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
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
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
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 +
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
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.
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 +
heart
.
I recognized it, because I'm a sucker for really cool graphs Very clever ^^
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.
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.
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
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 =)
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.
"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.
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
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 =)
Yeah my course has all STEM (science, technology, engineering, mathematics) majors, so they understand the relevance of some math topics we've reviewed
On November 20 2013 16:46 Chairman Ray wrote: Haha amazing job. I wish I had profs like you.
On November 20 2013 17:29 flamewheel wrote: This was cute~
On November 20 2013 18:39 Big J wrote: marvelous
Thanks!
On November 20 2013 17:00 Artisian wrote: Youtube video? with the slides and your ever so wonderful voice?
On November 20 2013 18:01 Danglars wrote: 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.
Yeah, I agree My students are starting to learn decent strategies for solving trigonometric proofs (e.g., turn all functions into sines and cosines first), and they're having a lot more success than they did in geometry.
There are plenty of interesting math proofs out there, but the only two instances where high school students really learn about implementing proof is to either prove that two triangles are congruent in geometry, or proving trigonometric identities in precalculus. It's a pity that we don't reinforce valid mathematical reasoning and proof with more fun or creative arguments (like the fact that every perfect square can be written as either 4k or 8k+1, where k is an integer).
On November 20 2013 19:27 jrkirby wrote: 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.
And the classic: I wish I were your derivative so I could lie tangent to your curves
Or perhaps: I wish I were your second derivative so I could explore your concavities
On November 20 2013 20:00 Recognizable wrote: 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
Haha well done! Now can you (or anyone else) actually explain why you can't divide by zero in this case? It's a classic mantra in mathematics, that *you can't divide by zero* ( ::cough:: until calculus and limits ::cough:: ), but... why not?
On November 20 2013 20:42 Paljas wrote: 1 = -(i²) 1= 1^(1/2) = (-(i²))^(1/2) = (-1)^(1/2)*(i²)^(1/2) =i*i = -1
This is also a great false proof I occasionally show this to my students after they cover imaginary numbers, although we don't really do any proofs with i.
On November 20 2013 20:42 Paljas wrote: 1 = -(i²) 1= 1^(1/2) = (-(i²))^(1/2) = (-1)^(1/2)*(i²)^(1/2) =i*i = -1
I was thinking of that one as well.
Also the one in the OP I know in this version: a = b + c |-2a -a = b+c-2a |+(b+c) b+c-a = 2(b+c-a) 1 = 2
And it always reminds me of the time when I took quantum theory 1 and the professor told us that "now we only have to calculate the determinant of this 2x2 matrix which is easy" and wrote det(A)=ad+bc on the blackboard... Physics...
On November 20 2013 20:42 Paljas wrote: 1 = -(i²) 1= 1^(1/2) = (-(i²))^(1/2) = (-1)^(1/2)*(i²)^(1/2) =i*i = -1
I was thinking of that one as well.
Also the one in the OP I know in this version: a = b + c |-2a -a = b+c-2a |+(b+c) b+c-a = 2(b+c-a) 1 = 2
And it always reminds me of the time when I took quantum theory 1 and the professor told us that "now we only have to calculate the determinant of this 2x2 matrix which is easy" and wrote det(A)=ad+bc on the blackboard... Physics...
What do your bars/ vertical lines ( denoted as | in your false proof) refer to? It doesn't look like division or conditional probability.
And he wrote ad+bc instead of ad-bc? Would you mind sharing why you're allowed to change the minus sign?
On November 20 2013 20:42 Paljas wrote: 1 = -(i²) 1= 1^(1/2) = (-(i²))^(1/2) = (-1)^(1/2)*(i²)^(1/2) =i*i = -1
I was thinking of that one as well.
Also the one in the OP I know in this version: a = b + c |-2a -a = b+c-2a |+(b+c) b+c-a = 2(b+c-a) 1 = 2
And it always reminds me of the time when I took quantum theory 1 and the professor told us that "now we only have to calculate the determinant of this 2x2 matrix which is easy" and wrote det(A)=ad+bc on the blackboard... Physics...
What do your bars/ vertical lines ( denoted as | in your false proof) refer to? It doesn't look like division or conditional probability.
And he wrote ad+bc instead of ad-bc? Would you mind sharing why you're allowed to change the minus sign?
Oh, the bars refer to doing the operation to both sides. E.g.: I have a = b + c and then substract 2a from both sides (so |-2a) which leads to -a on the left and b+c-2a on the right.
The joke is that the quantum guy was all like "2x2 determinant, that's so easy". And then calculated it wrong, since it is ad-bc ofc.
