When doing algebra and solving equations, there is this move we often make which is usually called “doing the same thing to both sides”. For many people it is their fundamental rule of algebra. (It’s not mine, but that’s a discussion for another day.) You use it when solving an equation like this:

4x-3=13

4x-3 + 3 = 13 + 3

4x = 16

4x / 4 = 16 / 4

x = 4

On the second line we “added 3 to both sides” and on the fifth line we “divided both sides by 4”. The “sides” being referred to here are the two sides of the = sign.

Quite recently this phrase of “both sides” has begun to bother me.

It started when I was trying to explain to students how they needed to be careful when they were solving equations using matrices. In their question, the matrix A was invertible and they solved for X like this:

XA = B

X = A^{-1}B

Imagine for a moment explaining to a student why this is not true. My first attempt was “When you’re solving this equation, you’re multiplying both sides by A^{-1}. With matrices you have to be careful which side you multiply the matrix on, so you have to make sure you multiply on the same side of both sides.” Crap. I need a different word for the sides of the equation and the sides of the matrix!

It got worse when I was helping students manipulate inequalities. Consider this one:

4 < x + 2 < 5

4 – 2 < x + 3 – 2 < 5 – 2

2 < x < 3

How would you describe what I did on the second line there? I subtracted 2 from all three … all three what? Certainly not all three *sides*! What should I call those things? Parts?

And then there were all those times when students did this with their differential equations:

ln(y) = 2x + C

y = e^{2x}+ C

How do you explain to students that the C belongs up in the power of e? You could say, “You have to do the ‘e to the power of’ to this whole side.” And at least some will reply, “I need to do it to everything on both sides, right…”

ln(y) = 2x + C

y = e^{2x}+ e^{C}

And now you have to dust off your emergency stash of extra patience.

Then suddenly it occurred to me. I should never call it a “side” because that’s not what it is. I’m not talking about locations relative to the = sign, I’m talking about two objects that the = sign is relating. The = sign or the < sign is telling me how things are related to each other. Those things are complete whole objects in their own right that may be written as some algebraic expression showing how they are constructed, but they really are single things. That is, it’s more (4x-3) = (13) rather than 4x-3 = 13.

So from now on I’m not saying “sides” of an equation or inequality. I will start calling them by what they are: “Divide both of these numbers by 4”, “Multiply both of these matrices by A^{-1} on their right”, “Subtract 2 from all three numbers”, “Raise e to the power of both expressions”. Failing that, I’m going to use the good old fashioned all-purpose word “thing”.

Is ‘sides’ just refering to location though?

Very interesting post, David. One of the phrases used by one of my more memorable teachers was, “when you change the side, you must change the sign”. I don’t use this with students either but do talk about equality quite a lot. It might not sink in straight away, but like your post, gets them thinking deeper about what they’re doing. Cheers

Thanks for the comment John. Anything I can do to help students focus on the meaning of = is a good thing. And often I have a total of 20 minutes ever with some students I work with!