Rethinking the Order of Operations (i.e. PEMDAS)

I was talking to my students the other day about the Order of Operations (i.e. PEMDAS). I quickly reviewed the Order of Operations as we worked through several algebraic expressions. Then, I asked my students why we followed the Order of Operations. They said, “It’s just what we’re supposed to follow.” I decided to investigate this further and I asked them, “But why does the Order of Operations follow the order that it does?” They didn’t know.

Why do we follow the order that makes up the Order of Operations? I asked several other people and got the same response — their teacher told them it’s what they were supposed to follow. Welcome to the era of education that focused primarily on the product. Why study the reason behind the Order of Operations, when teaching students that they’re supposed to follow it works? That is like asking, “Why teach an archer about trajectory physics, when you could just teach the archer to hit a specific target?”

So, I entertained the challenge of understanding the Order of Operations further with my students. I wrote PEMDAS vertically on my marker board and analyzed the directionality of each operation.

  • Parentheses only refers to embedded operations, so we left that aside.
  • Exponents were essentially a form of multiplication, since  x^2 is equivalent to  x \cdot x.
  • Multiplication and Division are usually computed together from left to right. I stopped my students at this point to consider the two different operations. Could multiplication be computed bidirectionally? Yes. What about division? If we think of division as multiplication, then  \frac x 2  could be written as  x \cdot \frac 1 2  (i.e. multiplying x by the multiplicative inverse of 2).
  • Addition and Subtraction are much like Multiplication and Division. While addition could be computed bidirectionally, subtraction won’t allow for it. Fortunately, we are able to write subtraction as the addition of an additive inverse. For example,  5-3  could be written as  5 + (-3).

Therefore, instead of PEMDAS, we have PMA. This simplifies things a little, but it doesn’t answer the question why we follow the order of PEMDAS (even if it’s written as PMA).

PEMDAS

 

Let’s visualize a situation where PEMDAS would be needed. In the above picture, we have four stacks of squares with nine circles in each square, a single square with nine circles in it, and four circles by themselves. If we had to put this in an expression, we could write it as  4(3^2) + 3^2 + 4. If we had to add up all of the blue circles, how would we do it? Would you want to count all the circles one-by-one? This may be a little time-consuming. What if we computed each of the squares as nine circles (from  3^2 = 9). Then, if one square equals nine circles, four squares must equal 36 circles. Thus, 36 circles + 9 circles + 4 circles. Interestingly, we just followed PEMDAS. Wait! I don’t remember bringing PEMDAS into this. It seems that the reason for PEMDAS is right in front of us.

Think about it.

Four squares of nine circles + one square of nine circles + four circles

4 ( 9 circles) + 9 circles + 4 circles

36 circles + 9 circles + 4 circles

Basically, we had to convert the original expression into an expression that allowed for addition to happen with quantities of similar terms. There was no way to add a square with four circles. Instead, we had to convert the square to nine circles to allow for addition to be possible. Ultimately, PEMDAS follows PMA, ranking the operations in priority from the embedded operation (P), to multiplication (M), and then to addition (A). If addition is the final operation performed, then all quantities must be similar in order to be added. Likewise, if multiplication is the final operation performed, then all quantities must be similar in order to be multiplied.

In summary, we follow the Order of Operations to allow for multiplication and addition with quantities of similar terms.

 

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