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In my precalculus class, my students have been working on trigonometric proofs for the past week or so. It's probably the most grueling section in the course; they have to memorize and be able to apply all those different identities and formulas (Pythagorean, reciprocal, even/ odd, sum/ difference, half-angle, double-angle, etc.). It's one of the necessary evils of trigonometry.
And so, in an effort to make proofs somewhat more enjoyable for my students, I told them all about false proofs in mathematics (I also give them brain teasers and logic puzzles). False proofs are basically just "proving" something silly or obviously wrong in mathematics, using trickery, cleverness, or just a misunderstanding of a math concept.
An example of a false proof would be: To simplify 16/64, you just cancel out the 6's and you get 1/4. It turns out that 16/64 actually does simplify to be 1/4, but obviously not via cancelling out digits. It's a coincidence, and can be easily shown not to work with many other fractions (e.g., 24/48 = 1/2, not 2/8 or 1/4). Simplifying fractions is all about dividing out common factors, not digits.
So anyways, there's a pretty famous false proof that 1 = 2. And usually, when I teach geometry or trigonometry (or any other course with proofs), I write out the false proof that 1 = 2, complete with seemingly valid mathematical reasons for every step, and ask the students to figure out where I went wrong. And the students generally enjoy analyzing the proof, even if they rarely figure out what's technically invalid about it (I purposely fast-talk and word the proof in such a way that my students will keep nodding in agreement until they realize I've "proved" that 1 = 2; then they get confused. This is about as devious as I get in real life.)
So this semester, I decided to try something new. I was thinking about math in the shower (because what better thing is there to think about in the shower?), and I started going off on a tangent (pun intended) and reciting the false proof. At first, I figured I'd probably just demonstrate the false proof for the students and have them analyze it as a whole, like usual. But as I said it aloud over and over again, I started to realize how easy it was to rhyme some of the key words. Then I figured: Why not turn it into some sort of song or slam poem? Surely that would make it more interesting.
So it took me about five minutes to write the poem, and then another five minutes to make a relevant powerpoint presentation so that students could see every step I was invoking in the false proof. As I got through nearly every line, I would click the next slide to express the next step in the proof. Unfortunately, I'm not able to post the powerpoint in here (and it would be weird to just show the images without changing them at the key times in the poem), so I'll just paste my poem and attempt to write out the proof, line by line, matching the lyrics. It's a pity you're unable to listen to my dreamy voice recite this lovely mathematics, but text is better than nothing, I suppose. And so, without further ado...
A false proof in math is incredibly fun,
And now I will show you that 2 equals 1.
So what is the Given? Let’s choose a and b. . . . . . . . . . .1. a = b.....Given
Please note that these letters are arbitrary.
Let’s multiply both of the sides by a, . . . . . . . . . . . . . .2. a^2 = ab.....Multiplication
Then subtract b-squared without delay. . . . . . . . . . . . .3. a^2 - b^2 = ab - b^2.....Subtraction
From here we can factor and leave a minus b, . . . . . . .4. (a + b) (a – b) = b (a – b).....Factoring
Divide that out now and let’s see what we see.
a plus b equals b, so let’s use substitution; . . . . . . . . . . .5. a + b = b.....Division
Turn a into b to work towards our solution. . . . . . . . . . . . .6. b + b = b.....Substitution
b plus b on the left, so let’s do the addition,
2b and 1b are a matching condition. . . . . . . . . . . . . . . . .7. 2b = 1b.....Addition
So now end with division of b and you’re through,
And we clearly see how 1 equals 2. . . . . . . . . . . . . . . . . . . . . . .8. 2 = 1.....Division
+ Show Spoiler +(Now don’t thank me, because I’m no hero;
Turns out that back in step 5 I divided by zero.)
The kids absolutely loved the poem  Cheers, everyone! 
~DPB
 
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Ahh that's pretty clever. I really want to come up with some sort of math sonnet now. For now this is the best that I can do. It works if you replace the equations with their shapes / simplified form.
I gave ∫ du my all, (pi/2 rad) from the start
r = 1 - sin(theta) hole, you shot an y = x through my (x^2 + y^2 -1)^3 - x^2 y^3 = 0
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On November 20 2013 14:20 Chocolate wrote:
Ahh that's pretty clever. I really want to come up with some sort of math sonnet now. For now this is the best that I can do. It works if you replace the equations with their shapes / simplified form.
I gave ∫ du my all, (pi/2 rad) from the start
r = 1 - sin(theta) hole, you shot an y = x through my (x^2 + y^2 -1)^3 - x^2 y^3 = 0
Hahahaha well done ^^
Unfortunately, I think you ignored the +C in the integral phrase though 
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I'm no math wiz, and I haven't done math since high school, but could you tell me if I'm right? + Show Spoiler +Isn't step 3 just 0 = 0? Because b^2 also = ab right?
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I didn't want to bother with latex to set the limits to 0 and u, but you're right.
Yours works much better in verbal form, though. I don't know anybody that would immediately recognize the equation of a curve that looks like a + Show Spoiler +.
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On November 20 2013 14:42 Just_a_Moth wrote:I'm no math wiz, and I haven't done math since high school, but could you tell me if I'm right? + Show Spoiler +Isn't step 3 just 0 = 0? Because b^2 also = ab right?
