Understanding Negative Numbers for Parents and Educators

[Gnubie]

]My 6 yr old saw negative numbers in my math book and wanted to know what they meant. Using pennies as counters didn't help. She didn't understand at all the idea of taking something from nothing (0). Explaing negatives as just distance from zero on the number line did work. Now she has asked "what's on the other side of zero." I asked if she meant negative zero (yes). Tried explaining that nothing was just nothing--there is nowhere for the zero to go. This answer is not enough. This kid has rolled it over in her head for a couple of days and is now back asking about the negative zero. Please pardon us if this question is a bit ridiculous. Her interest in the math has me feeling humbled by the responsibililty not to dampen that interest with some ham-handed response she won't get. That's why this question is harder than "Why can dog drink out of the toilet and we can't." Or the time @ the laundrymat a young guy called his girlfriend a "useless t**t", my innocent one wanted to know "what's t**t?" Told her it was the past tense of twit. Then she wanted to know what past tense meant, so we were back to square one, but at least she forgot about t**t. But how do I explain why there's no negative zero? Maybe tell her it's the past tense of zero? Any ideas on explaining this will be so appreciated. Thank you.

 

[Math Is Hard]

So does she think that zero is a positive number?

 

[lisab]

How about this. Find a sandbox. Make a hill, and a hole. Explain that the level sand is like zero. The hill above zero is like positive numbers, and the hole is like negative numbers.

If you don't have a sandbox, try drawing it.

 

[chiro]

With regard to positive and negative numbers you could talk about money in accounting.

Say you start off with 5 dollars and go to a shop and buy some candy for 3, then you are left with 2.

But suppose you buy candy for 6 dollars and you only have 5. In this case you owe the shop 1 dollar, but its 1 dollar that you don't have! so basically you have -1 dollars because you owe the shop one dollar that you don't have yet. When you pay the shop that one dollar you will go back to zero since you don't owe anybody anything.

I would say that accounting examples are good to explain arithmetic because they are more tangible in that we can see them easily in the real world.

 

[Astronuc]

Positive and negative have to do with direction or orientation.

Zero has a unique special property that it is neither negative or positive, otherwise it's both based on a = -a, or 2a = 0, or -2a = 0.

Zero, or for that matter, any number on the number line is simply a point, and points do not have direction. Positive and negative refer to direction on the number line, with negative being the oppositive direction of positive. Similarly for vectors, negative implies the opposite direction/orientation to positive. We have adopted conventions with respect to +/-, in/out, forward/backwards, up/down, right hand/left hand, etc.

In arithmetic, negative is the additive inverse. The positive operation means to add or accumulate in number. Negative means to subtract or take away or decrease in number.

On the number line, negative means counting units backward, or counting in the oppositive direction of positive.

Perhaps one can describe + and negative - as MIH suggested, or use an example of a see saw, where zero is the middle between positive (up) or negative (down). When you're up, you're up, when you're down, you're down, and when you're in between, you're neither up nor down - but zero.

 

[Redbelly98]

We need to keep the answers at a 6-year-old's level.

I would tell her that "negative zero" is the same as "positive" zero. You could follow that up by saying zero is the only number for which this is true. Or that adding and subtracting zero gives the same answer.

 

[DanP]

We need to keep the answers at a 6-year-old's level.

I would tell her that "negative zero" is the same as "positive" zero.

I think that would confuse the hell out of many 6 years old who have issues with negative numbers.

 

[Redbelly98]

I think that would confuse the hell out of many 6 years old who have issues with negative numbers.

Okay, I thought from the OP that his daughter finally understood negative numbers. Rereading it, it is no longer clear to me if that is true or not.

 

[leroyjenkens]

This is how I would explain negative numbers; I would relate it to something she may have experienced. Like if one of her friends owed her a cookie, but she didn't have a cookie to give her. That would mean her friend has -1 cookie, because when her friend does get a cookie, she has to give it away. If she had zero cookies, then if she did get a cookie, she would get to keep it, which is the difference between zero and negative one.
Then you could say that if her friend has zero cookies, she neither has cookies or owes cookies. If you have negative cookies, you owe cookies, if you have positive cookies you get to keep them.

That's how I would understand it. Sometimes simple things are hard to explain. It's like trying to define the word "the".

 

[mathematicsma]

I vote for the sandbox or the see-saw (perhaps both). I think if she sees those explanations, she might be satisfied. As she gets older, she'll gain a deeper understanding of it.

I'm not an expert in early childhood or anything, but my guess is that accounting (negative balance) or negative cookies, is too complicated for a six year old. It requires her to conceptualize the idea that owing someone a cookie is a negative, and that's not very tangible. At least not when you're six.

In any case, it depends on the child. I'd say she's relatively precocious, if she's asking about this stuff so early. Keep the fire burning inside her, don't let school make her jaded toward learning like it does for most people. Good luck!

 

[rootX]

I believe there is +0 and there is -0. I remember having trouble with conditions at 0- and 0+ or when at 0- we have very big negative current and 0+ we have very big positive current (discontinuity in something like cloudy to sunny in an instant). :shy: Using time would be ideal to explain 0. Everything before now is negative. Everything just being said now is 0+ while everything said just before now is 0-.

 

[mathematicsma]

Interesting point, but how can a six year old understand that? She needs something tangible.

 

[Dr Lots-o'watts]

How about with time? Today is day "0", yesterday was day "-1", and tomorrow is day "+1".

 

[Jimmy Snyder]

How about a thermometer?

