Area

Lesson 10

Math

Unit 4

3rd Grade

Lesson 10 of 14

Objective


Compose and decompose a rectangle, seeing and making use of the idea that the sum of the areas of the decomposed rectangle is equal to the area of the composed rectangle. 

Common Core Standards


Core Standards

  • 3.MD.C.7.C — Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.

Criteria for Success


  1. Look for structure to understand that the area of a rectangle with whole-number side lengths $$a$$ and $$b+c$$ is the sum of $$a\times b$$ and $$a\times c$$ (MP.7).
  2. Make use of structure by breaking apart a rectangle into two smaller rectangles, finding the area of those smaller rectangles, then adding those areas together to get the area of the composed rectangle (MP.7).

Tips for Teachers


Students might naturally connect work of this lesson to the distributive property. You can discuss that connection and use the notation for the distributive property if students seem ready for it, but this lesson has been designed so that students are not expected to use it and thus is not on the Problem Set. The next lesson will make the connection to the distributive property more explicit. 

Lesson Materials

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Anchor Tasks


Problem 1

The area model below shows the floor plan for a storage closet. The storage closet will have a tiled floor with grey tiles on the left and white tiles on the right. Each tile is 1 square foot.

a.   What is the area of the whole floor?

b.   What is the area of the floor covered with gray tiles? What is the area of the floor covered with white tiles?

c.   What do you notice about your answers to Parts (a) and (b) above? What do you wonder?

Guiding Questions

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References

Illustrative Mathematics Illustrative Mathematics task, “Introducing the Distributive Property,”

Adapted from Illustrative Mathematics task, “Introducing the Distributive Property,”, accessed on Sept. 23, 2022, 4:56 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by Fishtank Learning, Inc.

Problem 2

Use the following area model to answer Parts (a) and (b). 

a.   Decompose the area model into smaller rectangles whose products you know from memory to be able to find the total product represented by the area model.

b.   Find a second way to decompose the area model, then use that decomposition to find its area.

Guiding Questions

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Problem Set


Answer Keys

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Discussion of Problem Set

  • In #1(a), how does knowing the side lengths of the grid help you find the side lengths of the small rectangles without counting? 
  • Explain to a partner how you found the length and width for the new rectangle in Problem #3(b). 
  • If you labeled the width 13 and length 4, how would that change your drawing? How would that affect the area of the rectangle? 
  • Did anyone multiply the side lengths to solve #3(c)? What strategy did you use to multiply 4 x 13? 
  • How was #5 different from the other problems?
  • How is today’s work similar to the multiplication strategy of breaking a factor into a sum of addends we used in Unit 3? How is it different? 

Target Task


Use the two rectangles below to answer the questions.

a.   Write an equation to represent the area of Rectangle A. 

b.   Write an equation to represent the area of Rectangle B. 

c.   Rectangles A and B are pushed together to make a larger rectangle. What is the area of that larger rectangle? Explain. 

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 9

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Lesson 11

Lesson Map

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

Topic A: Understanding Concepts of Area

Topic B: The Distributive Property and Composite Area

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