There is nothing really new in this lesson. It is the cumulation of the other lessons in this chapter. We will do some reducing, a little multiplying, and then some combining. Of course, the regular order of operations needs to be followed. First do functions (roots and exponents). You might need to multiply unreducible roots to make a root that will reduce.
After reducing, you may see like roots (tags) that you can combine.
5@2*@6-8@3 5@12-8@3 5*2@3-8@3
10@3-8@3 2@3
3@5*@10+7@2 3@50+7@2 3*5@2+7@2
15@2+7@2 22@2
3@12+5@27 3*2@3+5*3@3
6@3+15@3 21@3
4@8-2@18 4*2@2-2*3@2
8@2-6@2
2@2
AlGeBrA: PolynoMIAls
Polynomial means “many terms.” In other words, many compound numbers combining with each other. These terms have certain limitations and must fit this pattern, axn. Any number can be “a,” but the exponent, n, of the variable, x, must be non-negative (0 or greater) and must be a whole number. Also, the terms can be combined or multiplied, but not divided. Therefore, a/x is not permitted. Don’t be frightened. These limitations actually make our job easier because we will have less to deal with.
Polynomial expressions will prepare us for equations, although most students find polynomials more difficult than equations. So, in a way, this chapter represents the final ascent up Math Mountain. After this, it gets easier, not downhill, but more like a gentle slope up a long ridge.
Because we will mostly be starting and stopping with variables, we will not be able to get a single number for our answer. Rather, our answers will look like 2x or x+4/6 We will take problems as far as they can go and then just stop. Don’t stretch this next statement too far, but our main goal in this chapter is not so much to get a single answer as it will be to learn the processes and use the Actions correctly. These problems are really part of larger problems for those who go further in math. Therefore, a student must correctly learn the steps along the way if s/he wants to arrive at the right answer.
And speaking of “right answer.” That phrase can take on a whole new meaning when we leave the world of arithmetic and enter the world of algebra.
Thinking in Algebra
To understand algebra you must think like an algebratician. Algebratician?! Is that even a word? No, but if it were, it would mean a mathematician generalized. Imagine Einstein and Sherlock learning 2+2, then they learn another problem, 2+3. Then they learn another problem, 3+3, and so on. What would happen when they encounter 200 + 200? If they did not know how to think in algebra, then they would have to learn about adding these two numbers as if they had nothing to do with all the adding problems they had already solved. So, you have halfway done algebra already by generalizing about adding and all the other operations, but now you need to understand adding x + x.
We have also been introduced to algebra in two other ways. When we worked with tags and exponents we used variables in the examples. We did algebra, but without understanding it from an algebra point of view. Also, when we worked with formulas, we were using a prime example of algebra. Rather than re-inventing a new formula for every word problem we meet, we use one that someone else has already figured out. The variables in a formula are the placeholders that anyone can use at anytime to drop in their own custom numbers for their own unique problem.
(Motivation alert: Too often students are solving somebody else’s problems instead of encountering and experiencing their own. A formula takes on more meaning when it helps you solve something important, rather than simply being a means to someone else’s end.)
So algebra helps us expand our thinking beyond this problem with this number to working with patterns and any number at all. Of course, this is a reflection of life.
What if my mother tells me to wash my hands before supper tonight? What if she told me to wash my hands before supper last night and the night before and the night before that? What if it never occurs to me to generalize her command from this night to all nights?
That would be all right for a night or two, but wouldn’t you consider me a slow learner if I never caught on after 5 nights or 50 nights?
Just as my own thinking abilities should extend my mother’s command from a few nights to any and all nights, so algebra extends arithmetic from common numbers to any number, even unknown numbers. Of course, the answer to such a question usually looks like “any number” instead of a specific number, but we try to get as close as we can.
Let’s look at a problem like 3+3. Since both numbers are the same, we can rename 3 to x and say x+x. Of course, we can do this with any other two numbers that match.
But now what? What do we do with x+x? Since x could be any number, I can’t arbitrarily say x+x=6 or 10 or -25! What if x is 7? Then x+x=14. My three earlier guesses would be wrong. In fact all my guesses (even 14) could be wrong if I don’t know what x equals.
So what good is algebra if my odds of guessing the right number is 1 in infinity?
Algebra is not about guessing the right number. It is all about figuring out the right pattern which sometimes happens to be a number.
Let’s get back to our example, 3+3 and x+x. What do we know about our example and what can we generalize about it? Isn’t 3+3 the same as 2^3? Yes, and isn’t 4+4 the same as 2^4, and 5+5 the same as 2^5, and so on? Because x+x fits that pattern, wouldn’t it fit the 2^x pattern as well? Wouldn’t ANY number fit those patterns? Wouldn’t ANY number added to itself equal two times itself?
