Most students were introduced in High School to Algebra where letters
substitute for numbers, Geometry where there weren’t any numbers at all, and Trigonometry,
which, in my day, meant using a lot of tables to get back to “numbers.” Some
have had the joy of Calculus. As the saying goes, “no one flunks the Calculus,
they just flunk the Algebra.” That refers to the fact that there’s a lot of Algebra
in the Calculus. (You may have noticed we just say “Algebra,” but we say “The
Calculus.” That’s just a funny little habit mathematicians have.)

Actually, my experience teaching adults mathematics is that
the problems start in Arithmetic. The place where the problems begin is with
fractions. If you really understand fractions, then the leap to Algebra isn’t
all that hard.

(I would also review “powers,” that’s exponents and
logarithms … that’s exponents too. Fractions and exponents were where people
who had been out of school for a few years needed a good review.)

So, let’s start with a little review. We will begin with the
rules for multiplying fractions. First a reminder of terminology: the part of a
fraction on the top is called the “numerator” and the part on the bottom is
called the “denominator.”

To multiply two fractions, you just multiply the numerators
and then you multiply the denominators and you have the new fraction that is
the “product.” For example, 1/3 * 3/4 = 1 * 3 over 3 * 4 or 3/12. You would
reduce that to 1/4 … the simplest form.

(I’ll have to write my fractions this way since this word
processor doesn’t let me write them above and below a line like I would prefer.
Mathematicians and word processors sometimes have to arm wrestle, and — in this
case — the word processor won.)

In words you could pronounce the multiply sign as “of.” So
the problem is spoken as one-third of three-quarters. Well, one-third of
three-quarters is a quarter. Sure, if it was left-over pizza, that’s how you
would divide it among three people. Note that somehow, almost mysteriously,
multiplication of fractions seems more like division. Well, it is. There is this thing called the "reciprocal." But let’s
not think about that right now. We’ve got some other fish to fry. (Hmmmm, fish
tacos, why not fish pizza? This is more than a math class. This is a cooking
class. And now we’re really cooking!)

It is a little harder adding of two fractions. To add two
fractions, they must have the same denominator. Adding 3/8 + 1/8 is easy since
they both have 8 for a denominator. The rule is that you add the numerator, and
then write that over the “common” denominator.

3/8 + 1/8 = 3 + 1 over the 8, which equals 4/8. Of course,
now you should do some simplification where you convert the answer to the
“lowest denominator.” For example, 4/8 = 2/4 = 1/2. Now you are at the lowest
denominator.

Let’s do another. 3/8 + 5/8. That’s 8/8 = 4/4 = 2/2 = 1.
Remember, when the numerator = denominator, that’s one. Are these rules coming
back to you?

Now, in math, just as in real life, once you figure out the
easy cases, they throw the hard ones at you. What if the denominators are
different? Well, they you convert the fractions until they have the same
denominator … a “common” denominator.

Let’s add 3/8 + 1/4. You probably notice, with this simple
problem, that the “4” in the second fraction is a factor of the “8” in the
first fraction. So what we could do is multiply the 4 in the denominator by “2”
and we would get “8.” But we must do this without changing the fractions value.
You may recall the trick was to multiply both the denominator AND the numerator
by 2. So you get 2/2 * 1/4 = 2/8.

Basically, 2/8 is the same as 1/4. What is really happening
is that you are multiplying the fraction by another fraction that has the same
number in the numerator and the denominator. Recall the rule that any number
divided by itself is 1, and 1 times any number is still that number. So,
effectively, you are just multiplying the original fraction by one and that
doesn’t change anything. I didn’t lose you, did I? Read it again. Get out a
pencil and try for yourself. Now you’ve got it.

Ah ha, just reminded you that a fraction is also a division
problem where the numerator is the “dividend” and the denominator is the “divisor.”
(The denominator “goes-inta” the numerator, as we use to say in the sticks.)

Remember the pizza example. Multiplying by a fraction is
really a form of division. (You may even recall that to divide fractions, you
invert and multiply. Slippery little devils aren’t they. Oh yes, it’s all
coming back now!)

So, 1/4 is 1 divided by 4 and 2/8 is 2 divided by 8. Now
back to our problem.

3/8 + 1/4 = 3/8 + 2/8 = 5/8 and that’s the “final answer.”

Well, that was easy. What if the denominators don’t have any
common factors? Suppose it was 3/4 + 1/3?

Well, then you just multiply the first denominator by the
second denominator and you multiply the second denominator by the first. That’s
actually the general rule that will always work. Of course, to keep from changing
the fraction, you have to multiply the numerator by the same amount. Get out
your pencil and follow along:

So you multiply the 3/4 by the denominator of the second
fraction: 3.

So you get 3/3 * 3/4 = 9/12.

Now multiply the second fraction by the denominator from the
first: 4.

That’s 4/4 * 1/3 = 4/12.

Now they’ll both have the same denominator. Works every
time. Now you can add:

3/4 + 1/3 = 9/12 + 4/12 = 13/12.

That’s actually the lowest form, but some would convert that to a mixed fraction of 1-1/12. (Spoken as one and one-twelfth.)

Now we’re ready for the jump to Algebra. Algebra is just
generalized Arithmetic where you work the calculations with letters that stand
for “variables.” Basically, a variable can be (almost) any number. The results
will be true if you then substitute actual values for the variables. That’s the
start of Algebra.

So, if you had to add a/b + c/d, you don’t know what values
b and d will have, so you take the basic rule I went over above and multiply
the first fraction by the denominator of the second and vice-versa.

That would be

(and I’ll use the Algebra shortcut that you don’t need the multiply sign (*), you can just write the letters next to each other to signify multiplication)

(and I’ll use the Algebra shortcut that you don’t need the multiply sign (*), you can just write the letters next to each other to signify multiplication)

d/d * a/b = ad/bd and b/b * c/d = cb/bd

(Notice the order of multiplication doesn’t matter: b * d = d * b. That’s called “commutativity.” Another rule of algebra.)

So the addition becomes ad/bd + cb/bd = (ad + cb) / bd. (I
added the parentheses because this editor won’t let me write the fractions as
above and below the line like I would prefer.)

I’ll try:

__ad + cb__

bd

Sometimes we used Algebra to create "formulas" that can then be used to simply solve problems. That last formula is the solution for adding two fractions, no matter what the denominators are. It is the first fraction numerator (a) times the second denominator (d), plus the second fraction numerator (c) times the first fraction denominator (b). Then all that is divided by the product of the two denominators. So this formula is the "general solution" for adding two fractions. Program it into your computer, and you can forget all the Arithmetic you ever knew.

So now you may have a better understanding of how Algebra can be considered generalized Arithmetic. Also notice that it is true that most students begin to have trouble with math at fractions. If you really know and understand fractions well, then you have a great advantage when you start studying Algebra. And good Algebra skills make Calculus easy — just a tip from the old master. Now wasn’t that fun? Who says math is hard?!!?

So now you may have a better understanding of how Algebra can be considered generalized Arithmetic. Also notice that it is true that most students begin to have trouble with math at fractions. If you really know and understand fractions well, then you have a great advantage when you start studying Algebra. And good Algebra skills make Calculus easy — just a tip from the old master. Now wasn’t that fun? Who says math is hard?!!?

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