On the left panel, I show the numbers in different colors. Zero is in the middle.

From the middle, moving to the right (Yellow) are the positive real numbers.
From the middle, moving to the left (Blue) are the negative real numbers.
From the middle, moving up(Red) are the positive multiples of i, the square root of -1. And down(Green) are the negative multiples of i.

The two panels on the right, show where each number ends up after you square it. As you can see, the negative Blues end up on the positive side (right) after you square them.

I could explain why exactly the right side has two panel instead of one... but I'll leave it at this.

An explanation of angle correspondences by Madison Cross Sugg. Color-coding (reg goes to red and so on) is separate from the above pictures.

I wrote a Java program to make this, so I could show it to my kids, and explain the wonders of complex multiplication.

My first point was really to show that multiplication has a rotational aspect.
In the complex plan, multiplying two numbers rotates the first number by the angle of the second number(angle with the positive x-axis).

Looking at it this way, we see that the angle of a negative number is PI (180 degrees)...
So when we multiply a negative times a negative, we rotate the the angle of PI by PI again, so it becomes 2PI (360 degrees) and lands back on the positive side.
Visually, we see that all the blue number in the left panel which represent negative numbers, end up on the positive side, of the two result panels on the right.

First thing to make sure you understand is that when you multiple a complex number by a complex number, the angle of the result with the x-axis is equal to the sum of the angles of the two multiplicands.

In this case, I am squaring each complex number, so I am doubling the angle.

So for the numbers in the top half of the plan (from pure yellow along the x-axis; though all of the shades of red; and up to pure blue along the x-axis), the angles start at 0 and increase to Pi (180 degrees).
If we double the angles from 0 to Pi, the results are from 0 to 2*Pi (360 degrees).
So the results cover the entire plan (leaving some holes in between, which is way we see black spots in the result plans).

Then we move on to the bottom half of the plan (from pure blue along the x-axis; through all the shades of green; then to the pure yellow along the x-axis)...the angles here go from Pi (180 degrees) to 2*Pi (360 degrees).
So the results are from 2*Pi to 4*Pi, which is equivalent to 0 to 2*Pi...
So I picture it on a separate result plan.
Again half the plan maps to cove the entire result plan, but leaving some holes in between, which are black.

The good, the bad, and hovercrafts

by Mike South

I think the appropriate answer is:

minus times minus is plus
the reason for this, we will not discuss

Just kidding.

With questions like these, I often wonder--do I understand it, or am I just used to it? Like the fact that picking a rational out of the interval from 0-1 is zero, even though there are infinitely many rationals in there, there are infinitely infinitely more irrationals. Things I really hated when I first heard them, but now...have accepted. Or maybe I have understood them. I'm sure in some cases I actually understand, because I can "explain" those....but I have to wonder on some other things.

Anyway, back on topic.

Here's a mnemonic that might aid with acceptance. Illustrations by Madison Cross Sugg.

Kind of thing

Happens to this kind of person

How we feel

bad

good

bad

good

bad

bad

good

good

good

bad

bad

good

That's only for the unenlightened among us who haven't learned universal love and forgiveness though. So for the other .0000001 % of the population you'll need something else.

I watched a very long thread on this once, and I didn't like most of the explanations. I think most explanations end up at a level a little too abstract for the person asking (at the time in their lives that most people first ask, I mean).

We talk about multiplication as repeated application of bunches of things, and I think people get that. I put three bags, of five apples each, in the cart; I have just repeated "add five" three times. It's not too tough to get a negative into that metaphor--I take three bags, of five apples each, from the cart. In many applications you get a natural negative. But it always seems to start making my head swim when you try to make them both negative. "I take three bags containing five antimatter apples from the cart"...

Say you have a hovercraft and it's sitting up there in the sky and it has some sandbags hanging off of it and some weather balloons pulling up on it, and the weather balloons pull up exactly the same amount that the sandbags pull down. Also say that if you add a weather balloon to the hovercraft, it will go up exactly one smidge. If you add a sandbag, it will go down exactly one smidge.

If you add three sets of five sandbags, then, three times you've added negative five to the height, that makes sense that it goes down fifteen smidges. If you add three groups of five balloons, it's up fifteen. If you take away three groups of five balloons, you're down fifteen. If you take away three groups of five sandbags, it's up fifteen again.

