The "Why" of Subtracting Negative Numbers

[Tiramisu]

I'm calling on experienced moms so I'm using this board even though the topic isn't a high school one.

 

Dd NEEDS to know WHY subtracting a negative number is adding.

 

2-(-2)=4 is the same as 2+2=4

 

She can memorize the procedure, but she is insisting on knowing WHY. I've gone through and watched Khan Academy and youtube videos on this topic but none seem to explain why, just that you do it.

 

I've explained to her that the signs indicate direction and how you change direction twice, and wind up going forward on the number line, but that's not enough for her.

 

Can anyone explain this or point me to a handy resource to help dd who is getting very hung up on this.

 

FWIW, dd is like this about new concepts. She needs to understand what's behind it or she hits a wall. Eventually, she'll get over it but the beginning of something new is very frustrating...for me.:glare:

 

Thanks.:001_smile:

[Teachin'Mine]

Have you seen the movie Stand and Deliver? Great movie!

 

Anyway, the math teacher, Jaime Escalante, in teaching about negative numbers, uses the analogy of digging a hole in the sand. So tell your daughter to imagine the flat sand as being 0. Now, you're going to represent -2 by digging out two buckets of sand. That -2 is represented by two buckets of air in the hole. Now you want to subtract -2. -2 = two buckets of air. So in order to take away two buckets of air from the hole, you have to fill it with sand. How much? Two buckets. So now you're adding two buckets of sand and get back to level sand or 0. So -2 - (-2) = 0

 

Might work. :lol:

Edited February 27, 2012 by Teachin'Mine

[usetoschool]

What I have told my kids is to stop thinking of the minus or plus as meaning to "do" something and think of it as an adjective telling you what kind of number it is. 3-2 is a list, instead of 3 minus 2, it is a positive 3 and a negative 2. Start at 0 on the number line, go positive 3 and then from there negative 2 and you end up at the sum. There really is a positive sign in front of the 3, it is just taken for granted.

 

We also talk about the negative sign meaning opposite. Every number has an opposite that, when added to the number gets you a zero. So, 3 - (-2) would be a positive 3 and the opposite of the opposite of 2. If you trace it on a number line you can see that you end up at positive 2.

 

Just some wording that helps sort things out at our house - probably sounds clear as mud trying to explain it here though :D

 

Now let the real math majors chime in...

[LisaKinVA]

You could also trek on over to the Art of Problem Solving website and watch their two videos on negation from the Pre-Algebra book :D No charge. Although, I like the sand explanation :D

[StephanieZ]

AoPS defines subtracting as adding the additive inverse . . .

 

SO,

 

2 - 3 = 2 +(-3) = -1

 

Or,

 

2 - (-3) = 2 + (-(-3)) = 2 + 3 = 5

 

The key to understanding it this way is to first explore/understand the concept of additive inverse.

 

I think it's a super cool way to understand why subtracting a negative is just like adding a positive.

 

HTH

[Tiramisu]

Thanks all. Watching the AoPS video made me think of Foerster's proofs.

 

Dd is starting to understand, I think, or maybe just doesn't want to deal with it anymore. Hopefully, watching the videos made something click.

 

I think my plan for future math lessons--we're using CLE--is to fast forward to the review material as soon as I see any sign of frustration with new concepts so dd doesn't lose it so badly that she can't recover enough to finish the lesson. At the end of the lesson, we can go back to the new concepts and practice problems, after she's warmed up. :001_smile:

[abacus2]

I explained this concept once with packets of jelly. I'm going to try with smilies.

 

For this example, :p is a positive 1 and :eek: is a negative 1.

 

Problem: 1 - (-1)

 

Start with: :p

 

You need to take away an :eek:, but there aren't any.

 

Since positive 1 and negative 1 together is zero, we can add one of each without changing anything.

 

Now we have: :p :p :eek:

 

We can now take away an :eek: and are left with 2.

 

1 - (-1) = 1 + 1 = 2.

[wapiti]

If you are looking for something on a simpler level than the AoPS videos, there's also a MM topic book.

[CaffeineDiary]

2-(-2)=4 is the same as 2+2=4

 

She can memorize the procedure, but she is insisting on knowing WHY. I've gone through and watched Khan Academy and youtube videos on this topic but none seem to explain why, just that you do it.

 

With a lot of mathematics, the "why" becomes clearer after practice. But asking "why" is good too.

