Counting On and Counting Back Games - 2nd Grade Mathematics

Engaging games and activities to master counting strategies for addition and subtraction within 20.

Games 1-10 with Solutions
1 Number Line Hop
Game 1
Player starts at 5 on the number line. Hop forward 3 spaces. Where do they land?
Difficulty: Beginner Time: ~3 minutes Skills: Counting On, Number Line Visualization
Definition:

Number Line Hop: A game where players move forward or backward on a number line to practice counting on and counting back strategies.

Note: This game helps students visualize addition as moving forward and subtraction as moving backward on a number line.

Step-by-step gameplay:
  1. Identify the starting position (5 in this case)
  2. Determine the direction and distance to move (forward 3 spaces)
  3. Move systematically, counting each space
  4. Land on the final position
Starting Position
5
Movement
→→→
Final Position
8
Step 1: Start at position 5

Player begins at the number 5 on the number line

Step 2: Move forward 3 spaces

Count: 6 (hop 1), 7 (hop 2), 8 (hop 3)

Step 3: Identify landing position

Player lands on number 8

5 + 3 = 8
Final answer:

The player lands on 8.

Applied rules:

• Forward movement: Represents addition (counting on)

• Sequential counting: Each hop counts as one unit

• Visual tracking: See the movement on the number line

• Practice Tip: Use physical movement or manipulatives to enhance learning

2 Counting Race
Game 2
Two players start at 10. First player counts back 4. Second player counts on 2. Who reaches their destination first?
Difficulty: Beginner Time: ~4 minutes Skills: Counting Back, Counting On, Comparison
Definition:

Counting Race: A competitive game where players practice counting on and counting back simultaneously, comparing results.

Note: This game reinforces both strategies while adding an element of competition and comparison.

Step-by-step gameplay:
  1. Both players start at the same number (10)
  2. Player 1 counts back by their number (10-4)
  3. Player 2 counts on by their number (10+2)
  4. Compare the results
Player 1: 10-4
10→9→8→7→6
Player 2: 10+2
10→11→12
Results
6 and 12
Step 1: Both players start at 10

Initial position for both players: 10

Step 2: Player 1 counts back 4

Count: 9, 8, 7, 6 (lands on 6)

Step 3: Player 2 counts on 2

Count: 11, 12 (lands on 12)

Player 1: 6, Player 2: 12
Final answer:

Both players finish their counting simultaneously. Player 1 reaches 6, Player 2 reaches 12.

Applied rules:

• Simultaneous play: Both players count at the same time

• Direction matters: Back for subtraction, forward for addition

• Equal time: Both players have equal opportunity to finish

• Practice Tip: Focus on accuracy over speed initially

3 Addition Adventure
Game 3
Hero starts with 7 gold coins. They find 6 more. How many coins do they have now?
Difficulty: Intermediate Time: ~3 minutes Skills: Counting On, Word Problems, Story Context
Definition:

Addition Adventure: A story-based game where students solve addition problems within an engaging narrative context.

Note: Story contexts help students connect mathematical concepts to real-world scenarios and make learning more meaningful.

Step-by-step gameplay:
  1. Identify the starting amount (7 coins)
  2. Identify the amount gained (6 coins)
  3. Count on from the starting amount
  4. Calculate the final total
Starting Amount
7 coins
Coins Found
+6 coins
Total Coins
13 coins
Step 1: Start with 7 coins

Hero begins the adventure with 7 gold coins

Step 2: Find 6 more coins

Count on from 7: 8, 9, 10, 11, 12, 13

Step 3: Calculate total

Hero now has 13 gold coins

7 + 6 = 13
Final answer:

The hero has 13 gold coins.

Applied rules:

• Story interpretation: Identify numbers and operations from the narrative

• Counting on: Start from the initial amount and count up

• Contextual meaning: Understand what the numbers represent

• Practice Tip: Act out the story to visualize the counting process

4 Subtraction Safari
Game 4
Explorer sees 15 animals. They lose sight of 8. How many animals can they still see?
Difficulty: Intermediate Time: ~3 minutes Skills: Counting Back, Word Problems, Story Context
Definition:

Subtraction Safari: A story-based game focused on subtraction problems, helping students visualize taking away or losing quantities.

Note: Safari contexts emphasize the concept of losing, taking away, or reducing quantities in subtraction.

Step-by-step gameplay:
  1. Identify the initial quantity (15 animals)
  2. Identify the amount lost (8 animals)
  3. Count back from the initial quantity
  4. Calculate the remaining amount
Initial Animals
15 animals
Animals Lost
-8 animals
Remaining Animals
7 animals
Step 1: Start with 15 animals

Explorer initially sees 15 animals

Step 2: Lose sight of 8 animals

Count back from 15: 14, 13, 12, 11, 10, 9, 8, 7

Step 3: Calculate remaining

Explorer can still see 7 animals

15 - 8 = 7
Final answer:

The explorer can still see 7 animals.

