Mental Math Strategies for Subtraction

Master mental subtraction strategies: counting back, making tens, compensation, and related facts through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Counting Back Strategy
Exercise 1
Calculate: 15 - 4 using the counting back strategy
Definition:

Counting back: Start with the larger number and count backwards by the smaller number

Strategy: Counting Back

Start at 15 and count back 4 numbers: 14, 13, 12, 11

The final number is your answer: 11

15
←
14
←
13
←
12
←
11
Step 1: Start with the larger number

Begin at 15

Step 2: Count backwards

Count back 4 numbers: 14, 13, 12, 11

Step 3: The final number is your answer

After counting back 4 from 15, we reach 11

15 - 4 = 11
Final answer:

15 - 4 = 11

Applied rules:

• Counting back: Start with the larger number and count backwards

• Efficiency: This is faster than starting from 1

• Number sense: Helps develop understanding of number relationships

2 Making Tens Strategy
Exercise 2
Calculate: 13 - 8 using the making tens strategy
Definition:

Making tens: Break apart the number being subtracted to make a ten with the starting number

Strategy: Making Tens

Think: 13 - 8 = 13 - (3 + 5) = (13 - 3) - 5 = 10 - 5 = 5

Break 8 into 3 and 5, where 3 makes 10 with 13

1
2
3
4
5
6
7
8
9
10
Original
13 - 8
Break apart
13 - (3 + 5)
Make ten
(13 - 3) - 5
Final
10 - 5 = 5
Step 1: Look for numbers close to 10

13 is 3 more than 10, so break 8 into 3 and 5

Step 2: Subtract to make 10

13 - 3 = 10

Step 3: Subtract the remainder

We took 3 from 8, leaving 5, so 10 - 5 = 5

Step 4: Verify with counting back

Start at 13, count back 8: 12, 11, 10, 9, 8, 7, 6, 5 ✓

13 - 8 = 5
Final answer:

13 - 8 = 5

Applied rules:

• Associative property: a - (b + c) = (a - b) - c

• Ten combinations: Know pairs that make 10

• Breaking apart: Decompose numbers strategically

3 Related Facts Strategy
Exercise 3
If 8 + 7 = 15, then 15 - 7 = ?
Definition:

Related facts: Addition and subtraction facts that use the same numbers

Strategy: Related Facts

If 8 + 7 = 15, then 15 - 7 = 8

Addition and subtraction are inverse operations

Knowing addition facts helps with subtraction

Given fact
8 + 7 = 15
Related subtraction
15 - 7 = ?
Answer
15 - 7 = 8
Step 1: Recognize the relationship

Addition and subtraction are inverse operations

Step 2: Use the known addition fact

Since 8 + 7 = 15, then 15 - 7 = 8

Step 3: Verify the answer

Check: 8 + 7 = 15 ✓

Step 4: Apply to other problems

This works for any addition/subtraction pair

15 - 7 = 8
Final answer:

15 - 7 = 8

Applied rules:

• Inverse operations: a + b = c means c - a = b and c - b = a

• Addition/subtraction connection: Knowing one helps with the other

• Fact families: Related addition and subtraction facts

Subtraction Strategies & Methods
a - b = c ↔ a = b + c
Inverse Operations Relationship
Counting Back
Start at a, count b steps back
Simple subtraction
Making Tens
a - b = (a - c) - (b - c) where a-c is round
Round to friendly numbers
Related Facts
If a + b = c, then c - a = b
Use addition to solve subtraction
Complete Subtraction Strategy Summary
Key Definitions:

Subtraction: Taking away or finding the difference between numbers

Minuend: The number being subtracted from

Subtrahend: The number being subtracted

Difference: The result of subtraction

Inverse operations: Operations that undo each other

Core Methods:
  1. Counting Back: Start with larger number, count backwards
  2. Making Tens: Break apart numbers to form round numbers
  3. Related Facts: Use addition facts to solve subtraction
  4. Compensation: Adjust and balance numbers
  5. Place Value: Subtract by place value
Tip 1: For subtracting 1, 2, or 3, use counting back.
Tip 2: Numbers close to 10 (like 8 or 9) work well with making tens.
Tip 3: Memorize addition facts to use for subtraction.
Tip 4: Practice with visual aids like number lines and ten frames.
Counting Back
Start big, count down
Making Tens
Round to 10
Related Facts
Use addition
Check Work
Add to verify
Essential Rules:

• Counting back: Start with minuend, count back by subtrahend

• Making tens: Break subtrahend to form round numbers

• Inverse relationship: a - b = c means a = b + c

• Verification: Add difference to subtrahend to get minuend

• Order matters: a - b ≠ b - a

Solution: Exercises 4 to 5
4 Compensation Strategy
Exercise 4
Calculate: 16 - 9 using the compensation strategy
Definition:

Compensation: Adjust both numbers by the same amount to make subtraction easier

Strategy: Compensation

Add 1 to both numbers: 16 + 1 = 17, 9 + 1 = 10

Now calculate: 17 - 10 = 7

Adding the same amount to both numbers doesn't change the difference

Original
16 - 9
Compensate
(16 + 1) - (9 + 1)
Simplify
17 - 10 = 7
Step 1: Look for easier numbers

9 is close to 10, which is easier to subtract

Step 2: Add the same amount to both numbers

Add 1 to both: 16 + 1 = 17, 9 + 1 = 10

Step 3: Subtract the new numbers

17 - 10 = 7

Step 4: Verify with counting back

Start at 16, count back 9: 15, 14, 13, 12, 11, 10, 9, 8, 7 ✓

16 - 9 = 7
Final answer:

16 - 9 = 7

Applied rules:

• Compensation principle: a - b = (a + c) - (b + c)

• Adjustment: Add same amount to both numbers

• Difference preservation: The difference stays the same

5 Place Value Strategy
Exercise 5
Calculate: 24 - 17 using the place value strategy
Definition:

Place value: Breaking apart numbers by tens and ones to subtract

Strategy: Place Value

Break apart: 24 = 20 + 4 and 17 = 10 + 7

Subtract tens: 20 - 10 = 10

Subtract ones: 4 - 7 = -3 (not possible, so regroup)

Regroup: 24 = 10 + 14, then 14 - 7 = 7, 10 - 10 = 0, total = 7

Break apart
24 = 20 + 4
17 = 10 + 7
Regroup
24 = 10 + 14
Subtract
14 - 7 = 7, 10 - 10 = 0
Total
7
Step 1: Break apart by place value

24 = 20 + 4 (2 tens and 4 ones)

17 = 10 + 7 (1 ten and 7 ones)

Step 2: Notice you can't subtract 7 from 4

Regroup 24 as 1 ten and 14 ones: 24 = 10 + 14

Step 3: Subtract ones

14 - 7 = 7

Step 4: Subtract tens

1 ten - 1 ten = 0

Step 5: Combine results

0 + 7 = 7

24 - 17 = 7
Final answer:

24 - 17 = 7

Applied rules:

• Place value decomposition: Break numbers into tens and ones

• Regrouping: When ones aren't enough, borrow from tens

• Subtraction with regrouping: Adjust values to subtract

Subtraction Practice Worksheet
Practice Problems
Problem
Strategy
Answer
12 - 3
Counting Back
9
14 - 6
Making Tens
8
18 - 9
Compensation
9
15 - 8
Related Facts
7
23 - 16
Place Value
7
Key Concepts:

Number sense: Understanding relationships between numbers

Strategic thinking: Choosing the best method for each problem

Memorization: Building automatic recall of basic facts

Verification: Checking answers using different methods

Problem-Solving Steps:
  1. Look: Examine the numbers for patterns
  2. Choose: Select the best strategy
  3. Calculate: Perform the mental computation
  4. Check: Verify using a different method
Tip 1: Practice with small numbers first, then increase difficulty.
Tip 2: Use your fingers or objects to visualize problems.
Tip 3: Practice daily for 10-15 minutes to build fluency.
Subtraction Goals:

• Subtract numbers up to 20 mentally

• Use multiple strategies flexibly

• Build computational fluency

• Develop number sense

• Verify solutions independently

Questions & Answers

Question: I don't understand when to use making tens versus counting back. How do I decide?

Answer: Here's how to decide:

  • Use counting back when: Subtracting a small number (1, 2, or 3)
  • Use making tens when: The number being subtracted is close to 10

Examples:

• 15 - 2 → Counting back (start at 15, count back 2: 14, 13)

• 13 - 8 → Making tens (think: 13 - 3 = 10, then 10 - 5 = 5)

The key is looking at the numbers and seeing which approach feels easier mentally!

Question: How can I help my child practice subtraction at home?

Answer: Here are effective ways to practice subtraction at home:

  • Use everyday situations: "We had 10 cookies and ate 3, how many left?"
  • Play subtraction games with cards or dice
  • Practice with visual aids like number lines or counters
  • Encourage explaining their thinking out loud
  • Make it fun with songs or subtraction rhymes
  • Practice for short, consistent sessions

Regular practice in small doses works better than long sessions!

Question: What if I make a mistake? Should I try again or ask for help?

Answer: Making mistakes is part of learning! Here's what to do:

  • First, try to catch your mistake by checking with a different strategy
  • If you're stuck, ask for help - teachers and parents want to assist you
  • Write down your error to learn from it
  • Practice similar problems to strengthen that skill

Remember, every mathematician makes mistakes - what matters is learning from them!