Solved Exercises on Multiplication using Arrays in Grade 3

Master multiplication using arrays: visual representations, rows and columns, and array models for multiplication through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Array Model
Exercise 1
Draw an array to represent 3 × 4 and find the product.
Definition:

Array: A rectangular arrangement of objects in rows and columns. For multiplication: Rows × Columns = Total Objects

Method for drawing arrays:
  1. Identify the first number as rows
  2. Identify the second number as columns
  3. Draw rectangles with specified rows and columns
  4. Count total objects
Rows
3
Columns
4
Product
12
Step 1: Identify rows and columns

3 × 4 means 3 rows and 4 columns

Step 2: Draw the array
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Step 3: Count the objects

Row 1: 4 objects

Row 2: 4 objects

Row 3: 4 objects

Total: 4 + 4 + 4 = 12 objects

Step 4: Verify with multiplication

3 × 4 = 12

3 × 4 = 12
Final answer:

The array with 3 rows and 4 columns has 12 objects, so 3 × 4 = 12.

Applied rules:

• Array Structure: Rows go horizontal, columns go vertical

• Multiplication: Number of rows × Number of columns = Total objects

• Repetition: Each row has the same number of objects

2 Identifying Arrays
Exercise 2
Look at this array and write the multiplication sentence it represents:
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Definition:

Reading arrays: Count the number of rows and columns, then multiply to find the total.

Rows
4
Columns
3
Product
12
Step 1: Count the rows

Looking horizontally: 4 rows of circles

Step 2: Count the columns

Looking vertically: 3 columns of circles

Step 3: Write the multiplication sentence

Rows × Columns = Total

4 × 3 = 12

Step 4: Verify by counting

Total circles: 12

4 × 3 = 12 ✓

4 × 3 = 12
Final answer:

The array represents 4 × 3 = 12.

Applied rules:

• Row Counting: Count horizontal lines

• Column Counting: Count vertical lines

• Multiplication Sentence: Rows × Columns = Product

3 Array Word Problem
Exercise 3
Sarah arranges her books in 5 rows with 6 books in each row. Draw an array and write the multiplication sentence to find the total number of books.
Definition:

Word problems with arrays: Identify rows and columns from the problem description.

Rows
5
Columns
6
Product
30
Step 1: Identify the problem

5 rows with 6 books in each row

Step 2: Draw the array
B
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B
Step 3: Write the multiplication sentence

5 rows × 6 books per row = 30 books

5 × 6 = 30

Step 4: Verify the answer

Count by 6s: 6, 12, 18, 24, 30

Or: 5 × 6 = 30 ✓

5 × 6 = 30
Final answer:

Sarah has 5 × 6 = 30 books in total.

Applied rules:

• Problem Translation: "5 rows with 6 in each" means 5 × 6

• Array Representation: Rows × Columns = Total

• Verification: Count by the number in each row

Rules and methods, laws,...
Rows × Columns = Total
Array Multiplication
Rows
Horizontal
Go across
Columns
Vertical
Go up/down
Total
Multiply
Rows × Columns
Key definitions:

Array: A rectangular arrangement of objects in rows and columns

Row: A horizontal line of objects

Column: A vertical line of objects

Multiplication: Repeated addition of equal groups

Factors: The numbers being multiplied (rows and columns)

Product: The result of multiplication (total objects)

Complete methodology:
  1. Identify rows and columns: From the problem or array
  2. Draw the array: Rectangular grid with correct dimensions
  3. Count objects: Verify by counting total items
  4. Write multiplication: Factor × Factor = Product
  5. Verify: Check that repeated addition matches
Tip 1: Count rows horizontally, columns vertically.
Tip 2: Each row has the same number of objects.
Tip 3: Multiplication is faster than repeated addition.
Tip 4: Arrays show the connection between addition and multiplication.
Common errors: Miscounting rows/columns, confusing rows with columns, skipping objects when counting.
Exam preparation: Practice drawing arrays, learn to identify rows and columns, master basic multiplication facts.
Formulas to know by heart:

• Array Model: Rows × Columns = Total Objects

• Multiplication: Factor × Factor = Product

• Repeated Addition: 3 × 4 = 4 + 4 + 4 = 12

• Commutative Property: 3 × 4 = 4 × 3 = 12

Solution: Exercises 4 to 5
4 Comparing Arrays
Exercise 4
Compare these two arrays:
Array A: 4 rows × 3 columns
Array B: 3 rows × 4 columns
Draw both arrays and explain what you notice.
Definition:

Commutative property: The order of factors does not change the product.

