Using Area Models for Multiplication - Practice Exercises

Master area models: visualize multiplication, partition rectangles, and find products with these 10 detailed exercises.

Exercises 1 to 5
1 Area Model for 12×3
Exercise 1
Draw an area model to represent 12×3 and find the product.
Definition:

Area model: A rectangle divided into sections to represent multiplication as area.

Area model method:
  1. Break apart the larger number by place value: 12 = 10 + 2
  2. Draw a rectangle with dimensions 12 (length) and 3 (width)
  3. Partition the rectangle into two sections: 10×3 and 2×3
  4. Calculate each section's area and add them together
Partition
12 = 10 + 2
Areas
10×3 + 2×3
Total
30 + 6
Step 1: Break apart 12 by place value

12 = 10 + 2 (1 ten and 2 ones)

Step 2: Draw rectangle with length 12 and width 3

Rectangle is 12 units long and 3 units wide

Step 3: Partition the rectangle

Draw a vertical line to separate 10 and 2 parts

Step 4: Calculate each section's area

Left section: 10 × 3 = 30, Right section: 2 × 3 = 6

Step 5: Add the areas

30 + 6 = 36

12 × 3 = 36
Final answer:

12 × 3 = 36

Applied rules:

• Area model: Length × Width = Area

• Distributive property: 12×3 = (10+2)×3 = 10×3 + 2×3

• Partitioning: Break apart by place value

2 Area Model for 15×4
Exercise 2
Draw an area model to represent 15×4 and find the product.
Definition:

Area model: A rectangle divided into sections to represent multiplication as area.

Partition
15 = 10 + 5
Areas
10×4 + 5×4
Total
40 + 20
Step 1: Break apart 15 by place value

15 = 10 + 5 (1 ten and 5 ones)

Step 2: Draw rectangle with length 15 and width 4

Rectangle is 15 units long and 4 units wide

Step 3: Partition the rectangle

Draw a vertical line to separate 10 and 5 parts

Step 4: Calculate each section's area

Left section: 10 × 4 = 40, Right section: 5 × 4 = 20

Step 5: Add the areas

40 + 20 = 60

15 × 4 = 60
Final answer:

15 × 4 = 60

Applied rules:

• Area model: Length × Width = Area

• Distributive property: 15×4 = (10+5)×4 = 10×4 + 5×4

• Partitioning: Break apart by place value

3 Area Model for 24×3
Exercise 3
Draw an area model to represent 24×3 and find the product.
Definition:

Area model: A rectangle divided into sections to represent multiplication as area.

Partition
24 = 20 + 4
Areas
20×3 + 4×3
Total
60 + 12
Step 1: Break apart 24 by place value

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

Step 2: Draw rectangle with length 24 and width 3

Rectangle is 24 units long and 3 units wide

Step 3: Partition the rectangle

Draw a vertical line to separate 20 and 4 parts

Step 4: Calculate each section's area

Left section: 20 × 3 = 60, Right section: 4 × 3 = 12

Step 5: Add the areas

60 + 12 = 72

24 × 3 = 72
Final answer:

24 × 3 = 72

Applied rules:

• Area model: Length × Width = Area

• Distributive property: 24×3 = (20+4)×3 = 20×3 + 4×3

• Partitioning: Break apart by place value

4 Area Model for 18×5
Exercise 4
Draw an area model to represent 18×5 and find the product.
Definition:

Area model: A rectangle divided into sections to represent multiplication as area.

Partition
18 = 10 + 8
Areas
10×5 + 8×5
Total
50 + 40
Step 1: Break apart 18 by place value

18 = 10 + 8 (1 ten and 8 ones)

Step 2: Draw rectangle with length 18 and width 5

Rectangle is 18 units long and 5 units wide

Step 3: Partition the rectangle

Draw a vertical line to separate 10 and 8 parts

Step 4: Calculate each section's area

Left section: 10 × 5 = 50, Right section: 8 × 5 = 40

Step 5: Add the areas

50 + 40 = 90

18 × 5 = 90
Final answer:

18 × 5 = 90

Applied rules:

• Area model: Length × Width = Area

• Distributive property: 18×5 = (10+8)×5 = 10×5 + 8×5

• Partitioning: Break apart by place value

5 Area Model for 32×4
Exercise 5
Draw an area model to represent 32×4 and find the product.
Definition:

Area model: A rectangle divided into sections to represent multiplication as area.

Partition
32 = 30 + 2
Areas
30×4 + 2×4
Total
120 + 8
Step 1: Break apart 32 by place value

32 = 30 + 2 (3 tens and 2 ones)

Step 2: Draw rectangle with length 32 and width 4

Rectangle is 32 units long and 4 units wide

Step 3: Partition the rectangle

Draw a vertical line to separate 30 and 2 parts

Step 4: Calculate each section's area

Left section: 30 × 4 = 120, Right section: 2 × 4 = 8

Step 5: Add the areas

120 + 8 = 128

32 × 4 = 128
Final answer:

32 × 4 = 128

Applied rules:

• Area model: Length × Width = Area

• Distributive property: 32×4 = (30+2)×4 = 30×4 + 2×4

• Partitioning: Break apart by place value

Exercises 6 to 10
6 Area Model for 27×3
Exercise 6
Draw an area model to represent 27×3 and find the product.
Definition:

Area model: A rectangle divided into sections to represent multiplication as area.

