Solved Exercises on Multiplying by 10 and 11 in Grade 3

Master multiplying by 10 and 11: place value shifts, patterns, and real-world applications through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Multiply by 10
Exercise 1
What is \(7 \times 10\)? Show how the digits move.
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
\(7\)
After ×10
\(70\)
Tens
7
Ones
0
↓ ×10 ↓
Hundreds
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 Multiply by 11
Exercise 2
What is \(6 \times 11\)? Show the pattern for multiplying by 11.
Definition:

Multiplication by 11: For single digits, the result is the digit repeated

6
6
Original
\(6\)
After ×11
\(66\)
Step 1: Identify the pattern

For single digits, multiplying by 11 repeats the digit

Step 2: Apply the pattern

\(6 \times 11 = 66\)

Step 3: Verify the result

\(6 \times 11 = 6 \times (10 + 1) = 60 + 6 = 66\)

\(6 \times 11 = 66\)
Final answer:

\(6 \times 11 = 66\)

Applied rules:

• Eleven rule: For single digits, repeat the digit

• Pattern recognition: \(n \times 11 = nn\) for single digits

• Distribution: \(6 \times 11 = 6 \times (10 + 1) = 60 + 6\)

3 Mixed operations
Exercise 3
Compare \(8 \times 10\) and \(8 \times 11\). Which is larger and why?
Definition:

Comparison: Determining which number is larger using mathematical reasoning

First
\(8 \times 10\)
Second
\(8 \times 11\)
Difference
\(8\)
Step 1: Calculate \(8 \times 10\)

\(8 \times 10 = 80\)

Step 2: Calculate \(8 \times 11\)

\(8 \times 11 = 88\)

Step 3: Compare the results

\(88 > 80\), so \(8 \times 11\) is larger

Step 4: Explain why

\(8 \times 11 = 8 \times (10 + 1) = (8 \times 10) + (8 \times 1) = 80 + 8 = 88\)

\(8 \times 11 = 88\) is larger than \(8 \times 10 = 80\)
Final answer:

\(8 \times 11 = 88\) is larger than \(8 \times 10 = 80\) by 8 units

Applied rules:

• Comparison: \(8 \times 11 > 8 \times 10\) because \(11 > 10\)

• Distribution: \(8 \times 11 = 8 \times (10 + 1) = 80 + 8\)

• Reasoning: Multiplying by a larger number gives a larger result

Multiplication by 11 Patterns and Tricks
\(n \times 11 = nn\)
Single Digit × 11
The Eleven Trick for Single Digits
For any single digit n: \(n \times 11 = nn\)
Multiplication by 11 Patterns
×
1
2
3
4
5
6
7
8
9
10
11
11
11
22
33
44
55
66
77
88
99
110
121
Key definitions:

Multiplication by 11: A mathematical operation that increases a number by a factor of 11

Pattern recognition: Identifying consistent relationships between numbers

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

Multiplication by 11 strategies:
  1. For single digits: Repeat the digit (e.g., \(5 \times 11 = 55\))
  2. For two digits: Add digits and place sum between them
  3. For larger numbers: Use distribution: \(n \times 11 = n \times (10 + 1) = 10n + n\)
Tip 1: Remember: multiplying by 11 is like multiplying by 10 and adding the original number.
Tip 2: For single digits, just repeat the digit (e.g., 7×11=77).
Tip 3: Use the pattern to check your work.
Tip 4: Practice with small numbers first, then increase complexity.
Key insight: 11 is 10 + 1, so multiplying by 11 is like multiplying by 10 and adding the original number.
Common error: Confusing the pattern for single digits. Remember: \(n \times 11 = nn\).
Comparison: Multiplying by 10 vs 11
\(n \times 10 \text{ vs } n \times 11\)
Side-by-Side Comparison
Comparison Table
Number
×10
×11
1
10
11
2
20
22
3
30
33
4
40
44
5
50
55
6
60
66
7
70
77
8
80
88
9
90
99
Multiply by 10
\(n \times 10 = n0\)
Add zero to end
Multiply by 11
\(n \times 11 = nn\)
Repeat digit (single)
Difference
\(n \times 11 - n \times 10 = n\)
Always equals original number
Multiplying by 10 and 11 Tips
\(\text{Multiply by 10} \leftrightarrow \text{Multiply by 11}\)
Inverse Operations
Key concepts:

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

Pattern recognition: Identifying consistent relationships between numbers

Digit movement: When multiplying by 10, digits shift left; when multiplying by 11, there's a different pattern

Steps for success:
  1. Identify the operation: Determine if multiplying by 10 or 11
  2. Apply the rule: Use the appropriate method for each
  3. Check your work: Use the inverse operation to verify
  4. State the answer: Write the final result
Tip 1: Remember: multiplying by 10 makes numbers 10 times larger.
Tip 2: Remember: multiplying by 11 makes numbers 11 times larger.
Tip 3: Use the difference pattern: \(n \times 11 - n \times 10 = n\)

Tip 4: Practice both operations together to see the relationship.

Common mistake: Confusing the rules for multiplying by 10 and 11.