Applied rules:

• Story interpretation: Identify quantities and operations from the safari narrative

• Counting back: Start from the initial amount and count down

• Subtraction concept: Taking away or losing reduces the total

• Practice Tip: Use physical objects to represent the animals

5 Mixed Operation Challenge
Game 5
Solve 9 + 4 and 12 - 3. Which answer is greater?
Difficulty: Intermediate Time: ~4 minutes Skills: Counting On, Counting Back, Comparison
Definition:

Mixed Operation Challenge: A game that combines both counting on and counting back strategies, requiring students to select the appropriate method for each operation.

Note: These challenges help students develop flexibility in choosing the right strategy for different operations.

Step-by-step gameplay:
  1. Solve the addition problem using counting on
  2. Solve the subtraction problem using counting back
  3. Compare the two results
  4. Identify which is greater
Addition: 9+4
9→10→11→12→13
Subtraction: 12-3
12→11→10→9
Comparison
13 > 9
Step 1: Solve 9 + 4 (counting on)

Start at 9, count up 4: 10, 11, 12, 13

Step 2: Solve 12 - 3 (counting back)

Start at 12, count back 3: 11, 10, 9

Step 3: Compare results

13 is greater than 9

9 + 4 = 13, 12 - 3 = 9, 13 > 9
Final answer:

9 + 4 = 13 is greater than 12 - 3 = 9.

Applied rules:

• Strategy selection: Choose counting on for addition, counting back for subtraction

• Operation identification: Recognize + vs - symbols

• Comparison: Use greater than/less than concepts

• Practice Tip: Practice both strategies separately before combining them

Games 6-10 with Solutions
6 Pattern Detective
Game 6
Detective needs to solve 8 + 5 and 13 - 4 to crack the case. What are the answers?
Definition:

Pattern Detective: A game that emphasizes pattern recognition and systematic counting to solve mathematical mysteries.

Note: Detective themes encourage systematic thinking and pattern recognition skills in mathematics.

Step-by-step gameplay:
  1. Identify the first problem (8 + 5)
  2. Apply counting on strategy
  3. Identify the second problem (13 - 4)
  4. Apply counting back strategy
First Case: 8+5
8→9→10→11→12→13
Second Case: 13-4
13→12→11→10→9
Case Solved
13 and 9
Step 1: Solve 8 + 5 (counting on)

Start at 8, count up 5: 9, 10, 11, 12, 13

Step 2: Solve 13 - 4 (counting back)

Start at 13, count back 4: 12, 11, 10, 9

8 + 5 = 13 and 13 - 4 = 9
Final answers:

8 + 5 = 13 and 13 - 4 = 9. Case solved!

7 Speed Counting
Game 7
Race against time! Solve 11 + 6 and 17 - 5 as quickly as possible using counting strategies.
Definition:

Speed Counting: A timed game that helps students develop fluency with counting strategies under time pressure.

Note: Speed games build automaticity and fluency with basic arithmetic operations.

Step-by-step gameplay:
  1. Quickly identify the operations needed
  2. Select the appropriate counting strategy
  3. Apply the strategy efficiently
  4. Provide answers rapidly
Addition: 11+6
11→12→13→14→15→16→17
Subtraction: 17-5
17→16→15→14→13→12
Results
17 and 12
Step 1: For 11 + 6, count on efficiently

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

Step 2: For 17 - 5, count back efficiently

Start at 17, count back 5: 16, 15, 14, 13, 12

11 + 6 = 17 and 17 - 5 = 12
Final answers:

11 + 6 = 17 and 17 - 5 = 12.

8 Memory Match
Game 8
Match the problem 6 + 7 with its solution, and 13 - 6 with its solution.
Definition:

Memory Match: A matching game that connects problems with their solutions to reinforce counting strategies.

Note: Matching games strengthen connections between problems and their solutions, building memory and recall.

Step-by-step gameplay:
  1. Solve each problem using appropriate strategy
  2. Match problem with correct solution
  3. Verify the matches
  4. Build memory connections
Problem: 6+7
6→7→8→9→10→11→12→13
Problem: 13-6
13→12→11→10→9→8→7
Matches
6+7→13, 13-6→7
Step 1: Solve 6 + 7 (counting on)

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

Step 2: Solve 13 - 6 (counting back)

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

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

Match 6 + 7 with 13, and 13 - 6 with 7.

9 Strategy Showdown
Game 9
Compete in strategy selection! Solve 14 + 5 and 19 - 7 using the most efficient counting strategies.
Definition:

Strategy Showdown: A competitive game focusing on selecting the most efficient counting strategy for each problem.

Note: Competition motivates students to choose the best approach and apply it efficiently.

Step-by-step gameplay:
  1. Analyze each problem to determine optimal strategy
  2. Apply the most efficient counting method
  3. Execute the strategy systematically
  4. Compare efficiency of approaches
Addition: 14+5
14→15→16→17→18→19
Subtraction: 19-7
19→18→17→16→15→14→13→12
Efficiency
Counting strategies
Step 1: For 14 + 5, use efficient counting on

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

Step 2: For 19 - 7, use efficient counting back

Start at 19, count back 7: 18, 17, 16, 15, 14, 13, 12

14 + 5 = 19 and 19 - 7 = 12
Final answers:

14 + 5 = 19 and 19 - 7 = 12. Efficient strategies used!

