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# A Few Thoughts on Why We Learn Algebra

I recently responded to a question on WyzAnt's “Answers” page from a very frustrated student asking why he should bother learning algebra. He wanted to know when he would ever need to use it in the “real world” because it was frustrating him to tears and “I'm tired of trying to find your x algebra, and I don't care y either!!!”

Now, despite that being a pretty awesome joke, I really felt for this kid. I hear this sort of complaint a lot from students who desperately want to just throw in the towel and skip math completely. But what bothered me even more were the responses already given by three or four other tutors. They were all valid points talking about life skills that require math, such as paying bills, applying for loans, etc., or else career fields that involve math such as computer science and physics. I hear these responses a lot too, and what bothers me is that those answers are clearly not what this poor student needed to hear. When you're that frustrated about math, the last thing you want to hear is that you'll “need” to be able to do math in order to live. That might cause you to take those responses as a sign that you should not venture anywhere near computer science or physics, or ever rent an apartment or own a car. What this student needed to hear was the larger picture of why math is useful even if you never touch another x or y in your life. So here are my thoughts, in the form of an example from my own life.

When you get right down to it, at its most basic level, algebra centers around the idea that you can add, subtract, multiply, or divide both sides of an equation by the same number and the equation will continue to be true. That very basic concept of “doing the same thing to both sides” has the implication of allowing you to rewrite the same equation a multitude of different ways without changing its value. In essence, algebra is problem solving at its most basic. You start with what you know, and by working step by step through rewriting the equation while maintaining its value, you arrive at a version of the equation that makes the unknown element clear. The whole time, the laws of algebra remind you that you're not changing what the equation means; you're just rewriting it in a way that's easier to work with and understand. This type of step-by-step problem solving has a multitude of uses in everyday life that don't involve a single number.

Here's my example. One evening in college, I arrived back at my dorm building after a long day of classes, only to find that my wallet was not in my bag. I had no idea how long I'd been without my wallet, and even less idea where it was. On top of that, I had a small window of time in which I was supposed to go home and change out my books before heading out again, so I needed to get into the dorm NOW, which I couldn't do without my student ID card, which was – you guessed it – in my wallet. So what do I do?

Well, I'll be honest, I began to panic slightly. But I worked through the panic and figured out my first plan: retrace my steps until I found my wallet. Fortunately, all of my classes that day had been in the same building, so I didn't have far to go. Unfortunately, my wallet was not anywhere on the path I'd taken from the dorm to the classroom, the path home, or anywhere in between. The wallet was lost.

Having hit a dead-end on that front, I decided to set that problem aside and deal with the second issue: I still needed to get into the dorm to change out my textbooks. I figured I'd work on getting into the dorm, and perhaps once I was there more options for the lost wallet would present themselves. So instead of heading for the back door, which required an ID swipe to get in, I walked around to the front entrance and went into the lobby (a public area). I then headed over to the door that led to my wing, and killed time by pretending to read the bulletin board on the wall nearby. Soon, another student came by and swiped her card to open the door. I hurriedly slipped in behind her before the door closed, knowing that most people ignored the signs saying to not let anyone else in after you. I ran up to my room, opened the door (thankfully I still had my keys!) and there was my wallet, lying on the floor in the middle of the room.

So what does any of this have to do with algebra? Well, compare my problem-solving strategies to the process of solving a system of equations. In my case, I had two variables: I needed to get into the dorm, and my wallet was gone. I started by trying to find my wallet – when solving a system, you start by solving one equation for one variable. I got as far as I could go on that path and eventually ended up with wallet = gone. I had to set that equation aside for a moment and deal with the other variable, just as you then switch equations in the system. I plugged “I don't have my ID” into the equation of “getting into the dorm” and solved that problem using what I knew about the building and the residents' laziness, and managed to get into my room (I solved for “I need to get into the dorm”). Once in my room, the first equation became solvable again, since my wallet turned out to be there – right where it had fallen out of my bag before I left the room that morning.

This may sound way too coincidental, but the truth is that algebraic reasoning is incredibly important for a lot of tasks that have nothing to do with numbers. The ability to rewrite an equation while maintaining its value until the answer presents itself is at the heart of all problem-solving abilities. I often remind my students of the larger usefulness of the skills learned in math class by encouraging them to “take the numbers out of it.” What exactly are you doing in a broader sense, and how might you be able to use those skills in other situations? Give it a try – you might find that you like math more than you thought.

Nice analogy and conception of equation manipulations. Thanks for sharing!
That is a great story. Here is another. I volunteered to help with the collection of funds for a talent show at my school where I teach mathematics. No one told me to keep track of how many "student" tickets or "adult" tickets were sold. The "adult" tickets sold for a higher price than the "student" tickets. I did know that in advance and charged the correct price for each ticket. After the performance I turned in all of the collected monies along with a head count (number of tickets sold.) to my surprise, I was asked for an exact breakdown of the ticket sales. I thought for about 5 seconds and announced, "No problem !". Using Algebra I was able to recreate the situation. Here is what I did.

I let the variable A represent the number of adult tickets. These sold for \$5 each. I let S represent the number of student tickets. These sold for only \$2 each. We had an attendance of 346 people and we collected a total of \$1022. I wrote the following equations:

A + S  = 346             (Total attendance)
5A + 2S = 1022         (Total sales)

So, multiplying the top equation by 2 yields 2A + 2S = 692
Then, subtracting this from the sales equation gives 3A = 330
So, dividing by sides by 3 gives A = 110
Substituting 110 for A in the first equation gives 110 + S = 346
So, by subtracting 110 from both sides yields S = 236.

Thus the talent show was attended by 110 adults and 236 students.
The adult receipts were a total of \$550.     (110 times \$5)
The student receiots were a total of \$472.  (236 times \$2)

The sum of \$550 and \$472 is \$1022.  (Check)    Also, the sum of 110 and 236 was 346.  (Check)

I (with the help of Algebra) was able to solve the problem in a matter of less than one minute. I was surprised by how many people were amazed that this was such a simple problem for me to solve.

I guess the treasurer of the PTA was pretty clever to put an Algebra teacher in charge of collecting the money for the talent show.
Awesome story! It also reminded me of similar stories from my own life when I had to solve problems. The truth is I use the problem solving strategies I learned in Algebra class everyday.