kitchen table math, the sequel: My Favorite Identities (In Context)

Monday, July 16, 2007

My Favorite Identities (In Context)

Back when I taught college algebra, I liked to tell students what I see when I look at a complicated equation. Perhaps it will help others.

The first thing I look for are the major "chunks" or terms that are separated by a plus sign, a minus sign, or an equals sign. I would write things like this on the board


3(x-1)^2/(45x) - (10x+4)/2 + 55 = 1/(x^2-1)


and tell them to circles the "chunks" or terms. I told them to always include the sign with the term. If a term doesn't have a sign, then the sign is '+'.


Identity [a = 1 * a]

25 = (+1) * 25

If you don't see a sign, it's a plus. Always remember that 1 is a factor.


The first term is

+3(x-1)^2/(45x)


The second term is

-(10x+4)/2


The third term is

+55


and the fourth term is

+1/(x^2-1)


This is the first level of breakdown of any equation. You have to see the terms.


You can move any term (no matter how complicated) from one side of the equals sign to the other by changing its sign. Remember to change and take the sign of the term with you. None of the other terms change at all when you do this.

I haven't said what I wanted to do with this equation. I'm just seeing what I can do if I want.


All terms are fractions (rational expressions) and have a numerator and a denominator.

For the first term, the numerator is +3(x-1)^2 and the denominator is (45x). The third term has a numerator of 55 and a denominator of 1. If you don't explicitly see a denominator, it is 1.


Identity [a = a/1]

This is the second level of breakdown for any equation. You have to see the fractions.


Next, look at each term (numerator and denominator) and find all of the factors. These are things that are multiplied together. Remember that exponents belong to the factor.
For the first term, the numerator has (3) and (x-1)^2 as factors. (I suppose that you could say that (x-1)^2 is really two factors, (x-1) and (x-1), but I don't think that kind of factoring is necessary. Keeping the exponent has its uses.) The denominator has the factors (45) and (x). If you see something like x-5 all by itself in a numerator or denominator, I really, really like to see it surrounded by parens. This makes it clear that it is a factor and must be treated as a whole. Students do weird things with something like "x-5" if you don't enclose it with parens.


This is the third level of breakdown for any equation. You have to see the factors.


You can change the position of any factors in the numerator or any factors in the denominator.

Identity [a*b = b*a]

Term 1 could be:

+(x-1)^2*3/(x*45)


Which brings up another issue. Mathematicians like to order factors in a particular way. This is not required, but it's a good thing to know. The simplest rule is to put the number factors (constants) first and the variables after that. Some like to sort the factors based on the size of the exponents, or they like to keep the variables in the same order for all terms.


All factors (actually, all numbers and variables) have exponents.

Identity [a = a^1] (This is not really part of the basic math identity set)

If you don't see an exponent, it's a one.


You can move any factor from the denominator to the numerator, or vice versa, by changing the sign of the exponent.


Identity [x^(-a) = 1/x^a]

1/(x-2) = (x-2)^(-1)


(x-2) is a factor. Its exponent is 1. I can move it to the numerator by changing the sign of the exponent. You need to be able to make these changes (for whatever reason) without a second thought. It has to be automatic.


With these basic identities and breakdown, you can push terms and factors around to meet your needs. This is the lowest level of equation manipulation.

12 comments:

Instructivist said...

Great explanation!

"If you see something like x-5 all by itself in a numerator or denominator, I really, really like to see it surrounded by parens. This makes it clear that it is a factor and must be treated as a whole. Students do weird things with something like "x-5" if you don't enclose it with parens."

This cannot be emphasized often enough.

I wonder if you could provide a clear explanation of how to drop parens when negatives are involved as in this example:

(6x + 3)-(2x + 4)

The distinction between a minus and neg sign can be confusing.

SteveH said...

(6x + 3)-(2x + 4)

I would still like the students to find the terms. If you remember to include the sign with the term, then the second term would be

-(2x + 4)

This is the same as

-1*(2x+4)

The full expression would be:

(6x+3) + (-1)*(2x+4)

A minus in front of any factor is the same as plus sign followed by -1 times that factor. In fact, I usually like to visualize a negative sign as -1.

Then, if they multiply it out, they get

-2x - 4


Next, the first term is really

+1*(6x+3)

which stays the same when multiplying through.

6x + 3

Then combine them together to get

6x + 3 - 2x -4


In some ways, these steps are trivial, but in other ways they really aren't. When I taught, I always tried to slow down and think about everything that goes through my head when I do a problem. Now that I am teaching my son, I've noticed that I still take some things for granted.

Anonymous said...

I will be saving this in the Steve folder for my son to look at in the fall.

Instructivist said...

Nice illustration. Thank you, Steve.

I can imagine that a student learning this might wonder if (6x+3) + (-1)*(2x+4) is a case of converting a subtraction problem to an addition problem by adding the opposite. Then what happened to the plus sign here 6x + 3 - 2x -4?

VickyS said...

Great post, Steve. You should have just convinced them to hire you as the math teacher at your son's old school!

Are you actually teaching this to your 6th grader? Man, we're way behind!

SteveH said...

"Are you actually teaching this to your 6th grader? Man, we're way behind!"

Not for complicated equations. We're doing it for simple equations, and we haven't gone very far yet with solving simple equations for 'X'. In coming back to our public school (which is now much more flexible), he will be in sixth grade for most classes, but seventh for math. He has a lot of work to do.

It will be interesting to see what happens. I'm trying to force him to tie everything to basic identities and not just say things like "it goes away" for canceling factors.

SteveH said...

"Then what happened to the plus sign here 6x + 3 - 2x -4?"

I'm always surprised by my son's questions. I've dealt a lot with what you might call remedial algebra issues, but not much with first time issues. I was working with my son on negative exponents and saying that

x^-2 can be written as 1/x^2

He wanted to know where the '1' came from. Even when I wrote:

x^-2 = 1*x^-2 = 1/x^-2

he wasn't fully convinced. I suppose it's hard to realize that there can be no other solution.

Catherine Johnson said...

You can move any term (no matter how complicated) from one side of the equals sign to the other by changing its sign.

I FIRST learned this in Mathematics 6 ("Russian Math")!

I've been trying to teach it to Christopher, but I probably need to break it down and do some formal instruction and practice....

Scratch "probably."

I need to do this.

Catherine Johnson said...

This is fantastic!

Catherine Johnson said...

If you see something like x-5 all by itself in a numerator or denominator, I really, really like to see it surrounded by parens.

Good advice.

I'm going to label this post "Greatest Hits," fyi - makes it easier to find.

Catherine Johnson said...

(x-2) is a factor. Its exponent is 1. I can move it to the numerator by changing the sign of the exponent. You need to be able to make these changes (for whatever reason) without a second thought. It has to be automatic.

Very helpful.

This is the kind of information I need at this point - what procedures are automated for algebra??

Catherine Johnson said...

I will be saving this in the Steve folder for my son to look at in the fall.

It will be in the "Steve" folder on ktm-2, as well.