Is there an intuitive appeal for multiplication of integers?
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Is there an intuitive appeal for multiplication of integers?
This is an abstract and superelementary question, but mathematicians are thoughtful people so perhaps it's something that people have thought about over time.
To put this question in context, I volunteer about ten hours a week with a local vocational training program, tutoring lowincome adults in basic math to prepare them for the GED (the highschool equivalency exam), entrance exams for nursing schools, and the like. Mostly I'm taking people who are starting with multiplication and division of natural numbers and taking them through decimals and percentages and proportions and the beginnings of algebra and plane geometry. It's an interesting experience, since it's five or six years of material that can get covered at any point, and it's rewarding to see people getting that math is something that they can understand with sufficient dedication and patience.
If there is one thing that seems to catch people very frequently, it is the rules for adding and multiplying integers. The part that sucks, of course, is that the rules for the two operations are different. To explain it in my own words (which are not the words I'd use in a class), if you are adding two integers with the same sign, then the sign of the sum is that common sign and the magnitude of the sum is the sum of the magnitudes of the two summands. But if you're adding two integers with different signs, then the sign of the sum is the sign of the summand with the larger magnitude and the magnitude of the sum is the difference between the magnitudes of the larger and smaller summand (unless the two summands have the same magnitude and different signs, in which case the sum is 0). But but with multiplication, the rule is that the magnitude of the product is always the product of the magnitudes of the factors and the sign of the product is positive if and only the signs of the two factors are the same.
Curiously, even though the rules for addition are more complex, it seems easier for the students to grasp, since there are a wealth of metaphors that I can use to describe the problem in realworld terms in which the sum has an obvious meaning. For instance, positive and negative integers could correspond to money that is won and lost in different rounds of gambling, or money that you either owe or are owed, and the sum is the amount of money that you have or owe at the end of the several transactions. Other metaphors are a person standing on the number line taking a series of steps in the positive and negative direction and the sum being his location at the end of the steps, or the momentum of a rope when you have a tug of war between teams of different sizes tugging on the different sides of the number line.
But try as I might, I can't think of a similar sort of intuitive metaphor for the multiplication of integers that would justify to an earnest student why the product of two negative numbers should be positive. The best I can come up with is a positive "scalar multiple" of an integer, like (3) x 4 could represent owing three dollars to each of four people or a robot that takes three steps to the left along the number line every time its button is pressed and calculating how far it moves when you press that button four times. But those framing devices don't extend to negative multiplicands, as it doesn't make any sense to press a button 2 times. Another thing that might have potential is the area of a rectangle whose sides extend however far along the xaxis and yaxis, but again I can't quite work out why one would expect a rectangle in the first or third quadrant to be a positive area while a rectangle in the second and fourth quadrant to be negative. Does anyone have any thoughts?
To be upfront, I'm not looking for a proof that (a)(b)=ab. I know that proof, but my students wouldn't understand it and are still at a level where all they need is a way to internalize the rules and faith in me that it all leads to consistent and useful results.
To put this question in context, I volunteer about ten hours a week with a local vocational training program, tutoring lowincome adults in basic math to prepare them for the GED (the highschool equivalency exam), entrance exams for nursing schools, and the like. Mostly I'm taking people who are starting with multiplication and division of natural numbers and taking them through decimals and percentages and proportions and the beginnings of algebra and plane geometry. It's an interesting experience, since it's five or six years of material that can get covered at any point, and it's rewarding to see people getting that math is something that they can understand with sufficient dedication and patience.
If there is one thing that seems to catch people very frequently, it is the rules for adding and multiplying integers. The part that sucks, of course, is that the rules for the two operations are different. To explain it in my own words (which are not the words I'd use in a class), if you are adding two integers with the same sign, then the sign of the sum is that common sign and the magnitude of the sum is the sum of the magnitudes of the two summands. But if you're adding two integers with different signs, then the sign of the sum is the sign of the summand with the larger magnitude and the magnitude of the sum is the difference between the magnitudes of the larger and smaller summand (unless the two summands have the same magnitude and different signs, in which case the sum is 0). But but with multiplication, the rule is that the magnitude of the product is always the product of the magnitudes of the factors and the sign of the product is positive if and only the signs of the two factors are the same.