On November 20 2013 20:42 Paljas wrote: 1 = -(i²) 1= 1^(1/2) = (-(i²))^(1/2) = (-1)^(1/2)*(i²)^(1/2) =i*i = -1
I was thinking of that one as well.
Also the one in the OP I know in this version: a = b + c |-2a -a = b+c-2a |+(b+c) b+c-a = 2(b+c-a) 1 = 2
And it always reminds me of the time when I took quantum theory 1 and the professor told us that "now we only have to calculate the determinant of this 2x2 matrix which is easy" and wrote det(A)=ad+bc on the blackboard... Physics...
What do your bars/ vertical lines ( denoted as | in your false proof) refer to? It doesn't look like division or conditional probability.
And he wrote ad+bc instead of ad-bc? Would you mind sharing why you're allowed to change the minus sign?
Oh, the bars refer to doing the operation to both sides. E.g.: I have a = b + c and then substract 2a from both sides (so |-2a) which leads to -a on the left and b+c-2a on the right.
The joke is that the quantum guy was all like "2x2 determinant, that's so easy". And then calculated it wrong, since it is ad-bc ofc.
On November 20 2013 20:00 Recognizable wrote: 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
Haha well done! Now can you (or anyone else) actually explain why you can't divide by zero in this case? It's a classic mantra in mathematics, that *you can't divide by zero* ( ::cough:: until calculus and limits ::cough:: ), but... why not?
[/QUOTE] Because R is a field, and in a field, and 0 is not in the group of units. \box.
Well you don't divide by 0 essentially because multipliying sth by 0 is not an bijective function, so taking the inverse doesn't make any sense.
On November 20 2013 20:42 Paljas wrote: 1 = -(i²) 1= 1^(1/2) = (-(i²))^(1/2) = (-1)^(1/2)*(i²)^(1/2) =i*i = -1
This is also a great false proof I occasionally show this to my students after they cover imaginary numbers, although we don't really do any proofs with i. [/QUOTE] Nice one.
I found one time a paper proving P=NP , and in the middle of the proof a beautifull bijection between Q and R. I didn't had to look further :D
Heres a peom for the proof of the Halting problem Ive always liked:
No general procedure for bug checks will do. Now, I won’t just assert that, I’ll prove it to you. I will prove that although you might work till you drop, you cannot tell if computation will stop.
For imagine we have a procedure called P that for specified input permits you to see whether specified source code, with all of its faults, defines a routine that eventually halts.
You feed in your program, with suitable data, and P gets to work, and a little while later (in finite compute time) correctly infers whether infinite looping behavior occurs.
If there will be no looping, then P prints out ‘Good.’ That means work on this input will halt, as it should. But if it detects an unstoppable loop, then P reports ‘Bad!’ — which means you’re in the soup.
Well, the truth is that P cannot possibly be, because if you wrote it and gave it to me, I could use it to set up a logical bind that would shatter your reason and scramble your mind.
Here’s the trick that I’ll use — and it’s simple to do. I’ll define a procedure, which I will call Q, that will use P’s predictions of halting success to stir up a terrible logical mess.
For a specified program, say A, one supplies, the first step of this program called Q I devise is to find out from P what’s the right thing to say of the looping behavior of A run on A.
If P’s answer is ‘Bad!’, Q will suddenly stop. But otherwise, Q will go back to the top, and start off again, looping endlessly back, till the universe dies and turns frozen and black.
And this program called Q wouldn’t stay on the shelf; I would ask it to forecast its run on itself. When it reads its own source code, just what will it do? What’s the looping behavior of Q run on Q?
If P warns of infinite loops, Q will quit; yet P is supposed to speak truly of it! And if Q’s going to quit, then P should say ‘Good.’ Which makes Q start to loop! (P denied that it would.)
No matter how P might perform, Q will scoop it: Q uses P’s output to make P look stupid. Whatever P says, it cannot predict Q: P is right when it’s wrong, and is false when it’s true!
I’ve created a paradox, neat as can be — and simply by using your putative P. When you posited P you stepped into a snare; Your assumption has led you right into my lair.
So where can this argument possibly go? I don’t have to tell you; I’m sure you must know. A reductio: There cannot possibly be a procedure that acts like the mythical P.
You can never find general mechanical means for predicting the acts of computing machines; it’s something that cannot be done. So we users must find our own bugs. Our computers are losers!
On November 21 2013 02:30 docvoc wrote: Very cool DPB. I like it haha. I can't verify the proof, but I can marvel at how well it rhymes haha.
