| Why do we need it? | How to use brackets | The basic rules | The complete rules | Using calculators | Quick quiz |
We know there are 14, but how do we write this calculation? If we just write
2 + 3 x 4
how does a reader know whether the answer is
2 + 3 = 5, then multiply by 4 to get 20 or
3 x 4 = 12, then 2 + 12 to get 14?
There are two steps needed to find the answer; addition and multiplication. Without an agreed upon order of when we perform each of these operations to calculate a written expression, we could get two different answers. If we want to all get the same 'correct' answer when we only have the written expression to guide us, it is important that we all interpret the expression the same way.
division first (correct) | subtraction first | blue indicates the operations being worked on first |
15 - 10 ÷ 5 | 15 - 10 ÷ 5 | |
= 15 - 10 ÷ 5 | = 15 - 10 ÷ 5 | |
= 15 - 2 = 13 | = 5 ÷ 5 = 1 |
want division first | want subtraction first | blue indicates the operations being worked on first |
15 - (10 ÷ 5) | (15 - 10) ÷ 5 | |
= 15 - (10 ÷ 5) | = (15 - 10) ÷ 5 | |
= 15 - 2 = 13 | = 5 ÷ 5 = 1 |
If we used brackets consistently we would not have to be concerned with the order of operations. We could just work from innermost brackets outwards to eventually get our answer. However using lots of brackets can become tedious and confusing, as in the following example, so we need some agreed rules.
You can check how to work out this monster by clicking here, but the next section tells you how to avoid the worst monsters.
YOU CAN ALWAYS USE BRACKETS TO SHOW HOW
A CALCULATION SHOULD BE DONE
• | RULE 1: Calculate anything in brackets first, then apply the other rules. (For further discussion about expressions with more than one set of brackets, see the next section.) |
• | RULE 2: If a calculation involves only addition and subtraction, work from left to right. |
• | RULE 3: If a calculation involves only multiplication and division, work from left to right. |
• | RULE 4: Do multiplication and division before addition and subtraction. |
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Rule for multiple brackets: If there are brackets in the expression, calculate them first. If there is more than one set of brackets then begin with the innermost brackets and work outwards. If there is more than one set of brackets but they are isolated from each other, then do them independently. |
Working Out | Thinking |
We work on the innermost brackets first. Here there are 2 isolated sets of inner brackets. Then we move to the next level of brackets and so on. |
Did you know that in other versions of the memory aide, such as BIDMAS and BEDMAS, the 'O' has been replaced by 'I' for indicies or 'E' for exponents respectively. This is useful as it extends the mnemonic to expressions which involve squares etc. See below. |
Actual rule: Multiplication and division are inverse operations and as such need to be treated equally. When confronted with multiplication and division, always work from left to right. |
Actual rule: Addition and subtraction are inverse operations and as such need to be treated equally. When confronted with addition and subtraction, always work from left to right. |
Actual rule: Fractions should be treated as if the numerator is in brackets and the denominator is in brackets and the fraction bar (the 'vinculum') is division. |
Brackets |
multiplication and division |
1 | 5 | 7 |
1. | Using the example 10 - 1 - 2 , show why you need to follow the correct order of operations. |
2. | Calculate the following expressions: |
(a) | 11 x (3 + 2) x 4 ÷ 2 |
(b) | 7 - 18 ÷ 2 x 3 + 5 |
(c) | 42 ÷ 3 x 7 |
3. | Calculate 9 + 4 ÷ 2 x 7 - 6 ÷ 3 - 4 x 2 + 8 ÷ 2 + 3 x 3 |
4. | Using the expression in question 3, make 3 alternative expressions and answers by inserting brackets. |
5. | Find the answer to, showing the method you have used to ensure you follow the correct order of operations, e.g. checklist, colour scheme, arrows etc. |
6. | Calculate the following expressions: |
(a) | 32 ÷ 42 x (3 - 8) |
(b) | 81 ÷ (4 - 7)3 |
(c) | |
(d) | |
(e) | |
7. | Find out how to use your calculator to evaluate the expressions in question 6. |
8. | Bernie is in the process of landscaping the gardens of two new townhouses. If he buys 30 bundles of 12 wooden planks for the fence for each house and 15 bundles of 10 hardwood planks for the decking for each house, write an expression for the total number of planks bought and then work it out. If Bernie then returned 2 bundles of the wooden fence planks but bought 5 extra bundles of the hardwood planks, write a new expression and then work out the answer. |
9. | Two thirds of all Year 8 students, one quarter of all Year 9 students, only 30 Year 10 students and two fifths of Year 11 and 12 students combined ride their bike to school. If there are 99 Year 8 students, 124 Year 9 students, 111 Year 10 students, 65 Year 11 students and 50 Year 12 students attending the school, how many students ride their bike to school. Write a mathematical expression for the number of students who ride to school and then find the answer. |
10. | (300 ÷ (10 x 2)) x 4. Create an appropriate worded problem from this mathematical expression. |
3 + ((4÷2)x7) - (6÷3) - ((4x2) + ((8÷2) + (3x3))) | |
= | 3 + ((4÷2)x7) - (6÷3) - ((4x2) + ((8÷2) + (3x3))) |
3 + ((2)x7)) - (2) - ((8) + ((4) + (9)) | |
= | 3 + (14) - (2) - (8 + (13)) |
3 + (14) - (2) - (21) | |
= | 3 + 14 - 2 - 21 |
17 - 2 - 21 | |
= | 15 - 21 |
= | -6 |