Mental Math Strategies for Subtraction - 2nd Grade Mathematics

Advanced mental math techniques for mastering subtraction within 20 through efficient mental calculation strategies.

Mental Math Strategies 1-10 with Solutions
1 Counting Back
Strategy 1
Solve 15 - 3 using the counting back strategy mentally.
Difficulty: Beginner Time: ~2 minutes Skills: Mental Calculation, Counting Back, Subtraction
Definition:

Counting back: A mental math strategy where you start with the minuend and count backwards by the subtrahend to find the difference.

Note: This strategy is particularly effective when subtracting a small number from a larger one.

Step-by-step mental strategy:
  1. Start with the minuend (15)
  2. Count backwards mentally by the subtrahend (3)
  3. Visualize each count in your head
  4. Arrive at the difference
Starting Number
15
Count Back
14, 13, 12
Final Answer
12
Step 1: Start with 15 in your mind

Hold 15 in your mental workspace

Step 2: Count back 3 numbers mentally

Count: 14 (that's 1), 13 (that's 2), 12 (that's 3)

Step 3: Identify the final number

The number you land on is your answer: 12

15 - 3 = 12
Final answer:

15 - 3 = 12

Applied rules:

• Mental visualization: Picture the numbers in your mind

• Sequential counting: Count each number in sequence

• Backwards counting: Each count subtracts one from the total

• Practice Tip: Use fingers initially to support mental visualization

2 Counting Up
Strategy 2
Solve 12 - 7 using the counting up strategy mentally.
Difficulty: Beginner Time: ~2 minutes Skills: Mental Calculation, Counting Up, Subtraction
Definition:

Counting up: A mental math strategy where you start with the subtrahend and count up to the minuend to find the difference.

Note: This strategy is particularly effective when the subtrahend is close to the minuend.

Step-by-step mental strategy:
  1. Start with the subtrahend (7)
  2. Count up mentally to the minuend (12)
  3. Keep track of how many numbers you count
  4. The count is the difference
Starting Number
7
Count Up
8, 9, 10, 11, 12
Steps Counted
5
Step 1: Start with 7 in your mind

Hold 7 as your starting point

Step 2: Count up to 12 mentally

Count: 8, 9, 10, 11, 12 (that's 5 numbers)

Step 3: Count the numbers you said

You counted 5 numbers, so the difference is 5

12 - 7 = 5
Final answer:

12 - 7 = 5

Applied rules:

• Start with smaller number: Begin with the subtrahend

• Count up mentally: Move forward to the minuend

• Track the steps: The number of steps equals the difference

• Practice Tip: This is often more efficient when numbers are close together

3 Making Tens
Strategy 3
Solve 14 - 8 using the making tens strategy mentally.
Difficulty: Intermediate Time: ~3 minutes Skills: Mental Calculation, Making Tens, Subtraction
Definition:

Making tens: A mental math strategy where you adjust numbers to make a ten, then adjust the result accordingly.

Note: This strategy leverages the fact that tens are easier to work with mentally.

Step-by-step mental strategy:
  1. Adjust the problem to make a ten
  2. Solve the adjusted problem mentally
  3. Compensate for the adjustment
  4. Arrive at the final answer
Original Problem
14 - 8
Make Ten
14 - 4 = 10, then 10 - 4 = 6
Final Answer
6
Step 1: Think of 8 as 4 + 4

Break 8 into 4 and 4 to make it easier

Step 2: Subtract 4 from 14 first

14 - 4 = 10 (making a ten)

Step 3: Subtract the remaining 4

10 - 4 = 6

14 - 8 = 6
Final answer:

14 - 8 = 6

Applied rules:

• Decompose the subtrahend: Break it into parts that make tens

• Use friendly numbers: Tens are easier to work with

• Sequential subtraction: Subtract in parts

• Practice Tip: Master combinations that make ten (6+4, 7+3, 8+2, 9+1)

4 Breaking Apart Numbers
Strategy 4
Solve 16 - 9 using the breaking apart strategy mentally.
Difficulty: Intermediate Time: ~3 minutes Skills: Mental Calculation, Number Decomposition, Subtraction
Definition:

Breaking apart: A mental math strategy where you decompose the subtrahend to make the calculation easier.

Note: This strategy involves breaking the number being subtracted into more manageable parts.

