Counting On and Counting Back Activities - 2nd Grade Mathematics

Master counting strategies for addition and subtraction within 20 through visual learning and hands-on practice.

Exercises 1-10 with Solutions
1 Counting On Strategy
Exercise 1
Solve 5 + 3 using the counting on strategy.
Difficulty: Beginner Time: ~2 minutes Skills: Counting Forward, Addition
Definition:

Counting on: Starting from the larger number and counting up by the amount of the smaller number to find the sum.

Note: This strategy helps children understand addition as moving forward on a number line and builds number sense.

Step-by-step counting on method:
  1. Start with the larger number (5 in this case)
  2. Count up by the smaller number (count up 3 times)
  3. Each count represents adding one more to the total
Starting Number
5
Count Up
6, 7, 8
Final Answer
8
Step 1: Start with the larger number

Begin counting from 5 (the first addend)

Step 2: Count up by the second number

Count: 6 (that's 1), 7 (that's 2), 8 (that's 3)

Step 3: Identify the final number

The last number counted is the sum: 8

5 + 3 = 8
Final answer:

5 + 3 = 8

Applied rules:

• Counting on: Always start from the larger number

• Sequential counting: Count each number in sequence

• One-to-one correspondence: Each count adds one to the total

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

2 Counting Back Strategy
Exercise 2
Solve 8 - 3 using the counting back strategy.
Difficulty: Beginner Time: ~2 minutes Skills: Counting Backward, Subtraction
Definition:

Counting back: Starting from the larger number and counting backwards by the amount being subtracted to find the difference.

Note: This strategy helps children understand subtraction as moving backward on a number line and develops number sense.

Step-by-step counting back method:
  1. Start with the larger number (8 in this case)
  2. Count backwards by the number being subtracted (count back 3 times)
  3. Each count represents subtracting one from the total
Starting Number
8
Count Back
7, 6, 5
Final Answer
5
Step 1: Start with the larger number

Begin counting from 8 (the minuend)

Step 2: Count backwards by the subtrahend

Count: 7 (that's 1), 6 (that's 2), 5 (that's 3)

Step 3: Identify the final number

The last number counted is the difference: 5

8 - 3 = 5
Final answer:

8 - 3 = 5

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 or objects to visualize the counting back process

3 Mixed Counting Strategies
Exercise 3
Solve 7 + 4 and 12 - 5 using appropriate counting strategies.
Difficulty: Intermediate Time: ~4 minutes Skills: Counting On, Counting Back, Strategy Selection
Definition:

Mixed counting strategies: Using counting on for addition and counting back for subtraction based on the operation needed.

Note: Students learn to identify which strategy to use based on whether they're adding or subtracting.

Step-by-step mixed strategy method:
  1. Identify the operation (addition or subtraction)
  2. Choose the appropriate strategy (counting on or counting back)
  3. Apply the strategy systematically
Addition: 7+4
7→8→9→10→11
Subtraction: 12-5
12→11→10→9→8→7
Results
11 and 7
Step 1: Solve 7 + 4 (counting on)

Start at 7, count up 4: 8, 9, 10, 11

Step 2: Solve 12 - 5 (counting back)

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

7 + 4 = 11 and 12 - 5 = 7
Final answers:

7 + 4 = 11 and 12 - 5 = 7

Applied rules:

• Operation identification: Addition uses counting on, subtraction uses counting back

• Strategy selection: Choose method based on operation

• Systematic approach: Apply strategy consistently

• Practice Tip: Practice both strategies separately before combining them

4 Number Line Practice
Exercise 4
Use a number line to solve 6 + 3 and 9 - 2.
Difficulty: Beginner Time: ~3 minutes Skills: Number Line Visualization, Counting Strategies
Definition:

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

Note: Number lines provide a concrete visual representation of counting 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 starting number
  3. Move right for addition, left for subtraction
  4. Count the jumps made
Addition: 6+3
→→→
Subtraction: 9-2
←←
Results
9 and 7
Step 1: For 6 + 3, start at 6 and move right 3 spaces

6 → 7 → 8 → 9 (3 jumps to the right)

Step 2: For 9 - 2, start at 9 and move left 2 spaces

9 ← 8 ← 7 (2 jumps to the left)

6 + 3 = 9 and 9 - 2 = 7
Final answers:

6 + 3 = 9 and 9 - 2 = 7

Applied rules:

• Direction matters: Right for addition, left for subtraction

• Jump counting: Each jump represents one unit

• Visual representation: Number lines make abstract concepts concrete

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

5 Word Problems
Exercise 5
Sarah has 10 apples. She gets 6 more. How many does she have now? Tom had 15 cookies and ate 4. How many are left?
Difficulty: Intermediate Time: ~4 minutes Skills: Problem Solving, Counting Strategies, Real-world Application
Definition:

Word problems: Real-world scenarios that require identifying operations 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. Apply appropriate counting strategy
  4. Check the answer makes sense
Sarah's Problem
10+6=16
Tom's Problem
15-4=11
Answers
16 and 11
Step 1: Sarah's problem (10 + 6, counting on)

Start at 10, count up 6: 11, 12, 13, 14, 15, 16

Step 2: Tom's problem (15 - 4, counting back)

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

Sarah has 16 apples, Tom has 11 cookies left
Final answers:

Sarah has 16 apples. Tom has 11 cookies left.

Applied rules:

• Context recognition: Getting more means adding, taking away 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 Visual Counting Patterns
Exercise 6
Solve 8 + 5 and 13 - 6 by recognizing counting patterns.
Definition:

Counting patterns: Recognizing sequences and relationships in numbers to make counting more efficient.

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

Step-by-step pattern method:
  1. Identify the starting number
  2. Recognize the pattern of counting
  3. Apply the pattern systematically
  4. Verify the result
Addition: 8+5
8→9→10→11→12→13
Subtraction: 13-6
13→12→11→10→9→8→7
Results
13 and 7
Step 1: For 8 + 5, count on from 8

Pattern: 8, 9, 10, 11, 12, 13 (5 counts)

Step 2: For 13 - 6, count back from 13

Pattern: 13, 12, 11, 10, 9, 8, 7 (6 counts)

8 + 5 = 13 and 13 - 6 = 7
Final answers:

8 + 5 = 13 and 13 - 6 = 7

7 Counting Back from Larger Numbers
Exercise 7
Solve 16 - 7 using the counting back strategy.
Definition:

Counting back from larger numbers: Applying the counting back strategy to subtraction problems with larger minuends.

Note: This helps students work with numbers beyond basic facts and develop fluency with subtraction within 20.

Step-by-step counting back method:
  1. Start with the larger number (minuend)
  2. Count backwards by the subtrahend
  3. Keep track of each count
  4. Arrive at the difference
Starting Number
16
Count Back
15, 14, 13, 12, 11, 10, 9
Final Answer
9
Step 1: Start at 16 (the minuend)

Begin the counting process from 16

Step 2: Count back 7 numbers

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

Step 3: Identify the result

The number reached after 7 counts back is 9

16 - 7 = 9
Final answer:

16 - 7 = 9

8 Counting On with Larger Addends
Exercise 8
Solve 9 + 8 using the counting on strategy.
Definition:

Counting on with larger addends: Applying the counting on strategy to addition problems with larger numbers to build fluency.

Note: This helps students work with numbers beyond basic facts and develop fluency with addition within 20.

Step-by-step counting on method:
  1. Start with the larger addend
  2. Count up by the smaller addend
  3. Track each count carefully
  4. Arrive at the sum
Starting Number
9
Count Up
10, 11, 12, 13, 14, 15, 16, 17
Final Answer
17
Step 1: Start at 9 (the first addend)

Begin counting from 9

Step 2: Count up 8 numbers

Count: 10, 11, 12, 13, 14, 15, 16, 17

Step 3: Identify the result

The number reached after 8 counts up is 17

9 + 8 = 17
Final answer:

9 + 8 = 17

9 Combined Strategies
Exercise 9
Solve 11 + 6 and 18 - 5 using the appropriate counting strategies.
Definition:

Combined strategies: Selecting and applying the appropriate counting strategy (on or back) based on the operation.

Note: Students learn to identify which operation is needed and apply the corresponding strategy automatically.

Step-by-step combination method:
  1. Identify the operation in each problem
  2. Select the appropriate strategy
  3. Apply the strategy systematically
  4. Verify the results
Addition: 11+6
11→12→13→14→15→16→17
Subtraction: 18-5
18→17→16→15→14→13
Results
17 and 13
Step 1: For 11 + 6, use counting on

Start at 11, count up 6: 12, 13, 14, 15, 16, 17

Step 2: For 18 - 5, use counting back

Start at 18, count back 5: 17, 16, 15, 14, 13

11 + 6 = 17 and 18 - 5 = 13
Final answers:

11 + 6 = 17 and 18 - 5 = 13

10 Advanced Counting Problems
Exercise 10
Solve 14 + 5 and 19 - 8 using efficient counting strategies.
Definition:

Advanced counting problems: More challenging problems within 20 that require efficient application of counting strategies.

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

Step-by-step efficiency method:
  1. Quickly identify the operation needed
  2. Select the most efficient counting strategy
  3. Apply the strategy with accuracy
  4. Verify the result efficiently
Addition: 14+5
14→15→16→17→18→19
Subtraction: 19-8
19→18→17→16→15→14→13→12→11
Results
19 and 11
Step 1: For 14 + 5, count on efficiently

Start at 14, count up 5: 15, 16, 17, 18, 19

Step 2: For 19 - 8, count back efficiently

Start at 19, count back 8: 18, 17, 16, 15, 14, 13, 12, 11

14 + 5 = 19 and 19 - 8 = 11
Final answers:

14 + 5 = 19 and 19 - 8 = 11

Counting On and Counting Back Strategies

Learning Objectives

  • Understand when to use counting on for addition
  • Master counting back for subtraction
  • Develop number sense within 20
  • Build mental math fluency
  • Apply strategies to real-world problems

Counting On

Start with the larger number and count up by the amount of the smaller number

Counting Back

Start with the larger number and count down by the amount being subtracted

Addition Uses Counting On
Subtraction Uses Counting Back
Start with Bigger Number
Practice Makes Perfect
5
6
7
8
9
10

Key Strategies

Always start counting from the larger number, use your fingers or objects to visualize the process, and practice regularly to build fluency with numbers within 20.

Counting Process Steps:

1
Identify the operation (addition or subtraction)
2
Choose the appropriate strategy (counting on or back)
3
Start with the larger number
4
Count by the appropriate amount
5
Identify the final number as your answer
Key Rules, Methods, and Strategies
Counting On: a + b = Start at a, count up b times
Counting On Strategy
Key definitions:

Counting on: Starting from one addend and counting up by the value of the other addend to find the sum.

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

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

Complete methodology:
  1. Identify the operation: Determine if it's addition or subtraction
  2. Choose the strategy: Counting on for addition, counting back for subtraction
  3. Start with the larger number: Begin counting from the appropriate number
  4. Count systematically: Count each unit carefully and accurately
Tip 1: Always start counting from the larger number to make the process more efficient.
Tip 2: Use your fingers or draw dots to help visualize the counting process.
Tip 3: Practice counting forward and backward regularly to build fluency.
Tip 4: Check your answer by counting in the opposite direction to verify.
Common errors: Starting from the wrong number, miscounting, confusing addition with subtraction, forgetting to count the starting number.
Success strategies: Practice regularly, use visual aids, verbalize the counting process, verify answers by checking.
Rules to remember:

• Addition: Count ON from the larger number

• Subtraction: Count BACK from the larger number

• Always start: With the larger number in the problem

• Count accurately: Each count represents one unit

• Verify results: By checking your answer

Questions & Answers

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

Answer: Great question! The key is to look at the operation symbol:

  • If you see the PLUS sign (+), you COUNT ON (move forward)
  • If you see the MINUS sign (-), you COUNT BACK (move backward)

Think of it like walking: if you're adding, you walk forward (+), and if you're subtracting, you walk backward (-). Always start counting from the larger number in the problem!

Question: My child is struggling with counting back. What can I do to help them understand this concept better?

Answer: Here are effective strategies to help your child with counting back:

  • Create a physical number line with objects they can move along
  • Use their fingers to count backwards from the starting number
  • Practice with real-life examples: "You had 10 cookies and ate 3, how many are left?"
  • Encourage them to say the numbers aloud as they count back

Remember to start with smaller numbers and gradually increase difficulty as they gain confidence.

Question: Why do we start counting from the bigger number instead of the smaller one?

Answer: Starting from the bigger number makes counting more efficient! Here's why:

  • It reduces the number of counts needed to reach your answer
  • For example: 5 + 3 is easier when you start at 5 and count 3 more: 6, 7, 8
  • Rather than starting at 3 and counting 5 more: 4, 5, 6, 7, 8
  • Both give the same answer, but starting with the larger number is faster

This strategy saves time and helps prevent miscounting!

Math Glossary

Counting On
A strategy for addition where you start with the larger number and count up by the amount of the smaller number.
Counting Back
A strategy for subtraction where you start with the larger number and count down by the amount being subtracted.
Addend
A number that is added to another number in an addition problem.
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.

Elementary Math Educational Team

Certified Elementary Mathematics Educators

Our team of experienced elementary school teachers specializes in developing age-appropriate math resources that build foundational skills. All content is aligned with Common Core State Standards and designed to support 2nd grade mathematical development through engaging, visual learning approaches.