Lesson 12

Rectangles with the Same Area

Warm-up: Number Talk: Divide in Parts (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.

When students use properties of operations and quotients with a value of 10 to find other quotients, they look for and make use of structure (MP7).

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.

• $$40 \div 4$$
• $$60 \div 4$$
• $$80 \div 4$$
• $$96 \div 4$$

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

• “How does knowing the first fact help you find other facts?”
• “Who can restate _____’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone approach the problem in a different way?”
• “Does anyone want to add on to _____’s strategy?”

Activity 1: Area of 24 (15 minutes)

Narrative

The purpose of this activity is for students to understand that rectangles with the same area do not necessarily have the same perimeter. The work here reinforces the idea that area and perimeter are two separate measures of shapes. An area of 24 was chosen because it has many factors and is a familiar number to students.

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.
Action and Expression: Develop Expression and Communication. Provide access to blank pre-formatted graph paper to create rectangles.
Supports accessibility for: Fine Motor Skills, Social-Emotional Functioning

Launch

• Groups of 2
• “Take a couple of minutes to draw some rectangles that have an area of 24 square 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 perimeter of each rectangle.”
• 6–8 minutes: partner work time
• Monitor for different rectangles students draw to share in the synthesis.

Student Facing

1. Draw as many different rectangles as you can with an area of 24 square units.
2. Find the perimeter of each rectangle you draw. Explain or show your reasoning.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

• Invite students to share the rectangles they drew and to explain how they knew the area was 24.
• “How did you decide what rectangles to draw?” (I thought about what numbers could be multiplied to give 24, like 6 times 4, and made those the side lengths of the rectangle.)
• “We just showed that rectangles with a certain area do not always have the same perimeter.”
• “How would you explain to someone how to draw rectangles with an area of 30 square units but different perimeters?” (Think about pairs of numbers that multiply to 30, use those numbers for the side lengths, and find the perimeter of each.)

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

Narrative

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

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 one of the following areas: 12 square units, 20 square units, 42 square units, 48 square units.
• Students will cut out and tape their rectangles on to one of the visual displays.

Launch

• Groups of 2
• Display the visual display labeled with each of the four areas 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 paper for drawing rectangles.

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

1. 12 square units
2. 20 square units
3. 42 square units
4. 48 square units
2. Cut out the rectangles you want to share and place 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, discuss something you notice and something you wonder.

Student Response

For access, consult one of our IM Certified Partners.

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.
• Consider asking: “What area did you and your partner choose to work with when you could choose your own area? Why did you choose that area?”

Lesson Synthesis

Lesson Synthesis

“Over the last few lessons, we’ve been learning about area and perimeter.”

“What have you learned about area and perimeter that you want to be sure to remember?” (Rectangles can have the same area and different perimeters. Rectangles can have the same perimeter, but different areas. Area is measured in square units. Perimeter is measured in length units. Perimeter and area are both measures of a shape.)

Cool-down: Area of 36 (5 minutes)

Cool-Down

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Student Section Summary

Student Facing

In this section, we drew rectangles with the same perimeter and different areas. We also drew rectangles with the same area and different perimeters.

rectangles with a perimeter of 16 units

rectangles with an area of 24 square units