tag:blogger.com,1999:blog-8748665016211866969.post866033279213246220..comments2022-11-19T20:58:11.158-08:00Comments on Coffee and Graph Paper: Completing the SquareH.http://www.blogger.com/profile/00155248585975222332noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-8748665016211866969.post-81143624847924687182011-11-01T14:25:37.497-07:002011-11-01T14:25:37.497-07:00Thanks for telling how that went, Rebecka - it mad...Thanks for telling how that went, Rebecka - it made my day:) I haven't taught completing the square in more than two years, and it's so much fun to think that someone found this useful now!H.https://www.blogger.com/profile/00155248585975222332noreply@blogger.comtag:blogger.com,1999:blog-8748665016211866969.post-10426809330973889722011-11-01T09:35:14.723-07:002011-11-01T09:35:14.723-07:00I just taught completing the square how you sugges...I just taught completing the square how you suggested today and it went sooooo well! Thank you! They saw and recognized the patterns on their own. So when I taught completing the square...I didn't have to teach it using a procedure, they already knew what to do. Awesome!Rebeckahttps://www.blogger.com/profile/01400123698812925804noreply@blogger.comtag:blogger.com,1999:blog-8748665016211866969.post-32094872472863291262008-04-04T20:14:00.000-07:002008-04-04T20:14:00.000-07:00The method that I have used for the last two years...The method that I have used for the last two years seems to work well for me.<BR/>I sort of "scaffold" their understanding from factoring a perfect trinomial square to completing the square.<BR/>I might begin with a perfect trinomial such as y = x^2 + 6X +9 and have them factor it. Then I write y = x^2 + 6X +9 + 7 and talk about writing it in vertex-graphing form (since this is when I introduce completing the square). Building on this, I may use y = x^2 + 6X + 16 and have them recognize that this is the same as the previous question, they just need to break the 16 up first. Then, after this type becomes comfortable, I use something like y = x^2 + 6X + 4 and wait until they realize that they need to make the 4 into a 9 so they add 5 to both sides of the equation, or they add then subtract 5 from the same side. I continue this way until they are comfortable with completing the spuare with y = ax^2 + bx + c where a = 1. <BR/>Then I use questions that have a common factor for a and b, then I move into ones that do not have a common factor.<BR/>I find that students following this process understand what they are doing and we don't need to "memorize steps".Unknownhttps://www.blogger.com/profile/06476559356550004089noreply@blogger.comtag:blogger.com,1999:blog-8748665016211866969.post-40506349289330812932008-03-28T16:38:00.000-07:002008-03-28T16:38:00.000-07:00Well shucks, thanks for the warm welcome :) I act...Well shucks, thanks for the warm welcome :) I actually post around the net as several names... Druin most of the time, mrs. temple is from my gmail account (hence blogger), and Shelli/HS Math on teachers.net... <BR/><BR/>I've enjoyed reading your posts and check in on your blog a few times a week to see what you've written. I am really into pattern recognition/discovery methods, so I love your method of teaching quadratics.<BR/><BR/>Keep up the great work! :)druinhttps://www.blogger.com/profile/12363634340959613461noreply@blogger.comtag:blogger.com,1999:blog-8748665016211866969.post-67020306031864380752008-03-27T21:57:00.000-07:002008-03-27T21:57:00.000-07:00Jonathan, I'll probably try that story on one of t...Jonathan, I'll probably try that story on one of the kids who needs to make up the quadratics test - I'll see how it goes.<BR/><BR/>Mrs Temple - <I>you're actually Druin?!</I> I'm honored to see you here! I love <A HREF="http://ilovemath.org" REL="nofollow">I Love Math</A>, and now posting to the site moved a few notches further up on my todo list. <BR/><BR/>I've long been hoping that one day I'll get to teach a stats course so that I can learn the stuff myself - just took a tiny course on it in college, not nearly enough, and have forgotten most of even that.<BR/><BR/>Sam - working backwards from vertex form, again and again, is worthwhile. And I decided that for Intermediate Algebra sacrificing some of the complicated cases (where you have to find half of a fraction, etc) was worthwhile in order to keep focus on the big picture. I also agree that deriving the quadratic formula by completing the square is very satisfying - hoping to post some notes on that soon.H.https://www.blogger.com/profile/00155248585975222332noreply@blogger.comtag:blogger.com,1999:blog-8748665016211866969.post-20664424879144445012008-03-27T12:25:00.000-07:002008-03-27T12:25:00.000-07:00i like your way of building their intuition. i try...i like your way of building their intuition. i try to do that in general, but i totally failed when teaching them how to "complete the square." <BR/><BR/>i thought it would be easy (silly me), but there are so many steps, and the book presents it in such a procedural manner, that my students lost exactly what they were doing. they saw it as a set of steps to memorize, which is exactly what i want to avoid. <BR/><BR/>next year, i'm going to do it like you. <BR/><BR/>i think that the big problem in presenting this thing is that the big picture gets lost... why are they doing it? at least that's what my problem was. so i'm going to try doing it backwards first. show them an equation like (x-2)^2-3 and tell them why this form is so useful (can be used to quickly find the vertex, graph, and find the zeros). THEN i'll go through the rest.<BR/><BR/>i did like teaching completing the square for the "a ha, math be cool" moment. students had learned or heard about the quadratic formula earlier, but it came from nowhere. i motivated the class by trying to excite them to show that math wasn't magic and that we were going to come up with this awesome thing ourselves. <BR/><BR/>i'm always amazed at how much i can get away with by saying it's "awesome."Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8748665016211866969.post-20506667031882768072008-03-25T19:21:00.000-07:002008-03-25T19:21:00.000-07:00Excellent :) I love how you kept them engaged and...Excellent :) I love how you kept them engaged and worked with pattern recognition rather than just "here's how you complete the square".<BR/><BR/>By the way, in an earlier post you mentioned about posting to ILoveMath. You can actually link your blog posts in the lesson plans section so that you retain the ability to update it easily. I tried it out with my own blog under the Statistics section.<BR/><BR/>Keep up the good work!!!druinhttps://www.blogger.com/profile/12363634340959613461noreply@blogger.comtag:blogger.com,1999:blog-8748665016211866969.post-78907810293861203352008-03-25T15:48:00.000-07:002008-03-25T15:48:00.000-07:00I draw a square, eg a + 5 by a + 5, divided into a...I draw a square, eg a + 5 by a + 5, divided into a big square, two rectangles (the 'wings') and a little square (25). We talk a bunch about the parts. I eventually get something like x^2 + 20x + 97 into the discussion, digress to a story***<BR/><BR/>when we were little, my sister liked snacking, but couldn't always make up her mind. Sometimes she would try grown up crackers, triscuits, but didn't like them, and put them back after nibbling them***<BR/><BR/>and draw the perfect square with my sister's nibble gone from the little square. (squeals and "oh gross"). And then we undo my sister's damage (in this case, but putting 3 back to fix the triscuit (or, as the mathematicians say, complete the square)<BR/><BR/>Not wonderful, but it seems to hold.<BR/><BR/>JonathanAnonymousnoreply@blogger.com