Curiously, even though the rules for addition are more complex, it seems easier for the students to grasp, since there are a wealth of metaphors that I can use to describe the problem in realworld terms in which the sum has an obvious meaning. For instance, positive and negative integers could correspond to money that is won and lost in different rounds of gambling, or money that you either owe or are owed, and the sum is the amount of money that you have or owe at the end of the several transactions. Other metaphors are a person standing on the number line taking a series of steps in the positive and negative direction and the sum being his location at the end of the steps, or the momentum of a rope when you have a tug of war between teams of different sizes tugging on the different sides of the number line.
But try as I might, I can't think of a similar sort of intuitive metaphor for the multiplication of integers that would justify to an earnest student why the product of two negative numbers should be positive. The best I can come up with is a positive "scalar multiple" of an integer, like (3) x 4 could represent owing three dollars to each of four people or a robot that takes three steps to the left along the number line every time its button is pressed and calculating how far it moves when you press that button four times. But those framing devices don't extend to negative multiplicands, as it doesn't make any sense to press a button 2 times. Another thing that might have potential is the area of a rectangle whose sides extend however far along the xaxis and yaxis, but again I can't quite work out why one would expect a rectangle in the first or third quadrant to be a positive area while a rectangle in the second and fourth quadrant to be negative. Does anyone have any thoughts?
To be upfront, I'm not looking for a proof that (a)(b)=ab. I know that proof, but my students wouldn't understand it and are still at a level where all they need is a way to internalize the rules and faith in me that it all leads to consistent and useful results.
Last edited by Tirian on Sun Jun 12, 2011 4:54 pm UTC, edited 1 time in total.
 jestingrabbit
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Re: Is there an intuitive appeal for multiplication of integ
Probably the easiest way to make it clear is with money. When money goes negative, you're in debt and that's an experience that we can all understand. So, say right at the moment your costs are bigger than your income by D dollars a week, so you're down D dollars a week, and its been that way for a while. You just now ended up with a balance of 0. From that you can work out that your balance n weeks from now is f(n) = n*(D). It works if n is negative too.
So, how much money did you have in the bank n weeks ago? (n)(D) = nD dollars. That's maybe not great, ie its hugely depressing, but that's the basic domain that I'd try to explain it in.
So, how much money did you have in the bank n weeks ago? (n)(D) = nD dollars. That's maybe not great, ie its hugely depressing, but that's the basic domain that I'd try to explain it in.
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Re: Is there an intuitive appeal for multiplication of integ
Hmmm, people are pretty good at finding patterns. You might find some people were happy enough to fill in boxes in a table like this where you can see that you take 10 away each time.
10*4 = 40
10*3 = 30
10*2 = 20
10*1 = 10
10*0 = 0
10*1 = ?
10*2 = ?
Then once they're happy with that, more complicated ones like
10*3 = 30
10*2 = 20
10*1 = 10
10*0 = 0
10*1 = ?
10*2 = ?
Doesn't really explain why two negatives multiply to a positive, but it might help people to see that they do.
As for real world examples, can't really do any better than jestingrabbit, just seen stuff like 'the temperature is dropping by 3° every day, how has the temperature changed since seven days ago', i.e. 3*7
Edit: Darn, my horizontal tables didn't work. Hopefully the vertical ones look ok.
10*4 = 40
10*3 = 30
10*2 = 20
10*1 = 10
10*0 = 0
10*1 = ?
10*2 = ?
Then once they're happy with that, more complicated ones like
10*3 = 30
10*2 = 20
10*1 = 10
10*0 = 0
10*1 = ?
10*2 = ?
Doesn't really explain why two negatives multiply to a positive, but it might help people to see that they do.
As for real world examples, can't really do any better than jestingrabbit, just seen stuff like 'the temperature is dropping by 3° every day, how has the temperature changed since seven days ago', i.e. 3*7
Edit: Darn, my horizontal tables didn't work. Hopefully the vertical ones look ok.
Re: Is there an intuitive appeal for multiplication of integ
As long as one of the factors is positive, then something like 2 * 6 can be thought of as repeated addition. So, 2 * 6 = 2 + 2 + 2 + 2 + 2 + 2. We're taking "two steps back" six times. Or we're paying 2 dollars 6 times.
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Re: Is there an intuitive appeal for multiplication of integ
6 * 2 = 12: Gaining 6 dollars twice gets you 12 dollars.
6 * 2 = 12: Gaining two 6 dollar debts leaves you down 12 dollars.
6 * 2 = 12: Losing 6 dollars twice leaves you down 12 dollars.
6 * 2 = 12: Losing two 6 dollar debts means you've gained 12 dollars.
6 * 2 = 12: Gaining two 6 dollar debts leaves you down 12 dollars.
