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In my workshop this week in Monterey, CA, a math teacher named Paul came up and said, “I ask everybody the same question: what is a numeric illustration of the fact that a negative number multiplied by a negative number is a positive number?”

I put his question out to Twitter and more than one hundred responses came in over the next few hours. You can click through to my tweet and see many of them. I’ve pulled out a sample here:

@ddmeyer I think of it as subtracting debt repeatedly, or reducing the money loss until you get a gain.

— Chris Adams (@MrAdamsProblems) June 17, 2013

@ddmeyer In a movie scene. If a car in the scene is moving backwards, playing the scene in reverse cause the car to go forward

— Chris Adams (@MrAdamsProblems) June 17, 2013

@ddmeyer Would this example work (See attached picture of an example)? Just a thought. pic.twitter.com/xZSLSkguaG

— Gary A. Petko (@GaryPetko) June 17, 2013

@ddmeyer If a negative charge is moved towards a negative electric potential, the electric potential energy increases positively by Vq

— Doug Smith (@bcphysics) June 17, 2013

@dcox21 @ddmeyer My student: "when you love love it's love, if you hate love it's hate but if you hate hate its love." #realworldexample

— Eric Benzel (@mrbenzel) June 17, 2013

@ddmeyer taking away a penalty in football moves the team forward.

— Jeffery Baugus (@baugusj) June 18, 2013

I appreciated a lot of these illustrations (as did Paul, though he pointed out that many of them aren’t numeric) but my heart belongs to Bryan Meyer’s response:

@PaiMath @ddmeyer Somewhat. From what I know neg #s were invented for debt. Once they exist, we might ask what happens when we multiply?

— Bryan Meyer (@doingmath) June 17, 2013

See, there are these things called negative numbers. Our students understand that they’re useful descriptions. They understand how to add and subtract them. (Perhaps with a metaphor like going into more or less debt.)

We know how to add and subtract *positive* numbers, sure, but we can also *multiply* and *divide* them. Is the same true of negative numbers? What would multiplying and dividing negative numbers look like? What are your theories?

In this post we have two very different organizing principles for a math class:

- Students will commit to difficult math work if we can cite some job that uses that math or some moment where it occurs in the world outside the math classroom.
- Students will commit to difficult math work if we can put our students in a position to experience what’s curious and perplexing about it.

There’s some overlap, sure, but not a lot. Over a year, those organizing principles create very different classrooms. Over a career, those organizing principles create very different teachers. Let’s talk about those differences in the comments.

**Always Related**:

- Samuel Otten’s Cornered by the Real World.

**Featured Pushback**

I am not sure why these need to be either/or or why they rise to organizing principles.

Just based on the history of mathematics, some parts of it are very practical and driven by the real world. Other parts are more abstract and were discovered and elaborated long before anybody found a practical use for them or a connection to the rest of mathematics.

The “because you need this” and “because it’s possible” are going to tap different kids, and shouldn’t I be trying to inspire them all?

Here is my response.

**Featured Comment**

Many of the examples you shared did not apparently address multiplication. For example, some of them gave an explanation for why the negative of a negative number is positive, which is not quite the same thing as explaining why a negative number times a negative number is positive.

Also:

On the present question, about Dan’s two alternatives for organizing a math class, I prefer the second. Creating perplexity and curiosity in students requires that they have some comfort and understanding that leads to a little intuition or projection that can appear to be contradicted by something, hence the surprise. If a student simply gets used to applying a formula that they don’t understand, then it is difficult to surprise them about a result related to that formula.

**2014 Mar 10**. James Key contributes a valuable entry to our project.

## 46 Comments

## Brian

June 19, 2013 - 6:26 pmI am not sure why these need to be either/or or why they rise to organizing principles.

Just based on the history of mathematics, some parts of it are very practical and driven by the real world. Other parts are more abstract and were discovered and elaborated long before anybody found a practical use for them or a connection to the rest of mathematics. Switching between these views of math as an almost literal representation of something on the one hand and a completely abstract world where things can be transformed outside of their original context on the other is an important idea, and choosing either #1 or #2 would skip over that point.

