Last week, in Number Sense 012, we got more definitions, discovering the difference between distance and displacement along a number line. We also defined absolute value, a concept that is ridiculously simple when dealing with counting numbers, but seems to throw students for a loop when applied to integers. This week, we will take a look at a number property.
What is a number property? Well, you could say it's the way numbers behave when no one is looking. Fortunately for mathematics, it is the same way they behave when someone is looking.
An example: we know that 2 + 3 is equal to 5. That's a simple addition fact. We also know that 3 + 2 is equal to 5. Another addition fact. But when we compare these two facts we can see that the order we add these numbers doesn't change the sum. It's just the way 2 and 3 behave when they are added.
So we turn this behavior into a property. We make it general: for any two integers, we can add them in either order and the sum will be the same. We can write it symbolically:
for integers x and y, if the sum of x + y = z, then the sum of y + x = z as well
We can demonstrate this on a number line:
We can add the numbers in either order, they always add up to five. Does it work with integers?
What about positive and negative integers?
What about subtraction?
Ahhh, no. Above the line we have 3 – 2, below the line is 2 – 3. These two combinations do NOT end up at the same place on the number line.
So this property works for addition, but it does not work for subtraction.
Is there anywhere else this does not work?
Sure, and not just in mathematics. Getting dressed, for instance.
(underwear, socks) + (shirt, pants, shoes) = (dressed acceptably)
(shirt, pants, shoes) + (underwear, socks) = (dressed weirdly)
Concatenation (that's the operation where you just jam stuff on the end, as in 22 & 2 = 222)
Sometimes it works: 2 & 22 is also 222, but most of the time it doesn't:
go & at is goat but at & go is atgo.
If we could give this property a name in English, we'd probably call it something like “the old switcheroo for addition.” Unfortunately for us English speakers, it was originally named by a Frenchman, who did call it “the old switcheroo” but did so in French, and we just borrowed the word he used, which is “commutative.” (literally: tending to switch, or switchable)
Since it is a property of addition, though, it tells us something about any addition, whether we know the numbers we are adding or not. We can represent numbers by lengths, so if we add two lengths, we know it doesn't matter which comes first, the total length or sum, is the same.
If we compare solving math problems to a game, then the commutative property tells us one of the moves we can make.
Have fun in the comments.