kitchen table math, the sequel: Backwards Identities and Identities in Context

Monday, February 2, 2009

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*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)


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

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.


concernedCTparent said...

Steve- this is fantastic. Thank you for taking the time to put it together. It's certainly a keeper.

Anonymous said...

--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'm not sure what you mean by "i'm not sure why"

do you mean e.g. why is the equality 2 = 3x - 1
the same as the equality
2 - 3x = - 1

the statement is an equality; the two expressions on the left and right of the equals sign are equal.

any validly defined operation you perform to both the right and left maintains the equality. (i'll get to what'd valid in a bit)

so you can add -3x to both sides and maintain equality. when you add it to the left, you go from 2 to 2 - 3x. when you add it to the right, you add -3x or that is, subtract 3x from 3x, cancelling it out.

3x -1 -3x is -1.

Anonymous said...

I'm noticing the same problems.

One of the big differences between my son's Gr. 7&8 experience and mine is the total lack of using the properties to explain how the student is factoring and simplifying equations.

For example if I had

2w+2l=P, let l = 2w+7, P=74 find w
I would have had to write:

2w+2(2w+7)= 74 substitution l=2w+7
and P=74

2w+4w+14=74 distributive property

6w+14=74 simplify

6w+14-14=74-14 addition property of equality

6w=60 additive inverse property

My son has no requirement at all to explain his work...which deprives him of the opportunity to exercise and improve his understanding of the properties.

SteveH said...

"i'm not sure why"

I meant that about the word "swinging". I have this image of the term following a curved path from one side to the other. I suppose I shouldn't have mentioned it, but I seem to translate things into movements. If you give me a list of states to remember, I will remember them by imagining myself flying from one state to another on the map. I can more easily remember the criss-cross flight pattern.

Barry Garelick said...

Yes, the whole number divided by 1 trick works well. My daughter has a problem in expanding expressions such as 1/2(x + 4). She can see that the first term is 1/2 (x), but has difficulty with the second term. When I remind her it's fration multiplication and write the 4 as 4/1 she then sees that the answer is 4/2.

Steve's post drives home the value of basic skills. When you work with adding and subtracting negative numbers, you can then apply such knowledge with ease when working with equations. My daughter had to brush up on what she learned about negative numbers when working with an equation and ending up with -15 - 2 on one side. She remembered the trick her teacher had taught her in 7th grade (and which Steve talks about here) and she wrote it as -15 + (-2) and got it right.

Anonymous said...

It is really important to remember that these things aren't obvious to students, even older students!

I still remember working with one algebra challenged college sophomore who could not solve a linear equation. I worked with her and got it to the point where it looked something like: 15 = 3x (but with ugly chemistry numbers).

She couldn't see how to solve it! I finally realized that her problem had to do with where x was -- if it wasn't on the left, she didn't know how to solve for it. So I taught her the identity x=y so y=x, which was apparently brand new to her. It is sometimes hard to realize how a hole like that makes the entire problem impossible.

SteveH said...

How ironic.

I just picked up my son from school and he told me that (in spite of the fact that I'm teaching him math at home) he has to put together a 1 - 3 minute SmartBoard presentation for math class. All 7th and 8th grade students have to do this. Do you want to know what his topic is?

Order of Operations

SteveH said...

Apparently PowerPoint isn't good enough anymore.

Anonymous said...

"Order of Operations"

The *best* part is that order of operations is pretty much arbitrary (unlike, say, how addition works).

So there won't be any (or at least not *much*) *WHY*. Because the answer for *WHY* is ... *because*.

-Mark Roulo

SteveH said...

I asked my son what the order of operations was for:

3/4 * 12/21

I don't want to hear anything about left-to-right.

SteveH said...

This is what I found at

on order of operations, and it's typical of most sites. Google the topic yourself and see.


1. Calculations must be done from left to right.

2. Calculations in brackets (parenthesis) are done first. When you have more than one set of brackets, do the inner brackets first.

3. Exponents (or radicals) must be done next.

4. Multiply and divide in the order the operations occur.

5. Add and subtract in the order the operations occur."

I'm struggling to undo this sort of damage to my son. This is what he was taught, and it's wrong. He is struggling with the idea that things being taught in school could be wrong. He thinks I'm being fussy or picky.

Imagine, order of operations is being taught in a rote fashion without understanding. Just memorize PEMDAS. Just follow the steps.

It goes beyond that, however, since nothing "must" be done left to right. It's wrong.

One of the chapters in the Arithmetricks book talks about flexible adding, such as adding the following numbers:

7 + 4 + 3 + 16

The point is that you shouldn't do it left to right.

Then, when you get to algebra and have multiple variables and constants, order of operations has much less meaning. You talk more about simplifying, expanding, or solving.

SteveH said...

3/4 * 12/21

OK. One more. When you write this in a proper 2D mathematical form with horizontal dividing lines, there is no such thing as left to right. The modern push and misuse of order of operations centers around the calculator and its linear input. Calculators are anti-algebra. Order of operations was never stressed when I was growing up. It just wasn't a big deal. What's the big change?

Calculators. Linear. Left to right.

Algebra. Two-dimensional. Freedom.

VickyS said...

Steve, what a wonderful post. You should tag it with something though (maybe just "algebra" at a minimum) so we can all find it again when we want.

I swear our 7th grade boys are twins separated at birth. We are at *exactly* this place in his home study algebra. I showed him the linear equation you posted and he couldn't get the slope either. I had to go through the same explanation. Algebra I is such a great subject to's so clear to me that *this* is the year that lays the foundation for all the later material.

Thankfully, lgm, his textbook (he is in an online course at Northwestern) does, in fact, require him to state the property or identity involved in each step. And like Steve I require him to show each step separately. The identities are the hardest. To him, it seems so lame to learn that a*1=a. But what he hasn't come to appreciate is that a good "1" is a mathematician's best friend! For example, my older son was working through trigonometric identities tonight and sin^2x/sin^x was a very convenient "1"!

Man I love math. Guess that's why I'm here.

Anonymous said...

"Order of operations was never stressed when I was growing up. It just wasn't a big deal. What's the big change?"

As unhappy as you are about this, teaching order of operations isn't new. I learned "Please Excuse My Dear Aunt Sally" 25+ years ago.

I never learned the "left to right", which may be new with calculators, but this:

    4 + 2 * 6

is ambiguous without rules for order of evaluation.

Even without a calculator.

Learning to regroup (commutivity? associativity? I think this is commutivity) is important ... no argument there. But the expression above needs rules for order of evaluation.

-Mark Roulo

SteveH said...

4 + 2 * 6

Order of operations is used when you compare two consecutive operations.

If I have

4 + 6 + 2*(3+4) + 4

why do I have to do what's inside the parentheses first? Also, the left to right seems to be all about calculators.

VickyS said...

I'm with Steve on order of operations. Some of you remember my posts about trying to teach it to my Math Masters competition students. It has little to nothing to do with understanding the math, and any ambiguities should be resolved with parentheses. Once you get to algebra, OOP pretty much disappears as a concept with any real utility anyway, doesn't it?

SteveH said...

"Once you get to algebra, OOP pretty much disappears as a concept with any real utility anyway, doesn't it?"

I checked my son's Glencoe Algebra book and the left to right is in there. This might be fine for third or fourth grade, but by the time kids get to pre-algebra, they should be moving on. Isn't it ironic that they really push such a rote process when their goal is supposedly understanding. Once variables replace numbers, OOP needs to take a back seat. It's not about how you do something, but what you can do. It's more like a toolbox of skills than a process.

LSquared32 said...

You see order of operations differently in algebra, but you really need it. Indeed, a lot of mistakes made in algebra are due to not understanding order of operations. I see 2*3^n=6^n on student papers a lot, which is a classic order of operations error.

Left to right is important (and only important) when you have addition and subtraction mixed or multiplication and division mixed. We forget this, because to us all subtraction is adding a negative, and all division is multiplying by a fraction.

For example:
6-3+2=3+2=5 is correct (left to right)
6-3+2=6-5=1 is incorrect (which is what happens if you do the operations right to left). The way out of this, of course, is to see everything as addition, in which case order doesn't matter.

The same can be done (and is true of) multiplication and division, as in Steve H's 3/4 * 12/21, or even just 3/4*12. You either need some grouping (such as the grouping that happens when you write 3/4 and 12/21 as fractions), or you have the left to right rule, or you do something similar in your head to what you do when you change 6-3+2 to 6+(-3)+2

r. r. vlorbik said...

the standard "order of operations"
appears to have been designed
(or to have evolved) to make
manipulation of polynomial expressions
as transparent as possible.
i seem never to have realized this until
reading it in wu somewhere.
but it sure makes sense.
boy he's great.

"left to right" is a vicious lie: the technical term
for anybody who would perform the calculation
1 +(-2) + 2 + (-3) + 3
from left to right is "imbecile"
(smarter thqn an idiot; dumber than a moron).