Year 9 maths lesson · GEMA (Order of Operations)

Order of Operations (GEMA)

Learn the correct order of operations using the GEMA rule — Groupings, Exponents, Multiplication (and division), Addition (and subtraction). Understand why multiplication must come before addition by seeing it as a “package” of repeated addition.

  • About 5–8 minutes
  • 7 quick questions as you go
  • Then about 5 minutes of practice
Practice questions → Set for Class

Worked examples

2 × 3 + √(5 + 4)

  1. G: Square-root bar is a grouping → 5 + 4 = 9
  2. E: √9 = 3
  3. M: 2 × 3 = 6
  4. A: 6 + 3 = 9

5 × 2 + 2 × 3 − 3 × 2²

  1. No groupings
  2. E: 2² = 4
  3. M (left to right): 5×2 = 10, 2×3 = 6, 3×4 = 12
  4. A (left to right): 10 + 6 − 12 = 4

√(10×2 − 2×2) − 3² × 2²

  1. G: Inside the root — M first: 10×2 = 20, 2×2 = 4, then A: 20 − 4 = 16
  2. E: √16 = 4, 3² = 9, 2² = 4
  3. M: 9 × 4 = 36
  4. A: 4 − 36 = −32

Before this lesson

It helps to be comfortable with:

Lesson transcript

Everything the lesson says, word for word

Student: Hey Alex, what are we working on today?

Teacher: Order of operations. Have a look at this one — what do you reckon?

Student: Easy — 2 plus 3 is 5, then times 4 is 20.

Teacher: Nice try, but the answer’s actually 14. Believe it or not, we don’t do it in that order.

Student: What? 14? But we just go left to right, surely?

Teacher: Fair question. The thing is, 3 times 4 is really just repeated addition — that's the whole point of multiplication. 3 times 4 means 4 plus 4 plus 4.

Teacher: So 2 plus 3 times 4 is really 2 plus 4 plus 4 plus 4. Which adds to 14.

Teacher: Multiplication is a package of repeated addition. If we had to put brackets around every package, maths would be a brackety mess. So we just do multiplication before addition — no brackets needed.

Teacher: Same idea with exponents. 2 to the power of 3 really means 2 times 2 times 2 — a package of repeated multiplication.

Teacher: So 3 times 2 cubed is really 3 times 2 times 2 times 2, which is 24. Again, we don't want to write brackets round every power — so exponents happen before multiplication.

Teacher: There's a nice way to remember all of this. It's called GEMA.

Teacher: G is for Groupings. Basically, any symbol that goes around a set of stuff makes it a group. Take a square root — it goes around and over an expression, locking it all together. Brackets work the same way: pop them around an expression and the whole thing becomes a group. Even a fraction line counts — anything sitting above or below the line gets bundled into a single group.

Teacher: E is for Exponents — which means powers, square roots, squared, cubed etc.

Teacher: M is for Multiplication — and that also covers division. They're the same operation, just backwards.

Teacher: And A is for Addition — which covers subtraction too. Same reason.

Student: G-E-M-A. Groupings, exponents, mult, add. Got it.

Teacher: Let's work through a few.

Teacher: First one. Have a think about how you would do this one.

Teacher: G first. The bar over the top of a square root is a grouping — anything under the bar, do that first. 5 plus 4 is 9.

Teacher: E next — a square root counts as an exponent. Root 9 is 3.

Teacher: M — two times three is six.

Teacher: And finally A — six plus three is nine.

Teacher: Here’s the next one. Have a think about this one.

Teacher: No groupings. E next — 2 squared is 4.

Teacher: M — three multiplications, work left to right. 5 times 2 is 10. 2 times 3 is 6. And 3 times 4 is 12.

Teacher: And A. Addition and subtraction, working left to right — 10 plus 6 minus 12 is 4.

Student: So when there’s more than one multiplication, you just go left to right?

Teacher: Last one. Have a think about this one.

Teacher: The root bar is a grouping — finish everything underneath first. And inside the group, GEMA still applies. Multiply first: 10 times 2 is 20, 2 times 2 is 4. Then subtract: 20 minus 4 is 16.

Teacher: E — three exponents to deal with. Root 16 is 4. 3 squared is 9. And 2 squared is 4.

Teacher: M — 9 times 4 is 36.

Teacher: And A — 4 minus 36 is negative 32.

Student: Wait — you did multiplication inside the root before doing the root itself?

Teacher: Yep. Grouping first means finish everything inside the group before you move on. And inside the group, GEMA still decides the order.

Teacher: Your turn. Try these.

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