Solved Exercises on Multiplication Fact Probe in Grade 3

Master multiplication facts: arrays, repeated addition, equal groups, and problem-solving strategies through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic multiplication facts
Exercise 1
Solve: \(4 \times 6 = ?\)
Definition:

Multiplication: Repeated addition of the same number. \(a \times b = a + a + a + ... + a\) (b times)

Method:
  1. Understand multiplication as repeated addition
  2. Add the first number repeatedly as many times as the second number
  3. Count or group systematically
Expression
\(4 \times 6\)
Repeated Addition
\(4 + 4 + 4 + 4 + 4 + 4\)
Result
\(24\)
Step 1: Understand the problem

\(4 \times 6\) means 4 groups of 6 items OR 6 groups of 4 items

Step 2: Convert to repeated addition

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

Step 3: Count systematically

\(4 + 4 = 8\), \(8 + 4 = 12\), \(12 + 4 = 16\), \(16 + 4 = 20\), \(20 + 4 = 24\)

Step 4: Verify using array model

Draw 4 rows with 6 dots in each row: 4 × 6 = 24 dots total

\(4 \times 6 = 24\)
Final answer:

\(4 \times 6 = 24\)

Applied rules:

• Commutative Property: \(4 \times 6 = 6 \times 4 = 24\)

• Associative Property: Grouping doesn't change the result

• Identity Property: Any number × 1 = itself

Array Model for 4 × 6:

● ● ● ● ● ● (Row 1)

● ● ● ● ● ● (Row 2)

● ● ● ● ● ● (Row 3)

● ● ● ● ● ● (Row 4)

Total: 24 dots

2 Array model multiplication
Exercise 2
Use arrays to solve: \(3 \times 7 = ?\)
Definition:

Array Model: Organizing objects in rows and columns to visualize multiplication

Expression
\(3 \times 7\)
Array
3 rows × 7 columns
Total
\(21\)
Step 1: Draw the array

Create 3 rows with 7 objects in each row

Step 2: Count by rows

Row 1: 7 objects, Row 2: 7 objects, Row 3: 7 objects

Step 3: Add the totals

\(7 + 7 + 7 = 21\)

Step 4: Count by columns

Column 1: 3 objects, Column 2: 3 objects, ..., Column 7: 3 objects

\(3 + 3 + 3 + 3 + 3 + 3 + 3 = 21\)

\(3 \times 7 = 21\)
Final answer:

\(3 \times 7 = 21\)

Applied rules:

• Array Structure: Rows × Columns = Total objects

• Visual Pattern: Helps recognize multiplication relationships

• Commutative Property: \(3 \times 7 = 7 \times 3\)

Array Model for 3 × 7:

● ● ● ● ● ● ● (Row 1)

● ● ● ● ● ● ● (Row 2)

● ● ● ● ● ● ● (Row 3)

Total: 21 dots

3 Equal groups multiplication
Exercise 3
There are 5 bags with 8 apples in each bag. How many apples in total?
Definition:

Equal Groups: Number of groups × Items per group = Total items

Problem
5 bags × 8 apples
Equation
\(5 \times 8 = ?\)
Result
\(40\)
Step 1: Identify the groups

Number of groups = 5 bags

Items per group = 8 apples per bag

Step 2: Set up the multiplication

Total = Number of groups × Items per group

Total = \(5 \times 8\)

Step 3: Solve using repeated addition

\(5 \times 8 = 8 + 8 + 8 + 8 + 8 = 40\)

Step 4: Verify with skip counting

Count by 8s: 8, 16, 24, 32, 40

\(5 \times 8 = 40\) apples
Final answer:

There are 40 apples in total

Applied rules:

• Grouping Model: Total = Groups × Items per group

• Real-world Application: Connect math to everyday situations

• Verification: Use multiple methods to confirm answers

Equal Groups Model:

Bag 1: ● ● ● ● ● ● ● ● (8 apples)

Bag 2: ● ● ● ● ● ● ● ● (8 apples)

Bag 3: ● ● ● ● ● ● ● ● (8 apples)

Bag 4: ● ● ● ● ● ● ● ● (8 apples)

Bag 5: ● ● ● ● ● ● ● ● (8 apples)

Total: 5 groups of 8 = 40 apples

Multiplication Facts & Properties
\(a \times b = c\)
Multiplication Formula
Commutative Property
\(a \times b = b \times a\)
Order doesn't matter
Identity Property
\(a \times 1 = a\)
Multiplying by 1
Zero Property
\(a \times 0 = 0\)
Multiplying by 0
Multiplication Models: Arrays, equal groups, repeated addition, number lines
Key Strategies: Skip counting, doubling, using known facts
Tip 1: Use arrays to visualize multiplication problems.
Tip 2: Practice skip counting (2s, 3s, 4s, etc.) to build fluency.
Tip 3: Remember that multiplication is repeated addition.
Tip 4: Use the commutative property to solve harder problems.
Solution: Exercises 4 to 5
4 Multiplication word problem
Exercise 4
Sarah has 6 boxes of crayons. Each box contains 9 crayons. How many crayons does Sarah have altogether?
Definition:

Word Problems: Identify groups and items per group, then multiply

Given
6 boxes, 9 crayons each
Equation
\(6 \times 9 = ?\)
Solution
\(54\)
Step 1: Read and identify key information

Number of boxes = 6 (groups)

Crayons per box = 9 (items per group)

Step 2: Set up the equation

Total crayons = Number of boxes × Crayons per box

Total = \(6 \times 9\)

Step 3: Solve using multiplication facts

\(6 \times 9 = 54\) (You can think of it as \(6 \times 10 - 6 = 60 - 6 = 54\))

