Solved Exercises on Distributive Property Using Area Model in Grade 3

Master distributive property using area models: visual multiplication strategies, area decomposition, and mathematical reasoning through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic area model
Exercise 1
Find the area of a rectangle that is 6 units wide and 7 units tall using the distributive property with area model.
Hint: Break 7 into 5 + 2
Definition:

Distributive Property: \(a(b+c) = ab + ac\)

Area Model: Visual representation of multiplication as rectangles

Area model method:
  1. Draw a rectangle representing the total area
  2. Break one dimension into parts (e.g., 7 = 5 + 2)
  3. Divide the rectangle into smaller rectangles
  4. Calculate each smaller area
  5. Add all areas together
Expression
\(6 \times 7\)
Decomposition
\(6 \times (5 + 2)\)
Distributed
\(6 \times 5 + 6 \times 2\)
Step 1: Draw the rectangle

Rectangle: 6 units wide, 7 units tall

Step 2: Break the height into parts

7 = 5 + 2

Step 3: Calculate each part

\(6 \times 5 = 30\) and \(6 \times 2 = 12\)

Step 4: Add the areas

\(30 + 12 = 42\)

\(6 \times 7 = 6 \times (5 + 2) = 6 \times 5 + 6 \times 2 = 30 + 12 = 42\)
Final answer:

The area of the rectangle is 42 square units

Applied rules:

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

• Area Formula: Length × Width

• Decomposition: Breaking numbers into easier-to-multiply parts

2 Two-digit multiplication
Exercise 2
Find the area of a rectangle that is 8 units wide and 12 units tall using the distributive property with area model.
Break 12 into 10 + 2
Definition:

Distributive Property: \(a(b+c) = ab + ac\)

Area Model: Breaking larger numbers into tens and ones

Expression
\(8 \times 12\)
Decomposition
\(8 \times (10 + 2)\)
Distributed
\(8 \times 10 + 8 \times 2\)
Step 1: Draw the rectangle

Rectangle: 8 units wide, 12 units tall

Distributive Property Using Area Model | 3rd Grade Math | Area and Multiplication

Distributive Property Using Area Model - Complete Guide for 3rd Grade

Master the distributive property using area models: visual representations that make multiplication easier through partitioning rectangles into smaller parts.

Distributive Property Rules & Methods
\(a(b+c) = ab + ac\)
Distributive Property
Property
\(a(b+c) = ab + ac\)
Distribute the first number to both parts inside parentheses
Area Model
Length × Width = Area
Break large rectangles into smaller ones
Partitioning
\(4 \times 7 = 4 \times (5+2) = 4 \times 5 + 4 \times 2\)
Split one dimension into easier numbers
Key Definitions:

Distributive Property: Multiplying a number by a sum equals multiplying the number by each addend and adding the results

Area Model: A visual representation showing multiplication as rectangular areas

Partitioning: Breaking apart numbers into smaller, easier-to-work-with parts

Step-by-Step Method:
  1. Identify the larger number: Find the number that's hard to multiply
  2. Partition it: Break the number into easier parts (usually tens and ones)
  3. Apply distribution: Multiply the outside number by each part
  4. Add the partial products: Combine the results
Tip 1: Always break numbers into tens and ones for easier multiplication.
Tip 2: Draw rectangles to visualize the area model when solving problems.
Tip 3: Check your work by adding partial products to get the total.
Tip 4: Use the distributive property to make mental math faster.
Common errors: Forgetting to multiply both parts, adding instead of multiplying, miscounting partial products.
Memory aid: Think "distribute" means to give out equally to everyone inside the parentheses.
Essential Rules:

• Distribution: Multiply the outside number by each term inside parentheses

• Area Addition: Total area equals sum of individual rectangle areas

• Partitioning Strategy: Break numbers into tens and ones for easier calculation

Solution: Exercises 1 to 3
1 Basic Area Model
Exercise 1
Find the area of a rectangle with length 6 and width 8 using the distributive property by breaking 8 into 5 + 3.
Concept:

Area Model: A rectangle with dimensions 6 × 8 can be split into two rectangles: 6 × 5 and 6 × 3

Solution method:
  1. Draw a rectangle representing 6 × 8
  2. Partition the width 8 into 5 + 3
  3. Calculate areas of the two smaller rectangles
  4. Add the areas together
Rectangle 1
6 × 5 = 30
Rectangle 2
6 × 3 = 18
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
Expression
\(6 \times 8\)
Partition
\(6 \times (5+3)\)
Distribute
\(6 \times 5 + 6 \times 3\)
Calculate
\(30 + 18 = 48\)
Step 1: Set up the distributive property

We want to find \(6 \times 8\), so we break 8 into 5 + 3

\(6 \times 8 = 6 \times (5+3)\)

