### 7 out of 8 Ain't Bad

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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)

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):

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!)

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

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

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