possibletodraw a polygon that is a scaled copy of both Polygon A and Polygon B?Either draw such a polygon, or explain how you know this is impossible.
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
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, e.g., 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 look 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? (Two sides can be enough to tell that polygons are not scaled copies; all sides are needed 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.
1.4: Cool-down - Scaling L (5 minutes)
CCSS Standards
Addressing
- 7.G.A.1
Cool-Down
For access, consult one of our IM Certified Partners.
