Area of Rectangle Using Distributive Property - 3rd Grade Math

Master area calculation with distributive property: decomposing rectangles, partial areas, and step-by-step solutions for 3rd grade students.

Distributive Property and Area
a(b + c) = ab + ac
Distributive Property
Key Definitions:

Distributive Property: A rule that allows us to break apart multiplication problems

Decompose: To break a number or shape into smaller parts

Partial Area: The area of a part of a rectangle

Rectangle: A 4-sided shape with 4 right angles

Total Area: The combined area of all parts

Width: The shorter side of a rectangle

Length: The longer side of a rectangle

Area: The space inside a shape measured in square units

Distributive Property Method:
  1. Break apart the length: Split one dimension into two or more parts
  2. Create sub-rectangles: Draw lines to separate the big rectangle
  3. Find partial areas: Calculate area of each smaller rectangle
  4. Add partial areas: Sum all the smaller areas together
  5. Verify: Check that the sum equals the total area
  6. Record: Include "square units" in your answer
Tip 1: Break numbers into tens and ones (like 12 = 10 + 2).
Tip 2: Draw lines to show the decomposition visually.
Tip 3: Always add the partial areas to get the total area.
Tip 4: Check your work by using regular multiplication.
Property Connection: a(b+c) = ab + ac applies to area calculation.
Visual Concept: A large rectangle equals the sum of its parts.
Essential Rules:

• Decompose one dimension of the rectangle

• Calculate area of each part separately

• Add all partial areas to get total area

• Units must be squared in final answer

• The sum of parts equals the whole

Exercise Solutions: 1 to 3
1 Basic Decomposition
Exercise 1
Find the area of a rectangle that is 6 units long and 4 units wide using the distributive property by breaking 6 into 4 + 2.
Problem Analysis:

Dimensions: Length = 6 units, Width = 4 units

Decomposition: 6 = 4 + 2

Method: Use distributive property: 6 × 4 = (4 + 2) × 4

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Part 1
(4×4)
Part 2
(2×4)
Break 6
6 = 4 + 2
Multiply
(4+2)×4
Distribute
4×4 + 2×4
Add
16 + 8 = 24
Step 1: Break apart the length

6 = 4 + 2

Step 2: Apply distributive property

6 × 4 = (4 + 2) × 4

Step 3: Distribute the width

(4 + 2) × 4 = 4 × 4 + 2 × 4

Step 4: Calculate partial areas

Part 1: 4 × 4 = 16 square units

Part 2: 2 × 4 = 8 square units

Step 5: Add partial areas

Total: 16 + 8 = 24 square units

Area = 24 square units
Final answer:

The area of the rectangle is 24 square units.

Applied rules:

• Distributive property: a(b + c) = ab + ac

• Area calculation: Length × Width for each part

• Sum of parts: Add all partial areas

2 Multiple Parts
Exercise 2
Find the area of a rectangle that is 9 units long and 3 units wide using the distributive property by breaking 9 into 5 + 4.
Problem Analysis:

Dimensions: Length = 9 units, Width = 3 units

Decomposition: 9 = 5 + 4

Method: Use distributive property: 9 × 3 = (5 + 4) × 3

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Part 1
(5×3)
Part 2
(4×3)
Break 9
9 = 5 + 4
Multiply
(5+4)×3
Distribute
5×3 + 4×3
Add
15 + 12 = 27
Step 1: Break apart the length

9 = 5 + 4

Step 2: Apply distributive property

9 × 3 = (5 + 4) × 3

Step 3: Distribute the width

(5 + 4) × 3 = 5 × 3 + 4 × 3

Step 4: Calculate partial areas

Part 1: 5 × 3 = 15 square units

Part 2: 4 × 3 = 12 square units

Step 5: Add partial areas

Total: 15 + 12 = 27 square units

Area = 27 square units
Final answer:

The area of the rectangle is 27 square units.

Applied rules:

• Distributive property: a(b + c) = ab + ac

• Area calculation: Length × Width for each part

• Sum of parts: Add all partial areas

3 Three Parts
Exercise 3
Find the area of a rectangle that is 8 units long and 5 units wide using the distributive property by breaking 8 into 3 + 3 + 2.
Problem Analysis:

Dimensions: Length = 8 units, Width = 5 units

Decomposition: 8 = 3 + 3 + 2

Method: Use distributive property: 8 × 5 = (3 + 3 + 2) × 5

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
Part 1
(3×5)
Part 2
(3×5)
Part 3
(2×5)
Break 8
8 = 3+3+2
Multiply
(3+3+2)×5
Distribute
3×5+3×5+2×5
Add
15+15+10=40
Step 1: Break apart the length

8 = 3 + 3 + 2

Step 2: Apply distributive property

8 × 5 = (3 + 3 + 2) × 5

Step 3: Distribute the width

(3 + 3 + 2) × 5 = 3 × 5 + 3 × 5 + 2 × 5

Step 4: Calculate partial areas

Part 1: 3 × 5 = 15 square units

Part 2: 3 × 5 = 15 square units

Part 3: 2 × 5 = 10 square units

Step 5: Add partial areas

Total: 15 + 15 + 10 = 40 square units

Area = 40 square units
Final answer:

The area of the rectangle is 40 square units.

