Solved Exercises on Area Model Partial Products Multiplication in Grade 4

Master area model partial products multiplication: decomposition, multiplication, addition, and visual representations through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic area model multiplication
Exercise 1
Multiply: \(24 \times 13\) using area model partial products
Definition:

Area model partial products: Decompose numbers by place value and multiply each part separately

Area model method:
  1. Decompose each number into tens and ones
  2. Create a rectangle divided into sections
  3. Multiply each section (partial products)
  4. Add all partial products together
Decomposition
\(24 = 20 + 4\), \(13 = 10 + 3\)
Partial products
\(20×10 + 20×3 + 4×10 + 4×3\)
Sum
\(200 + 60 + 40 + 12\)
Step 1: Decompose the numbers

\(24 = 20 + 4\) (2 tens and 4 ones)

\(13 = 10 + 3\) (1 ten and 3 ones)

Step 2: Create area model rectangle

Rectangle divided into 4 sections: \(20×10\), \(20×3\), \(4×10\), \(4×3\)

Step 3: Calculate each partial product

\(20 × 10 = 200\)

\(20 × 3 = 60\)

\(4 × 10 = 40\)

\(4 × 3 = 12\)

Step 4: Add all partial products

\(200 + 60 + 40 + 12 = 312\)

\(24 \times 13 = 312\)
Final answer:

\(24 \times 13 = 312\)

Applied rules:

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

• Place value decomposition: Separate numbers by tens, ones, etc.

• Partial products: Multiply each decomposed part separately

2 Three-digit by two-digit area model
Exercise 2
Multiply: \(123 \times 15\) using area model partial products
Definition:

Extended area model: Decompose into hundreds, tens, and ones for larger numbers

Decomposition
\(123 = 100 + 20 + 3\), \(15 = 10 + 5\)
Partial products
\(100×10 + 100×5 + 20×10 + 20×5 + 3×10 + 3×5\)
Sum
\(1000 + 500 + 200 + 100 + 30 + 15\)
Step 1: Decompose the numbers

\(123 = 100 + 20 + 3\) (1 hundred, 2 tens, 3 ones)

\(15 = 10 + 5\) (1 ten, 5 ones)

Step 2: Create area model rectangle

Rectangle divided into 6 sections: \(100×10\), \(100×5\), \(20×10\), \(20×5\), \(3×10\), \(3×5\)

Step 3: Calculate each partial product

\(100 × 10 = 1000\)

\(100 × 5 = 500\)

\(20 × 10 = 200\)

\(20 × 5 = 100\)

\(3 × 10 = 30\)

\(3 × 5 = 15\)

Step 4: Add all partial products

\(1000 + 500 + 200 + 100 + 30 + 15 = 1845\)

\(123 \times 15 = 1845\)
Final answer:

\(123 \times 15 = 1845\)

Applied rules:

• Extended distributive property: Each part of first number multiplies each part of second

• Systematic approach: Organize partial products to avoid missing any

• Place value understanding: Maintain correct place value during multiplication

3 Area model with zero in tens place
Exercise 3
Multiply: \(107 \times 24\) using area model partial products
Definition:

Zero handling: Even if tens digit is 0, include it in decomposition for completeness

Decomposition
\(107 = 100 + 0 + 7\), \(24 = 20 + 4\)
Partial products
\(100×20 + 100×4 + 0×20 + 0×4 + 7×20 + 7×4\)
Sum
\(2000 + 400 + 0 + 0 + 140 + 28\)
Step 1: Decompose the numbers

\(107 = 100 + 0 + 7\) (1 hundred, 0 tens, 7 ones)

\(24 = 20 + 4\) (2 tens, 4 ones)

Step 2: Create area model rectangle

Rectangle divided into 6 sections: \(100×20\), \(100×4\), \(0×20\), \(0×4\), \(7×20\), \(7×4\)

Step 3: Calculate each partial product

\(100 × 20 = 2000\)

