kitchen table math, the sequel: Sign Rules

Tuesday, October 2, 2007

Sign Rules

OK. Maybe it's me, but here is what's in my son's Glencoe Pre-Algebra book (2008 ed.), section 2-2 (Adding Integers).

- - - - - - - - -

Key Concept Number 1

Words:

To add integers with the same sign, add their absolute values.

The sum is:

- positive if both integers are positive
- negative if both integers are negative


Examples

-5 + (-2) = -7

6 + 3 = 9

- - - - - - - - - - - - -

Key Concept Number 2

To add integers with different signs, subtract their absolute values.

The sum is:

- positive if the positive integer's absolute value is greater
- negative if the negative integer's absolute value is greater

- - - - - - - - - - - - - - -

It seems to me that you can't make it much more complicated.


And he hasn't even gotten to subtracting integers, which states:

"To subtract an integer, add its additive inverse."

(Then use one of the rules above.)



It sounds like they are defining a foolproof method, but how about

4 + (-4) = ????

Neither integer is greater.

and how are these rules going to help when you get to variables, like

X +(3-2X) = ?????

You can't combine

X + (-2X)

because you don't know the absolute value of X?

Perhaps they will get new rules when they introduce variables.

And what about decimals? The sections only talk about integers!


- - - - - - - - - -

The simple rules I taught my son during the summer are:

1. When adding or subtracting, you often get two signs in a row, like:

5 + (-2)

If the signs are the same, combine them into a '+'

If the signs are opposite, combine them into a '-'

This works for numbers or variables. I told him that there is no difference between a sign and an operation, since: -5 is the same as (0 - 5).


2. When multiplying, you often get two signs that get multiplied together, like:

(-5)(-2)

Remember that this is the same as (-1)(+5)(-1)(+2) = (-1)(-1)(5)(2)

If the signs are the same, combine them into a '+'

If the signs are different, combine them into a '-'


3. When dividing, it's the same rule:

If the signs are the same, combine them into a '+'

If the signs are different, combine them into a '-'

[You just have to make sure that the signs are for factors. For (-3)/(-2+x), the sign in the denominator is a '+', not a '-'.]


The same rule applies to all three cases and for numbers or variables. It couldn't be easier. [I changed some flash cards to add minus signs to many of them.]


There are some things you have to watch out for and they need to understand variations, but it all boils down to one simple rule about combining signs.

- - - - - - - - - -
Things to feel comfortable about:

1. 5 = -(-5)

2. 2-5 = 2+(-5)

3. 5 = (+1)(+5)

4. -5 = (-1)(+5)

5. (-4)/(5) = (-1)(+4)/(+5) = -1(4/5) = 0 - 4/5


I told my son that there is a case where two positives equals a negative, and he said "Yeah, right!"

I'm sorry, I couldn't resist.

29 comments:

Anonymous said...

2+(-5)

I've used a game form with students by calling positive and negative integers good guys and bad guys respectively.

Then I ask: Are there more good or bad guys here? Answer: bad guys. Q: How many more bad guys? Invariably they shoot back the answer: 3

Anonymous said...

Yeah, I agree those are some pretty complicated rules. I'm looking at a 50 year old book that has a similar, if not even more complicated description of absolute value and addition of negative numbers...

"The sum of a postive number and a nagative number having unequal absolute values is positive if the positive number has the greater absolute value and nagatie if the negitive number..." blah blah

If I had a kid that could understand those rules I'd teach them the definition of additive inverse from the get-go.

And then...
If the kid understands that
0 - 5 = -5

Then by the defintion of addition and subtraction you can show that
If 0 - 5 = -5, then (5) + (-5) = 0

It's just another "fact family."

I'd use decomposition to show that

(5) +(-2)--->
(3 + 2) + (-2)---> (the additive inverse rears its head)--->
(3)+ (2 + (-2))

If they are going to skip "why" it is that their rules work out this way then the most efficient thing is to employ a more user friendly set of marching orders.

Catherine Johnson said...

If the signs are the same, combine them into a '+'

If the signs are opposite, combine them into a '-'


omg

you don't know the pain we went through on this topic

the 6th grade accelerated math course OPENS with this AND WITH THE DISTRIBUTIVE PROPERTY

Kids fall off the track RIGHT THEN AND THERE

it was a nightmare

still is, pretty much, but we're used to it

I'll show this to Chris.

Doug Sundseth said...

I wonder if a light-switch analogy might help make sense of the multiplicative effects of negatives:

Start with lights on if the base number is positive and off if the base number is negative. When you see no sign or a positive, you don't flip the switch. For each negative, you flip the switch once.

At the end, lights on (even number of switch flips/negatives) is positive and lights off (odd number of switch flips/negatives) is negative.

Anonymous said...

While most students' first introduction to absolute values seems incredibly easy (just drop the sign), actually doing any computation with absolute values quickly gets complicated. That's why the reference here to absolute values seems to me to complicate things.

I would tend to refer to the visual of a number line for adding and subtracting positive and negative integers. If you must bring the absolute values into it, the number line also helps with that. The distance of a point from zero along the number line is its absolute value.

Dan K.

le radical galoisien said...

Start with the axioms?

I assume they talk about commutative and distributive laws, etc.?


'I told my son that there is a case where two positives equals a negative, and he said "Yeah, right!"'

e^(pi*i) !

SteveH said...

"I would tend to refer to the visual of a number line for adding and subtracting positive and negative integers."

The book shows examples of this (I like number line explanations too), but then it goes right off to those definitions.

SteveH said...

"Start with the axioms? I assume they talk about commutative and distributive laws, etc.?"

Yes. They covered them in the last chapter, but then they use the rules shown above!?!

Anonymous said...

What horror and misery.

There's a reason you learn number theory AFTER you learn how to do arithmetic. Technically, of course, the absolute value answer is correct. When you are doing theory, or trying to prove how arithmetic works, and how the axioms work, then you need this level of detail.

But this level of detail obscures reality. It does not help elucidate anything. Only after you already know about positives and negatives, about how to add and subtract numbers to the point where it's cold can you then learn and say "aha! yes, what I'm really doing is...."

But by then you're at least so proficient with negative numbers that you would never ever ever get it wrong.

Skip the darn book. Teach moving on the number line. Teach how -5 is the same as "minus 5" and "minus 5" means the same thing when it's a noun or a verb--you can start at -5 on the number line, or you can start at zero and then you can move left 5 spaces.
So 0 -5 = -5. Teach that until it's cold. Then work left and right on the number line. Do it hundreds of times. Maybe you can make a bead and a string and a ruler to help him do it until it's totally completely rote.

SusanJ said...

I'm a bit uncomfortable teaching kids that there is no difference between a unary sign and a binary sign of operation.

I would teach that
-n = (-1)n

I do, however, think they should know that adding a negative number is the same as subtracting a positive number and adding a positive number is the same as subtracting a negative number.

That is,
5 - 6
= +5 - +6
= +5 + -6
= -6 + +5
= -6 - -5
= (-1)6 - (-1)5
= (-1)(6-5)

lgm said...

Are they treating it as review? Glencoe's Course 2 1999 version (reg. ed. 7th grade text) section 5-4 Adding Integers has a modeling w/counters lab, then defines the additive inverse property algebraically and with an arithmetic example, then shows the number line representation before stating the key concepts you typed out. It also suggests playing integer war with the card deck to facilitate mastery and gives a real life example of the position of a biker on a ramp (the kind that looks like a half pipe in cross section) relative to the starting point.

The counter thing is a little bizarre, but it does explicitly state the point that you can combine terms and remove them from an equation because 'adding or removing zero does not change the value of the counters on the mat'.

SteveH said...

"I'm a bit uncomfortable teaching kids that there is no difference between a unary sign and a binary sign of operation."

The issue I have had (including with my son) is that kids get confused about negative numbers, signs, unary operators, and binary operators. They think that negative symbols mean different things in different situations. My son really didn't like it when I used "minus" and "negative" interchangeably, so I had to stop and explain to him that when I look at

6-5

I automatically think of either:

6 minus 5

or

6 plus a negative 5

or

6 plus (-1)*5


[It probably didn't help that I would sometimes call "-5" "negative five", and sometimes "minus five".]


The third way is the best understanding for algebra. A minus sign is really a factor of (-1) for the number or term that follows. My point is that I don't want kids to get hung up on definitions of sign, negative, minus, and operation. When signs come together during basic operations or complex algebraic manipulations, they need to know how to combine them. I don't want them to think that there are different methods for different types of signs.

SteveH said...

"Are they treating it as review?"

No. (This is the 2008 edition.)

It does explain it using the number line (which is nice), but the rules I listed above are the ones they expect the kids to use. They make no attempt to explain how their rules match up with the number line explanation. The rules are just for integers, so the kids will have to learn something else when they get to variables. This is a complete waste of effort.

Anonymous said...

"They think that negative symbols mean different things in different situations."

But the "-" does mean different things in different situations! It's the mathematical version of a homophone.

We could just as easily adopt a different notation for negative numbers. Say (-3) could be represented by (inv 3). The problem that I see it is that adults are trying to explain some highly complicated concepts to children that are too young to understand the reasoning that it would take to demonstrate why it is that adding (inv 3)gives the same results as subtracting 3.

Telling people THAT it works usually seems to suffice, they work a million problems and then they get a feel for it, but you are in a pickle if have a kid that wants to know WHY it works.

To me that isn't an indication that the kid isn't "getting it", it takes a bright kid to be skeptical of the rules and require that the rules be justified.

The discussion about absolute value is a geometric one: it's the distance of these numbers from zero on a numberline (coordinates).

Mr. Person said...

If it would be of any conceptual help, I wrote up something on adding and subtracting with negative numbers here, and a brief one on multiplying here.

SteveH said...

"But the "-" does mean different things in different situations! It's the mathematical version of a homophone."

Perhaps I should have said "mean different rules". I'm talking about how kids get confused because they think there are different rules depending on how a "-" is used.


" but you are in a pickle if have a kid that wants to know WHY it works."

There are different levels of understanding, and there is always a chance that some kids want more, but I wouldn't always call it a "pickle". The goal is to find the right balance between mechanics and true mathematical understanding.


My complaint with the Glencoe book is that they made the wrong tradeoff.

SteveH said...

"I wrote up something on adding and subtracting with negative numbers .."

Thank you mr. person. It's like a number line explanation that has a useful meaning.


The problem I've run into is the next level of understanding and mechanics. Kids can say "yeah, OK" on number line descriptions, but they start to doubt themselves when it comes to variables.

For example,

Z - 2(3X - 5Y)

Is the minus sign after the Z a minus sign (operation), or does it belong to the 2 (a negative 2)? Do you multiply (3X - 5Y) by 2 or (-2)

Is the result:

Z - 2(3X) - 2(5Y)

or

Z + (-2)(3X) - (-2)(5Y)

Kids get confused about minus versus negative. This is a different level of understanding. You can talk about basic mathematical rules or identities, but you also have to learn how they work together. This type of understanding comes from practice. That's why I always talk about linkage between practice and understanding.

I've told students to always keep the minus sign with the number or term to the right. so:

Z - 2(3X - 5Y)

should be seen as:

Z + (-2)(3X - 5Y)

The negative 2 is just a factor in the second term. Then there is no confusion when you expand or do anything else in the second term. If there is any doubt, change a minus sign into a factor of the following term.

SusanJ said...

You can always tell the difference between a unary sign and a sign of operation from syntax. Kids need to learn to understand syntax!

Here are some of the syntax rules.

A single minus or plus sign between two terms is always a sign of operation.
a+b
3-6

If there are two signs between two terms, the first is a sign of operation and the second is a unary sign.

3--6

This is poor notation and would better be written as
3-(-6)

A single minus or plus sign before a number or symbol (but not between two terms) is a unary sign. It is also a unary sign if it together with the number are enclosed in parentheses.) If there is no unary sign, then a positive sign is implied.

SusanJ said...

Following on with Myrtle's excellent post, if I were dealing with a smart kid, I might have them actually convert all the unary + and - signs to some other symbols and insert any "missing" unary + signs. You might try # for unary + and ~ for unary minus. This representation should make it easy to understand the axioms (e.g. -~ is equivalent to +#) or why Steve's heuristics work.
-3+6 = ~3+6 = ~3+#6
~3+#6 = #6+~3 = #6-#3

Actually, there is only one axiom this way. You get an equivalent expression if and only if you change both the operation and the unary sign to its opposite.

Catherine Johnson said...

And he hasn't even gotten to subtracting integers, which states:

"To subtract an integer, add its additive inverse."


I wish to he** I'd had this post two years ago.

I don't think C. was EVER taught to subtract by adding the additive inverse, so I've been chronically trying to "show" him this, in between rescuing him from Cs, Ds, and Fs.

criminy

Catherine Johnson said...

If the signs are the same, combine them into a '+'

If the signs are opposite, combine them into a '-'


good lord

why didn't I think of that

C. struggled CHRONICALLY with adding & subtracting integers throughout his entire 6th grade year

I can't tell you how many times Ed reported to me: "He has no idea how to subtract a negative" or whatever.

C. finally more or less became able to add & subtract integers correctly (though he is still, to this day, "leaving out the minus sign" on some occasions) through sheer, brute repetition & NON-deliberate practice.

Catherine Johnson said...

Remember that this is the same as (-1)(+5)(-1)(+2) = (-1)(-1)(5)(2)

This I did tell him, mainly because I figured it out for myself as a kid.

I think Ms. K may have taught them this, too, but I don't remember.

Catherine Johnson said...

I've used a game form with students by calling positive and negative integers good guys and bad guys respectively.

I love that!

Catherine Johnson said...

If I had a kid that could understand those rules I'd teach them the definition of additive inverse from the get-go.

EXACTLY

Catherine Johnson said...

Start with lights on if the base number is positive and off if the base number is negative. When you see no sign or a positive, you don't flip the switch. For each negative, you flip the switch once.

At the end, lights on (even number of switch flips/negatives) is positive and lights off (odd number of switch flips/negatives) is negative.


That's pretty cool, because it's quite difficult to come up with an image or analogy for kids to use when remembering how to multiply & divide integers.

Catherine Johnson said...

I would tend to refer to the visual of a number line for adding and subtracting positive and negative integers. If you must bring the absolute values into it, the number line also helps with that. The distance of a point from zero along the number line is its absolute value.

right

I think that's what Prentice Hall Pre-Algebra did, so in practice they didn't spend a lot of time trying to memorize and use the absolute value definition of addition & subtraction of integers...

They were still horribly confused.

Catherine Johnson said...

The book shows examples of this (I like number line explanations too), but then it goes right off to those definitions.

yup

Saxon spends a LONG time doing "algebraic addition" using the number line.

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SusanJ said...

Is part of the problem that we thinking of adding as always moving to the right on the number line and subtracting as always moving to the left?

Those directions are actually only applicable when adding or subtracting a positive number. When adding or subtracting a negative number, you move in the opposite direction.

Add positive = move right.
Subtract positive = move left.
Add negative = move left.
Subtract negative = move right.