Multiplying by 11 and 12 for 3rd Grade Students

Master multiplying by 11 and 12: understand the patterns, practice with exercises, and learn visual strategies through these 10 detailed problems.

Multiplying by 11 and 12: Exercises 1 to 5
1 Multiplying by 11
Exercise 1
Calculate: 7 × 11 = ?
Definition:

Multiplying by 11: For single-digit numbers, multiplying by 11 means repeating the digit twice. For example, 7 × 11 = 77.

Multiplication method:
  1. Take the original number (7)
  2. Repeat the digit to form a two-digit number
  3. The result is 77
Problem
7 × 11
Method
Repeat digit
Result
77
Step 1: Identify the number to multiply

We have 7 as our number

Step 2: Apply the rule for multiplying by 11

For single-digit numbers, repeat the digit to form a two-digit number

Step 3: Write the final answer

Repeating the digit 7 gives us 77

7 × 11 = 77
Final answer:

7 × 11 = 77

Applied rules:

• Rule of 11: For single-digit numbers, repeat the digit

• Pattern recognition: All single-digit numbers multiplied by 11 create repeated digits

• Place value: The same digit appears in both tens and ones places

2 Multiplying by 12
Exercise 2
Calculate: 5 × 12 = ?
Definition:

Multiplying by 12: You can think of this as multiplying by 10 and then adding twice the original number.

Problem
5 × 12
Method
(5 × 10) + (5 × 2)
Result
50 + 10 = 60
Step 1: Break down 12 into 10 + 2

We can think of 5 × 12 as 5 × (10 + 2)

Step 2: Apply the distributive property

5 × (10 + 2) = (5 × 10) + (5 × 2)

Step 3: Calculate each part

5 × 10 = 50 and 5 × 2 = 10

Step 4: Add the results

50 + 10 = 60

5 × 12 = 60
Final answer:

5 × 12 = 60

Applied rules:

• Distributive property: a × (b + c) = (a × b) + (a × c)

• Breaking down: Think of 12 as 10 + 2

• Pattern recognition: 12 is 10 plus 2 times the number

3 Multiplying by 11 with 2-digit numbers
Exercise 3
Calculate: 12 × 11 = ?
Definition:

Multiplying by 11 with 2-digit numbers: For numbers less than 100, add the digits and place the sum between them.

Problem
12 × 11
Method
1(1+2)2
Result
132
Step 1: Identify the digits

We have 1 and 2 in the number 12

Step 2: Add the digits

1 + 2 = 3

Step 3: Place the sum between the digits

Put 3 between 1 and 2 to get 132

12 × 11 = 132
Final answer:

12 × 11 = 132

Applied rules:

• Rule of 11 for 2-digit numbers: Add the digits and place the sum between them

• Pattern recognition: 12 × 11 = 1(1+2)2 = 132

• Place value: Each position has a specific value in the result

Multiplying by 11 and 12: Exercises 6 to 10
4 Multiplying by 12 with larger numbers
Exercise 4
Calculate: 8 × 12 = ?
Definition:

Multiplying by 12: Think of this as multiplying by 10 and then adding twice the original number.

Problem
8 × 12
Method
(8 × 10) + (8 × 2)
Result
80 + 16 = 96
Step 1: Break down 12 into 10 + 2

We can think of 8 × 12 as 8 × (10 + 2)

Step 2: Apply the distributive property

8 × (10 + 2) = (8 × 10) + (8 × 2)

Step 3: Calculate each part

8 × 10 = 80 and 8 × 2 = 16

Step 4: Add the results

80 + 16 = 96

8 × 12 = 96
Final answer:

8 × 12 = 96

Applied rules:

• Distributive property: a × (b + c) = (a × b) + (a × c)

• Breaking down: Think of 12 as 10 + 2

• Pattern recognition: 12 is 10 plus 2 times the number

5 Multiplying by 11 with single digits
Exercise 5
Calculate: 9 × 11 = ?
Definition:

Multiplying by 11: For single-digit numbers, multiplying by 11 means repeating the digit twice.

Problem
9 × 11
Method
Repeat digit
Result
99
Step 1: Identify the number to multiply

We have 9 as our number

Step 2: Apply the rule for multiplying by 11

For single-digit numbers, repeat the digit to form a two-digit number

Step 3: Write the final answer

Repeating the digit 9 gives us 99

9 × 11 = 99
Final answer:

9 × 11 = 99

Applied rules:

• Rule of 11: For single-digit numbers, repeat the digit

• Pattern recognition: All single-digit numbers multiplied by 11 create repeated digits

• Place value: The same digit appears in both tens and ones places

6 Multiplying by 12 with single digits
Exercise 6
Calculate: 3 × 12 = ?
Definition:

Multiplying by 12: Think of this as multiplying by 10 and then adding twice the original number.

