tag:blogger.com,1999:blog-2433841880619171855.post5132516170447536114..comments2024-03-27T21:09:44.320+00:00Comments on Pat'sBlog: More on Vectors in the HS CurriculumUnknownnoreply@blogger.comBlogger3125tag:blogger.com,1999:blog-2433841880619171855.post-88299285209834405452010-06-09T21:20:07.837+01:002010-06-09T21:20:07.837+01:00I didn't spend much time in working with the c...I didn't spend much time in working with the concept (dividing a line segment in a given ratio), but rather I wanted to show how the proof his text gave for the midpoint formula easily leads to the more general situation (the congruent triangle set-up relaxes to a similar triangle set-up). I also showed the power of geometry (synthetic geometry, that is) by setting up the xy-coordinate equations for finding the midpoint, which is<br /><br />dist(P,Q) = dist(Q,R) = (1/2)*dist(P,R)<br /><br />for finding the midpoint Q of two given points P and R. [If you omit the right hand equals, you'll have one equation in two variables (the 2 coordinates of Q), which leads to finding the perpendicular bisector of segment PR.]<br /><br />Now I'm not all that keen on geometric methods (mainly because I'm awful at geometry), but the contrast in difficulty is enough to make even a geometry-phobe like myself take notice!Dave L. Renfrohttps://www.blogger.com/profile/00863074796446784081noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-63903683861477731982010-06-09T20:45:36.306+01:002010-06-09T20:45:36.306+01:00Thanks Dave, Unfortunatly my experience is that ma...Thanks Dave, Unfortunatly my experience is that many teachers never think to generalize the question (and some don't know how to do what you showed this student). I find that when I show students they find notation like 3/4 a + 1/4 b to be confusing (and often reverse the proportionals). I really prefer a+t(b-a), and it allows you to find not just a point between Pts A and B, but also points outside the segment by use of values of t>1. And they seem to find it pretty natural that to find points the other way, they go b+t(a-b).Pat's Bloghttps://www.blogger.com/profile/15234744401613958081noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-46241915741580337292010-06-09T20:25:28.599+01:002010-06-09T20:25:28.599+01:00"... the frequency with which Alg I and II st..."... the frequency with which Alg I and II students are asked to find the midpoint of a segment (there is probably even a “midpoint formula” in the book) but are almost never asked to find the point 2/3 of the way from A to B."<br /><br />It's interesting that you bring this up now! The last session I had with a certain algebra II student I'm tutoring involved working with the midpoint and distance formulas. He's a reasonably good 10th grade student and he's taking a pretty stiff honors algebra II course, but his parents think some outside work with a tutor will be worth their money (and I'm pretty cheap, at $20 an hour). In particular, they want me to supplement, extend, enrich, etc. what he covers in class. So naturally, when we met and he told me that the midpoint formula was one of the things covered in his class last week, I brought up a topic you can find in most any old analytic geometry text (see [1], a google-books search for "ratio" AND "point dividing a line segment"), namely that of a point dividing a line segment in a given ratio.<br /><br />[1] http://tinyurl.com/22qowkq<br /><br />Here's a higher-dimensional result you and your blog's readers may find interesting. For each positive epsilon > 0 and for each positive integer N, there exists a positive integer n such that at least N many n-dimensional balls of diameter (1 - epsilon) can be placed without overlap in an n-dimensional cube with edge length 1. See the solution to P.167 on pp. 605-606 of Canadian Mathematical Bulletin Vol. 14, 1971, which is freely available on the internet (at least in the U.S.) at [2]. The solution is actually not all that difficult.<br /><br />[2] http://tinyurl.com/2664k6oDave L. Renfrohttps://www.blogger.com/profile/00863074796446784081noreply@blogger.com