In this activity, students are given the result of folding a figure along one or more lines of symmetry and they are asked to reason about the original figure. No lines of symmetry are specified, so students must consider all sides of a folded figure 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-figures  on patty paper, to make cutouts of them, or to use other tools or techniques to reason about the original figure. 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 Figures A–F to facilitate the discussion. (The figures are provided in the blackline master.)

• Invite students to share their responses and strategies.
• When discussing the second question, ask students why B, C, and E are not possible figures 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 from corner to corner, 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, students 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).

• 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)