Solved Exercises on Counting On and Counting Back in 2nd Grade

Master counting on and counting back: addition and subtraction strategies within 20 through these 5 detailed exercises with visual methods.

Solution: Exercises 1 to 3
1 Counting On Addition
Exercise 1
Solve: 8 + 4 = ? Use counting on strategy.
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12
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Definition:

Counting on: Starting from the first number and counting forward the amount of the second number.

Counting on method:
  1. Start with the first number (the larger one is easier)
  2. Count forward the amount of the second number
  3. Keep track of how many numbers you've counted
  4. The number you land on is the answer
Start
8
Count
4
Forward
9,10,11,12
Step 1: Identify the numbers

We have 8 + 4. Start at 8.

Step 2: Count forward

Count 4 numbers after 8: 9, 10, 11, 12

Step 3: Land on the answer

After counting 4 numbers from 8, we land on 12

8 + 4 = 12
Final answer:

8 + 4 = 12

Applied rules:

• Counting on: Start with the first number and count forward

• Visual tracking: Use fingers, counters, or number line to keep count

• Efficiency: Start with the larger number to minimize counting

2 Counting Back Subtraction
Exercise 2
Solve: 12 - 3 = ? Use counting back strategy.
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10
11
12
11
10
9
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Definition:

Counting back: Starting from the first number and counting backward the amount of the second number.

Start
12
Count
3
Backward
11,10,9
Step 1: Identify the numbers

We have 12 - 3. Start at 12.

Step 2: Count backward

Count 3 numbers before 12: 11, 10, 9

Step 3: Land on the answer

After counting back 3 numbers from 12, we land on 9

12 - 3 = 9
Final answer:

12 - 3 = 9

Applied rules:

• Counting back: Start with the first number and count backward

• Visual tracking: Use fingers, counters, or number line to keep count

• Direction matters: Move left on number line for subtraction

3 Mixed Strategy Application
Exercise 3
Solve: 7 + 6 = ? Which strategy should you use?
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13
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Definition:

Strategy selection: Choosing the most efficient counting method based on the numbers involved.

Problem
7 + 6
Strategy
Counting On
Answer
13
Step 1: Analyze the problem

We have 7 + 6. Both numbers are relatively close in size.

Step 2: Choose strategy

Either number could be the starting point, but we'll start with 7 and count on 6.

Step 3: Execute the strategy

Start at 7 and count forward 6: 8, 9, 10, 11, 12, 13

7 + 6 = 13
Final answer:

7 + 6 = 13. Used counting on strategy.

Applied rules:

• Efficiency rule: Start with the larger number when possible

• Strategy choice: Addition uses counting on, subtraction uses counting back

• Flexibility: Either number can be the starting point

Counting Strategies Guide
Counting On: a + b = a, a+1, a+2, ..., a+b
Addition Strategy
Counting On
Addition
Move Forward
Counting Back
Subtraction
Move Backward
Efficiency
Start Large
Minimize Steps
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10
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Solution: Exercises 4 to 5
4 Advanced Counting Back
Exercise 4
Solve: 15 - 7 = ? Use counting back strategy.
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10
9
8
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Definition:

Advanced counting back: Applying the counting back strategy with larger numbers requiring more steps.

Start
15
Count
7
Backward
14,13,12,11,10,9,8
Step 1: Identify the numbers

We have 15 - 7. Start at 15.

Step 2: Count backward

Count 7 numbers before 15: 14, 13, 12, 11, 10, 9, 8

Step 3: Land on the answer

After counting back 7 numbers from 15, we land on 8

15 - 7 = 8
Final answer:

15 - 7 = 8

Applied rules:

• Counting back: Start with the first number and count backward

• Systematic counting: Keep track of each step to avoid errors

• Verification: Check by adding the answer to the subtracted number

5 Problem Solving Application
Exercise 5
Sarah had 14 stickers. She gave 5 to her friend. How many stickers does she have now? Use counting strategy.
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Definition:

Word problem solving: Translating real-world scenarios into mathematical operations and applying counting strategies.

