Year 7NumberNumber structures and operationsYear 7 Maths

Order of operations, rounding and remainders

What this lesson covers

  • Rounding, estimation, and using benchmarks support comparing numbers and checking whether findings are reasonable.
  • Division can result in a remainder expressed as a whole number, fraction, or decimal.
  • In expressions that have more than one operation, the order of operations is important.
  • A mnemonic, such as GEMA: grouped, exponents, multiplicative (× and ÷), and additive (+ and −) can be used to remember the order.
Key idea

Is 3 + 4 × 5 equal to 35 or 23?

  • Work left to right and you get 35.
  • The answer is 23.
  • And every calculator in the world agrees.
MethodWorkingGets
Left to right3 + 4 = 7, then 7 × 535
GEMA4 × 5 = 20, then 3 + 2023
Key idea

Why there has to be a rule

  • Two people, one expression, two answers.
  • That cannot be allowed to happen in maths.
  • So everybody agrees on one order.
MethodWorkingGets
Left to right3 + 4 = 7, then 7 × 535
GEMA4 × 5 = 20, then 3 + 2023
Key idea

GEMA

  • Grouped. Exponents. Multiplicative. Additive.
  • Four steps, always in that order.
  • The last two are pairs, not single operations.
Means
GGrouped: brackets first
EExponents: powers next
MMultiplicative: × and ÷, left to right
AAdditive: + and −, left to right
Key idea

G: anything in brackets

  • Brackets say "do me first, whatever I am".
  • They are how you override the rest of the rule.
  • (3 + 4) × 5 really is 35.
ExpressionAnswer
3 + 4 × 523
(3 + 4) × 535
Key idea

E: powers next

  • An exponent is repeated multiplication.
  • So it happens before ordinary multiplying.
  • 2 × 3² is 2 × 9, which is 18. Not 6² = 36.
ExpressionWorkingAnswer
2 × 3²2 × 918
(2 × 3)²6²36
Key idea

M: × and ÷ share a step

  • They do not take turns. They are one category.
  • Work through them left to right.
  • 12 ÷ 3 × 2 is 4 × 2 = 8, not 12 ÷ 6.
ExpressionWorkingAnswer
12 ÷ 3 × 2left to right8
all ÷ first12 ÷ 62 (wrong)
Key idea

A: + and − share a step too

  • Same arrangement. One category, left to right.
  • This is where the old rhymes go wrong.
  • BEDMAS puts S before A. That is not the rule.
Means
GGrouped: brackets first
EExponents: powers next
MMultiplicative: × and ÷, left to right
AAdditive: + and −, left to right
Worked example

20 − 3 × 4 + 2

  • Multiplicative first: 3 × 4 = 12.
  • Now 20 − 12 + 2, left to right.
  • 20 − 12 = 8, then 8 + 2 = 10.
StepExpressionWhy
Start20 − 3 × 4 + 2
Multiplicative20 − 12 + 23 × 4 = 12
Additive, left to right8 + 220 − 12 = 8
Answer10
Worked example

(2 + 3)² × 2

  • Grouped first: 2 + 3 = 5.
  • Exponent next: 5² = 25.
  • Then multiply: 25 × 2 = 50.
StepExpressionWhy
Start(2 + 3)² × 2
Grouped5² × 22 + 3 = 5
Exponents25 × 25² = 25
Answer50
Key idea

How would you know if you got it wrong?

  • Both wrong answers looked perfectly fine.
  • You need something to compare against.
  • That is what estimating is for.
MethodWorkingGets
Left to right3 + 4 = 7, then 7 × 535
GEMA4 × 5 = 20, then 3 + 2023
Key idea

Rounding a whole number

  • Find the column you are rounding to.
  • Look at the digit just to its right.
  • 5 or more rounds up. Less than 5 stays.
NumberTo theBecomes
47,382nearest ten47,380
47,382nearest hundred47,400
47,382nearest thousand47,000
47,382nearest ten thousand50,000
Key idea

Watch the ten thousands one

  • 47,382 to the nearest ten thousand is 50,000.
  • The 7 rounds the 4 up to 5.
  • Everything to the right becomes zero.
Watch the ten thousands one
Key idea

Rounding a decimal

  • Exactly the same rule, further right.
  • 3.267 to the nearest hundredth is 3.27.
  • To the nearest whole number it is just 3.
NumberTo theBecomes
3.267nearest hundredth3.27
3.267nearest tenth3.3
3.267nearest whole3
Worked example

Estimating 0.73 + 0.8 + 0.19

  • Round each one to the nearest tenth.
  • 0.7 + 0.8 + 0.2 is 1.7.
  • So the real answer should be near 1.7.
WorkingAnswer
Estimate0.7 + 0.8 + 0.21.7
Exact0.73 + 0.8 + 0.191.72
Close?yes, so the answer is sensible
Key idea

What an estimate catches

  • It will not catch a small slip.
  • It catches the big ones: a lost decimal point.
  • If you got 17.2, the estimate says look again.
AnswerAgainst 1.7
1.72closesensible
17.2ten times outcheck it
0.172ten times smallcheck it
Worked example

327 ÷ 5

  • Five goes into 32 six times, with 2 left.
  • Five goes into 27 five times, with 2 left.
  • 65, and 2 left over.
StepWorking
32 ÷ 56, remainder 2
carry the 227 ÷ 5
27 ÷ 55, remainder 2
So far65 remainder 2
Key idea

Three ways to say what is left

  • 65 remainder 2. Or 65 and two fifths.
  • Or 65.4, because two fifths is 0.4.
  • All the same answer. The question picks the form.
FormAnswerGood for
Whole number65 remainder 2how many buses
Fraction65 and 2/5how much pizza
Decimal65.4how many dollars
Your turn

Your turn: four questions

  • Pause here and try all four.
  • On question 1, do the exponent before anything else.
  • On question 3, estimate before you decide.
Question
1Work out 6 + 2 × 3².
2Round 58,461 to the nearest thousand.
3Estimate 2.9 × 4.1. Is 11.89 reasonable?
4Work out 419 ÷ 8 as a decimal.
Show answer

1. 24. Exponents first: 3² = 9. Then 2 × 9 = 18. Then 6 + 18 = 24. 2. 58,000, because 461 is less than 500. 3. About 3 × 4 = 12, so 11.89 is very reasonable, and it is exact. 4. 52.375. 8 × 52 = 416, remainder 3, and 3/8 = 0.375.

Worked example

The answers

  • 3² = 9 first, so 6 + 18 = 24.
  • 58,461 rounds to 58,000. And 11.89 is fine: 3 × 4 is 12.
  • 419 ÷ 8 is 52.375.
Answer
16 + 2 × 9 = 24
258,000
3estimate 12, so yes, 11.89 is reasonable
452.375
Summary

What to take away

  • GEMA: grouped, exponents, multiplicative, additive.
  • The last two are pairs worked left to right.
  • Estimate first, and let it check the answer you get.
Remember
GEMAgrouped, exponents, multiplicative, additive
× and ÷together, left to right; not all ÷ first
+ and −together, left to right
Estimate firstthen check your answer is close
Remainderswhole, fraction or decimal: the context decides

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