Haha thanks I tried writing each step out on the side so you can follow along ^^
Here it is again, in its entirety:
1. a = b.............................................Given 2. a^2 = ab........................................Multiplication 3. a^2 - b^2 = ab - b^2........................Subtraction 4. (a + b) (a – b) = b (a – b)................Factoring 5. a + b = b.......................................Division 6. b + b = b......................................Substitution 7. 2b = 1b........................................Addition 8. 2 = 1............................................Division QED.
No general procedure for bug checks will do. Now, I won’t just assert that, I’ll prove it to you. I will prove that although you might work till you drop, you cannot tell if computation will stop.
For imagine we have a procedure called P that for specified input permits you to see whether specified source code, with all of its faults, defines a routine that eventually halts.
You feed in your program, with suitable data, and P gets to work, and a little while later (in finite compute time) correctly infers whether infinite looping behavior occurs.
If there will be no looping, then P prints out ‘Good.’ That means work on this input will halt, as it should. But if it detects an unstoppable loop, then P reports ‘Bad!’ — which means you’re in the soup.
Well, the truth is that P cannot possibly be, because if you wrote it and gave it to me, I could use it to set up a logical bind that would shatter your reason and scramble your mind.
Here’s the trick that I’ll use — and it’s simple to do. I’ll define a procedure, which I will call Q, that will use P’s predictions of halting success to stir up a terrible logical mess.
For a specified program, say A, one supplies, the first step of this program called Q I devise is to find out from P what’s the right thing to say of the looping behavior of A run on A.
If P’s answer is ‘Bad!’, Q will suddenly stop. But otherwise, Q will go back to the top, and start off again, looping endlessly back, till the universe dies and turns frozen and black.
And this program called Q wouldn’t stay on the shelf; I would ask it to forecast its run on itself. When it reads its own source code, just what will it do? What’s the looping behavior of Q run on Q?
If P warns of infinite loops, Q will quit; yet P is supposed to speak truly of it! And if Q’s going to quit, then P should say ‘Good.’ Which makes Q start to loop! (P denied that it would.)
No matter how P might perform, Q will scoop it: Q uses P’s output to make P look stupid. Whatever P says, it cannot predict Q: P is right when it’s wrong, and is false when it’s true!
I’ve created a paradox, neat as can be — and simply by using your putative P. When you posited P you stepped into a snare; Your assumption has led you right into my lair.
So where can this argument possibly go? I don’t have to tell you; I’m sure you must know. A reductio: There cannot possibly be a procedure that acts like the mythical P.
You can never find general mechanical means for predicting the acts of computing machines; it’s something that cannot be done. So we users must find our own bugs. Our computers are losers!
Haha that's awesome! I don't have experience with computability theory or computer programming, but I read about the Halting problem here: http://en.wikipedia.org/wiki/Halting_problem Nice rhymes!
On November 20 2013 20:00 Recognizable wrote: 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
Haha well done! Now can you (or anyone else) actually explain why you can't divide by zero in this case? It's a classic mantra in mathematics, that *you can't divide by zero* ( ::cough:: until calculus and limits ::cough:: ), but... why not?
Because R is a field, and in a field, and 0 is not in the group of units. \box.
Well you don't divide by 0 essentially because multipliying sth by 0 is not an bijective function, so taking the inverse doesn't make any sense.
Agreed. When I explain this to my high school/ early college students, they don't have any knowledge of fields (or even what it means to be bijective), so I just show them a very simplistic version of the errors that can occur by dividing by zero, starting with 0 = 0. For example:
0 = 0..............Given 4(0) = 3(0).......Factoring 4 = 3..............Division of the common zero factor (error here) QED
Obviously, 4 and 3 can be replaced with any other two numbers you want lol...
Our professor gave us this proof that all numbers in an n-tuple are the same:
The proof is by mathematical induction. For n = 1, it's obvious since there's only one number (a1) so all numbers are the same. Now let's assume that the statement holds for some n natural number, that for every n-tuple (a1,...,an) a1=a2=...=an holds. We choose an arbitrary (n+1)-tuple (a1,...,a(n+1)). Now we notice we can construct two n-tuples (a1,...,an) and (a2,...,a(n+1)) so now by assumption we get a1=a2=..=an and a2=a3=...=a(n+1), thus a1=a2=...=an=a(n+1). PMI QED
On November 21 2013 05:09 CoughingHydra wrote: Our professor gave us this proof that all numbers in an n-tuple are the same:
The proof is by mathematical induction. For n = 1, it's obvious since there's only one number (a1) so all numbers are the same. Now let's assume that the statement holds for some n natural number, that for every n-tuple (a1,...,an) a1=a2=...=an holds. We choose an arbitrary (n+1)-tuple (a1,...,a(n+1)). Now we notice we can construct two n-tuples (a1,...,an) and (a2,...,a(n+1)) so now by assumption we get a1=a2=..=an and a2=a3=...=a(n+1), thus a1=a2=...=an=a(n+1). PMI QED
Base case isn't covered because you need at least n=2 for your induction (for infering a1=a2) It's a good exemple for people that don't master inductions
Works all the time, especially if you draw it on a black board.
Hahahaha. There's a running joke in my math department regarding the mindset of students (especially those who take standardized tests and rely too heavily on triangle diagrams that aren't drawn to scale):
Prove that the triangle is equilateral. Well, it looks equilateral in the diagram, so therefore it is!
Works all the time, especially if you draw it on a black board.
Hahahaha. There's a running joke in my math department regarding the mindset of students (especially those who take standardized tests and rely too heavily on triangle diagrams that aren't drawn to scale):
Prove that the triangle is equilateral. Well, it looks equilateral in the diagram, so therefore it is!
hahaha
on the blog: Very nice proof indeed. I missed it the first i read through it, so i'm a bit embarassed. But i smile everytime i see one of these, thanks for the blog^^
On November 20 2013 18:01 Danglars wrote: 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.
I'm working with a flipped classroom idea for a Precalculus class this semester. It's actually quite a bit of fun. We have some great class activities and lots of time to work one-on-one with students until they understand what's going on. You don't really see that at a big university.
The teaching style is focused on understand concepts yourself and not just memorizing how to do things. Nearly 3 months in and I feel we've been pretty successful. I might show our lecturer / lead teacher the poem here. I bet he would like it.
Works all the time, especially if you draw it on a black board.
Hahahaha. There's a running joke in my math department regarding the mindset of students (especially those who take standardized tests and rely too heavily on triangle diagrams that aren't drawn to scale):
Prove that the triangle is equilateral. Well, it looks equilateral in the diagram, so therefore it is!
hahaha
on the blog: Very nice proof indeed. I missed it the first i read through it, so i'm a bit embarassed. But i smile everytime i see one of these, thanks for the blog^^
Glad you enjoyed it! I'm still in the honeymoon phase of teaching, so I'm constantly motivated to try out new things with the students
On November 21 2013 08:08 LaNague wrote: I like the induction one where you create a false proof that all sheep are black.
oh, coughinghydra was faster ^^, except its funnier with sheeps.
I do like that one too (that all sheep are black... or that all sheep are white, etc.).
Here's the argument, taken from an exam I found online:
Find the mistake in the following argument and explain why it’s a mistake. The following is a proof that there exist no black sheep. First, we will prove that in any group of sheep, every sheep has the same color by doing induction on the number of sheep in the group. It is obvious that in any group of sheep which consists of exactly one sheep, every sheep has the same color. This establishes the base case. The inductive hypothesis is that in every group of n sheep, every sheep has the same color. Now look at a group of n + 1 sheep. Let’s pick one, set it aside, and look at the rest of the animals. They form a group of n sheep, therefore they all have the same color by the inductive hypothesis. Now we will prove that the sheep we set aside has the same color too. Let’s pick another sheep and switch it with the sheep we set aside. We still have a group of n sheep, therefore they all have the same color by the inductive hypothesis. Hence the sheep we first set aside has the same color as all the others. The above argument shows that every sheep on earth has the same color. I suppose that you’ve seen a white sheep before. Now you know that every other sheep must also be white. Hence there exist no black sheep despite any rumor you might have heard to the contrary. No, the mistake is not that the result contradicts reality. The fact that there indeed exist both black and white sheep–see the herds grazing along Highway 111 just north of Calexico–only tells you that the argument must have a mistake in it, but is not itself the mistake. Since it is obviously false that in any set of sheep all the animals are the same color, there must be an error in the inductive argument. It is not that I didn’t prove the inductive hypothesis. As its name suggests, the inductive hypothesis is only meant to be assumed, not proved. You can find the mistake by trying how the induction supposedly goes from the base case to n = 2. Of course, before you go looking for a mistake, make sure you understand why the argument seems to work in going from n sheep to n + 1 sheep.
On November 20 2013 18:01 Danglars wrote: 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.
I'm working with a flipped classroom idea for a Precalculus class this semester. It's actually quite a bit of fun. We have some great class activities and lots of time to work one-on-one with students until they understand what's going on. You don't really see that at a big university.
The teaching style is focused on understand concepts yourself and not just memorizing how to do things. Nearly 3 months in and I feel we've been pretty successful. I might show our lecturer / lead teacher the poem here. I bet he would like it.