 

One way to think about it -- certainly not the only way, but one way -- is to think about the inverse relationship between addition and subtraction. Specifically, adding and subtracting one. In the natural numbers, you can always add one, forever, ad infinitum. No matter what number you choose, I can always add one to it. Subtraction is defined as the inverse of addition. However, if you limit yourself to the counting numbers, then you can't subtract 1 from 0: this makes subtraction a non-symmetrical inverse function. Since mathematicians view this as a problem, we posit the existence of the negative numbers as a way of solving the problem.

 

Once subtraction becomes a perfectly symmetrical function, then you must be able to describe it in terms of addition, and be able to describe addition in terms of subtraction. Any function that meets certain tests can always be described in terms of its inverse. So if I want to describe subtraction in terms of addition, I can say

 

6 - 3 = 6 + -3

 

Or, if I want to describe addition in terms of subtraction, I can say:

 

10 + 5 = 10 - -5

 

So the short answer to your daughter's question "why?" is "The fact that subtracting a negative numbers makes the minuend increase in value follows from the definition of subtraction as the inverse of addition." If this were not true, then the examples I gave above would not be true.

Edited February 27, 2012 by peterb

[Beth in SW WA]

With a lot of mathematics, the "why" becomes clearer after practice. But asking "why" is good too.

 

:iagree: and welcome to the Hive, Peter. :)

 

To the OP...Dd visualized this concept when we started Hands On Equations last year. Perhaps you could watch the free HoE videos on youtube to get an idea. Those clever green & red cubes are worth their weight in gold around my house. :)

[CaffeineDiary]

Thanks for the welcome.

 

Following up to my earlier post, you can also think about it this way:

 

Imagine you have two numbers, a and b, that add up to c.

 

a + b = c

 

The definition of subtraction is as the operation that, when given the minuend c and the subtrahend a, returns you the number b.

 

A good exercise for your kid might be to ask her "What are the alternatives if we have the problem 1 - (-3) = x? What could it mean, and what happens if we adopt each of those definitions?"

 

(1) We could disallow the subtraction of negative numbers completely. That's not terribly useful, and would mean that subtraction was no longer the inverse of addition.

 

(2) We could say that it acts like subtracting a positive number, so that 1 - (-3) = -2. But this would mean that 1 + -2 = -3, which is false. This is a contradiction, so this is impossible.

 

(3) We could say that it acts like adding a positive number: 1 - (-3) = 4. This is consistent with the additive inverse: -3 + 4 = 1, and so is the definition we choose.

 

David Berlinski talks about this very issue in chapter 19 of his excellent book One, Two, Three, which I'd highly recommend to anyone interested in knowing a little bit of simple number theory. It's available in eBook form in iBooks (http://itunes.apple.com/us/book/one-two-three/id422521699?mt=11) and also on Amazon.

 

Good luck!

[lauracolumbus]

Geesh, such a great explanations, and I still don't get it. Hopefully the "One, Two, Three" book will do the job.

 

Laura

[Teachin'Mine]

Oooh - here's another way to look at it.

 

I say something that is seen in a negative light. But then I take it back (subtract) and

now you are left with a positive impression. Take away the negative and

it becomes a positive.

 

Yeah I know - nothing to do with math. :tongue_smilie::lol:

[lauracolumbus]

Oooh - here's another way to look at it.

 

I say something that is seen in a negative light. But then I take it back (subtract) and

now you are left with a positive impression. Take away the negative and

it becomes a positive.

 

Yeah I know - nothing to do with math. :tongue_smilie::lol:

 

Ok, now that makes sense to me!!!!

Laura

[Teachin'Mine]

Ok, now that makes sense to me!!!!

Laura

 

:D

[Bootsie]

I find the easiest way to think about this as:

 

How much worth do you have? The amount of money in your pocket minus how much you owe to someone.

 

If you have $5 in your pocket and owe someone $3 you have only $2 left.

 

If you have $5 in your pocket and someone owes you $3 (you owe $-3) you really have:

 

$5 - (-$3) = $8

[CaffeineDiary]

I find the easiest way to think about this as:

 

How much worth do you have? The amount of money in your pocket minus how much you owe to someone.

 

If you have $5 in your pocket and owe someone $3 you have only $2 left.

 

If you have $5 in your pocket and someone owes you $3 (you owe $-3) you really have:

 

$5 - (-$3) = $8

 

This is totally a valid way to think of how it "works" in real life, and would be a great answer for a kid that wanted to know how it works. I answered the way I did, though, because the kid was asking "why", and in math "why" has a very specific meaning: "prove it to me." So I'm a big believer in using analogies when they help, but I try to avoid them when answering "why".

 

Negative numbers are extremely convenient and modern mathematics couldn't function without them. But, in a very real sense, they don't exist: analogies notwithstanding, you just can't point to "-8" of some tangible item without resorting to a shared fiction such as a bank account. So the question "why does it work this way?" is a really great one.

