Counting Up and Counting Back to Subtract - 2nd Grade Mathematics

Strategies for mastering subtraction within 20 using counting up and counting back methods.

Exercises 1-10 with Solutions
1 Counting Up to Subtract
Exercise 1
Solve 10 - 7 using the counting up strategy.
Difficulty: Beginner Time: ~3 minutes Skills: Counting Up, Subtraction, Number Sense
Definition:

Counting up to subtract: A strategy where you start with the smaller number (subtrahend) and count up to the larger number (minuend) to find the difference.

Note: This strategy is particularly useful when the subtrahend is much smaller than the minuend, making it easier to count up than to count back.

Step-by-step counting up method:
  1. Start with the smaller number (7 in this case)
  2. Count up until you reach the larger number (10)
  3. Count the number of steps taken
  4. The number of steps is the difference
Starting Number
7
Count Up
8, 9, 10
Steps Taken
3
Step 1: Start with the subtrahend (7)

Begin counting from 7

Step 2: Count up to the minuend (10)

Count: 8 (that's 1), 9 (that's 2), 10 (that's 3)

Step 3: Count the steps taken

It took 3 steps to go from 7 to 10

10 - 7 = 3
Final answer:

10 - 7 = 3

Applied rules:

• Counting up: Start with the smaller number

• Sequential counting: Count each number in sequence

• Step counting: The number of steps equals the difference

• Practice Tip: Use fingers or objects to visualize the counting up process

2 Counting Back to Subtract
Exercise 2
Solve 12 - 5 using the counting back strategy.
Difficulty: Beginner Time: ~3 minutes Skills: Counting Back, Subtraction, Number Sense
Definition:

Counting back to subtract: A strategy where you start with the larger number (minuend) and count backwards by the amount being subtracted (subtrahend) to find the difference.

Note: This strategy is useful when subtracting a smaller number from a larger one, especially when the difference is relatively small.

Step-by-step counting back method:
  1. Start with the larger number (12 in this case)
  2. Count backwards by the number being subtracted (5)
  3. Each count represents subtracting one from the total
  4. Identify the final number as the difference
Starting Number
12
Count Back
11, 10, 9, 8, 7
Final Answer
7
Step 1: Start with the minuend (12)

Begin counting from 12

Step 2: Count backwards by the subtrahend (5)

Count: 11 (that's 1), 10 (that's 2), 9 (that's 3), 8 (that's 4), 7 (that's 5)

Step 3: Identify the final number

The last number counted is the difference: 7

12 - 5 = 7
Final answer:

12 - 5 = 7

Applied rules:

• Counting back: Start from the larger number

• Backwards counting: Count each number in reverse sequence

• One-to-one correspondence: Each count subtracts one from the total

• Practice Tip: Use number lines to visualize the counting back process

3 Choosing the Best Strategy
Exercise 3
For 15 - 8 and 14 - 3, decide which strategy (counting up or counting back) is more efficient and solve.
Difficulty: Intermediate Time: ~5 minutes Skills: Strategy Selection, Subtraction, Efficiency Analysis
Definition:

Strategy selection: Choosing the most efficient subtraction strategy based on the numbers involved in the problem.

Note: Counting up is often more efficient when the subtrahend is much smaller than the minuend. Counting back is often more efficient when the subtrahend is closer to the minuend.

Step-by-step strategy selection method:
  1. Examine the numbers in the subtraction problem
  2. Determine which strategy would require fewer steps
  3. Apply the more efficient strategy
  4. Solve the problem using the chosen strategy
Problem 1: 15-8
Counting up: 8→9→10→11→12→13→14→15 (7 steps)
Problem 2: 14-3
Counting up: 3→4→5→6→7→8→9→10→11→12→13→14 (11 steps) vs Counting back: 14→13→12→11 (3 steps)
Better Choice
15-8: Either, 14-3: Counting back
Step 1: For 15 - 8, analyze the numbers

Counting up: 8 to 15 (7 steps). Counting back: 15 to 8 (7 steps). Both require the same number of steps.