 

[Chi Meson]

The two most common uses of negative numbers are regarding temperature and money. But I'm not sure if either are going to satisfy your 6 year old's question. I just went though the same kind of discussion with my 7 year old son. He doesn't get it either, but my 9year old daughter has no problem with it.

I tried explaining it as "things you don't have." Chairs for example. Your kid can count the chairs you have around the table (6 for example). Then say, 10 people need a place to sit, how many chairs "don't we have"? Then I changed that into money: "You have 6 dollars, but you buy something that cost 10 dollars, how much money do you have?" My daughter understood this as meaning you "owed" someone 4 bucks, but my son still couldn't understand what $-4 looked like.

The more pure mathematical understanding of negative numbers will have to come later. I think the conceptual grasp of "negativity" requires that the brain develop the right set of neurons. Until then, point out the thermometer.

Edit: goldangit Jimmy! Thermometer's my idea, mine! i tell ye wot boy...

 

[story645]

How about a thermometer?

Cool idea. I also like the sandbox.

How about a number line? Kids can count at six, so you can show the positive numbers as going forward, the negative numbers as going back. Draw the number line on paper or with chalk.

You can even turn it into a "mother may I" game. Mark the 0 point as the start (chalk) and talk about two steps back as negative two steps. (It'll teach her about the idea of +/- indicating direction, if nothing else.)

 

[TylerH]

I believe there is +0 and there is -0. I remember having trouble with conditions at 0- and 0+ or when at 0- we have very big negative current and 0+ we have very big positive current (discontinuity in something like cloudy to sunny in an instant). :shy: Using time would be ideal to explain 0. Everything before now is negative. Everything just being said now is 0+ while everything said just before now is 0-.

Do you realize how big a pile you're stepping in? The +/-0 is only important in deciding +/-∞ at the asymptote of a function. So know he has to explain both +/-0 and limits?

IMHO the number line one a piece of paper would be best, at least it was for me. If that doesn't work then one of the more physical explanations listed here may be better.

 

[rootX]

Do you realize how big a pile you're stepping in? The +/-0 is only important in deciding +/-∞ at the asymptote of a function. So know he has to explain both +/-0 and limits?

IMHO the number line one a piece of paper would be best, at least it was for me. If that doesn't work then one of the more physical explanations listed here may be better.

Yes, it is bit hard to explain discontinuity. But that's the only place where I see both a positive and a negative zero. I cannot recall any other simpler use of positive and negative zero at this moment.

 

[cobalt124]

I'll plummet for the Drs suggestion. That seems to be the easiest and most accessible for a child. I asked my ten year old son how he would define a negative number to get his perspective. He came up with some ideas and we had a bit of a discussion, and the best we could come up with was they are a sequence that goes 1,2,3... "forwards" and 1,0,-1... backwards. This is very basic and doesn't explain the points raised in the thread, but it's states something about 0 and the negative numbers that can be built on, and then just look around and if you see an example of something that sheds light on negative numbers mention it and further facts will be learned (you sound like an expert at this given the "past tense of twit" incident).

 

[cobalt124]

Oh, apologies to story645. I read the first page of the thread too quickly and saw a lot of stuff a 6 year old might not grasp, and I didn't read your post, so just add "Like story645 says" somewhere in my post.

 

[chronon]

The important property of zero is that it's the additive identity. This would suggest that the sandbox is the best way of thinking about it - positive zero is no hill, i.e. level sand, and negative zero is no hole, also level sand. The trouble with the number line/temperature etc is that zero is just an arbitrary point on it.

She seems to have the idea of the negative numbers as 'looking glass numbers', and the looking glass world could be entirely separate from the actual world, so there's no reason to identify +0 with -0. However, the two worlds would typically be seen as joined by mirror, so +0 and -0 would then be at the same place - i.e. the surface of the mirror.

 

[PhDorBust]

Just show her it is additive inverse.

You have a 4 that you want to turn into a zero. So you add -4. If you want to turn a zero into a zero, you can add either -0 or 0 so there is no distinction.

I think analytic approach is nicer since real-world explanations always involve a bit of fuzziness. You do calculation first and then discuss interpretation later.

 

[Proton Soup]

Positive and negative have to do with direction or orientation.

Zero has a unique special property that it is neither negative or positive, otherwise it's both based on a = -a, or 2a = 0, or -2a = 0.

Zero, or for that matter, any number on the number line is simply a point, and points do not have direction. Positive and negative refer to direction on the number line, with negative being the oppositive direction of positive. Similarly for vectors, negative implies the opposite direction/orientation to positive. We have adopted conventions with respect to +/-, in/out, forward/backwards, up/down, right hand/left hand, etc.

In arithmetic, negative is the additive inverse. The positive operation means to add or accumulate in number. Negative means to subtract or take away or decrease in number.

On the number line, negative means counting units backward, or counting in the oppositive direction of positive.

Perhaps one can describe + and negative - as MIH suggested, or use an example of a see saw, where zero is the middle between positive (up) or negative (down). When you're up, you're up, when you're down, you're down, and when you're in between, you're neither up nor down - but zero.

yes, i like direction and orientation. you just stretch out a string on the grass, or lay some chalk dust or paint in a line. mark one spot at the center as the start (zero). face one direction along the line (positive). take a steps forward for positive, take steps backwards for negative. you could even hash off the step sizes. do this a few days as a game, and i think it becomes intuitive.

edit: oh, n/m, i see story645 posted this idea already.

 

[DBTS]

I learned about negative numbers by looking at a number line.

 

[Jimmy Snyder]

negative numbers darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple

I assume she has read Maseres.