multiply a negative by a negative to get a positive

Discussion in 'Mathematics' started by maths_man_, Sep 8, 2011.

1. maths_man_

Are there any nice resources or video clips that anyone may have on this topic?
I want to visually show students in my year 9 next week. They are working at level 5 - 7

2. AnonymousNew commenter

I seem to remember my teacher using two fingers to represent this. Dead easy and very visual.

3. penchoNew commenter

I believe there has been at least one discussion thread about this in the past, which came up with some interesting responses. Might be worthwhile doing a search.

4. dydx

I save &pound;5 per day = +5
I gamble &pound;5 per day = -5
Time in the future is +ve; time in the past is -ve.

I save &pound;5 per day for 10 days: (+5) x (+10) = +50 better off than today
I save &pound;5 per day, 10 days ago: (+5) x (-10) = -50 worse off than today
I gamble &pound;5 per day, in 10 days time: (-5) x (+10) = -50 worse off than today
I gamble &pound;5 per day, 10 days ago: (-5) x (-10) = +50 better off than today

Waits for all the 'purists' to attack........

5. AnonymousNew commenter

You must be a bad gambler to lose &pound;5 everyday. No winnings at all??

6. jixatron

I always wave arms around, joke and get laughs going - when doing such maths stuff with any age group.....
Directions are a good illustration. The number line does it of course (good old Descartes). Also, if you say, 'Hi Jack (or Jill etc.) give us seven of your sweets - I've earnt 'em as your teacher haven't I?"
"But Mr. X, I've only got five of 'em."
"Oh that'll do, give us the five, get a couple off Tom next to you, you'll owe him two then."
"Oh Sir," Jack says, "I'll own minus two sweets then! When I get some more, I'll have to give up two before I can eat one myself at all!"
"Yes of course, you'll own minus two and with plus two, you'll break even, -2+2=0. Tell you what, I'll be kind and TAKE AWAY your MINUS TWO you owe, then you'll owe nothing. -(-2) =+2"
"Wow, you're kind Mr. X cancelling my debt. You've given me two sweets! Great."

7. coyoteNew commenter

(10-1)(10-1) = 10*10 -10 -10 + (-1)*(-1) = 80+(-1)*(-1)
But 9*9=81 so (-1)*(-1)=1

You can also construct a diagram to show this.

- from a recent issue of the ATM Journal, sorry can't remember which one!

8. DMNew commenter

Trouble is this assumes that expanding brackets is obvious (it isn't) and that no-one would ever dispute that 10 x -1 = -10 (many would). Never mind though.

9. Maths_MikeNew commenter

I am not a purist and I hope you dont regard this as an attack but isnt this just making it far more difficult than it actually is?
+ X + is positve
+ X - is negative (5 lots of negative 4 is negative 20)
- x + is the same as above becuase orde of muliplication does not matter.
40 divided by -5 must be -8 so -8 x -5 =40
Surely any attempt to relate to real life etc. just makes it more complicated?

10. googolplexOccasional commenter

Pretty much with Mike on this. I just put consecutive equations on the board in columns... 2x2=4, 2x1=2, 2x0=0, 2x(-1)=-2 etc. Seems to work...

11. Polecat

Like most mathematicians I can't explain it.
But then I thought that is a bit negative. It sounds a bit pathetic
to not be able to explain the rules of multiplication of negative
numbers.
Then I thought, if I can't not explain it then perhaps I have explained it.
I'll go back to sleep and try something easier.

12. coyoteNew commenter

You're far too quick to dismiss this. What I've shown here is a concept
that one can use, not a polished resource. All I'm assuming is that the
people on this forum can understand what I've written. Implementation is
left as an exercise for the reader...

13. algebraist

You can construct *some* algebraic structure in which -1.-1 = -1 (it won't be "an algebra" technically) but it won't have the behaviour you expect numbers to have. I laid out in my post above the things you use to prove that -1.-1 = 1. Here they are again:
(0) a + (b + c) = (a + b) + c for all a,b,c
(1) 1.x = x for any x
(2) x + 0 = x for any x
(3) x + -x = 0 for any x
(4) (a + b)c = ac + bc for all a, b, c
(5) 0.x = 0 for any xIf you want-1.-1 to be something other than 1, you can't have all of these being true. Which ones are you happy to lose?
It isn't just a convention - please don't tell your pupils that. It's a consequence of the basic rules we choose for manipulating the symbols we call numbers. We can choose different symbols and different rules, and mathematicians delight in doing this - but the usual rules are what they are because they are a good match for the way we want to use numbers in the world.

14. Polecat

The only example I can think of is a bit silly: the (ring) of integers modulo 2.
So it just consists of 0 and 1 with addition and multiplication mod 2.
Now 1 + 1 = 0 (mod 2) (sorry I can't do the 'equivalent to' symbol).
So -1 = 1 (mod 2).
-1 x -1 = 1 x 1 = 1 = -1 (mod 2)

15. reilloc

Of course you can create such algebraic structures but they will not behave like "normal arithmetic" in other ways too. I suppose I was making the assumption that we were looking for a proof or explanation that would satisfy a pupil.

I take your point about non-Euclidean geometries, but, then, I usually state that I am talking about a flat surface when introducing the "180degree law" (and indeed the angles in other shapes). I might mention other surfaces to G&T pupils.

16. jstone98

1x1=1
Therefore (2-1)x(2-1)=1
Therefore (2+(-1))x(2+(-1))=1
4+2(-1)+2(-1)+(-1)(-1)=1
(-1)(-1)=1
You only need to be able to expand brackets and convince them that a positive times a negative is a negative (which most will accept). Then negxneg has to be pos.

17. shiyani

When good things happen to good people thats good

good being +
I've usually tell them that and it helped the students remember it, but not understand the concept.

18. mathsteacher1953

Love your post Shiyani! I suppose it's like"The enemy of my enemy is my friend" Even as a mathematician I prefer this approach to expanding brackets etc which I have read above. Some of you must teach at some amazing schools if you have classes where any more than a very few top setters would be at all interested in such long winded algebraic approaches. Don't get me wrong, I do give concrete examples to show why it works rather than just say,"Learn this". My example is a school I worked at where pupils were given red slips for bad behaviour and yellow slips for good behaviour. If a girl was given 30 yellow slips in a week and 4 red slips each day for a week she would have a net + 10 points. If the teacher who had given the red slips then realised he had given them to the wrong pupil and took them away she would go back up to + 30 points. i.e. - 5 x -4 =+ 20

19. maths_man_

I ended up talking about having an negative overdraft!
I don't want you to be impolite - what am I saying?

I'd still like to give a better explanation.

20. TillyWarden

Before I write about my favourite explanation of this I should say that I would never teach it this way, I just like it.

I like to see operations as transformations of points on the number line, so addition and subtraction are just translations to the left or right. Multiplication is an enlargement of the scale factor (3 x 4 is 3 enlarged by scale factor 4 in x direction). A multiplication by -1 is a rotation by 180 degrees. Multiplication by -3 is a rotation by 180 degrees followed by an enlargement.

So -3 x -4: Start at -3, enlarge by scale factor 4, and rotate by 180 degrees = 12

Again, I don't teach it this way as it would confuse them no end, but I do like it a lot.