Solved Exercises on Area Using Multiplication in Grade 3

Master area using multiplication: connecting multiplication arrays to area concepts, finding area by multiplying length and width, and solving area word problems through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Connecting multiplication to area
Exercise 1
Show how 3 × 4 = 12 connects to the area of a 3-unit by 4-unit rectangle.
Definition:

Multiplication connection: The area of a rectangle equals the multiplication of its length and width

Multiplication to area method:
  1. Identify the dimensions (length and width)
  2. Recognize that the rectangle forms an array of unit squares
  3. Apply multiplication to find the total number of unit squares
  4. Express the result as area in square units
Multiplication
3 × 4 = 12
Area
3 × 4 = 12
Square Units
12 sq units
Step 1: Identify dimensions

Length = 3 units, Width = 4 units

Step 2: Visualize the array

The rectangle has 3 rows and 4 columns of unit squares

Step 3: Apply multiplication

3 rows × 4 columns = 12 unit squares

3 × 4 = 12 square units
Final answer:

The area of the rectangle is 12 square units, which equals 3 × 4

Applied rules:

• Array connection: Rectangle forms a multiplication array

• Area formula: Area = Length × Width

• Unit squares: Count total unit squares

2 Area using multiplication
Exercise 2
Find the area of a rectangle that is 5 units long and 3 units wide using multiplication.
Definition:

Area using multiplication: Finding area by multiplying the length and width of a rectangle

Length
5 units
Width
3 units
Area
5 × 3 = 15
Step 1: Identify dimensions

Length = 5 units, Width = 3 units

Step 2: Apply the formula

Area = Length × Width = 5 × 3

Step 3: Calculate

Area = 15 square units

Area = 15 square units
Final answer:

The area of the rectangle is 15 square units

Applied rules:

• Area formula: Area = Length × Width

• Multiplication: 5 × 3 = 15

• Area units: Always express in square units

3 Square area with multiplication
Exercise 3
Find the area of a square that is 4 units on each side using multiplication.
Definition:

Square area: A special rectangle where length equals width, using the same multiplication principle

Side Length
4 units
Length
4 units
Width
4 units
Area
4 × 4 = 16
Step 1: Identify the side length

All sides = 4 units (square has equal sides)

Step 2: Apply rectangle area formula

Area = Length × Width = 4 × 4

Step 3: Calculate

Area = 16 square units

Step 4: Verify with array

4 rows × 4 columns = 16 unit squares ✓

Area = 16 square units
Final answer:

The area of the square is 16 square units

Applied rules:

• Square area: Area = Side × Side

• Multiplication: 4 × 4 = 16

• Area units: Always express in square units

Area Using Multiplication Fundamentals
\text{Area} = \text{Length} \times \text{Width}
Area Formula Using Multiplication
Rectangle
L × W
Length × Width
Square
S × S
Side × Side
Array Model
Rows × Columns
Connecting to area
Key definitions:

Area: The amount of space inside a shape, measured in square units

Multiplication connection: Area equals the number of unit squares in an array

Array model: A rectangular arrangement of objects in rows and columns

Square unit: A unit used to measure area

4-step multiplication method:
  1. Identify: Find the length and width of the rectangle
  2. Connect: Recognize the array structure (rows and columns)
  3. Multiply: Apply the formula: Area = Length × Width
  4. Verify: Check if the answer makes sense
Tip 1: Think of area as filling a rectangle with unit squares.
Tip 2: Draw a grid to visualize the array model.
Tip 3: Remember: Area = Length × Width for all rectangles.
Tip 4: A square is a special rectangle where length = width.
Common errors: Forgetting to multiply, confusing length with width, not including square units.
Memory aid: "Area is Length times Width, always add 'square' to the unit!"
Key multiplication facts:

• Area of rectangle = Length × Width

• Area of square = Side × Side

• Area is always measured in square units

• Multiplication shows rows × columns in area

Solution: Exercises 4 to 5
4 Multiplication array verification
Exercise 4
Verify that a 2×7 rectangle has an area of 14 square units by both multiplication and counting.
Definition:

Verification: Confirming that multiplication and counting give the same result

Dimensions
2 × 7
Multiplication
2 × 7 = 14
Counting
14 squares
Verification
✓ Match
Step 1: Apply multiplication

