Lesson 9

This warm-up prompts students to carefully analyze and compare attributes of two-dimensional 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. 

• 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

• “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

• 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.”

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 half-shapes 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.)

• Groups of 2
• Give a ruler or a straightedge to each student.
• Provide access to protractors, patty paper, scrap paper, and scissors.

• 5 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for the different strategies students use to identify the original shape of the half-shapes (as noted in the activity narrative).

• 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.)

Previously, students reason about line-symmetric 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 twice-folded figure, looking for and making use of structure (MP7) as they do so.

• Groups of 2–4
• Give a ruler or a straightedge to each student.
• Provide access to patty paper, scrap paper, and scissors.

• 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.

• 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)