Here is a quick set of vids on completing the square. You can find all 5 parts here on the Advanced Algebra page. Basically we take playing with squares to the next level and out of necessity we have to cram a lot of info into a short period of time because he has grades to make and tests to take.
If he had been playing with squares and building square numbers for a while previous to this, everything would come a lot easier and with less stress. Get your students started early...and by "early" I mean as early as possible, five or six year olds can do a lot of this math as seen on my website and other posts on this blog. For them, you go slow and make a game of it...but they learn and discover all kinds of things that will come in handy here.
Take a look at this picture:
In a moment it will make more sense. You can see that instead of a red square and blue bars which are the x we have orange twos...the red square is now a two square and the 9 has twos around it because each x = 2 (and of course x² = 4), and 25 = 25...when x = 2. What would get for x² + 6x + 9 if x = 3?
Watch this vid:
All we are doing is some simple problem solving, trying to figure out what x is. Then we immediately begin adding more meaning and information with graphing...but first we practice without the blocks:
And as you can see he has a bit of difficulty so we go back to the blocks and practice some more. Check out this page on completing the square [link not built yet] for the full story, the page contains a link to both the entire 23:30 minute video and the 5 separate videos on a page very similar to this but with a little more explanation and pictures and video and links...that vid cuts off abruptly and you will need a password to get into the pages that contain the rest of it.
Here is Part 5 where we do one more problem that looked a little daunting but many of my young students have built that very square just playing on their own. Self directed students often get out four or even nine red squares and put x's and units around them...building bigger squares is fun. They can also see the similarities between x² + 6x + 9 and 169 and see the square root is 13 in the case of 169 and (x + 3) in the case of x² + 6x + 9 and certainly if each x is two then x² + 6x + 9 = 25. In fact substituting for 1, 2, 3 and 4 is no big deal and not scary at all for a 6 year old. But the teen-agers who are being exposed to this for the first time do not agree. Too bad they didn't get to play with these blocks when they were kids.
Little kids would just build it draw a picture of it and that would be it. Later we could set it equal to zero or another square number and solve for x. As I mention in the vid, first you introduce the positive root then as they become more familiar with square numbers and integers you can introduce the negative root. 5² = 25 and so does -5².
±5 = 25 is just economy of symbol.
Here is the 5th part, in part 3 and 4 we cover graphing and more completing the square with a negative middle term...which is also ridiculously easy and visually obvious. If you have a set of blocks see if you can figure it out for yourself without using your password to look at the video...
Being familiar with the concepts ahead of time solves a lot of problems. For students who are struggling to pass algebra the blocks make the concepts assessable and easy to SEE and therefore understand and remember.
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