Solved Exercises on Multiplying by 10 with Factors of 3 in Grade 3

Master multiplying by 10 with factors of 3: 30, 60, 90, and their applications through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Multiplying 3 by 10
Exercise 1
What is \(3 \times 10\)? Show how the place values shift.
Definition:

Multiplication by 10: Each digit moves one place value position to the left

Multiply by 10 method:
  1. Take the original number
  2. Add a zero to the end
  3. Or move each digit one place to the left
Original
\(3\)
After ×10
\(30\)
Tens
3
Ones
0
↓ ×10 ↓
Tens
3
Ones
0
Step 1: Start with the original number

3 in the ones place

Step 2: Multiply by 10

Each digit moves one place to the left

Step 3: Fill empty places with zeros

3 moves to tens place, ones place gets 0

\(3 \times 10 = 30\)
Final answer:

\(3 \times 10 = 30\)

Applied rules:

• Place value shift: Each digit moves one place to the left

• Zero rule: Add a zero to the end of the number

• Digit movement: 3 ones become 3 tens

2 Multiplying 6 by 10
Exercise 2
Calculate \(6 \times 10\). Show the factor relationship.
Definition:

Factor of 3: A number that can be divided evenly by 3 (e.g., 3, 6, 9, 12...)

Factor of 3 Relationship
\(6 = 3 \times 2\)
So \(6 \times 10 = (3 \times 2) \times 10\)
Expression
\(6 \times 10\)
Factor form
\((3 \times 2) \times 10\)
Result
\(60\)
Step 1: Identify the factor relationship

6 is a factor of 3 since \(6 = 3 \times 2\)

Step 2: Apply the multiplication

\(6 \times 10 = 60\)

Step 3: Verify the result

60 is 10 times larger than 6

\(6 \times 10 = 60\)
Final answer:

\(6 \times 10 = 60\)

Applied rules:

• Factor identification: Recognize 6 as a multiple of 3

• Multiplication by 10: Add zero to end of number

• Place value: Digits move one place left

3 Multiplying 9 by 10
Exercise 3
Find \(9 \times 10\). Show how this relates to other factors of 3.
Definition:

Factors of 3: Numbers divisible by 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30...

Multiplication Pattern
\(3 \times 10 = 30\)
\(6 \times 10 = 60\)
\(9 \times 10 = 90\)
Original
\(9\)
After ×10
\(90\)
Step 1: Recognize 9 as a factor of 3

\(9 = 3 \times 3\)

Step 2: Multiply by 10

Add a zero to the end of 9

Step 3: Observe the pattern

Results are 30, 60, 90 - all multiples of 30

\(9 \times 10 = 90\)
Final answer:

\(9 \times 10 = 90\)

Applied rules:

• Factor recognition: 9 is the third factor of 3 (\(3 \times 3\))

• Multiplication by 10: Adds one zero to end

• Pattern identification: Results follow 30, 60, 90 sequence

Factors of 3 and Multiplication by 10 Chart
\(3n \times 10 = 30n\)
General Formula
3
6
9
12
Factors of 3 Multiplied by 10
×
1
2
3
4
5
6
7
8
9
10
3
3
6
9
12
15
18
21
24
27
30
6
6
12
18
24
30
36
42
48
54
60
9
9
18
27
36
45
54
63
72
81
90
3×10=30
First factor of 3
6×10=60
Second factor of 3
9×10=90
Third factor of 3
Key definitions:

Factor of 3: A number that can be divided evenly by 3 with no remainder

Multiple: The product of a number and an integer

Pattern: A consistent relationship between numbers

Multiplication by 10 strategy:
  1. Identify the factor: Confirm it's divisible by 3
  2. Apply the rule: Add a zero to the end of the number
  3. Verify: Check that the result is 10 times the original
Tip 1: Remember: multiplying any factor of 3 by 10 results in a multiple of 30.
Tip 2: Use the pattern: 30, 60, 90, 120... for factors of 3 multiplied by 10.
Tip 3: Check divisibility by 3: if digits sum to a multiple of 3, the number is divisible by 3.
Tip 4: Practice with larger factors of 3 like 12, 15, 18 multiplied by 10.
Key insight: When multiplying factors of 3 by 10, the result is always a multiple of 30.
Common error: Forgetting that the factor relationship still applies after multiplication.
Multiplication Patterns and Properties
\(a \times 10 = a0\)
Basic Multiplication by 10
Commutative Property
\(a \times b = b \times a\)
Order doesn't matter
Identity Property
\(a \times 10 = a0\)
Multiplying by 10
Zero Property
\(a \times 0 = 0\)
Multiplying by 0
Factors of 3 Pattern
3×10=30
6×10=60
9×10=90
12×10=120
15×10=150
Multiplication by 10 with Factors of 3
Pattern
Example
Rule
Why
Visual
×10
3×10=30
Add zero
Place value shift
●●● → ●●●○
×10
6×10=60
Add zero
Place value shift
●●●●●● → ●●●●●●○
×10
9×10=90
Add zero
Place value shift
●●●●●●●●● → ●●●●●●●●●○
Multiplication by 10 with Factors of 3 Tips
\(3n \times 10 = 30n\)
Factor of 3 Multiplied by 10
Key concepts:

Factor of 3: A number that can be evenly divided by 3

Place value: The value of a digit based on its position in the number

Pattern recognition: Identifying consistent relationships between numbers

Problem-solving steps:
  1. Identify the factor: Confirm the number is divisible by 3
  2. Apply multiplication by 10: Add a zero to the end
  3. Verify the result: Ensure it's 10 times the original number
  4. State the answer: Write the final result
Tip 1: To check if a number is a factor of 3, add its digits - if the sum is divisible by 3, so is the number.
Tip 2: Remember: multiplying by 10 always adds one zero to the end of the number.
Tip 3: The result of multiplying a factor of 3 by 10 is always a multiple of 30.
Tip 4: Practice with larger factors of 3 to build fluency.
Common mistake: Confusing the rules for multiplying by 3 versus multiplying by 10.
Success strategy: First confirm it's a factor of 3, then apply the multiply by 10 rule.
Important multiplication facts to remember:

• Any factor of 3 multiplied by 10 results in a multiple of 30

• Multiplying by 10 adds one zero to the end of the number • The factor relationship is preserved after multiplication • These operations can be done mentally with practice • Check divisibility by adding digits (for factor of 3 test)

Questions & Answers

Question: How can I tell if a number is a factor of 3? Is there a trick?

