tag:blogger.com,1999:blog-5964889903484807623.post227044816011888706..comments2021-10-13T13:00:30.262-07:00Comments on Questions?: VolumeDavid Coxhttp://www.blogger.com/profile/06277427735527075341noreply@blogger.comBlogger14125tag:blogger.com,1999:blog-5964889903484807623.post-67853205554905279672012-05-31T20:54:50.715-07:002012-05-31T20:54:50.715-07:00Hey I saw http://www.youtube.com/watch?v=whYqhpc6S...Hey I saw http://www.youtube.com/watch?v=whYqhpc6S6g from one of my students and thought of your post and our suggestion of using ropes.<br /><br />Their proof is simpler.untilnextstophttps://www.blogger.com/profile/15285583728476473117noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-88350438557879566682010-12-21T07:29:43.078-08:002010-12-21T07:29:43.078-08:00Awesome! Check out http://www.youtube.com/watch?v...Awesome! Check out http://www.youtube.com/watch?v=aLyQddyY8ikKrishttps://www.blogger.com/profile/18112238121767903592noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-21713945194337383972010-12-20T10:17:36.772-08:002010-12-20T10:17:36.772-08:00There's the other way of getting the area of a...There's the other way of getting the area of a circle from the circumference--take circles and cut them into smaller and smaller wedges (fourths, eighths and sixteenths work well). Rearrange the wedges top to tail to get shapes that are closer and closer to parallelograms. Then your calculus is just imagining what happens if you could cut into infinitely small wedges to do this with, and you conclude that your rearranged shape (in the limit) is a parallelogram (or a rectangle) with base = 1/2 circumference (because the circumference is distributed half on top and half on bottom) and height = radius.<br /><br />It does sound like it's going to be a great semester!LSquared32https://www.blogger.com/profile/00858524638866166691noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-24242017777627660062010-12-18T22:34:01.831-08:002010-12-18T22:34:01.831-08:00The two year loop is one of the things that really...The two year loop is one of the things that really drew me to take this position. <br /><br />I really like your idea for the area of a circle. Looks like a trip to the hardware store is in order. Gonna need some rope.David Coxhttps://www.blogger.com/profile/06277427735527075341noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-59675110444003877702010-12-18T05:13:51.675-08:002010-12-18T05:13:51.675-08:00This comment has been removed by the author.untilnextstophttps://www.blogger.com/profile/15285583728476473117noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-41841164811832097242010-12-18T05:09:58.959-08:002010-12-18T05:09:58.959-08:00Cool, David. How inspiring. :) It's really int...Cool, David. How inspiring. :) It's really interesting that you get to keep them for two years. I've taught some kids back-to-back before, but usually it's a fluke (ie. I happen to be shuffling my schedule after the year is done). It'd be awesome to be allowed to keep the same kids for two years.<br /><br />By the way, I was thinking about the inquiry of 2*pi*r --> pi*r^2. You might be able to show it using Calculus concepts but presented in a fairly intuitive way. The inquiry part can be for kids to take a bunch of ropes (with width ~ 1cm) to form concentric rings in a circle, and then lay the ropes out side by side. What they should notice is that: <br /><br />1. All the concentric rings have lengths (circumferences) that are increasing linearly with their distance away from the center of the circle.<br /><br />2. All of the rings/ropes together covered the entire inside of the circle, so you must be able to figure out the circular area using those ropes (sum of length of each rope X width).<br /><br />3. From there, you can somehow guide them to figuring out that to add the length of all the strings is the same as adding the shortest and the longest, times n/2, where n = number of concentric circle ropes.<br /><br />(ie. the arithmetic series formula. For example, adding 1 + 2 + 3 + ... 100 is the same as adding 1 + 100, times 50 pairs. Since 1 + 100 = 2 + 99 = 3 + 98 = ... = 49 + 50 = 101, and there are 50 "pairs" of those numbers each with a pair sum = 101. This series formula is very intuitive, even though it's part of Precalc. My dad taught me this algorithm in like the 3rd or 4th grade, so I'm pretty sure your kids could understand it as middle-schoolers.)<br /><br />4. This simplifies to (0 + 2*pi*r)(1)(r/2) = pi*r^2. Here, 0 is the theoretical length of the innermost (shortest) ring. 2*pi*r is the length of the outermost (longest) ring. 1 is the thickness of each rope. r/2 is how many ropes there are, divided by 2to get how many "pairs" you have. Tada!untilnextstophttps://www.blogger.com/profile/15285583728476473117noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-82121257728234715482010-12-17T13:10:54.626-08:002010-12-17T13:10:54.626-08:00Mimi
