(0.5)!

So, it is AP Calculus prep time, my least favorite time of the year.

This year I am lucky to have one student, just one, who will NOT take the AP exam. So, I asked him, “Do you think it would be a boring waste of time to do the old AP exam your classmates are doing?”

“Yes,” he answered. So, I pulled down the old Calculus textbook by Paul Foerster–whose explorations are essential to my class–and thumbed through it for a good problem that would take him some time to do…

The factorial function and the Gamma function!

He set to it, doing some improper integrals, and some integration by parts and then he used his calculator to integrate the factorial function to get 0.5! = 0.886.

The book said, find a simple expression involving pi that equals 0.5! and after a little time, he found.

Moreoever, he learned that (-1)! and (-2)! were infinite but (-0.5)! and (-1.5)! were finite.

And I learned these facts, too!

Back in college, I saw the gamma function in my textbooks but spent zero minutes using it, much less investigating its definition. So having a student who was not constrained by the AP test prep (and I’m only spending a week and a half on prep) gave me the opportunity to broaden and deepen my knowledge of mathematics… as well as this one student, and after the exam is over and done with, I’ll use this experience to push the rest of the students to learn (0.5!). It will fit in with the “Integrals and Statistics” unit I’ve composed for the month & a half after May 6.

This is what studying math brings you: new discoveries, new connections, a sense of wonder at how the world of numbers makes sense.

Thank you, Paul Foerster, for your wonderful books!

An Intro to Work (in a Calculus course)

I’m trying out this WorkLesson today. In the past, I’ve just demonstrated the example at the board, of a leaky bucket being pulled upward a certain distance.

The goal of this is to develop the integral of F(x)dx as an analog to motion integrals, like integral of a(t)dt, and to develop it in connection with Riemann Sums (my kids have a tough time with those), and to pull away the differing context to show the identical structure of the two integrals.

I’ll record some thoughts here after the lessons, and the homework is checked tomorrow.

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?

Classroom Action: Explore Rotation with Compass, Straightedge

Today I had my kids work on this little exploration as a fun way to keep up their compass & straightedge skills.  The one question this exploration is lacking is the final generalization:  WHY does this work?  (Hint: the perpendicular bisector theorem…)

And then when I checked out Sam J. Shah’s blog, I found out he’s interested in the same thing: Discovering the center of a rotation.