I think developing curiosity about the world is more important in some sense than turning students into savvy cell phone plan customers, but they are both good outcomes from studying math.

## Rebecca Phillips

June 19, 2013 - 6:33 pmI’m with Brian. If I look not just at a lesson, but at a unit, can I not attempt to incorporate both at different points? The “because you need this” and “because it’s possible” are going to tap different kids, and shouldn’t I be trying to inspire them all?

## Chris Adams

June 19, 2013 - 6:49 pmI think the problem with the first organizing principle is that students will disengage when the math doesn’t relate or apply to what they are going to do.

When a student asks, “when am I ever going to use this?” They don’t want to hear about applications in engineering or computer science if they plan on being a biologist.

I think because a lot of what we teach doesn’t always have nice and pretty applications, it’s important to find the overarching reasons to know this. Focusing more on the thinking skills and the big ideas, and less on the forced disingenuous “applications”.

Not all math needs to have a carrot, and students shouldn’t need to be manipulated into chasing it. Math is enough for its own sake, and that what more where I fall.

## Lucy

June 19, 2013 - 7:02 pmI think number 2 should be the priority. We often try to convince students that they need all of these skills for the future, when in realty, they may not.

Now, I understand the importance if planning for the future, but what about right now? Students should be excited to learn about difficult math, because it’s fun and interesting to learn new things, not just because it will serve them well later on. I want my students to love learning, love the process, so they will ask their own questions and pursue their own curiosity.

## Michael

June 20, 2013 - 1:04 amI know that I have reached toward #1 because I have one or more students in almost every class who frequently ask for an example of #1 (either out of genuine curiosity or a hope that I’ll say “you’re right; you’ll never see this outside of a math classroom; therefore you don’t need to do your homework”), and preemptively answering it is less embarrassing than coming up with a weak response or no response on the spot.

## Timfc

June 20, 2013 - 3:30 amI’m busy analyzing some data right now where we interviewed prospective secondary teachers about a capstone course that they were taking in spring semester.

As soon as the content wasn’t something that they recognized as being explicitly part of the K12 curriculum they basically tuned out. Part of the course was to show other ways of thinking about the content… So, even for people who are about to be teachers it seems like #1 might fail as an organizing principle due to the fact that if the content isn’t presented in a way that aligns with their (traditional) learning experience, well, then it’s not really needed for their (image of their) future job.

Representative quote:

I think it’d be really cool if we could see, if we could review more. I mean if we’re reviewing more of the actual content, what we’re going to be teaching in a high school, … rather than getting into all the theoretical stuff that I can’t ever really imagine needing to use in a Freshman Algebra class.

## Andrew Shauver

June 20, 2013 - 3:47 amThe most success (by that, I mean the highest engagement and the most authentic collaboration and problem-solving) that I’ve had are with perplexing problems. But, it tends to work best when the problem exists in an everyday context. Sharpening pencils, peanuts in a Snickers, maps of the Great Lakes (we are from Michigan), for example.

Also, when there is a variety of possible problem-solving approaches and techniques possible, that usually pays off well.

I think Lucy makes some terrific points in that focusing on application, especially when all (or most) applications are somewhere in the future, gives our students the impression that this stuff isn’t useful today. And it should be useful today.

## Santosh Zachariah

June 20, 2013 - 4:37 am@ddmeyer “There’s some overlap, sure, but not a lot. Over a year, those organizing principles create very different classrooms. Over a career, those organizing principles create very different teachers.”

I think the same can be said of students – each of whom passes through those different classrooms sequentially in the course of their schooling.

## Jennifer

June 20, 2013 - 5:15 amI like the removing holes metaphor, but I haven’t gotten much support for it!

Anyway, here are two ways to visualize the products in a coordinate grid. http://bit.ly/14422z7 and http://bit.ly/17obh3I

## Ashley

June 20, 2013 - 5:42 amOn a basic number line, think of numbers as both distance and direction. The number is how far away you go. The sign is in what direction you go. What makes something change in a positive direction? In a negative direction? How do you direct someone to go the other way?