6 * 2 = 12: Losing 6 dollars twice leaves you down 12 dollars.
6 * 2 = 12: Losing two 6 dollar debts means you've gained 12 dollars.
 agelessdrifter
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Re: Is there an intuitive appeal for multiplication of integ
Tell them that a negative sign indicates a reflection on the number line, then remind them that they can always assign negatives to any number in the series of multiplications.
So if 2*3 means "take two steps to the right three times", and (2*3) means "take two reflected (eg leftward) steps three times, (2)(3) is the same as 1(2)(3), so that's the same as (2*3), which means "take two reflected steps three times and then reflect the result"
I dunno how helpful that is, though  I know that "factoring out" that negative one would terrify some of the students I work with.
So if 2*3 means "take two steps to the right three times", and (2*3) means "take two reflected (eg leftward) steps three times, (2)(3) is the same as 1(2)(3), so that's the same as (2*3), which means "take two reflected steps three times and then reflect the result"
I dunno how helpful that is, though  I know that "factoring out" that negative one would terrify some of the students I work with.
Re: Is there an intuitive appeal for multiplication of integ
lightvector wrote:6 * 2 = 12: Gaining 6 dollars twice gets you 12 dollars.
6 * 2 = 12: Gaining two 6 dollar debts leaves you down 12 dollars.
6 * 2 = 12: Losing 6 dollars twice leaves you down 12 dollars.
6 * 2 = 12: Losing two 6 dollar debts means you've gained 12 dollars.
This looks like the best example given so far, but I think it's important to focus on the more abstract rules first.
multiplying even numbers of negatives = positive result
multiplying odd numbers of negatives = negative result
You can use practical examples to illustrate that the rules work, as long as they don't try to see all multiplication problems in terms of money. It's a lot easier to go from practical to abstract than to go from practical to abstract back to practical. The analogy may make things even more confusing, since you still have to get your head around all the negatives in the english text.
Re: Is there an intuitive appeal for multiplication of integ
Good suggestions for appeals to intuition here. I'm tempted to suggest an appeal to associativity/commutativity/distributivity as well (you can prove that the product of negatives is positive), but I know from experience how well that usually goes.
Try this: Call the first number "how many hats you sell," and the second "the profit on a hat". Be clear that a negative number of hats means you bought, not sold, hats (maybe they were returned?). And that a negative cost means you actually make a loss on every hat sold (maybe there's a sale on?).
If you sell hats (+), sold at a profit (+), you will gain wealth (+).
If you sell hats (+), sold at a loss (), you will lose wealth ().
If you buy hats (), sold at a profit (+), you will lose wealth ().
If you buy hats (), sold at a loss (+), you will gain wealth (+).
This isn't really integers unless you consider money to come in discrete units. For most practical purposes there is a quantum of money (the cent), so this isn't a serious problem.
Try this: Call the first number "how many hats you sell," and the second "the profit on a hat". Be clear that a negative number of hats means you bought, not sold, hats (maybe they were returned?). And that a negative cost means you actually make a loss on every hat sold (maybe there's a sale on?).
If you sell hats (+), sold at a profit (+), you will gain wealth (+).
If you sell hats (+), sold at a loss (), you will lose wealth ().
If you buy hats (), sold at a profit (+), you will lose wealth ().
If you buy hats (), sold at a loss (+), you will gain wealth (+).
This isn't really integers unless you consider money to come in discrete units. For most practical purposes there is a quantum of money (the cent), so this isn't a serious problem.
Last edited by SunAvatar on Fri Sep 09, 2011 6:38 pm UTC, edited 1 time in total.
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Re: Is there an intuitive appeal for multiplication of integ
Signs are additive. (+) is 0, () is 1. If the sum of the signs is even, then it's (+), if it's odd, then it's (). Give them a calculator and let them play with this for a while to see that this is true, no matter how many things they're multiplying. With a simple and intuitive enough rule, most people will rarely ask why something is what it is, they will just accept it as a clever quirk those crazy mathemagicians came up with and use it as they need it.
Or, are they confident with the idea of subtracting a negative? If so, then instead approach it from the angle that multiplication by a positive is repeat addition, and multiplication by a negative is instead repeat subtraction (even though this distinction is essentially immaterial), so if subtracting a negative number gives you a positive number, subtracting it more will give you a larger positive number.
Then again, if this is for anyone going into the medical industry, they really should be capable of intuiting Bayesian inference at some point, so maybe these suggestions are in the wrong direction.