10 Ultimate Challenge
Game 10
Master the ultimate challenge! Solve 16 + 4 and 18 - 9 with precision and speed.
Definition:

Ultimate Challenge: Advanced problems that test mastery of counting strategies with larger numbers within 20.

Note: These challenges push students toward mastery and build confidence with numbers approaching 20.

Step-by-step gameplay:
  1. Apply mastered counting strategies
  2. Execute with precision and accuracy
  3. Complete with speed and confidence
  4. Demonstrate full understanding
Challenge: 16+4
16→17→18→19→20
Challenge: 18-9
18→17→16→15→14→13→12→11→10→9
Mastery
20 and 9
Step 1: For 16 + 4, use precise counting on

Start at 16, count up 4: 17, 18, 19, 20

Step 2: For 18 - 9, use precise counting back

Start at 18, count back 9: 17, 16, 15, 14, 13, 12, 11, 10, 9

16 + 4 = 20 and 18 - 9 = 9
Final answers:

16 + 4 = 20 and 18 - 9 = 9. Ultimate challenge completed!

Detailed Summary: Counting On and Counting Back Games
Core Concepts and Definitions:

Counting On: A strategy where you start with the larger number and count up by the amount of the smaller number to find the sum.

Counting Back: A strategy where you start with the larger number and count down by the amount being subtracted to find the difference.

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

Mathematical Fluency: The ability to perform calculations quickly and accurately.

Core Rules and Principles:
  1. Operation Identification: Addition uses counting on, subtraction uses counting back
  2. Starting Point: Always begin counting from the larger number in the problem
  3. Direction Matters: Forward for addition, backward for subtraction
  4. Sequential Counting: Count each number in sequence without skipping
Step-by-Step Procedures: 1) Identify operation, 2) Choose strategy, 3) Start from larger number, 4) Count systematically, 5) Identify final answer.
Examples (Simple → Advanced): Simple: 3+2=5, Advanced: 17-9=8. Games progress from basic to complex problems.
Tips: Use fingers or objects to visualize counting, start with smaller numbers, practice regularly, check answers by counting in reverse.
Common Pitfalls: Starting from the wrong number, miscounting, confusing addition with subtraction, forgetting to count the starting number.

Key Takeaways for Memorization

  • Remember: Addition = Count ON, Subtraction = Count BACK
  • Always start counting from the larger number
  • Use visual aids like number lines or fingers
  • Practice regularly to build fluency
  • Check your work by verifying with the opposite operation
Essential Teaching Points:

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

• Consistent Practice: Regular exposure builds automaticity and fluency

• Contextual Learning: Games and stories make abstract concepts concrete

• Progressive Difficulty: Start with small numbers, gradually increase complexity

Counting On and Counting Back Games

Learning Objectives

  • Master counting on for addition problems
  • Develop counting back for subtraction problems
  • Build number sense within 20
  • Improve mathematical fluency
  • Apply strategies to various contexts

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
Play to Learn

Interactive Game Elements

These games incorporate visual, auditory, and kinesthetic learning styles to reinforce counting strategies through engaging, memorable activities.

Game Success 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 and verify 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 like the games but sometimes I get confused about when to count on and when to count back. How can I remember?

Answer: Great question! Here's an easy way to remember:

  • Look for the sign: PLUS (+) means COUNT ON (like going forward)
  • MINUS (-) means COUNT BACK (like going backward)
  • Think of it like a video game: + moves you forward, - moves you backward

Practice with the games and it will become automatic! The more you play, the more natural it feels.

Question: My child enjoys the games but seems to slow down with larger numbers. How can I help them build confidence?

Answer: Here are strategies to build confidence with larger numbers:

  • Start with smaller numbers and gradually increase difficulty
  • Use visual aids like number lines or counters
  • Encourage estimation before counting to predict the answer
  • Celebrate small improvements and progress
  • Practice the same problems multiple times for familiarity

Remember, building fluency takes time and consistent practice!

Question: Why do we play games to learn math? Can't we just do regular problems?

Answer: Games make learning more effective because:

  • They make practice fun and engaging
  • They reduce anxiety about making mistakes
  • They help you remember concepts longer
  • They allow you to practice without feeling like work
  • They build confidence as you succeed in playful challenges

Games help your brain learn better while having fun!

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.
Mathematical Fluency
The ability to perform calculations quickly and accurately.
Number Sense
Understanding the relationships between numbers and their quantities.

Elementary Games Educational Team

Certified Elementary Mathematics Educators

Our team specializes in creating engaging, game-based learning experiences for elementary mathematics. All content is aligned with Common Core State Standards and designed to promote mathematical fluency through interactive, enjoyable activities that make learning memorable and effective.