Partition
27 = 20 + 7
Areas
20×3 + 7×3
Total
60 + 21
Step 1: Break apart 27 by place value

27 = 20 + 7 (2 tens and 7 ones)

Step 2: Draw rectangle with length 27 and width 3

Rectangle is 27 units long and 3 units wide

Step 3: Partition the rectangle

Draw a vertical line to separate 20 and 7 parts

Step 4: Calculate each section's area

Left section: 20 × 3 = 60, Right section: 7 × 3 = 21

Step 5: Add the areas

60 + 21 = 81

27 × 3 = 81
Final answer:

27 × 3 = 81

Applied rules:

• Area model: Length × Width = Area

• Distributive property: 27×3 = (20+7)×3 = 20×3 + 7×3

• Partitioning: Break apart by place value

7 Area Model for 45×2
Exercise 7
Draw an area model to represent 45×2 and find the product.
Definition:

Area model: A rectangle divided into sections to represent multiplication as area.

Partition
45 = 40 + 5
Areas
40×2 + 5×2
Total
80 + 10
Step 1: Break apart 45 by place value

45 = 40 + 5 (4 tens and 5 ones)

Step 2: Draw rectangle with length 45 and width 2

Rectangle is 45 units long and 2 units wide

Step 3: Partition the rectangle

Draw a vertical line to separate 40 and 5 parts

Step 4: Calculate each section's area

Left section: 40 × 2 = 80, Right section: 5 × 2 = 10

Step 5: Add the areas

80 + 10 = 90

45 × 2 = 90
Final answer:

45 × 2 = 90

Applied rules:

• Area model: Length × Width = Area

• Distributive property: 45×2 = (40+5)×2 = 40×2 + 5×2

• Partitioning: Break apart by place value

8 Area Model for 19×6
Exercise 8
Draw an area model to represent 19×6 and find the product.
Definition:

Area model: A rectangle divided into sections to represent multiplication as area.

Partition
19 = 10 + 9
Areas
10×6 + 9×6
Total
60 + 54
Step 1: Break apart 19 by place value

19 = 10 + 9 (1 ten and 9 ones)

Step 2: Draw rectangle with length 19 and width 6

Rectangle is 19 units long and 6 units wide

Step 3: Partition the rectangle

Draw a vertical line to separate 10 and 9 parts

Step 4: Calculate each section's area

Left section: 10 × 6 = 60, Right section: 9 × 6 = 54

Step 5: Add the areas

60 + 54 = 114

19 × 6 = 114
Final answer:

19 × 6 = 114

Applied rules:

• Area model: Length × Width = Area

• Distributive property: 19×6 = (10+9)×6 = 10×6 + 9×6

• Partitioning: Break apart by place value

9 Area Model for 56×3
Exercise 9
Draw an area model to represent 56×3 and find the product.
Definition:

Area model: A rectangle divided into sections to represent multiplication as area.

Partition
56 = 50 + 6
Areas
50×3 + 6×3
Total
150 + 18
Step 1: Break apart 56 by place value

56 = 50 + 6 (5 tens and 6 ones)

Step 2: Draw rectangle with length 56 and width 3

Rectangle is 56 units long and 3 units wide

Step 3: Partition the rectangle

Draw a vertical line to separate 50 and 6 parts

Step 4: Calculate each section's area

Left section: 50 × 3 = 150, Right section: 6 × 3 = 18

Step 5: Add the areas

150 + 18 = 168

56 × 3 = 168
Final answer:

56 × 3 = 168

Applied rules:

• Area model: Length × Width = Area

• Distributive property: 56×3 = (50+6)×3 = 50×3 + 6×3

• Partitioning: Break apart by place value

10 Area Model for 38×4
Exercise 10
Draw an area model to represent 38×4 and find the product.
Definition:

Area model: A rectangle divided into sections to represent multiplication as area.