Step 2: For 14 - 3, analyze the numbers

Counting up: 3 to 14 (11 steps). Counting back: 14 to 11 (3 steps). Counting back is more efficient.

Step 3: Solve using the most efficient strategy

15 - 8 = 7 (using either strategy), 14 - 3 = 11 (using counting back)

15 - 8 = 7, 14 - 3 = 11
Final answers:

15 - 8 = 7 (either strategy), 14 - 3 = 11 (counting back is more efficient)

Applied rules:

• Efficiency consideration: Choose the strategy that requires fewer steps

• Strategy analysis: Compare both strategies before deciding

• Flexibility: Be willing to adapt based on the numbers

• Practice Tip: Practice both strategies to know when to use each

4 Number Line Subtraction
Exercise 4
Use a number line to solve 11 - 4 and 13 - 6 using both counting up and counting back strategies.
Difficulty: Beginner Time: ~5 minutes Skills: Number Line Visualization, Counting Strategies, Subtraction
Definition:

Number line strategy: Using a visual number line to represent counting up (moving right) and counting back (moving left) for subtraction.

Note: Number lines provide a concrete visual representation of subtraction strategies and help students understand the relationship between numbers.

Step-by-step number line method:
  1. Draw or visualize a number line
  2. Mark the relevant numbers
  3. For counting up: start at subtrahend, move right to minuend
  4. For counting back: start at minuend, move left by subtrahend
Counting Up: 11-4
4→5→6→7→8→9→10→11 (7 steps)
Counting Back: 11-4
11→10→9→8→7 (4 steps)
More Efficient
Counting back (4 steps vs 7 steps)
Step 1: For 11 - 4, compare strategies

Counting up: 4 to 11 (7 steps). Counting back: 11 to 7 (4 steps). Counting back is more efficient.

Step 2: For 13 - 6, compare strategies

Counting up: 6 to 13 (7 steps). Counting back: 13 to 7 (6 steps). Counting back is slightly more efficient.

11 - 4 = 7, 13 - 6 = 7
Final answers:

11 - 4 = 7 (counting back is more efficient), 13 - 6 = 7 (counting back is more efficient)

Applied rules:

• Direction matters: Right for counting up, left for counting back

• Jump counting: Each jump represents one unit

• Visual representation: Number lines make abstract concepts concrete

• Practice Tip: Draw number lines to visualize the subtraction process

5 Word Problems
Exercise 5
Sarah had 16 balloons. She gave 9 to her friends. How many does she have left? Tom started with 18 marbles and lost 4. How many does he have now?
Difficulty: Intermediate Time: ~6 minutes Skills: Problem Solving, Subtraction, Real-world Application
Definition:

Word problems: Real-world scenarios that require identifying subtraction and applying counting strategies to solve.

Note: Word problems help students connect mathematical concepts to everyday situations and develop problem-solving skills.

Step-by-step word problem method:
  1. Read the problem carefully
  2. Identify key information and operation needed
  3. Choose the most efficient counting strategy
  4. Apply the strategy systematically
  5. Check the answer makes sense
Sarah's Problem
16-9=7 (counting back: 16→15→14→13→12→11→10→9→8→7)
Tom's Problem
18-4=14 (counting back: 18→17→16→15→14)
Answers
7 and 14
Step 1: Sarah's problem (16 - 9)

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

Step 2: Tom's problem (18 - 4)

Start at 18, count back 4: 17, 16, 15, 14

Sarah has 7 balloons, Tom has 14 marbles
Final answers:

Sarah has 7 balloons left. Tom has 14 marbles now.

