Previously, students reasoned about the perimeter of two-dimensional figures based on given side lengths and known attributes of the figures, including symmetry. In this activity, students find unknown side lengths given the perimeter, some side lengths, and information about the symmetry of the figures. Students have opportunities to practice adding, subtracting, and multiplying numbers with fractions, as not all of the given measurements are whole numbers.

• Select students to share their responses and reasoning.
• “How do the lines of symmetry in P, R, and S help you find the unknown side lengths?” (The lines of symmetry tell us the lengths of unlabeled sides that mirror labeled sides, making it possible to find the length of the side with a question mark.)
• “What about the lines of symmetry in Q?” (The vertical line of symmetry tells us the left and right sides have the same length and the horizontal one tells us the top and bottom sides are of equal length. So, all sides have the same length.)

The purpose of this activity is for students to practice completing a geometric drawing given half of the drawing and a line of symmetry, and reason about the perimeter of a line-symmetric figure. While a precise drawing is not an expectation here, if no students considered using tools and techniques—such as using patty paper or by folding—to complete the drawing precisely, consider asking how it could be done (MP5).

Monitor for and select students with the following approaches for determining whether Lin has enough tape to share in the Activity Synthesis:

• Add 5 segments on one side of the line of symmetry and double it: .
• Multiply each segment by 2 and find the sum of those products: .
• List the length of each segment in an addition expression or vertical list and use the associative property to find sums that are easier (for example, add 19 and 6 to get 25, then multiply 25 and by 4: ).

The approaches will later be displayed side by side to help students connect ways of reasoning about side lengths and symmetry to different expressions for the perimeter of the design. As students practice adding and multiplying fractional side lengths, they will use properties of operations in their reasoning. Students will also have opportunities to look for and make use of structure (MP7) to expedite their calculation. Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven't shared recently.

• Invite previously selected students to display their equations for the perimeter side by side without sharing their thinking.
• “Take a minute to look at each of these ways of determining whether Lin has enough tape.”
• Connect students’ approaches by asking:
  • “How are the different equations the same and different?” (They show the same side lengths. They show doubling side lengths. The order and how the numbers are grouped is different. The numbers added or multiplied are different.)
• Connect students’ approaches to the learning goal by asking:
  • “How does having sides of the same length help us find the perimeter?” (You can look for ways to use multiplication instead of having to add a bunch of different lengths.)