Partition
38 = 30 + 8
Areas
30×4 + 8×4
Total
120 + 32
Step 1: Break apart 38 by place value

38 = 30 + 8 (3 tens and 8 ones)

Step 2: Draw rectangle with length 38 and width 4

Rectangle is 38 units long and 4 units wide

Step 3: Partition the rectangle

Draw a vertical line to separate 30 and 8 parts

Step 4: Calculate each section's area

Left section: 30 × 4 = 120, Right section: 8 × 4 = 32

Step 5: Add the areas

120 + 32 = 152

38 × 4 = 152
Final answer:

38 × 4 = 152

Applied rules:

• Area model: Length × Width = Area

• Distributive property: 38×4 = (30+8)×4 = 30×4 + 8×4

• Partitioning: Break apart by place value

Complete Summary: Using Area Models for Multiplication

Area Model Concepts 🧮

📊
Step 1: Partition
Break apart by place value: 23 = 20 + 3
Step 2: Draw Rectangle
Length = Multi-digit number, Width = Single-digit
Step 3: Partition Rectangle
Divide into sections based on place values
Step 4: Calculate Areas
Area = Length × Width for each section
Step 5: Add Areas
Sum all partial products to get final answer
Area Model Process
12
Partition
10+2
Break
×3
Multiply
30+6
Areas
36
Total
Visual Example
Rectangle: 12×3
┌─────────────┬───┐
│ 30 │ 6 │
│ │ │
│ │ │
└─────────────┴───┘
10×3 + 2×3 = 36
Visualize Multiplication! 🌟
Area = Length × Width
Area Formula
Key Definitions:

Area Model: A rectangle divided into sections to represent multiplication as area

Partitioning: Breaking apart a number into its place values

Partial Products: Products found by multiplying each partition separately

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

Length: The longer dimension of a rectangle

Width: The shorter dimension of a rectangle

Area Model Method:
  1. Partition: Break apart the multi-digit number by place value
  2. Draw: Create a rectangle with dimensions of the factors
  3. Partition: Divide the rectangle according to place values
  4. Calculate: Find the area of each section (partial products)
  5. Combine: Add all partial products to get the final product
Tip 1: Always break apart by place value first.
Tip 2: Draw rectangles neatly to visualize clearly.
Tip 3: Label each section with its dimensions.
Tip 4: Always verify your answer makes sense.
Common Errors: Incorrect partitioning, miscalculating areas, forgetting to add partial products.
Success Strategy: Practice place value understanding, label clearly, check work systematically.
Area Model Properties:

• Visual Representation: Shows multiplication as area of rectangle

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

• Place Value: Each partition represents place value

• Flexibility: Works for any size numbers

Area Model Concepts & Strategies
a(b+c) = ab + ac
Distributive Property
Area Model Applications:

Single-Digit by Multi-Digit: 3 × 24 = (3 × 20) + (3 × 4)

Two-Digit by Two-Digit: 23 × 14 (requires more partitions)

Real-World: Calculating areas of rooms, gardens, etc.

Advanced Strategies:
  1. Grid Paper: Use to draw precise rectangles
  2. Color Coding: Use different colors for each section
  3. Labeling: Write dimensions and areas clearly
  4. Verification: Check by using standard algorithm
Strategy 1: Start with smaller numbers to build understanding.
Strategy 2: Use manipulatives to build concrete understanding.
Strategy 3: Connect to real-world area problems.
Strategy 4: Practice with various number combinations.
Key Concepts: Area models connect multiplication to geometric area.
Real World: Area models apply to room dimensions, garden plots, etc.
Real-World Applications:

• Construction: Calculating materials needed for rectangular areas

• Gardening: Planning plots with different sections

• Manufacturing: Calculating production based on area coverage

• Finance: Calculating costs for different sized areas

Questions & Answers

Question: Why do we need to draw rectangles for multiplication? Can't we just multiply the numbers?

Answer: Drawing rectangles (area models) helps you understand WHY multiplication works:

  • Understanding: It shows why the standard algorithm works
  • Visualization: You can see the parts that make up the whole
  • Error detection: Easier to spot mistakes when you see the parts
  • Foundation: Builds understanding for more complex multiplication later

While you can multiply numbers directly, area models help you understand the "why" behind the "how." It's like learning to walk before running - the understanding makes you stronger in math!

Plus, area models are especially helpful when multiplying larger numbers.

Question: My child is struggling to draw the rectangles accurately. How can I help them practice area models?

Answer: Here are practical ways to practice area models:

  • Grid paper: Use graph paper to draw precise rectangles
  • Manipulatives: Use tiles or blocks to build actual rectangles
  • Color coding: Use different colors for each section
  • Start simple: Begin with numbers like 12×3 before moving to larger numbers
  • Verbal practice: Describe the area model without drawing

Focus on the concept rather than perfect drawing. The key is understanding that the rectangle represents the multiplication and can be broken into parts.

Encourage your child to label each section clearly even if the drawing isn't perfect.

Question: How can I connect area models to the distributive property?

Answer: Area models are the visual representation of the distributive property:

  • Concrete example: 12×3 = (10+2)×3 = 10×3 + 2×3
  • Visual model: The rectangle shows how (a+b)×c = a×c + b×c
  • Real-world connection: If you have 3 rows of 12 items, you have 3 rows of 10 plus 3 rows of 2
  • Pattern recognition: Show that this works for any numbers

Emphasize that partitioning the rectangle is just applying the distributive property. This makes multiplication of larger numbers manageable and helps students understand the mathematical principle behind the visual model.

The area model provides a visual proof of the distributive property!