Step 2: Apply the distributive property

Multiply 6 by each part: \(6 \times (5+3) = 6 \times 5 + 6 \times 3\)

Step 3: Calculate each partial product

\(6 \times 5 = 30\) and \(6 \times 3 = 18\)

Step 4: Add the partial products

Total area: \(30 + 18 = 48\)

\(6 \times 8 = 6 \times (5+3) = 6 \times 5 + 6 \times 3 = 30 + 18 = 48\)
Final answer:

The area of the rectangle is 48 square units

Applied rules:

• Distribution: Multiply each term in parentheses by the outside factor

• Area Addition: Total area equals sum of partial areas

• Partitioning: Break complex multiplication into simpler parts

2 Medium Difficulty
Exercise 2
Use the area model to solve \(7 \times 12\) by breaking 12 into 10 + 2.
Concept:

Area Model: A rectangle with dimensions 7 × 12 can be split into two rectangles: 7 × 10 and 7 × 2

Solution method:
  1. Break 12 into 10 + 2 (tens and ones)
  2. Apply distributive property: \(7 \times (10+2)\)
  3. Calculate \(7 \times 10\) and \(7 \times 2\)
  4. Add the results together
Rectangle 1
7 × 10 = 70
Rectangle 2
7 × 2 = 14
Expression
\(7 \times 12\)
Partition
\(7 \times (10+2)\)
Distribute
\(7 \times 10 + 7 \times 2\)
Calculate
\(70 + 14 = 84\)
Step 1: Partition the harder number

Break 12 into 10 + 2 because 10 and 2 are easier to multiply

\(7 \times 12 = 7 \times (10+2)\)

Step 2: Apply the distributive property

Multiply 7 by each part: \(7 \times (10+2) = 7 \times 10 + 7 \times 2\)

Step 3: Calculate each partial product

\(7 \times 10 = 70\) and \(7 \times 2 = 14\)

Step 4: Add the partial products

Total: \(70 + 14 = 84\)

\(7 \times 12 = 7 \times (10+2) = 7 \times 10 + 7 \times 2 = 70 + 14 = 84\)
Final answer:

\(7 \times 12 = 84\)

Applied rules:

• Partitioning Strategy: Break numbers into tens and ones

• Distribution: Multiply each term in parentheses separately

• Partial Products: Add results of individual multiplications

3 Advanced Application
Exercise 3
Solve \(4 \times 15\) using the area model by breaking 15 into 10 + 5, then draw the visual representation.
Concept:

Area Model: A rectangle with dimensions 4 × 15 can be split into two rectangles: 4 × 10 and 4 × 5

Solution method:
  1. Visualize a rectangle of 4 rows and 15 columns
  2. Partition the 15 columns into 10 and 5 sections
  3. Calculate areas of the two sub-rectangles
  4. Sum the areas to get the total
Rectangle 1
4 × 10 = 40
Rectangle 2
4 × 5 = 20
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
Expression
\(4 \times 15\)
Partition
\(4 \times (10+5)\)
Distribute
\(4 \times 10 + 4 \times 5\)
Calculate
\(40 + 20 = 60\)
Step 1: Identify the complex multiplication

We need to solve \(4 \times 15\), which is difficult to compute directly

Step 2: Break 15 into easier parts

Partition 15 into 10 + 5: \(4 \times 15 = 4 \times (10+5)\)

Step 3: Apply the distributive property

\(4 \times (10+5) = 4 \times 10 + 4 \times 5\)

Step 4: Calculate each partial product

\(4 \times 10 = 40\) and \(4 \times 5 = 20\)

Step 5: Add the partial products

Total area: \(40 + 20 = 60\)

\(4 \times 15 = 4 \times (10+5) = 4 \times 10 + 4 \times 5 = 40 + 20 = 60\)
Final answer:

\(4 \times 15 = 60\)

Applied rules:

• Partitioning Strategy: Break complex numbers into tens and ones

• Area Model: Represent multiplication as rectangular areas

• Distribution: Apply the distributive property to simplify calculations

Solution: Exercises 4 to 5
4 Word Problem Application
Exercise 4
Sarah has 6 boxes of cookies. Each box has 14 cookies. How many cookies does Sarah have in total? Use the distributive property by breaking 14 into 10 + 4.
Concept:

Real-world Application: Use the distributive property to solve multiplication word problems

Solution method:
  1. Identify the multiplication: 6 boxes × 14 cookies per box
  2. Break 14 into 10 + 4 for easier computation
  3. Apply distributive property: \(6 \times (10+4)\)
  4. Calculate and sum the partial products
Boxes 1-6 with 10 cookies each
6 × 10 = 60 cookies
Boxes 1-6 with 4 cookies each
6 × 4 = 24 cookies
Problem
\(6 \times 14\)
Partition
\(6 \times (10+4)\)
Distribute
\(6 \times 10 + 6 \times 4\)
Calculate
\(60 + 24 = 84\)
Step 1: Set up the multiplication

