# Lesson 10

Problem Solving With Perimeter and Area

## Warm-up: True or False: Divide in Parts (10 minutes)

### Narrative

The purpose of this True or False is to elicit strategies and understandings students have for dividing within 100. It also prompts them to rely on properties of operations and familiar division facts to facilitate division.

When students think about how to decompose larger dividends using facts about 10 to make the division easier, they look for and make use of structure (MP7).

### Launch

- Display one statement.
- “Give me a signal when you know whether the statement is true and can explain how you know.”
- 1 minute: quiet think time

### Activity

- Share and record answers and strategy.
- Repeat with each statement.

### Student Facing

Decide if each statement is true or false. Be prepared to explain your reasoning.

- \(60 \div 6 = 10\)
- \(72 \div 6 = (60 \div 6) + (12 \div 6)\)
- \(78 \div 6 = (60 \div 10) + (18 \div 6)\)
- \(96 \div 8 = (80 \div 8)-(16 \div 8)\)

### Student Response

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### Activity Synthesis

- “How can you explain your answer without finding the value of both sides?”

## Activity 1: Rope Off the Garden (15 minutes)

### Narrative

The purpose of this activity is for students to differentiate methods for finding perimeter from those for finding area. While addition and multiplication are both involved in various ways, students need to understand the problem situation and think about whether the operations performed will provide the desired information. As in earlier problems, students can find perimeter in various ways. The emphasis should be on how understanding the problem situation and the information given should inform the solution method.

When students analyze claims about how to use addition and multiplication to find the perimeter of a rectangle they construct viable arguments (MP3).

*MLR1 Stronger and Clearer Each Time.*Synthesis: Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to “Who do you agree with? Explain or show your reasoning.” Invite listeners to ask questions, to press for details and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation based on the feedback they receive.

*Advances: Writing, Speaking, Listening*

### Launch

- Groups of 2 and 4
- Display the situation: “Andre wants to know how much rope is needed to rope off the new rectangular school garden. The length of the garden is 30 feet. The width of the garden is 8 feet.”
- “Take a minute to read this situation about Andre and the school garden. How would you figure out how many feet of rope is needed?”
- 1 minute: quiet think time

### Activity

- “With your partner, look at Clare and Diego’s thinking and answer the questions.”
- 3–5 minutes: partner work time

### Student Facing

Andre wants to know how much rope is needed to enclose the new rectangular school garden. The length of the garden is 30 feet. The width of the garden is 8 feet.

- Clare says she can use multiplication to find the length of rope Andre needs.
- Diego says he can use addition to find the length of rope Andre needs.

Who do you agree with? Explain or show your reasoning.

### Student Response

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### Activity Synthesis

- Invite students to share who they agreed with and why. Record their reasoning for all to see.
- “How did you know that multiplying \(8 \times 30\) would not give you the total amount of rope needed?” (Multiplying \(8 \times 30\) would give us the area of the rectangle, not the distance around the rectangle.)
- “How did you know that Diego’s strategy would work?” (Diego is finding the perimeter by adding the side lengths of the garden.)
- “When might it be appropriate to use multiplication to find perimeter?” (When there are two or more sides that are the same length. When we know half of the perimeter, we can double that number to find the whole perimeter.)

## Activity 2: Info Gap: A Garden and a Playground (20 minutes)

### Narrative

This info gap activity gives students a chance to understand that given the area and one side length of a rectangle, the perimeter can be found, and that given the perimeter and one side length of a rectangle, the area can be found. In both cases, students need to find the missing side length to solve the problem. There are several ways students might find the missing side length and then the perimeter or area once the missing side length is known.

This activity uses MLR4 Information Gap.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is an image of the cards for reference:

*Representation: Internalize Comprehension.*Synthesis: Invite students to identify which details were needed to solve the problem. Display the sentence frame, “The next time I use the area of a rectangle to find the perimeter, I will look for . . . .“

*Supports accessibility for: Memory, Visual-Spatial Processing*

### Required Materials

Materials to Copy

- Info Gap: A Garden and a Playground

### Required Preparation

- Each group of 2 will need a copy of the 2 data and problem card sets. Keep set 1 separate from set 2.

### Launch

- Groups of 2

**MLR4 Information Gap**

- Display the task statement, which shows a diagram of the info gap structure.
- 1 minute: quiet think time
- Read the steps of the routine aloud.
- “I will give you either a problem card or a data card. Silently read your card. Do not read or show your card to your partner.”
- Distribute cards.
- 1–2 minutes: quiet think time
- Remind students that after the person with the problem card asks for a piece of information the person with the data card should respond with “Why do you need to know (restate the information requested)?”

### Activity

- 3–5 minutes: partner work time
- After students solve the first problem, distribute the next set of cards. Students switch roles and repeat the process with Problem Card 2 and Data Card 2.

### Student Facing

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

Pause here so your teacher can review your work.

Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.

### Student Response

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### Activity Synthesis

- Share and record responses.
- Display the info gap cards.
- “What do you need to know in order to find the perimeter or area of these rectangles?” (The missing side length.)
- “How did you use the area and one side you knew to find the missing side length?”
- “How did you use the perimeter and one side to find the missing side length?”

## Lesson Synthesis

### Lesson Synthesis

“Today we saw some problems that asked us to think about area and perimeter together.”

“How are perimeter and area alike?” (They are both measurements of shapes. We need side lengths to find both the area and perimeter of rectangles.)

“How are they different?” (Perimeter is about distance, so it is measured in length units. Area is about the amount of space within a shape, so it is measured in square units.)

If the different types of units used to measure area and perimeter don’t come up, ask, “How are the units we use to measure area and perimeter different? Why?”

Consider recording students’ ideas in two columns labeled “alike” and “different.”

## Cool-down: Lin’s Garden Fence (5 minutes)

### Cool-Down

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