Success strategy: Remember that multiplying by 11 is like multiplying by 10 and adding the original number.

Important facts to remember:

• Multiplying by 10 adds a zero to the end of a number

• Multiplying by 11 repeats the digit for single-digit numbers

• \(n \times 11 = n \times 10 + n\) (distribution property)

• Multiplication by 11 makes numbers 11 times larger

• The difference between \(n \times 11\) and \(n \times 10\) is always \(n\)

Questions & Answers

Question: Why does multiplying by 11 repeat the digit for single numbers? Is this always true?

Answer: Yes, for single digits (1-9), multiplying by 11 repeats the digit! This happens because of the distributive property:

For example: \(6 \times 11 = 6 \times (10 + 1) = 60 + 6 = 66\)

This works because:

  • The original digit goes in the tens place
  • The same digit goes in the ones place
  • So 6 becomes 66, 7 becomes 77, etc.

This pattern only works for single digits!

Question: My child is struggling to remember which rule applies to 10 vs 11. How can I help them distinguish?

Answer: Here are helpful strategies to distinguish the rules:

  • For 10: "Add a zero" - think of the number getting 10 times bigger
  • For 11: "Repeat the digit" for single digits - think of the number getting 11 times bigger
  • Connection: Remember that 11 = 10 + 1, so \(n \times 11 = n \times 10 + n\)
  • Practice: Do side-by-side comparisons to see the difference

Practice with simple numbers first and emphasize the difference in results!

Question: What happens if I multiply a two-digit number by 11? Does the same pattern work?

Answer: Great question! For two-digit numbers, the pattern is different:

For numbers like 12, 23, 34, etc., the pattern is:

  • \(12 \times 11 = 132\) (first digit, sum of digits, last digit)
  • \(23 \times 11 = 253\) (2, 2+3=5, 3)
  • \(34 \times 11 = 374\) (3, 3+4=7, 4)

This is a more advanced pattern that works for two-digit numbers where the sum of digits is less than 10!

Detailed Summary: Multiplying by 10 and 11

Definitions of Key Concepts

  • Multiplication by 10: A mathematical operation that increases a number by a factor of 10
  • Multiplication by 11: A mathematical operation that increases a number by a factor of 11
  • Place value: The value of a digit based on its position in the number (ones, tens, hundreds, etc.)
  • Digit shift: Movement of digits when multiplying by powers of 10
  • Pattern recognition: Identifying consistent relationships between numbers

Core Rules and Principles

  • Multiplication by 10 Rule: When multiplying by 10, add one zero to the end of the number
  • Multiplication by 11 Rule: For single digits, repeat the digit (e.g., \(n \times 11 = nn\))
  • Digit Movement: Multiplication by 10 moves digits one place left
  • Relationship: \(n \times 11 = n \times 10 + n\) (distribution property)

Step-by-Step Methods and Procedures

  1. For Multiplication by 10:
    • Take the original number
    • Add a zero to the end of the number
    • Alternatively, move each digit one place value to the left
  2. For Multiplication by 11 (single digits):
    • Take the original single digit
    • Repeat the digit to form a two-digit number
    • Alternatively, calculate \(n \times 10 + n\)
  3. For Verification:
    • Use the inverse operation to check your answer
    • Confirm that the result makes logical sense

Concrete Examples (Simple to Advanced)

  • Multiplying by 10: \(1 \times 10 = 10\), \(5 \times 10 = 50\), \(9 \times 10 = 90\), \(12 \times 10 = 120\)
  • Multiplying by 11 (single digits): \(1 \times 11 = 11\), \(5 \times 11 = 55\), \(9 \times 11 = 99\)
  • Multiplying by 11 (two digits): \(12 \times 11 = 132\), \(23 \times 11 = 253\), \(34 \times 11 = 374\)
  • Comparison: \(8 \times 10 = 80\), \(8 \times 11 = 88\), difference = 8

Tips, Tricks, and Common Pitfalls

  • Tip 1: Remember: multiplying by 10 is like counting by tens
  • Tip 2: Remember: for single digits, multiplying by 11 repeats the digit
  • Tip 3: Use the relationship: \(n \times 11 = n \times 10 + n\)
  • Common Pitfall: Confusing the rules for multiplying by 10 and 11
  • Common Mistake: Forgetting that the 11 pattern only works for single digits

Key Notes for Memorization

  • Rhyme: "To multiply by 10, add a zero at the end!"
  • Rhyme: "To multiply by 11, repeat the digit (for single digits)!"
  • Memory Aid: 11 is 10 + 1, so multiply by 10 and add the original number
  • Pattern: The difference between \(n \times 11\) and \(n \times 10\) is always \(n\)

Explanation for Students

When you multiply by 10, you're making the number 10 times bigger by adding a zero at the end. When you multiply by 11, you're making it 11 times bigger. For single digits, there's a cool pattern: just repeat the digit! So 6×11=66, 7×11=77, and so on. This happens because 11 is really 10+1, so you're taking the number 10 times and adding the number one more time. The key is to remember that multiplying by 10 adds one zero, while multiplying by 11 has a special pattern for single digits.