Last week, in Fundamental Understanding of Mathematics LI we made a model for thinking about algebraic equations with a simple balance scale. We demonstrated two rules for keeping the scale in balance:
- add or subtract the same thing from both sides
- multiply or divide each side by the same thing
But what do we do if we want to take bananas away from both sides of the scale, and only one pan contains bananas?
Let's suppose our set up looks like this. We can say, well, even though there are no bananas on the right side, we can still subtract bananas from 42. But this is confusing. How can we take away something that isn't there. Yes, we know it works, but why does it work?
For this, we go back to our number line, and the idea of opposites.
So, here we have a number line, and a mark at the value of one banana. If we were to add the banana to its opposite, we'd end up with zero.
Lets look at a simple addition family:
We are going to substitute the banana for 1, the banana's opposite for 2, and zero for 3
The first two members of the family show that
if we add something to its opposite, we get zero
This is the mathematical definition of opposite.
This is what it looks like on the number line
An opposite is the same distance from zero, but in the other direction. So, the yellow one is the banana, and the red one is the opposite banana! Well, not quite. Opposite is a relationship. The red banana is the opposite of the yellow banana, but the yellow banana is the opposite of the red banana. Which one is "the opposite" depends on which one we start with.
Rather than worry about which one is the opposite, we will combine both the red banana and the yellow banana, into an opposing pair of numbers we call a zero pair, because, taken together, they add up to zero.
Back in Fundamental Understanding of Mathematics VIII we took a look at the additive identity property, that adding zero to a number doesn't change that number:
Take a close look at that additive identity: there is the same number (a) on both sides of the equal sign (a = a) which we know is true and call the reflexive property of equality.
But we also have zero added to only one side of the equation. This is the exception to our earlier rule that we had to add the same thing to both sides to keep it balanced. We can add zero to only one side, and the balance won't be unbalanced, because adding zero doesn't change the value of the expression.
Combine this with opposites adding up to zero, and we see that we can add a zero pair to only one side of the equation, and still have a true equation.
This is still balanced, but now we have a banana on the right side (along with its opposite red banana.) We can now use our earlier rule of taking away the same thing from both sides, because we now have a yellow banana on the right side to take away.
If we can do it once, we can do it again, and again. Add two more zero pairs to the right side, take away two bananas from both sides.
We have removed all the bananas from the left side, and are left with 42 plus three red bananas on the right side. We return to our number family to remind ourselves that
zero minus a yellow banana is a red banana. Instead of adding a red banana, we can subtract a yellow banana (or three of them.)
So, there it is: a demonstration that we can subtract an unknown from an expression that does not initially contain that unknown. Of course, in practice, we wouldn't go through the trouble of adding zero pairs first, we'd just jump straight to the conclusion, and "subtract three bananas from both sides."
This, then, is the kind of thing you'd do to prove why something works, rather than just accept that teachers know what they're talking about. Because just accepting that teachers know what they are talking about means that learning math involves copious note taking and lots of memorizing. In other words, it turns math from a fun, puzzle solving game into tedious drudgery.
If math is boring, you're not doing it right.
Have fun in the comments.