tag:blogger.com,1999:blog-1222092101762956112.post3941877976790288479..comments2024-03-12T23:03:35.556-07:00Comments on Infinigons, etc.: Putting myself in my kids' shoesAllison Johnsonhttp://www.blogger.com/profile/01731073560005744198noreply@blogger.comBlogger30125tag:blogger.com,1999:blog-1222092101762956112.post-16404996658749257272022-04-07T02:36:05.250-07:002022-04-07T02:36:05.250-07:00Succeed! It could be one of the most useful blogs ...Succeed! It could be one of the most useful blogs we have ever come across on the subject. Excellent info! I’m also an expert in this topic so I can understand your effort very well. Thanks for the huge help. <a href="https://shasaralala.blogspot.com/2022/02/laus.html" rel="nofollow">안전놀이터</a><br /><br /><br />안전놀이터noreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-84711948606013299422021-10-22T05:59:37.180-07:002021-10-22T05:59:37.180-07:00Excellent site you’ve got here.. It’s hard to find...Excellent site you’ve got here.. 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Anonymoushttps://www.blogger.com/profile/08114677772547494751noreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-26928123961367377002013-08-09T13:00:58.217-07:002013-08-09T13:00:58.217-07:00Allison, I'm making the answer key for the puz...Allison, I'm making the answer key for the puzzles in the book (<i>Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers</i>), so I *finally* did this problem, just now. I loved it. I did the portion above the diagonal, so the integration would be easier. My only mistake was in simplifying the square root messes. Wolfram wouldn't do that for me. (Very interesting to bump up against its limitations.)<br /><br />I'll send you my write-up.<br /><br />Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-26806024720651158762011-11-09T12:06:37.487-08:002011-11-09T12:06:37.487-08:00Sweet. Ms. Cookie - my solution was almost identic...Sweet. Ms. Cookie - my solution was almost identical to yours, except my calculus skills are a bit rusty so I only integrated y=-x^2/4 + 1 from 0 to -2+2sqrt(2), and then subtracted that integral by the triangular area A=0.5(-2+2sqrt(2))^2 which does not lie inside the 1/8 slice. Then I divided by the area of the 1/8 slice to get the same probability. <br /><br />I like this problem.untilnextstophttps://www.blogger.com/profile/15285583728476473117noreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-36198356753930030592011-09-13T06:23:51.065-07:002011-09-13T06:23:51.065-07:00If you figure out how to teach this skill of riski...If you figure out how to teach this skill of risking failure and trying really hard, please come teach me? I'm working on writing my first paper and I'm still not very good at it. I occasionally try to teach it to my students, and it almost always goes poorly!<br />I'm glad to see that you seem to be happy and doing well!<br />Annaannahttps://www.blogger.com/profile/09079758718513998328noreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-67942776060826455712011-07-12T00:24:47.662-07:002011-07-12T00:24:47.662-07:00Damn it, I got "one third of (13 - 8*sqrt(2))...Damn it, I got "one third of (13 - 8*sqrt(2))". Must have done something wrong in my three pages somewhere. Had fun, though, and will try to correct it later.<br /><br />My dartboard went from (0,0) to (2,2) so that the center was at (1,1) and I used an upright parabola, which I think is easier to integrate. Mind you, I did have to deal with a *lot* of surds, so maybe it's not easier.<br /><br />Hopefully I can extract some parts of the problem for the delectation of students. Thanks for the stimulus!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-2760656832125199932011-07-09T13:17:29.258-07:002011-07-09T13:17:29.258-07:00This is the (very positive) direction in which we ...This is the (very positive) direction in which we are taking Math instruction over the next several years. By giving students challenging tasks and allowing them to struggle, collaborate, and revise ideas based on feedback, we are creating mathematical *thinkers* that are far more adaptable than "procedure followers." The Common Core standards will require educators at all levels to embrace this methodology, and the Shell Center is putting together a set of tasks like these that are aligned to the Common Core standards.<br /><br />I know it's the right way to go because, when I attended a PD on this idea 2 weeks ago, they gave us sample math tasks that *I* enjoyed solving (like this one!). I got the same answer as you and used basically the same methodology except that I used the "upper" triangle so I didn't need to split the integral.<br /><br />- Julienjaimelesmathshttps://www.blogger.com/profile/07766887057966758554noreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-46724353979578436232011-07-01T02:14:09.539-07:002011-07-01T02:14:09.539-07:00Here is a the full process I went through to solve...Here is a the full process I went through to solve this problem:<br /><br />Step 1: See a reference to the problem on Dan Meyer's blog and thought it sounded interested. Find this page<br /><br />Step 2: Grumble to myself about pet peeve with probability problems using dart boards since most people actually aim for the middle, so have strong aversion to using a dark board as a model for randomness. Consider adding line "I am so bad at darts that I have an equal chance of hitting any spot on the target" to turn problem into joke instead.<br /><br />Step 3: Spend 1 minute running through and throwing out trivial solutions.<br /><br />Step 4: Realize that it is some kind of curve, and that problem can be simplified by looking at smaller triangles (gotta love symmetry).<br /><br />Step 5: Consider a generalized algebraic solution - then realize the problem has a high chance of containing messy powers and roots. Lose motivation, go with simple numerics. Setting square area to 1 means just finding area under curves with no need for division at end.<br /><br />Step 6: Go out on my inlines and forget about problem. Nothing on paper at this point.<br /><br />Step 7: Unable to forget about problem while skating. Run through obvious curve types while sakting: parabolic, hyperbolic, trig, etc. Almost run into a biker while distracted. Realize I need to just sit down and solve the simple equation.<br /><br />Step 8: Get home and draw a square on an XY axis around the origin, about 1 minute of simple algebra give the equation of the parabola.<br /><br />Step 9: Smack myself on the head and say "D'oh" for not remembering that this is actually one of the definitions of a parabola - all the points equidistant from a point and a line.<br /><br />Step 10: Spend a few minutes drawing a nice picture in GeoGebra mainly because GeoGebra is really cool and really fun to use.<br /><br />Step 11: Pick on of the many options for integrating to find the area under the parabolas.<br /><br />Step 12: Roll my eyes at ending up with messy powers and roots. Do them anyway.<br /><br />Step 13: Type my integral into Wolfram Alpha to check the answer - but mostly as an excuse to use that site since it is even cooler than GeoGebra.<br /><br />Step 14: Check my answer against Allison's and feel satisfied that it is the same.<br /><br />Step 15: Want to continue with this since it was fun. Think about Mr. H's extension to N-sided polygons - see the basic idea but realize the calculations are likely to be a bit messy. Attention span wavers.<br /><br />Step 16: Lament privately that I can't give this to my students since I am doubtful there is a way to do this without integals (I would be curious if someone found one) and I don't currently teach that level of math.<br /><br />Step 17: Realize that I could give the first step of the problem to advanced students - have them just find the equation of the parabolas. Possibly have the really good ones describe the area and put some sort of clever upper and lower bound on the area.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-76806345042830688492011-06-25T11:49:17.188-07:002011-06-25T11:49:17.188-07:00Am I missing something ... if the dartboard is cir...Am I missing something ... if the dartboard is circular then the the dart is closer to the center than the edge if it lands within half the radius of the circle. Can't we just compare the ratio of the areas then? If the circular dartboard has a square frame around it then the denominator changes, but the numerator stays the same.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-66790153947614255802011-06-23T00:45:13.409-07:002011-06-23T00:45:13.409-07:00(4*(2^.5)-5)/5 ... take the difference of two inte...(4*(2^.5)-5)/5 ... take the difference of two integrals (one of which is the area of a triangle) and divide it by the area of another triangleAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-56360595505966300252011-06-21T09:02:26.285-07:002011-06-21T09:02:26.285-07:00There is a doubly interesting relationship between...There is a doubly interesting relationship between this problem, and the one I posted yesterday:<br />http://blog.matthen.com/post/6738340459/this-is-a-good-example-of-a-curve-that-can-be-made<br /><br />(check out the comments), it all boils down to this picture:<br /><br />http://www.matthen.com/misc/twosquares.gifMatt Hendersonhttp://blog.matthen.comnoreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-1833420491575664892011-06-21T06:51:04.485-07:002011-06-21T06:51:04.485-07:00@Kate - So do I! I also love problems that can be ...@Kate - So do I! I also love problems that can be stated in one simple sentence that anyone can understand. You could pose this problem to a middle schooler and they'd understand the question, and maybe even come up with some clever approach to solving it that doesn't use integrals!Allison Johnsonhttps://www.blogger.com/profile/01731073560005744198noreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-35176809086676966362011-06-21T06:36:28.783-07:002011-06-21T06:36:28.783-07:00Have you ever started reading a book, and then slo...Have you ever started reading a book, and then slowly realize that maybe you already read it, like maybe 5-10 years ago? That just happened to me for the first time with a math problem! Weird feeling! I <3 problems like this that use all the parts of your math brains. Thank you!Katehttps://www.blogger.com/profile/14229054922453438248noreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-3523106378602418232011-06-20T14:31:10.482-07:002011-06-20T14:31:10.482-07:00Thanks for sharing this problem. I had a lot of fu...Thanks for sharing this problem. I had a lot of fun/frustration while working this out, mostly fun. I had quite a few false starts and made arithmetic and algebraic errors; it took a bit of patience. Makes me think that maybe my students need to see me struggle with a question more often and how much time and persistence it takes to solve problems.