This slideshow is the Smartboard I used in this class that I describe below:
The day my students got back from spring break, this week, I started things off with a bang! At least, that was the intention. Instead of doing the whole “Paris was great, where did you go for your vacation?” time waster, I decided to dive right in. They left for break having learned some trigonometry basics, and I had blown their minds with the idea that you can measure angles with something other than degrees (radians). Monday’s lesson was to be on central angles, arcs, and angular velocity.
On Sunday night, I downloaded GoogleEarth onto my school laptop. In an ideal world, I wanted to say:
So kiddos, I flew from Newark to Paris for spring break. Let’s all get out our laptops and find out their longitude and latitude, and the distance between them. [I do it in front of them on GoogleEarth]. Now let’s use this info to the find the radius of the earth.
Unfortunately, after a few minutes of playing around with this problem, it becomes very clear to me that it is not as simplistic and trivial as I had hoped, because the latitude and longitude are different. Picture the following. Draw a line from the center of the earth to Newark, from the center of the earth to Paris, and a curved line (along the globe) from Newark to Paris. You have a sector of a circle. You know the arc length. But what’s the central angle (the angle between the radii)? It’s not a trivial problem for students to calculate that angle.
So instead, I copped out and found two random spots on the equator (one in Brazil, one in the Democratic Republic of Congo), and the distance between them. And then I had students come up with what to do next. It took very little time before they stumbled upon the idea of setting up a proportion: angle/full circle angles = arc/full circle perimeter.
From that, we found the circumference of the globe (about 40,046km). And then we compared it to what Wikipedia said (40,075km), marveled at our own acumen, and talked about the sources of error. And because I truly am a big nerd, I talked about the origin of the meter (see this book).
This warmup took about 10 minutes, and was a perfect segway into a discussion of arc length and central angles.
We set up the same proportion as we did above, but for a general circle of radius r, central angle , and arc length s, to discover the wonderful formula . Then I asked them if it made sense… (The larger the circle’s radius, the larger the arc length… check! The smaller the angle, the smaller the arc length… check!) And when I asked them when they had seen this before, most of my students noticed: “duh Mr. Shah, the Earth problem…”
Finally, I got to the final topic, angular speed. I remember when I was first taught this, I was taught it as a formula. There was no conceptual understanding behind it. And students tend to love to rely on formulas without understanding them. So I introduced the ideas of “linear speed” and “angular speed” — and then asked them to calculate (without showing them any problems) the linear speed and angular speed of someone in Brazil or the Democratic Republic of Congo as the earth spun on it’s axis. And someone’s hand shot in the air, then another, and soon the problem was solved.
In addition, one student noticed that the angular velocity remained the same no matter where on the globe you stood, but your linear velocity decreased the further away you got from the equator.
Which led to the final equation of the day: .
Some more practice problems were done, and that was that. With that concluded one of my better, more cohesive lessons. It all tied together with the globe thing, and they really left with a sense of the concepts, more than the simply memorizing the formulas.
 One class noticed we could calculate the radius of the earth with this formula, since in our earlier problem, we knew the distance between the places in Brazil and the Democratic Republic of Congo, and the angle between them… So we did that, and got another wonderfully accurate answer too!