Combining Like Terms & The Order of Operations

Be fair-warned…this might seem pedantic, and I agree, it is.  But, with good cause. The point of this discussion is to cause students to pause and think about something they’ve always done in math without any understanding.  It’s true, you can combine like terms, but why?  To that point, and I do not say this to put anybody down, but I bet most math teachers have never wondered this on their own.

Why Can’t We Combine Unlike Terms?

I pose this question to Algebra 1 students every year, for over a decade now.  The responses are always with an air of incredulity.  Their looks say to me:

Come on Mr. Brown, you should be smarter than to ask such a dumb question!

When I press for an answer they’re suddenly tongue-tied.  The most common answer, often spit at me with some venom is:

Because they’re not like terms.

Okay then, why can you add things that ARE like terms?

Are We in Violation of PEMDAS

The order of operations is the set of instructions by which math is to be performed.  It’s kind of a big deal!  Mess it up and you’re in a lot of trouble.  And yet, how is it that we can combine like terms if there’s multiplication occurring first?

Consider 3x + 5x – 2a.  Even the most average of 9th grade, Algebra 1 students will tell you that obviously equals 8x – 2.  My question is this:  How can you add 3and 5x if there’s multiplication taking place.  The three and the x are multiplying, right?  And in the order of operations, doesn’t multiplication come before addition?

Student Responses

When I ask students why like terms can be combined even though there’s multiplication taking place and they add before multiplying, they just say, “cuz.”  This is a classic example of something that’s been taught and accepted without understanding.  It’s not a complicated issue, but promoting mathematical literacy is my crusade of late.

Consider the product of four and five.  I remember being a young child and counting groups of five to find the product of five and a number.  4 x 5 = 5 + 5 + 5 + 5

Good to go on that point?

Then 3x = x + x  + x

And 5x = x + x + x + x + x

Then 3x + 5x =  x + x  + x  + x + x + x + x + x8x

Because multiplication is repeated addition, and we’re adding the same thing repeatedly, we can combine the coefficients of x.

Why Can’t You Combine Unlike Terms?

Consider our original expression: 3x + 5x – 2a.  Two times a is a + a.  Three and five makes 8x.  But x and a are probably not the same thing (we don’t REALLY know because they’re unknown), so we cannot combine them with addition.

8x + 2

x + x  + x  + x + x + x + x + x + a + a

This cannot be simplified further than 8x + 2a.

To combine unlike terms would be in violation of the order of operations.  Since and a are different, we would need to multiply 8 and 2 by them respectively, before adding them.

I hope this short discussion has caused you to pause and think about what you thought you knew about Algebra.  Let me know what you think in the comments.  Thank you, again, for reading.

Philip