This Changes My Game

I have been trying to create moodle questions that get my students to construct something in GeoGebra, get some sort of information from the construction, and answer the question. I have been struggling with the idea that any calculated moodle questions I come up with is ultimately reduced to an algorithm. Nevertheless, if my students are unaware of the algebra behind the algorithm, and need to think of what to construct, then perhaps that is O.K.

One question I have used recently is this:

A circle has a center (17,15) and a radius 59. Write the equation of the circle. Then, find the larger of the two y-values when x=33.

By algorithm, the better algebra students will easily be able to solve this circle equation for the two y-values, but the students in this particular class are not my better algebra students. What I hope would happen is for them to graph the circle in GeoGebra (meaning they must write the equation correctly...#1 goal),

find a good window setting (which is essentially scrolling out until you can see the entire circle...do they know what they are looking for? Goal #2),

graph the line x = 33 (which is difficult to do on my calculator of choice, the Nspire, or other graphing calculators),

constructing the two intersection points, and taking the larger of the two y-values (recognizing that this is the desired answer to the question),

To answer the question correctly, there is still a lot of intuitive mathematical understanding that is being displayed, especially when you ask them why the answer must be 71.79.

I have been encouraging my students to find different web-based resources, lately, and one of my students found a web-based calculator web2.0calc.Imagine my surprise when one of my students did this to solve this problem:

producing this result

Not a CAS, but a pretty nifty numerical solver. And intuitive! When I asked the student what made him try this, he told me that he just wrote the equation, and substituted 33 in for x like the question asked him to do. There were two answers for y, so he used the larger of the two. Intuitive.

THIS changes my game.

Go ahead and try this for yourself on the calculator below.