 

The other thing that's worth mentioning is that if and when you get to higher math, analogies break down and can actually get in the way of understanding. So I think it's always worth trying to go for the gold and give the number theoretical explanation, when it's available to you.

[LucyStoner]

I use the signs as part of the number and explain that in the absence of a sign the number is presumed to be positive.

 

7-5 is then a +7 and a -5

 

When you have a positive 7 and a negative 5, you have 2.

 

Starting this way from the beginning makes the whys of working with negative numbers intuitive.

 

7-(-7) is really a positive 7 combined with the product of a negative 1 and a negative 7. We know from the order of operations that multiplication comes before addition/subtraction (the combining step). So -1*-7 is 7. Combining 7 and 7 we get 14.

Edited February 28, 2012 by kijipt

[Ellie]

Negative numbers never, ever make sense to me. The "rule" for subtracting them sounds like the rules for that Star Trek game, Fizzbin, where each player gets six cards, except for the player on the dealer's right, who gets seven. The second card is turned up, except on Tuesdays. Wtih two jacks, you need a king and a deuce, except at night, when you need a queen and a four. And in some cases, if you get a king instead of a jack, the player would get another card, except when it's dark, in which case he'd have to give it back. [Thanks to Wiki for that, lol.]

 

I've seen the digging-a-hole explanation, and using a number line, and I don't know what-all. I don't know how I ever made it through algebra...

[CaffeineDiary]

For every x and y, the operation x – y designates the one and only number z such that y + z = x. The every is important. Because it means that y can be larger than x.

 

Without negative numbers, you can't have full, universal subtraction. Simple as that.

[Ellie]

For every x and y, the operation x – y designates the one and only number z such that y + z = x. The every is important. Because it means that y can be larger than x.

 

Without negative numbers, you can't have full, universal subtraction. Simple as that.

I have no idea what you just said. It's a good thing I no longer have to worry about teaching algebra, lol.

[LucyStoner]

For every x and y, the operation x – y designates the one and only number z such that y + z = x. The every is important. Because it means that y can be larger than x.

 

Without negative numbers, you can't have full, universal subtraction. Simple as that.

 

 

Very nice way of putting it.

[angela in ohio]

A little levity, since you are frustrated...

 

A physicist, a biologist, and a mathematician are sitting by the window in a bar, passing time by watching what is going on up and down the street. They notice two people go into the house across the street. A little while later, three people come out. The physicist says, "Our measurement must have been off." The biologist says, "They must have reproduced." The mathematician says, "If one more person goes into that house, it will be empty." :D

 

Seriously, though, :iagree: with StephanieZ.

[Poke Salad Annie]

We have been working with Eli's Magic Peanuts from CSMP. Playing with the stones which I made to represent positive and negative numbers illustrates this beautifully!

[Ellie]

A little levity, since you are frustrated...

 

A physicist, a biologist, and a mathematician are sitting by the window in a bar, passing time by watching what is going on up and down the street. They notice two people go into the house across the street. A little while later, three people come out. The physicist says, "Our measurement must have been off." The biologist says, "They must have reproduced." The mathematician says, "If one more person goes into that house, it will be empty." :D.

:lol::lol::lol:

[Tiramisu]

:lol::lol::lol:

 

I like this, too. :D

[Tiramisu]

I just want to send out a thank you to everyone who helped with this yesterday. Dd just sailed through these problems this morning. No issue at all. When I started her off gently to gauge her reaction, she took over and insisted, "I can do this!" And, yes, she did them without any problem.

 

It wasn't just the subtraction of a negative number; it was the whole additive inverse concept that she seems to have mastered!

 

Usually, when we hit a spot like this, we'll have at least a couple of days of frustration, but with your help dd got over the hump very quickly.

 

You are the best!!!

[Charles Wallace]

I'm calling on experienced moms so I'm using this board even though the topic isn't a high school one.

 

Dd NEEDS to know WHY subtracting a negative number is adding.

 

2-(-2)=4 is the same as 2+2=4

 

She can memorize the procedure, but she is insisting on knowing WHY. I've gone through and watched Khan Academy and youtube videos on this topic but none seem to explain why, just that you do it.

 

I've explained to her that the signs indicate direction and how you change direction twice, and wind up going forward on the number line, but that's not enough for her.

 

Can anyone explain this or point me to a handy resource to help dd who is getting very hung up on this.

 

FWIW, dd is like this about new concepts. She needs to understand what's behind it or she hits a wall. Eventually, she'll get over it but the beginning of something new is very frustrating...for me.:glare:

 

Thanks.:001_smile:

 

Think of it like money.

