Area of Rectangle Using Distributive Property Worksheet - 3rd Grade Math

Complete worksheet with exercises on decomposing rectangles, partial areas, and step-by-step solutions for 3rd grade students.

Area of Rectangle Using Distributive Property
Worksheet for 3rd Grade Students

Instructions:

  • For each problem, break apart one dimension using the distributive property
  • Calculate the area of each part separately
  • Add the partial areas to find the total area
  • Show all your work and write the final answer in square units
  • Check your answer by multiplying the original dimensions
a(b + c) = ab + ac
Distributive Property Formula
Key Concepts:

Distributive Property: Breaking apart multiplication problems to make them easier

Decompose: To split a number or shape into smaller parts

Partial Area: The area of a portion of the rectangle

Total Area: The combined area of all parts

Rectangle: A 4-sided shape with 4 right angles

Area: The space inside a shape measured in square units

Problem Solving Steps:
  1. Identify dimensions: Find length and width of rectangle
  2. Choose number to break: Select which number to decompose
  3. Apply distributive property: Rewrite multiplication using distribution
  4. Calculate partial areas: Multiply each part separately
  5. Add partial areas: Sum all parts to get total area
  6. Verify answer: Check by direct multiplication
Tip 1: Break numbers into tens and ones (like 12 = 10 + 2).
Tip 2: Always draw a visual to help understand the problem.
Tip 3: Add partial areas carefully to avoid mistakes.
Tip 4: Check your work by multiplying original dimensions.
Problem Set: Problems 1 to 3
1 Basic Decomposition
1
Find the area of a rectangle that is 7 units long and 3 units wide using the distributive property by breaking 7 into 5 + 2.
Solution:
Write the original multiplication: 7 × 3
Break apart 7: 7 = 5 + 2
Apply distributive property: (5 + 2) × 3 = 5 × 3 + 2 × 3
Calculate partial areas: 5 × 3 = 15, 2 × 3 = 6
Add partial areas: 15 + 6 = 21
7 × 3 = (5 + 2) × 3 = 5 × 3 + 2 × 3 = 15 + 6 = 21
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Answer:
The area of the rectangle is 21 square units.
Verification:

Direct multiplication: 7 × 3 = 21 ✓

Distributive property: (5 + 2) × 3 = 5 × 3 + 2 × 3 = 15 + 6 = 21 ✓

2 Two-Part Division
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Find the area of a rectangle that is 9 units long and 4 units wide using the distributive property by breaking 9 into 6 + 3.
Solution:
Write the original multiplication: 9 × 4
Break apart 9: 9 = 6 + 3
Apply distributive property: (6 + 3) × 4 = 6 × 4 + 3 × 4
Calculate partial areas: 6 × 4 = 24, 3 × 4 = 12
Add partial areas: 24 + 12 = 36
9 × 4 = (6 + 3) × 4 = 6 × 4 + 3 × 4 = 24 + 12 = 36
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Answer:
The area of the rectangle is 36 square units.
Verification:

Direct multiplication: 9 × 4 = 36 ✓

Distributive property: (6 + 3) × 4 = 6 × 4 + 3 × 4 = 24 + 12 = 36 ✓

3 Three-Part Division
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Find the area of a rectangle that is 12 units long and 3 units wide using the distributive property by breaking 12 into 5 + 5 + 2.
Solution:
Write the original multiplication: 12 × 3
Break apart 12: 12 = 5 + 5 + 2
Apply distributive property: (5 + 5 + 2) × 3 = 5 × 3 + 5 × 3 + 2 × 3
Calculate partial areas: 5 × 3 = 15, 5 × 3 = 15, 2 × 3 = 6
Add partial areas: 15 + 15 + 6 = 36
12 × 3 = (5 + 5 + 2) × 3 = 5 × 3 + 5 × 3 + 2 × 3 = 15 + 15 + 6 = 36
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Answer:
The area of the rectangle is 36 square units.
Verification:

Direct multiplication: 12 × 3 = 36 ✓

Distributive property: (5 + 5 + 2) × 3 = 5 × 3 + 5 × 3 + 2 × 3 = 15 + 15 + 6 = 36 ✓

Problem Set: Problems 4 to 5
4 Real-world Application
4
A rectangular garden is 15 feet long and 4 feet wide. The gardener wants to plant vegetables in a 10-foot section and flowers in a 5-foot section. What is the total area of the garden using the distributive property?
Solution:
Identify the dimensions: Length = 15 feet, Width = 4 feet
Break apart 15: 15 = 10 + 5
Apply distributive property: (10 + 5) × 4 = 10 × 4 + 5 × 4
Calculate partial areas: 10 × 4 = 40, 5 × 4 = 20
Add partial areas: 40 + 20 = 60
15 × 4 = (10 + 5) × 4 = 10 × 4 + 5 × 4 = 40 + 20 = 60
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Answer:
The total area of the garden is 60 square feet.
Verification:

Direct multiplication: 15 × 4 = 60 ✓

Distributive property: (10 + 5) × 4 = 10 × 4 + 5 × 4 = 40 + 20 = 60 ✓

5 Advanced Problem
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A rectangular room is 14 feet long and 6 feet wide. It costs $3 per square foot to install flooring. If the room is divided into a 10-foot section and a 4-foot section, how much will it cost to install flooring in the entire room using the distributive property?
Solution:
Find the area using distributive property: 14 × 6 = (10 + 4) × 6
Break apart 14: 14 = 10 + 4
Apply distributive property: (10 + 4) × 6 = 10 × 6 + 4 × 6
Calculate partial areas: 10 × 6 = 60, 4 × 6 = 24
Add partial areas: 60 + 24 = 84 square feet
Calculate cost: 84 × $3 = $252
Area: 14 × 6 = (10 + 4) × 6 = 10 × 6 + 4 × 6 = 60 + 24 = 84 sq ft
Cost: 84 × $3 = $252
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Answer:
It will cost $252 to install flooring in the entire room.
Verification:

Direct multiplication: 14 × 6 = 84 ✓

Distributive property: (10 + 4) × 6 = 10 × 6 + 4 × 6 = 60 + 24 = 84 ✓

Cost calculation: 84 × $3 = $252 ✓

Questions & Answers

Question: I don't understand why we need to break apart the numbers when we can just multiply them normally. Isn't that harder?

Answer: Great question! The distributive property can seem harder at first, but it's actually very helpful for several reasons:

  • Making hard problems easier: For example, 13 × 7 might be hard, but (10 + 3) × 7 = 10×7 + 3×7 = 70 + 21 = 91 is easier
  • Understanding multiplication: It helps you see what multiplication really means
  • Real-world applications: Sometimes spaces are divided into sections, so you need to calculate each part separately
  • Building math skills: It prepares you for more advanced math in higher grades

Think of it like taking a big problem and breaking it into smaller, easier pieces to solve!

Question: How can I help my child practice the distributive property at home?

Answer: Here are some practical ways to practice at home:

  • Use everyday objects: Arrange blocks, toys, or snacks in rectangular arrays and break them apart
  • Draw pictures: Have your child draw rectangles and color-code the different parts
  • Practice with grid paper: Use graph paper to visualize the rectangles and their parts
  • Make up story problems: Create real-life scenarios involving areas of rooms, gardens, or other spaces
  • Play games: Create flashcards with area problems that use the distributive property

Start with simple numbers and gradually increase complexity. The key is to make it visual and hands-on!

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

Answer: Here are some helpful tips for choosing which number to break apart:

  • Break the bigger number: It's usually easier to work with smaller numbers
  • Think about tens and ones: Like breaking 14 into 10 + 4, or 27 into 20 + 7
  • Look for numbers that are easy to multiply: Numbers that make multiplication facts you know well
  • Follow the problem: Sometimes the problem tells you how to break it apart

For example, with 16 × 5, you might break 16 into 10 + 6: (10 + 6) × 5 = 10×5 + 6×5 = 50 + 30 = 80. This is often easier than multiplying 16 × 5 directly!