Algebra without the Letters

This document was originally an appendix to my Master's Thesis at Minot State University entitled "Algebra Anxiety".

An Introductory

Lesson for Students

Who are confused

by the letters

used in

Algebra

Art Rude

Copyright 1997

Algebra, Without

the Letters

To the Student:

For some students, the presence of letters in Algebra is confusing. All the way through the elementary grades, Math (or Arithmetic) meant numbers, and letters were used for the construction of words. That was true every day of every year, but now this math teacher wants to use letters in a way, . . . well, quite frankly, they just don't belong.

This lesson will teach you the processes of beginning Algebra, without using the letters where they don't belong. By the end of the lesson, it will become clearer to you why letters are used, and you may even come away with a real appreciation for why Algebra is used at all.

What's in the box?

We will play a little game on paper. The game is called, "What's in the box?" To play the game, let's back up to first grade. You may have seen math problems like this:

This is easy. The number 2 would fit in the box because 2+3 = 5. In grade school you probably answered a great number of these problems, and thought them easy. But let's use this as an example of how to play the game. "The game" is based on something that you learned on the playground at an early age. Some may have called it a see-saw, but I grew up calling it a teeter-totter. It was my favorite. There were so many things you could do with that teeter-totter. For instance, to keep it balanced, if there was a big kid on one end you could load it up with little kids on the other side. If one more kid would get on one side, you could balance it again by putting the same size kid on the other side. In general I think all of us have experienced what I call the teeter-totter rule:

The equal sign (=) means exactly the same thing as a balanced teeter-totter when you think about it. Just as the weight (more correctly, torque) has to be the same on both sides of the balance point of the teeter-totter , the values on each side of the equal sign must have the same weight (or value). I picture it like this:

I picture the equal sign as having the fulcrum, or the balance point of a teeter-totter immediately below it, because that is the way an equal sign works, like a balanced teeter-totter.

There is one more rule from nature that we need to apply. That everything usually has an opposite, and usually the opposite will reverse the effects of the original event. For instance, if we walk down three flights of stairs, we can reverse the effects of walking down, by walking up three flights of stairs, which returns us to the original position. If we drive our car 5 miles west of town, we can get back to town by driving 5 miles east. In general,

now lets look at our original problem ( the easy one we could have done in our heads ) and see how we can make use of these two ideas to get the problem to solve itself. First of all if I could get the box all by itself on one side of the teeter-totter, then what ever is on the other side of the teeter-totter must be the same as what is in the box.

In general:

Now let's do another problem:

Ask yourself, "What is keeping the box from being by itself"?

Answer: the twelve, and the twelve is being added to the box. So to get rid of the 12 we must do the opposite of adding twelve, which is to subtract 12.

I must remember to keep the teeter-totter balanced, so I must subtract 12 from each side.

Because adding 12 and subtracting 12 are opposites, they cancel each other out (or are equal to adding zero), so we get:

Now if you go back to the original problem, and put 15 in the box, it works! 15 + 12 = 27. We didn't even have to think about it, just follow the rules of the game and it is like the problem answered itself.

The game also works with subtraction:

Once again, ask yourself, "What is keeping the box by itself"?

The answer is the five, but notice that this time the five is being subtracted. Don't panic, it works the same way, we do the opposite of subtracting is adding, so we add 5 to both sides to keep the teeter-totter balanced.

Once again subtracting 5 and adding 5 are opposites and cancel each other out, or are the same as adding zero, so:

If 23 is put in the box of the original problem, it works because 23 - 5 = 18.

You might have noticed that the teeter-totter principle works regardless of what side of the teeter-totter you happen to be standing on. Likewise the teeter-totter rule works if we turn around our problem.

To play the game , ask yourself; "What is keeping the box from being by itself"?

Answer: the 18. Because the eighteen is being added to the box, we do the opposite or subtract 18.

Now is when we must remember that in order to keep the teeter-totter balanced we must do the same thing to both sides of the equal sign.

So:

So, because (18 - 18) is zero, or adding 18 and subtracting 18 are opposites, they cancel each other out, and the box is left by itself, with the answer on the other side.

or,

If you put 16 back in the original box, it will work, because 34 = 16 + 18.

Once you know how to "play the game", simply by treating the equal sign as a balanced teeter-totter, and doing the opposite, these problems answer themselves!

Here are a few more problems for you to practice on. Go ahead and draw a triangle beneath the equals sign to remind yourself it is like a teeter-totter, then do the opposite of whatever is keeping the box from being by itself. Remember to do the same thing to both sides of the equal sign, so the teeter-totter remains balanced. When you finish the box should be by itself, and what is on the other side is automatically the correct answer.

Now that you have an idea of how "solving " works with simple problems, let's move on to problems where the answer is not so obvious. To do so, we must first recall some of the math that you learned in elementary school. Usually, in third grade is where most students are first taught multiplication. It is usually taught as if it is something new to the students, but really it is simply a shortcut for adding the same number over and over again.

For example:

5 + 5 = 10 = 2 times 5

5 + 5 + 5 = 15 = 3 times 5

5 + 5 + 5 +5 = 20 = 4 times 5

or we could say, 2 five's = 10 which implies 2(5) = 10. This is the way we usually write 2 times 5 in upper level mathematics.

etc.

So if we follow this pattern, 5 would imply 5 times the box, (or 5 boxes).

So 5 = 15, or five boxes equals 15.

Can we use the teeter-totter rule to solve this problem? The answer is yes, if we remember that the opposite of multiplying is dividing.

So, we divide both sides by 5,

however, if you look at a computer keyboard, or on many calculators, you will not find a divide sign like you used in elementary school, instead you will find a (/) (slash) to indicate division.

So in our example we write:

On the left side, 5 boxes divided by 5, should give us what is in just one box, (or multiplying by 5 and dividing by 5 are opposites, and "undo" each other)

and 15/5 (fifteen divided by 5) is 3

Once again the teeter-totter principle works, as 5 boxes with 3 in each would equal 15, or 5(3) = 15

Try:

Again, we look at the problem and ask ourselves, "What is keeping the box from being by itself?"

Answer: the four. But now we have to recall what the four is "doing" to the box. It is multiplying the value. In other words, four boxes must be equal to four times what is in one box.

and 28/4 (twenty eight divided by 4) equals 7 so,

and if we try 7 in the original problem it will work because

28 = 4(7)

twenty eight = four sevens (or 4 times 7)

Try some more:

So now what happens when we have more than one thing being done to the box at one time? As in:

I ask the same question, "What is keeping the box from being by itself?"

The answer is a 5 and a 3. Now if I remember to do the easy part first (add or subtract), I say, "let's get rid of the 3 first". The problem becomes:

On the left side, adding 3 and subtracting 3 cancel each other out and have the same effect as adding zero or nothing.

but 18 - 3 = 15, so our problem becomes:

We now ask ourselves the same question we have all along, "What is keeping the box from being by itself?" The answer is the 5, and the operation is multiply. So now we do the opposite of multiplying by 5, which is dividing by 5, and we do it to both sides to keep things balanced.

On the left side, 5 boxes divided by 5 should show us what is in one box, (or, multiplying by 5 and dividing by 5 are exact opposites and will "undo" each other) so:

but 15 divided by 5 is 3 (15/5=3), so

Let's do another one:

Ask the question, "What is keeping the box from being by itself?" Answer: the eight and the six. Remember we will make it easy on ourselves by doing the easiest operation (add & subtract) first. That means that I will get rid of the six first, (it is currently being subtracted) so I will add 6 to both sides first.

On the right side, subtracting 6 and adding 6 cancel each other out, and 58 + 6 is something I can do so,

On the left, 64/8 means 64 divided by 8, which is 8. On the right 8 boxes divided by 8 should be what is in one box (or multiplying by 8 and dividing by 8 are opposites and will cancel each other out)

If you try 8 in the original box it will work because 58 = 8(8) - 6.

Try some more on your own. Remember to draw in the teeter-totter, and refer to the rules if you have trouble remembering what to do.

Remember the

1. Ask, what is keeping the box from being by itself?

2. Always do the opposite to get rid of something (subtract is the opposite of add, and divide is the opposite of multiply).

3. Always do the same thing to both sides, to keep the teeter-totter balanced.

4. If there is more than one thing to get rid of, start with the easiest (add or subtract)

Here are some practice problems:

I hope you see that this can be used to solve some difficult problems with ease. You might have noticed that at the end of each of these problems the box is usually equal to a different value each time. In other words the boxes are not the same.

That is what the letters are used for in algebra. The letters are simply names for the boxes, a way of telling them apart. If you would go back through the problems you just solved, and each time you come to a new problem, write a letter in the box for that problem. For instance, for the first problem on this section, we wrote:

In algebra we write:

or

In Algebra we call the letters variables, which just means they are like the boxes, some value can go in where the variable (or letter) is, just as in elementary school we learned to put the answer in the box. If it looks to you as if the letters don't belong, just remember that the letter (or variable) is just a name for a box that you are trying to find the answer to.

In fact when you start in a traditional algebra book, if the letters bother you, just draw a box around the letter to remind yourself what you are doing. By doing so, the problem:

becomes

which we can use the teeter-totter game to solve.

Please remember this is not all there is to algebra. This is just a method of seeing how algebra works without having to deal with the letters. Now that you have an idea how algebra can be used to solve a problem, and hopefully you see a purpose to the letters, we should be ready to start a more traditional algebra course, and it should make sense to you. If it is still confusing to you, please read through the lesson again.

The examples used in this lesson are designed to show you how the process works, they are not designed to be all inclusive. There are several basic ideas from beginning algebra that need to be learned that are not dealt with in this lesson. If you can now approach the course with a knowledge of what variables are, and not be intimidated by their presence, you can probably learn algebra. If you can see there is a purpose to solving algebra problems, we can now progress to a traditional, introductory algebra course.

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