Shapes and Their Perimeter

Lesson 13

Math

Unit 5

3rd Grade

Lesson 13 of 16

Objective


Find rectangles with the same area and different perimeters.

Common Core Standards


Core Standards

  • 3.MD.D.8 — Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

Foundational Standards

  • 3.MD.C.5
  • 3.MD.C.6
  • 3.MD.C.7

Criteria for Success


  1. Determine the dimensions of all possible rectangles with a given area (e.g., a 1-by-12 rectangle, 2-by-6 rectangle, and 3-by-4 rectangle all have an area of 12 square units) (MP.1).
  2. Understand that rectangles with inverted dimensions are not unique because of the commutative property (i.e., a 1-by-12 rectangle is not distinctly different from a 12-by-1 rectangle).
  3. Determine the perimeter of all possible rectangles with a given area and whole-number side lengths. 
  4. Understand that a given area does not imply a particular perimeter (i.e., a rectangle with a particular area could have a variety of perimeters).
  5. Determine the rectangle with the greatest perimeter given a particular area.
  6. Understand that given a particular area, the most square-like rectangle will have the least perimeter, and the most oblong rectangle will have the greatest perimeter (MP.8).

Tips for Teachers


  • You might decide to spend two days on this lesson, depending on pacing and students’ need to explore this idea even more in-depth.
  • “With strong and distinct concepts of both perimeter and area established, students can work on problems to differentiate their measures. For example, they can find and sketch rectangles with the same perimeter and different areas or with the same area and different perimeters and justify their claims” (MP.3) (MD Progression, p. 18). Further, by distinguishing perimeter from area, students' understanding of each concept is enhanced, thus connecting the additional cluster content of perimeter to the major cluster content of area.

Lesson Materials

  • Optional: Square inch tiles (24 per student or small group) — Students might not need these depending on their reliance on concrete materials. See Anchor Task 1 for more information.
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Anchor Tasks


Problem 1

a.   Create as many rectangles as you can with an area of 12 square units. Find the perimeter of each rectangle that you created. Then record the length, width, area, and perimeter of each rectangle in the table below:

Length (units) Width (units) Area (square units) Perimeter (units)
       
       
       
       
       
       
       

b.   What do you notice about the rectangles you created? What do you wonder?

Guiding Questions

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References

EngageNY Mathematics Grade 3 Mathematics > Module 7 > Topic D > Lesson 19Concept Development

Grade 3 Mathematics > Module 7 > Topic D > Lesson 19 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.
EngageNY Mathematics Grade 3 Mathematics > Module 7 > Topic D > Lesson 18Concept Development

Grade 3 Mathematics > Module 7 > Topic D > Lesson 18 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

What is the greatest perimeter you can make with a rectangle that has an area of 24 square units?

Guiding Questions

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References

Open Middle Rectangles: Maximizing Perimeter

Rectangles: Maximizing Perimeter by is made available on Open Middle under the CC BY-NC-SA 4.0 license. Accessed Feb. 10, 2019, 5:16 p.m..

Problem Set


Answer Keys

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Discussion of Problem Set

  • How did you find a rectangle with the same area but a smaller perimeter in #3? Did you think you’d need a longer and skinnier rectangle or a more square-like one? Why?
  • Can you think of other numbers of unit squares, like 13 and 17, that only have one possible rectangle? How did you come up with them?
  • For which number of unit squares in #4 can a square be drawn? How do you know? Can you brainstorm other numbers of unit squares from which a square can be drawn?
  • Why do you think a square has a smaller perimeter than any other rectangle with the same area?
  • Can you think of another number of unit squares that would allow us to make four rectangles?
  • Can you think of another number of unit squares that would allow us to make five rectangles? What’s the smallest number for which this is true?

Target Task


Leslie is building a dog park in her neighborhood. She has a few different options for sizes, one of which is shown below.

a.   What is the perimeter of Dog Park A above?

b.   Another option, Dog Park B, has the same area as Dog Park A but a different perimeter. Draw one possible shape for Dog Park B below. Explain how you found your answer.

Student Response

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Additional Practice


The Extra Practice Problems can be used as additional practice for homework, during an intervention block, etc. Daily Word Problems and Fluency Activities are aligned to the content of the unit but not necessarily to the lesson objective, therefore feel free to use them anytime during your school day.

Extra Practice Problems

Answer Keys

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Word Problems and Fluency Activities

Word Problems and Fluency Activities

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Lesson 12

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Lesson 14

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Attributes of Two-Dimensional Shapes

Topic B: Understanding Perimeter

Topic C: Distinguishing Between Area and Perimeter

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