\(100 × 4 = 400\)

\(0 × 20 = 0\)

\(0 × 4 = 0\)

\(7 × 20 = 140\)

\(7 × 4 = 28\)

Step 4: Add all partial products

\(2000 + 400 + 0 + 0 + 140 + 28 = 2568\)

\(107 \times 24 = 2568\)
Final answer:

\(107 \times 24 = 2568\)

Applied rules:

• Zero multiplication: Any number times 0 equals 0

• Complete decomposition: Include zero placeholders to maintain structure

• Accuracy: Don't skip parts even if they equal zero

Rules and methods, laws,...
\((a + b)(c + d) = ac + ad + bc + bd\)
Distributive Property for Area Model
Rule 1
\((a + b)(c + d) = ac + ad + bc + bd\)
Basic area model expansion
Rule 2
\((a + b + c)(d + e) = ad + ae + bd + be + cd + ce\)
Extended area model
Rule 3
\(n \times 0 = 0\)
Zero multiplication property
Key definitions:

Area model: Visual representation of multiplication using rectangles divided into sections

Partial products: Individual multiplication results from decomposed numbers

Decomposition: Breaking numbers into place value parts (tens, ones, etc.)

Distributive property: Mathematical principle that governs area model multiplication

Complete methodology:
  1. Identify the numbers: Recognize the multi-digit numbers to be multiplied
  2. Decompose each number: Break into place value components
  3. Create area model: Draw rectangle and divide into sections
  4. Calculate partial products: Multiply each combination of decomposed parts
  5. Sum partial products: Add all individual results together
  6. Verify answer: Check for accuracy and reasonableness
Tip 1: Always decompose systematically: hundreds, tens, ones, etc.
Tip 2: Draw a grid to organize partial products visually
Tip 3: When adding partial products, align by place value to avoid errors.
Common errors: Missing partial products, incorrect place value, miscalculating individual products.
Key insight: Area model shows why standard algorithm works by breaking it into smaller parts.
Essential formulas to remember:

• Distributive property: \((a + b)(c + d) = ac + ad + bc + bd\)

• Place value decomposition: \(abc = a×100 + b×10 + c×1\)

• Partial products sum: Total = all individual multiplication results added

Solution: Exercises 4 to 5
4 Word problem with area model multiplication
Exercise 4
A rectangular garden has dimensions 34 feet by 26 feet. What is the area of the garden?
Definition:

Area calculation: Length × Width = Area, using area model partial products method

Dimensions
\(34\) feet × \(26\) feet
Decomposition
\(34 = 30 + 4\), \(26 = 20 + 6\)
Partial products
\(30×20 + 30×6 + 4×20 + 4×6\)
Sum
\(600 + 180 + 80 + 24 = 884\)
Step 1: Identify what needs to be calculated

Area = Length × Width = 34 × 26 square feet

Step 2: Decompose the numbers

\(34 = 30 + 4\) (3 tens and 4 ones)

\(26 = 20 + 6\) (2 tens and 6 ones)

Step 3: Calculate each partial product

\(30 × 20 = 600\)

\(30 × 6 = 180\)

\(4 × 20 = 80\)

\(4 × 6 = 24\)

Step 4: Add all partial products

\(600 + 180 + 80 + 24 = 884\)

The area is 884 square feet
Final answer:

The area of the garden is 884 square feet

Applied rules:

• Area formula: Rectangle area = length × width

• Word problem setup: Identify what operation is needed

• Real-world context: Include appropriate units in the final answer

5 Four-digit by two-digit area model
Exercise 5
Multiply: \(1,234 \times 18\) using area model partial products
Definition:

Large number multiplication: Extending area model to four digits using systematic decomposition

Decomposition
\(1234 = 1000 + 200 + 30 + 4\), \(18 = 10 + 8\)
Partial products
\(1000×10 + 1000×8 + 200×10 + 200×8 + 30×10 + 30×8 + 4×10 + 4×8\)
Sum
\(10000 + 8000 + 2000 + 1600 + 300 + 240 + 40 + 32\)
Total
\(22212\)
Step 1: Decompose the numbers