Or, are they confident with the idea of subtracting a negative? If so, then instead approach it from the angle that multiplication by a positive is repeat addition, and multiplication by a negative is instead repeat subtraction (even though this distinction is essentially immaterial), so if subtracting a negative number gives you a positive number, subtracting it more will give you a larger positive number.
Then again, if this is for anyone going into the medical industry, they really should be capable of intuiting Bayesian inference at some point, so maybe these suggestions are in the wrong direction.
should be: If you buy hats (), sold at a loss (), you will gain wealth (+).If you buy hats (), sold at a loss (+), you will gain wealth (+).
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Re: Is there an intuitive appeal for multiplication of integ
lightvector wrote:6 * 2 = 12: Gaining 6 dollars twice gets you 12 dollars.
6 * 2 = 12: Gaining two 6 dollar debts leaves you down 12 dollars.
6 * 2 = 12: Losing 6 dollars twice leaves you down 12 dollars.
6 * 2 = 12: Losing two 6 dollar debts means you've gained 12 dollars.
I had a friend that just did not want to accept that a negative times a negative was a positive without hearing some real life reason for it. I gave him this example with different numbers and he seems relatively satisfied. It seems like people can understand money better than abstract numbers...
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Re: Is there an intuitive appeal for multiplication of integ
To ease the introduction, I'd advise you to focus on showing that (x) = x. I believe multiplication follows pretty easy from that, as it's pretty much repeating additions. I'd do that by interpreting the outer minus sign as "taking out stuff", and the inner one as "stuff that has inverse value". EG: removing a debt means gaining money.
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Re: Is there an intuitive appeal for multiplication of integ
These are wonderful ideas. I'll have to work a few of them into my lesson plans and see how they go over. Thanks to everyone for tossing in your two cents!
Now that I read through these suggestions, I do recall that I did once try to explain x*y as the location of a train on the number line at "y o'clock" that moved at a constant velocity of x per hour and passed through the origin at exactly noon, and that therefore 3 * 4 would be considering a train that moved three units to the left and asking where it was four hours before noon. I wasn't particularly happy with the way it went, but now I'm wondering if it's because I was ad libbing the example. I think I'm also growing fond of the example of buying/selling hats at a profit/loss, but I'll have to work out the script in my own words.
I've not thought about talking about the "opposite" of a number as being an intuitive concept and obviously an involution and that multiplying a positive number and a negative number would be the opposite of the product of the positive number and the opposite of the negative number, and therefore that the product of two negative numbers is the opposite of the opposite of the product of the opposites of the two numbers. Even though it's not a realworld concept, it seems like it might resonate with some students.
It is indeed curious how formalistic arguments can seem completely elegant in my mind and fall flat on an audience that perhaps doesn't perceive math as axiomatic (although I don't know enough about mathematical education theory to speak with authority on the issue). An example of that is that I've explained that x^y can be considered for our purposes as taking y copies of x and multiplying them together. Therefore, by that definition we could see that x^3 * x^4 = (x*x*x) * (x*x*x*x) = x*x*x*x*x*x*x, which is seven copies of x multiplied together and therefore x^3 * x^4 = x^7. I have a really rotten batting average with that argument. Sometimes I think that math education is best spread out over twelve years because you can't schedule all of the breakthrough insights, especially the jump from arithmetic to algebra.
Now that I read through these suggestions, I do recall that I did once try to explain x*y as the location of a train on the number line at "y o'clock" that moved at a constant velocity of x per hour and passed through the origin at exactly noon, and that therefore 3 * 4 would be considering a train that moved three units to the left and asking where it was four hours before noon. I wasn't particularly happy with the way it went, but now I'm wondering if it's because I was ad libbing the example. I think I'm also growing fond of the example of buying/selling hats at a profit/loss, but I'll have to work out the script in my own words.
I've not thought about talking about the "opposite" of a number as being an intuitive concept and obviously an involution and that multiplying a positive number and a negative number would be the opposite of the product of the positive number and the opposite of the negative number, and therefore that the product of two negative numbers is the opposite of the opposite of the product of the opposites of the two numbers. Even though it's not a realworld concept, it seems like it might resonate with some students.
It is indeed curious how formalistic arguments can seem completely elegant in my mind and fall flat on an audience that perhaps doesn't perceive math as axiomatic (although I don't know enough about mathematical education theory to speak with authority on the issue). An example of that is that I've explained that x^y can be considered for our purposes as taking y copies of x and multiplying them together. Therefore, by that definition we could see that x^3 * x^4 = (x*x*x) * (x*x*x*x) = x*x*x*x*x*x*x, which is seven copies of x multiplied together and therefore x^3 * x^4 = x^7. I have a really rotten batting average with that argument. Sometimes I think that math education is best spread out over twelve years because you can't schedule all of the breakthrough insights, especially the jump from arithmetic to algebra.
