Addition | Fourth Grade

Complete Notes & Formulas

1. Add Numbers Up to Five Digits

Definition: Adding numbers up to five digits means combining numbers from 10,000 to 99,999 using column addition with regrouping.

📝 Steps for Adding Large Numbers:

Step 1: Write the numbers in columns, aligning place values (ones under ones, tens under tens, etc.)
Step 2: Start adding from the ONES place (rightmost column)
Step 3: If the sum is 10 or more, write the ones digit and carry the tens digit to the next column
Step 4: Move left to TENS, then HUNDREDS, then THOUSANDS, then TEN THOUSANDS
Step 5: Continue carrying over when needed until all columns are added

✏️ Example with Regrouping:

    ¹ ¹ ¹ ¹         (carries)
    4 5 , 6 7 8
  + 2 8 , 4 5 9
  ___________
    7 4 , 1 3 7

Step-by-step:

Ones: 8 + 9 = 17 → Write 7, carry 1
Tens: 1 + 7 + 5 = 13 → Write 3, carry 1
Hundreds: 1 + 6 + 4 = 11 → Write 1, carry 1
Thousands: 1 + 5 + 8 = 14 → Write 4, carry 1
Ten Thousands: 1 + 4 + 2 = 7 → Write 7

🔑 Key Formula:

Sum = Addend 1 + Addend 2 + ... + Addend n

2. Addition Word Problems

Definition: Real-world problems that require adding large numbers to find solutions.

📋 Steps to Solve Word Problems:

Step 1: Read the problem carefully and identify what you need to find
Step 2: Identify the numbers (addends) given in the problem
Step 3: Look for keywords: total, altogether, in all, sum, combined
Step 4: Add the numbers using column addition
Step 5: Write the answer with proper units (if applicable)

✏️ Example Problem:

Problem: A school library has 45,678 books. They received 23,459 new books as a donation. How many books does the library have now?

Solution:

Books already in library: 45,678

New books received: + 23,459

Total books = 45,678 + 23,459 = 69,137 books

💡 Common Addition Keywords:

3. Addition: Fill in Missing Digits

Definition: Finding unknown digits in addition problems by using place value and carrying logic.

🔍 Strategy to Find Missing Digits:

Step 1: Start from the ONES place (rightmost column)
Step 2: Add known digits in that column
Step 3: Determine what digit is needed to get the given sum (including carry)
Step 4: Check if a carry is needed for the next column
Step 5: Move to the next column and repeat

✏️ Example:

    3 5 □ 8
  + 2 □ 4 9
  __________
    5 8 6 7

Solution:

Ones: 8 + 9 = 17 → 7 in sum ✓ (carry 1)
Tens: □ + 4 + 1(carry) = 6 → □ = 1
Hundreds: 5 + □ = 8 → □ = 3
Thousands: 3 + 2 = 5 ✓

Answer: Missing digits are 1 and 3

💡 Key Tip:

Always check for carries! They affect the next column's calculation.

4. Properties of Addition

Definition: Mathematical rules that describe how addition behaves with different numbers.

📐 Five Properties of Addition:

1. Commutative Property

Changing the order of addends does NOT change the sum.

a + b = b + a

Example: 7 + 5 = 5 + 7 = 12

2. Associative Property

Grouping of addends does NOT change the sum.

(a + b) + c = a + (b + c)

Example: (4 + 2) + 3 = 4 + (2 + 3) = 9

3. Identity Property (Additive Identity)

Adding ZERO to any number gives the same number.

a + 0 = 0 + a = a

Example: 145 + 0 = 145

4. Closure Property

The sum of two whole numbers is ALWAYS a whole number.

If a and b are whole numbers, then a + b is also a whole number

Example: 25 + 38 = 63 (all are whole numbers)

5. Additive Inverse Property

A number plus its opposite equals ZERO.

a + (-a) = 0

Example: 8 + (-8) = 0

5. Add Three or More Numbers

Definition: Adding multiple large numbers (three or more addends) using column addition.

📝 Steps for Adding Multiple Numbers:

Step 1: Align all numbers by place value in columns
Step 2: Add all digits in the ONES column
Step 3: If sum ≥ 10, carry to next column
Step 4: Repeat for TENS, HUNDREDS, THOUSANDS, and TEN THOUSANDS
Step 5: Don't forget to add the carries!

✏️ Example with 4 Numbers:

    1 2 , 3 4 5
    2 3 , 6 7 8
    3 4 , 9 8 2
  + 1 5 , 2 0 0
  ___________
    8 6 , 2 0 5

Column-by-column:

Ones: 5 + 8 + 2 + 0 = 15 → Write 5, carry 1
Tens: 1 + 4 + 7 + 8 + 0 = 20 → Write 0, carry 2
Hundreds: 2 + 3 + 6 + 9 + 2 = 22 → Write 2, carry 2
Thousands: 2 + 2 + 3 + 4 + 5 = 16 → Write 6, carry 1
Ten Thousands: 1 + 1 + 2 + 3 + 1 = 8 → Write 8

💡 Quick Tip:

Look for pairs that make 10 to speed up addition: 7+3, 6+4, 8+2, 9+1

6. Addition Patterns Over Increasing Place Values

Definition: Recognizing patterns when adding numbers with the same digits but different place values.

