Today I led a highly structured class that made effective use of technology, and finally got some skeptical students on board. Several students told me when they got to class that they were having a hard time with rationalizing denominators, and fortunately I had my lesson set up to begin there. We started by rationalizing a one-term denominator and then a two-term denominator by hand. I explained how and why we began rationalizing denominators in the first place, and although this can be a useful tool in trigonometry (trig ratios) and calculus (limits) it would not be a major topic of focus for us. I then showed them how to rationalize a denominator using Wolfram|Alpha by simply typing “rationalize 12/sqrt(75)”. The focus shifted to determining which of the results seemed best suited as an answer for us.
We moved on to graphing square root functions by hand using transformations. After 4 by-hand examples, we opened up Desmos and learned to speed the process up. They felt graphing by hand wasn’t too difficult, but did appreciate the speed and precision of Desmos.
We then pivoted to solving radical equations in Desmos by graphing two functions and looking for the point(s) of intersection. We first solved two equations by hand before turning to Desmos, and it was fun watching them try to figure out what happened with the extraneous solution. They then moved on to solve 7 more difficult equations, including one where a square root was equal to an absolute value. (That would have been challenging to solve by hand – definitely more of a college algebra level problem!)
We finished with some applications of pendulums and skid mark analysis, and we used Desmos to solve the equations by graphing after we solved them by hand.
I think students left feeling that these two sites are quite powerful and can really help them to focus on conceptual understanding. A good day!