7 out of 8 Ain't Bad

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2 Comments
Standards for Mathematical Practice (SMP's) in one problem that is! I had so much fun interacting with the students on this one PreCalculus question that I tweaked my neck!

Here it is...Solve this baby over [0, 2pi)



That's it, that little thing. Such beauty and so much discussion. (SMP 1: make sense of problems and perservere in solving them)

We first went after it like this: cos^2(2x) + 2cos2xsin2x + sin^2(2x)=1
The students suspected that cos^2(2x) +sin^2(2x) was a pythagorean identity for one and would leave the equation to be solved 2cos2xsin2x=0. It was pleasant to divide out the 2, and look at the zero product property that was left: cos2x = 0 or sin2x = 0 yaddah, yaddah (SMP 6: attend to precision)...nice discussion about where is cosxsinx = 0 (SMP 8: look for and express regularity in reasoning) and which identity would be nice to use for cos2x.
Eventually they got to : 0, 45, 90, 135, 180, 225, 270, and 315. (SMP 7: look for amd make use of structure)  And the question remained: are these the same results for the original equation? (SMP 3: critique the reasoning of others) How to check without all that work?

 DESMOS of course: (SMP 4: Use appropriate tools strategically)













The students noticed immediately that both the original equation and the simplified equation had the exact same x-intercepts. Wow, cool! Yay! Done!

Then Mr. Clever (The only student truly awake at 8 am) said, Zim, can't we just square root both sides? Well, yes we can! Let's do it!

cos2x + sin 2x =1 or cos2x + sin 2x = -1

 They wanted to see those graphs in Desmos too. (I made them beg just a little):

They were a little sad when the graphs did not have any common intercepts. Until of course, we remembered that the equation had the word OR right there. (SMP 2: reason abstractly and quantitatively)  So much dancing, so much skipping! So fun! (so much neck tweaking!) 


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2 comments :

  1. Five TrianglesJanuary 23, 2015 at 10:47 AM

    After that problem, they may be ready to tackle this one:

    http://bit.ly/1wqqIBx

    ReplyDelete
  2. Amy zimmerJanuary 23, 2015 at 11:00 AM

    Thank you for sharing this problem. I look forward to doing it with my students and having THEM relate it to the practices.

    ReplyDelete

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Hi there, I'm Amy! A high school math teacher, poet, crafter, athlete, and mom. This blog features both math and teaching strategies, as well as some of my favorite recipes and everyday triumphs and struggles. Thanks for stopping by, feel free to share on Twitter, Facebook, or email me if you have any questions!

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