Understanding Algebra: What’s it all about?

Algebra 1 Tutorial

Understanding Algebra: What's it all about?

Intro

Algebra is a magnificent, yet scary and often confusing step in learning mathematics. It is the point where math ceases to be just a tedious sequence of mechanical steps; instead, math becomes creative. It is the study of observing and identifying patterns, discovering the properties of those patterns, and then using everything you’ve learned to create your own patterns. These patterns, the ones we create, are what allow our modern society to run: they are the formulas that determine the stock market value of a company. They are the code running in your computer that displays this text to the screen. And they are even the physical laws that describe the entirety of the universe itself.

So, yeah, understanding algebra is pretty important! It is a huge, exciting new world to explore, but discovering new things isn’t always easy. It takes a lot of patience, experience, and lateral thinking to really understand each new concept that comes your way, some more than others. But I’ve been talking in really general terms so far. Let’s look at an everyday problem to see what algebra is really about.

Sample Problem

Say you’re at a store and you want to buy some apples and bananas. An apple costs $0.65 and a bunch of bananas costs $1.51. Which of the algebraic expressions below models the total cost of my purchase if I buy A apples and B bananas?

(0.65 + A) * (1.51 + B)


1.51 * A + 0.65 * B


(1.51 + A) * (0.65 + B)


0.65 * A + 1.51 * B


Solution

The difference between elementary arithmetic and algebra is the difference between 2+3 and a+b. Arithmetic was about giving you an expression, like 2+3 or 12/4 or 6-5+2, and then using the rules of addition, subtraction, etc. to find the solution. Algebra, on the other hand, is about abstraction. Instead of looking at a single problem with a single answer, algebra takes a single problem and generalizes it.

Let’s take the above problem for example. I could’ve instead asked “What is the cost if I buy 3 apples ($0.65 each) and 1 bunch of bananas ($1.51 each)?” This is an arithmetic question because you’re given all of the information you need to answer the question. 1 apple costs $0.65, so 2 apples is going to cost that plus another $0.65, and 3 apples is another $0.65. Altogether, that’s $0.65 + $0.65 + $0.65 = $1.95 for 3 apples (or $0.65 * 3 = $1.95). Then, you add that cost to the cost of the bananas, which is just $1.51 since there is only one bunch, and you get a total of $3.46.

And that’s the important part in arithmetic: the total, the final product you get after operating on all the numbers. But that’s not what algebra is about. Algebra asks “Well, what if I don’t want 3 apples? What if I want 5 apples, or even 10? What if I want 3 bunches of bananas?” And more importantly, algebra asks “Is there a pattern here? Is there a way I can model the cost so I can quickly find an answer?” And the answer is almost always an astounding YES. You just have to look for it, and that’s where the hard, creative part of math comes in.

So how do we go about finding a pattern in our fruit-buying problem? To begin, let’s take one part of our arithmetic solution, like the total cost of apples. We said the cost for 3 apples was $0.65 added together 3 times, or $0.65 * 3, which is $1.95. First things first, forget the $1.95, because remember, algebra isn’t about the total. Instead, focus on the expression $0.65 * 3 and think about how this expression changes if we want, say, 5 apples instead of 3. We simply replace the 3 for 5, giving $0.65 * 5. Nothing else in the expression changed, not the price nor the multiplication, only the number of apples.

We can go further. Instead of replacing 3 or 5 with some random number, let’s replace it with something else, a symbol that can represent any number. For example, the symbol “A”. What we get now is an algebraic expression, $0.65 * A, where “A” represents the number of apples I want to buy. “A” can be 5, it can be 10, it can be 300 if I really like apples, it doesn’t matter. What matters is that I’ve made a model that represents the cost of apples no matter what number I choose. All I need to do is plug in whatever number I want for A, say 6, and I can calculate that the cost of 6 apples is $0.65 * 6. Then, now that I have all the information I need, it’s just a matter of using that elementary arithmetic to find out what my precise answer is, which happens to be $3.90.

Of course, the cost of apples was just one part of the original problem. We still have to account for the cost of bananas and then how we combine the costs of both fruit to get the total. I’ll leave that as an exercise for the reader. Once you think you’ve got it figured out or you gave it your best shot, take a look at the answer above and try to see how it works.

What’s important is that we understand the abstract, pattern-finding essence of algebra. As one final example, let’s take a quick look at how a cash register might work (and computers in general). When I go up and try to buy my 234 apples, the person behind the register doesn’t have to figure out the algebraic expression and then type out that expression into a calculator. Rather, the expression is programmed into the computer. There might be a button that you press to calculate “price-of-item * number-of-items” (the algebraic expression), and from there all the cashier has to do is input the specific price of the item ($0.65) and how many I bought (234). Then, voila, the computer spits back $152.1, no work needed on the cashier’s part. The programmer on the other hand? LOTS of mathematical and algebraic thinking required.

So hopefully the importance and power of algebra is at least a little bit clear. It’s a big change in perspective and the transition can be tough, but with more and more time and practice and experience, you’ll be reading and writing and manipulating algebraic expressions and equations like it’s a second language.



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