Solved Exercises on Multiplying by 1 to 9 in Grade 3

Master multiplying by 1 to 9: multiplication facts, patterns, and real-world applications through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Basic multiplication facts
Exercise 1
Complete the multiplication table for 6: \(6 \times 1\), \(6 \times 2\), \(6 \times 3\), \(6 \times 4\), \(6 \times 5\), \(6 \times 6\), \(6 \times 7\), \(6 \times 8\), \(6 \times 9\)
Definition:

Multiplication table: A list of products for a specific number multiplied by 1 through 9

Multiplication table method:
  1. Start with the number and multiply by 1
  2. Increase the multiplier by 1 each time
  3. Continue until reaching 9
  4. Notice the pattern in results
Expression
\(6 \times 1\)
Result
\(6\)
6×
1
2
3
4
5
6
7
8
9
=
6
12
18
24
30
36
42
48
54
Step 1: Multiply 6 by 1

\(6 \times 1 = 6\)

Step 2: Multiply 6 by 2

\(6 \times 2 = 12\)

Step 3: Multiply 6 by 3

\(6 \times 3 = 18\)

Step 4: Multiply 6 by 4

\(6 \times 4 = 24\)

Step 5: Multiply 6 by 5

\(6 \times 5 = 30\)

Step 6: Multiply 6 by 6

\(6 \times 6 = 36\)

Step 7: Multiply 6 by 7

\(6 \times 7 = 42\)

Step 8: Multiply 6 by 8

\(6 \times 8 = 48\)

Step 9: Multiply 6 by 9

\(6 \times 9 = 54\)

\(6 \times 1 = 6\), \(6 \times 2 = 12\), \(6 \times 3 = 18\), \(6 \times 4 = 24\), \(6 \times 5 = 30\), \(6 \times 6 = 36\), \(6 \times 7 = 42\), \(6 \times 8 = 48\), \(6 \times 9 = 54\)
Final answer:

\(6 \times 1 = 6\), \(6 \times 2 = 12\), \(6 \times 3 = 18\), \(6 \times 4 = 24\), \(6 \times 5 = 30\), \(6 \times 6 = 36\), \(6 \times 7 = 42\), \(6 \times 8 = 48\), \(6 \times 9 = 54\)

Applied rules:

• Sequential multiplication: Multiply by consecutive numbers

• Pattern recognition: Results increase by the original number

• Repetition: Builds fluency through practice

2 Pattern recognition
Exercise 2
Identify the pattern in the 9s multiplication table: \(9 \times 1\) to \(9 \times 9\)
Definition:

Numerical pattern: A sequence of numbers following a specific rule

9
18
27
36
45
54
63
72
81
9×
1
2
3
4
5
6
7
8
9
=
9
18
27
36
45
54
63
72
81
Special 9s Pattern: In the 9s table, the digits of each product sum to 9
Pattern
9s table
Special Rule
Digits sum to 9
Step 1: List the 9s multiplication table

\(9 \times 1 = 9\), \(9 \times 2 = 18\), \(9 \times 3 = 27\), etc.

Step 2: Identify the pattern

For each product, the digits add up to 9: \(1+8=9\), \(2+7=9\), \(3+6=9\), etc.

Step 3: Verify the pattern

\(9 \times 4 = 36\): \(3+6=9\) ✓

\(9 \times 7 = 63\): \(6+3=9\) ✓

9, 18, 27, 36, 45, 54, 63, 72, 81
Final answer:

In the 9s table, the digits of each product sum to 9

Applied rules:

• Pattern identification: Look for consistent relationships

• Digit sum rule: For 9s table, digits always sum to 9

• Verification: Check the pattern holds for all cases

3 Real-world application
Exercise 3
A school has 8 classrooms, and each classroom has 7 students. How many students are there in total?
Definition:

Real-world application: Using math to solve practical problems

Given
8 classrooms × 7 students
Operation
\(8 \times 7\)
Result
\(56\)
Step 1: Identify the key information

8 classrooms, each with 7 students

Step 2: Determine the operation

We need to find the total number of students: 8 groups of 7

Step 3: Apply multiplication

\(8 \times 7 = 56\)

Step 4: State the answer

There are 56 students in total

There are 56 students in total
Final answer:

There are 56 students in total

Applied rules:

• Problem-solving: Identify key information and operation needed

• Equal groups: Multiplication represents equal-sized groups

• Real-world connection: Connect math to everyday situations

Multiplication Chart (1 to 9)
Multiplication Table
×
1
2
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
2
2
4
6
8
10
12
14
16
18
3
3
6
9
12
15
18
21
24
27
4
4
8
12
16
20
24
28
32
36
5
5
10
15
20
25
30
35
40
45
6
6
12
18
24
30
36
42
48
54
7
7
14
21
28
35
42
49
56
63
8
8
16
24
32
40
48
56
64
72
9
9
18
27
36
45
54
63
72
81
Square Numbers (1 to 9)
1×1=1, 2×2=4, 3×3=9, 4×4=16, 5×5=25, 6×6=36, 7×7=49, 8×8=64, 9×9=81
Key definitions:

Multiplication: Adding equal groups together quickly

Factors: Numbers being multiplied together

Product: Result of a multiplication problem

Multiplication strategies:
  1. Repeated addition: \(4 \times 3 = 4 + 4 + 4\)
  2. Arrays: Organizing items in rows and columns
  3. Number lines: Skip counting
  4. Groups: Equal sets of objects
Tip 1: Practice with small numbers first, then increase gradually.
Tip 2: Use arrays or grouping to visualize multiplication.
Tip 3: Remember that multiplication is commutative: \(4 \times 3 = 3 \times 4\).
Tip 4: Use the multiplication table to find patterns and shortcuts.
Important property: Any number multiplied by 0 equals 0.
Important property: Any number multiplied by 1 equals itself.
Multiplication Concepts and Properties
\(a \times b = c\)
Basic Multiplication
Commutative Property
\(a \times b = b \times a\)
Order doesn't matter
Associative Property
\(a \times (b \times c) = (a \times b) \times c\)
Grouping doesn't matter
Identity Property
\(a \times 1 = a\)
Multiplying by 1
Zero Property
\(a \times 0 = 0\)
Multiplying by 0
Patterns in Multiplication (1-9)
Pattern
Example
Rule
Why
Visual
×2
2×4=8
Double
Add to itself
●●●● + ●●●●
×5
5×6=30
Ends in 0 or 5
Skip count by 5s
5,10,15,20,25,30
×10
10×7=70
Add zero
Place value shift
7→70
×9
9×8=72
Digits sum to 9
7+2=9
72
×4
4×3=12
Double twice
2×(2×3)=12
(2×3)×2
Multiplication Practice and Tips
\(3 \times 4 = 12\)
Three Groups of Four
Key concepts:

Equal groups: Multiplication shows equal-sized groups

Arrays: Visual representation in rows and columns

Repeated addition: Another way to think about multiplication

Problem-solving steps:
  1. Read the problem: Understand what is being asked
  2. Find key information: Identify numbers and relationships
  3. Choose operation: Decide if multiplication is needed
  4. Solve: Calculate the multiplication
  5. Check: Make sure the answer makes sense
Tip 1: Practice skip counting (2, 4, 6, 8...) to learn multiplication facts.
Tip 2: Draw pictures or use objects to represent multiplication problems.
Tip 3: Notice patterns in the multiplication table to help with memorization.
Tip 4: Use known facts to figure out unknown ones (e.g., 6×7 = 5×7 + 7).
Common error: Confusing multiplication with addition. Remember: multiplication is repeated addition.
Practice tip: Start with smaller numbers and work up. Master 1-5 before moving to 6-9.
Important multiplication facts to remember:

• Any number × 0 = 0

• Any number × 1 = that number

• Multiplication is commutative: 3×4 = 4×3

• Multiplication by 10 adds a zero to the end

• Multiplication by 5 ends in 0 or 5

Questions & Answers

Question: Why is the 9s multiplication table so special? The digits always add up to 9!

Answer: That's a fascinating pattern! The 9s table has special properties because of how our number system works.

Here's why it happens:

  • 9 is one less than 10 (our base number system)
  • When you multiply by 9, you're essentially subtracting from a multiple of 10
  • This creates a unique digit pattern

For example: 9×7 = 63, and 6+3=9. This pattern holds for all 9s multiplications up to 9×10!

Question: My child is struggling with the 7s and 8s tables. Are there any tricks for these harder numbers?