\(1234 = 1000 + 200 + 30 + 4\) (1 thousand, 2 hundreds, 3 tens, 4 ones)

\(18 = 10 + 8\) (1 ten, 8 ones)

Step 2: Calculate each partial product

\(1000 × 10 = 10000\)

\(1000 × 8 = 8000\)

\(200 × 10 = 2000\)

\(200 × 8 = 1600\)

\(30 × 10 = 300\)

\(30 × 8 = 240\)

\(4 × 10 = 40\)

\(4 × 8 = 32\)

Step 3: Add all partial products

\(10000 + 8000 + 2000 + 1600 + 300 + 240 + 40 + 32 = 22212\)

Step 4: Verify the result

Check reasonableness: \(1200 × 20 = 24000\), which is close to our answer of 22212

\(1234 \times 18 = 22212\)
Final answer:

\(1234 \times 18 = 22212\)

Applied rules:

• Systematic decomposition: Work from largest to smallest place values

• Pattern recognition: Each part of first number multiplies each part of second

• Verification: Estimate to check reasonableness of large number results

Complete Guide to Area Model Partial Products Multiplication
\((a + b)(c + d) = ac + ad + bc + bd\)
Distributive Property Foundation
Key definitions:

Area model: Visual representation of multiplication showing the relationship between factors and product as rectangular areas

Partial products: Individual multiplication results when each part of one factor is multiplied by each part of the other factor

Decomposition: Breaking numbers into their place value components (hundreds, tens, ones)

Distributive property: Mathematical principle stating that \(a(b + c) = ab + ac\)

Place value: The value of a digit based on its position in the number (ones, tens, hundreds, etc.)

Complete methodology:
  1. Identify the multiplication problem: Recognize the two numbers to be multiplied
  2. Decompose both numbers: Break each number into place value components
  3. Set up area model: Draw a rectangle and divide it based on the number of components
  4. Calculate all partial products: Multiply each component of the first number by each component of the second number
  5. Organize partial products: Arrange in a systematic way to avoid missing any
  6. Sum all partial products: Add the results together to get the final answer
  7. Verify the result: Check for accuracy and reasonableness
Tip 1: Draw a grid to keep track of partial products: for \(24 × 13\), create a 2×2 grid
Tip 2: Always decompose systematically from largest to smallest place value
Tip 3: When adding partial products, group by place value (add thousands together, hundreds together, etc.)
Tip 4: Use estimation to verify your final answer makes sense
Commutative property: \(a × b = b × a\) - order of factors doesn't change the product
Distributive property: \(a(b + c) = ab + ac\) - basis of area model
Zero property: Any number multiplied by 0 equals 0
Formulas to know by heart:

• Distributive property: \((a + b)(c + d) = ac + ad + bc + bd\)

• Place value decomposition: \(abcd = a×1000 + b×100 + c×10 + d×1\)

• Partial products formula: Sum of all individual component multiplications

• Area model verification: Total area = sum of all partial areas

Visualizing Area Model Multiplication
Understanding Partial Products with Visual Models
Visual representation of \(24 × 13\) using area model with partial products shown

Concept: Area model multiplication shows how the total area equals the sum of all partial areas.

  • \(24 × 13\) creates 4 partial products: \(20×10\), \(20×3\), \(4×10\), \(4×3\)
  • Total area = \(200 + 60 + 40 + 12 = 312\)
  • Each partial product corresponds to a rectangular section of the area model

Questions & Answers

Question: Why do we need to break apart the numbers? Isn't the regular multiplication method faster?

