Lesson 9
Perimeter Problems
Warm-up: Estimation Exploration: Statue of Liberty (10 minutes)
Narrative
The purpose of an Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information.
Launch
- Groups of 2
- Display the image.
- “What is an estimate that’s too high?” “Too low?” “About right?”
- 1 minute: quiet think time
Activity
- “Discuss your thinking with your partner.”
- 1 minute: partner discussion
- Record responses.
Student Facing
The Statue of Liberty has two square bases—one larger than the other. The larger base has side lengths of 132 feet each.
Estimate the perimeter of the smaller square base.
Record an estimate that is:
| too low | about right | too high |
|---|---|---|
| \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) | \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) | \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) |
Student Response
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Activity Synthesis
- “If you wanted to know the perimeter of the star-shaped base, how would you find it?” (We’d need to know the length of each side and add the lengths together. If the lengths were the same we could count them and multiply the length by number of sides.)
- Consider asking:
- “Based on this discussion does anyone want to revise their estimate?”
Activity 1: Missing Measurements (15 minutes)
Narrative
The purpose of this activity is for students to find the length of a missing side of a shape when the perimeter is given, using any strategy that makes sense to them. The synthesis highlights the variety of methods students used to solve the problem.
Launch
- Groups of 2
- “In an earlier lesson, we found the perimeter of shapes when not all the side lengths were labeled. Now, let’s find some missing side lengths when we know the perimeter.”
Activity
- 5–7 minutes: partner work time
- Monitor for students who:
- subtract each side length from the perimeter
- double a given side length, subtract the result from the perimeter, and divide to find the other two sides
- divide the perimeter when the side lengths are all equal
Student Facing
-
This pentagon has a perimeter of 32 cm. What is the length of the missing side? Explain or show your work.
-
This rectangle has a perimeter of 56 feet. What are the lengths of the unlabeled sides? Explain or show your work.
-
This pentagon has a perimeter of 65 inches. All the sides are the same length. What is the length of each side? Explain or show your work.
Student Response
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Activity Synthesis
- Select previously identified students to share their strategies. Be sure to share at least one method (more if possible) for each problem.
- Consider asking:
- “When would this strategy be most useful?”
- “Did anyone think about it in a different way?”
Activity 2: Can I Use Perimeter? (20 minutes)
Narrative
The purpose of this activity is for students to solve problems in situations that involve perimeter (MP2). Students may draw diagrams with length labels or simply reason arithmetically. They also explain how each problem does or does not involve perimeter. The activity synthesis provides an opportunity to begin discussing the difference between area and perimeter, which will be fully explored in upcoming lessons.
Advances: Reading, Representing
Supports accessibility for: Language, Visual-Spatial Processing
Launch
- Groups of 2
Activity
- “Take some time to solve these problems on your own.”
- 5 minutes: independent work time
- “Share with your partner your reasoning on your favorite problem.”
- 2 minutes: partner discussion
- Monitor for a variety of ways students solve these problems, such as by drawing a diagram or writing expressions or equations. Identify one student to share for each problem, with a variety of ways shown across the problems.
Student Facing
Solve each problem. Explain or show your reasoning.
- A rectangular park is 70 feet on the shorter side and 120 feet on the longer side. How many feet of fencing is needed to enclose the boundary of the park?
- Priya drew a picture and is framing it with a ribbon. Her picture is square and one side is 9 inches long. How many inches of ribbon will she need?
- A rectangular flower bed has a fence that measures 32 feet around. One side of the flower bed measures 12 feet. What are the lengths of the other sides?
- Kiran took his dog for a walk. Their route is shown. How many blocks did they walk?
- A room is 10 feet by 8 feet. How many tiles will be needed to cover the floor if each tile is 1 square foot?
Student Response
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Advancing Student Thinking
If students say they aren’t sure how to get started on a problem, consider asking:
- “What is the problem about?”
- “How could you represent the problem?”
Activity Synthesis
- Select previously identified students to share their reasoning for each problem. After each problem, consider asking: “Did anyone solve this problem in a different way?”
- If possible, keep the student work displayed for the lesson synthesis.
Lesson Synthesis
Lesson Synthesis
“Look back through the problems you solved in the last activity. Discuss with your partner whether each problem involves perimeter.”
“How do you know if a situation involves perimeter?” (If it’s about finding the distance around something. If answering the question means adding up all the side lengths of a shape.)
“Why was perimeter not useful in the last problem about tiling a floor?” (The perimeter would give the length around the outside of the room, not how many tiles covered the whole room. To know how many tiles cover the whole room is to find the area of the room.)
“What is the difference between perimeter and area?” (Perimeter is the distance around the outside of a shape. Area is the amount of space a shape covers.)
Cool-down: Sides of a Pool (5 minutes)
Cool-Down
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Student Section Summary
Student Facing
In this section, we learned that perimeter is the boundary of a flat shape.
We can find the length of a perimeter by adding the lengths of all the sides, or by using multiplication when there are sides with the same length.
\(9+9+21+21\)
\((2 \times 9) + (2 \times 21)\)
We used our knowledge of shapes to find the perimeter even when some side lengths were missing, and to use the perimeter to find missing side lengths.
For example, if we know the perimeter of this rectangle is 32 feet, we can find the lengths of the three unlabeled sides.
