Backwards Identities and Identities in Context

I ran into an understanding problem with my seventh grade son a while back. (I teach him algebra at home because it would really screw up his
schedule if he took the 8th grade class at school. The school was flexible about this and it was better than the online course the school
offered.) He had to take an equation and change it into the slope-intercept form of a line:

y = mx + b

(where 'm' is the slope and 'b' is the y-intercept)

and then graph it. When he rearranged the equation, he came up with:

y = 3 - x/2

He couldn't figure out what 'm' was. We've been over this in the past, but not so much in context. He couldn't "see" the equation properly. I
know that I instantly see things in equations that he doesn't see, but I found (once again) that I have to reinforce the message. I think a big
problem is that the identities are so simple. What could be simpler than:

a/1 = a

What about what I call backwards identities, like:

a = a/1

My son know this because I've drilled it into his head. Everything is a fraction or a rational expression; if you don't see a dividing line, you
can put one in there. This was big back when I taught him about dividing fractions. What if you have something like this:

5 / (3/4)

The invert and multiply works fine if you know that 5 = 5/1.


Another simple identity is: a*1 = a

What about: a = 1*a ?

for something like x/2, you can look at it as (1*x)/2

Although this :

(a/b) * (c/d) = (a*c)/*(b*d)

is not a basic identity, many don't see it in reverse.

I could rewrite

(1*x)/2

as

(1*x)/(2*1) = (1/2)*(x/1) = 1/2 * x

By the time his textbook gets to the slope-intercept problems, they assume that these simple(?) things are not an issue. But, I've mentioned before that it took me until about my junior year (trig class) in high school before I really mastered all of these things.


(Side note: I hate text math even though I'm forced to do that in my programs. Math is 2D. It's graphics, not text, and I can "see" so much more when I write equations on a piece of paper. Dividing lines are horizontal, not slashes, and there is no place for a '^' in math. Kids need to do
math with a pencil and paper. OK. I got that out of my system, almost.)


So, lately, I've been trying to teach my son how I "see" equations and all of my tricks or understandings I had to figure out myself. I've talked
about these things a little bit in the past. I'll see if I can hit the highlights.

When you look at an equation:

1. Look for the terms. These are the big "chunks" of the equation that are separated only by a '+', '-' , or '=' sign. It doesn't matter how complicated the terms are, circle them and ALWAYS include the preceeding '-' sign with the term.

All equations look like this:

... (term) + (term) = (term) + (term) ...

Since the sign belongs with the term, there will always be a '+' between terms. I've told students to look at it this way:

25 - 3x is really 25 + (-3x)

and

15 - x is really 15 + (-1*x)

This helps when you get to something like:

15 - 3(x-2) or 15 - (x - 2)

Many students just don't know what to do with the '-' sign. If you think of it as:

15 + (-3)(x-2) or 15 + (-1)(x-2)

Then there should be less mistakes.


2. You can change the position of any term on one side of the "=" sign just by moving it. (a+b = b+a) The key is that you have to keep the sign
with the term. (If you don't see a sign, it's a '+'.)

Some students have a hard time doing this:

3 - 10x = -10x + 3

They just don't know what to do with the '-' sign. I automatically write equations so that they don't start with a minus sign, but I should probably not do that.


3. You can change the position of a term from one side of an equal sign to the other by changing the sign of the term. Even if you see
complicated integral signs floating around, this still works. I'm not sure why, but I've always thought of it as swinging (?) a term from one
side of an equation to the other and changing the sign. I've noticed that because my son can be sloppy, he makes simple mistakes when he formally subracts or adds terms from both sides. My "swinging" method reduces writing and his sloppy mistakes.

This brings up another topic. I've been really strict with my son about clearly writing each algebraic change on a separate line. Doesn't
everyone remember this from 7th and 8th grade? I've told my son that this prevents silly mistakes and makes it easier for the teacher to read. He
doesn't like to rewrite equations, so he makes the next change to the same equation. If he makes a silly mistake and the teacher can't see it,
then it will be marked completely wrong. If it's clear, he might get partial credit.

The next level is to "see" the terms.

4. Find and circle all of the factors in each term. these are the things that are simply multiplied and divided together. If you don't see the
dividing line, divide the term by 1. (a = a/1) If you see multiple dividing lines, make sure you know what is the overall numerator and
denominator. All terms are fractions or rational expressions. Note that it's harder to "see" the factors of a rational expression in text using
'/' as the dividing line.


(Side Note 2: Order of operations is really all about math as text. The first time it was formally taught to me was when I started to program.
You should never see anything like 4/8/2 in math. Even with programs, nobody should ever write something like that even if you know exactly how the compiler handles order of operations. It should be written as (4/8)/2. But that's really ugly. Paper and pencil math is beautiful.)

Factors are easy to see if you're talking about somehting like 3x/2. The numerator has (3)(x) and the denominator has (2).

But what about [3x^2(3x-5)^1/3] / [(x-5)(y+1)]

Note that I can't "see" this term as well as if it was in a nice 2D, graphic form.

The exponent always belongs to the factor just to the left. My son has had trouble with this. It's (3) * (x^2), not (3x)^2.

You have to see the factors instantly: (3), (x^2), (3x-5)^1/3 for the numerator, and (x-5), (y+1) for the denominator.


5. Any factor in the numerator can change position with any other factor in the numerator. Likewise for the denominator. It's surprising that
when terms get more complicated, students freeze up. They don't see it as a*b = b*a.


I think that order of operations affects this. It's driven into kids heads. They are given problems where they have to evaluate expressions in one particular order. What they think of as "understanding" is really a rote process. Order of operations is mostly for computer language compilers because they have to have some rote process to correctly evaluate expressions from left to right.

When I look at a complicated term (or equation), I never think of order of operations. I think about what I can do with it. I can rearrange factors however I want in the numerator or denominator as long as I know the rules, and the rules aren't defined by order of operations.


Here is something else to "see".

6. You can move any factor to or from the numerator or denominator by changing the sign of the exponent. If you don't see an exponent for a
factor, then it's 1. These sorts of steps come from what I call backwards identities or rules. Students have to know that the rules work both
ways.

My son knows very well the distributive rule, but he still has a hard time factoring:

(3x - 12x^2) = 3x(1 - 4x)

He doesn't feel comfortable with what to do when the 3x is factored out. He has to realize that 3x = (3x)*(1), a backward identity.


Most learn something like this:

x^(-1) = 1/x

but can't see that this means that you can bring a factor up from the denominator into the numerator by changing the sign of the exponent; the
backwards form of this rule.

One might talk about order of operations when you have two operations next to each other, but to apply them to a term or an equation is wrong.

This is a little bit of how I see equations. I hope it helps.