Lesson 30: Order of Operations
Brackets and the Four Rules of number
Do Now
Task Sheet 30a can be used at this stage.
This Do Now will help you assess the current understanding of your students. You may decide to
encourage discussion of each step of the task, in particular in questions 3 and 4.
The intention is that students will recognise that the operations are the same but that changing
the order gives a different answer.
Coach Input
The Do Now introduced students to the fact that sometimes the order in which calculations are
carried out affects the answer. In calculations with more than one operation there is a consensus
on the order in which to complete them. This is referred to as the order of operations.
If you have access to Dienes Blocks or Multilink cubes they can be helpful with this explanation.
First let’s look at just addition, calculate:
1+3+5
3+1+5
5+1+3
You will find that regardless of the order the sum is 9. We call this property commutative. It is
not important that students recall the word, however they should understand the concept.
Next let’s consider just multiplication, calculate:
2×5×3
3×5×2
Again you will find that multiplication is commutative (the order does not affect the answer)
Independent Learning A
Task sheet 30b can be used at this stage.
Whilst this is an independent task, coaches should ask students to explain their answers, perhaps
choosing one question in each section to discuss.
Develop Learning A
Students have now investigated the commutative property and should have discovered that division is not commutative (changing the order changed the answer). The next step is to introduce
more than one operation in the same calculation. Some students will have been exposed to the
acronym BIDMAS/BODMAS for remembering the order of operations. We recommend caution if
using this as students need to understand that in this convention multiplication and division are of
equal precedence, as are addition and subtraction. Use the following example:
These calculations involve operations of different precedence:
3+5×6
6÷3-1
5×43
3+8÷2
Multiply before
Adding
Divide before
Subtracting
Multiply before
Subtracting
Divide before
Adding
5 × 6 = 30
6÷3=2
5 × 4 = 20
8÷2=4
3 + 30 = 33
21=1
20  3 = 17
3+4=7
These calculations involve operations of equal precedence, in each case work from left to right:
7+1-6
76+1
6÷3×2
6×2÷3
Work left to right
Work left to right
Work left to right
Work left to right
7+1=8
76=1
6÷3=2
6 × 2 = 12
86=2
1+1=2
2×2=4
12 ÷ 3 = 4
Talk Task
Task sheet 30c can be used at this stage.
These “Follow Me” cards form a continuous chain. Each question links to an answer on a different
piece. Some questions have more than one possible answer. Make sure you only cut the thicker
lines (there should be a question and an answer on all the pieces)
Develop Learning B
It is important to keep reinforcing the concept that multiplication and division must be done before addition and subtraction, however there is an exception. Brackets can be used to dictate
that a certain operation or set of operations is completed first. Consider the example below:
4 × (2 + 3)
Ordinarily we would Multiply first, but the Brackets dictate that we complete the operation inside
them first. The answer to the above calculation is 20.
The calculations below contain the same terms and operations but the answers change when
brackets are added.
21  5 × 2 + 6 ÷ 3
(21  5) × 2 + 6 ÷ 3
21  5 × (2 + 6 ÷ 3)
Multiply and Divide first
Contents of the Brackets first
Contents of the Brackets first
21  5 = 16
5 × 2 = 10 and 6 ÷ 3 = 2
Then Add and Subtract
21  10 + 2 = 13
Then Multiply and Divide
16 x 2 = 32 and 6 ÷ 3 = 2
2+6÷3=4
Then Multiply
5 x 4 = 20
Then Subtract
Then Add
32 + 2 = 34
21  20 = 1
Independent Learning B
Task sheet 30d can be used at this stage.
Whilst this is an independent task, coaches should ask students to explain their answers, perhaps
choosing one question in each section to discuss.
Plenary
Coaches can summarise findings or extend the strategies learnt today by exploring order of operations
further. For example students can investigate how many different answers they can create with the
same set of numbers. Task sheet 30e could also be used as a plenary if there is time left in the lesson.
The pieces link together to form a recognisable shape.
Any of the task sheets could be used for interim homework allowing students to practise their skills
and consolidate their learning. Do share any useful resources with members of the Mathematics Mastery community via the online toolkit.
Teacher notes: