One question I have used recently is this:

By algorithm, the better algebra students will easily be able to solve this circle equation for the twoA 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.

*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.

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