Sarah has 6 boxes with 14 cookies each: \(6 \times 14\)

Step 2: Partition the harder number

Break 14 into 10 + 4: \(6 \times 14 = 6 \times (10+4)\)

Step 3: Apply the distributive property

\(6 \times (10+4) = 6 \times 10 + 6 \times 4\)

Step 4: Calculate each partial product

\(6 \times 10 = 60\) and \(6 \times 4 = 24\)

Step 5: Add the partial products

Total cookies: \(60 + 24 = 84\)

\(6 \times 14 = 6 \times (10+4) = 6 \times 10 + 6 \times 4 = 60 + 24 = 84\)
Final answer:

Sarah has 84 cookies in total

Applied rules:

• Word Problem Translation: Convert words into mathematical expressions

• Partitioning Strategy: Break complex numbers into tens and ones

• Distribution: Apply the distributive property to simplify multiplication

5 Complex Area Model
Exercise 5
A garden has dimensions 8 feet by 16 feet. The gardener wants to divide it into two sections: one for vegetables (8 × 10) and one for flowers (8 × 6). Calculate the total area using the distributive property.
Concept:

Area Model Application: Real-world scenarios where the distributive property helps calculate combined areas

Solution method:
  1. Recognize the total area: 8 × 16
  2. Identify the partition: 16 = 10 + 6
  3. Apply distributive property: \(8 \times (10+6)\)
  4. Calculate individual areas and sum them
Vegetable Section
8 × 10 = 80 sq ft
Flower Section
8 × 6 = 48 sq ft
Total Area
\(8 \times 16\)
Partition
\(8 \times (10+6)\)
Distribute
\(8 \times 10 + 8 \times 6\)
Calculate
\(80 + 48 = 128\)
Step 1: Identify the total area

The garden is 8 feet by 16 feet: \(8 \times 16\)

Step 2: Recognize the partition

The garden is divided: 10 feet for vegetables + 6 feet for flowers = 16 feet

Step 3: Apply the distributive property

\(8 \times 16 = 8 \times (10+6) = 8 \times 10 + 8 \times 6\)

Step 4: Calculate each section's area

Vegetable area: \(8 \times 10 = 80\) square feet

Flower area: \(8 \times 6 = 48\) square feet

Step 5: Sum the areas

Total area: \(80 + 48 = 128\) square feet

\(8 \times 16 = 8 \times (10+6) = 8 \times 10 + 8 \times 6 = 80 + 48 = 128\)
Final answer:

The total area of the garden is 128 square feet

Applied rules:

• Real-world Application: Connect mathematical concepts to practical scenarios

• Area Addition: Total area equals sum of individual areas

• Distribution: Use the distributive property to break complex multiplications

Questions & Answers

Question: Why do we need to break the number into parts when we can just multiply normally? It seems like more work.

Answer: Great question! The distributive property makes multiplication easier in several ways:

  • It breaks difficult problems into easier ones: \(7 \times 13\) becomes \(7 \times 10 + 7 \times 3\)
  • It helps you understand why multiplication works the way it does
  • It builds skills for future math concepts like algebra
  • It makes mental math faster once you practice it

Think of it like this: Would you rather solve \(6 \times 15\) all at once, or solve \(6 \times 10\) and \(6 \times 5\) separately and add them? Most people find the second way easier!

Question: How do I know which number to break apart when using the area model? Should I always break the bigger number?

Answer: Generally, you want to break apart the number that's harder to multiply. Usually this is the larger number, but not always! Here's the strategy:

  • Look for numbers that end in 0, 5, or 2 - these are easier to multiply
  • Break the number that doesn't have a familiar multiplication fact
  • Always try to break into tens and ones when possible (like 13 = 10 + 3)

Example: For \(4 \times 17\), break 17 into 10 + 7. For \(9 \times 12\), break 12 into 10 + 2. The goal is to make each multiplication easier than the original!

Question: What if I break a number differently? Like for 8 × 14, could I break 14 into 7 + 7 instead of 10 + 4?

Answer: Yes, you absolutely can! The distributive property works with any way you break apart a number. Let's compare:

Method 1 (10 + 4): \(8 \times 14 = 8 \times (10+4) = 8 \times 10 + 8 \times 4 = 80 + 32 = 112\)

Method 2 (7 + 7): \(8 \times 14 = 8 \times (7+7) = 8 \times 7 + 8 \times 7 = 56 + 56 = 112\)

Both methods give the same answer! However, breaking into tens and ones (10 + 4) is usually easier because multiplying by 10 is simple. The key is that the distributive property always works, no matter how you partition the number.