Applied rules:

• Context recognition: Giving away or losing means subtracting

• Strategy application: Use appropriate counting method

• Reasonableness check: Ensure answer fits the context

• Practice Tip: Act out problems with objects to understand the situation

Exercises 6-10 with Solutions
6 Pattern Recognition
Exercise 6
Identify the pattern and solve: 14 - 7 = ?, 17 - 8 = ?, 15 - 6 = ?, 16 - 9 = ?
Definition:

Pattern recognition: Identifying subtraction patterns and relationships in numbers to make counting more efficient.

Note: Pattern recognition helps students develop mental math skills and understand number relationships in subtraction.

Step-by-step pattern method:
  1. Identify the minuend and subtrahend
  2. Recognize the pattern of subtraction
  3. Apply the appropriate counting strategy
  4. Verify the result
Problem 1: 14-7
14→13→12→11→10→9→8→7 (7 steps back)
Problem 2: 17-8
17→16→15→14→13→12→11→10→9 (8 steps back)
Problem 3: 15-6
15→14→13→12→11→10→9 (6 steps back)
Problem 4: 16-9
16→15→14→13→12→11→10→9→8→7 (9 steps back)
Step 1: For 14 - 7, count back from 14

Pattern: 14, 13, 12, 11, 10, 9, 8, 7 (7 counts back)

Step 2: For 17 - 8, count back from 17

Pattern: 17, 16, 15, 14, 13, 12, 11, 10, 9 (8 counts back)

Step 3: For 15 - 6, count back from 15

Pattern: 15, 14, 13, 12, 11, 10, 9 (6 counts back)

Step 4: For 16 - 9, count back from 16

Pattern: 16, 15, 14, 13, 12, 11, 10, 9, 8, 7 (9 counts back)

14 - 7 = 7, 17 - 8 = 9, 15 - 6 = 9, 16 - 9 = 7
Final answers:

14 - 7 = 7, 17 - 8 = 9, 15 - 6 = 9, 16 - 9 = 7

7 Advanced Counting Up
Exercise 7
Use counting up to solve: 18 - 5 = ?, 19 - 7 = ?, 17 - 4 = ?, 20 - 6 = ?
Definition:

Advanced counting up: Applying the counting up strategy to subtraction problems with larger differences where it's more efficient.

Note: Counting up is particularly efficient when the subtrahend is much smaller than the minuend.

Step-by-step counting up method:
  1. Start with the subtrahend (smaller number)
  2. Count up to the minuend (larger number)
  3. Count the number of steps taken
  4. The number of steps is the difference
Problem 1: 18-5
5→6→7→8→9→10→11→12→13→14→15→16→17→18 (13 steps)
Problem 2: 19-7
7→8→9→10→11→12→13→14→15→16→17→18→19 (12 steps)
Problem 3: 17-4
4→5→6→7→8→9→10→11→12→13→14→15→16→17 (13 steps)
Problem 4: 20-6
6→7→8→9→10→11→12→13→14→15→16→17→18→19→20 (14 steps)
Step 1: For 18 - 5, count up from 5 to 18

Count: 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 (13 steps)

Step 2: For 19 - 7, count up from 7 to 19

Count: 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 (12 steps)

Step 3: For 17 - 4, count up from 4 to 17

Count: 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 (13 steps)

Step 4: For 20 - 6, count up from 6 to 20

Count: 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 (14 steps)

18 - 5 = 13, 19 - 7 = 12, 17 - 4 = 13, 20 - 6 = 14
Final answers:

18 - 5 = 13, 19 - 7 = 12, 17 - 4 = 13, 20 - 6 = 14

8 Advanced Counting Back
Exercise 8
Use counting back to solve: 16 - 8 = ?, 15 - 7 = ?, 14 - 6 = ?, 13 - 5 = ?
Definition:

Advanced counting back: Applying the counting back strategy to subtraction problems where it's more efficient than counting up.

Note: Counting back is particularly efficient when the subtrahend is closer to the minuend.

