Add and Subtract Within 100

Understanding Addition and Subtraction Within 100

Fluency with two-digit numbers means being able to add and subtract numbers from 0 to 100 quickly, accurately, and with understanding. This skill is essential for everyday life and forms the foundation for working with larger numbers and more complex mathematics.

What Does "Within 100" Mean?

"Within 100" means: - Numbers range from 0 to 100 - Both addends (numbers being added) are under 100 - The sum (answer) stays at or under 100 - For subtraction, start with a number 100 or less

Examples: - 45 + 32 = 77 ✓ (within 100) - 89 - 56 = 33 ✓ (within 100) - 67 + 18 = 85 ✓ (within 100)

Why This Skill Matters

Fluent addition and subtraction within 100 helps you: - Solve everyday problems - money, time, measurements - Build mental math skills - calculate without paper - Prepare for larger numbers - foundation for three-digit math - Develop number sense - understand magnitude and relationships - Make quick decisions - estimate and verify - Succeed in school - essential for all future math

Strategies for Adding Within 100

There are multiple strategies for adding two-digit numbers. Choose the one that makes most sense to you!

Strategy 1: Count On

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

Example: 47 + 5 = ? - Start at 47 - Count up 5: "48, 49, 50, 51, 52" - Answer: 52

Best for: Adding small numbers (1-10) to larger numbers

Strategy 2: Make a Ten

Break one addend to make a multiple of 10, then add the rest.

Example: 38 + 7 = ? - 38 needs 2 more to make 40 - Break 7 into 2 + 5 - 38 + 2 = 40 - 40 + 5 = 45 - Answer: 45

Visual:

38 + 7
  ↓  ↓
38 + 2 + 5
  ↓
  40 + 5 = 45

Best for: One addend is close to a multiple of 10

Strategy 3: Decompose by Place Value

Add tens and ones separately, then combine.

Example: 34 + 52 = ? - Tens: 30 + 50 = 80 - Ones: 4 + 2 = 6 - Combine: 80 + 6 = 86 - Answer: 86

Visual:

  34 = 30 + 4
+ 52 = 50 + 2
─────────────
      80 + 6 = 86

Best for: Any two-digit addition without regrouping

Strategy 4: Add Tens, Then Ones

Add the tens first, then add the ones.

Example: 45 + 23 = ? - Start with 45 - Add 20: 45 + 20 = 65 - Add 3: 65 + 3 = 68 - Answer: 68

Step-by-step:

45 + 23
  ↓
45 + 20 = 65
  ↓
65 + 3 = 68

Best for: Keeping track mentally, one step at a time

Strategy 5: Compensation (Adjust and Balance)

Round one number to make it easier, then adjust.

Example: 58 + 24 = ? - Round 58 to 60 (easier to work with) - 60 + 24 = 84 - But we added 2 extra, so subtract 2 - 84 - 2 = 82 - Answer: 82

Visual:

58 + 24
  ↓
(58+2) + 24 - 2
  ↓
60 + 24 - 2 = 82

Best for: One number is close to a friendly number

Strategy 6: Standard Algorithm (Vertical)

Stack the numbers and add column by column.

Example: 45 + 37 = ?

  45
+ 37
────
  82

Steps: 1. Add ones: 5 + 7 = 12 (write 2, carry 1) 2. Add tens: 4 + 3 + 1 = 8 3. Answer: 82

Best for: More complex problems, paper-and-pencil work

Addition with Regrouping

Regrouping (or carrying) happens when a column's sum is 10 or more.

Understanding Regrouping

When ones add up to 10 or more: - Keep the ones digit in the ones place - "Carry" the ten to the tens column

Example: 48 + 35 = ?

Step-by-step:

    ¹   (carry the 1)
  48
+ 35
────
  83

1. Ones: 8 + 5 = 13 (write 3, carry 1)
2. Tens: 4 + 3 + 1(carry) = 8
3. Answer: 83

More Regrouping Examples

Example 1: 57 + 26 = ?

    ¹
  57
+ 26
────
  83

• Ones: 7 + 6 = 13 (write 3, carry 1)
• Tens: 5 + 2 + 1 = 8
• Answer: 83

Example 2: 39 + 48 = ?

    ¹
  39
+ 48
────
  87

• Ones: 9 + 8 = 17 (write 7, carry 1)
• Tens: 3 + 4 + 1 = 8
• Answer: 87

Example 3: 76 + 18 = ?