Re: Is there an intuitive appeal for multiplication of integ
If you're looking for a metaphor, I always use the idea of a double negative in English.
If I say "I have an apple", then it means I have an apple (no negatives)
If I say "I don't have an apple, or I have no apple", then it means I don't have an apple (1 negative)
If I say "I don't have no apple", it really means I have at least 1 apple (2 negatives)
Though sometimes there are students who don't understand double negatives... So that's an issue. Usually, I'm just teaching one student though, so when they don't get it I'll explain: If I have no apples, then I have 0 apples. So "I don't have no apples" means the same as "I don't have 0 apples". So If I don't have 0 apples, I must have more than that. That's a double negative. One of the negatives cancels the other one out.
If I say "I have an apple", then it means I have an apple (no negatives)
If I say "I don't have an apple, or I have no apple", then it means I don't have an apple (1 negative)
If I say "I don't have no apple", it really means I have at least 1 apple (2 negatives)
Though sometimes there are students who don't understand double negatives... So that's an issue. Usually, I'm just teaching one student though, so when they don't get it I'll explain: If I have no apples, then I have 0 apples. So "I don't have no apples" means the same as "I don't have 0 apples". So If I don't have 0 apples, I must have more than that. That's a double negative. One of the negatives cancels the other one out.
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Re: Is there an intuitive appeal for multiplication of integ
I would be really careful with that, since a lot of people use "I don't have no apple" to mean they have no apple, especially in the US.undecim wrote:If you're looking for a metaphor, I always use the idea of a double negative in English.
If I say "I have an apple", then it means I have an apple (no negatives)
If I say "I don't have an apple, or I have no apple", then it means I don't have an apple (1 negative)
If I say "I don't have no apple", it really means I have at least 1 apple (2 negatives)
Though sometimes there are students who don't understand double negatives... So that's an issue. Usually, I'm just teaching one student though, so when they don't get it I'll explain: If I have no apples, then I have 0 apples. So "I don't have no apples" means the same as "I don't have 0 apples". So If I don't have 0 apples, I must have more than that. That's a double negative. One of the negatives cancels the other one out.
Re: Is there an intuitive appeal for multiplication of integ
achan1058 wrote:I would be really careful with that, since a lot of people use "I don't have no apple" to mean they have no apple, especially in the US.
That reminds me of a lecture I once attended where the speaker said "In some languages, a double negative makes a positive, like in most English dialects, while in other languages, a double negative is still a negative. To my knowledge though, there is no language in which a double positive makes a negative". From the back of the room, someone said "Yeah, right!"
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Re: Is there an intuitive appeal for multiplication of integ
How about analogy with a photographic "negative?" Making a negative of a normal picture flips the colors (sign), and making a negative of a negative would turn the colors back to normal (= positive sign).
Re: Is there an intuitive appeal for multiplication of integ
Here's a slightly higher brow conceptual explanation, which may be helpful when thinking about how to teach this stuff, though I know nothing about Math.Ed.
There are at least two ways that students think about mathematics. First, they may think of mathematics as a game of symbols with certain rules regarding how they may push these symbols around. I am told (personal accounts) that this often happens with kids when they start learning their multiplication tables and how to solve equations, i.e. to manipulate quantities without units, which they interpret as manipulating symbols without meaning (partly because we insist that pure numbers have no units?). Kids like these usually fare well until they hit a subject whose notation is inconsistent (such as Calculus), in which case the formal manipulation of symbols begins to break down because the subject is rife with abuse of notation.
Second is the impression that the numbers themselves are objects of some kind, albeit a abstract or thoughtobjects, and that the operations of addition and multiplication actually act on these objects to produce new ones. This looks exactly the same as the first approach, when written on paper, but the thought process and intuitions behind the paper are significantly different. Students that end up with this approach do better in the long run is my impression (e.g. Calculus is nowhere near as much of a hurdle because the NOTATION doesn't matter to these students as much).
The correct way to think about these things (albeit perhaps one that takes slightly more maturity in matters of abstract thought than a beginning student would have) is, in my view, subtly different from the second view. Mathematical objects, as actually used are abstractions of certain experiences and what we do with them is a formalization of certain intuitions. Addition (of integers!), for example, corresponds to the fact that we can quantize discrete objects (knowing that two collections of real world objects are equinumerous, when you think about it, does come about establishing a 11 and onto correspondence between the two collections), and our intuition tells us that quantizing the disjoint union of collections of real world objects depends only on quantizing the two individual collections. Negative numbers come about from formalizing our intuition about removing subcollections from collections, and in particular I like to think that defining negative numbers as equivalence classes of pairs of nonnegative numbers whose differences are equal is actually a reasonable explanation if done right. The key idea here is that we need not shy away from what problem mathematics solves: it gives us a language with which to model or parametrize or simplify the world and the elements of that language come from formalizing our intuitions (the tricky part is that mathematics is also meta in that it can then model or parametrize or simplify its own objects).
The point is that our counting numbers are evidently invariant under choice of units and that we can only ever add quantities of a common type (3 apples and 5 oranges are neither 8 apples nor 8 oranges, but they are 8 pieces of fruit!), and furthermore, subtraction and negative numbers are natural things (especially nowadays with the notion of debt).
Multiplication is a different beast though since we almost never actually multiply two quantities with the same units in an intuitively meaningful way. Consider lightvector's post:
The numbers on the left and on the right in these multiplications mean different things. Both are quantized by integers, but morally we have two (isomorphic) sets of integers here: one set that parametrizes dollars and debt, and one set that parametrizes the number of times you gain or lose! The latter happens to ACT on the former, that is, you always gain or lose SOMETHING a certain number of times, and in this case it is dollars and debt.
Similarly, one may think of the usual natural number multiplication model: that of measuring how many trees you have in a rectangular orchard  to get the total number of trees you multiply the number of trees in a row by the number of trees in a column. In other words # rowtrees * # columntrees = # alltrees ("#"="number of"). Notice that these quantities have units in the sense that you cannot add # columntrees to # rowtrees, but you can multiply them: in some sense the #columntrees acts on the #rowtrees to produce the #alltrees. This is our intuition made precise.
Multiplying pure numbers then happens when we forget the units of our quantities. More precisely, what happens when we forget the units of our quantities that are parametrized (quantized) by two copies of the same object (the integers) is that we identify the two copies of the integers with each other, that is, we establish an isomorphism between them, and then action of one quantity on the other gets pulled by the isomorphism to an action of the integers on themselves, which we call multiplication when it happens to satisfy certain axioms (associativity, commutativity, existence of unity). Whether or not you actually get multiplication depends on the situation, however.
The point I'm trying to make here is that in the real world, the two negatives that you are "multiplying together" MEAN different things, and it just so happens that in most situations where we have on quantity acting on another, the negative of one quantity kills the negative of the other quantity, which is why our common object for modeling situations like these (e.g. the integers) have a multiplication that treats negatives the way that it does.
Now, what the axiomatic treatment tells you is that if the action of one quantity on the other is distributive, then the multiplication that results from forgetting the units of our quantities must be such that negative times a negative is a positive, that is, that the negatives of the one quantity have to kill the negative of the other quantity (distributivity gives 0x=(0+0)x=0x+0x, hence by cancelation (existence of unique negatives, which squares with intuition) we have 0=0x; then 0=0y=(xx)y=xy+(x)y, so xy=xy; (x)(y)=(x(y))=(((xy))=xy ( a=a by unqieness of negatives giving us that negation is an involution/reflection).
To summarize, in situations where one quantity acts on another (different units!), if the action distributes, then negative quantity of one sort acting on a negative quantity of the other sort, would result in a positive quantity of the third sort, if all three quantities have an addition with inverses and 0.
There are at least two ways that students think about mathematics. First, they may think of mathematics as a game of symbols with certain rules regarding how they may push these symbols around. I am told (personal accounts) that this often happens with kids when they start learning their multiplication tables and how to solve equations, i.e. to manipulate quantities without units, which they interpret as manipulating symbols without meaning (partly because we insist that pure numbers have no units?). Kids like these usually fare well until they hit a subject whose notation is inconsistent (such as Calculus), in which case the formal manipulation of symbols begins to break down because the subject is rife with abuse of notation.
Second is the impression that the numbers themselves are objects of some kind, albeit a abstract or thoughtobjects, and that the operations of addition and multiplication actually act on these objects to produce new ones. This looks exactly the same as the first approach, when written on paper, but the thought process and intuitions behind the paper are significantly different. Students that end up with this approach do better in the long run is my impression (e.g. Calculus is nowhere near as much of a hurdle because the NOTATION doesn't matter to these students as much).