That sounds really cool I've found that- although it seems interesting in theory, implementing the actual practice of a flipped classroom can be pretty difficult! What's your opinion of the effectiveness of a flipped classroom set-up? What grades (high school? college?) are you implementing it in, and are they honors/ math-driven students, or not so much?
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
Division by 0 is not allowed. Since you have stated a = b, we can deduce that (a - b) = 0
edit: oh sorry didn't see spoiler.
An astute observation, nonetheless
A few years ago, I gave this proof to one particular geometry class, as a break from their tedious triangle congruence proofs... I knew most of them would struggle to find the zero division error... what I didn't expect is for this proof to actually convince some students that 1 really does equal 2! A few of them were like "Oh, well I guess they really are the same number then", and I had to re-explain to them that the proof was false.
It takes a cool teacher to do something like that to teach his students. I know enough Geometry to get the gist of what you said although 5. confuses me. Division? But you said substitution... uhh.
On November 22 2013 01:18 3FFA wrote: It takes a cool teacher to do something like that to teach his students. I know enough Geometry to get the gist of what you said although 5. confuses me. Division? But you said substitution... uhh.
edit: nvm. Although I do know that 1 != 2.
Yeah, I didn't want to post two steps on the same line in the blog, for the sake of organization and clarity
On November 20 2013 18:01 Danglars wrote: 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.
I'm working with a flipped classroom idea for a Precalculus class this semester. It's actually quite a bit of fun. We have some great class activities and lots of time to work one-on-one with students until they understand what's going on. You don't really see that at a big university.
The teaching style is focused on understand concepts yourself and not just memorizing how to do things. Nearly 3 months in and I feel we've been pretty successful. I might show our lecturer / lead teacher the poem here. I bet he would like it.
That sounds really cool I've found that- although it seems interesting in theory, implementing the actual practice of a flipped classroom can be pretty difficult! What's your opinion of the effectiveness of a flipped classroom set-up? What grades (high school? college?) are you implementing it in, and are they honors/ math-driven students, or not so much?
If it wasn't clear, I mean flipped classroom in the sense that the students listen to lectures at home and work on learning activities / homework in class (not the other meaning where you have students teaching students).
There was a lot of work that went into it at the beginning, and the videos aren't nearly where we want them at yet. But for the most part the class is going really well.
The class is a Precalculus class at the college level meant for people who will eventually be taking Calculus. Hence, we've boiled most of the material down to what we really think they need to know to succeed in Calculus.
I think one of the most important aspects of how we set up the class is the grading system, which works really well. We are using online software (which we are not a fan of, but that's another story) for students to work homework problems and take "quizzes" on the problems.
The students must score 100% on each and every one of these assignments in order to pass the class! The quizzes are set up so that if they get 1 question wrong they need to retake the entire quiz again. The students don't like it, but unbeknownst to them they are getting a lot of good repetition by doing it this way. Each week there is also a short in-class quiz that is exactly like their online quiz problems. This is to ensure they can do the important stuff without notes and what-not. Again, they must "Pass" all of these quizzes in order to pass the class. "Passing" consists of getting very nearly 100%, though we allow for some minor errors. The cool thing is, the quizzes are so short that I grade them on the spot, and if the student did something wrong I have them walk me through how they did the problem and explain to me (with some prodding on my part) what went wrong.
A few of the classes are also set aside for just working on problems and getting help, and a few more we have group work activities that get them to be able to generalize (write down an algorithm for completing the square, then do this algorithm with letters instead of numbers), something we are finding Calculus students struggle with in our department.
The big kicker is that if the students do all of this, they pass the class! We do still have exams, and the students have to not completely bomb the exam. But by the strict requirements elsewhere we are saying the students have demonstrated a satisfactory understanding and will pass the class. Their exact grade between A and C- is determined based on all their exam scores.
What we are finding so far is great news! We are getting the same number of A's and B's, but we are getting a lot more students to pass the course. Typically this class has about 40% of the students failing or dropping out, but we are far lower than that so far. The reason is that the students who are already getting A's and B's are getting them on their own and not needing to spend very much time in class, but the students who are struggling have lots of opportunity to get help and 1 on 1 teaching.
That's pretty cool, Sherlock I'm glad to hear you guys are getting a good passing percentage, despite having very high standards. Our precalculus courses also have super-high standards, but we don't flip the classroom.
We've also integrated online software (our program is called ALEKS) which constantly assesses students' knowledge on essentially everything, as well as creates assessments too. Mixed results there though lol...
EPT
Cheers, everyone! 
. I could start on the programming, computer science, and computer graphics ones next though.
