Some weeks you get the bear, some weeks the bear gets you. That is to say, me. After last weeks massive fail, Fundamental Understanding of Mathematics L's misguided attempt to try some abstract math (two comments, including the tip jar) I'm going to go back into concrete math.
This week I thought we'd take a look at a balance model for understanding the symbolic manipulation of equations.
Time was, scales didn't have any numbers on them. They weren't full of electronic parts, didn't have displays, didn't have springs, didn't have calibrated marks on them, in fact, they were little more than a couple of pans balanced on a stick. The only thing you could tell from one of those scales is whether one pan was heavier than the other, or whether they balanced.
And that limitation of ancient, scaleless balances makes them a good model for mathematical equations, because when we are presented with an equation, say,
all we really know about it is that the two expressions are equal. The value of the stuff to the right of the equal sign is the same as the value of the stuff on the left side of the equal sign. We have no idea what that value is, but we do know the scale is balanced: that's what the equal sign tells us.
Scales, of course, don't always balance. We have mathematical notation for that, too. If all we know is that the stuff on the left is NOT the same as the stuff on the right, we'd use a not equal sign.
which is an equal sign with a slash through it, showing that it is not an equal sign. If we knew which side was heavier than the other, we have symbols for that as well.
means the right side outweighs the left side. If the inequality sign is pointing the other way,
then the left side is the heavy side. We don't have fancy names for these signs, we just call them "less than" and "greater than."
There's not much manipulation we can do with the not equal sign, it really doesn't tell us much. But we can manipulate inequalities, very much the same way we manipulate equalities. After all, if we know the left side is heavier, if we do manipulations that would have kept equal sides the same, those manipulations will also keep the heavy side heavy. Think about it like this: we can set aside some value from the heavy side so that what we leave in the pan balances the lighter side. Then we can do manipulations that maintain equality. Whatever we do that would maintain equality, there will still be that extra bit on the heavy side which, when we add it back on the scale at the end, will make that side heavier again.
So, we really don't need to worry about learning any extra stuff to understand inequalities. If we learn how to manipulate equalities, the inequalities come along for the ride.
So, let's make a simple balance:
Now we'll put something on it.
and it's no longer in balance. And so we come to the first rule of keeping balances balanced: add the same thing to both sides.
Now it's back in balance. Three bananas is the same as three bananas. Likewise, we can subtract a banana from one side, if we also subtract a banana from the other side at the same time.
So we can add or subtract the same thing from both sides.
Notice that the pans in this drawing have only two bananas apiece, while the drawing before that there were three. The rules don't guarantee that the pans will always have the same value -- only that they will balance.
Speaking of values, we can combine known values with the bananas (unknown values)
as long as we add or subtract the same thing on both sides.
If we multiply or divide, we must make sure to multiply or divide everything on the pan. To see why this is important, suppose we did not have exactly the same thing on each side.
If we multiplied, say, just the bananas by 2, we would end up with six bananas on the left side and four bananas on the right side.
If we now subtract the original contents (remember, the pans balanced) from each side, that is, two apples and three bananas from the left side, and one apple, four, and two bananas from the right side, we wind up with the result
three bananas is the same as two bananas. Since these are not the same, we made a mistake in multiplying just the bananas by two.
Instead, let's suppose we replace the contents of the left pan with a weight that has the same value as two apples and three bananas.
We don't know the value of the weight, but we do know it has the same value as everything on the right hand pan. So if we multiply the weight by 2, we must multiply everything in the right hand pan by 2 as well, in order for the pans to stay balanced.
Since the weight represents the original contents of the left hand pan, two weights represent four apples and six bananas, double the original contents of that side.
So, we see that multiplication of everything on each side by the same number does not throw the scale out of balance.
We can use a similar argument for division, cutting the bananas and apples in half, or in thirds, and so on, but we won't show that. However, notice that if we start with the last drawing,
and divide both sides by two, we end up with our original drawing, which balanced.
So, here is another model to use when thinking about manipulating symbolic equations. We have two rules for keeping the scales balanced:
- add or subtract the same thing from both sides
- multiply or divide each side by the same thing
Next week, we'll add the idea of opposites and zero pairs to the model, and wind up with a very powerful tool for solving equations.
Have fun in the comments.