The correct way to think about these things (albeit perhaps one that takes slightly more maturity in matters of abstract thought than a beginning student would have) is, in my view, subtly different from the second view. Mathematical objects, as actually used are abstractions of certain experiences and what we do with them is a formalization of certain intuitions. Addition (of integers!), for example, corresponds to the fact that we can quantize discrete objects (knowing that two collections of real world objects are equinumerous, when you think about it, does come about establishing a 11 and onto correspondence between the two collections), and our intuition tells us that quantizing the disjoint union of collections of real world objects depends only on quantizing the two individual collections. Negative numbers come about from formalizing our intuition about removing subcollections from collections, and in particular I like to think that defining negative numbers as equivalence classes of pairs of nonnegative numbers whose differences are equal is actually a reasonable explanation if done right. The key idea here is that we need not shy away from what problem mathematics solves: it gives us a language with which to model or parametrize or simplify the world and the elements of that language come from formalizing our intuitions (the tricky part is that mathematics is also meta in that it can then model or parametrize or simplify its own objects).
The point is that our counting numbers are evidently invariant under choice of units and that we can only ever add quantities of a common type (3 apples and 5 oranges are neither 8 apples nor 8 oranges, but they are 8 pieces of fruit!), and furthermore, subtraction and negative numbers are natural things (especially nowadays with the notion of debt).
Multiplication is a different beast though since we almost never actually multiply two quantities with the same units in an intuitively meaningful way. Consider lightvector's post:
lightvector wrote:6 * 2 = 12: Gaining 6 dollars twice gets you 12 dollars.
6 * 2 = 12: Gaining two 6 dollar debts leaves you down 12 dollars.
6 * 2 = 12: Losing 6 dollars twice leaves you down 12 dollars.
6 * 2 = 12: Losing two 6 dollar debts means you've gained 12 dollars.
The numbers on the left and on the right in these multiplications mean different things. Both are quantized by integers, but morally we have two (isomorphic) sets of integers here: one set that parametrizes dollars and debt, and one set that parametrizes the number of times you gain or lose! The latter happens to ACT on the former, that is, you always gain or lose SOMETHING a certain number of times, and in this case it is dollars and debt.
Similarly, one may think of the usual natural number multiplication model: that of measuring how many trees you have in a rectangular orchard  to get the total number of trees you multiply the number of trees in a row by the number of trees in a column. In other words # rowtrees * # columntrees = # alltrees ("#"="number of"). Notice that these quantities have units in the sense that you cannot add # columntrees to # rowtrees, but you can multiply them: in some sense the #columntrees acts on the #rowtrees to produce the #alltrees. This is our intuition made precise.
Multiplying pure numbers then happens when we forget the units of our quantities. More precisely, what happens when we forget the units of our quantities that are parametrized (quantized) by two copies of the same object (the integers) is that we identify the two copies of the integers with each other, that is, we establish an isomorphism between them, and then action of one quantity on the other gets pulled by the isomorphism to an action of the integers on themselves, which we call multiplication when it happens to satisfy certain axioms (associativity, commutativity, existence of unity). Whether or not you actually get multiplication depends on the situation, however.
The point I'm trying to make here is that in the real world, the two negatives that you are "multiplying together" MEAN different things, and it just so happens that in most situations where we have on quantity acting on another, the negative of one quantity kills the negative of the other quantity, which is why our common object for modeling situations like these (e.g. the integers) have a multiplication that treats negatives the way that it does.
Now, what the axiomatic treatment tells you is that if the action of one quantity on the other is distributive, then the multiplication that results from forgetting the units of our quantities must be such that negative times a negative is a positive, that is, that the negatives of the one quantity have to kill the negative of the other quantity (distributivity gives 0x=(0+0)x=0x+0x, hence by cancelation (existence of unique negatives, which squares with intuition) we have 0=0x; then 0=0y=(xx)y=xy+(x)y, so xy=xy; (x)(y)=(x(y))=(((xy))=xy ( a=a by unqieness of negatives giving us that negation is an involution/reflection).
To summarize, in situations where one quantity acts on another (different units!), if the action distributes, then negative quantity of one sort acting on a negative quantity of the other sort, would result in a positive quantity of the third sort, if all three quantities have an addition with inverses and 0.
