Who Put Letters In Math 

I’m not one much for the “beauty” of math, per se.  Sure, I can certainly see where that idea comes from, but my mind is more like an engineer’s than a mathematician’s, so I’m more about the “utility” of math.  Which is why my mind was absolutely blown when I took calculus in college.  I was so excited because I could immediately see what a powerful tool it is for figuring out how things work and such.  You can build an airplane without it, but if you use it, you can optimize that airplane without a bunch of trial and error.  And while trial is fun, error isn’t so great with airplanes.

As it turns out you can’t do calculus without algebra.  And, as expected in this space, we’d have to say that pre-algebra is where we pick up the tools with which we can do the algebra.

Pre-algebra is where we learn about various properties in math, and it’s also where we start all that nonsense about adding letters to our equations.  Who in the world would put letters in math?  Well, if you think engineers know everything then you might be onto something with that question.  But if you think there may occasionally be something that an engineer doesn’t know, then you’ll need some way to express “the unknown” in your math.  We’ll get to the unknowns eventually, but for the moment let’s just start with one of those properties I mentioned.

This one is called the Distributive Property because in math all is fair, and distributing equally seems fair enough.  (This is math, not economics, after all.)  Here’s the essential idea:  5 x 23 = 5 x (20 + 3) = 5 x 20 + 5 x 3 = 100 + 15 = 115.  You can try this with any pair of numbers and you’ll see that it works.  But that doesn’t really help you understand why it works, right?  Let’s see if we can clear that up a bit.

I was so excited because I could immediately see what a powerful tool it is for figuring out how things work and such.

Now the first and last steps seem fairly clear, but the middle steps are tricky, right?  But if you simply look at your pencil bag in your binder you can see what’s happening here.  See it has 20 pencils (P) and 3 erasers (E).  Now if you had five of those bags, and we represent the bag like this, (), then we have 5 x (20P + 3E) = 100P + 15E.  Yes, five bags would mean a grand total of 100 pencils and 15 erasers.  So if you’re multiplying stuff inside parentheses, you have to multiply each thing in there individually. 

The Jones’ garage has two cars, one lawn mower, five bicycles and a floor mat.  Six of their neighbors are keeping up with the Jones’, so they have the identical things in their garages.  How many of each is on the block?  That’s seven (including the Jones’), so 7 x (2C + 1L + 5B + 1M) = 14 cars, 7 mowers, 35 bikes, and 7 mats.

Now we have but one more step to finish this thing off.  And it’s in a grocery bag.  Dad and his three brothers have identical lists and end up with two apples, one jar of peanut butter and three packages of sliced cheese.  So altogether they have 4 x (2A + 1P + 3C) for eight apples, four jars, and 12 packs of cheese.  And there’s the rub.  How many slices of cheese do they have?  AH!  The unknown!  How many slices are there in each package?  16?  24?  So the C can represent the number of packages, and we call it a variable because the number of slices can vary depending upon the value of C – the number of slices per package.

So there’s a little introduction to pre-algebra.  Sure, there’s more to take on, but hopefully you can see you have nothing to fear if that’s where you’re headed next year!