Answer: The area model serves important purposes beyond just getting the answer:

  • Understanding: It shows WHY multiplication works, not just HOW to do it
  • Visualization: Students can see the multiplication as actual areas being combined
  • Error prevention: Helps catch mistakes by breaking the problem into smaller, manageable parts
  • Foundation: Builds understanding for more advanced topics like algebra

While the standard algorithm may be faster for experienced users, the area model helps develop number sense and conceptual understanding. Once students fully understand the concept, they can transition to the standard algorithm with a deeper understanding of what they're doing.

Think of it as learning to walk before running - the area model teaches the "why," which makes the "how" more meaningful.

Question: How do I know I haven't missed any partial products when using the area model?

Answer: There are several systematic ways to ensure you don't miss any partial products:

  1. Count method: If the first number has m parts and the second has n parts, you should have m × n partial products
  2. Grid method: Draw a rectangle grid where one dimension represents parts of first number and other represents parts of second number
  3. Systematic approach: Multiply each part of the first number by each part of the second number in order

For example, with \(24 × 13\) where \(24 = 20 + 4\) and \(13 = 10 + 3\):

  • First number has 2 parts, second number has 2 parts
  • Should have 2 × 2 = 4 partial products
  • Calculate: \(20×10\), \(20×3\), \(4×10\), \(4×3\) (4 products total) ✓

Drawing the actual area model rectangle and labeling each section also helps ensure you calculate every combination.

Question: What happens when I have zeros in my numbers? Do I still include them in the area model?

Answer: Yes, you should include zero parts in your decomposition, though the partial products involving zero will equal zero:

For example, with \(107 × 24\):

  1. Decompose: \(107 = 100 + 0 + 7\) and \(24 = 20 + 4\)
  2. Partial products: \(100×20\), \(100×4\), \(0×20\), \(0×4\), \(7×20\), \(7×4\)
  3. Results: \(2000\), \(400\), \(0\), \(0\), \(140\), \(32\)
  4. Sum: \(2000 + 400 + 0 + 0 + 140 + 32 = 2572\)

Including the zero parts:

  • Keeps your work organized and systematic
  • Ensures you don't accidentally skip place values
  • Maintains the correct structure of the area model

Even though \(0 × 20 = 0\) and \(0 × 4 = 0\), including these steps helps maintain the proper mathematical structure and prevents errors.

Question: How can I check if my area model multiplication is correct? Sometimes I'm not sure if I made a mistake.

Answer: Here are several verification methods for area model multiplication:

  1. Standard algorithm check: Solve the same problem using the traditional multiplication method
  2. Estimation: Round numbers to nearby convenient values and check if your answer is reasonable
  3. Calculator verification: Use a calculator to confirm your final answer
  4. Partial product verification: Double-check each individual partial product calculation

For example, checking \(24 × 13 = 312\):

  • Estimate: \(25 × 10 = 250\), which is close to 312 ✓
  • Standard algorithm: \(24 × 13 = 72 + 240 = 312\) ✓
  • Partial products: \(200 + 60 + 40 + 12 = 312\) ✓

Also count your partial products - for \(24 × 13\) you should have 4 partial products (2 parts × 2 parts = 4).

Question: What's the easiest way to understand the area model? I learn better with pictures.

Answer: Visual models are perfect for understanding area model multiplication! Here's how to picture it:

  1. Draw a rectangle: For \(24 × 13\), draw a large rectangle
  2. Split the sides: Split one side into 20 and 4, the other into 10 and 3
  3. Divide the rectangle: Draw lines to create 4 smaller rectangles inside
  4. Label each section: Each small rectangle represents one partial product

For \(24 × 13\):

  • Top-left rectangle: \(20 × 10 = 200\)
  • Top-right rectangle: \(20 × 3 = 60\)
  • Bottom-left rectangle: \(4 × 10 = 40\)
  • Bottom-right rectangle: \(4 × 3 = 12\)

The total area of the large rectangle equals the sum of all small rectangles: \(200 + 60 + 40 + 12 = 312\).

This visual approach connects the abstract mathematical concept to a concrete geometric idea!