Problem
3 × 12
Method
(3 × 10) + (3 × 2)
Result
30 + 6 = 36
Step 1: Break down 12 into 10 + 2

We can think of 3 × 12 as 3 × (10 + 2)

Step 2: Apply the distributive property

3 × (10 + 2) = (3 × 10) + (3 × 2)

Step 3: Calculate each part

3 × 10 = 30 and 3 × 2 = 6

Step 4: Add the results

30 + 6 = 36

3 × 12 = 36
Final answer:

3 × 12 = 36

Applied rules:

• Distributive property: a × (b + c) = (a × b) + (a × c)

• Breaking down: Think of 12 as 10 + 2

• Pattern recognition: 12 is 10 plus 2 times the number

7 Multiplying by 11 with 2-digit numbers
Exercise 7
Calculate: 13 × 11 = ?
Definition:

Multiplying by 11 with 2-digit numbers: Add the digits and place the sum between them.

Problem
13 × 11
Method
1(1+3)3
Result
143
Step 1: Identify the digits

We have 1 and 3 in the number 13

Step 2: Add the digits

1 + 3 = 4

Step 3: Place the sum between the digits

Put 4 between 1 and 3 to get 143

13 × 11 = 143
Final answer:

13 × 11 = 143

Applied rules:

• Rule of 11 for 2-digit numbers: Add the digits and place the sum between them

• Pattern recognition: 13 × 11 = 1(1+3)3 = 143

• Place value: Each position has a specific value in the result

8 Multiplying by 12 with 2-digit numbers
Exercise 8
Calculate: 7 × 12 = ?
Definition:

Multiplying by 12: Think of this as multiplying by 10 and then adding twice the original number.

Problem
7 × 12
Method
(7 × 10) + (7 × 2)
Result
70 + 14 = 84
Step 1: Break down 12 into 10 + 2

We can think of 7 × 12 as 7 × (10 + 2)

Step 2: Apply the distributive property

7 × (10 + 2) = (7 × 10) + (7 × 2)

Step 3: Calculate each part

7 × 10 = 70 and 7 × 2 = 14

Step 4: Add the results

70 + 14 = 84

7 × 12 = 84
Final answer:

7 × 12 = 84

Applied rules:

• Distributive property: a × (b + c) = (a × b) + (a × c)

• Breaking down: Think of 12 as 10 + 2

• Pattern recognition: 12 is 10 plus 2 times the number

9 Multiplying by 11 with single digits
Exercise 9
Calculate: 6 × 11 = ?
Definition:

Multiplying by 11: For single-digit numbers, multiplying by 11 means repeating the digit twice.

Problem
6 × 11
Method
Repeat digit
Result
66
Step 1: Identify the number to multiply

We have 6 as our number

Step 2: Apply the rule for multiplying by 11

For single-digit numbers, repeat the digit to form a two-digit number

Step 3: Write the final answer

Repeating the digit 6 gives us 66

6 × 11 = 66
Final answer:

6 × 11 = 66

Applied rules:

• Rule of 11: For single-digit numbers, repeat the digit

• Pattern recognition: All single-digit numbers multiplied by 11 create repeated digits

• Place value: The same digit appears in both tens and ones places

10 Multiplying by 12 with single digits
Exercise 10
Calculate: 4 × 12 = ?
Definition:

Multiplying by 12: Think of this as multiplying by 10 and then adding twice the original number.

Problem
4 × 12
Method
(4 × 10) + (4 × 2)
Result
40 + 8 = 48
Step 1: Break down 12 into 10 + 2

We can think of 4 × 12 as 4 × (10 + 2)

Step 2: Apply the distributive property

4 × (10 + 2) = (4 × 10) + (4 × 2)

Step 3: Calculate each part

4 × 10 = 40 and 4 × 2 = 8

Step 4: Add the results

40 + 8 = 48

4 × 12 = 48
Final answer:

4 × 12 = 48

Applied rules:

• Distributive property: a × (b + c) = (a × b) + (a × c)