Re: Is there an intuitive appeal for multiplication of integ
how about Coulomb's law:
F =  bla * q1 * q2
that is, if you look at some charged spheres some distance apart, the force between them is proportional to both charges.
q1 is the charge on the leftmost object, q2 is the charge on the rightmost object, and F is the force on the leftmost object (the force on the other one is equal and opposite, by Newton's 3rd law).
we can tell the behaviour of the system just from looking at the minuses
+ve and ve means they attract, because the negatives cancel and F is positive, so object 1 moves towards the right
+ve and +ve means they repel, because F is negative
ve and ve also means they repel, because two of the negatives cancel, leaving F negative again
Note that which you call positive and which negative is an arbitrary measurement choice, what matters is that whenever the charges are the same kind (same sign) you get one set of behaviour, and when they are different you get another.
F =  bla * q1 * q2
that is, if you look at some charged spheres some distance apart, the force between them is proportional to both charges.
q1 is the charge on the leftmost object, q2 is the charge on the rightmost object, and F is the force on the leftmost object (the force on the other one is equal and opposite, by Newton's 3rd law).
we can tell the behaviour of the system just from looking at the minuses
+ve and ve means they attract, because the negatives cancel and F is positive, so object 1 moves towards the right
+ve and +ve means they repel, because F is negative
ve and ve also means they repel, because two of the negatives cancel, leaving F negative again
Note that which you call positive and which negative is an arbitrary measurement choice, what matters is that whenever the charges are the same kind (same sign) you get one set of behaviour, and when they are different you get another.
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Re: Is there an intuitive appeal for multiplication of integ
Mat wrote:how about Coulomb's law:
F =  bla * q1 * q2
that is, if you look at some charged spheres some distance apart, the force between them is proportional to both charges.
q1 is the charge on the leftmost object, q2 is the charge on the rightmost object, and F is the force on the leftmost object (the force on the other one is equal and opposite, by Newton's 3rd law).
we can tell the behaviour of the system just from looking at the minuses
+ve and ve means they attract, because the negatives cancel and F is positive, so object 1 moves towards the right
+ve and +ve means they repel, because F is negative
ve and ve also means they repel, because two of the negatives cancel, leaving F negative again
Note that which you call positive and which negative is an arbitrary measurement choice, what matters is that whenever the charges are the same kind (same sign) you get one set of behaviour, and when they are different you get another.
This might not work so well because it requires the concept of a "negative" force, which doesn't work very well until you can visualize force as a vector. Since we're talking about basic arithmetic here, bringing in vectors just makes things needlessly complicated.
...And that is how we know the Earth to be bananashaped.
Re: Is there an intuitive appeal for multiplication of integ
Last time I tried to reason this to my students by separating the multiplication. Like a*b= (1)*(a)*(1)*(b)=(1)*(1)*a*b. After separating you come up with explanations for situations of 1*1, 1*1, 1*(1) and (1)*(1) using the number line and/or the debt.
Although this created confusion among some of why 1*3=3 if you explain the negative numbers with subtraction like 03=3. It can seem like the negative numbers don't match or that they mean a different thing.
Although this created confusion among some of why 1*3=3 if you explain the negative numbers with subtraction like 03=3. It can seem like the negative numbers don't match or that they mean a different thing.
Re: Is there an intuitive appeal for multiplication of integ
Tirian wrote:Curiously, even though the rules for addition are more complex, it seems easier for the students to grasp, since there are a wealth of metaphors that I can use to describe the problem in realworld terms in which the sum has an obvious meaning. For instance, positive and negative integers could correspond to money that is won and lost in different rounds of gambling, or money that you either owe or are owed, and the sum is the amount of money that you have or owe at the end of the several transactions. Other metaphors are a person standing on the number line taking a series of steps in the positive and negative direction and the sum being his location at the end of the steps, or the momentum of a rope when you have a tug of war between teams of different sizes tugging on the different sides of the number line.
But try as I might, I can't think of a similar sort of intuitive metaphor for the multiplication of integers that would justify to an earnest student why the product of two negative numbers should be positive.
I portray multiplying by a negative as flipping. A hill flipped upside down is a hole. A hole flipped upside down is a hill. A big hill with a little hole becomes a a big hole with a little hill
Re: Is there an intuitive appeal for multiplication of integ
If you can convice them that (a+b)*c = a*c + b*c, you can try:
0 * (6) = 0
(5  5) * (6) = 30 + ? = 0
But I think that tables like at greengiant's post are even better to show that. Any other values than the right ones would really break the pattern.
0 * (6) = 0
(5  5) * (6) = 30 + ? = 0
But I think that tables like at greengiant's post are even better to show that. Any other values than the right ones would really break the pattern.
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