🔢 Pattern Rule:

If you know a basic addition fact, you can solve larger additions with the same digits!

Simply add the same number of zeros to your answer!

✏️ Pattern Examples:

Pattern 1: Basic fact 3 + 4 = 7

3 + 4 = 7 (ones)
30 + 40 = 70 (tens)
300 + 400 = 700 (hundreds)
3,000 + 4,000 = 7,000 (thousands)
30,000 + 40,000 = 70,000 (ten thousands)

Pattern 2: Basic fact 5 + 6 = 11

5 + 6 = 11
50 + 60 = 110
500 + 600 = 1,100
5,000 + 6,000 = 11,000
50,000 + 60,000 = 110,000

📐 Formula for Pattern:

If a + b = c, then (a × 10ⁿ) + (b × 10ⁿ) = c × 10ⁿ

(where n = number of zeros added)

7. Choose Numbers with a Particular Sum

Definition: Selecting specific numbers from a given set that add up to a target sum.

🎯 Strategy to Find Numbers:

Step 1: Identify the target sum you need
Step 2: Look at all available numbers
Step 3: Start with the largest number (if target is large)
Step 4: Calculate what's still needed: Target - First Number
Step 5: Find other numbers that make up the difference

✏️ Example Problem:

Available numbers: 12,345 | 23,456 | 34,567 | 10,000 | 25,000

Target Sum: 58,023

Solution:

Try: 34,567 + 23,456 = 58,023 ✓

Answer: 34,567 and 23,456

💡 Key Tips:

Use estimation to eliminate unlikely combinations
Round numbers to nearest thousand to check quickly
Look at the ones digit first - they must match the target's ones digit

8. Estimate Sums

Definition: Finding an approximate answer by rounding numbers before adding them.

📝 Steps to Estimate Sums:

Step 1: Decide what place value to round to (nearest 10, 100, 1000, etc.)
Step 2: Round each addend to that place value
Step 3: Add the rounded numbers
Step 4: Write "≈" (approximately equals) symbol

✏️ Examples of Estimating Sums:

Example 1: Estimate to nearest 10

Add: 38 + 52

Round: 40 + 50

Estimated Sum: 90

(Actual sum = 90, perfect match!)

Example 2: Estimate to nearest 100

Add: 387 + 512

Round: 400 + 500

Estimated Sum: 900

(Actual sum = 899, very close!)

Example 3: Estimate to nearest 1,000

Add: 45,678 + 23,459

Round: 46,000 + 23,000

Estimated Sum: 69,000

(Actual sum = 69,137)

📐 Estimation Formula:

Estimated Sum ≈ Rounded Addend 1 + Rounded Addend 2 + ...

🎯 When to Use Estimation:

When you need a quick answer
To check if your exact answer is reasonable
When exact values are not needed (shopping, budgeting)
To compare large numbers quickly

9. Estimate Sums: Word Problems

Definition: Real-life problems where you estimate sums by rounding to find approximate answers.

📋 Steps for Word Problems:

Step 1: Read the problem and identify numbers to add
Step 2: Look for keywords: about, approximately, estimate, around
Step 3: Decide which place value to round to
Step 4: Round each number
Step 5: Add rounded numbers and write answer with units

✏️ Example Word Problems:

Problem 1: A cinema had 38,452 visitors in January and 41,789 visitors in February. About how many total visitors did the cinema have?

Solution:

Round to nearest thousand:

38,452 ≈ 38,000

41,789 ≈ 42,000

38,000 + 42,000 = 80,000

Answer: About 80,000 visitors

Problem 2: A school collected 12,367 bottles for recycling in March and 18,845 bottles in April. Estimate the total number of bottles collected.

Solution:

Round to nearest thousand:

12,367 ≈ 12,000

18,845 ≈ 19,000

12,000 + 19,000 = 31,000

Answer: About 31,000 bottles

💡 Estimation Keywords:

Addition Quick Reference Chart

Concept	Key Formula/Rule
Basic Addition	Sum = Addend 1 + Addend 2 + ...
Commutative Property	a + b = b + a
Associative Property	(a + b) + c = a + (b + c)
Identity Property	a + 0 = a
Pattern Rule	If 3+4=7, then 30+40=70, 300+400=700
Estimation	Round each number, then add
Regrouping/Carrying	When sum ≥ 10, carry to next place value
Word Problem Keywords	Total, Sum, Altogether, Combined, In All