It might be best to wait until the explainee understands why "take away three groups of five balloons" is modeled as -3 * 5, as well as why "add three groups of five sandbags" is 3 * -5. Then you might be able to ask what -3 * -5 would mean in this language. And then, only after they are completely comfortabel with the answer "that's taking away three sets of five sandbags", would you ask them what direction the hovercraft moves if you take away sandbags. Maybe a light will go on there. I have higher hopes for this one than explanations with negative financial balances or going back in time, because those feel to me like you have an extra layer of abstraction that you have to keep in your head at the same time. I don't think I've ever tried it on live people yet, though, this is purely theoretical.

(BTW, this idea is just a modification of something that my wife learned about in a math ed class, a guy named Hy in a hot air balloon (Hy in the Sky), and adding sandbags made him go down. I don't know if there was anything else in it other than that, but I wanted to give credit where credit was due.)

Disk model

by Colleen King

Think of the expression, -2 x -3, as taking away 2 groups of -3.
You have no + or - discs but you need to take away 2 groups of -3.
From what will you remove the groups of -3?
We're starting with zero but zero can be 1 + disc and 1 - disc, or 2 of each, or three of each, and so on.
Model zero with 6 +- pairs.
Now you can take away 2 groups of -3.
And when you do, you're left with 2 groups of +3 which is positive 6.

if you take away 2 groups of +3 (-2 x 3), you're left with -6.

This works really well as a hands-on visual if you first do a lot of zero pair modeling. Zero pairs also work well for modeling addition and subtraction of integers.

Response by Mike South:

Well, if you want to do that kind of thing you can consider a circular hovercraft of infinite diameter with equally spaced weather balloon/sandbag pairs... :)

actually I have sketched out something like that (it took forever!) where you have a laser beam coming in from one side with a balloon/sandbag pair holding this mirrorbox in balance so that the beam goes straight through it. If you cut the sandbag, the balloon pulls up an the mirrors angle (like a periscope) and move the beam up one step. Then the idea of the activity would be that you would have a certain number of scissorbirds you could release and you have a target you want to hit with the laser. The point of the activity is to get kids comfortable at a gut level with the idea of take away this, the number goes down, take away that, the number goes up.

really needs an illustration :)

Football field model

by Key Curriculum Press

Time*Money, Time*Temperature, Skips*front/back

By Maria Droujkova

My favorite way to explain multiplication phenomena is through models. My favorite models for negative number multiplication: time*money, time*temperature, skips*front/back on the stairs. They come from the multiplication models poster.

Suppose today is day zero, and I have zero money (the ground level, so to speak).
If I make $2 every day, a week (7 days) in the future I will have 2*7=14 dollars
If I make $2 every day, 7 days ago I was IN DEBT - I had 2*(-7)=-14 (or fourteen dollars LESS than now - the negative)
If I pay someone $2 every day, 7 days in the future I will be in debt - I will have (-2)*7=-14 (fourteen dollars less than now)
If I pay someone $2 every day, 7 days ago I had MORE money than now: (-2)*(-7)=14 (positive fourteen)

Another example:

It is the same essential thing with the temperature; I will just make one example:

If the temperature drops 2 degrees every hour, 10 hours ago the temperature was HIGHER than now: (-2)*(-10)=20 (twenty degrees higher - the positive twenty).

Another example:

With skips*front/back, you can actually play on real stairs - or draw a number line on a sidewalk with chalk, or, failing that, at least draw on a large piece of paper and skip with your fingers. It usually helps kids, even older kids, to understand math through their whole body movement. This one calls for good spatial skills, or can help to develop them. In the previous model, you only had one space variable, and then one time variable. We can have each skip be positive - skipping so many steps ahead, to the side we face, or negative - skipping back (be careful on the stairs, hehe). Then we can face upstairs, in the positive direction, or downstairs, in the negative direction. Start from stair you name zero. This one is so spatial, it's kind of hard to explain in words, but if you actually draw a picture and do it, you will see.

Three skips ahead, times each skip of two stairs while you are facing up the stairs: 3*2=6 (you will be six stairs higher than ground zero)
Three skips ahead, times each skip of two stairs while you are facing down the stairs: 3*(-2)=-6 (you will be six stairs lower than ground zero)
Three backwards skips, times each skip of two stairs while you are facing up the stairs: (-3)*2=-6 (you will be six stairs lower)
Three backwards skips, times each skip of two stairs while you are facing down the stairs: (-3)*(-2)=6 (you will be six stairs higher - BINGO!)

## Multiply Signed Numbers

## Table of Contents

## The complex answer

by Raymond FaberThis might seem too abstract for some:

On the left panel, I show the numbers in different colors. Zero is in the middle.