Step-by-step counting back method:
  1. Start with the minuend (larger number)
  2. Count backwards by the subtrahend (amount being subtracted)
  3. Keep track of each count
  4. Arrive at the difference
Problem 1: 16-8
16→15→14→13→12→11→10→9→8 (8 steps back)
Problem 2: 15-7
15→14→13→12→11→10→9→8 (7 steps back)
Problem 3: 14-6
14→13→12→11→10→9→8 (6 steps back)
Problem 4: 13-5
13→12→11→10→9→8 (5 steps back)
Step 1: For 16 - 8, count back from 16

Count: 15, 14, 13, 12, 11, 10, 9, 8

Step 2: For 15 - 7, count back from 15

Count: 14, 13, 12, 11, 10, 9, 8

Step 3: For 14 - 6, count back from 14

Count: 13, 12, 11, 10, 9, 8

Step 4: For 13 - 5, count back from 13

Count: 12, 11, 10, 9, 8

16 - 8 = 8, 15 - 7 = 8, 14 - 6 = 8, 13 - 5 = 8
Final answers:

16 - 8 = 8, 15 - 7 = 8, 14 - 6 = 8, 13 - 5 = 8

9 Combined Strategies
Exercise 9
Choose the most efficient strategy for: 17 - 9 = ?, 18 - 6 = ?, 16 - 7 = ?, 19 - 4 = ?
Definition:

Combined strategies: Selecting and applying the most efficient counting strategy (up or back) based on the numbers in the problem.

Note: Students learn to evaluate which strategy will be most efficient for each specific problem.

Step-by-step combination method:
  1. Examine the numbers in each problem
  2. Determine which strategy would be more efficient
  3. Apply the chosen strategy systematically
  4. Verify the results
Problem 1: 17-9
Counting back: 17→16→15→14→13→12→11→10→9→8 (9 steps) vs Counting up: 9→10→11→12→13→14→15→16→17 (8 steps) - Counting up is more efficient
Problem 2: 18-6
Counting back: 18→17→16→15→14→13→12 (6 steps) vs Counting up: 6→7→8→9→10→11→12→13→14→15→16→17→18 (12 steps) - Counting back is more efficient
Problem 3: 16-7
Counting back: 16→15→14→13→12→11→10→9 (7 steps) vs Counting up: 7→8→9→10→11→12→13→14→15→16 (9 steps) - Counting back is more efficient
Problem 4: 19-4
Counting back: 19→18→17→16→15 (4 steps) vs Counting up: 4→5→6→7→8→9→10→11→12→13→14→15→16→17→18→19 (15 steps) - Counting back is more efficient
Step 1: For 17 - 9, compare strategies

Counting up (8 steps) is more efficient than counting back (9 steps)

Step 2: For 18 - 6, compare strategies

Counting back (6 steps) is more efficient than counting up (12 steps)

Step 3: For 16 - 7, compare strategies

Counting back (7 steps) is more efficient than counting up (9 steps)

Step 4: For 19 - 4, compare strategies

Counting back (4 steps) is more efficient than counting up (15 steps)

17 - 9 = 8, 18 - 6 = 12, 16 - 7 = 9, 19 - 4 = 15
Final answers:

17 - 9 = 8, 18 - 6 = 12, 16 - 7 = 9, 19 - 4 = 15

10 Comprehensive Practice
Exercise 10
Comprehensive practice: 15 - 8 = ?, 19 - 5 = ?, 14 - 9 = ?, 18 - 7 = ?, 16 - 4 = ?, 17 - 8 = ?
Definition:

Comprehensive practice: A variety of subtraction problems that require efficient application of counting up or counting back strategies.

Note: These problems help students develop fluency and confidence with subtraction strategies as they approach mastery.

Step-by-step efficiency method:
  1. Quickly analyze the numbers in each problem
  2. Select the most efficient counting strategy
  3. Apply the strategy with accuracy
  4. Verify the result efficiently
Problem 1: 15-8
Counting back: 15→14→13→12→11→10→9→8→7 (8 steps) vs Counting up: 8→9→10→11→12→13→14→15 (7 steps) - Counting up is more efficient
Problem 2: 19-5
Counting back: 19→18→17→16→15→14 (5 steps) - More efficient
Problem 3: 14-9
Counting up: 9→10→11→12→13→14 (5 steps) vs Counting back: 14→13→12→11→10→9→5 (9 steps) - Counting up is more efficient
Problem 4: 18-7
Counting back: 18→17→16→15→14→13→12→11 (7 steps) - More efficient
Step 1: For 15 - 8, compare strategies

Counting up (7 steps) is more efficient than counting back (8 steps)

Step 2: For 19 - 5, counting back is efficient

Count: 18, 17, 16, 15, 14 (5 steps)

Step 3: For 14 - 9, compare strategies

Counting up (5 steps) is more efficient than counting back (9 steps)

Step 4: For 18 - 7, counting back is efficient

Count: 17, 16, 15, 14, 13, 12, 11 (7 steps)

15 - 8 = 7, 19 - 5 = 14, 14 - 9 = 5, 18 - 7 = 11, 16 - 4 = 12, 17 - 8 = 9
Final answers:

15 - 8 = 7, 19 - 5 = 14, 14 - 9 = 5, 18 - 7 = 11, 16 - 4 = 12, 17 - 8 = 9

Detailed Summary: Counting Up and Counting Back to Subtract
Core Concepts and Definitions:

Counting Up to Subtract: A strategy where you start with the smaller number (subtrahend) and count up to the larger number (minuend) to find the difference.

Counting Back to Subtract: A strategy where you start with the larger number (minuend) and count backwards by the amount being subtracted (subtrahend) 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.

Number Sense: Understanding the relationships between numbers and their quantities.

Core Rules and Principles:
  1. Counting Up Rule: Start with the subtrahend and count up to the minuend; the number of steps is the difference.
  2. Counting Back Rule: Start with the minuend and count back by the subtrahend; the final number is the difference.
  3. Strategy Selection: Choose the strategy that requires fewer steps for efficiency.
  4. Sequential Counting: Count each number in sequence without skipping.
  5. Verification: Always check your answer by using the opposite operation.
Step-by-Step Procedures: 1) Identify minuend and subtrahend, 2) Choose the most efficient strategy, 3) Apply the strategy systematically, 4) Count steps or numbers, 5) Identify the difference, 6) Verify the result.
Examples (Simple → Advanced): Simple: 10-7=3 (counting up: 7→8→9→10), Advanced: 19-4=15 (counting back: 19→18→17→16→15). Problems progress from basic to complex differences.
Tips: Use fingers or objects to visualize counting, practice both strategies, choose the one that requires fewer steps, verify answers by adding the difference back to the subtrahend.
Common Pitfalls: Confusing minuend and subtrahend, miscounting steps, choosing the less efficient strategy, forgetting to count the starting number, not verifying answers.

Key Takeaways for Memorization

  • Counting up: Start with the smaller number and count up to the larger number
  • Counting back: Start with the larger number and count back by the smaller number
  • Choose the strategy that requires fewer steps
  • Counting up is often better when the subtrahend is much smaller
  • Counting back is often better when the subtrahend is closer to the minuend
  • Always verify by adding the difference back to the subtrahend
Essential Teaching Points:

• Strategy Flexibility: 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

Counting Up and Counting Back to Subtract

Learning Objectives

  • Master counting up for subtraction
  • Develop counting back for subtraction
  • Build number sense within 20
  • Improve mathematical fluency
  • Select the most efficient strategy
  • Apply strategies to various problem types

Counting Up

Start with subtrahend and count up to minuend (number of steps = difference)

Counting Back

Start with minuend and count back by subtrahend (final number = difference)

Counting Up: 7→8→9→10 = 3 steps
Counting Back: 10→9→8→7 = 3 steps
10 - 7 = 3
Choose Efficient Strategy

Strategy Selection Tips

Counting up is efficient when the subtrahend is much smaller than the minuend. Counting back is efficient when the subtrahend is closer to the minuend.

