-->

## Tuesday, January 10, 2017

### Triangle of Time

Sometimes GeoGebra spurs me on to a new way of looking at something old and familiar.

In this case, a clock. Yesterday I posted the clock I made in GeoGebra. Today I have tweaked it a bit to help pose a question.

The three hands are all pointing to 3 points on the circumference of the circle. These 3 points form a triangle. Can we work with the area of that circle?

1) The smallest area of the circle is zero. How often does that happen in a 12 hour time span?
2) What is the largest area? How often does that happen in 12 hours?

A stretch for trig or precalculus students might be generating a graph of the area as a function of time of day. A stretch for calculus students might involve determining exactly when the area is largest by maximizing that function.

A stretch for younger students might be generating this graph on their own. It is too bad that the politics of education make it virtually impossible for a math teacher to take time to work with students on questions like these.

I suspect textbook publishers do not like questions such as this!