Solved Exercises on Multiplying by 10 in Grade 3

Master multiplying by 10: place value shifts, patterns, and real-world applications through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Place value shifting
Exercise 1
What is \(7 \times 10\)? Show how the digits move.
Definition:

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

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
\(7\)
After ×10
\(70\)
Tens
7
Ones
0
↓
Tens
7
Ones
0
Step 1: Start with the original number

7 in the ones place

Step 2: Multiply by 10

Each digit moves one place to the left

Step 3: Fill empty places with zeros

7 moves to tens place, ones place gets 0

\(7 \times 10 = 70\)
Final answer:

\(7 \times 10 = 70\)

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: 7 ones become 7 tens

2 Pattern recognition
Exercise 2
Complete the pattern: \(3 \times 10 = 30\), \(30 \times 10 = ?\), \(300 \times 10 = ?\)
Definition:

Numerical pattern: A sequence of numbers following a specific rule

3
×
10
=
30
30
×
10
=
300
300
×
10
=
3000
First
\(3 \times 10\)
Second
\(30 \times 10\)
Third
\(300 \times 10\)
Step 1: Identify the pattern

Each number is multiplied by 10

Step 2: Apply the rule to next number

\(30 \times 10\): Move each digit one place left

Step 3: Continue the pattern

\(300 \times 10\): Move each digit one place left

\(30 \times 10 = 300\), \(300 \times 10 = 3000\)
Final answer:

\(3 \times 10 = 30\), \(30 \times 10 = 300\), \(300 \times 10 = 3000\)

Applied rules:

• Pattern continuation: Same rule applies to each term

• Place value shift: Each digit moves one place left

• Exponential growth: Powers of 10

3 Real-world application
Exercise 3
A school has 8 classrooms, and each classroom has 10 desks. How many desks are there in total?
Definition:

Real-world application: Using math to solve practical problems

Given
8 classes × 10 desks
Operation
\(8 \times 10\)
Result
\(80\)
Step 1: Identify the key information

8 classrooms, each with 10 desks

Step 2: Determine the operation

We need to find the total number of desks: 8 groups of 10

Step 3: Apply the multiply by 10 rule

\(8 \times 10 = 80\) (add zero to 8)

Step 4: State the answer

There are 80 desks in total

There are 80 desks in total
Final answer:

There are 80 desks in total

Applied rules:

• Problem-solving: Identify key information and operation needed

• Multiply by 10: Add a zero to the original number

• Real-world connection: Math applies to everyday situations

Place Value and Multiply by 10 Patterns
Position
Thousands
Hundreds
Tens
Ones
Value
1000
100
10
1
↓ ×10 ↓
Position
Ten Thousands
Thousands
Hundreds
Tens
Value
10000
1000
100
10

Multiply by 10 Rule

When you multiply any whole number by 10, you add one zero to the end of the number.

OR each digit moves one place to the left.

\(1 \times 10 = 10\)
\(10 \times 10 = 100\)
\(100 \times 10 = 1000\)
\(5 \times 10 = 50\)
\(50 \times 10 = 500\)
\(500 \times 10 = 5000\)
Key definitions:

Multiplication by 10: A shortcut for adding a number ten times

Place value: The value of a digit based on its position

Digit shift: Movement of digits when multiplying by powers of 10

Two ways to multiply by 10:
  1. Add a zero: Simply append a 0 to the original number
  2. Shift digits: Move each digit one place to the left
Tip 1: Think of it as each digit becoming 10 times larger.
Tip 2: Practice with single digits first, then move to larger numbers.
Tip 3: Use place value charts to visualize the digit movement.
Tip 4: Remember: multiplying by 10 is the same as counting by tens.
Key insight: Each place value is 10 times the value of the place to its right.
Common error: Forgetting to add the zero or shift all digits.
Multiplication by 10 Concepts and Properties
\(n \times 10 = n0\)
Any Number × 10
Zero Rule
\(n \times 10 = n0\)
Add zero to end
Place Value Shift
Ones→Tens
Digit moves left
Repeated Addition
\(n \times 10 = n+n+...+n\)
Ten times
Associative Property
\(a \times (b \times 10) = (a \times b) \times 10\)
Grouping doesn't matter

Visual Digit Movement

7
→
7
0

7 ones become 7 tens

More Examples

\(2 \times 10 = 20\)
\(6 \times 10 = 60\)
\(9 \times 10 = 90\)
\(15 \times 10 = 150\)
\(23 \times 10 = 230\)
\(47 \times 10 = 470\)
Multiplication by 10 Summary and Tips
\(n \times 10 = n \text{ with zero added}\)
The Zero Rule
Key concepts:

Place value: Each position represents 10 times the value of the position to its right

Digit movement: When multiplying by 10, each digit shifts one place to the left

Zero rule: Adding a zero to the end of a number multiplies it by 10

Step-by-step process:
  1. Identify the number: Look at the number being multiplied by 10
  2. Apply the rule: Add a zero to the end OR move each digit one place left
  3. Check your work: Ensure the answer is 10 times larger than the original
  4. State the answer: Write the final result
Tip 1: Practice with single digits first (1×10, 2×10, etc.) then move to larger numbers.
Tip 2: Remember: multiplying by 10 is the same as counting by tens.
Tip 3: Use place value charts to visualize how digits move when multiplying by 10.
Tip 4: Check your answer by dividing by 10 to see if you get back to the original number.
Common mistake: Forgetting to add the zero or only adding it to single digits. The rule works for any number!
Memory aid: "To multiply by 10, just add a zero at the end!" This works for whole numbers.
Important multiplication by 10 facts to remember:

• Any number × 10 = that number with a zero added to the end

• Each digit moves one place value to the left

• The result is always 10 times larger than the original number

• This rule works for any whole number, big or small

• Multiplying by 10 increases the value by a factor of 10

Questions & Answers

Question: Why do we add a zero when we multiply by 10? Why not add a different number?

Answer: Great question! It's because of our base-10 number system. In base-10, each place value is 10 times the value of the place to its right.

When you multiply by 10, each digit becomes worth 10 times more. For example:

  • 5 in the ones place (value = 5) becomes 5 in the tens place (value = 50)
  • Since we moved everything left, the ones place is now empty
  • We fill that empty place with a zero to show it's empty

We could think of it as "moving each digit left and filling the empty spot with 0" - that's why we add a zero specifically!

Question: My child is confused about multiplying larger numbers by 10. Does the same rule apply to numbers like 123?

Answer: Yes, absolutely! The same rule applies to any whole number, regardless of how large it is.

Examples:

  • \(123 \times 10 = 1230\) (add zero to the end)
  • \(456 \times 10 = 4560\) (each digit moves one place left)
  • \(1000 \times 10 = 10000\) (even works with numbers that already have zeros!)

The rule "add a zero" works for ANY whole number. You can think of it as each digit (1, 2, 3) moving one place to the left, and a zero filling the empty ones place.

Question: What happens when we multiply by 100 or 1000? Is there a pattern?

Answer: Yes! There's a wonderful pattern with powers of 10:

  • Multiply by 10: Add 1 zero (move digits 1 place left)
  • Multiply by 100: Add 2 zeros (move digits 2 places left)
  • Multiply by 1000: Add 3 zeros (move digits 3 places left)

Examples:

  • \(5 \times 10 = 50\)
  • \(5 \times 100 = 500\)
  • \(5 \times 1000 = 5000\)

The number of zeros you add matches the number of zeros in the multiplier!

Detailed Summary: Multiplying by 10

Definitions of Key Concepts

  • 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 a number (ones, tens, hundreds, etc.)
  • Digit Shift: The movement of digits to the left when multiplying by powers of 10
  • Base-10 System: Our number system where each place value is 10 times the value of the place to its right

Core Rules and Principles

  • Zero Rule: When multiplying any whole number by 10, add one zero to the end of the number
  • Place Value Shift: Each digit moves one place value position to the left
  • Value Increase: The result is always 10 times larger than the original number
  • Universal Application: This rule works for any whole number, regardless of size

Step-by-Step Methods and Procedures

  1. Identify the number: Look at the number you want to multiply by 10
  2. Apply the rule: Add a zero to the end of the number
  3. Alternative method: Move each digit one place value position to the left
  4. Fill empty spaces: Put a zero in any place values that become empty
  5. Verify your answer: Check that the result is 10 times the original number

Concrete Examples (Simple to Advanced)

  • Simple: \(1 \times 10 = 10\), \(2 \times 10 = 20\), \(5 \times 10 = 50\)
  • Moderate: \(12 \times 10 = 120\), \(25 \times 10 = 250\), \(99 \times 10 = 990\)
  • Advanced: \(123 \times 10 = 1230\), \(456 \times 10 = 4560\), \(1000 \times 10 = 10000\)

Tips, Tricks, and Common Pitfalls

  • Tip 1: Remember: multiplying by 10 is the same as counting by tens
  • Tip 2: Use place value charts to visualize digit movement
  • Tip 3: Check your work by dividing the answer by 10 to get back to the original number
  • Common Pitfall: Forgetting to add the zero or only applying the rule to single digits
  • Common Mistake: Not moving all digits, especially when dealing with larger numbers

Key Notes for Memorization

  • Rhyme: "To multiply by 10, just add a zero at the end!"
  • Pattern: Each place value is 10 times the one to its right
  • Connection: This connects to our base-10 number system
  • Application: This skill is foundational for working with decimals later

Explanation for Students

When you multiply by 10, you're essentially making the number 10 times bigger. In our number system, we group things in tens, so when something gets 10 times bigger, each digit needs to move to a place that's worth 10 times more. Since we moved everything left, we put a zero in the empty spot to show it's empty. It's like upgrading each digit to a more valuable place!