Initial
14
Given Away
5
Operation
Subtraction
Step 1: Understand the problem

Sarah starts with 14 stickers and gives away 5.

Step 2: Identify the operation

Giving away means subtraction: 14 - 5

Step 3: Apply counting back strategy

Start at 14 and count back 5: 13, 12, 11, 10, 9

Step 4: State the answer

Sarah has 9 stickers left.

Sarah has 9 stickers remaining
Final answer:

Sarah has 9 stickers remaining.

Applied rules:

• Problem translation: Convert words to mathematical operations

• Strategy application: Choose appropriate counting method

• Contextual answer: Include units and context in final answer

Counting Strategies Properties
a + b = Count On: a → a+1 → a+2 → ... → a+b
Counting On Formula
Key definitions:

Counting on: An addition strategy where you start with the first number and count forward the amount of the second number.

Counting back: A subtraction strategy where you start with the first number and count backward the amount of the second number.

Efficiency: The principle of starting with the larger number to minimize counting steps.

Strategy selection guide:
  1. Addition problems: Use counting on strategy
  2. Subtraction problems: Use counting back strategy
  3. Choose starting number: Pick the larger number when possible
  4. Track your steps: Keep count of how many numbers you've moved
Tip 1: Use your fingers to keep track when counting on or back.
Tip 2: Start with the larger number to make counting faster.
Tip 3: Use a number line to visualize the counting process.
Tip 4: Always double-check your answer by doing the opposite operation.
Common errors: Counting the starting number as part of the count, miscounting steps.
Key insight: Counting on and back builds mental math skills and number sense.
Important strategies:

• Counting on for addition: 5 + 3 = 5, 6, 7, 8

• Counting back for subtraction: 8 - 3 = 8, 7, 6, 5

• Start with larger number: 4 + 7 is easier as 7 + 4

• Use landmark numbers: Count to 10 first, then continue

• Verify answers: Check 8 + 4 = 12 by seeing if 12 - 4 = 8

Strategy Visualization Chart
Counting On vs Counting Back
Visual comparison of counting strategies:
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2
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10
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12
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Counting On: 6 + 4 = 10
Counting Back: 10 - 4 = 6

Observation: Addition moves right on the number line, subtraction moves left.

  • Counting on: Start at first number, move right by second number
  • Counting back: Start at first number, move left by second number
  • Both strategies build number sense and mental math skills
  • Visual representations help connect abstract concepts to concrete understanding

Questions & Answers

Question: My child sometimes counts the starting number when using the counting on strategy. How can I help them avoid this mistake?

Answer: This is a very common error! Here are strategies to address it:

  • Physical cues: Have your child touch the starting number and say it, then move to the next number to begin counting
  • Verbal strategy: Teach them to say "start at [number], then count [amount]"
  • Visual markers: Use a pointer or finger to clearly indicate when counting begins
  • Practice with small numbers: Start with +1 and +2 problems to build the habit

Example: For 5 + 3, say "Start at 5, then count 3 more: 6, 7, 8."

Question: How can I help students who are still using finger counting instead of transitioning to mental counting strategies?

Answer: Here's a gradual transition approach:

  • Bridge strategy: Allow finger counting initially but encourage students to visualize the process
  • Verbal rehearsal: Have students say the counting sequence out loud without fingers
  • Gradual reduction: Start with finger support, then gradually reduce dependence
  • Positive reinforcement: Praise efforts to use mental strategies

Remember, the transition takes time. Counting on and back strategies build naturally from finger counting.

Question: Are there any games or activities that can help practice counting on and back strategies?

Answer: Absolutely! Here are engaging activities:

  • Number line hop: Have children physically move forward/backward on a large floor number line
  • Race to 20: Roll dice and add/subtract to reach exactly 20
  • Counting songs: Use rhythmic counting songs that emphasize forward/backward patterns
  • Story problems: Create fun scenarios like "monsters eating cookies" or "birds flying away"

Make it playful and hands-on - children learn best when they're actively engaged!