New Year’s Resolution: Opening Questions, Developing Questions, Extending Questions

So, in the new year, I resolve to open up my questions more and to allow students to develop the question more in class.  I’m going to invite more conjectures from them.

Here’s a glimmer of an idea I ripped off from an AMC question:

Five central angles are drawn in a circle.  Their degree measures are integers in an arithmetic sequence.  What could their measures be?Could you have only one set of five angles?  Could you have more than one set? If so, what’s the smallest degree measure that the first (lowest) angle could be?  If you are stuck, change the question.  Suppose that the five angles are equal and solve.  Could you use anything from that to answer the original question?What if it is a geometric sequence?

This thinking is making me wish that the definitions of the terms geometric sequence and arithmetic sequence get taught earlier–guess I can do that in about ten minutes in my own class.  There’s a lot of problems we can make slightly more challenging by replacing an assumption of a set of equally numbered values with an arithmetic (or geometric) sequence.

Another thought, is that I should be including questions about 3D analogs to much of the 2D geometry I teach.  I picked this suggestion up from Judah Schwartz, who has had and continues to have many good ideas about what math education ought to be.  For instance, many geometry classes teach how to make a circumcircle of any triangle… but Judah has a good challenge:  Can you draw a sphere through the 4 vertices of any tetrahedron?

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