I have had many people say to me over the years, “But algebra is easy: just tell them to do the same thing to both sides!” This is wrong in several ways, not least of which is the word “easy”. The particular way it’s wrong that I want to talk about today is the idea that doing the same thing to both sides is somehow the only move in algebra, because it’s not even the most important or the most common move.
I think the most important and most common move in algebra is this:
Replace something with something else you know it’s equal to.
This rule isn’t even an algebra rule, it’s a rule you’ve used in plain arithmetic. Look at this working:
All of those steps were replacing something with something else I know it’s equal to. First, I replaced the (4+3) with 7, and then I replaced 7×5 with 35. Finally I replaced 35-6 with 29. The reason I was able to write the “=” signs there was because I knew each expression was produced by replacing something with another thing it’s equal to. Of course they are the same. If you have ever written your working in the way I did there, then you were using the replacement move.
Indeed, this move is the heart of how mathematicians write calculations. We always move from one step to another by replacing something with something it’s equal to. Knowing that this is what we do, we read other people’s working by comparing each expression in a chain to the next to see what it is that has been replaced.
It’s interesting that nobody has ever told me explicitly that this is how to read maths working that’s written with a chain of equal signs. I just somehow figured it out. I am sure quite a few of my students don’t actually know this strategy.
This idea that anything can be replaced with something it’s equal to is for me the major thrust of all the algebra laws and identities.
For example, the distributive law―a(b+c) = ab+ac―doesn’t tell me how to “expand brackets”. Instead it tells me any time I see a(b+c), I’m allowed to replace it with ab+ac, and every time I see ab+ac, I’m allowed to replace it with a(b+c). The same goes for all the algebra laws, even though some of them seem very complicated.
For me, completing the square illustrates this very well. For example:
3x² + 24x + 7
= 3(x² + 8x) + 7
= 3(x² + 8x + 16 – 16) + 7
= 3((x+4)² -16) + 7
= 3(x+4)²- 48 + 7
= 3(x+4)² – 41
The first move was to replace 3x² + 24x with 3(x² + 8x) by the distributive law.
Then I replaced 0 with +16-16.
Then I replaced x² + 8x + 16 with (x+4)².
Then I replaced 3((x+4)² -16) with 3(x+4)² – 48.
Then I replaced -48 + 7 with -41.
The second pair of moves are usually extremely surprising to students, and even I have to stop and look closely when I read already-written-down completing the square working. But they make sense when I compare the line of working to the one above and see which parts are the same, so that I can deduce that the parts that change must have been replaced because they are equal.
Incidentally, this is why I am such a stickler for keeping things lined up when you do maths working if you can. Changing the order makes it hard to deduce what’s been replaced from one line to the next. Of course, you can change orders, but I prefer to do that as its own move so you can see when it happened.
Finally, the replacing move is actually used when solving equations too, even though it’s usually hidden. Look at this working that I did in a chemistry lab with a student once.
ρ = m/V
ρ × V = m/V × V
ρ × V = m
The move from line 1 to line 2 was multiplying both parts of the equation by V, but the move from line 2 to line 3 was replacing m/V × V with m because you know they are equal.
The student who I did this with was ok with the multiplying both parts by the same thing, but they were not ok with the replacing. They complained that I hadn’t done the same thing to both sides. And this was when I realised that there was another move in algebra and it’s much more fundamental than doing the same thing to both parts of an equation. And we need to tell students explicitly about the existence of this move.
So there you have it. Algebra is not all about doing the same thing to both sides, it’s very very often about replacing something with something else it’s equal to. Keep an eye out for it next time you read or do any maths working, and maybe explicitly remind your students every so often when it happens.
Actually just a bit of an epilogue: the replacing move is really rather fun to do “in reverse” as it were. Usually we do it in arithmetic by replacing an expression with a single number, but there’s nothing stopping you replacing a single number with an expression and so finding some rather complicated expressions for familiar numbers. This first one is based on the fact that if 1 = 2-1, then any 1 can be replaced with 2-1 at any time. (If you click on the tweet you can see me replacing 1 to 9.)
Replacing 1. pic.twitter.com/ojsYcthQt9
— David Butler (@DavidKButlerUoA) August 19, 2019
May you enjoy your replacing too.