Concrete example for showing that the product of two negative integers is positive

I am trying to get teach my kids WHY this works. I don’t want them to just learn the rule for anything we go over, so I am seeking concrete ideas that will enlighten them. It has to be easy enough for 12 year old kids to understand. I have shown them how it would work using the Distributive Property, and that seemed to help. What I really want is a more “hands on” type of example, something that will really stick with them for some time. I have poured over the internet to no avail, so I would really appreciate any help.

moejuck

It’s a matter of the definition of negative numbers and multiplication. I don’t believe there is a concrete example. Now, if we were talking about addition, that would be easy (and I assume you already figured that one out).

You have three credit cards (liability accounts, as opposed to saving accounts [assets]) that each have $-10 balances (debt). What’s your total debt? -3 x -10 = 30, or $30.

That’s not cheating. Expand it: you have 2 credit cards with balances of -15 each, and a savings account overdrawn at -25, and two separate money market accounts with 12 each: (2 x -15) + -25 + (2 x 12), netting a worth of -31.

I tried to think up something concrete using oranges, but it just wasn’t working.

When a good(+) thing happens to a good(+) person, that’s good(+).

When a good(+) thing happens to a bad(-) person, that’s bad(-).

When a bad(-) thing happens to a good(+) person, that’s bad(-).

And finally, when a bad(-) thing happens to a bad(-) person, that’s good(+).

If someone is beating up someone (-), and you come in and stop them (-), thats good (+).

But why is that multiplication as opposed to addition? I took the OP to mean a concrete example of an actual multiplication, not just a sort of anaogy that might apply to anything.

John Mace is right. I can tell them that in grammar a double negative is the same as a positive, and so on. I am looking for something that applies to multiplication, and not an analogy of negatives and positives on their own.

Balthisar: The only problem with your first example is that debt is represented as a negative integer in math, so I don’t want to confuse them by telling them to express the debt as a total of 30. The second example uses addition and multiplication together. Thanks for the effort though, you are on the right track as far as providing some sort of concrete example.

Thanks everyone for trying, this one is really starting to get on my nerves.

The problem with Balthisar’s first example is that you don’t have -3 credit cards, you have 3 credit cards. And (unfortunately) the sum of many debts is a greater debt, not a credit. The correct equation is 3 x -10 = -30

I’ve always thought a geometric interpretation is a natural way to think about it. Part of multiplication can be described as “scaling”–if a multiply a number by 2, 3, or 4, what I’ve done is double, triple, or quadruple, respectively, that number’s distance from zero.

Multiplication by a negative number “reflects” that number across zero. What I mean is that if I start with, say, 3.5, and multiply it by -1, then my new number is still 3.5 units from zero, only on the opposite side from where it started. If I do that reflection twice, I’m back where I started.

So, breaking it down step by step, multiplying -2 * -3:

Start with 3, think of it as being three units to the “right” of zero. The “-” in -3 reflects it across zero, so now we’re three units to the “left” of zero. Now multiplying it by 2 doubles that distance, so we’re six units to the left of zero. The final “-” (in -2) flips it again, so we’re now six units to the right of zero: -2 * -3 = +6.

If you want to toss in complex numbers, I think it becomes even nicer to think of. A complex number has a real part and an imaginary part; picture a plane with two coordinate axes, the horizontal axis denoting the real part, the vertical axis denoting the imaginary part. This is the complex (Argand) plane, an extension of the usual real number line. For example, the complex number 4 - 5i would correspond to the point on the plane that is four units to the right, and five units down from the origin (= the complex number zero).

Multiplying in the complex plane has a very pretty picture. I can indicate any point in the plane by identifying two things: 1. How far it is from the origin, and 2. What angle does it make with the positive real axis, measured counterclockwise from the positive real axis.

If I take two points (complex numbers) in the plane, and want to multiply them, the result will be 1. Its distance from the origin is the product of the original numbers’ distances from the origin, and 2. Its angle (as described above) is the sum of the original numbers’ angles.

So, for example, if I take one number that is 3 units from the origin, at an angle of 37[sup]o[/sup], and another number that is 2.5 units from the origin, at an angle of 113[sup]o[/sup], the product of those twon numbers is 7.5 (=3*2.5) units from the origin, at an angle of 150[sup]o[/sup] (=37[sup]o[/sup] + 113[sup]o[/sup]).

