I am so excited! I taught completing the square today in my "low" grade 9 class for the first time, and I decided to try a geometric approach this time as inspired by my friend from PCMI (the awesome Danielle Soderberg*), and the kids loved it!! After just two examples on the board, they had no trouble at all working through cases with integer leading coefficients and integer vertices. It was fantastic.
Here was the worksheet I made for completing the square, and here was the warmup sheet I used prior to introducing complete the square (cut into half-sheet strips; the kids skipped the conversion to vertex form at the beginning of class and came back to that column at the end, after the lesson).
Enjoy!
*Although, as a small disclaimer, I never saw Danielle's worksheets, so it's quite possible her approach is different and/or it is far superior. She vaguely mentioned something about algebra tiles and completing the square.
I love this approach, and the worksheet you have here introduces it in a clear fashion. If you don't mind, I'd love to share it with some colleagues here. Great work!
ReplyDeleteNo problem. Please share it!
ReplyDeleteI had to walk the kids through two examples on the board, obviously, because the worksheet explanation is always not as clear as when you just do the problem at the board. I ended up just circling the diagonal corners and asking them how many x's go into the diagonal total (6x in this example), and therefore how many x's must go into each box (3x), which helps us figure out that it's (x + 3)^2 + ...
When they got to #7 on the worksheet, I showed them that if you draw a 2-by-2 box and put 2x^2 in the upper left corner, it's not possible to break it down evenly, so we must instead draw TWO copies of the boxes, each with one x^2 ... Hopefully that is clear to you! If you and your colleagues cannot envision this then I can try to pull a explanatory video together.
Love completing the square, and love completing the square geometrically! You might me interested in this completing the square idea that I first started playing around with at PCMI years ago. A colleague and I gave a presentation about it to the Metropolitan Mathematics Club of Chicago, and there is a link to a paper we had published in Mathematics Teacher a couple of years back.
ReplyDeletehttp://edwardsphelpsmmc.pbworks.com/w/page/2692268/FrontPage
Love the geometric approach - it connects a bunch of different concepts together, but is still efficient and elegant. Will be stealing for next year - thanks!
ReplyDeleteI understand how to use this method to rewrite the function in vertex form, but how or can this method be used to solve quadratics?
ReplyDeleteOnce they have a vertex form, for example 4(x - 3)^2 = 20, they can solve it from there like: (x - 3)^2 = 5, so x - 3 = + or - sqrt(5), x = 3 + or - sqrt(5). It's the same way the quadratic formula was derived is via completing the square.
ReplyDeleteOh, what I mean is if they're given 4x^2 - 24x + 36 = 20 they can just complete the square on one side first and then continue solving from there.
ReplyDelete