Applied rules:

• Distributive property: a(b + c + d) = ab + ac + ad

• Area calculation: Length × Width for each part

• Sum of parts: Add all partial areas

Exercise Solutions: 4 to 5
4 Real-world Application
Exercise 4
A rectangular garden is 12 feet long and 4 feet wide. The gardener wants to plant vegetables in sections. She divides the length into 10 feet and 2 feet. What is the total area of the garden using the distributive property?
Problem Analysis:

Context: Garden divided into planting sections

Dimensions: Length = 12 feet, Width = 4 feet

Decomposition: 12 = 10 + 2

Method: Use distributive property for area calculation

V
V
V
V
V
V
V
V
V
V
F
F
V
V
V
V
V
V
V
V
V
V
F
F
V
V
V
V
V
V
V
V
V
V
F
F
V
V
V
V
V
V
V
V
V
V
F
F
Vegetables
(10×4)
Flowers
(2×4)
Break 12
12 = 10+2
Multiply
(10+2)×4
Distribute
10×4+2×4
Add
40+8=48
Step 1: Break apart the length

12 = 10 + 2 feet

Step 2: Apply distributive property

12 × 4 = (10 + 2) × 4

Step 3: Distribute the width

(10 + 2) × 4 = 10 × 4 + 2 × 4

Step 4: Calculate partial areas

Veg area: 10 × 4 = 40 square feet

Flower area: 2 × 4 = 8 square feet

Step 5: Add partial areas

Total: 40 + 8 = 48 square feet

Garden area = 48 square feet
Final answer:

The total area of the garden is 48 square feet.

Applied rules:

• Distributive property: a(b + c) = ab + ac

• Real-world context: Area represents physical space

• Sum of parts: Total area equals sum of sections

5 Advanced Problem
Exercise 5
A rectangular room is 15 feet long and 6 feet wide. It costs $2 per square foot to carpet the room. If the room is divided into a 10-foot section and a 5-foot section, how much will it cost to carpet the entire room using the distributive property?
Problem Analysis:

Given: Length = 15 feet, Width = 6 feet, Cost = $2 per sq ft

Decomposition: 15 = 10 + 5

Steps: Find area using distributive property, then multiply by cost

S
S
S
S
S
S
S
S
S
S
P
P
P
P
P
S
S
S
S
S
S
S
S
S
S
P
P
P
P
P
S
S
S
S
S
S
S
S
S
S
P
P
P
P
P
S
S
S
S
S
S
S
S
S
S
P
P
P
P
P
S
S
S
S
S
S
S
S
S
S
P
P
P
P
P
S
S
S
S
S
S
S
S
S
S
P
P
P
P
P
Section 1
(10×6)
Section 2
(5×6)
Step 1
Area = (10+5)×6
Step 2
10×6 + 5×6 = 90
Step 3
Cost = 90×$2 = $180
Step 1: Break apart the length

15 = 10 + 5 feet

Step 2: Apply distributive property

15 × 6 = (10 + 5) × 6

Step 3: Distribute the width

(10 + 5) × 6 = 10 × 6 + 5 × 6

Step 4: Calculate partial areas

Part 1: 10 × 6 = 60 square feet

Part 2: 5 × 6 = 30 square feet

Step 5: Add partial areas and calculate cost

Total area: 60 + 30 = 90 square feet

Cost: 90 × $2 = $180

Carpeting cost = $180
Final answer:

It will cost $180 to carpet the entire room.

Applied rules:

• Distributive property: a(b + c) = ab + ac

• Area calculation: Length × Width for each part

• Cost calculation: Total area × Cost per unit area

Questions & Answers

Question: Why do we need to use the distributive property for area when we can just multiply length times width?

Answer: Great question! The distributive property is helpful for several reasons:

  • Breaking down complex problems: It makes multiplication easier when dealing with larger numbers (like 12×7 = (10+2)×7 = 70+14 = 84)
  • Understanding area: It helps you see that a large rectangle is made up of smaller rectangles
  • Real-world applications: Sometimes spaces are divided into sections, and you need to calculate each part separately
  • Foundation for algebra: It prepares you for more advanced math concepts

You can always multiply length times width directly, but the distributive property gives you another tool and helps you understand how multiplication works!

Question: How do I know which number to break apart when using the distributive property?

Answer: You can break apart either number, but here are some helpful guidelines:

  • Break the larger number: Usually easier to work with smaller numbers
  • Break into tens and ones: Like 14 = 10 + 4 or 27 = 20 + 7
  • Look for easy multiplication: Break into numbers that are easy to multiply mentally
  • Consider the context: If the problem mentions sections, break it that way

For example, with 13×6, you might break 13 into 10+3: (10+3)×6 = 10×6 + 3×6 = 60+18 = 78. This is often easier than trying to multiply 13×6 directly!

Question: How can I help my child visualize the distributive property with rectangles?

Answer: Here are several effective visualization strategies:

  • Draw rectangles: Sketch a large rectangle and divide it with a vertical line to show the parts
  • Use manipulatives: Use blocks or tiles to build the rectangle and physically separate the parts
  • Color coding: Color different parts of the rectangle differently to show the decomposition
  • Grid paper: Draw rectangles on graph paper to count squares in each part
  • Real objects: Arrange objects like books or toys in rectangular arrays

You can also use the phrase "the whole rectangle equals the sum of its parts" to reinforce the concept. The visual representation makes the abstract concept more concrete and memorable.