Array A
4×3=12
Array B
3×4=12
Same Total
Yes
Step 1: Draw Array A (4 × 3)
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4 rows × 3 columns = 12 total

Step 2: Draw Array B (3 × 4)
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3 rows × 4 columns = 12 total

Step 3: Compare the arrays

Both arrays have 12 objects

4 × 3 = 3 × 4 = 12

This demonstrates the commutative property of multiplication

Step 4: Explain the observation

Even though the arrays look different (different shape), they have the same total number of objects.

4 × 3 = 3 × 4 = 12
Final answer:

Array A (4×3) and Array B (3×4) both equal 12. This shows that multiplication is commutative: 4 × 3 = 3 × 4.

Applied rules:

• Commutative Property: Order of factors doesn't change the product

• Array Shape: Different dimensions can have same area

• Visual Proof: Arrays demonstrate mathematical properties

5 Advanced Array Problem
Exercise 5
A farmer plants apple trees in an orchard. He creates 6 rows with 7 trees in each row. Then he adds 2 more rows with 7 trees in each row. How many trees are there altogether? Use arrays to solve.
Definition:

Combining arrays: When arrays have the same number of columns, you can add the rows.

First Part
6×7=42
Added Part
2×7=14
Total
56
Step 1: Draw the first array (6 × 7)

6 rows × 7 columns = 42 trees

6 × 7 = 42

Step 2: Draw the additional array (2 × 7)

2 rows × 7 columns = 14 trees

2 × 7 = 14

Step 3: Combine the arrays

Total rows: 6 + 2 = 8 rows

Total columns: 7 columns (same for both)

8 × 7 = 56 trees

Step 4: Verify with addition

First part + Added part = 42 + 14 = 56 trees

8 × 7 = 56 ✓

8 × 7 = 56 trees
Final answer:

There are 8 × 7 = 56 trees altogether.

Applied rules:

• Combining Like Arrays: When columns are the same, add rows

• Distributive Property: (6 + 2) × 7 = 6×7 + 2×7

• Breaking Down Problems: Solve in parts, then combine

Key Concepts: Definitions, Formulas, Methods, Tips
Rows × Columns = Total
Array Multiplication
Key definitions:

Array: A rectangular arrangement of objects in rows and columns

Row: A horizontal line of objects (goes across)

Column: A vertical line of objects (goes up and down)

Multiplication: A way to add equal groups quickly

Factors: The numbers being multiplied (rows and columns)

Product: The answer to a multiplication problem

Complete methodology:
  1. Read the problem: Identify how many rows and columns
  2. Draw the array: Make a rectangle with correct dimensions
  3. Count objects: Verify the total by counting
  4. Write multiplication: Express as rows × columns = total
  5. Check your work: Verify by counting or repeated addition
Tip 1: Use dots, squares, or X's to make neat arrays.
Tip 2: Count by the number in each row to check your answer.
Tip 3: Arrays help you see multiplication as repeated addition.
Tip 4: The order of numbers doesn't matter: 3×4 = 4×3.
Common errors: Mixing up rows and columns, miscounting objects, forgetting to multiply.
Exam preparation: Practice with different array sizes, memorize basic facts, recognize patterns.
Formulas to know by heart:

• Array Model: Rows × Columns = Total Objects

• Multiplication: Factor × Factor = Product

• Repeated Addition: 5 × 3 = 3 + 3 + 3 + 3 + 3 = 15

• Commutative Property: a × b = b × a

• Distributive Property: (a + b) × c = a×c + b×c

Detailed Summary: Multiplication using Arrays
Core Definitions

Array: A rectangular arrangement of objects in equal rows and columns. It's a visual model that shows multiplication as a rectangular grid. For example, 3 × 4 is shown as 3 rows with 4 objects in each row.

