Order of Operations (PEMDAS)

🧮 Order of Operations (PEMDAS)

Part 1 of 5 — Why Order Matters

Topics in This Part

🔑 Key Concept: A single expression like $2 + 3 \times 4$ must give everyone the same answer. The order of operations is the worldwide rulebook that makes math unambiguous.

One Expression, Two Answers

Suppose you evaluate $2 + 3 \times 4$. There are two tempting paths:

These disagree! If everyone picked their own path, a calculator, a textbook, and a teacher could all report different answers for the same problem.

⚠️ The fix: Mathematicians agreed on one fixed order. Multiplication is done before addition, so $2 + 3 \times 4 = 14$ — always.

Concept Check 🎯

Reading an Expression Left to Right

When operations are of the same rank, you work strictly left to right — like reading a sentence.

$10 - 4 - 3 = (10 - 4) - 3 = 6 - 3 = 3$

Doing the $4 - 3$ first would give $10 - 1 = 9$, which is wrong. Subtraction is not done right to left.

💡 We will meet this "left-to-right tie-breaker" again for $\times$ vs. $\div$ and for $+$ vs. $-$. For now, remember: same rank ⟹ go left to right.

Left to Right 🧮

Evaluate each expression by working left to right.

1) $20 - 7 - 5 = \,?$ 2) $36 \div 6 \div 2 = \,?$ 3) $8 - 3 + 4 = \,?$

Same Rank, Left to Right 🔽

Each expression has only same-rank operations. Choose its correct value.

What You Now Know

• A bare expression must have one universal value.
• Some operations outrank others (we formalize this in Part 2).
• When operations tie in rank, evaluate left to right.

🔑 In Part 2 we name the full ranking with the memory word PEMDAS — Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

🧮 Order of Operations (PEMDAS)

Part 2 of 5 — The PEMDAS Rule

🔑 The Rule: Parentheses → Exponents → Multiplication & Division → Addition & Subtraction. The two middle pairs share a rank and are settled left to right.

The Ranking

⚠️ The #1 misconception: PEMDAS does not mean "all multiplication before all division." M and D are the same rank — whichever comes first as you read left to right is done first. The same is true for A and S.

Some people learn this as BODMAS or GEMS instead — they describe the exact same ranking, just with different words.

The MD and AS Tie-Breaker

Because multiply and divide share a rank, read left to right:

$24 \div 4 \times 2 = (24 \div 4) \times 2 = 6 \times 2 = 12$

Doing $4 \times 2 = 8$ first would give $24 \div 8 = 3$ — wrong, because $\times$ here sits to the right of $\div$.

Likewise for add and subtract:

$15 - 6 + 2 = (15 - 6) + 2 = 9 + 2 = 11$

💡 Picture M/D as one combined step and A/S as another. Inside each combined step, just sweep left to right.

Concept Check 🎯

Pick the First Step 🔽

For each expression, choose the operation that PEMDAS does first.

Apply the Ranking 🧮

Evaluate using PEMDAS (no parentheses or exponents yet).

1) $9 + 6 \div 3 = \,?$ 2) $48 \div 8 \times 3 = \,?$ 3) $10 - 2 \times 4 = \,?$

🧮 Order of Operations (PEMDAS)

Part 3 of 5 — Grouping Symbols & Nesting

🔑 The Power of Parentheses: Grouping symbols let you override the natural ranking. Anything inside is evaluated first, as a self-contained mini-problem.

Grouping Symbols Come First

Parentheses $(\;)$, brackets $[\;]$, and braces $\{\;\}$ all mean the same thing: do me first.

$2 \times (3 + 4) = 2 \times 7 = 14$

Compare that to the ungrouped version:

$2 \times 3 + 4 = 6 + 4 = 10$

The parentheses changed the answer from $10$ to $14$ by forcing the addition ahead of the multiplication.

💡 A fraction bar is a hidden grouping symbol: in $\dfrac{8 + 4}{3}$ you must add first, getting $\dfrac{12}{3} = 4$. Same idea for the contents under a $\sqrt{\;}$ sign.

Concept Check 🎯

Nested Grouping: Inside Out

When grouping symbols sit inside other grouping symbols, work from the innermost outward.

Worked Example: $3 \times [2 + (10 - 6)]$

1. Innermost parentheses: $10 - 6 = 4$
2. Now the brackets: $2 + 4 = 6$
3. Finally multiply: $3 \times 6 = 18$

$3 \times [2 + (10 - 6)] = 3 \times [2 + 4] = 3 \times 6 = 18$

⚠️ Never reach past an outer bracket to touch something outside before the inside is finished. Innermost always resolves first.

Inside Out 🔽

You are evaluating $4 + [12 \div (1 + 2)]$. Fill in each stage.

Grouping Practice 🧮

Evaluate each expression.

1) $(7 + 5) \div 4 = \,?$ 2) $2 \times (9 - 3) + 1 = \,?$ 3) $\dfrac{20 - 4}{2 \times 2} = \,?$

🧮 Order of Operations (PEMDAS)

Part 4 of 5 — Exponents & Full Expressions

🔑 Adding the E: Exponents rank just below grouping symbols — done after parentheses but before any multiply, divide, add, or subtract.

Where Exponents Fit

After grouping symbols, evaluate every exponent, then proceed to M/D and A/S.

Worked Example: $2 + 3^2 \times 4$

1. Exponent first: $3^2 = 9$
2. Multiply: $9 \times 4 = 36$
3. Add: $2 + 36 = 38$

$2 + 3^2 \times 4 = 2 + 9 \times 4 = 2 + 36 = 38$

⚠️ Watch the base! In $-2^2$ the exponent attaches only to the $2$: $-2^2 = -(2^2) = -4$. But $(-2)^2 = 4$, because the parentheses make $-2$ the base. The grouping symbol changes everything.

Concept Check 🎯

A Full PEMDAS Expression

Worked Example: $4 + 2 \times (6 - 1)^2 \div 5$

$4 + 2 \times (6 - 1)^2 \div 5 = 14$

💡 Notice steps 3–4: with $\times$ then $\div$ at the same rank, we still swept left to right.

Pick the First Step 🔽

For each expression, choose the operation PEMDAS does first.

Exponents in Context 🧮

Evaluate each expression with full PEMDAS.

1) $3^2 + 4 \times 2 = \,?$ 2) $(2 + 3)^2 - 10 = \,?$ 3) $50 - 2 \times 4^2 = \,?$

🧮 Order of Operations (PEMDAS)

Part 5 of 5 — Mixed Mastery & Exit Quiz

You can now handle (1) ranking, (2) the left-to-right tie-breaker, (3) nested grouping, and (4) exponents. Time to combine them all.

Quick Reference

⚠️ Three traps to avoid:

1. Doing addition before multiplication.
2. Forcing all $\times$ before all $\div$ (they tie — go left to right).
3. Forgetting that a fraction bar or $\sqrt{\;}$ groups what's inside it.

Spot the Error 🔽

For each expression, choose the correct value.

Mixed Mastery 🧮

Evaluate each expression completely.

1) $2 \times [3 + (4^2 - 6)] = \,?$ 2) $\dfrac{(9 - 5)^2}{2} + 7 = \,?$ 3) $100 \div (2 + 3)^2 - 1 = \,?$

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.