Order of operations inquiry
Mathematical inquiry processes: identify properties; generate more examples; extend a pattern. Conceptual field of inquiry: Order of operations.
The prompt was designed to introduce the order of operations to a year 7 class. Students knew that "you do the brackets first" and some had internalised the acronym BIDMAS (also known as PEDMAS), although without knowing what the 'I' stands for. However, none had met the idea of a hierarchy of operations in which addition and subtraction are on the lowest level with division and multiplication on a higher level. Many expressed surprise that the order of operations based on the hierarchy can contradict the order implied by BIDMAS.
The teacher can use the equations in the prompt to introduce the concept of a hierarchy. The second equation presents an opportunity to start the discussion: 4 x 3 ÷ 2 equals 6 in whichever order you carry out the calculations. That is not the case for the first equation. Calculations are carried out from left to right (division coming before multiplication) and would give a different answer if carried out the other way round.
The concept can be developed further by changing the second equation. As it stands, the equation would read 5 + 6 - 1 = 10 after multiplying and dividing. Again, the answer is the same in whichever order you carry out the operations. However, if we switch the operations to give 5 - 6 + 1, the equation equals 0 and not (-2) - that is, operations on the same level of the hierarchy are carried out from left to right, not in the order given by BIDMAS.
The inquiry starts with students noticing, questioning and wondering. If the class is inexperienced in inquiry, they may not notice the properties of the prompt. In that case, the teacher should point them out. This might lead to a structured inquiry that the teacher directs, at least in the initial stages.
The equations are made up of five consecutive numbers in descending order;
The numbers are the first five positive integers.
The four operations appear in the equations with each operation appearing once.
The first and third equations contain brackets.
The equations equal the first three multiples of five.
Why does the second equation not have brackets?
Could you use two pairs of brackets?
Could you use brackets around 4 numbers?
Can we make more multiples of five?
Can we use the first four positive integers in descending order to make multiples of four? Or the first six positive integers to create multiples of six? What operation would we exclude (or include) in these cases?
How could we include indices in the equations?
Could we change the constraints? For example, could we arrange the numbers in any order rather than consecutively?
Multiples of 5
In a short inquiry carried out by a year 9 class, students attempted to find an arrangement that gave 20. They had to use the addition sign to signify a positive integer as a way of achieving their goal.
(5 x 4)((+3) - 2) ÷ 1 = 20
5 x 4(3 - ((+2) ÷ 1) = 20
Other solutions that gave multiples of five are:
5 x (4 + 3 - 2) ÷ 1 = 25
5 x (4 + 3) ÷ (2 - 1) = 35
Other students in the class attempted to find multiples of four using the first four positive integers and any three of the four operations. These are some of their solutions, although eight could only be achieved by changing the order of the numbers: