In this activity, students find the perimeter of several shapes and write expressions that show their reasoning. Each side of the shape is labeled with its length, prompting students to notice repetition in some of the numbers. The perimeter of all shapes can be found by addition, but students may notice that it is efficient to reason multiplicatively rather than additively (MP8). For example, they may write instead of .

• Invite students to share the expressions they wrote for the perimeter of each shape. Record the expressions.
• For each shape, ask: “What is the same about these expressions? What is different?” (They each use the side lengths in the shape. They each show the total length around the shape. Some use addition, some use multiplication, some use both multiplication and addition.)
• Highlight that when two or more sides have the same length, the perimeter can be found by adding, multiplying, or a combination of both. 

In the previous activity, students found the perimeter of polygons when all the side lengths were given. In this activity, only some of the sides are labeled with their length, but students are given some information about the attributes of the shapes (presence or absence of parallel sides and symmetry). Students use that information to determine the length of the unlabeled sides and the perimeter. They may also conclude that the length of the perimeter cannot be determined.

When students interpret and analyze Mai's and Andre's reasoning about the quadrilateral perimeters, they critique the reasoning of others (MP3).

This activity uses MLR3 Critique, Correct, Clarify. Advances: reading, writing, representing

• Display Mai's response for students to consider:
• “We can’t find the perimeter of any quadrilateral because each one is missing one or more side lengths.”
• Read Mai’s reasoning aloud.
• “What parts of this response are unclear, incorrect, or incomplete?” (It's incorrect that you can't find the perimeter of any of the quadrilaterals. Missing one side length doesn't mean you can't find the perimeter of shapes.)
• 1 minute: quiet think time
• 2 minute: partner discussion
• Invite 2–3 groups share what they discussed. Record for all to see.
• “With your partner, work together to write a revised explanation for Mai.”
• Display and review the following criteria:
  • “We can’t find the perimeter of a quadrilateral if one or more side lengths are missing and _____.”
• 3–5 minute: partner work time

• Select 1–2 groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each group shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

• “How did you know that the perimeter of C can be found by combining twice 4 cm and twice cm or ?” (We saw earlier that when quadrilaterals have two pairs of parallel sides, opposite sides have the same length. If the sides are parallel, they are the same distance apart.)
• “How did you know the unlabeled side in Shape D is also cm long?” (The figure has a line of symmetry through the midpoint of the top and bottom segment.)
• “What expression can we write for the perimeter of D?” ( , or equivalent)

This optional activity gives students an additional opportunity to use the attributes and their emerging understanding of the properties of some categories of figures to find their perimeter, or to conclude that the perimeter cannot be determined.

• “How do you know these two sides have the same length?”
• “What do you need to know about a figure to be sure that any unlabeled sides have the same length as the labeled sides?”

• Invite students to share how they knew for which figures it is possible to find the perimeter and for which it is not.
• Select other students to share how they matched the expressions and the figures.