Tiling and the Distributive Property in Area - Third Grade Mathematics

When tiling a large floor or calculating the area of a large rectangle, multiplying large numbers directly can seem intimidating. Fortunately, you can slice a large rectangle into two smaller, friendlier rectangles, find the area of each section separately, and add them back together. This geometric strategy is the visual embodiment of the Distributive Property. In this chapter, you will master decomposing rectangles and connecting geometric area models to algebraic expressions.

Visualizing the Distributive Property with Area

Recall what the Distributive Property states: Multiplying a number by a sum equals the sum of the individual products: a x (b + c) = (a x b) + (a x c)

Look at how a rectangle represents this equation visually: Suppose you have a rectangle with a width of 5 units and a length of 12 units. Its total area is 5 x 12.

Let us slice the length of 12 into friendly parts: (10 + 2)!

               <---- 10 units ---->  <-- 2 units -->
              +--------------------+----------------+
              |                    |                |
     5 units  |    Rectangle 1     |  Rectangle 2   |
              |     5 x 10 = 50    |   5 x 2 = 10   |
              |                    |                |
              +--------------------+----------------+
               <--------------- 12 units -------------->

Watch what happened:
- Rectangle 1 has an area of: 5 x 10 = 50 square units.

- Rectangle 2 has an area of: 5 x 2 = 10 square units.

- Combine the two areas: 50 + 10 = 60 square units!
Therefore: 5 x 12 = (5 x 10) + (5 x 2) = 50 + 10 = 60 square units.

Why This Strategy Is So Powerful

Breaking apart area models allows you to solve multi-digit multiplication problems completely in your head using easy landmark numbers like 5 and 10!

Example 1: Solving 7 x 8

Suppose you are unsure of 7 x 8. Break the 8 into (5 + 3):
- Section A: 7 x 5 = 35

- Section B: 7 x 3 = 21

- Add them: 35 + 21 = 56!
Total Area = 56 square units.

                <--- 5 --->  <-- 3 -->
               +-----------+----------+
       7 units |  Area: 35 | Area: 21 |
               +-----------+----------+

Example 2: Solving 6 x 14

Break 14 into 10 + 4:
- Section A: 6 x 10 = 60

- Section B: 6 x 4 = 24

- Add them: 60 + 24 = 84!
Total Area = 84 square units.

Writing Mathematical Equations from Decomposed Area Models

To write an equation from a tiled area diagram:
1. Identify the common side length shared by both sections (the width).

2. Identify the two decomposed parts of the other side.

3. Write the distributive expression:
Shared Side x (Part 1 + Part 2) = (Shared Side x Part 1) + (Shared Side x Part 2).

Diagram:
A rectangle with height 4 is split into lengths of 7 and 3.
Equation: 4 x (7 + 3) = (4 x 7) + (4 x 3)
                      = 28 + 12
                      = 40 square units.

Chapter Practice Exercises

  1. Use the Distributive Property and area models to solve: a. 4 x 13: Break 13 into (10 + 3) b. 6 x 12: Break 12 into (10 + 2) c. 8 x 15: Break 15 into (10 + 5)
  2. Fill in the missing numbers to make the area equation true: a. 7 x 14 = (7 x 10) + (7 x ) b. 5 x 16 = (5 x ) + (5 x 6) c. 8 x (5 + 4) = (8 x ) + (8 x )
  3. Draw or describe an area model that shows 6 x 9 broken into (6 x 5) + (6 x 4). What is the total area?
  4. A tiled patio is 8 feet wide and 13 feet long. The homeowner tiles a section of 8 by 10 feet with red brick, and the remaining 8 by 3 feet with grey stone. a. What is the area of the red brick section? b. What is the area of the grey stone section? c. What is the total area of the patio?
  5. A student wants to solve 9 x 8 using the distributive property. Name two different friendly ways the student could break apart the factor 8.

Solutions and Step-by-Step Answers

  1. Distributive property calculations: a. 4 x 13 = (4 x 10) + (4 x 3) = 40 + 12 = 52 square units. b. 6 x 12 = (6 x 10) + (6 x 2) = 60 + 12 = 72 square units. c. 8 x 15 = (8 x 10) + (8 x 5) = 80 + 40 = 120 square units.
  2. Missing values: a. 7 x 14 = (7 x 10) + (7 x 4) b. 5 x 16 = (5 x 10) + (5 x 6) c. 8 x (5 + 4) = (8 x 5) + (8 x 4)
  3. Area model: A rectangle with height 6 and total base 9 is split into a 6 by 5 part (area = 30) and a 6 by 4 part (area = 24). Total area = 30 + 24 = 54 square units.
  4. Patio area: a. Red brick: 8 ft x 10 ft = 80 square feet. b. Grey stone: 8 ft x 3 ft = 24 square feet. c. Total area: 80 + 24 = 104 square feet.
  5. Two ways to break apart 8:
  6. Way 1: Break 8 into (5 + 3) -> (9 x 5) + (9 x 3) = 45 + 27 = 72.
  7. Way 2: Break 8 into (4 + 4) -> (9 x 4) + (9 x 4) = 36 + 36 = 72.