Step-by-step mental strategy:
  1. Break the subtrahend into convenient parts
  2. Subtract each part sequentially
  3. Combine the results mentally
  4. Arrive at the final answer
Original Problem
16 - 9
Break Apart
16 - 6 - 3 = 10 - 3 = 7
Final Answer
7
Step 1: Break 9 into 6 + 3

Since 16 - 6 = 10, this makes the problem easier

Step 2: Subtract 6 from 16 first

16 - 6 = 10

Step 3: Subtract the remaining 3

10 - 3 = 7

16 - 9 = 7
Final answer:

16 - 9 = 7

Applied rules:

• Decompose strategically: Break into parts that create friendly numbers

• Sequential subtraction: Subtract in logical order

• Mental organization: Keep track of partial results

• Practice Tip: Look for parts that create tens or multiples of five

5 Compensation
Strategy 5
Solve 17 - 8 using the compensation strategy mentally.
Difficulty: Intermediate Time: ~3 minutes Skills: Mental Calculation, Compensation, Subtraction
Definition:

Compensation: A mental math strategy where you adjust both numbers to make the calculation easier, then compensate for the adjustment.

Note: This strategy involves making both numbers more manageable while keeping the difference constant.

Step-by-step mental strategy:
  1. Adjust both numbers to make them easier to work with
  2. Solve the adjusted problem mentally
  3. Keep track of any compensation needed
  4. Arrive at the final answer
Original Problem
17 - 8
Compensate
(17 + 2) - (8 + 2) = 19 - 10 = 9
Final Answer
9
Step 1: Add 2 to both numbers

17 + 2 = 19 and 8 + 2 = 10 (now we have 19 - 10)

Step 2: Solve the easier problem

19 - 10 = 9

Step 3: The difference remains the same

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

17 - 8 = 9
Final answer:

17 - 8 = 9

Applied rules:

• Equal adjustment: Add or subtract the same amount from both numbers

• Difference preservation: The result stays the same

• Mental simplification: Create easier numbers to work with

• Practice Tip: Look for adjustments that create tens or round numbers

Mental Math Strategies 6-10 with Solutions
6 Doubles Plus/Minus One
Strategy 6
Solve 12 - 6 using the doubles strategy mentally.
Definition:

Doubles strategy: A mental math strategy that uses knowledge of doubles (like 6 + 6 = 12) to solve subtraction problems.

Note: This strategy connects addition and subtraction facts and builds on known doubles.

Step-by-step mental strategy:
  1. Recognize the doubles relationship
  2. Recall the related addition fact
  3. Apply the inverse relationship
  4. Arrive at the final answer
Original Problem
12 - 6
Doubles Relationship
6 + 6 = 12, so 12 - 6 = 6
Final Answer
6
Step 1: Recognize that 6 + 6 = 12

This is a doubles fact you should know

Step 2: Apply the inverse relationship

If 6 + 6 = 12, then 12 - 6 = 6

12 - 6 = 6
Final answer:

12 - 6 = 6

7 Number Line Visualization
Strategy 7
Solve 13 - 5 using mental number line visualization.
Definition:

Number line visualization: A mental math strategy where you visualize a number line in your mind and move along it to solve subtraction problems.

Note: This strategy helps develop spatial understanding of numbers and their relationships.

Step-by-step mental strategy:
  1. Visualize a number line in your mind
  2. Start at the minuend position
  3. Move backward by the subtrahend amount
  4. Arrive at the final answer
Original Problem
13 - 5
Mental Movement
13 → 12 → 11 → 10 → 9 → 8
Final Answer
8
Step 1: Visualize the number line in your mind

Picture numbers from 0 to 20 in sequence

Step 2: Start at 13 on your mental number line

Place yourself at the number 13

Step 3: Move back 5 positions mentally

Count back: 12, 11, 10, 9, 8

13 - 5 = 8
Final answer:

13 - 5 = 8

8 Related Facts
Strategy 8
Solve 11 - 4 using related facts mentally.
Definition:

Related facts: A mental math strategy that uses known addition facts to solve subtraction problems.

Note: This strategy connects addition and subtraction as inverse operations.

Step-by-step mental strategy:
  1. Think of the related addition fact
  2. Ask: "What number added to 4 equals 11?"
  3. Recall the addition fact
  4. Apply the inverse relationship
Original Problem
11 - 4
Related Fact
4 + ? = 11, so ? = 7, therefore 11 - 4 = 7
Final Answer
7
Step 1: Think: 4 + ? = 11

What number added to 4 gives 11?

Step 2: Recall that 4 + 7 = 11

This is a known fact

Step 3: Apply the inverse relationship

If 4 + 7 = 11, then 11 - 4 = 7

11 - 4 = 7
Final answer:

11 - 4 = 7

9 Decomposition
Strategy 9
Solve 15 - 7 using decomposition mentally.
Definition:

Decomposition: A mental math strategy where you break apart numbers into their component parts to make subtraction easier.

Note: This strategy involves breaking numbers into tens and ones, or other convenient groupings.

Step-by-step mental strategy:
  1. Break the numbers into convenient parts
  2. Subtract the parts separately
  3. Combine the results mentally
  4. Arrive at the final answer
Original Problem
15 - 7
Decompose
15 = 10 + 5, so (10 + 5) - 7 = 10 - 7 + 5 = 3 + 5 = 8
Final Answer
8
Step 1: Break 15 into 10 + 5

This makes the problem easier to handle

Step 2: Think of (10 + 5) - 7

This equals 10 - 7 + 5

Step 3: Calculate 10 - 7 + 5

3 + 5 = 8

15 - 7 = 8
Final answer:

15 - 7 = 8

10 Comprehensive Practice
Strategy 10
Solve 18 - 9 using the most efficient mental strategy.
Definition:

Comprehensive practice: A variety of problems that require efficient application of mental math strategies.

Note: This strategy helps students develop fluency and choose the most efficient approach for each problem.

Step-by-step mental strategy:
  1. Examine the numbers in the problem
  2. Choose the most efficient mental strategy
  3. Apply the strategy with accuracy
  4. Verify the result efficiently
Original Problem
18 - 9
Best Strategy
Counting up: 9→10→11→12→13→14→15→16→17→18 (9 steps)
Final Answer
9
Step 1: For 18 - 9, counting up is efficient

Count from 9 to 18: 10, 11, 12, 13, 14, 15, 16, 17, 18 (9 steps)

Step 2: Count the numbers you said

You counted 9 numbers, so the difference is 9

18 - 9 = 9
Final answer:

18 - 9 = 9

Detailed Summary: Mental Math Strategies for Subtraction
Core Concepts and Definitions:

Mental Math: Performing calculations in your head without writing anything down or using physical aids.

Subtraction: The mathematical operation of taking away one number from another to find the difference.

Minuend: The number from which another number is subtracted.

Subtrahend: The number that is subtracted from another number.

Difference: The result of a subtraction problem.

Mental Calculation: The ability to perform arithmetic operations using only your mind.

Core Rules and Principles:
  1. Counting Back: Start with the minuend and count backwards by the subtrahend.
  2. Counting Up: Start with the subtrahend and count up to the minuend.
  3. Making Tens: Adjust numbers to create friendly tens for easier calculation.
  4. Decomposition: Break apart numbers into more manageable parts.
  5. Compensation: Adjust both numbers equally to simplify the problem.
Step-by-Step Procedures: 1) Examine the numbers, 2) Choose the most efficient strategy, 3) Apply the strategy mentally, 4) Verify the result, 5) State the answer.
Examples (Simple → Advanced): Simple: 8 - 3 = 5 (counting back), Advanced: 17 - 8 = 9 (compensation: 17 - 10 + 2). Problems progress from basic to complex mental calculations.
Tips: Practice regularly, start with smaller numbers, visualize numbers in your head, choose the most efficient strategy for each problem, verify answers by adding back.
Common Pitfalls: Losing track of numbers in your head, choosing inefficient strategies, forgetting to adjust for compensation, miscounting steps, not verifying answers.

Key Takeaways for Memorization

  • Counting back is efficient for small subtrahends
  • Counting up is efficient when numbers are close together
  • Making tens simplifies calculations with difficult numbers
  • Decomposition breaks complex problems into simpler parts
  • Compensation adjusts numbers to create easier calculations
  • Practice related facts to strengthen number relationships
  • Always verify by adding the difference back to the subtrahend
Essential Teaching Points:

• Strategy Selection: Teach students to evaluate which strategy is most efficient for each problem

• Concrete to Abstract: Begin with physical objects, progress to visual representations, then mental math

• Consistent Practice: Regular exposure builds automaticity and fluency

• Verification Skills: Teach students to check their work by using the opposite operation

Mental Math Strategies for Subtraction

Learning Objectives

  • Master mental subtraction strategies
  • Build number sense within 20
  • Improve mathematical fluency
  • Develop efficient calculation methods
  • Choose appropriate strategies
  • Build confidence in mental math
  • Develop verification skills

Counting Back

Start with minuend, count back by subtrahend

Counting Up

Start with subtrahend, count up to minuend

Making Tens

Adjust to create friendly tens

Breaking Apart

Decompose numbers into manageable parts

Counting Back: 15→14→13→12
Counting Up: 7→8→9→10→11→12
Making Tens: 14-8 = 14-4-4
Verify Your Answer

Mental Math Practice Elements

These strategies provide efficient methods for performing subtraction mentally, building computational fluency and number sense.

