The latest meme to hit the mathematics blog world is to explain why a negative number times a negative number is a positive number. All the posts below are worth reading. (EDIT: I have included extra posts made after this one.)
Crossed Streams: When a minus times a minus equals a plus
Walking Randomly: Why is a ‘minus times a minus equal to a plus’?
Math Less Traveled: Minus times minus is plus
Hans Gilde’s weblog: Why a negative times a negative is a positive
mathrecreation: beautiful negatives
code monk: Minus times a minus is a plus
Points of Intersection: A negative times a negative
I thought I’d make my own contribution. Click the image for a larger version.
Filed under: Education, Mathematics |
The Number Warrior 
When I discuss this conversationally with adults, I posit the problem that they have $100, and are spending $7 of that per week. They get that the decreasing amounts in the future are negative, and they get that they have to multiply by the number of weeks to see how much they’ve spent. From there it’s a short jump to the going back in time.
My kids don’t seem to have quite the same sense of budgeting/debt, so even though they get money, they have trouble following that example.
For them, I put together this lesson.
The video idea does make me wonder if it’d be possible to do some sort of multimedia package: 10 ways to show a negative times a negative is a positive; sort of like the 25 ways to multiply from the 360 blog.
Oh, and I find it interesting that all of those posts (but not yours) freely substitute “minus” for “negative” and “plus” for “positive”.
I’m guessing the original post was being colloquial, and the rest were just following suit since they were inspired by the original post.
But yes, I personally avoid mixing terms of unary operators (positive, negative) with binary operators (plus/addition, minus/subtraction).
Students have issues with equations like -5x=15 where students will think “ah, inverse of minus is plus” and add 5 to both sides. Making sure they use the binary operation and think “the inverse of multiplying by negative five is dividing by negative five” helps a little.
What’s odd is a case along the lines of 5-x=3, where students will subtract 5 on both sides, getting -x=-2, implicitly swapping a binary with a unary operator.
Yes, we were being colloquial. I was also reminded of the poem by W.H.Auden (or possibly Ogden Nash – The net seems to disagree)
A minus times a minus is equal to a plus.
The reasons for this, we need not discuss.
It’s fun to see so many people getting in on the explanation though. Paul’s finacee (the lady who originally asked us) will probably be amazed that a non-mathematician could inspire so many mathematicians to write about a problem she was having.
Maybe we WILL convince her to take up maths after all!
If a quote is attributed to two people it is generally by neither. Likely it’s just some anonymous wag.
Instead of explaining it to the students, we ask them to create a story for some of the integer operations ( pos + pos; pos + neg; neg – neg; pos x neg; neg x neg). They have to explain the significance of the numbers (why are then positive or negative) and why that operation is being used. The hardest, by far, is the neg x neg. Occassionally, you get a gem: http://ow.ly/i/1Nb.
Once the posters are made, we aren’t finished. We then ask the students to choose from their peers work the best neg x neg poster in their opinion and explain it to us.
It’s a lot of time on a topic but it really gets down to the conceptual level and it brings in a good amount of storytelling, communication, critical thought and reflection.
Do you have any more pictures?
That was probably the best one from last year. The students this year have just created and turned in their posters/explanations. I’ll see if I can find some more (I’m not teaching grade 8 math this year) and put some links up here when I do.
This year they’ve also made screencasts to justify their choices. I’ll see if I can dig one of those up too!
Here’s another way to look at it:
Starting from 1 on the real number line:
Rotate by 90° = multiplying by i
Rotate by 180° = rotating by 90° twice, or i * i, lands on -1, so i^2 = -1
Rotate by 360° = rotating by 180° twice, or -1 * -1, lands on 1, so (-1)^2 = 1
I posted about this lesson a while back on my blog.
Barry Mazur’s book Imagining Numbers (predictably) uses the same notion.
I wouldn’t normally try this lesson until Pre-Calc; it worked fine with the Algebra II students?
It actually does work pretty well in Algebra 2. Since they are required to know powers of i and graphing complex numbers on a complex plane, it fits right in. I don’t use the lesson to teach about negative times negative, though – it’s just an interesting by-product of thinking of i as a 90° rotation.
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I was digging around for more old posts referencing the subject:
Danvk: A minus times a minus
Me and Me’s Story: Minus Times Minus??
(Nothing really new, although the second post brings up a pattern method.)
I also found this post which mentions a book where the author “seems to think minus times minus giving minus is more obvious, and perhaps should be standard.” (???)
Negative Math: How Mathematical Rules Can Be Positively Bent
I’ve used the pattern before (that type of thing also works well with negative exponents) but it gives the students a chance to make an observation rather than the ability to give an explanation. It’s good for determining what the rule is, but not why the rule is that way. It leads to the memorization of a rule and not the conceptual understanding of that rule.
BTW, after reading through all of the other posts on this subject, am I the only one who thinks most of these authors and commentors – while obviously quite mathematically intelligent – do not teach middle school or high school students?
I don’t think there’s enough context to tell; I presume the more technical explanations are not meant for the high school level.
A negative times a negative is a positive….
because I said so, that’s why!
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Sin(-A)=-Sin(A)
Sin(90)=1
-Sin(90)=-1
-1*-1=-Sin(90)*-Sin(90)
-1*-1=Sin(-90)*Sin(-90)
-1*-1=Sin^2(-90)
-1*-1=1-Cos^2(-90)
Since Cos(-90)=0==>Cos^2(-90)=0
-1*-1=1
Proved 🙂