# Lesson 11

Rectangles with the Same Perimeter

## Warm-up: Number Talk: Multiply to Divide (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 will be helpful later in this lesson when students will need to be able to divide fluently within 100.

### 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

• Record answers and strategy.
• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$5 \times 5$$
• $$10 \times 5$$
• $$2 \times 5$$
• $$85 \div 5$$

### Activity Synthesis

• “How did knowing the first 3 facts help you find the value of $$85 \div 5$$?” (The first 3 products added up to 85, which was the number I was dividing in the last problem. So, I was able to use those to figure out $$85 \div 5$$.)

## Activity 1: Perimeter of 16 Units (15 minutes)

### Narrative

The purpose of this activity is for students to understand that rectangles with the same perimeter do not necessarily have the same area. In the synthesis, students begin to consider how to systematically draw different rectangles with the same perimeter.

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
• “Take a couple of minutes to draw some rectangles that have a perimeter of 16 units.”
• 2–3 minutes: independent work time

### Activity

• “Share your rectangles with your partner and see if there are any other rectangles you can think of together. Then, find the area of each rectangle.”
• 6–8 minutes: partner work time
• Monitor for different rectangles students draw.

### Student Facing

1. Draw as many different rectangles with a perimeter of 16 units as you can.
2. Calculate the area of each rectangle you draw. Explain or show your reasoning.

### Activity Synthesis

• Select students to share their rectangles and to explain how they knew the perimeter was 16 and how they found the area.
• “We just showed that rectangles with a certain perimeter do not always have the same area.”
• “How would you explain to someone how to draw rectangles with a perimeter of 30 that had different areas?” (Choose a length for two of the sides, like 10, and then double that to get 20. There’s 10 left for the other two sides, so each side will be 5. Split 30 in half to get 15. The two different side lengths need to add up to 15, so we can use different pairs of numbers with the sum of 15.)

## Activity 2: Same Perimeter, Different Area (20 minutes)

### Narrative

The purpose of this activity is for students to draw rectangles with the same perimeter and different areas. Students draw a pair of rectangles for each given perimeter, then display their rectangles and make observations about them in a gallery walk.

Students may notice new patterns (MP7) in the rectangles with the same perimeter (for instance, that as two sides each increase by 1 unit, the other two sides each decrease in length by 1 unit). They may also notice that, so far, all the perimeters are even numbers. Students may wonder if it is possible for a perimeter to be an odd number. If these observations arise, consider discussing them in the synthesis.

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select to complete 3 of the 5 perimeter problems in task 1.
Supports accessibility for: Organization, Attention, Social-Emotional Skills

### Required Materials

Materials to Gather

Materials to Copy

• Square Dot Paper Standard

### Required Preparation

• Create 4 visual displays. Each visual display should be labeled with a different perimeter. Use the following perimeters: 12 units, 20 units, 26 units, 34 units).
• Students cut out and tape their rectangles on one of the visual displays during this activity.

### Launch

• Groups of 2
• Display the visual display labeled with each of the four perimeters in the first problem.
• Give each group 2 sheets of dot paper, scissors, and access to tape.

### Activity

• “Work with your partner to complete the first problem.”
• 6–8 minutes: partner work time
• “Choose which rectangles you want to share and put them on the appropriate poster. Try to look for rectangles that are different from what other groups have already placed.”
• 3–5 minutes: partner work time
• Monitor to make sure each visual display has a variety of rectangles.
• When all students have put their rectangles on the posters, ask students to visit the posters with their partner and discuss one thing they notice and one thing they wonder about the rectangles.
• 5 minutes: gallery walk

### Student Facing

Your teacher will give you some dot paper for drawing rectangles.

1. For each of the following perimeters, draw 2 rectangles with that perimeter but different areas.

1. 12 units
2. 20 units
3. 26 units
4. 34 units
5. Choose your own perimeter.
2. Cut out 1 or 2 rectangles you want to share and put them on the appropriate poster. Try to look for rectangles that are different from what other groups have already placed.
3. Gallery Walk: As you visit the posters with your partner, discuss something you notice and something you wonder.

### Activity Synthesis

• “As you visited the posters, what did you notice? What did you wonder?”
• Discuss observations or questions that can reinforce the connections between side lengths, perimeter, and area of rectangles.
• “What perimeter did you and your partner choose to work with when you could choose your own perimeter? Why did you choose that perimeter?”

## Lesson Synthesis

### Lesson Synthesis

Refer to the posters from the previous activity.

“How is it possible that many rectangles can have the same perimeter, but not have the same area?” (The perimeter is the distance around the rectangle, it does not determine the amount of space the rectangle covers.)

“How did you know the areas were different? Can you tell by looking at the rectangles whether they have the same area?” (Some you can tell just by looking at them that one takes up more space than the other. I would find the area to be sure. Even if the rectangles look different, they could have the same area.)