tag:blogger.com,1999:blog-8231784566931768362.post3452878837758014553..comments2021-06-16T05:56:38.112-04:00Comments on MathNotations: "On The Road Again" With 'TC' -- A Real World Application of GeometryDave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-8231784566931768362.post-43328409839884106112009-06-19T08:19:23.573-04:002009-06-19T08:19:23.573-04:00Eric,
Fascinating article -- the example provided ...Eric,<br />Fascinating article -- the example provided was very helpful. I would have probably taken that "short-cut" too! Besides, it was worth reading just to see the Nash Equilibrium concept mentioned.Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-85172070386580033022009-06-19T07:47:49.967-04:002009-06-19T07:47:49.967-04:00For more fun with traffic problems, algebraic this...For more fun with traffic problems, algebraic this time, you can demonstrate Braess' Paradox to students. The paradox demonstrates that <i>adding</i> a road to a travel network can <i>increase</i> the average travel time for the system's drivers. The Wikipedia article is a good guide.Eric Jablowhttps://www.blogger.com/profile/16327238795785012303noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-5795310590797248102009-06-17T15:12:24.864-04:002009-06-17T15:12:24.864-04:00Nice! I have seen that problem and it's defini...Nice! I have seen that problem and it's definitely worth sharing it with our students to demonstrate not only a counterintuitive idea but also the power of mathematics! Thank you for sharing that and the link...<br /><br />Both problems depend on the difference of the radii. I have also seen a few SAT problems which test this same idea. I seem to recall publishing a post on something like this before but I'll have to search to find it!Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-85581306150244591512009-06-17T14:56:00.196-04:002009-06-17T14:56:00.196-04:00Yes Dave, I was surprised with this counter intuit...Yes Dave, I was surprised with this counter intuitive result.<br /><br />This remind me of another counter intuitive result which is also involving two circles. It was about tying the earth with rope tightly and then extend this rope by 1 meter and then the gap produced is enough to make a cat pass under the rope.<br />Have you seen this problem? Ok I found one blog that discuss this. Here is the link: http://www.blogcatalog.com/discuss/entry/rope-around-the-worldwatchmathhttp://www.watchmath.comnoreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-9856699811512266772009-06-17T12:44:17.069-04:002009-06-17T12:44:17.069-04:00Excellent points, watchmath.
Giving students, pa...Excellent points, watchmath. <br /><br />Giving students, particularly at the middle school level, the time to experiment with a few pairs of radii would help them to make a conjecture before turning to an algebraic solution. My guess is that many students would come to a conclusion after just a couple of trials. <br /><br />I definitely could have added additional parts to this investigation, moving in smaller increments, but that's an individual teacher decision (based on the needs of the class). <br /><br />Certainly, it makes sense in Part III to have them try angles like 45, 90 and 120 degrees to form the basis for a conjecture before attempting a general proof.<br /><br />TC's problem is a jumping off point for us. The rest is up to the creativity of the teacher and the characteristics of the class.<br /><br />Watchmath, were you at all surprised by the result? What I would want my students to come to recognize is that the conclusions are based on the <i>difference</i> of the radii, not the radii themselves! This is because we were not really interested in the individual distances but, rather, how the distances compared! In other words, we are really looking at the difference between the two paths.Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-88472825383844854982009-06-17T10:30:47.990-04:002009-06-17T10:30:47.990-04:00In order the student to find themselves that this ...In order the student to find themselves that this problem is independent of the radii maybe after problem 1 ask the student to play around with various radii and tell them if they can find two radii that makes option 1 is shorter than option 2 and let them guess the general result.watchmathhttp://www.watchmath.comnoreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-45564009910285979882009-06-17T08:58:12.553-04:002009-06-17T08:58:12.553-04:00Hi Dave,
Thanks for your comments, and for the ex...Hi Dave,<br /><br />Thanks for your comments, and for the extensions to the problem. I think I agree that the progression with definite values for the parameters, moving on to a more abstract case would cater well to students of different levels. <br /><br />And, of course, one possible answer to all the variants is, to paraphrase Calvin (of Hobbes fame) is 'Given the traffic here, who knows which option is shorter!'<br /><br />TCUnknownhttps://www.blogger.com/profile/06449079338919787252noreply@blogger.com