'Carnival' of MathNotations 5-25-07

New feature...

I apologize for stealing the 'Carnival' term - I couldn't resist it. If the Carnival people want equal time, here's my plug for one of the best: Visit the home page of the Carnival of Mathematics - we are all indebted to Alon Levy (I hope I got that right!) for developing and maintaining this wonderful venue for the math-minded out there who may not have time to discover or visit all of the outstanding math blogs on the web. Considering that I'm hosting the Carnival the week of June 15th, I thought this attempt could also be a dry run...

It's easy to miss some interesting dialogue when we jump from site to site on the superhighway. I watch how my students and my children surf the net- if they don't see what they want on a page within 5-10 seconds, they're gone!

Since some of the most interesting 'stuff' often takes place off-post in the comments section, I thought it would make sense to summarize some of this at the end of the week. I'll probably tire of this after a couple of weeks, or choose to do this every other week like the real Carnival of Mathematics, but I'd like to try it anyway.

I apologize in advance to my other faithful readers and expert commentators for any errors of omission. Their contributions enrich this blog and their continual support inspire me to maintain these efforts. I will add more as I sift through my posts from the past two weeks. For now here are some highlights from the past 8 days.

• A series of posts (Part I and Part II) this past week have focused on developing meaning and application for exponential functions (and their relation to geometric sequences and series) designed for Advanced Algebra classes. These extensive classroom investigations use the concepts of mortgages, stressing the ideas behind those esoteric formulas. There are full-blown activities with open-ended questions and data analysis using the graphing calculator. Screen shots of data tables from the TI-84 enhance the activity. Further related activities to follow...
  
• A set of Thinking Outside The Box problems came up earlier that stimulated some great comments and ideas from Denise (author of the excellent site letsplaymath) and others. This has led to an ongoing discussion of the sticky problem regarding the smallest positive integer having a given number, N, of factors. The problem has been solved if N is prime, or if N is a semiprime (N = pq, where p,q are distinct primes), but it's not clear that we have a general solution. Someone out there must know the algorithm for finding this number - pls share it! I haven't yet posted answers or detailed solutions to some of the questions, so stay tuned.
  
• TC, another of my dedicated readers, posed the pi = 2 'proof' introduced first in the comments here and continued in the comments section of Part I of the mortgage problems. I challenged my Calculus class to explain the fallacy in this limit problem and one student, Matthew, came up with a pretty nice solution. This problem could easily be a post in and of itself with many possible extensions. TC presented an excellent 'curvature' argument and I added some thoughts of my own.
• Eric Jablow, our math/computer science expert, suggested 3 excellent algorithms (bisection method, method of false position and Newton's Method) for approximating roots of an equation. These are worth sharing with our algebra and precalculus students as well as those in Calculus and Computer Science. These comments are found under the post entitled Catching Up and Preview of Coming Attractions.
  

Pls give me your feedback. If you think the title is too presumptuous, say so! If you think it's a waste of time to summarize only a week or so of posts and comments, let me know that too. If you have something nice to say, I'll accept that as well! Signing off for now...