• Breaking down: Think of 12 as 10 + 2

• Pattern recognition: 12 is 10 plus 2 times the number

Complete Summary: Multiplying by 11 and 12
Key Concepts & Definitions
Multiplication by 11 Rule:
For single-digit numbers, multiplying by 11 creates a two-digit number with the same digit repeated. For two-digit numbers, add the digits and place the sum between them.
Multiplication by 12 Rule:
To multiply by 12, think of it as multiplying by 10 and then adding twice the original number. This uses the distributive property: a × 12 = (a × 10) + (a × 2).
Distributive Property:
This property allows us to break down multiplication into simpler parts: a × (b + c) = (a × b) + (a × c).
Core Rules & Principles
Multiplication by 11:
  • Single digits: n × 11 = nn (repeat the digit)
  • Two digits: ab × 11 = a(a+b)b (add digits and place sum between)
  • If sum is 10 or more: carry over to the left
Multiplication by 12:
  • n × 12 = (n × 10) + (n × 2)
  • This breaks the problem into easier multiplications
  • Always multiply by 10 first, then add twice the number
Step-by-Step Methods
Method for Multiplying by 11:
  1. For single-digit numbers: repeat the digit (5 × 11 = 55)
  2. For two-digit numbers: add the digits together
  3. Place the sum between the original digits
  4. If the sum is 10 or greater, carry the tens digit to the left
Method for Multiplying by 12:
  1. Break down 12 into 10 + 2
  2. Multiply the number by 10
  3. Multiply the number by 2
  4. Add the two results together
Examples & Applications
Multiplying by 11 Examples:
  • 3 × 11 = 33 (single digit rule)
  • 7 × 11 = 77 (single digit rule)
  • 12 × 11 = 132 (1+2=3, place between 1 and 2)
  • 14 × 11 = 154 (1+4=5, place between 1 and 4)
Multiplying by 12 Examples:
  • 5 × 12 = (5×10) + (5×2) = 50 + 10 = 60
  • 8 × 12 = (8×10) + (8×2) = 80 + 16 = 96
  • 3 × 12 = (3×10) + (3×2) = 30 + 6 = 36
  • 7 × 12 = (7×10) + (7×2) = 70 + 14 = 84
Tips, Tricks & Common Mistakes
Helpful Tips:
  • For 11 × single digits: just repeat the digit
  • For 12 × any number: think "times 10 plus double the number"
  • Practice the 11 pattern with different numbers to build confidence
  • Memorize the 11 and 12 multiplication facts for faster recall
Common Mistakes to Avoid:
  • Forgetting to add the middle digit when multiplying two-digit numbers by 11
  • Mistakenly calculating 12 as 10 + 1 instead of 10 + 2
  • Getting confused with carrying when the sum of digits is 10 or more
  • Mixing up the order of operations when using the distributive property
Memory Aids & Takeaways
Key Points to Remember:
  • "For 11 × single digits, repeat the digit"
  • "For 12 × any number, multiply by 10 and add double the number"
  • "11 makes twins" - helps remember the repeated digit pattern
  • "12 is 10 + 2" - use this breakdown for easier calculation
n × 11 = nn (for single digits)
Multiplying by 11 (single digits)
n × 12 = (n × 10) + (n × 2)
Multiplying by 12
Tip 1: Remember that multiplying by 11 with single digits creates repeated digits (like 6 × 11 = 66).
Tip 2: For 12, think of it as "multiply by 10 and add twice the number."
Tip 3: Practice with small numbers first, then move to larger ones.
Tip 4: Use the distributive property to break down harder problems.
Visual Guide: Multiplying by 11 and 12
Multiplication Patterns
🔢
Rule for multiplying by 11 (single digits):

Repeat the digit: n × 11 = nn

Rule for multiplying by 12:

Think of it as: (n × 10) + (n × 2)

5
→
11
=
55
3
→
12
=
(30 + 6) = 36
1 × 11
= 11
2 × 11
= 22
3 × 11
= 33
4 × 11
= 44
1 × 12
= 12
2 × 12
= 24
3 × 12
= 36
4 × 12
= 48
Key Concept: Distributive Property

For 12: n × 12 = (n × 10) + (n × 2)

This makes difficult multiplication easier!

Challenge 1
8 × 11
88
Challenge 2
9 × 12
108
Challenge 3
15 × 11
165
Challenge 4
6 × 12
72
Multiplication Method Result
1 × 11 Repeat digit 11
2 × 11 Repeat digit 22
3 × 11 Repeat digit 33
4 × 11 Repeat digit 44
5 × 12 (5×10)+(5×2) 60
6 × 12 (6×10)+(6×2) 72
7 × 12 (7×10)+(7×2) 84
8 × 12 (8×10)+(8×2) 96
n × 11 = nn
Multiplying by 11
n × 12 = (n × 10) + (n × 2)
Multiplying by 12
Tip 1: Remember that multiplying by 11 with single digits creates repeated digits.
Tip 2: For 12, think of it as multiply by 10 and add twice the number.
Tip 3: Practice with small numbers first, then move to larger ones.
Tip 4: Use the distributive property to break down harder problems.

Questions & Answers

Question: Why does multiplying by 11 repeat the digit for single-digit numbers? Like why does 7 × 11 equal 77?

Answer: Great question! When you multiply 7 by 11, you're really doing 7 × (10 + 1). Using the distributive property: 7 × (10 + 1) = (7 × 10) + (7 × 1) = 70 + 7 = 77.

So you're taking the number 7, putting it in the tens place (70), and adding the same number in the ones place (7), which gives you 77.

This creates the pattern where the digit repeats itself!

Question: My child is having trouble with multiplying by 12. What's the easiest way to explain this?

Answer: The easiest way to explain multiplying by 12 is to break it down:

  • First, multiply the number by 10 (just add a zero)
  • Then, multiply the number by 2 (double it)
  • Finally, add those two results together

For example: 8 × 12

  • 8 × 10 = 80
  • 8 × 2 = 16
  • 80 + 16 = 96

So 8 × 12 = 96!

Question: Does the 11 trick work for bigger numbers too? Like 23 × 11?

Answer: Yes! The 11 trick works for two-digit numbers too, but it's slightly different:

For 23 × 11:

  • Take the first digit: 2
  • Add the digits together: 2 + 3 = 5
  • Take the last digit: 3
  • Put them together: 253

So 23 × 11 = 253!

If the sum of the digits is 10 or more, you carry the extra digit to the left. For example: 57 × 11 = 627 (5+7=12, so put 2 in the middle and carry 1 to the 5, making it 6).