How to Explain Order of Operations to Fifth Graders

"But I got a different answer!"

When two students evaluate 3 + 4 × 2 and get different answers (11 and 14), you've discovered why order of operations exists. It's the mathematical agreement that ensures everyone gets the same result.

Why Order of Operations Matters

Without agreed-upon rules, math would be chaos:

3 + 4 × 2 = ?

• If you go left to right: 3 + 4 = 7, then 7 × 2 = 14
• If you multiply first: 4 × 2 = 8, then 3 + 8 = 11

Which is right? We need a rule!

The convention: Multiplication comes before addition.
The answer: 11

This matters for:

• Calculators and computers (they need consistent rules)
• Scientific formulas
• Real-world calculations
• Preparing for algebra

PEMDAS: The Order

P - Parentheses
E - Exponents
M/D - Multiplication and Division (left to right)
A/S - Addition and Subtraction (left to right)

The Memory Trick

"Please Excuse My Dear Aunt Sally"

But here's the CRITICAL point most students miss:

M and D are EQUAL - do them left to right
A and S are EQUAL - do them left to right

It's really: P → E → (M or D, left to right) → (A or S, left to right)

Step-by-Step Examples

Example 1: Basic Order

Problem: 8 + 3 × 4 - 2

Step 1: No parentheses or exponents
Step 2: Multiplication first: 3 × 4 = 12
Step 3: Left to right: 8 + 12 - 2 = 20 - 2 = 18

8 + 3 × 4 - 2
= 8 + 12 - 2
= 20 - 2
= 18

Example 2: With Parentheses

Problem: (8 + 3) × 4 - 2

Step 1: Parentheses first: 8 + 3 = 11
Step 2: Multiplication: 11 × 4 = 44
Step 3: Subtraction: 44 - 2 = 42

(8 + 3) × 4 - 2
= 11 × 4 - 2
= 44 - 2
= 42

See how parentheses changed the answer from 18 to 42!

Example 3: With Exponents

Problem: 2 + 3² × 4

Step 1: Exponents first: 3² = 9
Step 2: Multiplication: 9 × 4 = 36
Step 3: Addition: 2 + 36 = 38

2 + 3² × 4
= 2 + 9 × 4
= 2 + 36
= 38

Example 4: The M/D Left-to-Right Rule

Problem: 24 ÷ 4 × 2

This is where students stumble! Multiplication DOESN'T come before division.

Work left to right:

24 ÷ 4 × 2
= 6 × 2
= 12

NOT: 24 ÷ 8 = 3 (wrong!)

Example 5: The A/S Left-to-Right Rule

Problem: 15 - 8 + 3

Work left to right:

15 - 8 + 3
= 7 + 3
= 10

NOT: 15 - 11 = 4 (wrong!)

Example 6: Complex Expression

Problem: 5 + 2 × (6 - 2)² ÷ 4

Step 1: Parentheses: 6 - 2 = 4

5 + 2 × 4² ÷ 4

Step 2: Exponents: 4² = 16

5 + 2 × 16 ÷ 4

Step 3: Multiplication and Division (left to right):

5 + 32 ÷ 4
5 + 8

Step 4: Addition: 5 + 8 = 13

Nested Parentheses and Brackets

When parentheses are inside other parentheses, work from the inside out:

Problem: 3 × [4 + (2 + 1)²]

Step 1: Innermost parentheses: 2 + 1 = 3

3 × [4 + 3²]

Step 2: Exponents inside brackets: 3² = 9

3 × [4 + 9]

Step 3: Brackets: 4 + 9 = 13

3 × 13

Step 4: Multiplication: 3 × 13 = 39

Why Does This Order Make Sense?

Multiplication Before Addition: The Real-World Reason

Imagine: "I have 3 apples, and I buy 4 bags with 2 apples each."

3 + 4 × 2 represents this situation:

• 4 bags × 2 apples = 8 apples bought
• 3 apples I had + 8 = 11 total

The multiplication HAS to happen first to make sense!

Exponents Before Multiplication: Powers Grow Fast

2 × 3² means "2 groups of 3²" = 2 groups of 9 = 18

If we did 2 × 3 first, we'd get 6² = 36. Different meaning!

Parentheses First: Override the Default

Parentheses let us say "no, do THIS first!"

Without them: 3 + 4 × 2 = 11
With them: (3 + 4) × 2 = 14

Hands-On Activities

The Calculator Test

Type 3 + 4 × 2 into different calculators:

• Scientific calculators: 11
• Basic calculators: 14 (they often go left to right)

Discuss: Why do they disagree? Which follows PEMDAS?

Expression Building

Give an answer. Build different expressions:

"Make 14 using 2, 3, and 4:"

• (2 + 3) × 4 - 6
• 4 × 3 + 2
• 2 × (3 + 4)

Parentheses Power

Start with: 2 + 3 × 4 = 14

"Where can you add parentheses to change the answer?"

• (2 + 3) × 4 = 20
• 2 + (3 × 4) = 14 (no change—this is already the default)

Human Calculator

Line up students as an "expression": 3 + 4 × 2

• Student holding "4 × 2" steps forward first
• Says "8"
• Then "3 + 8" happens
• Final answer: 11

Four 4s Challenge

Using exactly four 4s and any operations, make numbers 1-20:

• (4 + 4) ÷ (4 + 4) = 1
• 4 ÷ 4 + 4 - 4 = 1
• 4 × 4 ÷ (4 + 4) = 2
• (4 + 4 + 4) ÷ 4 = 3
• ... and so on!

Common Mistakes and How to Fix Them

Mistake 1: Multiplication Always Before Division

Wrong: 16 ÷ 2 × 4 = 16 ÷ 8 = 2

Right: Work left to right!

16 ÷ 2 × 4 = 8 × 4 = 32

Fix: "M and D are TWINS. They have equal power. Go left to right."

Mistake 2: Addition Always Before Subtraction

Wrong: 10 - 3 + 2 = 10 - 5 = 5

Right: Work left to right!

10 - 3 + 2 = 7 + 2 = 9

Fix: "A and S are TWINS. Equal power. Left to right."

Mistake 3: Forgetting Parentheses Change Everything

Wrong: Treating (3 + 4) × 2 the same as 3 + 4 × 2

Fix: "Parentheses are like VIPs—they get served first, no matter what's inside."

Mistake 4: Exponents Only on One Number

Wrong: 2 × 3² = 6² = 36

Right: The exponent only applies to what it's directly attached to.

2 × 3² = 2 × 9 = 18

If you want 6², you need: (2 × 3)² = 36

Mistake 5: Skipping Steps

Wrong: Trying to do too much mentally

Fix: Write each step on its own line:

5 + 2 × 3²
= 5 + 2 × 9    (exponent)
= 5 + 18       (multiplication)
= 23           (addition)

Practice Ideas for Home

Expression of the Day

Post a daily expression on the fridge. First correct answer wins!

Day 1: 4 + 6 ÷ 2
Day 2: (4 + 6) ÷ 2
Day 3: 3 × 2 + 4²

Create the Expression

"The answer is 20. Create three different expressions that equal 20."

Error Hunt

Give expressions with wrong work. Find the mistake!

12 ÷ 3 + 1 × 2
= 12 ÷ 4 × 2     ← What's wrong?
= 3 × 2
= 6

(Error: Added 3 + 1 first. Should be: 4 + 2 = 6)

Parentheses Challenge

"Make the largest possible answer using 2, 3, 4, +, ×, and one pair of parentheses."

• 2 + 3 × 4 = 14
• (2 + 3) × 4 = 20
• 2 × (3 + 4) = 14
• 2 × 3 + 4 = 10

Winner: (2 + 3) × 4 = 20

Connecting to Future Concepts

Algebraic Expressions

In algebra, 3x + 2 means "multiply 3 times x, THEN add 2"—same order of operations!

Formulas

Science formulas require order of operations:

• E = mc² (square c first, then multiply)
• A = πr² (square r first, then multiply by π)

Programming

Every programming language uses order of operations. Getting it wrong means buggy code!

Complex Calculations

"Find the average: (85 + 90 + 78 + 92) ÷ 4"

The parentheses ensure you add FIRST, then divide.

The Bottom Line

Order of operations isn't arbitrary—it's a mathematical agreement that ensures expressions have exactly one correct answer. Without PEMDAS, "3 + 4 × 2" would be ambiguous.

The key insights for fifth graders:

1. Parentheses override everything—they're the "do this first" command
2. Exponents come next—they're super-powerful operations
3. M and D are equal—work left to right
4. A and S are equal—work left to right
5. Write each step—don't try to do too much in your head

When your fifth grader can look at 5 + 2 × (3 + 1)² and calmly work through it step by step, they've mastered a skill that will serve them through algebra, science, and any field that uses mathematical formulas.