Technically right but that's not the part he did that he wasn't allowed to do you're allowed to say 0 = 0 as much as you want, there is a step in there where he did something that's actually just NOT allowed ^^
e: got so caught up in showing off my high school math skills that I forgot to actually say how fun the poem was =) as someone who tutors high school math myself I've seen this a few times in my life and explained it a couple as well, but I've never seen it so eloquent =D
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On November 20 2013 14:42 Just_a_Moth wrote:I'm no math wiz, and I haven't done math since high school, but could you tell me if I'm right? + Show Spoiler +Isn't step 3 just 0 = 0? Because b^2 also = ab right?
It says so in the spoiler, but in the step that he + Show Spoiler +divided by (a - b) he was actually dividing by zero
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On November 20 2013 14:43 Chocolate wrote:I didn't want to bother with latex to set the limits to 0 and u, but you're right. Yours works much better in verbal form, though. I don't know anybody that would immediately recognize the equation of a curve that looks like a + Show Spoiler +.
I recognized it, because I'm a sucker for really cool graphs Very clever ^^
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On November 20 2013 14:42 Just_a_Moth wrote:I'm no math wiz, and I haven't done math since high school, but could you tell me if I'm right? + Show Spoiler +Isn't step 3 just 0 = 0? Because b^2 also = ab right?
As Cyx. pointed out, it is indeed the case that 0 = 0, but that's not the part that invalidates the proof. The spoiler I posted in my OP explains it... and here's more info on it: + Show Spoiler +I divided by 0 when I divided out the (a - b) from both sides.
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On November 20 2013 14:46 Chocolate wrote:Show nested quote +On November 20 2013 14:42 Just_a_Moth wrote:I'm no math wiz, and I haven't done math since high school, but could you tell me if I'm right? + Show Spoiler +Isn't step 3 just 0 = 0? Because b^2 also = ab right? It says so in the spoiler, but in the step that he + Show Spoiler +divided by (a - b) he was actually dividing by zero was the actual problem. All of the steps before that one were true statements as well. Simply stating a succession of true statements is not wrong, really. It is actually one of the fundamental premises of algebra.
Oh, okay I see thanks. I never really did much (nothing?) with proofs, so that sequence of statements was a little confusing, but now I see how the whole thing works.
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On November 20 2013 14:43 Cyx. wrote:Show nested quote +On November 20 2013 14:42 Just_a_Moth wrote:I'm no math wiz, and I haven't done math since high school, but could you tell me if I'm right? + Show Spoiler +Isn't step 3 just 0 = 0? Because b^2 also = ab right? Technically right but that's not the part he did that he wasn't allowed to do you're allowed to say 0 = 0 as much as you want, there is a step in there where he did something that's actually just NOT allowed ^^ e: got so caught up in showing off my high school math skills that I forgot to actually say how fun the poem was =) as someone who tutors high school math myself I've seen this a few times in my life and explained it a couple as well, but I've never seen it so eloquent =D
Haha well thank you I may try to do more of these in the future
Are you studying mathematics besides tutoring it?
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28076 Posts
Maths too hard for me (I'm an econ major after all), but I still enjoyed the blog
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On November 20 2013 14:59 DarkPlasmaBall wrote:Show nested quote +On November 20 2013 14:43 Cyx. wrote:On November 20 2013 14:42 Just_a_Moth wrote:I'm no math wiz, and I haven't done math since high school, but could you tell me if I'm right? + Show Spoiler +Isn't step 3 just 0 = 0? Because b^2 also = ab right? Technically right but that's not the part he did that he wasn't allowed to do you're allowed to say 0 = 0 as much as you want, there is a step in there where he did something that's actually just NOT allowed ^^ e: got so caught up in showing off my high school math skills that I forgot to actually say how fun the poem was =) as someone who tutors high school math myself I've seen this a few times in my life and explained it a couple as well, but I've never seen it so eloquent =D Haha well thank you I may try to do more of these in the future Are you studying mathematics besides tutoring it?
Software engineering, heavy on the graphics and physics ^^ so no, not studying it, but I use a SHITLOAD of it lol =)
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Haha amazing job. I wish I had profs like you.
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Youtube video? with the slides and your ever so wonderful voice?
please?
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FREEAGLELAND26780 Posts
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Clever.
So much good mathematics/engineering science is done in the shower!
I'm a surrogate math teacher to 3 struggling trig/algebra students whose public school teachers are uninterested in their student's success. Let me tell you ... we drill trig identities and common 30/60/90 45/45/90 trig functions until the cows come home! Once they have that down, its the unit circle and odd/even followed by 1/2 angles and double angle identities. That sense of pride they have when they can solve elementary proofs ... nobody can take that away, it's so good.
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marvelous
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I like my math pickup lines:
"Hey girl; Are you a matrix? Because you make my vector go through linear transformations."
or,
"Hey baby. Are you looking for a guy with a large set of vectors to span your null-space?"
and,
"How's it going? That dress must be a 4x4 matrix, cause you are going through affine transformation."
not to mention,
"I wanna find your local derivative, cause I'd do gradient decent on that function all night long."
it never ends!
"Is your name Taylor? Cause if I did my calculations right, you're an infinite series that is converging on my function."
I would make a bunch about tensors, but I never took enough math  . I could start on the programming, computer science, and computer graphics ones next though.
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From step 4 to 5 something has to be wrong... a-b=0 so you can't divide both sides by (a-b) right?
Edit: yay i got it right. Cool all that math education is paying off :D
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EPT
