Shapes and Their Perimeter

Lesson 14

Math

Unit 5

3rd Grade

Lesson 14 of 16

Objective


Find rectangles with the same perimeter and different areas.

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 perimeter (e.g., a 1-by-7 rectangle, 2-by-6 rectangle, 3-by-5 rectangle, and 4-by-4 rectangle all have a perimeter of 16 units) (MP.1).
  2. Determine the area of all possible rectangles with a given perimeter.
  3. Understand that a given perimeter does not imply a particular area (i.e., a rectangle with a particular perimeter could have a variety of areas).
  4. Determine the rectangle with the greatest area given a particular perimeter.
  5. Understand that given a particular perimeter, the most square-like rectangle will have the greatest area, and the most oblong rectangle will have the least area (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 (25 per student or small group) — Students might not need these depending on their reliance on concrete materials. See Anchor Task 1 for more information.
  • Optional: Paper clips (25 per student or small group) — These should measure 1 inch in length. 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 a perimeter of 16 units. Find the area of each rectangle that you found. Then record the length, width, perimeter, and area of each rectangle in the table below:

Length (units) Width (units) Perimeter (units) Area (square 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 21Concept Development

Grade 3 Mathematics > Module 7 > Topic D > Lesson 21 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 22Concept Development

Grade 3 Mathematics > Module 7 > Topic D > Lesson 22 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 20Concept Development

Grade 3 Mathematics > Module 7 > Topic D > Lesson 20 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 area you can make with a rectangle that has a perimeter of 20 units?

Guiding Questions

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References

Open Middle Rectangles: Maximizing Area

Rectangles: Maximizing Area by is made available on Open Middle under the CC BY-NC-SA 4.0 license. Accessed Feb. 10, 2019, 5:14 p.m..

Problem Set


Answer Keys

Unlock the answer keys for this lesson's problem set and extra practice problems to save time and support student learning.

Discussion of Problem Set

  • If you know a rectangle’s perimeter, do you know for sure what its area is? What about the other way around? How does #1 support your thinking?
  • How did you find a rectangle with the same perimeter but a smaller area in #3? Did you think you’d need a longer and skinnier rectangle or a more square-like one? Why?
  • Why are all of the perimeters even?
  • For which perimeters in #4 can a square be drawn? What do they all have in common? Can you brainstorm other numbers of unit squares from which a square can be drawn?
  • Why do you think a square has a greater area than any other rectangle with the same perimeter?
  • For what perimeters would you expect to be able to make seven rectangles? Why?

Target Task


Use the rectangle shown below to answer the question. 

A new rectangle has the same perimeter but a greater area than the rectangle above. Find the length and width of the new rectangle.

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 13

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Lesson 15

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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