I’ve been using Andrew Stadel’s estimation180 with my 9^{th} grade geometry students regularly this semester. Yesterday, I gave them an “estimation challenge” of my own.

This picture was taken on Sunday. At what time of day did I take the picture?

Since this was so different from other estimates they’ve encountered so far, we worked as a class to establish a window of reasonable answers. Someone noted the picture was taken during daylight hours, so between 6am and 7pm seemed like a good place to start. Someone else observed that shadows appear much longer very early and very late in the day. This was probably some time in the middle of the day. Someone remembered that the sun rises in the East and sets in the West, so which direction is he facing? Isn’t that how a sundial works (yes)?

When pressed, I conceded that the left-hand side of the picture was “West-ish.”

After deciding on a window of 1 pm-5:30pm, students wrote down their estimates. And then I showed them this:

If they could determine the angle of the sun, they could determine the time the picture was taken!

The students have been introduced to definitions of sine, cosine, and tangent, and they have been putting those definitions to use solving for unknown right triangle parts using an old-fashioned trig table.

Why a table? Well, it seems to reinforce the idea that “sin 75” is a *number* and not just some fancy piece of calculator magic… And, then on the day (today) I show them a picture of my son and his shadow, I can also ask them to figure out what time the picture was taken and have them discover inverse trig functions for themselves. And they did! They measured my son’s height, they measured his shadow, they determined the angle of the sun, and they determined the time the photo was taken. They can see so clearly on a table the relationship between (in this case) “tan” and “tan^{-1}” and, in fact, define it for themselves. Beautiful. Sometimes that old technology (paper) is the best teaching tool.