The number -1 is one unit from the origin, and has an angle of 180[sup]o[/sup]. So multiplying by a negative number rotates the plane by 180[sup]o[/sup]–if I do this twice (a negative times a negative), I rotate the plane by 180[sup]o[/sup] + 180[sup]o[/sup] = 360[sup]o[/sup], and we’re back where we started.

I know this is kind of verbose, and that probably gets in the way of clarity. Don’t let that throw you, it’s my fault for not making this any clearer; a picture would make it clear that it’s really a simple idea; this site might be worth playing around with:

http://www.accesscom.com/~lillge/pgc/pgc-gallery2.shtml

moejuck: When you said you’ve “shown them how it would work using the Distributive Property”, did you mean something like this:

Let X = (-1)[sup]2[/sup]

Hence X - 1 = (-1)[sup]2[/sup] - 1 = -1[-1 + 1] (distributive property)

Therefore X - 1 = 0
Therefore X = 1

Therefore (-1)[sup]2[/sup] = 1 (which is clearly a positive number)

Someone on a Montessori listserv I frequent had the same question. An elementary teacher posted his way of introducing the topic to his students.

He has the child stand up and take steps:

(+1) x (+1): forward step… is a positive step (+1) facing in a positive (+)direction;

(–1) x (+1): backward step is a negative (–1)step facing in a positive (+) direction.

(+1) x (–1): If you turn 180 degrees and take a step forward, you take a positive step (+1) while facing in a negative(–) direction so the result is a –1.

(–1) x (–1): Starting again at the beginning, if you turn 180 degrees and take a backward step, you have taken a negative step (–1) while facing in a negative direction (–) and you ended up moving “forward” from your point of origin.

Very concrete, and according to other elementary teachers (I teach younger students), it works well as an introduction.

Lets draw a line down the middle of the table, things to the left are negative, things to the right are positive. Then lets use a small block of wood to illustrate positive and negative multiplication.

First put the block of wood so the left edge is on the line (representing a positive number), if we multiply the block by 3, we move the block 3 block-lengths to the right. If we start over and multiply the block by -2, we move the block 2 block lengths to the left.

Now we put the block of wood so the right edge is on the line (representing a negative number). In the negative zone, the block will move to the left if we multiply by a positive number and to the right if we multiply by a negative number. It’s just the mirror-image of what happens if the block is in the positive zone.

Two wrongs do not make a right, but three rights make a left.

The way I thought of it was like this. Suppose I’ve been losing $1000 a month.(A negative amount of money.) Now if I think how much has my money changed if I go from now to 6 months back. If I do that I can see that my change in money is actually positive.(6 months ago I $6000 more than I do now.)

Suppose I have a chair worth $10, and another chair so ruined I need to pay someone $10 to take it to the tip. I have a net worth of $0, as I can sell one chair, and use that money to hire someone to chuck the other, leaving me wioth $0 and no chairs.

Suppose I had 3 good chairs, and someone stole them. I would lose 3 things (-3) worth 10 each, ie. I would have -310=$-30.

Suppose a friend came round and left three bad chairs. I would have gained (*3) chairs worth $-10 each, ie. I would have 3*-10=-30.

Suppose I had three bad chairs, and someone stole them. I would lose (-3) things with -10 each, ie. I would be -3-10=$30 better off!

For the age level I teach, this would work perfectly. Thanks everyone for the suggestions, and my students will thank you too.

Karl Gauss: That would probably be too complicated at this point for my students. I simply showed them how to distribute multiplication with two positive numbers ( 6 x 14 = 6(10 + 4)). Then we made one of the numbers negative and did the same thing. We deduced that you would always get the opposite result when changing the “sign” of one of the numbers. This led the students to believe that if you changed both numbers to their opposite, that the product would become negative, but switch back to positive again. Not as concrete as I wanted, which is why I posted here.

Thanks.

Doubleplus ungood.

I have three credit cards, each has -$1000 balance. How much money do I have?
Well, 3 x -1000 = -3000, so I have -$3000.

Some foolish crook steals my identity and takes my cards, assuming my debts. How much money more money do I have?
Well, -3 x -1000 = 3000. I am $3000 better off for having minus three credit cards.

But it isn’t. If you say “No, never,” you are not agreeing to anything.

A double negative is nonstandard English, but it is never a positive. “He doesn’t know no English” has the same meaning as “He doesn’t know any English.”

Yeah. Right. :rolleyes:

:wink: