tag:blogger.com,1999:blog-8231784566931768362.post3496834989407834293..comments2023-09-09T08:21:55.454-04:00Comments on MathNotations: Take any number, Add Three, Divide the Result by -1. Now Repeat this! Recursive Sequences and Functions Part I: Grades 7-12Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-8231784566931768362.post-64712910648118310162007-06-29T21:40:00.000-04:002007-06-29T21:40:00.000-04:00I like to use Excel when teaching about recurrence...I like to use Excel when teaching about recurrences. Generating recursive sequences in Excel is quick and easy, and I think it helps to make the concept more intuitive and visual. Also, students realize that being able to use Excel is a useful job skill, and this helps to hold their interest.<BR/><BR/>While preparing a lesson on the logistic map, I found that I could generate a reasonable picture of the bifurcation diagram in Excel, but I left this out because I was short on time.Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-54310105147408946952007-06-24T08:52:00.000-04:002007-06-24T08:52:00.000-04:00Dave,can you create (and update) an index? Then yo...Dave,<BR/><BR/>can you create (and update) an index? Then you would just need a link to that index (really, a post of links to worksheets). <BR/><BR/>This might be something like the contract tabs I have at the top of my page.<BR/><BR/>Your other ideas might work as well.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-70421018033231430672007-06-23T08:21:00.000-04:002007-06-23T08:21:00.000-04:00jonathan and mathmom--thanks for the generous comm...jonathan and mathmom--<BR/>thanks for the generous comments... Let me know if students find this problem reasonable next term.<BR/>Of course by then this problem will be hidden in the archives and most will not even see it. How do I keep some of these posts active long after other posts have moved them off the screen? I have a few ideas (like setting up links on the sidebar to some of my favorites) or creating a post consisting of links, but I would like other ideas. Blogger may not be the right medium for this or I am simply clueless, which is far more likely!<BR/>Take a look at the Geometry Challenge I just posted. Either it's a good problem or I messed up and it's really simple! I'm thinking of sending it to the next Carnival along with this post. I'd love your opinions!Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-21476010773036487062007-06-22T17:14:00.000-04:002007-06-22T17:14:00.000-04:00Thanks, Dave, I've bookmarked this to use with my ...Thanks, Dave, I've bookmarked this to use with my middle school group next year.mathmomhttps://www.blogger.com/profile/05869925405540832241noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-37424989485879071972007-06-21T08:29:00.000-04:002007-06-21T08:29:00.000-04:00I had a very smart, very lazy student in combinato...I had a very smart, very lazy student in combinatorics this past fall. You've seen my "Ghost the Bunny" problem?? The bunny hops up one or two steps at a time, how many ways can he reach the top of a flight of 12 steps. <BR/><BR/>The lazy student stared and stared and stared at the problem (while his groupmates scribbled vigorously). And after, I don't know, 15? minutes, called me over to offer the sum of some combinations (IOW, he found the closed form for n=12 without noticing the recursiveness)<BR/><BR/>Strange kid. Very smart.<BR/><BR/>I should add that, at that level, the problem is far less enaging, as the kids already have ALL the tools to analyze and find the closed form. There is no "stretch," just application - all of your worksheets are designed to provide stretch.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-68450696985098442072007-06-21T05:56:00.000-04:002007-06-21T05:56:00.000-04:00Eric--Yes, Towers of Hanoi is a wonderful problem ...Eric--<BR/>Yes, Towers of Hanoi is a wonderful problem to get the students hooked on a recursive procedure. BTWm will the Clay Institute award $1 million if a student solves the Collatz Problem!<BR/>Actually giving some students an 'unsolved' problem would really pique their interest. I know I want to try it. The fact that such an innocent recursive definition as this one can lead to uncertainty is fascinating. That's why I like this topic.<BR/>The repetition in the problem in my post is easy to grasp. If we change the relationship to <BR/>a_[n+1] = C(a_n) + D, we obtain a more general linear recurrence relation. If C is not equal to -1, the sequence is more complicated of course. Part II of this investigation is leading in this direction and will develop a general solution (closed) to this inhomogeneous case. It will take me some time to develop a logical sequence of questions that students can manage. I'm not sure some readers appreciate how labor intensive it is to write these activities. It's considerably harder than writing an exposition of the method.<BR/>Anyway, what I'm actually leading up to is the closed solution of the most famous recursively defined sequence: 1,1,2,3,5,8,...<BR/>Getting students to this will be a challenge for me. The easy way of course is to give them the formula and have them check it but...<BR/><BR/>I feel like very few will appreciate this series of activities, particularly in the hot summer months. Do you think it will be worth the effort?Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-13032916331506362022007-06-20T21:56:00.000-04:002007-06-20T21:56:00.000-04:00Here's some other activities for your students (or...Here's some other activities for your students (or your successor's students).<BR/><BR/>1) Prove that the n-disk Tower of Hanoi problem can be solved in 2^{n} − 1 steps.<BR/><BR/>You can start by showing that (call the number of steps required to solve the n-disk problem h_n),<BR/><BR/>h_{n+1} = h_n + 1 + h_n.<BR/><BR/>2) Explain the Collatz problem, which is unsolved:<BR/><BR/>Start with any positive integer n, and define the sequence {a_k} as:<BR/><BR/>a_0 = n<BR/><BR/>a_{k+1} = a_k / 2 if a_k is even,<BR/><BR/>a_{k+1} = 3a_k + 1 if a_k is odd.<BR/><BR/>Does this sequence always end up with the '1-cycle', 1,4,2,1,…? Is there a starting point n such that the sequence forms another cycle, or is there a starting point n such that the sequence never repeats but remains bounded from below.<BR/><BR/>We believe that every n eventually leads to the 1,4,2,1 cycle. No one has proven it yet. Every n < 2.88×10^{18} leads to the 1-cycle. But it can take a long time for any given number. Wikipedia notes that 27 leads to 1 after 111 steps.Anonymousnoreply@blogger.com