<br /><br />Here's a post where I pose an extension to the problem about n-sided regular polygons: <br /><a href="http://mrhodotnet.blogspot.com/2011/06/square-dartboard-probability.html" rel="nofollow">http://mrhodotnet.blogspot.com/2011/06/square-dartboard-probability.html</a><br /><br />I hope I didn't make mistakes. Here I attempt a solution and verify your and Ms. Cookie's answer for n=4 (square dartboard).<br /><a href="http://mrhodotnet.blogspot.com/2011/06/ngon-dartboard-probability.html" rel="nofollow">http://mrhodotnet.blogspot.com/2011/06/ngon-dartboard-probability.html</a>Mr. Hhttps://www.blogger.com/profile/12620847580362451503noreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-16550212224551793692011-06-20T11:52:11.291-07:002011-06-20T11:52:11.291-07:00Also, I'm really curious what your method was ...Also, I'm really curious what your method was vs. your boyfriend's method. What was different about them? Did your two different insights help you figure out the glitches in each others' problem solving?Maxhttps://www.blogger.com/profile/16935784635103701185noreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-34165651092570344972011-06-20T10:47:34.682-07:002011-06-20T10:47:34.682-07:00Thanks for this problem. I started with solving it...Thanks for this problem. I started with solving it if the dartboard were a circle, and that got me wondering... what if the dartboard were an equilateral triangle.<br /><br />Here's my solution. It's a lot like yours, I think...<br />http://mathforum.org/blogs/max/problem-solving-journal/Maxhttps://www.blogger.com/profile/16935784635103701185noreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-34146803906263301312011-06-19T22:05:15.773-07:002011-06-19T22:05:15.773-07:00Ms. Cookie - how cool! I would've never though...Ms. Cookie - how cool! I would've never thought to use parabolas like that.Allison Johnsonhttps://www.blogger.com/profile/01731073560005744198noreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-51760030132396157382011-06-19T17:50:26.123-07:002011-06-19T17:50:26.123-07:00The 'intellectual masturbation' comment ab...The 'intellectual masturbation' comment about grad school is exactly how I felt. I started a PhD program at UCSD, but quit it as quick as I could.<br /><br />I love your description of the frustration, and avoiding something you think you might fail at anyway. The hardest part about math might just be keeping enough faith in yourself to sludge through the muck until you get some insight.Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-52392880169117530042011-06-19T17:06:01.436-07:002011-06-19T17:06:01.436-07:00Okay, here is my solution to the target problem: h...Okay, here is my solution to the target problem: http://www.box.net/shared/jpnqyy8dcc5opux9gqbm<br /><br />It's not as neatly written as yours, but this is basically what I did. And I couldn't quickly find a way to combine my 5 jpegs into one other than a word document .... and it's 2meg!Shireen Dadmehrhttps://www.blogger.com/profile/16282965851939089408noreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-44157845353214120322011-06-19T15:27:36.649-07:002011-06-19T15:27:36.649-07:00You might want to check these out as well if you h...You might want to check these out as well if you haven't yet. They might serve you well in your new position:<br />http://www.mathcircles.org/<br />http://wildaboutmath.com/2011/06/18/review-mathematics-education-for-a-new-era-video-games-as-a-medium-for-learning/Phil Wagnerhttps://www.blogger.com/profile/08938707552495871086noreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-30461409372246753932011-06-19T12:25:00.431-07:002011-06-19T12:25:00.431-07:00@Jenny - Thank you for reading my blog and for sup...@Jenny - Thank you for reading my blog and for supporting me always. You are the BEST :)Allison Johnsonhttps://www.blogger.com/profile/01731073560005744198noreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-61068185494068156612011-06-19T12:05:37.062-07:002011-06-19T12:05:37.062-07:00@Ms. Cookie, I'm curious about this "para...@Ms. Cookie, I'm curious about this "parabola knowledge"<br /><br />@Everyone - Eureka! sqrt(3-2sqrt(2)) = sqrt(2)-1!!! That simplifies the answer significantly. Revised solution here: http://tinyurl.com/3bhxz58Allison Johnsonhttps://www.blogger.com/profile/01731073560005744198noreply@blogger.comtag:blogger.com,1999:blog-1222092101762956112.post-61028921718508067172011-06-19T11:57:24.363-07:002011-06-19T11:57:24.363-07:00I completely agree with Pat's comment. One of ...I completely agree with Pat's comment. One of the best ways to help your students is to share your experiences with them. Knowing that you still struggle at times will help them connect to since as much as they want you to be perfect, they also want to see that you are vulnerable too.<br /><br />Share this with all students and all teachers. When we open up to our kids we allow the relationship to grow deeper and the connection becomes stronger, and in my opinion, the learning too. Thank you for sharing this and, of course, your incredible thoughtfulness.Jennynoreply@blogger.com