Lesson 7
Same Perimeter, Different Shapes
Warmup: True or False: Sums of Four Numbers (10 minutes)
Narrative
The purpose of this True or False is to elicit strategies and understandings students have for adding multidigit numbers. It prompts students to rely on their understanding of the properties of operations and place value. The strategies used here will be helpful as students find the perimeter of shapes with repeated side lengths later in the lesson.
Launch
 Display one equation.
 “Give me a signal when you know whether the equation is true and can explain how you know.”
 1 minute: quiet think time
Activity
 Share and record answers and strategy.
 Repeat with each equation.
Student Facing
Decide whether each statement is true or false. Be prepared to explain your reasoning.
 \(123 + 75 + 123 + 75 = 100 + 100 + 70 + 70 + 5 + 5 + 3 + 3\)
 \(123 + 75 + 123 + 75= (2 \times 123) + (2 \times 75)\)
 \(123 + 75 + 123 + 75 = 208 + 208\)
 \(123 + 75 + 123 + 75 = 246 + 150\)
Student Response
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Activity Synthesis
 “How can you explain your answer without finding the value of both sides?”
Activity 1: All Kinds of Shapes (15 minutes)
Narrative
The purpose of this activity is for students to understand that many different shapes can have the same perimeter. Students start to focus more specifically on shapes with repeated side lengths, so they can leverage the efficient addition strategies elicited in the warmup (MP7).
Advances: Representing, Conversing
Supports accessibility for: Memory, VisualSpatial Processing
Launch
 Groups of 2
 Display the shapes.
 “Which shape do you think has the longest perimeter and which has the shortest?” (I think shape J has the longest—it looks like a really long shape. I think D has the shortest. It is small.)
 1–2 minutes: partner discussion
 Share responses.
Activity
 “Work with your partner to find the perimeter of 3 shapes.”
 “Then, work independently to find at least one shape that has the same perimeter as a shape you chose.”
 5 minutes: partner work time
 2–3 minutes: independent work time
 Monitor for students who use sides of the same length or symmetry of the shape to find the perimeter in efficient ways.
Student Facing
 Choose any 3 shapes and find the perimeter of each shape. Explain or show your reasoning.

Find one shape that has the same perimeter as one of the shapes you chose earlier. Be prepared to explain your reasoning.
Student Response
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Advancing Student Thinking
 “How did you find the perimeter of this shape?”
 “What other strategies could you use so you wouldn’t have to count one unit at a time?”
Activity Synthesis
 Invite students to share a variety of methods for finding the perimeter of the different shapes. Ask students who found the perimeter of shapes in efficient ways to share their reasoning.
 Consider asking: “Did anyone find the perimeter in a different way?”
 “Shapes A and G look very different but have the same length for their perimeter. How could that be?” (The shapes may look different but the distance around each is the same number of units. Different shapes can have the same perimeter.)
Activity 2: Draw Your Own (20 minutes)
Narrative
The purpose of this activity is for students to draw shapes with specific perimeters. Students may create any shape that uses horizontal and vertical lines. Since diagonal lines that connect the dots are not one length unit, students cannot find the perimeter of shapes that include diagonal sides. Encourage students to be creative in drawing their shapes to reinforce the idea that different shapes can have the same perimeter.
Launch
 Groups of 2
Activity
 “Work independently to draw two shapes that have each perimeter. Be prepared to explain how you drew your shapes.”
 8–10 minutes: independent work time
 Monitor for students who:
 draw shapes other than rectangles or squares
 can explain their method for drawing shapes with a specific perimeter, such as drawing sides of the same length first or drawing sides one at a time around the shape
 “Share the shapes you drew with your partner. Think about how your shapes are alike and how they are different. Afterwards, work on the last problem together.”
 5 minutes: partner work time
Student Facing

Draw 2 shapes with each perimeter.
12 units
26 units
48 units


With your partner, choose a length. Then, draw your own shape with that perimeter.

Share the shapes you drew and discuss how they are alike and how they are different.

Student Response
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Activity Synthesis
 Select previously identified students to share their strategies for drawing shapes for one of the perimeters in the first problem.
 “What perimeter did you and your partner choose to draw a shape for? Why did you pick that number?”
Lesson Synthesis
Lesson Synthesis
“Today we learned that different shapes can have the same perimeter.”
“How would you explain to someone how this is possible?” (The perimeter is the total length of all the sides of a shape, and there are different ways to add numbers to get the same sum.)
Consider using a string of interconnected paper clips to form different shapes. The shapes would have the same perimeter because the length of the string (or the number of paper clips) hasn’t changed.
Cooldown: Create Your Own Shapes (5 minutes)
CoolDown
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