Lesson 9
Symmetry in Action (optional)
Warmup: Which One Doesn’t Belong: Figures (10 minutes)
Narrative
This warmup prompts students to carefully analyze and compare attributes of twodimensional figures with attention to the number of sides, symmetry, and presence of parallel and perpendicular lines. The activity enables the teacher to observe the attributes that students notice intuitively and hear the terminologies they feel comfortable using.
Launch
 Groups of 2
 Display the image.
 “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
 1 minute: quiet think time
Activity
 “Discuss your thinking with your partner.”
 2–3 minutes: partner discussion
 Share and record responses.
Student Facing
Which one doesn’t belong?
Student Response
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Activity Synthesis
 If no students mention parallel sides as an attribute to consider, ask them about it.
 Consider asking: “Let’s find at least one reason why each one doesn’t belong.”
Activity 1: Before and After (15 minutes)
Narrative
In this activity, students are given the result of folding a shape along one or more lines of symmetry and asked to reason about the original shape. No lines of symmetry are specified, so students must consider all sides of a folded shape as a possible line of symmetry and visualize the missing half accordingly.
The first question offers opportunities to practice choosing tools strategically (MP5). Some students may wish to trace the halfshapes on patty paper, to make cutouts of them, or to use other tools or techniques to reason about the original shape. Provide access to the materials and tools they might need.
During the activity synthesis, discuss the different ways students approach the second question. Consider preparing cutouts of shapes A–F to facilitate the discussion. (The shapes are provided in the blackline master.)
Supports accessibility for: Conceptual Processing, Organization, Attention
Required Materials
Materials to Gather
Materials to Copy
 Before and After
Launch
 Groups of 2
 Give a ruler or a straightedge to each student.
 Provide access to protractors, patty paper, scrap paper, and scissors.
Activity
 5 minutes: independent work time
 2–3 minutes: partner discussion
 Monitor for the different strategies students use to identify the original shape of the halfshapes (as noted in the activity narrative).
Student Facing

Mai has a piece of paper. She can get two different shapes by folding the paper along a line of symmetry. What is the shape of the paper before it was folded?

Diego folded a piece of paper once along a line of symmetry and got this right triangle.
Which shapes could the paper have before it was folded? Explain or show how you know.
Student Response
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Activity Synthesis
 Invite students to share their responses and strategies.
 When discussing the second question, ask students why B, C, and E are not possible shapes of the original piece of paper even though there’s a line that breaks each figure into two right triangles that match the given triangle. (B and E have no line symmetry. C has lines of symmetry but not diagonally from corner to corner. If you fold there, the triangles would not be on top of one another.)
Activity 2: Before and After, Perimeter Edition (20 minutes)
Narrative
Previously, students reason about linesymmetric figures that have been folded once along a line of symmetry. In this activity, they encounter figures that have been folded more than once, each time along a line of symmetry, and reason about the perimeter of the original figure. They think about how a given set of expressions could represent the original perimeter of a twicefolded figure, looking for and making use of structure (MP7) as they do so.
Advances: Writing, Representing
Required Materials
Materials to Gather
Launch
 Groups of 2–4
 Give a ruler or a straightedge to each student.
 Provide access to patty paper, scrap paper, and scissors.
Activity
 3–4 minutes: independent work time for the first set of questions
 Pause for a brief class discussion on possible shapes of the original piece of paper and possible expressions for its perimeter.
 5 minutes: independent work time for the second question
 2–3 minutes: group discussion
 Monitor for students who use the drawings or expressions from the first question to help them reason about the second question. Select them to share during synthesis.
Student Facing

Jada folded a piece of paper along a line of symmetry and got this rectangle.
 What could the paper look like before being folded? Draw one or more sketches.
 Write an expression for the perimeter of the unfolded paper.

Kiran folded a piece of paper twice—each time along a line of symmetry—and got the same rectangle as Jada did.
Show that each expression could represent the perimeter of the paper Kiran folded.
 \((4 \times 182) + (4 \times 105)\)
 \((2 \times 182) + (8 \times 105)\)
 \((8 \times 182) + (2 \times 105)\)
Student Response
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Activity Synthesis
 Select students to share their responses and reasoning to the second question.
 Consider asking: “Aside from the three figures whose perimeters are represented here, are there other possible figures that the original piece of paper could have?” (No)
 “How do you know?” (The folded rectangle has two pairs of sides of the same length. There are only three possible pairs of lines of symmetry: both along the 182 mm side, both along the 105 mm side, and once along each 182 and 105 mm side. All three are already represented by the given expressions.)
 “If different original figures can be folded into the same figure, does that mean the original figures have the same perimeter?” (No)
Lesson Synthesis
Lesson Synthesis
“Today we practiced visualizing shapes that have been folded along a line of symmetry and reasoning about the perimeter of the original shapes.“
Display:
“Suppose this right triangle is a result of folding once along a line of symmetry. What strategies could we use to determine the possible shapes before they were folded?” (Reflect the triangle along each of the sides—mentally, using tracing paper, or cutting out two copies of the triangle and arranging them so they mirror each other.)
“To find the perimeter of the original shape, could we just double the perimeter of the folded shape? Why or why not?” (No, because there is one side—along the folding line—that is not part of the perimeter of the original shape.)
“What could be the perimeters of the original shapes that fold into this triangle?” (The perimeter varies depending on line of symmetry or line of folding:
 If folded along the longest side: it will be 28, or \((2 \times 8) + (2 \times 6)\).
 If folded along the side that is 8 units long, it will be 32, or \((2 \times 6) + (2 \times 10)\).
 If folded along the shortest side, it will be 36, or \((2 \times 8) + (2 \times 10)\).)
Cooldown: Fold It Once (5 minutes)
CoolDown
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