From the middle, moving to the right (Yellow) are the positive real numbers.

From the middle, moving to the left (Blue) are the negative real numbers.

From the middle, moving up(Red) are the positive multiples of i, the square root of -1. And down(Green) are the negative multiples of i.

The two panels on the right, show where each number ends up after you square it. As you can see, the negative Blues end up on the positive side (right) after you square them.

I could explain why exactly the right side has two panel instead of one... but I'll leave it at this.

An explanation of angle correspondences by Madison Cross Sugg. Color-coding (reg goes to red and so on) is separate from the above pictures.

See also "imaginary carousel" chapter from "The Magister of Absent-Minded Sciences" and other materials we have on imaginary numbers for young kids.

Add your carousels here or email them to maria@naturalmath.com

More detailed explanations from Raymond...

I wrote a Java program to make this, so I could show it to my kids, and explain the wonders of complex multiplication.

My first point was really to show that multiplication has a rotational aspect.

In the complex plan, multiplying two numbers rotates the first number by the angle of the second number(angle with the positive x-axis).

Looking at it this way, we see that the angle of a negative number is PI (180 degrees)...

So when we multiply a negative times a negative, we rotate the the angle of PI by PI again, so it becomes 2PI (360 degrees) and lands back on the positive side.

Visually, we see that all the blue number in the left panel which represent negative numbers, end up on the positive side, of the two result panels on the right.

First thing to make sure you understand is that when you multiple a complex number by a complex number, the angle of the result with the x-axis is equal to the sum of the angles of the two multiplicands.

In this case, I am squaring each complex number, so I am doubling the angle.

So for the numbers in the top half of the plan (from pure yellow along the x-axis; though all of the shades of red; and up to pure blue along the x-axis), the angles start at 0 and increase to Pi (180 degrees).

If we double the angles from 0 to Pi, the results are from 0 to 2*Pi (360 degrees).

So the results cover the entire plan (leaving some holes in between, which is way we see black spots in the result plans).

Then we move on to the bottom half of the plan (from pure blue along the x-axis; through all the shades of green; then to the pure yellow along the x-axis)...the angles here go from Pi (180 degrees) to 2*Pi (360 degrees).

So the results are from 2*Pi to 4*Pi, which is equivalent to 0 to 2*Pi...

So I picture it on a separate result plan.

Again half the plan maps to cove the entire result plan, but leaving some holes in between, which are black.

## The good, the bad, and hovercrafts

by Mike SouthI think the appropriate answer is:

minus times minus is plus

the reason for this, we will not discuss

Just kidding.

With questions like these, I often wonder--do I understand it, or am I just used to it? Like the fact that picking a rational out of the interval from 0-1 is zero, even though there are infinitely many rationals in there, there are infinitely infinitely more irrationals. Things I really hated when I first heard them, but now...have accepted. Or maybe I have understood them. I'm sure in some cases I actually understand, because I can "explain" those....but I have to wonder on some other things.

Anyway, back on topic.

Here's a mnemonic that might aid with acceptance. Illustrations by Madison Cross Sugg.

Kind of thingHappens to this kind of personHow we feelThat's only for the unenlightened among us who haven't learned universal love and forgiveness though. So for the other .0000001 % of the population you'll need something else.

I watched a very long thread on this once, and I didn't like most of the explanations. I think most explanations end up at a level a little too abstract for the person asking (at the time in their lives that most people first ask, I mean).

We talk about multiplication as repeated application of bunches of things, and I think people get that. I put three bags, of five apples each, in the cart; I have just repeated "add five" three times. It's not too tough to get a negative into that metaphor--I take three bags, of five apples each, from the cart. In many applications you get a natural negative. But it always seems to start making my head swim when you try to make them both negative. "I take three bags containing five antimatter apples from the cart"...

Say you have a hovercraft and it's sitting up there in the sky and it has some sandbags hanging off of it and some weather balloons pulling up on it, and the weather balloons pull up exactly the same amount that the sandbags pull down. Also say that if you add a weather balloon to the hovercraft, it will go up exactly one smidge. If you add a sandbag, it will go down exactly one smidge.

If you add three sets of five sandbags, then, three times you've added negative five to the height, that makes sense that it goes down fifteen smidges. If you add three groups of five balloons, it's up fifteen. If you take away three groups of five balloons, you're down fifteen. If you take away three groups of five sandbags, it's up fifteen again.

It might be best to wait until the explainee understands why "take away three groups of five balloons" is modeled as -3 * 5, as well as why "add three groups of five sandbags" is 3 * -5. Then you might be able to ask what -3 * -5 would mean in this language. And then, only after they are completely comfortabel with the answer "that's taking away three sets of five sandbags", would you ask them what direction the hovercraft moves if you take away sandbags. Maybe a light will go on there. I have higher hopes for this one than explanations with negative financial balances or going back in time, because those feel to me like you have an extra layer of abstraction that you have to keep in your head at the same time. I don't think I've ever tried it on live people yet, though, this is purely theoretical.

(BTW, this idea is just a modification of something that my wife learned about in a math ed class, a guy named Hy in a hot air balloon (Hy in the Sky), and adding sandbags made him go down. I don't know if there was anything else in it other than that, but I wanted to give credit where credit was due.)

## Disk model

by Colleen KingThink of the expression, -2 x -3, as taking away 2 groups of -3.

You have no + or - discs but you need to take away 2 groups of -3.

From what will you remove the groups of -3?

We're starting with zero but zero can be 1 + disc and 1 - disc, or 2 of each, or three of each, and so on.

Model zero with 6 +- pairs.

Now you can take away 2 groups of -3.

And when you do, you're left with 2 groups of +3 which is positive 6.

if you take away 2 groups of +3 (-2 x 3), you're left with -6.

This works really well as a hands-on visual if you first do a lot of zero pair modeling. Zero pairs also work well for modeling addition and subtraction of integers.

Response by Mike South:

Well, if you want to do that kind of thing you can consider a circular hovercraft of infinite diameter with equally spaced weather balloon/sandbag pairs... :)

actually I have sketched out something like that (it took forever!) where you have a laser beam coming in from one side with a balloon/sandbag pair holding this mirrorbox in balance so that the beam goes straight through it. If you cut the sandbag, the balloon pulls up an the mirrors angle (like a periscope) and move the beam up one step. Then the idea of the activity would be that you would have a certain number of scissorbirds you could release and you have a target you want to hit with the laser. The point of the activity is to get kids comfortable at a gut level with the idea of take away this, the number goes down, take away that, the number goes up.

really needs an illustration :)

## Football field model

by Key Curriculum Press## Time*Money, Time*Temperature, Skips*front/back

By Maria DroujkovaMy favorite way to explain multiplication phenomena is through models. My favorite models for negative number multiplication: time*money, time*temperature, skips*front/back on the stairs. They come from the multiplication models poster.

Suppose today is day zero, and I have zero money (the ground level, so to speak).

If I make $2 every day, a week (7 days) in the future I will have 2*7=14 dollars

If I make $2 every day, 7 days ago I was IN DEBT - I had 2*(-7)=-14 (or fourteen dollars LESS than now - the negative)

If I pay someone $2 every day, 7 days in the future I will be in debt - I will have (-2)*7=-14 (fourteen dollars less than now)

If I pay someone $2 every day, 7 days ago I had MORE money than now: (-2)*(-7)=14 (positive fourteen)

Another example:

It is the same essential thing with the temperature; I will just make one example:

If the temperature drops 2 degrees every hour, 10 hours ago the temperature was HIGHER than now: (-2)*(-10)=20 (twenty degrees higher - the positive twenty).

Another example:

With skips*front/back, you can actually play on real stairs - or draw a number line on a sidewalk with chalk, or, failing that, at least draw on a large piece of paper and skip with your fingers. It usually helps kids, even older kids, to understand math through their whole body movement. This one calls for good spatial skills, or can help to develop them. In the previous model, you only had one space variable, and then one time variable. We can have each skip be positive - skipping so many steps ahead, to the side we face, or negative - skipping back (be careful on the stairs, hehe). Then we can face upstairs, in the positive direction, or downstairs, in the negative direction. Start from stair you name zero. This one is so spatial, it's kind of hard to explain in words, but if you actually draw a picture and do it, you will see.

Three skips ahead, times each skip of two stairs while you are facing up the stairs: 3*2=6 (you will be six stairs higher than ground zero)

Three skips ahead, times each skip of two stairs while you are facing down the stairs: 3*(-2)=-6 (you will be six stairs lower than ground zero)

Three backwards skips, times each skip of two stairs while you are facing up the stairs: (-3)*2=-6 (you will be six stairs lower)

Three backwards skips, times each skip of two stairs while you are facing down the stairs: (-3)*(-2)=6 (you will be six stairs higher - BINGO!)