### Starry Nights and Another Argument for Simplified Radical Form--sort of

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Left this activity for my Geometry students with a sub the other day: Please be patient, there is a WOW at the very end, really.

**Find the area of the triangle in two different ways.**

Too many students wrote 1/2(6x5) = 15 sq units

Upon my return, and I inquired how did you know that AB was 6 units and and BC was 5 units.

"I counted them." Oh, really? I mused. How can you be sure. Saddy faces, big bottom lips. "Yeah, I guess I needed to be more accurate. I asked what would you get if you counted by square units instead of linear units? Heads went down, tongues out instead of big bottom lips and pencils scratched across paper. "12!" shouts an otherwise unengaged kid. (yay for counting!)

Which is more accurate I ask, and why? A fun banter moves along, and finally the the gal who is ALWAYS on it, but doesn't want to stick out as a know-it-all, can't take it any longer: This is what I did: Pythagorean Theorem and some subtraction:

Boy, did kids like that. We all were still in the first way because the students who tried 1/2bxh quickly abandoned it when h was really hard to figure out. Now I know you must be thinking, really, this is all you got?

Brandon, my love bug, decides that the triangle is a right triangle and he likes the Pythag stuff, so for his second attempt he writes: A =1/2(sqrt 29)(sqrt 20) = 12.04. Nice I say. How do you know the triangle is right triangle? "Because it looks..." Whoa Stop! Paaaleeesseee. How can we make sure? So we check out the slopes, Ah...not perpendicular. But close right? It is a good approximation, and we High Five.

Meanwhile, across the room the same notion is being utilized, but with this sweet, sweet, amazing and fabulous student work:

He deserves extra large, right? I could weep. And we get the opportunity for a conversation about rounding, how big of a purchase do you need to make when ordering materials to off by .15 of unit per unit, a case for simple radical form (even though it is about slope). What a day!

After doing a lot of searching for how to make square roots meaningful and I decided to give good ol' Theodorus another whirl.

Day 1: Ladies and Gentleman, this is the bar: (As I am cutting and pasting and resizing, I realize this student's name is Estrella!)

(yeah ignore that her right angles aren't right angles, she was inspired and did all that math!)

The next day, I get Jonathan's and I am sobbing. (he told me he thinks he did it incorrectly, and he is correct in his thinking, which is very cool, so he fixed it with pencil underneath) I am so lucky. Kids are just amazing!

Love the lime and cilantro.

What other questions would you ask? What Standards of Math Practice did I hit? Did I miss any opportunities? Please let me know!

Boy, did kids like that. We all were still in the first way because the students who tried 1/2bxh quickly abandoned it when h was really hard to figure out. Now I know you must be thinking, really, this is all you got?

Brandon, my love bug, decides that the triangle is a right triangle and he likes the Pythag stuff, so for his second attempt he writes: A =1/2(sqrt 29)(sqrt 20) = 12.04. Nice I say. How do you know the triangle is right triangle? "Because it looks..." Whoa Stop! Paaaleeesseee. How can we make sure? So we check out the slopes, Ah...not perpendicular. But close right? It is a good approximation, and we High Five.

Meanwhile, across the room the same notion is being utilized, but with this sweet, sweet, amazing and fabulous student work:

He deserves extra large, right? I could weep. And we get the opportunity for a conversation about rounding, how big of a purchase do you need to make when ordering materials to off by .15 of unit per unit, a case for simple radical form (even though it is about slope). What a day!

**Something to make you smile:**After doing a lot of searching for how to make square roots meaningful and I decided to give good ol' Theodorus another whirl.

Day 1: Ladies and Gentleman, this is the bar: (As I am cutting and pasting and resizing, I realize this student's name is Estrella!)

(yeah ignore that her right angles aren't right angles, she was inspired and did all that math!)

The next day, I get Jonathan's and I am sobbing. (he told me he thinks he did it incorrectly, and he is correct in his thinking, which is very cool, so he fixed it with pencil underneath) I am so lucky. Kids are just amazing!

Love the lime and cilantro.

What other questions would you ask? What Standards of Math Practice did I hit? Did I miss any opportunities? Please let me know!