Order of operations
FoundationMultiply/divide first, then add/subtract; brackets first.
After this lesson, you can:
- Order: brackets, then ×÷, then +−
- 2 + 3 × 4 = 14, never 20
- ×÷ go left to right: 18 ÷ 3 × 2 = 12
1Summary 重點摘要
Multiply/divide first, then add/subtract; brackets first.
2Vocabulary cards Vocabulary
o
order of operations運算次序
The rules for mixed calculations: brackets, then ×÷, then +−.
b
brackets括號
( ), the part calculated first.
m
multiplication / division first先乘除
× and ÷ come before + and −, worked left to right.
a
addition / subtraction last後加減
+ and − are worked last, also left to right.
m
mixed operations四則混合
One expression with both +− and ×÷, e.g. 2 + 3 × 4.
e
expression算式
A line of numbers and operation signs, e.g. 18 ÷ 3 + 2.
l
left to right由左至右
Equal-level operations go in order: in 8 ÷ 2 × 3, do 8÷2 first.
w
word problem應用題
A written problem needing a mixed expression to solve correctly.
3Step by step Steps
- Part 1: Three rules cover everythingTo work out (5 + 2) × 3, first do the brackets: 5 + 2 = 7. Then come × and ÷, and finally + and −. So 7 × 3 = 21. However long the expression is, follow the order: brackets, then multiply/divide, then add/subtract. With the order memorised, even a long mixed expression can be worked through step by step without confusion.
- Part 2: Without brackets, ×÷ firstFor 2 + 3 × 4, never just work left to right. Multiply first: 3 × 4 = 12, then add: 2 + 12 = 14. The answer is 14. Adding 2 + 3 first would give 20, which is the most common mistake of all. Multiplication and division outrank addition and subtraction in an expression — keep this rule firmly in mind.
- Part 3: Multiply and divide share a levelIn 18 ÷ 3 × 2, multiplication and division belong to the same level, so work left to right: 18 ÷ 3 = 6 first, then 6 × 2 = 12. Doing the right-hand 3 × 2 first would wrongly give 18 ÷ 6 = 3. Operations on the same level follow a fixed left-to-right order; multiplication is not 'first' just because it looks stronger.
- Part 4: Brackets change everything(2 + 3) × 4 = 5 × 4 = 20, but 2 + 3 × 4 = 14. The digits are exactly the same; only the bracket position differs, yet the answers are far apart. Brackets mean do this part first, so never leave them out when copying a question. Conversely, when you want to change the order on purpose, brackets are how you say so.
- Part 5: Multi-step expressionsFor 5 + 6 × 2 − 8 ÷ 4, do the ×÷ first: 6 × 2 = 12 and 8 ÷ 4 = 2, leaving 5 + 12 − 2. Then add and subtract from the left: 5 + 12 = 17, then 17 − 2 = 15. Writing every step clearly keeps mistakes away. Handle one operation per step and jot the middle results, and checking afterwards becomes easy.
- Part 6: Writing expressions for stories3 pens at 4 dollars each, paid for with 20 dollars: how much change? Write 20 − 3 × 4. First 3 × 4 = 12, then 20 − 12 = 8, so the change is 8 dollars. Think about the order of operations before you start calculating. Writing 'whole minus the sum of the parts' as one expression is both tidy and safe from missed steps.
4Diagrams Diagrams
5Worked example Example
Example: work out (18 − 6) ÷ 4 + 2 × 5.
- Brackets first: (18 − 6) = 12; brackets always come before everything.
- The expression becomes 12 ÷ 4 + 2 × 5, ready for the ×÷ level.
- ×÷ left to right: 12 ÷ 4 = 3 and 2 × 5 = 10, both finished.
- + and − last: 3 + 10 = 13, and the whole expression is done.
- Answer: Answer: (18 − 6) ÷ 4 + 2 × 5 = 13. The order is brackets first, then multiply and divide, then add and subtract; within one level, work from left to right, one step at a time.
6Common mistakes Mistakes
✗ Doing 2 + 3 × 4 left to right = 20
✓ ×÷ first: 2 + 12 = 14
✗ Doing 3 × 2 first in 18 ÷ 3 × 2
✓ Same level, left to right: 18 ÷ 3 = 6, then 6 × 2 = 12
✗ Ignoring brackets and multiplying first
✓ Brackets always win: in (2+3) × 4, add first
✗ Doing 10 − 4 + 2 as 10 − 6 = 4
✓ +− left to right: 10 − 4 = 6, then 6 + 2 = 8
7Quick check Quick check
Q110 − 2 × 3 = ?
Answer: 4 (do 2×3 = 6 first)
Did you get it right?
Q2(2 + 3) × 4 = ?
Answer: 20
Did you get it right?
Q316 ÷ 4 × 2 = ?
Answer: 8 (left to right)
Did you get it right?
Q45 + 2 × 6 = ?
Answer: 17
Did you get it right?
Q5(9 − 5) ÷ 2 + 1 = ?
Answer: 3 (9−5=4, then 4÷2=2, then +1)
Did you get it right?
8Recap Recap
- Order: brackets, then ×÷, then +−
- 2 + 3 × 4 = 14, never 20
- ×÷ go left to right: 18 ÷ 3 × 2 = 12
- Moving brackets changes the answer
- Word-problem expressions follow the same order