Answer: Yes, here are some helpful strategies for the 7s and 8s:

For 7s:

  • Use the commutative property: 7×3 is easier than 3×7
  • Build from known facts: 7×6 = 6×7 = 42

For 8s:

  • Think of it as double, double, double (8 = 2×2×2)
  • So 8×4 = double of 4×4 = double of 16 = 32

Practice with visual models and connect to easier facts!

Question: How do I know if I should multiply or add when reading a word problem?

Answer: Look for key words and phrases:

Words that suggest multiplication:

  • "each" - "5 bags with 3 apples each" = 5×3
  • "times" - "3 times as many" = 3×
  • "product of" - means multiplication
  • "groups of" - "4 groups of 6" = 4×6

Words that suggest addition:

  • "total" (without "groups")
  • "sum" or "altogether"
  • "combined"

Think about whether you have equal groups (multiply) or different amounts to combine (add).

Detailed Summary: Multiplying by 1 to 9

Definitions of Key Concepts

  • Multiplication: A mathematical operation that represents repeated addition of equal groups
  • Factor: A number that is multiplied by another number in a multiplication problem
  • Product: The result of a multiplication problem
  • Multiplication Table: A grid showing the products of numbers multiplied together
  • Array: A rectangular arrangement of objects in rows and columns
  • Equal Groups: Sets of items that contain the same number of elements
  • Pattern: A consistent relationship between numbers that repeats in a predictable way

Core Rules and Principles

  • Identity Property: Any number multiplied by 1 equals itself (\(a \times 1 = a\))
  • Zero Property: Any number multiplied by 0 equals 0 (\(a \times 0 = 0\))
  • Commutative Property: The order of factors doesn't change the product (\(a \times b = b \times a\))
  • Associative Property: Grouping of factors doesn't change the product
  • Pattern Recognition: Multiplication facts follow predictable patterns
  • Sequential Learning: Master smaller numbers before advancing to larger ones

Step-by-Step Methods and Procedures

  1. Understanding Multiplication:
    • Recognize multiplication as repeated addition
    • Understand it represents equal groups of items
  2. Learning Multiplication Facts:
    • Start with 1 and 2 facts
    • Progress to 3-5, then 6-9
    • Practice each set until automatic
  3. Using Patterns:
    • Notice that multiplying by 10 adds a zero
    • Recognize that 9s digits sum to 9
    • Use known facts to derive unknown ones

Multiple Concrete Examples (Simple → Advanced)

  • Simple (1-3): \(2 \times 3 = 6\), \(3 \times 4 = 12\), \(1 \times 5 = 5\)
  • Moderate (4-6): \(4 \times 6 = 24\), \(5 \times 7 = 35\), \(6 \times 8 = 48\)
  • Advanced (7-9): \(7 \times 8 = 56\), \(8 \times 9 = 72\), \(9 \times 9 = 81\)
  • Word Problems: "If 7 friends each have 4 stickers, total = \(7 \times 4 = 28\)"

Tips, Tricks, and Common Pitfalls

  • Tip 1: Use the commutative property - if you know \(7 \times 3 = 21\), then \(3 \times 7 = 21\)
  • Tip 2: For 9s, notice that digits in the product always sum to 9
  • Tip 3: For 8s, think of it as doubling three times
  • Common Pitfall: Confusing multiplication with addition
  • Common Mistake: Forgetting that order doesn't matter in multiplication

Key Notes and Takeaways for Memorization

  • Memory Aid: "Times 1 keeps the number, times 0 gives zero"
  • Pattern: 10s always end in zero, 5s end in 0 or 5
  • Strategy: Use known facts to build unknown ones (e.g., \(7 \times 8 = 7 \times 7 + 7\))
  • Practice: Daily review of facts helps with automatic recall

Clear Explanations Written for Students

Multiplication is like having equal groups of things and counting them all at once. Instead of adding 4 + 4 + 4 + 4 + 4, you can just multiply 4 × 5 = 20. The numbers you multiply are called factors, and the answer is called the product. When you multiply by 1, the number stays the same. When you multiply by 0, you always get 0. There are cool patterns in multiplication - like when you multiply by 9, the digits in the answer always add up to 9! Learning multiplication facts takes practice, but once you know them, math becomes much easier.