Get students to just go a certain distance from some starting point (should it always be zero?) Figure out how far you are from where you started. How do you know you are the right distance away? (endless possibilities for difficult math) In what direction did you go? How do you know? Would someone else from a different location agree with your description of the change in distance and direction? (wait are we doing calculus now?)

Now think about traveling in the opposite direction? (multiplying by a negative). In what direction did you travel? (relative to your first direction?) How far did you travel in the opposite direction?

Chart some of it! (Distance, direction, change in distance, chance in direction, change in location, etc. (come up with a way to represent all of this relative change.) Do patterns emerge?

How do you do the same thing in the real world (360 degrees and 3 dimensions)? How do you describe direction then? What does it mean to just go a distance away? What does it mean to go a distance in a particular direction? in the opposite direction? an opposite distance?

Distance is easy – just how far. Opposite is easy – just go the other way. Multiplying and dividing them is easy too – (multiples of distance/ fractions of distance), and signs are just direction (further away in the same direction / back towards the start ). But modeling and describing all that is wonderful difficult engaging math!

## Ben

June 20, 2013 - 6:00 amMultiplying negative numbers makes perfect sense to me if you convert it into addition — which it where it seems like negative numbers really work on an intuitive level, because the negative sign just means take away instead of add.

2 x 3 means add up 2, 3 times. 2 + 2 + 2 = 6.

-2 x 3 means add up -2, 3 times. -2 + -2 + -2 = -6.

2 x -3 means *take away* 2, 3 times. – 2 – 2 – 2 = -6.

-2 x -3 means *take away* -2, 3 times. – (-2) – (-2) – (-2) = 6.

## Mary Dooms

June 20, 2013 - 6:02 amIn my early career as a math teacher I thought engagement meant caring more about real world application than mathematical inquiry and the theoretical, so I was in camp 1. That was a bad place to permanently park because if the situation didn’t directly affect the student the math would never be relevant. Sure I made connections, but some students responded, “Big deal” or said far worse.

Fast forward a few years and I’ve acquired a new definition of engagement to mean intellectually stimulating. For me that includes mathematical inquiry as well as exploring the theoretical. I attempt to straddle two or three worlds because I have to keep in mind where my students are at developmentally.

You ask “What would multiplying and dividing negative numbers look like?”

My 7th graders are treated to Tanton’s video http://www.youtube.com/watch?v=eV6iYvd4KS0 of the rules applied to “Why a negative times a negative is a positive”. Some are entranced by the logic. Others Scratch (pardon the pun) their heads and can only handle this less than mathematical explanation: http://scratch.mit.edu/projects/2073151/

## Sue VanHattum

June 20, 2013 - 8:49 amOne more in the same spirit as James Tanton: https://www.youtube.com/watch?v=8CGAjzU5M70

Also, I suppose it’s a subcategory of #2, but I like connecting the history, “Here’s what it was originally good for.” That sounds more like #1, doesn’t it? But it also gets at how we expanded our number system, or our thoughts about functions, to include something new. To see math as changing helps with the curious and perplexing part for me.

I want to start calculus out with more reference to the problems Newton and Leibniz were trying to solve.

## Paul Bogdan

June 20, 2013 - 9:37 amI tried tweeting this example that I thought of, but apparently didn’t do it correctly. I think it is pretty good. It’s not as simple as I would like, but I think an eighth grader could follow it.

Let’s say that when you dig a hole, the dug hole is a negative size deep. It is negative because a hole is below the ground. So, if you dig a hole 3 feet deep the result is negative 3. If you dig 5 of these holes the result is negative 15. This is a model for positive times a negative is negative.

Now think about negative digging. What might that be? If positive digging is the result below the ground then negative digging is the pile of dirt above the ground. If you dig the 5 holes negatively the result is positive 15. This is a model for negative times a negative is positive.

In this model the 3 is negative because the dirt came from below the ground. A positive 5 means we are talking about the 5 holes. A negative 5 means we are talking about the 5 piles of dirt.

The examples about negating a penalty and negating a debt are weak because they only get you back to zero. Language examples abound and I’ve used them for years this is my favorite:

We all know what it means to be not talking (or some of us anyway). What does it mean to be not not talking?

I really like the example of like magnetic charges repelling each other. This is not numeric, but will make a good demonstration.

## Dan Meyer

June 20, 2013 - 9:51 amBrian:Like I said, there’s some overlap and both moments can occur in the same class. But there exists a class where the real-world moment serves the larger purpose of intellectual need and another class where the real-world moment is an end unto itself, where the teacher heaves a relieved sigh and says, “I’m glad I figured out that third-degree polynomials can be used to model skateboarding usage during the 1990s,” where the real world is invoked

withoutintellectual need.Rebecca Phillips:We’re going to have a lot of trouble if a kid only responds to appeals of the “because you’ll need it for a job” sort.

If we train a kid to think that the only useful math is math that someone needs in their vocation, we’re going to lose all the kids who plan not to pursue jobs that require math. (And a ton of jobs don’t.) And we’re going to force ourselves to concoct increasingly bizarre vocations the more math we require to students to take. (eg. the statistician that model skateboard usage with third-degree polynomials.)

Our goal should be instead to produce

curiouskids, and that curiosity can be invoked in different ways, sometimes by a real-world connection and other times by the odd ways that numbers and shapes interact with each other.## James Key

June 20, 2013 - 9:56 amI like what Chris Adams had to say — learning math is not about “chasing carrots.” I know that Dan Meyer’s personal favorite construct for “what math education is all about” is rooted in the notion of creating moments where the student feels *perplexed.* This is a great notion, and I’ve learned a lot from trying it on. My own construct goes something like this: math education is all about *establishing connections to prior knowledge.* In other words, this is my theory of what math students are thinking: “So you want to teach me this new thing. How is it related to the things I already understand?” If students are not asking themselves this every day, every lesson, I’m going to suggest it’s because “the system killed them.” If math teachers go around saying that “a negative number times a negative number is a positive number, because if you hate hate, that’s love,” the student is likely to go “Great! That makes a lot of sense.” But then they develop the notion that mathematics is all about these metaphors the teacher offers with each new lesson, but which bear no real connection to each other. (Sorry if I just picked on your favorite explanatory device.)

So I’m going to come down on the side that says we should treat this problem like *every other problem we encounter in math* — let’s start by asking a few logical questions, and see where it leads.

I want to know what (-3)(-2) gives. Observations:

1. I should be able to use what I know about negative numbers.

2. I should be able to use what I know about multiplication.

My own answer to this question requires about 800 words, and you can read them at my blog over here:

http://iheartgeo.wordpress.com/wp-admin/post.php?post=65&action=edit&message=6&postpost=v2

Hint: it’s all about the distributive property, baby.

If you want a sneak preview in the size of a Tweet, here it is: climbing stairs. Negative numbers are like climbing *down* the stairs. (-3)(-2) is “the opposite of 3 groups of -2,” which is the opposite of -6, which is +6.

## Jason Dyer

June 20, 2013 - 10:24 amre: the statistician that model skateboard usage with third-degree polynomials, that’s not a very good statistician. He/she is doing overfitting. Taking a small set of data points and matching an equation to it just because you can is a fruitless excercise BOTH mathematically AND in the real world.

If a real-world application is genuine, it will have perplexity by nature. (That doesn’t automatically mean “interesting to students”, of course, but the catalogue of things that applies to that is so contextual it can’t be be abstracted into a single rule.)

## Jason Dyer

June 20, 2013 - 10:34 amAlso, one other approach is to ask “what happens if we go ahead and let a negative times a negative equal a negative”?

There’s a book that derives what the system would look like:

Negative Math: How Mathematical Rules Can Be Positively Bent

There’s no reason we can’t mess with the axioms to have the math come out to however we want. For axioms to be consistent with real-life mathematical models, however, we define them in a specific way. In this sense your #1/#2 split is false, because if you completely divorce one from the other then we’re just playing a game with symbols where we can change the rules at any time. (Comment from a review for the book above: “this book argues that the distributive rule is no more special than the commutative rule, and that accordingly it too can be restricted or rejected just like the universality of the commutative rule was rejected when the theories of quaternions and vectors were invented.”)

## Dan Meyer

June 20, 2013 - 11:17 amJason Dyer:Shouldn’t the rule on perplexity be “that which is perplexing to

yourstudents.” Students in Kansas aren’t as perplexed by giant wave surfing as students along the Pacific Rim. That doesn’t mean we throw up our hands and say, “Who can know?”## Jason Dyer

June 20, 2013 - 11:50 amStudents in Kansas aren’t as perplexed by giant wave surfing as students along the Pacific Rim.That’s what I meant by contextual. I agree with your entire comment right there.

## Christine

June 20, 2013 - 4:15 pmI draw a number line and as we go through each answer I show a hop in the negative direction so that the pattern is obvious:

4 x 3 = 12

4 x 2 = 8

4 x 1 =4

4 x 0 = 0

4 x -1 = -4 (I am saying: you owe your Mum a $1, you owe your Mum a $1 , you owe your Mum a $1, you owe your Mum a $1 means you owe your Mum $4 (ie -4)

4 x -2 = -8

4 x -3 = -12

then again on a number line this time starting at -12 and going in the positive direction.

4 x -3 = -12

3 x -3 = -9

2 x -3 = -6

1 x -3 = -3

0 x -3 = 0

-1 x -3 = 3

-2 x -3 = 6

-3 x -3 = 9

-4 x -3 =12

## Anne

June 20, 2013 - 4:59 pmTo me, we have to start by defining our overall objective as math teachers. Are we trying to prepare students for jobs in STEM or foster lifelong learning? Certainly both. However, there seem to be many who feel that the first is a goal for some students whereas the second is a goal for all students, which I disagree with. I entered mathematics in college because I had great math teachers in secondary school. Had I not had those teachers, I likely would have picked something different. We can’t expect that all students will go into STEM, but we also can’t assume that we know which ones will and which ones won’t. We should be teaching them all as if we could see them as part of the next generation of scientists, engineers and mathematicians. If they choose something else upon entering college, the workforce, etc., then it should be because they wanted to do something else, not because they didn’t feel prepared to be in STEM. Giving them a taste of how math is applied is essential if we want them to feel prepared to enter those fields.

## Manuel

June 21, 2013 - 7:17 amHere’s an accessible proof for most students:

if we accept 5-(-8)=13 as true,

Then we have that

5-(-8)=13 is equivalent to

5-1(-8)=13 and following order of operations this only works if

5+8=13 which implies that

-1(-8) = +8

Obviously different numbers can be used.

## Roger Gemberling

June 21, 2013 - 4:09 pmI used round disks with yellow on one side and red on the other side of the disk. The yellow disk, when face up, represents a positive value of one and the red disk, when face up, represents a negative value of one. When a yellow disk and red disk are paired, the value is zero.

Lets says you have a field of A Lot of yellow and red disks that represent a value of zero.

For the problem -3 x -5, students would remove 3 sets of 5 red disks (for a total of 15 red disks). The value of the field would be +15.

Beginning with a field that has a value of zero. For the problem -4 x -2, students would remove 4 sets of 2 red disks (for a total of 8 red disks). The value of the field would be +8.

## Benjamin A. Smith

June 22, 2013 - 10:54 amI assume that most readers here know this already.

Certain definitions in mathematics tend to follow from the need to be complete and thorough in a conceptual/intellectual sense, not from the ability to draw myriad real-world pictures of the situation. I think operations on integers and rationals falls under this.

We define negative numbers because they are useful in so many real-life contexts (elevation, temperature, finance, relative position, electricity, etc.), and in all these contexts the operations of addition and subtraction are used constantly.

But then we (should) naturally begin to wonder what happens when we multiply and divide with negatives. If we allow/define a new kind of number, we need to define operations on it. We want to have a complete, thoroughly defined system.

So real-world modeling motivates to define negative numbers, but INTELLECTUAL virtues (curiosity, thoroughness) drive us to define the less obviously useful operations on said numbers.

We recall that for natural numbers and fractions, division is really just multiplication. So we really just need to figure out how we should multiply negatives.

We are able to convince ourselves that a negative times a positive results in a negative. Four $30 dollar debits results in a change of -$120., etc.

But it remains to answer the problem at hand: To explain why two negatives have a positive product.

Since we are wanting to define the same operations for negative numbers that we defined for natural numbers, it follows that we should expect the operations to have the same properties: association, commutativity, distribution.

The distributive property is the relevant one. This is hinted at by the CCSS (7.NS.2.a) though not really explained there.

When defining the product of two negatives, we try the simplest case: (-1)(-1). There are really only two intuitively appealing possibilities: (-1)(-1)=1 and (-1)(-1)=(-1). The latter tends to appeal to students because they don’t want the negatives to “disappear”.

So we try both options in a simple distribution expression. First, the most appealing, (-1)(-1)=(-1).

If (-1)(-1)=(-1), then:

(-1)[1+(-1)]=(-1)(1)+(-1)(-1)

and therefore

(-1)(0)=(-1)+(-1) <– the rightmost product by our tentative definition

So that

0=(-2), which is a contradiction. So (-1)(-1) cannot = (-1).

It remains to try the other, less obvious possibility, (-1)(-1)=1.

(-1)[1+(-1)]=(-1)(1)+(-1)(-1)

so that

(-1)(0) = (-1)+1

which can be simplified to

0 = 0.

So (-1)(-1) equals 1, not negative 1, by careful reasoning based on the the requirement that operations on negatives have the same properties that operations on natural numbers did.

## David Radcliffe

June 22, 2013 - 3:30 pmPositive and negative numbers are used to describe the increase or decrease of a quantity. Related rates give a natural context for multiplication of signed quantities.

Imagine that you are in a hot air balloon. The air temperature decreases by 4 degrees for every 1000 feet of altitude. If you descend 2000 feet, then the air temperature will increase by 8 degrees. (-2000) * (-4/1000) = +8.

It is easy to think of examples from daily life. Using electricity costs money (a negative), so if you use less electricity then you will save money.

## Scott Farrand

June 22, 2013 - 8:15 pmMany of the examples you shared did not apparently address multiplication. For example, some of them gave an explanation for why the negative of a negative number is positive, which is not quite the same thing as explaining why a negative number times a negative number is positive.

The other day I spoke with a teacher who had been teaching addition of negative numbers, and had a student whose older sister had told her that two negatives make a positive, so she was certain that -3+-4=7.

I believe that explanations should clearly include multiplication (as opposed to addition) to be helpful. If you can read the context, you should be able to see why it is multiplication and not addition that is at play (e.g., loving or hating love doesn’t really scream addition or multiplication to me — what quantities are being added or multiplied?).

On the present question, about Dan’s two alternatives for organizing a math class, I prefer the second. Creating perplexity and curiosity in students requires that they have some comfort and understanding that leads to a little intuition or projection that can appear to be contradicted by something, hence the surprise. If a student simply gets used to applying a formula that they don’t understand, then it is difficult to surprise them about a result related to that formula.

The other piece to the second organizing principle that I find implicit in it is the idea that students should be provided opportunity to learn new mathematics in a way that allows it to become real to them (whether or not that is in a real context). Then they are ready to be surprised, to ask questions, to be perplexed, to make conjectures, and to give expression to their curiosity.

## Dan Meyer

June 23, 2013 - 3:56 amScott Farrand:Thanks for chiming in here,

Scott, both with the multiplication caveat and the implicit citation of Freudenthal right above. As I work with teachers, I encounter more theories of engagement than I can count. eg. “the math needs to be connected a job”; “the math needs to be real world”; “the math needs to be taught in its purest form so students can appreciate its applications later.”While it’s easy to find examples of student engagement that contradict those theories above, Freudenthal’s theory is tougher to assail, in part because it doesn’t actually reach all that far. His theory of engagement is possibly the largest true statement about engagement any of us can make, and it isn’t all that large!

It leaves the rather tough job of enactment – of finding ways to make all kinds of math real to students – to practitioners. You’ve pitched in a good one there – getting students enough intuition about a mathematical object that they can experience something surprising about it. We should find others.

## Lee Trampleasure

June 23, 2013 - 3:52 pmI was working on a solution along the line of Christine’s number line as the visual representation.

I find this similar to explaining how x^0 can equal 1. Unless you look at it as part of a sequence (x^2, x^1, x^0, x^-1), there seems to be no way to explain how something to the zero power can equal anything :-)

## Dan Meyer

June 23, 2013 - 4:59 pmHey, it’s Lee!

## Christopher Perry

June 23, 2013 - 7:00 pmAs an aspiring teacher in mathematics I agree with Rebecca and Brian. I don’t think we should limit ourselves to just one idea through a unit. I think to say that, “because you need this..” and “because it’s possible..”, you can not limit that to one lesson or one point. I think that in order to develop curiosity among students both principles should not be chosen as an either/or situation.

I invite you to come check out my blog sometime and see what is going on in my EDM310 class at the University of South Alabama.

the url is: http://www.perrychristopheredm310.blogspot.com

Also my twitter name is: @perry_chris098

## James Key

June 24, 2013 - 5:19 am@Lee: I think there is a key difference between the issue you raise — how we define the 0th power of a number — and the issue of multiplying negative numbers.

In the case of x^0, we simply *define* the value as 1. As you point out, the pattern argument supports this decision. That being said, there is nothing in the logical structure that requires us to define x^0 as 1, or even to define that expression at all.

But in the case of multiplying negative numbers, as soon as we set down our postulates for numbers, especially the distributive property, we are logically forced to require that neg times neg equal pos.

(5)(-3) + (-5)(-3) = (5+-5)(-3) = (0)(-3) = 0, from which it follows that (-5)(-3) is the additive inverse of (5)(-3).

## Scott Farrand

June 24, 2013 - 8:36 amHere’s a perspective from which there is a way to establish that x^0 is 1, when x represents a non-zero number, and it is akin to the way that you showed that a negative multiplied by a negative results in a positive number. I would argue that to accomplish that result, you took properties of the whole numbers, such as the distributive property, and wanted to see what happens when the number system is extended to include negative numbers. The integers arrive as an extension of the whole numbers, and this arithmetic property is then valid because we ask that the distributive property, for example, also should hold when negative integers are involved. Otherwise, what permits the use of the distributive property when there are negative numbers involved?

The property of interest for showing that x^0=1 (say when x is a positive integer) is the additive law of exponents x^(a+b)=(x^a)*(x^b), which is a property of the whole numbers (for a and b both positive integers). Extending this property to require that it hold on a larger set of numbers (including the possibility that a and/or b would be 0), then requires that x^(0+b)=(x^0)*(x^b). Since x^(0+b)=x^b, we are left with x^b=(x^0)*(x^b), and because x^b is non-zero, we can conclude that x^0 must be 1.

From this perspective, it is the extension of properties to a larger set of numbers, as we move to larger number systems (whole numbers to integers to rational numbers to real numbers to complex numbers) that provides many interesting mathematical junctures.

## Lee Trampleasure

June 24, 2013 - 12:52 pmNot to highjack the thread, but x^0=1 does not have to be a definition. When my science students are confused about it, I just point out that when you decrease an exponent by 1, you divide the previous solution by the the base. For example, (x^3)/x = x^2, (x^2)/x = x^1, and (x^1)/x = x^0, and x/x = 1, therefore x^0 = 1.

I’ll write on the board

10^2 = 100

10^1 = 10

10^0 = 1

10^-1 = 0.1

10^-2 = 0.01

This can also be shown by graphing y=a^x (plug in any number for a, including negative numbers!)

@ James: I try to avoid “defining” anything that can be shown based on examples/other rules :-)

## James Key

June 24, 2013 - 12:52 pm@Scott, I like your point here. I would also add that x^(b+-b) = x^b * x^-b, and if we already require that x^-b be the reciprocal of x^b, then it follows that x^0 = 1. i.e. once we commit to the definition of x^-n = 1/x^n, we are forced to choose x^0 = 1, provided (as you mention) we want the laws of exponents for pos integers to apply to neg integers as well.

## James Key

June 24, 2013 - 12:58 pm@Lee: I think we are really saying the same thing here. The argument you use in comment #35 just makes the case for the definition of x^0, but it is nevertheless a definition. In other words, you make a “pattern argument,” but who is to say the pattern can’t work like this:

10^2 = 100

10^1 = 10

10^0 = 53

There is no *logical* reason for refusing to admit 53 in the “pattern” above. The fact that you cite a particular numerical pattern and a particular graphical pattern just means that we *really, really want* to define x^0 as 1, but that doesn’t cancel the fact that it’s a definition, in a technical sense.

## Paul Bogdan

June 24, 2013 - 2:33 pmI posed the original question to elicit real-word numeric examples.

The zero power is tough. Intuitively the answer is zero, but that makes problems. Here’s an example, or train of thought.

Suppose we are talking about things that exist. Then an exponent of three has a lot of existence it represents a cube. Two is less (only length and width). One is less still. An exponent of zero still exists; it still has position. So, it is one for that reason. Making it zero would take it out of existence.

Higher exponents would mean more existence; we would add time, color, etc. Exponents that are less than zero imply less existence which is why they are fractions (and smaller the more negative they get.

Has anyone asked about zero factorial?

## David Radcliffe

June 25, 2013 - 8:53 pmPlace value could be a good way to introduce zero and negative exponents. If the hundreds place is 10^2 and the tens place is 10^1, what is the ones place? What is the hundredths place?

## George Bigham

July 9, 2013 - 11:54 amStudents are diverse, some will be motivated by the practical others by the more esoteric and abstract. As teachers we also have our leanings about which side we find more important and naturally want to teach in favor of that slant. Personally, I’ve always been more fascinated in the abstract and ‘pure’ side of math, but when teaching I think its most important to gauge what type of class and what type of students you have and teach to them and their motives (and of course the requirements of the standards,sigh.)

I think its important when students challenge the notion of (-1)*(-1)=1 we encourage this questioning and do not just shut them down. To appeal to both types of students and motivations I think I would answer that (-1)*(-1) doesn’t have to equal 1, but we defined it this way as a convention. It is not an arbitrary convention, it was done for many practical reasons for modeling real world behaviors described above. It was also done for abstract reasons such as preserving the distributive property also mentioned above. Deciding that (-1)*(-1)= -1 is within the powers of a mathematician, however, he or she has to be aware of the consequences that follow… such as no distributive property and many real world modeling problems that won’t make sense. Perhaps it would be good to address the problems with (-1)*(-1) = -1 to shine light on the virtues of (-1)*(-1)=1.

## Melissa Luzano

December 2, 2013 - 2:36 amI tried something new this year, so it obviously needs work, but I thought it’d be worth sharing (even though it’s several months late). I think it addresses both the negatives times negative and the real world issues.

Before introducing integers, we watched the movie Upside Down (trailer: http://www.youtube.com/watch?v=3veONCcRWbw). In it there’s an elevator with integers. Floor zero is in the middle of the two planets. The positive floors lead to the top planet, and the negative floors go to the planet below.

We make an Integer Elevator, and the students use it to do integer operations. It’s basically a vertical number line.

examples:

4-7 means you (starting at floor zero) go up 4 floors and then down 7 floors. You end on floor -3.

2(3) means you go up 3 floors twice.

2(-3) means you go down 3 floors twice.

(-2)(-3) means you go down 3 floors but twice in the opposite direction, which takes you to floor 6.

*The elevator thing doesn’t work for (-1/2)(-3.07), but I’m pretty satisfied that it helps the students understand (-2)(-3) first.

Is it a real world application? Yes, but from a different world, so I think it leans more to the perplexing side.

Mr. Meyer, any chance you attended an AP Institute in San Diego?