Order of Operations

🧮 Order of Operations

Part 1 of 5 — Why Order Matters

Topics in This Part

🔑 Key Concept: A math expression like $4 + 3 \times 2$ has only one correct answer. The order of operations is the agreed-upon set of rules that tells everyone in the world to solve it the same way.

Two People, Two Answers?

Look at this expression:

$4 + 3 \times 2$

Maya works left to right: $4 + 3 = 7$, then $7 \times 2 = 14$.

Liam multiplies first: $3 \times 2 = 6$, then $4 + 6 = 10$.

They got different answers — but a math problem can't have two right answers! So mathematicians made a rule: multiplication is done before addition.

That makes Liam correct: the answer is $\mathbf{10}$.

⚠️ Watch out: "Left to right" is not the first rule. You only work left to right within a step. The real boss is the order of operations.

The Rule: GEMDAS

Solve operations in this order — top of the list first:

You may have heard PEMDAS ("Please Excuse My Dear Aunt Sally"). It's the same rule — "P" for Parentheses is just another name for Grouping.

💡 The trick: Multiply/Divide are a team on the same level — do whichever comes first reading left to right. Add/Subtract are a team too. They are not "M before D" or "A before S."

Concept Check 🎯

Which Comes First? 🔽

For each expression, choose the operation that GEMDAS says to do first.

You've Got the Map

GEMDAS is the map for every expression you'll ever simplify:

$\textbf{G} \to \textbf{E} \to \textbf{MD} \to \textbf{AS}$

In Part 2 we'll zoom in on the trickiest part for most students: the Multiply/Divide team and the Add/Subtract team, and that little phrase "left to right."

🧮 Order of Operations

Part 2 of 5 — The Left-to-Right Teams

🔑 The Idea: Multiply and Divide share one level. Add and Subtract share one level. Inside a level, you go left to right — whichever symbol you reach first, you do first.

The Multiply / Divide Team

When an expression has both $\times$ and $\div$ with nothing higher in the way, do them left to right.

Worked Example: $20 \div 5 \times 2$

Reading left to right, division comes first:

$20 \div 5 \times 2 = 4 \times 2 = 8$

⚠️ Common mistake: Doing the multiply first gives $20 \div 10 = 2$ — wrong! Because $\times$ and $\div$ are equal, you must take them in the order they appear: division was on the left, so it goes first.

The Add / Subtract Team

Same idea for $+$ and $-$: equal level, so go left to right.

Worked Example: $15 - 8 + 3$

$15 - 8 + 3 = 7 + 3 = 10$

If you wrongly added first ($8 + 3 = 11$, then $15 - 11 = 4$), you'd get the wrong answer.

💡 Why left to right works: $15 - 8 + 3$ really means "start at 15, take away 8, then add 3." The $-$ belongs to the 8 and the $+$ belongs to the 3. Going left to right keeps each sign with its own number.

Concept Check 🎯

Putting the Two Teams Together

When an expression has multiply/divide and add/subtract, the higher team ($\times \div$) goes first — then the lower team ($+ -$).

Worked Example: $7 + 12 \div 4 - 1$

Step 1 — Multiply/Divide first: $12 \div 4 = 3$.

$7 + 3 - 1$

Step 2 — Now add/subtract, left to right: $7 + 3 = 10$, then $10 - 1 = 9$.

$7 + 12 \div 4 - 1 = 9$

Your Turn 🧮

Simplify each expression. Enter just the final number.

1) $36 \div 9 \times 3 = \,?$ 2) $20 - 6 + 5 = \,?$ 3) $10 + 8 \div 2 = \,?$

🧮 Order of Operations

Part 3 of 5 — Grouping Symbols

🔑 Why grouping wins: Parentheses are like a VIP pass — whatever is inside gets done first, no matter what operation it is. They let us override the normal order.

Parentheses Go First

Compare these two expressions — same numbers, different answers:

The parentheses force the $4 + 3$ to happen first.

Worked Example: $5 \times (9 - 6)$

Step 1 — Inside the parentheses: $9 - 6 = 3$.

$5 \times 3 = 15$

💡 Think of parentheses as a fence: solve everything inside the fence down to a single number before you do anything outside it.

Concept Check 🎯

Nested Grouping: Brackets

Sometimes a group lives inside another group. We use brackets $[\;]$ on the outside so the symbols are easier to read.

🔑 Rule: Work from the inside out — the innermost parentheses first.

Worked Example: $2 \times [ (3 + 1) \times 5 ]$

Step 1 — Innermost parentheses: $3 + 1 = 4$.

$2 \times [\, 4 \times 5 \,]$

Step 2 — Inside the brackets: $4 \times 5 = 20$.

$2 \times 20 = 40$

Order the Steps 🔽

You're simplifying $40 - [ (2 + 3) \times 4 ]$. Choose what happens at each stage.

Your Turn 🧮

Simplify each expression. Enter just the final number.

1) $(7 + 5) \div 4 = \,?$ 2) $3 \times (10 - 6) = \,?$ 3) $[ (8 - 2) \times 2 ] + 1 = \,?$

🧮 Order of Operations

Part 4 of 5 — Exponents & Full Expressions

🔑 Big Payoff: Now we add the last piece — exponents — and run complete GEMDAS problems from start to finish.

Exponents Come Right After Grouping

An exponent is a small raised number that tells you how many times to multiply the base by itself:

$3^2 = 3 \times 3 = 9 \qquad 2^3 = 2 \times 2 \times 2 = 8$

In GEMDAS, exponents are step 2 — done after grouping but before any multiply, divide, add, or subtract.

Worked Example: $5 + 2^3$

Step 1 — Exponent first: $2^3 = 8$.

$5 + 8 = 13$

⚠️ Common mistake: $2^3$ is not $2 \times 3 = 6$. It is $2 \times 2 \times 2 = 8$. The exponent counts how many copies you multiply, not what you multiply by.

Concept Check 🎯

A Full GEMDAS Problem

Let's run every step on one expression.

Worked Example: $2 \times (3 + 4)^2 - 10$

Step 1 — Grouping: $3 + 4 = 7$. $2 \times 7^2 - 10$

Step 2 — Exponent: $7^2 = 49$. $2 \times 49 - 10$

Step 3 — Multiply/Divide: $2 \times 49 = 98$. $98 - 10$

Step 4 — Add/Subtract: $98 - 10 = 88$.

$2 \times (3 + 4)^2 - 10 = 88$

✅ Check the order: G → E → MD → AS. We never skipped a level.

Walk Through It 🔽

Simplify $1 + 6 \times 2^2$. Choose what happens at each stage.

Your Turn 🧮

Simplify each expression. Enter just the final number.

1) $3^2 + 4 \times 2 = \,?$ 2) $(5 - 2)^2 + 1 = \,?$ 3) $50 - 2 \times 5^2 = \,?$

🧮 Order of Operations

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) follow GEMDAS, (2) handle the left-to-right teams, (3) work grouping symbols inside out, and (4) include exponents. Let's put it all together.

Quick Reference

⚠️ Two traps to remember: (1) $\times$ and $\div$ are equal — go left to right, don't always multiply first. (2) $2^3$ means $2 \times 2 \times 2 = 8$, not $2 \times 3$.

Mixed Practice 🎯

Mixed Drill 🧮

Simplify each expression. Enter just the final number.

1) $(4 + 6) \times 2 - 5 = \,?$ 2) $24 \div (8 - 2) + 3^2 = \,?$

Exit Quiz ✅

Answer all three to finish the lesson.