Tuesday, May 12, 2015
Graphing polynomials with a 10 year old.
Here is more graphing of a couple polynomials...each time we get a little more "advanced" but as you should clearly see the concepts are exactly the same every time the math just gets a little more complicated as the numbers change.
I did both of these quick and dirty as it were and the second one I forgot to turn the camera on so I just breezed over it after the fact. Upon inspection I find the seriously notation lacking. Also want to drive home the point that you must remember that when you are completing the square in a problem that has more than one x² you have to remember you divided by that number, in this case 2, so we have to remember to add 2x 3.0625...we didn't just add it once...if I had divided by 3 then it would be 3x..etc.
Lets do an easier one just for example. And we'll divide by 3 even to make the point.
3x² +18x + 24 = 0
factoring it should be EASY. Get out your blocks.
Now divide by three so we can complete the square and put it into vertex form...
3x² +18x + 24 / 3 = x² + 6x + 8 = 0 Even though it's already a perfect square we still have to complete the square AND the 9 we added is added THREE TIMES because we are multiplying the part we are completing the square with 3 times...we just divided by 3 to make it easy.
Then 3(x² + 6x + 8 = 0) ⇒ 3(x² + 6x + __ ) = -24 (that's 3 x 8 not just one 8) ⇒ 3(x² + 6x + 9) = -24 + 27 (that THREE 9s not just one 9).
Vertex form is now easy:
3(x² + 6x + 9) = -24 + 27 ⇒ 3(x² + 6x + 9) = 3 ⇒ 3(x + 3)² = 3 ⇒
3(x + 3)² - 3 = 0
Check out DESMOS for easy fast graphing of these polynomials...or wolfram alpha. Here is a graph:
You can see that the y-intercept is WAY up there...but the other points are easy to get once you know what you are doing. So here is a chance for you to sketch a graph of your own label the points and make sure you put in the axis of symmetry too.
The rule or formula -b/2a should be obvious too, and not something we have to memorize.
Anyhow now we are well on our way to being able to graph any polynomial. It's so easy a seven year old can do it with very little help...soon he won't need ANY help at all.
So now you see putting it together yet again. We start off with very basic completing the square. We play with square numbers (you will note I have tons of videos on square numbers as well as radicals) and learn why it's so easy to square numbers that end in 5. We do all of this via play and fooling with blocks not worksheets and and work books and wrote memorization. My students have FUN. Yours should too.
They ASK for algebra because it's fun...
Later we can talk about "F of X" [f(x) = y] and so on...for now quick and dirty math with emphasis on the concepts...not rules and process.
I make trig easy too, as well as subtraction, division...all of it. You can get hours more video, more explanation and a pile of PDFs (with more explanation and exercises and methodology) for $8.50, which gives access for a year. People are shocked and amazed at what they get for free...more so at how much they get for a paltry 8 and a half bucks a month.
***If you want yet more on this topic here is another blog post on graphing polynomials by now the problems should be EASY...and you can watch it from the perspective of what problems I use to teach which concepts.
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This might seem advanced for your less than ten year old now, so tell you what, let's just start with subtraction and go from there.