# Lesson 8

Find the Perimeter

## Warm-up: Number Talk: Decreasing Dividend (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for dividing within 100. These understandings help students develop fluency and are helpful as students use division to solve problems involving perimeter.

### Launch

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$90 \div 9$$
• $$81 \div 9$$
• $$45 \div 9$$
• $$54 \div 9$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “How could knowing $$90 \div 9$$ help you find the value of the other expressions?” (Once I knew $$90\div9$$ I was able to take away a group of 9 to find $$81\div9$$. I was able to find $$45\div9$$ by splitting the value of $$90 \div 9$$ because 45 is half of 90.)

## Activity 1: Ways to Find Perimeter (20 minutes)

### Narrative

The purpose of this activity is for students to practice finding the perimeter of shapes that have labeled side lengths. The synthesis focuses on methods students have for efficiently finding the perimeter of shapes with some or all side lengths having equal length. As students discuss and justify their decisions, they share a mathematical claim and the thinking behind it (MP3).

Monitor and select students who find the perimeter of the hexagon by:

• adding the individual side lengths around the shape
• adding the two 8–inch side lengths together and the four 4–inch side lengths together and then adding those sums together
• multiplying like side lengths, then adding, such as $$2 \times 8$$ for the long sides and $$4 \times 4$$ for the short sides and then adding those products together
• using symmetry to split the shape in half horizontally and adding $$4 + 8 + 4 = 16$$ for the top half of the shape and then doubling that for the sides on the bottom half of the shape
MLR8 Discussion Supports. Synthesis: Provide students with the opportunity to rehearse what they will say with a partner before they share with the whole class.

### Launch

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?” (Students may notice: One rectangle has numbers on the sides. One rectangle has tick marks on the sides. The rectangles are the same size. Students may wonder: Why are the sides of the rectangles marked differently? Could we find the distance around the rectangle with the numbers on the sides?)
• 1 minute: quiet think time
• 1 minute: partner discussion
• Share and record responses.

### Activity

• “Work with your partner to find the perimeter of each shape.”
• 5–7 minutes: partner work time
• As students work, consider asking:
• “How does having sides of the same length help us find the perimeter?”
• “Can you multiply to find the perimeter?”

### Student Facing

What do you notice? What you wonder?

Find the perimeter of each shape. Explain or show your reasoning.

1.

2.

3.
4.

5.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Ask previously identified students to share their strategies for finding the perimeter of the hexagon. Arrange the presentations in the order listed in the activity narrative.
• Give students a chance to ask questions about each strategy as it is shared.
• “How does having sides of the same length help us find the perimeter?”
• “Why does this strategy work with this shape?”
• “Did anyone else find the perimeter of this shape in a different way?”
• “Was it easier to find the perimeter of some shapes in this activity than others? Why?” (Yes, some of the shapes had several sides that are the same length, so we could multiply. In a rectangle, we can add two sides and then double the result to find the whole perimeter.)

## Activity 2: Something is Missing (15 minutes)

### Narrative

The purpose of this activity is for students to find the perimeter of shapes when some of the side lengths are not given. Students use their knowledge of shapes to reason about the length of the missing sides before they find the perimeter of the shape (MP7).

This activity uses MLR7 Compare and Connect. Advances: representing, conversing

Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were most useful to solve the problem. Display the sentence frame: “The next time I find the perimeter of a shape where some side lengths are not given, I will pay attention to . . . .“
Supports accessibility for: Memory, Visual-Spatial Processing

### Launch

• Groups of 2
• Display the rectangle from the first problem.
• “Find the perimeter of this rectangle.“
• 1–2 minutes: independent work time
• “Discuss with your partner how you found the perimeter of this rectangle even though some of the side lengths were not labeled.” (Since the shape is a rectangle, we know opposite sides of a rectangle are the same length.)
• 1 minute: partner discussion
• Share and record responses.
• Give each group tools for creating a visual display.

### Activity

• “Work with your partner to find the perimeter of the other two shapes. Be sure to record your reasoning to share with the class.”
• 6–8 minutes: partner work time
• Consider asking: “How did you know the length of that side?”

MLR7 Compare and Connect

• “Create a visual display that shows your thinking about the second problem. You may want to include details such as notes, diagrams, drawings, and so on, to help others understand your thinking.”
• 3–5 minutes: partner work time
• 5 minutes: gallery walk

### Student Facing

1. Find the perimeter of this rectangle. Explain or show your reasoning.

2. All the short sides of this figure are the same length, and all the angles are right angles. Find the perimeter. Explain or show your reasoning.

3. All the sides of the octagon are the same length. Find the perimeter. Explain or show your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “We had to find a lot of missing side lengths in this shape before we could find the perimeter.”
• “As you visited the displays, what did you notice about how others found the missing side lengths?” (I noticed some groups counted the number of short sides and multiplied by 40. I noticed some put the short side lengths into smaller groups before finding their combined lengths.)
• “Did anyone find the missing side lengths in a different way than you and your partner?”
• “Did anyone find the perimeter in a different way than you and your partner?”

## Lesson Synthesis

### Lesson Synthesis

“When you are finding the perimeter of a shape, you can always add the lengths of the sides one at a time. What other methods do you have for finding the perimeter of shapes?” (We can look for side lengths that are the same and group them together. In a square, we can multiply one side length by 4 since they are all the same length. In a rectangle, we can add a long side to a short side and then double that for the whole perimeter.)

Display a rhombus with side lengths that are the same length, but only one side labeled 7 in, such as:

“How can we find the perimeter of this rhombus if only one side is labeled?” (We know that a rhombus has four equal sides, so we can find $$4 \times 7$$, which is 28.)

## Cool-down: A Triangle and a Square (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.