    ¹
  76
+ 18
────
  94

• Ones: 6 + 8 = 14 (write 4, carry 1)
• Tens: 7 + 1 + 1 = 9
• Answer: 94

Strategies for Subtracting Within 100

Subtraction also has multiple strategies!

Strategy 1: Count Back

Start with the larger number and count back by the smaller amount.

Example: 52 - 4 = ? - Start at 52 - Count back 4: "51, 50, 49, 48" - Answer: 48

Best for: Subtracting small numbers (1-10)

Strategy 2: Count Up (Find the Difference)

Start at the smaller number and count up to the larger.

Example: 73 - 68 = ? - Start at 68 - Count up to 73: "69, 70, 71, 72, 73" (5 jumps) - Answer: 5

Visual on number line:

68 → 69 → 70 → 71 → 72 → 73
  +1  +1   +1   +1   +1  = 5

Best for: Numbers are close together

Strategy 3: Decompose by Place Value

Subtract tens and ones separately.

Example: 67 - 32 = ? - Tens: 60 - 30 = 30 - Ones: 7 - 2 = 5 - Combine: 30 + 5 = 35 - Answer: 35

Visual:

  67 = 60 + 7
- 32 = 30 + 2
─────────────
      30 + 5 = 35

Best for: No regrouping needed

Strategy 4: Subtract in Parts

Subtract tens first, then ones.

Example: 75 - 23 = ? - Start with 75 - Subtract 20: 75 - 20 = 55 - Subtract 3: 55 - 3 = 52 - Answer: 52

Step-by-step:

75 - 23
  ↓
75 - 20 = 55
  ↓
55 - 3 = 52

Best for: Mental math

Strategy 5: Add Up (Missing Addend)

Think of subtraction as "what do I add?"

Example: 81 - 56 = ? - Think: 56 + ? = 81 - 56 + 4 = 60 (to next ten) - 60 + 21 = 81 - Total added: 4 + 21 = 25 - Answer: 25

Visual:

56 ─→ 60 ─→ 81
   +4    +21
Total: 25

Best for: Finding differences, number sense

Strategy 6: Standard Algorithm (Vertical)

Stack the numbers and subtract column by column.

Example: 84 - 37 = ?

  84
- 37
────
  47

Steps: 1. Ones: 4 - 7 (can't do, so regroup) 2. Regroup: Borrow 1 ten → 14 - 7 = 7 3. Tens: 7 - 3 = 4 (one was borrowed) 4. Answer: 47

Best for: Paper-and-pencil work, complex problems

Subtraction with Regrouping (Borrowing)

Regrouping in subtraction happens when the top digit is smaller than the bottom digit.

Understanding Borrowing

When you can't subtract the ones: - "Borrow" 1 ten from the tens place - 1 ten = 10 ones - Add those 10 ones to the ones column

Example: 52 - 28 = ?

Step-by-step:

  ⁴₁₂   (borrowed 1 ten)
  52
- 28
────
  24

1. Ones: Can't do 2 - 8, so borrow
2. Borrow 1 ten: 5 tens → 4 tens, and 2 ones → 12 ones
3. Ones: 12 - 8 = 4
4. Tens: 4 - 2 = 2
5. Answer: 24

More Borrowing Examples

Example 1: 73 - 48 = ?

  ⁶₁₃
  73
- 48
────
  25

• Borrow: 7 tens → 6 tens, 3 ones → 13 ones
• Ones: 13 - 8 = 5
• Tens: 6 - 4 = 2
• Answer: 25

Example 2: 61 - 37 = ?

  ⁵₁₁
  61
- 37
────
  24

• Borrow: 6 tens → 5 tens, 1 one → 11 ones
• Ones: 11 - 7 = 4
• Tens: 5 - 3 = 2
• Answer: 24

Example 3: 90 - 46 = ?

  ⁸₁₀
  90
- 46
────
  44

• Borrow: 9 tens → 8 tens, 0 ones → 10 ones
• Ones: 10 - 6 = 4
• Tens: 8 - 4 = 4
• Answer: 44

Real-World Applications

Addition and subtraction within 100 appear constantly in daily life!

