Dec 2, 2022 | Red Deer

Why do we need to know the Order of Operations in math? The answer is similar to why we need to know the rules in grammar or traffic regulations; if you don’t comply with the rules, it would create confusion or accidents!

Think about saying “let’s eat, dad!” versus “let’s eat dad!” or when you make a right turn on a red light, you must yield the right-of-way to the vehicles traveling with the green light.

Take a basic operation here: 5 + 3 x 2. The very common and “logical” steps when a student hasn’t learned or gotten familiar with Order of Operations, they would answer 16 and not 11.



The Origin of Order of Operations

The concept of order of operations existed when symbolic algebra was developed (Quora forum by David Joyce, Ph.D.).

During the beginning of symbolic algebra in the 1500s, plus and minus were written as p and m, and the symbol + and – were used later on. Multiplication was shown by putting things next to each other i.e. juxtaposition, and division was represented as fractions, similar to now: the numerator as the top number and the denominator as the bottom. 

So multiplication was visually closer to something like 5a + 4;  5 is multiplied by a first and then 4 is added. If you meant to add 4 first, then you’d group a + 4 first before multiplying it by 5. A symbol of using a line over a+4 was used, but eventually, parentheses or brackets became the symbol of grouping subexpressions. So the operation would look like this: 5 x (a+4).

By the way, in the U.S., where the term PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) is commonly used, some prefer to change the mnemonic to “GEMDAS”. In this case, the G stands for Groupings, and this is because the upper levels of math include more than just parentheses. But whether the acronym is GEMDAS, PEMDAS, BEDMAS, BODMAS, the concept is the same: grouping first, followed by exponents, then multiplication/division (left to right), addition/subtraction (left to right). 


Why Do Some Kids Have Problems with Order of Operations?

Here are some of our experiences with students who find the Order of Operations challenging. By knowing what type of problem they have, we (parents, teachers) can identify what kind of individual help we need to provide for them.

  1. Lack of basic number facts. They have no problems with ordering the operations, they know which ones come first and which comes after that, but they lack basic number facts. It is hard for them to solve what is 49 : 7. The solution for this type of student is obvious – they need to strengthen their basic number facts first before they can do the operation efficiently.
  2. Lack of concern for order or detail. They are by nature disorganized; they may be good at basic number facts, but for some reason, things are jumbled in their head. Recommended solution: ask them to do it line by line, operation by operation (see below, right column). Be patient, don't rush. Even one of our best students often makes (careless) mistakes when doing things like the one on the left. 



Doing it line by line is also a good habit and really helps when the operation becomes more complicated in higher math:

Just beware when using a calculator. A simpler calculator will perform from left to right, and a more sophisticated one would give you the way the operation is intended to be.


Why Order of Operations is Essential?

For word problems, we can use our logical thinking to solve which operation comes first. For example, if I want to hang out with 5 friends and buy us all 2 burgers each, then I need to buy (1 + 5) x 2 = 12 burgers, not 11. But when creating formulas like in computer programming or in an excel spreadsheet, you need to be aware of the order of operations, otherwise you would have to clutter formulas and computer programs with lots of brackets to avoid ambiguity.

So, there you go. There is no ambiguity in math. Remember the simple order of operation problem that went viral in the internet some years ago: what is 8 : 2 (2 + 2) = ? Some answer 16, some 1. But there should have never really been a debate in the first place; this is math –  not social science, not politics. The real reason this simple math operation broke the internet stems from the practice of omitting the multiplication symbol, which was inappropriately brought to arithmetic from algebra. Do not omit multiplication symbol! (The Conversation: The simple reason a viral math equation stumped the internet). Had the problem been correctly presented as 8 : 2 x (2 + 2) = ? there would be no heated debate. With that, the debate over 1 versus 16 is now over. The answer is 16. Case closed.