# Lesson 7

Same Perimeter, Different Shapes

## Warm-up: 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 multi-digit 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.

• $$123 + 75 + 123 + 75 = 100 + 100 + 70 + 70 + 5 + 5 + 3 + 3$$
• $$123 + 75 + 123 + 75= (2 \times 123) + (2 \times 75)$$
• $$123 + 75 + 123 + 75 = 208 + 208$$
• $$123 + 75 + 123 + 75 = 246 + 150$$

### 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 warm-up (MP7).

MLR7 Compare and Connect. Synthesis: After all strategies have been presented, lead a discussion comparing, contrasting, and connecting the different approaches for finding the perimeter of one of the shapes. Ask, “How did the same perimeter show up in each method?” and “Why did the different approaches lead to the same outcome?”
Action and Expression: Develop Expression and Communication. Synthesis: Identify connections between strategies that result in the same outcomes but use differing approaches.
Supports accessibility for: Memory, Visual-Spatial 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

1. Choose any 3 shapes and find the perimeter of each shape. Explain or show your reasoning.
2. Find one shape that has the same perimeter as one of the shapes you chose earlier. Be prepared to explain your reasoning.

### Student Response

If students count the individual units around each shape, consider asking:
• “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.

• 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

1. Draw 2 shapes with each perimeter.

12 units

26 units

48 units

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

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

### 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.