EPToh, coughinghydra was faster ^^, except its funnier with sheeps.
Math Poetry - Page 3
| Blogs > DarkPlasmaBall |
|
LaNague
Germany9118 Posts
oh, coughinghydra was faster ^^, except its funnier with sheeps. | ||
|
Sherlock117
United States40 Posts
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. | ||
|
DarkPlasmaBall
United States45914 Posts
On November 21 2013 07:42 Yorbon wrote: 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 | ||
|
DarkPlasmaBall
United States45914 Posts
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. ~ http://www-rohan.sdsu.edu/~ituba/math521af07/math521aexam2sol.pdf | ||
|
DarkPlasmaBall
United States45914 Posts
On November 21 2013 09:42 Sherlock117 wrote: 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? | ||
|
mishimaBeef
Canada2259 Posts
| ||
|
mishimaBeef
Canada2259 Posts
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. | ||
|
DarkPlasmaBall
United States45914 Posts
On November 21 2013 21:46 mishimaBeef wrote: Dunno if this was said already but: 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. | ||
|
Big J
Austria16289 Posts
Not sure if it is your sense of humor (and I have made some mistakes; Analysis is Calculus in English). | ||
|
DarkPlasmaBall
United States45914 Posts
On November 21 2013 23:04 Big J wrote: A little unrelated, but maybe you like it... https://www.youtube.com/watch?v=JqB6cBzUHuQ Not sure if it is your sense of humor (and I have made some mistakes; Analysis is Calculus in English). Hahahaha I can appreciate the calculus humor here | ||
|
3FFA
United States3931 Posts
edit: nvm. Although I do know that 1 != 2.  | ||
|
DarkPlasmaBall
United States45914 Posts
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 | ||
|
DefMatrixUltra
Canada1992 Posts
On November 21 2013 21:46 mishimaBeef wrote: Dunno if this was said already but: 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. Double posted and didn't read the thread at all. Model poster. | ||
|
Sherlock117
United States40 Posts
On November 21 2013 21:27 DarkPlasmaBall wrote: 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. | ||
|
DarkPlasmaBall
United States45914 Posts
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... | ||
|
mishimaBeef
Canada2259 Posts
On November 22 2013 03:38 DefMatrixUltra wrote: Double posted and didn't read the thread at all. Model poster. it is expected to read entire threads now? | ||
|
DarkPlasmaBall
United States45914 Posts
On November 23 2013 03:57 mishimaBeef wrote: it is expected to read entire threads now? My blogs, yes But no worries | ||
|
Recognizable
Netherlands1552 Posts
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. Your reasoning about multiplying something by 0 is not bijective I understand. However I don't understand the bolded part. Could you eloborate? | ||
|
Sherlock117
United States40 Posts
On November 27 2013 20:04 Recognizable wrote: Your reasoning about multiplying something by 0 is not bijective I understand. However I don't understand the bolded part. Could you eloborate? Basically the same thing. In a field, the units are those elements which are invertible, or put another way multiplication by a unit is bijective. | ||
| ||