Area = 2 × 7 = 14 square units

Step 2: Count unit squares

Count each square: 1, 2, 3, ..., 14

Step 3: Verify both methods match

Multiplication: 14, Counting: 14 ✓

Area = 14 square units (2 × 7 = 14)
Final answer:

The area of the rectangle is 14 square units

Applied rules:

• Multiplication method: Length × Width

• Counting method: Count each unit square

• Verification: Both methods should yield the same result

5 Area comparison using multiplication
Exercise 5
Rectangle A is 3 units by 6 units. Rectangle B is 4 units by 5 units. Which rectangle has the larger area?
Definition:

Area comparison: Using multiplication to find and compare areas of different rectangles

Rectangle A
3 × 6 = 18
Rectangle B
4 × 5 = 20
Comparison
20 > 18
Step 1: Calculate Rectangle A's area

Area = 3 × 6 = 18 square units

Step 2: Calculate Rectangle B's area

Area = 4 × 5 = 20 square units

Step 3: Compare the areas

20 > 18, so Rectangle B is larger

Step 4: State the answer

Rectangle B has the larger area

Rectangle A: 18 sq units | Rectangle B: 20 sq units
Final answer:

Rectangle B has the larger area (20 square units compared to Rectangle A's 18 square units)

Applied rules:

• Rectangle area: Length × Width

• Comparison: Compare calculated areas

• Multiplication: Apply formula to both rectangles

Complete Area Using Multiplication Guide
\text{Area} = \text{Length} \times \text{Width}
Essential Area Formula
Key definitions:

Area: The amount of space inside a 2D shape, measured in square units

Multiplication connection: Area equals the total number of unit squares in an array

Array model: A rectangular arrangement showing rows and columns

Rectangle: A four-sided shape with four right angles

Advanced multiplication techniques:
  1. Array visualization: See the rectangle as a multiplication array
  2. Formula application: Area = Length × Width
  3. Verification method: Count unit squares to confirm multiplication
  4. Comparison approach: Use multiplication to compare areas
Tip 1: Practice with physical manipulatives to visualize the connection.
Tip 2: Always include "square units" in your final answer.
Tip 3: Draw a grid to help visualize the array model.
Tip 4: Connect multiplication facts to area concepts.
Common errors: Forgetting to multiply, confusing length with width, not using proper units.
Success strategies: Regular practice, visual aids, real-world applications.
Essential multiplication facts:

• Area measures the inside space of a rectangle

• Area is always expressed in square units

• Rectangle area = Length × Width

• Square area = Side × Side

Multiplication and Area Connection
Array Model Visualization
Understanding how multiplication arrays connect to area:
Rows × Columns = Total Unit Squares

Connection Between Multiplication and Area:

  • Rows: Represent the width of the rectangle
  • Columns: Represent the length of the rectangle
  • Total squares: Equal to the area in square units
  • Formula: Area = Length × Width = Rows × Columns

Questions & Answers

Question: How does multiplication connect to area? I don't see the relationship.

Answer: The connection is beautiful! Think of a rectangle as an array of unit squares:

  • Length: How many squares fit across the rectangle (columns)
  • Width: How many squares fit down the rectangle (rows)
  • Area: Total number of unit squares inside the rectangle

So if you have 3 rows and 4 columns, you have 3 × 4 = 12 unit squares inside!

The multiplication tells you how many unit squares fit in the rectangle!

Question: Can I use addition instead of multiplication to find area?

Answer: Yes, but multiplication is much more efficient! For example:

  • Addition: 3 + 3 + 3 + 3 = 12 (adding 4 groups of 3)
  • Multiplication: 3 × 4 = 12 (much faster!)

Multiplication is actually repeated addition, so 3 × 4 means "3 added 4 times" or "4 added 3 times".

For larger rectangles, multiplication saves a lot of time!

Question: What happens if I mix up length and width? Will I get the wrong answer?

Answer: No, you won't get the wrong answer! This is because multiplication is commutative:

  • 3 × 4 = 12 and 4 × 3 = 12
  • It doesn't matter which dimension you call length or width
  • As long as you multiply both dimensions together, you get the same area

However, it's good practice to think of length as the longer side and width as the shorter side.

The important thing is to multiply both dimensions!