Answer: Yes! There's a great trick for checking if a number is divisible by 3:

Add up all the digits in the number. If the sum is divisible by 3, then the original number is also divisible by 3.

Examples:

  • Is 24 a factor of 3? \(2 + 4 = 6\), and 6 is divisible by 3, so YES!
  • Is 35 a factor of 3? \(3 + 5 = 8\), and 8 is not divisible by 3, so NO!
  • Is 123 a factor of 3? \(1 + 2 + 3 = 6\), and 6 is divisible by 3, so YES!

This works for any number, no matter how large!

Question: My child is confused about why we multiply by 10 by adding a zero. Can you explain this simply?

Answer: Think of it this way: our number system is based on groups of 10.

When you multiply by 10, you're essentially taking each unit and turning it into a group of 10. For example:

  • 3 ones becomes 3 tens (which we write as 30)
  • 6 ones becomes 6 tens (which we write as 60)
  • 9 ones becomes 9 tens (which we write as 90)

Adding the zero is just a shortcut way of showing that each digit moved one place value position to the left!

Question: What happens if I multiply a bigger factor of 3 by 10, like 12 or 15?

Answer: The same rule applies to any factor of 3!

Examples:

  • \(12 \times 10 = 120\) (12 is a factor of 3 since \(12 = 3 \times 4\))
  • \(15 \times 10 = 150\) (15 is a factor of 3 since \(15 = 3 \times 5\))
  • \(18 \times 10 = 180\) (18 is a factor of 3 since \(18 = 3 \times 6\))

The result is always 10 times larger than the original number, and since the original was a multiple of 3, the result will be a multiple of 30!

Detailed Summary: Multiplying by 10 with Factors of 3

Definitions of Key Concepts

  • Factor of 3: A number that can be divided evenly by 3 without a remainder (e.g., 3, 6, 9, 12, 15, 18...)
  • Multiplication by 10: A mathematical operation that increases a number by a factor of 10
  • Place value: The value of a digit based on its position in the number (ones, tens, hundreds, etc.)
  • Multiple: The result of multiplying a number by an integer
  • Pattern: A consistent relationship or sequence between numbers

Core Rules and Principles

  • Multiplication by 10 Rule: When multiplying any number by 10, add one zero to the end of the number
  • Factor of 3 Rule: A number is divisible by 3 if the sum of its digits is divisible by 3
  • Combined Effect: Multiplying a factor of 3 by 10 results in a multiple of 30
  • Place Value Shift: Each digit moves one place value position to the left
  • Digit Sum Test: Add all digits; if sum is divisible by 3, the number is divisible by 3

Step-by-Step Methods and Procedures

  1. Identify the factor: Check if the number is divisible by 3 using the digit sum test
  2. Apply the multiply by 10 rule: Add a zero to the end of the number
  3. Verify the result: Ensure the answer is 10 times the original number
  4. Confirm the pattern: Check that the result is a multiple of 30

Multiple Concrete Examples (Simple → Advanced)

  • Simple: \(3 \times 10 = 30\), \(6 \times 10 = 60\), \(9 \times 10 = 90\)
  • Moderate: \(12 \times 10 = 120\), \(15 \times 10 = 150\), \(18 \times 10 = 180\)
  • Advanced: \(21 \times 10 = 210\), \(24 \times 10 = 240\), \(27 \times 10 = 270\)
  • Pattern Recognition: All results (30, 60, 90, 120, 150, 180...) are multiples of 30

Tips, Tricks, and Common Pitfalls

  • Tip 1: Use the digit sum test to quickly identify factors of 3
  • Tip 2: Remember: multiplying by 10 always adds one zero to the end
  • Tip 3: The result will always be a multiple of 30 when multiplying a factor of 3 by 10
  • Common Pitfall: Confusing the rules for multiplying by 3 versus multiplying by 10
  • Common Mistake: Forgetting that the factor relationship is preserved after multiplication

Key Notes for Memorization

  • Memory Aid: "Factors of 3 times 10 always end in 0 and are multiples of 30"
  • Pattern: 30, 60, 90, 120, 150, 180, 210... (increases by 30 each time)
  • Digit Sum Rule: If digits sum to 3, 6, 9, 12, 15, 18, etc., it's divisible by 3
  • Multiplication Shortcut: Multiply by 10 = add one zero

Explanation for Students

When you multiply a factor of 3 by 10, you're making the number 10 times bigger. The easy way to do this is to add a zero to the end. For example, 3 becomes 30, 6 becomes 60, and 9 becomes 90. These special numbers (3, 6, 9, 12, 15, etc.) are called factors of 3 because they can be divided evenly by 3. When you multiply any of these by 10, you get numbers like 30, 60, 90, 120, 150, etc. - all of which are multiples of 30. The key is to first make sure your starting number is a factor of 3 (by adding its digits), then just add a zero!