Last week, in Number sense 026, we took a look at equality, the equal sign in mathematics, meaning nothing more or less than a statement that one thing is the same as another thing. In arithmetic, the equal sign often gets an operational definition of: do a calculation and put results here. This operational definition often interferes with learning algebra, because algebra is not really about calculating, but about exploring and discovering.
Algebra is a game, and, like most games, there are rules, and moves that are allowed, and moves that are not allowed. Games have goals, a way to win, so does algebra.
This week, I'm going to outline the rules of the game.
Sometimes the goal in algebra is "isolate the variable" which means getting the unknown number on one side of the equal sign, and everything else on the other side. If our unknown number is a goat, our goal is to get the goat out of the garden and into its pen.
Sometimes the goal in algebra is to simplify things. We begin with a complicated mess, and want to make it look simpler. We do this so we might be able to understand the complicated mess better, or because we want to compare it to something else.
Sometimes the goal in algebra is to write an elegant expression. Something pretty, and true, and memorable. For example, we want to find the length of some line. We have coordinates for its endpoints. We arrange things so the line is the hypotenuse of a right triangle, and we can use
to find the length.
All these goals can be reached by starting with some basic mathematical principles, and following certain rules. None of these rules are mysterious. Most of them are obvious when stated baldly. All of them are easily demonstrated by referring to balance scales or simple diagrams.
But, like rules in most games, just knowing the rules won't make you an expert player. In order to become a good player, you have to actually sit down and play the game for a while. And, the more you play, the better you become.
The basis of most of the rules. Equality. The stuff on the left side of the equal sign is the same as the stuff on the right side of the equal sign.
a = a
This is what makes the balance scale a good analogy, with “stuff” being “weight” and same meaning the scales balance.
Equality is not directional. If a thing is the same as another thing, the other thing is also the same as the first thing.
If a = b then b = a
Equality is not particular. It applies to everything. This principle lets us draw conclusions about things that were initially unrelated.
If a = c and b = c, then a = b
Then there is a set of principles that speak to the order in which we do things
for multiplication, order does not matter
ab = ba and a(bc) = (ab)c
for addition, order does not matter
a+b = b+a and a+(b+c) = (a+b) +c
We can take apart multiplication by a sum, and do a sum of two multiplications
a(b+c) = ab + ac
There are some principles regarding zero and one, the identity elements for addition and multiplication
Adding zero doesn't change things
a + 0 = a
Subtracting a thing from itself equals zero
a – a = 0
Multiplying by one doesn't change things
Dividing anything by itself equals 1 (except for zero, dividing by zero is undefined.)
Subtraction is the same as adding the opposite number
a – b = a + (-b)
Definition of subtraction
If a + b = c then c – b = a and c – a = b
Definition of division
Here are some “legal moves” we can make when exploring. They are legal because, even though they change the value of the expressions on both side of the equal sign, they do not change the balance. Both sides remain equal, even though they both have a new value.
Adding something to both sides does not affect equality.
If a = b then a + c = b + c
Subtracting something from both sides does not affect equality
If a = b then a – c = b – c
Multiplying both sides by something does not affect equality
If a = b then ac = bc
Dividing both sides by something does not affect equality
Finally, another legal move that is sometimes useful has to do with substitution.
If f(x) = (expression) then f(x) can be substituted for (expression) where it occurs in an equality, and vice versa.
Let's work a problem to get a feeling for how these rules work.
Suppose we have this equality 4 – 6y = 8(1+2)
Our goal is to get the goat out of the garden. In this case, the goat is called 'y' and 4 minus six goats is the same as eight times the sum 1 plus 2.
I would start my exploration by doing any ordinary arithmetic I could find in the equation. In this case, (1+2) is easily done, and the equality becomes
There is still more ordinary arithmetic I can do: 8 times 3 is 24
4 – 6y = 24
Now there is no ordinary arithmetic left in the equation, so I'll take a look at the rules, and see if I can use one of them. The goal is to get the goat by itself.
Many legal moves begin with the condition “If a = b”
Can we meet this condition? If a = b ? In other words, is 4 – 6y really equal to 24? At the beginning of the problem, we were told that 4 – 6y = 8(1+2) was true. Ordinary arithmetic tells us that 8(1+2) = 24
our principle
If a = b then b = a
lets us write that as 24 = 8(1+2)
and our principle
So we see that a does indeed equal b, and we can use our rules that have that beginning condition.
The goat is in the expression 4 – 6y . There are two terms related by addition or subtraction. We can use one of the legal moves involving addition or subtraction to remove one of the terms.
If a = b then a – c = b – c
Subtracting something from both sides does not affect equality. Since I want to get rid of the term 4 on the left side, I'll use this legal move to subtract 4 from both sides
4 – 6y – 4 = 24 – 4
The right side of the equation is ordinary arithmetic. 24 – 4 = 20. I can simplify the equation by doing that arithmetic
4 – 6y – 4 = 20
We have a principle that tells us the order we add things doesn't matter. But this is subtraction. In order to use our addition principle, we must change those subtractions into additions. Fortunately, we have a legal move for that
a – b = a + (-b)
Subtraction is the same as adding the opposite number
So, 4 + (-6y) + (-4) = 20
Now, since they are all additions, we can rearrange them any way we like, following our two addition principles
a+b = b+a and a+(b+c) = (a+b) +c
for addition, order does not matter.
Which can give us
(-6y) + (4 + -4) = 20
Either our principles, or ordinary arithmetic will give us the result
(4 + -4) = 0
and one of our identity properties
a + 0 = a
Adding zero doesn't change things
means
(-6y) + (4 + -4) = 20 can change to (-6y) + (0) = 20 which can change to -6y = 20
Wow! All that work to get rid of a single number on one side of an equation. Mathematics is HARD!
Here's a secret: the principles and rules we used to do that are all so obvious and straightforward, that after doing this a few times, most people will skip all the steps and just do this part in their head, and will go
4 – 6y = 24
(subtract 4 from both sides)
-6y = 20
and, using the analogy of a balance scale, it doesn't really matter where you pluck things up from the pan, if you take 4 away, it's gone. If you take 4 away from both sides, the scale is still in balance.
We are very close to isolating the goat. If we could somehow get goat multiplied by one on the left side, our multiplication identity would give us a goat by itself.
How can we change -6 into 1? One identity principle says
Dividing anything by itself equals 1 (except for zero, dividing by zero is undefined.)
so if we divide -6y by -6 we will get y by itself. Can we do that?
There is a legal rule that says
Dividing both sides by something does not affect equality
So it seems we can do that.
And, since anything divided by itself is 1, and multiplying by 1 doesn't change things
So, there's our goat.
Have fun in the comments.