Last week, in Number Sense 028 we put some goats on a scale to solve an algebra problem. This week, we are going to play tricks on the scale, by adding and subtracting nothing.
If we look at a number line, we notice that all numbers have an opposite number on the other side of zero.
The red arrow points to a number, the green arrow points to the opposite number. Both arrows are the same length. If we add these two arrows
we get zero.
This is a tough idea to show using our balance scale. Recall that when we put something on one side of the balance scale
that side of the scale went down. So putting the opposite of brown goat (call it blue goat) on the scale should have the opposite effect.
And so it does, but it looks awkward. Putting things on balance scales should make the pan go down, not up.
We could pretend the blue goat was full of helium, like a balloon, and was somehow tied to the pan... Still awkward.
So, we must replace our trusty balance scale for something a bit more abstract:
Now we represent opposites like this:
A goat and its opposite on one side is the same as nothing on the other side.
Our other rules all involved placing or removing something from both sides at once. Opposites (used in pairs) allow us to place or remove something on only one side, without affecting the balance, since the pair adds up to zero.
Let's see how this works with a simple problem: 2x = 3 + (-x)
We can't use our rules about both sides, since we don't have the same thing on both sides. Somehow we have to get a blue goat on the left side, or two brown goats on the right side.
We add an opposite pair to the left side (which includes a blue goat)
It is still equal, because the opposite pair adds up to zero, and adding zero to one side doesn't change the value of that side. But now we can use our “remove the same thing from both sides” rule, to get
Three brown goats is three. Obviously one brown goat is equal to one. Problem solved.
Notice, I'm putting the removed goats below the line. No particular reason, I just put them there to keep track of what I've removed.
But what was that mysterious blue goat?
Lets work the problem the other way
Same beginning, but now we will add some brown goats to the right side (along with their opposites)
It's getting crowded, but now we can remove brown goats from both sides.
We want the blue goats by themselves, so subtract three from both sides
Three blue goats equals negative three, so one blue goat equals negative one.
Makes sense. A brown goat is 1, so its opposite blue goat is -1. 1 + (-1) = 0.
Opposites are not always negative. If we make a minor change to our original problem
2x = (-x) – 3
We discover that three brown goats
Equal negative three
Which means the blue goats are positive one, and the brown goats are negative one. If we assume the opposite goats are negative, we might make mistakes. All we know in the beginning is the two color goats are opposites. We do NOT know which side of zero they are on until we solve the equation.
Have fun in the comments.