Step 4: Verify with repeated addition

\(9 + 9 + 9 + 9 + 9 + 9 = 54\) ✓

\(6 \times 9 = 54\) crayons
Final answer:

Sarah has 54 crayons altogether

Applied rules:

• Problem Solving: Identify groups and items per group

• Mental Math: Use known facts to find unknown facts

• Verification: Check work using alternative methods

5 Mixed multiplication facts
Exercise 5
Complete the multiplication table pattern: \(7 \times 4 = ?\), \(7 \times 5 = ?\), \(7 \times 6 = ?\)
Definition:

Multiplication Patterns: Recognizing sequences and relationships in multiplication facts

Pattern
\(7 \times 4, 7 \times 5, 7 \times 6\)
Results
\(28, 35, 42\)
Pattern
+7 each time
Step 1: Solve the first problem

\(7 \times 4 = 7 + 7 + 7 + 7 = 28\)

Step 2: Solve the second problem

\(7 \times 5 = 7 + 7 + 7 + 7 + 7 = 35\)

Step 3: Solve the third problem

\(7 \times 6 = 7 + 7 + 7 + 7 + 7 + 7 = 42\)

Step 4: Identify the pattern

Each time we increase the second number by 1, the result increases by 7

This is because we're adding one more group of 7

\(7 \times 4 = 28\), \(7 \times 5 = 35\), \(7 \times 6 = 42\)
Final answer:

\(7 \times 4 = 28\), \(7 \times 5 = 35\), \(7 \times 6 = 42\)

Applied rules:

• Pattern Recognition: Multiplication facts follow predictable patterns

• Relationship Building: Use known facts to derive unknown facts

• Sequential Learning: Build on previous knowledge

Multiplication Facts Mastery Guide
\(a \times b = c\)
Basic Multiplication
Key definitions:

Multiplication: Adding equal groups together quickly

Factors: Numbers being multiplied together

Product: The result of multiplication

Array: Objects arranged in rows and columns

Complete methodology:
  1. Recognize the Problem: Identify if it's a multiplication situation
  2. Identify Groups: Determine number of groups and items per group
  3. Set Up Equation: Write the multiplication expression
  4. Solve: Use repeated addition, arrays, or known facts
  5. Verify: Check answer using a different method
Tip 1: Practice with concrete objects like counters or blocks.
Tip 2: Use skip counting to memorize multiplication tables.
Tip 3: Draw arrays to visualize multiplication problems.
Tip 4: Remember that multiplication is commutative: \(a \times b = b \times a\).
Common errors: Forgetting zero property, confusing addition with multiplication, miscounting groups.
Success strategies: Use multiple representations, practice regularly, connect to real-world situations.
Properties to know by heart:

• Identity Property: \(a \times 1 = a\)

• Zero Property: \(a \times 0 = 0\)

• Commutative Property: \(a \times b = b \times a\)

• Associative Property: \((a \times b) \times c = a \times (b \times c)\)

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

Multiplication Table Reference
Essential Multiplication Facts
Master these multiplication facts for quick recall:
\(2 \times 2 = 4\) to \(2 \times 10 = 20\)
\(3 \times 3 = 9\) to \(3 \times 10 = 30\)
\(4 \times 4 = 16\) to \(4 \times 10 = 40\)
\(5 \times 5 = 25\) to \(5 \times 10 = 50\)
\(6 \times 6 = 36\) to \(6 \times 10 = 60\)
\(7 \times 7 = 49\) to \(7 \times 10 = 70\)
\(8 \times 8 = 64\) to \(8 \times 10 = 80\)
\(9 \times 9 = 81\) to \(9 \times 10 = 90\)

Patterns to notice:

  • Any number × 10 ends in 0
  • Any number × 5 ends in 0 or 5
  • Products of even numbers are always even
  • Products of odd numbers follow alternating patterns

Questions & Answers

Question: I don't understand why \(4 \times 6\) equals \(6 \times 4\). Aren't they different problems?

Answer: Great question! This is called the commutative property of multiplication. Even though the problems look different, they have the same answer:

  • \(4 \times 6\) means 4 groups of 6, which equals 24
  • \(6 \times 4\) means 6 groups of 4, which also equals 24

Think of it this way: if you have 4 bags with 6 marbles each, that's the same as having 6 bags with 4 marbles each - you still have 24 marbles total!

This property helps you because if you know \(4 \times 6 = 24\), you automatically know \(6 \times 4 = 24\) too!

Question: My child is struggling with multiplication facts. What are some effective ways to practice?

Answer: Here are several effective strategies for practicing multiplication facts:

  • Visual aids: Use arrays, grouping objects, or drawing pictures to represent multiplication
  • Skip counting: Practice counting by 2s, 3s, 4s, etc. to build fluency
  • Games: Multiplication flashcards, dice games, or online math games make practice fun
  • Patterns: Help children notice patterns (like any number × 10 ends in 0)
  • Short sessions: Practice for 10-15 minutes daily rather than long sessions

Start with easier facts (×2, ×5, ×10) and gradually work up to more challenging ones. Consistency is key!

Question: How do I know if I should multiply or add when solving word problems?

Answer: Look for key words and situations that indicate multiplication:

  • Multiplication clues: "each," "per," "every," "total," "altogether" (when referring to equal groups)
  • Examples: "5 boxes with 8 apples each" → multiply 5 × 8
  • Visual clue: If you can organize items in rows and columns, it's multiplication

Ask yourself: "Am I combining equal groups?" If yes, multiply. If you're combining different amounts, you might add.

For example: "Tom has 5 apples and Mary has 8 apples" → add (5 + 8). But "Tom has 5 bags with 8 apples each" → multiply (5 × 8).