Order of Operations: 3.25

Indicator of Progress
Teaching Strategies
Further Resources

Supporting materials

Indicator of Progress

Success depends on students recognising that the order in which operations are carried out can affect the answers that you get. It is not always correct to perform calculations working from left-to-right. They understand that order is particularly important when using a calculator.

Illustration 1: Correct order vs left-to-right

Students without an understanding of the importance of order of operations will consider 3 + 2 × 5 to be equivalent to (3 + 2) × 5. They expect an answer of 25 (rather than the correct answer of 13) because if they perform the operations as they come to them, they will add 3 + 2 to get 5 (i.e. do the addition before the multiplication) to give 5 × 5.

For a problem such as 3 + 2 × 5 students may show their working as 3 + 2 = 5 × 5 = 25, with incorrect use of both order of operations and the equals sign.

NOTE: If a student is using a four-function calculator or a calculator on a mobile phone, and they enter the expression as it is written, then it is likely that they will get an answer of 25 which reinforces the error.

Illustration 2

Examples of the types of tasks that would be illustrative of ‘order of operations’ concepts, aligned from the Mathematics Online Interview:

Question 22 - Mentally solve addition and subtraction using derived strategies – near doubles, commutativity, build to next ten, fact families
Question 32 - Multiplication tasks
Question 33 - Division tasks

Teaching Strategies

The tasks below illustrate the necessity of using correct order of operations in calculations. At this level, only simple combinations are treated (see note on BODMAS). The teaching strategies are to establish the need for the rules (in this case to overcome ambiguity), then to provide practice in both open and structured activities.

Activity 1: Can the answers be different? is an awareness raising activity. Students learn better when they understand the need.
Activity 2: Do brackets help?uses an open ended task for students to explore the phenomenon creatively. It caters for a range of abilities.
Activity 3: Got it! provides practice, with feedback coming when students check each others' work.
Activity 4: Fewest buttons provides a challenge for students.

Activity 1: Can the answers be different?

This activity shows that there can appear to be two different answers to the one calculation, whereas in fact only one is acceptable and that the order of operations is determined by conventions.

Ask students to input 4 + 5 × 6 = directly into a calculating device (calculator, mobile phone, etc). Compare the answers they get. Discuss the fact that different devices work in different ways; which is the correct answer (34); whether there can be more than one correct answer (mathematics needs rules so that everyone knows what is meant); and how we can avoid the ambiguity (agree on an order of operations i.e. multiplication/division before addition/subtraction).

Discuss the need to know the rules being used by their own calculating device.

Activity 2: Do brackets help?

In this activity students place brackets and operations between numbers to make expressions of different values.

Start with 1 + 2 × 3 + 4

Ensure that students understand that, as it is written without brackets, 1 + 2 × 3 + 4 = 11.

Now put in a pair of brackets somewhere eg (1 + 2) × 3 + 4 and work out the answer (13). Continue, putting in one set of brackets in different places, and seeing the many different answers that can be obtained. Display the list of expressions and answers. If necessary, set the task of putting in brackets to give a specified answer eg 15.

expression	answer	expression	answer
1 + 2 × 3 + 4	11	1 + (2 × 3) + 4	11
(1 + 2) × 3 + 4	13	1 + (2 × 3 + 4)	11
(1 + 2 × 3) + 4	11	1 + 2 × (3 + 4)	15

Next ask the students to use the numbers 3, 7 and 9, in that order, with addition and multiplication and brackets to make as many different answers as possible. Encourage the students to be systematic. Note that it is important in a task such as this to correct the students' answers carefully, because it is possible for students to use an incorrect order of operations.

Advanced students can do the same activity using any operation (+, - , ×, ÷), but this is not required at level 3.25.

Activity 3: Got it!

This activity allows students to practise the skills of using correct order of operations and encourages translation from words into mathematical symbols.

Use a pack of cards for a group of students. Remove the picture cards and tens (at least at first). Aces show 1. Shuffle and deal five to each player. Turn up one of the remaining cards as target.

Players should then try to use as many of their cards as possible, brackets and operations to make a correct expression for the target number.

Once this is done they should lay out the cards in the correct order and explain their solution (which cards go together in brackets and the missing operations). Then the solver has to write down the expression correctly. The rest of the group judge its correctness, and correct any errors.

Activity 4: Fewest buttons

In this activity students are encouraged to think about efficient ways of entering expressions into calculators, reinforcing correct order of operations. It is also a good opportunity to ensure students can use the calculator memory.

Ask the students to use their own calculator to calculate the following pressing the fewest buttons possible. Explain that you know they can do the calculations in their heads, but you want to know how they would use a calculator.

Sample calculations

10 + 2 × 5

12 × 5 - 3 × 4 + 1

1 + 2 × 3 + 4 + 5 × 6

Compare their strategies. This task helps to show that the order that calculations are done and written is important.

Increase the complexity of the expression for students who have finished.

Note to teachers on BODMAS, BOMDAS and others.

These acronyms are useful to stress the basic idea that (MD) comes before (AS). B (brackets) is first, then M or D done from left-to-right and then A or S from left-to right.

For example, 3 × 10 - 5 + 2 is equal to 27. There are no brackets (B). There is one multiplication so do that first (3 × 10 = 30). This leaves the expression 30 - 5 + 2 which has a subtraction and an addition. The subtraction is done before the addition, since you have to work from left-to-right with AS. A full treatment of BODMAS is part of working towards level 5 Standard.

Further Resources

The following resource contains sections that may be useful when designing learning experiences:

Digilearn object *

Exploring order of operations – students work through mathematical operations in an algorithm. Students perform calculations with integers where the order of operations is important. Students start with two-step operations such as 6+10x2. Students move on to numerical expressions involving several operations and notation such as parentheses and indices. Students solve numerical expressions quickly to earn a time bonus for a point-scoring game.
(https://www.eduweb.vic.gov.au/dlr/_layouts/dlr/Details.aspx?ID=5201)

* Note that Digilearn is a secure site; SMART::tests login required.

Mathematics Developmental Continuum P-10