 

If I have a debt to Chase Visa of $500, and I get a voucher from a rich uncle that says, "This voucher subtracts $350 from your debt," then the first thing I will do is add it to my Chase Visa account. I now have a debt of $150.

 

Another example: My Chase Visa debt is still $500. I still get my voucher from my rich uncle saying, "This voucher subtracts $350 from your debt," but my rich aunt -- taking pity on me -- sends me a similar voucher saying, "This voucher subtracts $400 from your debt." I add both vouchers to my Chase Visa account and now I actually have a surplus of $250.

 

Think of adding negative numbers as "canceling debt," and it helps a lot.

[Teachin'Mine]

A little levity, since you are frustrated...

 

A physicist, a biologist, and a mathematician are sitting by the window in a bar, passing time by watching what is going on up and down the street. They notice two people go into the house across the street. A little while later, three people come out. The physicist says, "Our measurement must have been off." The biologist says, "They must have reproduced." The mathematician says, "If one more person goes into that house, it will be empty." :D

 

Seriously, though, :iagree: with StephanieZ.

 

 

:lol:

 

Mathematicians don't do their best thinking sitting in a bar. :D

[shann]

 

Think of adding negative numbers as "canceling debt," and it helps a lot.

 

That's a good one. I think of the double negative symbols in terms of language with the equation telling me to literally 'negate the negative'

[Hot Lava Mama]

if you have your problem 2 - (-3) = 5

 

Have her think of the first number (2) as made up as [ +5 and -3 ] since she understands adding a positive and negative number. Now, have her "take out" the -3, that leave the +5. It's one of the techniquest taught by the hands on equations program.

 

Hope that helps. It's a difficult concept, I still have to think about it every time.

Good luck!

[Julie in MN]

I hope I didn't miss this in the other posts, but has she thought about it in terms of money? Negatives would be debts or money spent, and those could be added to the books or taken away from the running total (taken away as in, you paid someone back or gave them something in exchange for taking away your debt or obligation to them, so your negative debt is taken away)? Then compare this to adding a debt, adding a deposit, and subtracting a deposit.

 

To me, money is a way kids can understand the why. Then, because its bulky, transition the ideas onto the number line for a more streamlined how (as was mentioned).

Edited February 29, 2012 by Julie in MN

[songbirdie]

Points. That is what finally made sense to us. And when I say "us", I mean dd AND me. LOL

 

Family XYZ gives each child points for doing each daily chore, and they can turn in their points at the end of the week for a reward. Mom gives Child A 5 points day one, 5 points day 2, but subtracts 3 points on day 3 since 3 chores were skipped. Child A has 7 points. 5 + 5 - 3 = 7

BUT. . .then she realized that on Wednesday she subtracted the 3 points from this Child A, but it was actually Child B's 3 chores that were skipped. So she has to *subtract the -3* from Child A's points. In essence, she's "giving back" the 3 points so that 7 - (-3) = 10. She "took away" the minus 3 points by adding 3 points back to her score.

 

Hope that helps!

[ktgrok]

The way I learned it was to have somone stand on a numberline that was drawn in chalk on the floor. Then you walk forwards for adding, and backwards for subtracting. But....you face one way for positive numbers and another way for negative numbers. So subtracting a negative is walking backwards but turned around, which has you go forwards.

[GingerPoppy]

I explained this concept once with packets of jelly. I'm going to try with smilies.

 

For this example, :p is a positive 1 and :eek: is a negative 1.

 

Problem: 1 - (-1)

 

Start with: :p

 

You need to take away an :eek:, but there aren't any.

 

Since positive 1 and negative 1 together is zero, we can add one of each without changing anything.

 

Now we have: :p :p :eek:

 

We can now take away an :eek: and are left with 2.

 

1 - (-1) = 1 + 1 = 2.

 

Points. That is what finally made sense to us. And when I say "us", I mean dd AND me. LOL

 

Family XYZ gives each child points for doing each daily chore, and they can turn in their points at the end of the week for a reward. Mom gives Child A 5 points day one, 5 points day 2, but subtracts 3 points on day 3 since 3 chores were skipped. Child A has 7 points. 5 + 5 - 3 = 7

BUT. . .then she realized that on Wednesday she subtracted the 3 points from this Child A, but it was actually Child B's 3 chores that were skipped. So she has to *subtract the -3* from Child A's points. In essence, she's "giving back" the 3 points so that 7 - (-3) = 10. She "took away" the minus 3 points by adding 3 points back to her score.

 

Hope that helps!

These are two great explanations that resonate with me, and I'll definitely use them to explain to students in the future. Thanks!