Mental Math Success Steps:

1
Examine the numbers in the problem
2
Choose the most efficient mental strategy
3
Apply the strategy mentally
4
Keep track of your mental calculations
5
Identify and state your answer
6
Verify by adding the difference to the subtrahend
Key Rules, Methods, and Strategies
Counting Back: a - b = Start at a, count back b units
Counting Back Strategy
Counting Up: a - b = Start at b, count up to a (difference = steps)
Counting Up Strategy
Key definitions:

Mental math: Calculating in your head without writing or using tools.

Subtraction: Taking away one number from another to find the difference.

Number sense: Understanding the relationship between numbers and their quantities.

Complete methodology:
  1. Examine the problem: Look at the minuend and subtrahend
  2. Choose the strategy: Select the most efficient mental approach
  3. Execute mentally: Perform the calculation in your head
  4. Verify the result: Check by using the opposite operation
Tip 1: Counting back is efficient when the subtrahend is small.
Tip 2: Counting up is efficient when numbers are close together.
Tip 3: Making tens simplifies calculations with difficult numbers.
Tip 4: Always check your answer by adding the difference back to the subtrahend.
Common errors: Losing track of numbers, choosing inefficient strategies, forgetting to verify answers, miscounting steps.
Success strategies: Practice regularly, use visualization, choose appropriate strategies, verify answers by checking.
Rules to remember:

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

• Counting up: Start with subtrahend, count up to minuend

• Making tens: Adjust to create friendly numbers

• Verify results: Always check by adding difference to subtrahend

• Choose efficiency: Select the strategy that requires the fewest steps

Questions & Answers

Question: I'm learning mental math strategies but sometimes I get confused about which one to use. How can I remember?

Answer: Great question! Here's how to choose:

  • For small numbers being subtracted (like 15-3), use counting back
  • When numbers are close together (like 12-8), use counting up
  • When one number is close to 10 (like 14-8), consider making tens
  • Practice with different strategies to see which feels easiest

The more you practice, the more natural it becomes to choose the right strategy!

Question: My child is struggling with mental math and seems to slow down when calculating. How can I help them build confidence?

Answer: Here are strategies to help your child with mental math:

  • Start with smaller numbers and gradually increase difficulty
  • Allow them to use fingers or paper initially as they build confidence
  • Practice for short periods regularly rather than long sessions
  • Celebrate small improvements and progress
  • Encourage them to explain their thinking out loud

Remember, building mental math fluency takes time and consistent practice!

Question: Why do we need to learn mental math? Can't we just write everything down?

Answer: Mental math is important because:

  • It helps you think faster and solve problems quickly
  • It builds your number sense and understanding
  • It's useful in everyday life when you don't have paper
  • It strengthens your brain and makes harder math easier later
  • It builds confidence in your math abilities

Mental math makes you a stronger, faster problem solver!

Math Glossary

Mental Math
Performing calculations in your head without writing anything down or using physical aids.
Subtraction
The mathematical operation of taking away one number from another to find the difference.
Minuend
The number from which another number is subtracted in a subtraction problem.
Subtrahend
The number that is subtracted from another number in a subtraction problem.
Difference
The result of a subtraction problem.
Counting Back
A mental math strategy where you start with the minuend and count backwards by the subtrahend.
Counting Up
A mental math strategy where you start with the subtrahend and count up to the minuend.
Making Tens
A mental math strategy that uses knowledge of combinations that make ten.
Decomposition
Breaking apart numbers into their component parts to make calculations easier.
Compensation
A strategy where you adjust both numbers to make the calculation easier.
Related Facts
Using known addition facts to solve subtraction problems.
Mathematical Fluency
The ability to perform calculations quickly and accurately.
Number Sense
Understanding the relationships between numbers and their quantities.
Verification
Checking your answer using the opposite operation to ensure accuracy.

Elementary Mental Math Educational Team

Certified Elementary Mathematics Educators

Our team specializes in developing mental math strategies that build computational fluency and number sense. All content is aligned with Common Core State Standards and designed to promote mathematical understanding through efficient calculation methods that prepare students for advanced mathematics.