I get them for two years. As 7th graders, mo...<b>Mimi</b><br />I get them for two years. As 7th graders, most are good at school and a few are really inquisitive. We just keep on encouraging inquiry and by the time they get to 8th grade, the majority of the class has a solid respect for learning and one another. So to answer your question:<br /><br />I think they come to me inquisitive, but some just don't remember how to do it because it's been trained out of them.David Coxhttps://www.blogger.com/profile/06277427735527075341noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-24723859882002832962010-12-17T12:40:44.432-08:002010-12-17T12:40:44.432-08:00Wait! I remembered this:
http://www.maa.org/pubs...Wait! I remembered this:<br /><br />http://www.maa.org/pubs/Calc_articles/ma018.pdf<br /><br />It uses area of a triangle. So, we'd have to know where that comes from, too (maybe you've already done that one, though).CalcDavehttps://www.blogger.com/profile/14039458440867020542noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-71944266098627342262010-12-17T12:38:03.130-08:002010-12-17T12:38:03.130-08:00I am curious: Do your kids arrive in your class wi...I am curious: Do your kids arrive in your class with this level of inquisitiveness, or do you single-handedly breed it?untilnextstophttps://www.blogger.com/profile/15285583728476473117noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-38631055328721449862010-12-17T12:21:09.823-08:002010-12-17T12:21:09.823-08:00Oh. Yeah, that's a bit less intuitive, I gues...Oh. Yeah, that's a bit less intuitive, I guess. I'm not coming up with a good way to do it without calculus. =(CalcDavehttps://www.blogger.com/profile/14039458440867020542noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-66126371307857108722010-12-17T11:46:58.615-08:002010-12-17T11:46:58.615-08:00Right, but how do we take 2*pi*r and end up with p...Right, but how do we take 2*pi*r and end up with pi*r^2? It seems like the formula should be 2*pi*r^2. It doesn't seem as intuitive as multiplying the area of the circle by h for volume.David Coxhttps://www.blogger.com/profile/06277427735527075341noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-59829368408877771902010-12-17T11:38:33.435-08:002010-12-17T11:38:33.435-08:00Well, it's the same concept, just taking it do...Well, it's the same concept, just taking it down a dimension. A bunch of concentric rings/strings to make the circle. And once you've gone that way, though, you have to be prepared for going the other...hyperspheres in 4D! <br /><br />Maybe introduce with a picture of a tree stump?CalcDavehttps://www.blogger.com/profile/14039458440867020542noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-10664759621961520702010-12-17T11:32:54.494-08:002010-12-17T11:32:54.494-08:00These are 8th grade algebra students and I never t...These are 8th grade algebra students and I never thought of the area of a circle as being the sum of a bunch of circumferences. Any suggestions for putting them in a position to try to grind that out?David Coxhttps://www.blogger.com/profile/06277427735527075341noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-67259753392898600032010-12-17T08:34:49.687-08:002010-12-17T08:34:49.687-08:00What grade do you teach? And this is calculus!
...What grade do you teach? And this is calculus! <br /><br />Now we need to know why area of a circle is pi r^2. And once we get that down to adding a bunch of circumferences, then why is C = pi * d? Do we even remember where pi comes from? Who would make up a number like that unless it came from a weird definition.CalcDavehttps://www.blogger.com/profile/14039458440867020542noreply@blogger.com