Row: A horizontal line of objects. Think of it as going "across" like the rows in a movie theater.

Column: A vertical line of objects. Think of it as going "up and down" like the columns of a building.

Multiplication: A mathematical operation that represents repeated addition of equal groups. For example, 3 × 4 means 3 groups of 4, or 4 + 4 + 4.

Factors: The numbers being multiplied together (the rows and columns in an array).

Product: The answer to a multiplication problem (the total number of objects in the array).

Core Rules and Laws

Array Rule: Number of rows × Number of columns = Total objects

Commutative Property: The order of factors doesn't change the product. For example: 3 × 4 = 4 × 3 = 12

Repeated Addition Connection: Multiplication is a shortcut for adding equal groups. For example: 3 × 4 = 4 + 4 + 4 = 12

Equal Groups: Each row in an array has the same number of objects.

Rectangular Shape: Arrays always form rectangles (or squares).

Step-by-Step Methods

Creating an Array:

  1. Look at the multiplication problem (e.g., 4 × 5)
  2. Identify the first number as rows (4 rows)
  3. Identify the second number as columns (5 columns)
  4. Draw rectangles arranged in the specified rows and columns
  5. Count the total number of objects

Reading an Array:

  1. Count the number of rows (horizontal lines)
  2. Count the number of columns (vertical lines)
  3. Write the multiplication sentence: rows × columns = total
  4. Verify by counting all objects
Examples and Applications

Simple Example: 2 × 3

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2 rows × 3 columns = 6 total objects

Intermediate Example: 4 × 5

4 rows of 5 objects each = 20 total objects

Advanced Example: Adding arrays

(3 × 4) + (2 × 4) = (5 × 4) = 20 total objects

Tips and Common Pitfalls

Tips:

  • Draw neat rows and columns to avoid miscounting
  • Use dots or small squares to make clear arrays
  • Count by the number in each row to check your answer
  • Remember: rows go across, columns go up and down
  • Arrays help visualize why 3 × 4 = 4 × 3

Common Mistakes:

  • Mixing up rows and columns
  • Not making equal groups in each row
  • Skipping objects when counting
  • Forgetting to multiply, just adding
  • Not seeing the connection to repeated addition
Key Notes for Memorization

ARRAY = ROWS × COLUMNS

ROWS GO ACROSS

COLUMNS GO UP/DOWN

MULTIPLICATION = FAST ADDITION

ORDER DOESN'T MATTER: 3×4 = 4×3

EACH ROW HAS SAME NUMBER

Questions & Answers

Question: I always mix up rows and columns. How can I remember which is which?

Answer: Here are some helpful memory tricks:

  • Rows = "Across": Think of rows in a movie theater - you walk across to find your seat
  • Columns = "Up and Down": Think of the columns of a building - they go up and down
  • Rhyme: "Rows go sideways, like when you ride sidesaddle"
  • Picture: R for Rows = Right to Left movement

When you look at an array, count rows by going across, and count columns by going up and down. Practice with small arrays like 2×3 until it becomes natural!

Question: Why do we need arrays when we can just multiply the numbers?

Answer: Great question! Arrays help us understand WHY multiplication works:

  • Visualization: Arrays show you exactly what multiplication means - equal groups
  • Understanding: Instead of just memorizing 3×4=12, you see 3 groups of 4
  • Foundation: Arrays build the foundation for bigger math concepts
  • Problem Solving: When you forget a fact, you can draw an array to figure it out

Think of arrays as a picture that shows what multiplication really is. Once you understand the concept, you'll remember the facts better!

Question: How can I help my child practice arrays at home?

Answer: Here are fun ways to practice arrays at home:

  • Snack Arrays: Arrange crackers, grapes, or cheese cubes in arrays
  • Toothpick Arrays: Use toothpicks or small sticks to make arrays
  • Grid Paper: Have your child color in arrays on graph paper
  • Real Objects: Use coins, buttons, or small toys to create arrays
  • Games: Roll two dice and have your child draw an array for those numbers

Make it hands-on and visual! The more your child sees and manipulates arrays, the better they'll understand multiplication.