Looking for Feedback - Radical Roots

I'm looking for a bit of feedback on an investigation. I am teaching an algebra heavy geometry course, we just tackled right angle trig. The kids have seen Pythagorean theorem and such... I need to teach radicals and exponents at some point and this seemed like a reasonable time with areas of polygons and coordinate geometry around the corner.

The objectives of the investigation is for students to:

• Develop geometrical meaning of square roots 
• Develop geometrical meaning for equivalent radical expressions such as:

I must admit most of what is posted is not my idea, but an adaptation of content from Math Alive! Course III.

Link to the Google doc investigation.

Any and all feedback is welcome.

Your math teacher might be a sleep deprived idiot...

This made me laugh. Yes, I am the teacher who wrote this question... Probably a clear indication that I need more sleep.

I loved the honest feedback from the student.

I love it when a plan comes together!

I was working my way through right angle trig. It was going pretty alright, but I was looking for an exit strategy that would tie in RAT to the new topic... Polygons, Surds, Exponentials. I saw connections to all of them and I was flip flopping between topics unable to choose. Seventeen ideas running through my head... Where's my Ritalin?

Then Kate Nowack of f(t) fame tweeted a link to a video:

There is genius in this video. Maybe not original math genius, but genius all the same. The connection was obvious. Pythagorean theorem and maybe some trig. Best of all students would need to think about lines, angles and triangles. Perfect.

I didn't really like the way the question was framed, felt like he gave away a bit too much (even in the original video, the one above presents the solution), but it does make for a good video. Before class I drew a series of 4 dots on all of the whiteboards, providing spaces for the students to present their results with some semblance of order. Each time an improved result was written up the class was drawn to it and they madly checked each other's math.

I heard things like, "that side is a hypotenuse and the other side is 1, so that side (the hypotenuse) is more than 1!" Or, "you used 0.333 instead of 1 over 3 that's why it didn't work!" Freaking awesome.

30 minutes in, I stopped class and asked how many wanted more time? Almost unanimously they wanted more time. I gave them the option of seeing the answer, but they didn't want it. What more can a teacher ask for? Finally, I ended the 65 minute period with the video and a brief discussion. During which one student asked, "Do mathematicians really play with soap bubbles?" With a twinkle in my eye, I replied, "Yep."

Math is real. Soap bubbles are real. The learning was real. No pseudo-context needed.

So close...

I am teaching Geometry for the first time this year. Which means I get to teach right angle trig from scratch! In the past I have taught trig (circular) functions, but never the more basic bits.

I was supposed to teach "triangles" which they already knew and had for some time (my school seems to have accepted that math will be boring and repetitive). So rather than belabor the point, I did a quick review of terms that led to the idea of similarity. I had them apply the idea of similarity to bigger things, like the school building... or the playground equipment... or the local mountain top.

Within a class period most if not all understood the idea that the ratio of two sides of a triangle can only be changed if the angle changes, in fact when I brought it up half the class gave me that look of "duh mister, we figured that out." This was all done with no mention of sine or cosine.

With the real learning done, I started into a bit of formalization. With a short back and forth discussion that required students to draw a few more triangles to convince themselves of a the details we had the 3 main trig functions on the board in just a few minutes, but still unlabeled.

With nothing more to teach I finally labeled the functions...  I gave them some typical problems to practice with, only one of which I did on the board. Then they were off and running. I felt so close to a masterful introduction of trig. So very close.

Next on my list was inverse trig functions...

I posed the question (with a drawing on the board), given two sides of a right triangle how do we find the angle? I figured there would be lots of drawing and struggle to draw the correct triangle in order to measure the angle. So close, I could almost taste it. Somewhere somehow during this time someone shared that they could use the calculator to find the angle, i.e. inverse trig functions. They don't understand inverse functions, but they have buttons to push...

I was so close. Sometimes I despise the power that buttons seem to have over students.

Should I have given them a table of trig functions instead of allowing them to use their calculators? That seems so contrived, so fake, so false. How do I avoid this next year?

Transformations Take 2

This year my usual transformation of function investigation didn't seem to work. I suspect it was not the investigation itself, but rather my students adjusting to a new classroom culture, one where the teacher doesn't tell you everything...

We took a break from functions and function notations to work on an investigation with some toy cars (post of that may be coming if the results are half decent), but I promised the students that we would revisit the topic of transformations. Its simply too core of a topic to move on with half the students not grasping the idea.

Searching for a new way to tackle transformations, I found one it was brilliant! Then I forgot the idea and came up with this:

http://www.geogebratube.org/material/show/id/708

Using Geogebra I created a simple transformation applet (link just below the pic) allowing them to graph any function and adjust the 4 parameters, do reflections, plus the nasty bits of absolute values. Paired up with the applet was a worksheet to guide them through.

The results seem to be far more positive and constructive. They seem to understand that they will need to ask questions and do their best to organize the answers and their understanding. I will of course do a debrief afterwards and to try and scoop up one or two lost students, but "take 2" seems to be working