# Lesson 22

Problems About Perimeter and Area

## Warm-up: How Many Do You See: Shaded Squares (10 minutes)

### Narrative

The purpose of this How Many Do You See routine is to prompt students to decompose a rectilinear figure to find its area and to recognize that there are many ways to do so. Students are also reminded that area is additive. The reasoning here prepares students to reason flexibly about the area of rectilinear figures later in the lesson.

### Launch

- Groups of 2
- “How many do you see? How do you see them?”
- Flash the image.

### Activity

- 30 seconds: quiet think time
- Display the image.
- 1 minute: partner discussion
- Record responses.

### Student Facing

How many shaded squares do you see? How do you see them?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

- “We’ve seen how helpful it is to decompose the figure into rectangles. In how many ways could we do that here?” (Many ways)
- As students share each way, record the thinking for all to see.
- “Are there ways to partition that are more helpful than others?” (Partitioning into larger rectangles is more efficient than smaller ones. The latter would mean more multiplication and more partial areas to add up.)

## Activity 1: Create a Class Banner (15 minutes)

### Narrative

In this activity, students solve geometric problems by reasoning about length and area, decomposing and recomposing of rectangles, considering units of measurements, and performing operations.

Each question can be approached in a variety of ways. Consider asking students to create a visual display of their approach and to share it with the class.

The first problem offers students an opportunity to make sense of a problem and persevere in solving it (MP1). They may focus on the area of the banner and poster paper or start thinking about cutting up the poster paper into pieces that can be used for the banner. They will also need to convert between feet and inches at some point in their solution.

*Representation: Internalize Comprehension.*Activate background knowledge. Say, “We are going to solve problems related to measurement and rectangles today. What are some words that we might come across?” Prompt for students to include words such as length, area, perimeter, inches, feet, yards, and square units. Then ask, “What are some relationships between these words that might be helpful for us to remember?” Prompt for students to include both conceptual relationships (for example, area is measured in square units) and conversions (for example, there are 12 inches in 1 foot). Create a visual display to record responses and invite students to examine some tools for measuring length, such as a ruler, yardstick, or tape measure.

*Supports accessibility for: Conceptual Processing, Language, Memory*

### Required Materials

Materials to Gather

### Launch

- Explain what a banner is or show an example, if needed.
- Give students access to grid paper and inch tiles.

### Activity

- 5–7 minutes: independent work time
- Monitor for different ways students reason about decomposing the 24 by 36 rectangle:
- students who discuss the side length that is 8 feet long and consider how many inches long it would be
- students who use tiles or a drawing to help reorganize the area

### Student Facing

Jada’s teacher bought a poster paper that measures 36 inches by 24 inches. Her plan is to cut it into pieces, rearrange them, and tape them back together to create a welcome banner that is 8 inches tall and 8 feet long.

- Does she have enough paper to make the banner? Show your reasoning.
- How many square inches is the poster paper?

### Student Response

For access, consult one of our IM Certified Partners.

### Advancing Student Thinking

Students may see that there are several things to consider about the situation but may be unsure how to begin. Consider asking:

- “What do you know about the problem and the situation? What don't you know?”
- “What are some ways to visualize or show what Jada's teacher plans to do?”
- “How might the relationship between feet and inches help us think about this problem?”

### Activity Synthesis

- Select students who use different strategies to share their reasoning. Record and display their strategies.
- If it does not come up as a strategy, consider asking, “How could Jada's teacher cut the paper up and rearrange it to make a banner?”
- Consider discussing any benefits or potential challenges of the different approaches. “Are some strategies more efficient or more prone to error than others?” (When cutting the paper into more pieces, there are more measurements to account for, making it more likely to miss something. Cutting the paper into more pieces is also less efficient for Jada's teacher, as it means more taping as well.)

## Activity 2: Replace the Classroom Carpet (20 minutes)

### Narrative

In this activity, students perform operations on multi-digit numbers to solve situations about perimeter and area. They use operations to convert units of measurements along the way. Converting inches to feet could be done by dividing by 12, but this is not an expectation at this point. Students could perform the conversion with multiplicative reasoning. To convert 180 inches into feet, for example, they could reason \(12 \times {?} = 180\), or \(12 \times 10 = 120\) and \(12 \times 5 = 60\).

In grade 3, students learned that area is additive, and that the area of rectilinear figures can be found by decomposing them into non-overlapping rectangles. Students apply that understanding here, after converting lengths in different units into the same unit.

*MLR8 Discussion Supports.*Prior to solving the problems, invite students to make sense of the situation. Monitor and clarify any questions about the context.

*Advances: Reading, Representing*

### Launch

- Groups of 2
- 1 minute: quiet time to read the opening paragraphs and look at the diagram in the task statement
- “What do you notice? What do you wonder?”
- Explain what a baseboard is or show an example.
- Make sure students recognize that all measurements need to be in the same unit before finding perimeter and area.

### Activity

- 6–8 minutes: independent work time, problems 1–2
- 2 minutes: partner discussion
- Monitor for students who:
- choose to consider all units in terms of inches, feet, or yards strategically
- find a way to convert and keep track of the values systematically

- Pause for a whole-class discussion on the first two questions before students answer the last question.
- “How did you decide which unit to use? How do we convert from _____ to _____?”
- “How did you keep track of all the conversions?”

- As students work on the last question, monitor for:
- different ways students decompose the diagram of the room to find its area
- equations that show how the area is computed

### Student Facing

A classroom is getting new carpet and baseboards. Tyler and a couple of friends are helping to take measurements.

Here is a sketch of the classroom and the measurements they recorded.

For each question, show your reasoning.

- How many feet of baseboard will they need to replace in the classroom? How many inches is that?
- 1,200 inches of baseboard material was delivered. Is that enough?
- How many square feet of carpet will be needed to cover the floor area?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

- Select students to share how they reasoned about the area of the room. Record and display their reasoning.
- If some students found the area using different units, ask how they would find out if the two answers represent the same amount.

## Lesson Synthesis

### Lesson Synthesis

“Today we used all kinds of operations on large numbers to solve problems about measurements.”

“Can you find examples where it was helpful or necessary to multiply, divide, add, or subtract multi-digit numbers?” (Multiplication and division were handy for converting from one unit to another. Multiplication was needed to find the area of rectangular shapes. Addition helped us find a total length or area. Subtraction was useful for finding a difference or amount left over.)

“What challenges did you come across when solving the problems? When multiplying or dividing?”

## Cool-down: Paper for a Banner (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.