I love Desmos!

Blog 2 of #MTBos30

Desmos has really expanded how I can explain topics and problems in the classroom.
Today in pre-calculus, we were solving some trig equations.  We are up to equations that involve quadratics.
The equation in question was 3sin²x - 8sinx - 3 = 0. One of my students did the old "my calculator keeps giving me an error!" thing.
"See, I keep doing 2nd sine 3 and it gives me an error!"
So I asked her where the sine was equal to 3 on the unit circle.  She thought about it for a second and said "oh yeah, duh!"
Then she wondered if that meant there were no solutions and I told her there weren't any for that factor.  I got to thinking about what that would look like, so I turned on the projector and graphed it on Desmos.  Then I graphed the two factors she had.  It was pretty cool.  She was able to see that one factor (3sinx + 1) generated all of the solutions for the equation, while the other factor (sinx - 3) did not generate any solutions because it has no solutions itself.
Here is a link to our graph: https://www.desmos.com/calculator/g5phkl23sw
More than that, though, you can see how the nature of the graph is determined by the graphs of the two factors.  When the first factor is negative, the graph becomes positive because the second factor is always negative.  When the first factor is positive, the graph becomes negative.  The shape is not perfectly symmetrical like it is for a "normal" looking sine graph because the positive portion is "scrunched" into a smaller segment.  The high and low points of the composite graph are 8 and -8 because they are determined by the high and low points of the factors.  These are all observations I couldn't have made as easily when just solving the equation algebraically.
These are also observations that are easier to make using Desmos than on the TI-84's.  I don't have an overhead set-up for the TI's.  I can put them under the ELMO, but then lighting is a problem.  I also don't have one of the new TI's that graphs in color so it is harder to distinguish the graphs of the function and factors from each other.
Desmos adds an extra dimension to my teaching!