Subtraction Strategy Steps:

1
Identify minuend and subtrahend in the problem
2
Compare the numbers to determine the most efficient strategy
3
Apply the chosen strategy systematically
4
Count steps or numbers accurately
5
Identify the difference
6
Verify by adding the difference to the subtrahend
Key Rules, Methods, and Strategies
Counting Up: a - b = Count from b to a (steps = difference)
Counting Up Strategy
Counting Back: a - b = Start at a, count back b steps (final number = difference)
Counting Back Strategy
Key definitions:

Counting up: Starting from the subtrahend and counting up to the minuend to find the difference.

Counting back: Starting from the minuend and counting down by the subtrahend to find the difference.

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

Complete methodology:
  1. Identify components: Determine the minuend and subtrahend
  2. Choose the strategy: Counting up or counting back based on efficiency
  3. Start with the appropriate number: Begin from the correct starting point
  4. Count systematically: Count each unit carefully and accurately
  5. Verify the result: Check by using the opposite operation
Tip 1: Counting up is often more efficient when the subtrahend is much smaller than the minuend.
Tip 2: Counting back is often more efficient when the subtrahend is closer to the minuend.
Tip 3: Use your fingers or draw dots to help visualize the counting process.

Tip 4: Check your answer by adding the difference back to the subtrahend to get the minuend.
Common errors: Confusing minuend and subtrahend, miscounting steps, choosing the less efficient strategy, forgetting to verify the answer.
Success strategies: Practice both strategies, analyze the numbers before choosing, use visual aids, verify answers by checking.
Rules to remember:

• Counting up: Start with subtrahend, count to minuend (steps = difference)

• Counting back: Start with minuend, count back by subtrahend (final number = difference)

• Strategy selection: Choose the method that requires fewer steps

• Count accurately: Each count represents one unit

• Verify results: Add difference to subtrahend to check

Questions & Answers

Question: I'm confused about when to count up and when to count back. How do I know which one to use?

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

  • Look at the two numbers in your subtraction problem
  • If the number you're subtracting (subtrahend) is much smaller, counting UP is usually easier
  • If the number you're subtracting is closer to the first number (minuend), counting BACK is usually easier
  • For example: 15-3 → count up from 3 to 15 (12 steps), but 15-12 → count up from 12 to 15 (3 steps)

Try both methods with small numbers to see which one takes fewer steps!

Question: My child is struggling with these subtraction strategies. How can I help them understand the difference between counting up and counting back?

Answer: Here are strategies to help your child understand:

  • Use physical objects like blocks or counters to demonstrate both methods
  • Draw number lines to visualize the movement in both directions
  • Practice with simple numbers first, then gradually increase difficulty
  • Emphasize that both methods give the same answer, but one may be more efficient
  • Encourage them to try both methods and see which feels easier for each problem

Remember, building fluency takes time and consistent practice!

Question: Why do we need two different ways to subtract? Can't we just use one way?

Answer: We use two ways because they help us solve problems more quickly! Here's why:

  • Sometimes counting up is faster (like 15-3: count up from 3 to 15)
  • Sometimes counting back is faster (like 15-12: count back from 15 by 12)
  • Having both tools helps us pick the easiest way for each problem
  • It's like having different tools in a toolbox - you pick the best one for the job

Both methods give the same answer, but one is usually easier depending on the numbers!

Math Glossary

Counting Up
A subtraction strategy where you start with the subtrahend and count up to the minuend; the number of steps is the difference.
Counting Back
A subtraction strategy where you start with the minuend and count back by the subtrahend; the final number is 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.
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 Subtraction Educational Team

Certified Elementary Mathematics Educators

Our team specializes in developing flexible subtraction strategies that build mathematical fluency through multiple approaches. All content is aligned with Common Core State Standards and designed to promote strategic thinking and mathematical reasoning through varied subtraction methods that accommodate different learning preferences.