Money

Addition: - Item 1: $34 - Item 2: $28 - Total: $34 + $28 = $62

Subtraction: - You have: $75 - You spend: $48 - Left: $75 - $48 = $27

Time

Addition: - Movie starts: 3:15 - Movie length: 45 minutes - Ends at: 3:15 + 45 min = 4:00

Subtraction: - School ends: 3:30 - Now: 2:45 - Time left: 3:30 - 2:45 = 45 minutes

Measurements

Addition: - Walk Monday: 23 minutes - Walk Tuesday: 37 minutes - Total: 23 + 37 = 60 minutes

Subtraction: - Goal: 80 pages - Read so far: 53 pages - Left to read: 80 - 53 = 27 pages

Games and Scores

Addition: - Round 1: 45 points - Round 2: 38 points - Total score: 45 + 38 = 83 points

Subtraction: - Start: 100 points - Lost: 34 points - Current: 100 - 34 = 66 points

Problem-Solving Examples

Example Problem 1: Addition

Problem: "You have 47 baseball cards. Your friend gives you 25 more. How many do you have now?"

Solution: - Start: 47 cards - Add: 25 cards - 47 + 25 = 72 - Answer: 72 baseball cards

Example Problem 2: Subtraction

Problem: "There are 83 students on the playground. 56 go inside. How many are still outside?"

Solution: - Start: 83 students - Subtract: 56 students - 83 - 56 = 27 - Answer: 27 students

Example Problem 3: Two-Step

Problem: "You had $65. You spent $28 on a game and $15 on snacks. How much money is left?"

Solution: - Step 1: Find total spent: 28 + 15 = 43 - Step 2: Subtract from start: 65 - 43 = 22 - Answer: $22 left

Example Problem 4: Comparison

Problem: "Team A scored 54 points. Team B scored 67 points. How many more points did Team B score?"

Solution: - Team B: 67 points - Team A: 54 points - Difference: 67 - 54 = 13 - Answer: Team B scored 13 more points

Practice Activities

Activity 1: Number Line Calculations

Materials: Large number line (0-100)

Activity: - Show 45 + 32 by jumping forward - Show 73 - 28 by jumping backward - Visualize the operation!

Activity 2: Mental Math Races

Partners compete: - Teacher calls: "47 + 25" - First to answer correctly wins - Build speed and accuracy!

Activity 3: Story Problem Creation

Create your own problems: - Shopping scenarios (money) - Collection problems (cards, stamps) - Sports scores - Time calculations

Activity 4: Card Game: Make 100

Materials: Deck of cards (1-10)

Rules: 1. Draw two cards (make two-digit number) 2. Draw two more cards (another two-digit number) 3. Add them 4. Closest to 100 without going over wins!

Activity 5: Two-Step Challenge

Complex problems: 1. Start with a number 2. Add something 3. Subtract something else 4. What's left?

Example: 50 + 35 - 18 = ?

Common Mistakes and Solutions

Mistake 1: Adding when subtracting (or vice versa)

Problem: 75 - 32 = 107 (added instead)

Solution: Underline the operation sign. Check: should answer be bigger or smaller?

Mistake 2: Forgetting to regroup

Problem: 48 + 35 = 713 (wrote both digits: 7 and 13)

Solution: When ones sum is 10+, carry the ten! 48 + 35 = 83

Mistake 3: Incorrect borrowing

Problem: 73 - 48 = 35 (didn't borrow correctly)

Solution: Practice the borrowing steps. Check with addition: 48 + 35 = 83 ≠ 73

Mistake 4: Place value errors

Problem: 34 + 52 = 514 (added vertically without aligning)

Solution: Line up place values! Tens under tens, ones under ones.

Assessment Checkpoints

You've achieved fluency when you can: - ✓ Add two-digit numbers with and without regrouping - ✓ Subtract two-digit numbers with and without borrowing - ✓ Choose appropriate strategies for different problems - ✓ Solve problems mentally when possible - ✓ Explain your thinking and methods - ✓ Check answers for reasonableness - ✓ Apply to real-world situations - ✓ Solve problems accurately and with confidence

Looking Ahead

Mastering addition and subtraction within 100 prepares you for: - Three-digit addition and subtraction: Extending to hundreds - Mental math strategies: Faster calculations - Estimation: Approximate answers - Word problems: Multi-step scenarios - Fractions: Part-whole relationships - Algebra: Variables and equations

Conclusion

Fluency with addition and subtraction within 100 is a cornerstone of mathematical competence. By mastering multiple strategies—from mental methods like making tens and decomposing to formal algorithms with regrouping—you develop flexibility and confidence. Practice these skills in real-world contexts daily: calculating money, measuring time, tracking scores, and solving everyday problems. With consistent practice, these operations become automatic, freeing your mind to tackle more complex mathematics. Remember: accuracy first, then speed, and always understanding before memorization!