Therefore, we have discovered something. x+x=2x. We can now say the answer to the problem, x+x, is 2x. If “answer” seems like a strong word to use, we could also say 2x is another way of looking at x+x, or we could also say, x+x can be rewritten as 2x. In and of itself this discovery seems small, but it could be the key to helping us crack a bigger
problem. More importantly, this simple problem is helping us right now think in algebra.
Let’s pause for a moment and think about what an “answer” is. We have been conditioned to think only one way. That is, we think an answer can only be a single number. Usually, that is the case, but why? More often than not, but not always, a single number is desirable because we want to know how many cups of flour to put in the cake batter, or how many tickets were sold, or what the net worth of a company is at the present moment. One number is easy, concrete, and we can nail it down.
However, is it always best to have just one number? A high school graduate takes the SAT and earns a score of 1200. Did he do reasonably well on both math and English, or did he do superb on one and bomb the other portion of the test? 1200, a single number, does not, and cannot, tell us. Something like 800+400 or 600+600 would be more helpful.
Another example, what if I want to write out a gazillion math problems for my eager math students to do in one night? I would write a computer program that would work something like the following.
1) Generate a list of random numbers. 2) Pick two numbers at a time (a and b), print them with a + sign in between. 3) On the answer key, reprint the problem and the calculated answer.
This brings us back to the use of variables as placeholders, but notice that the answer to my problem are the two numbers that are a problem for the student. Just printing an answer would be printing a list of random numbers. Instead, I need a list of random problems in the form a+b. The “answer” is a matter of perspective. The “answer” is what is most helpful, useful, and informative.
So now let’s go way back to the x+x example. Let’s say you have collected a lot of data from doctors around the world about eye exams. You then need to make a report about the number of eyes examined. Simple, you think, just add the number of exams to itself to figure the number of eyes.
However, in this realistic, but greatly simplified scenario, you discover that your computer will take 5 days to generate the report if you add all your numbers. Your CPU is not optimized for adding, but it excels at multiplication. Thinking algebraically, you reason that 2x is the same as x+x. The computer can generate the report in 5 hours, in time to meet the deadline. What did not seem like an “answer” before is now your lifesaver.
There is a cost to everything we do. It pays to discover new ways of thinking and doing because one day the reward will be greater than the cost. We have no idea now how our ingenious insights might build a bridge, launch a rocket, or save a life, but all the technology we have came from looking at problems and answers differently. Think outside the box of arithmetic. That is thinking in algebra.
As Few Variables as Possible
It takes little comprehension to see what is going on with 3+3. Count out three dots *
* *, then extend it with three more dots * * * for a total of 6 dots * * * * * *. There it is, right there before my very eyes.
But what if the numbers are different? What if we don’t know either of the numbers?
We can’t say x+x, because we can’t be sure they are the same. We need to say x+y, because it might be 5+8 or 2+7 or -1+4. Therefore, I can’t say the answer is 2x or 2y. I am forced by simple logic to say x+y=z. (z could equal x or y if either of them is 0, but the odds of that happening are low.)
So what do I do with x+y=z? Nothing. Too many unknowns. Too little specific information. I can’t nail anything down.
Let’s say a student of yours thinks they can finally stump you. He asks you, “What is x+y?” You ask, “What is x and y?” He says, “I don’t know.” Then you say, “Oh, I know the answer then! It is z.” He asks, “Really? What is z?” You say, “I don’t know numerically, but I know it is the right answer to your question!”
The point is that there are limits to algebra and we are the ones that need to put limits on it. If our problem does not have enough concrete and unchangeable numbers in it, then we can’t deal with it.
The basic rule of thumb is to avoid making a new variable whenever possible, because for each additional variable you will need another equation, if you want to find its value.
This dilemma is the topic of the last chapter on systems of equations.
Until then, in our math problems and word problems, we work with just what we are given and relate the information we have to itself. For example, a common word problem goes something like this. The second box has twice as many widgets as the first box. If their sum is 24, how many widgets are in each box?
The first inclination is to write, x+y=24. However, if we are ever able to write the problem with fewer variables, we should. We can relate the second box to the first box by describing it as 2x. Why not?! After all, the problem said the second box has twice the amount of the first!
So now our problem can be written as x+2x=24. This problem we can solve when we get to the next chapter on linear equations. For now, the answers are 8 and 16.
Avoiding making new variables applies to the type of problem which we will delve into next, multiplying with variables. We have already multiplied variables by merging them.
5x*2y=10xy, but what about variables in groups? (x+5)(x-2)
Without inventing any new variables, we can do this operation using the distribution.