Saturday, May 08, 2010

Circles and Tangents and Horizons

We're on the "tangents to circles" section in geometry, and I'd assigned a standard problem of "the satellite is ____ miles from the earth ... how far is it from the horizon?" Too many kids drew goofy pictures and didn't get the concept, so I started class on Friday with the following task.

1. We had a discussion about how when you're at the (ocean) beach, and you can see the horizon, maybe you wonder how far away it is .... any guesses? And do you think tall people and short people will see the same distance away?

2. Then I had them draw the "circle" earth, and we discussed drawing them standing on the earth (as a math stick figure with a big eyeball dot on top) and made sure they were standing up straight (an extension to the radius of the earth).

3. Then I asked them to just PLACE their ruler as if it was their line of sight to the horizon and walked around to make sure they knew what they were doing and corrected some misconceptions.

4. After they drew the line of sight to the horizon, they had "AH!"s of the right triangle possibility of finding the answer.

5. I asked them what information they knew or could find out, and we looked up the radius of the earth in miles on the Internet, and we recalled the number of feet in a mile.

6. Then before I set them on their way, I asked that if I was 5'4", how many feet tall am I and show me on your calculator as I walk around. I heard good conversations and them correcting each other. Then I got another kid to tell her height, and we did the same thing to make sure everyone got it.

7. Then I set them on their way to find how far each of them individually with THEIR OWN HEIGHT could see to the horizon. As I walked around, I saw too many kids doing everything in feet (because "I don't know HOW TO DO IT in miles")... they converted the radius of the earth (about 3960 miles) to feet!!! We had a discussion about how these are too large of numbers.

8. I stopped the class temporarily and did a mini-lesson. Let's say I have 53", how many feet is that. What about 38 feet, how many yards is that. Okay, let's say I have 5.25 feet, how many miles is that? They caught on. Then we had to have another mini-lesson on using our store button on the calculator instead of rounding early or writing EVERY BLESSES DIGIT out on paper and then retyping it in.

9. THEN when they'd calculated the length from their eyeball to the horizon, we asked the shortest kid what their answer was, and the tallest kid their answer. And we had to fix some kids' work and computation until they did it correctly.

10. Interesting span of possibilities of how far we all could see .... which I won't divulge in case you want to try it.


  1. OK, so if you can do the conversion, and you are on the west coast of something with a lot of ocean in front of you (sorry for you, sorry for me), there is a massively cool experiment.

    Don't start with the Earth's circumference or radius or anything interesting, just with the idea that it is spherical. We know it takes 24 hours (your kids now might say 1440 minutes or 86400 seconds) to make a full revolution.

    Near sunset. Lie down, face forward, and watch the ball dip. When the top goes under the horizon, jump to your feet, and count seconds until the ball dips again. And then do the math - I think getting the radius within 10% is more than realistic.

    Refinement: if you can climb up a lifeguard chair, cooler. If you have a little strip of beach, and can quickly run up to an overlook even cooler. Or you could do that with two people. By increasing the difference in heights, the time increases, and the percentage error decreases.


  2. That sounds like a great extension ... seems like it's ripe for someone to shoot a YouTube video with a timer on it ... or maybe have different heighted people with the times ..... Hmmmm, maybe a summer field trip to the west coast coming up for me :)

    Thanks for the idea,

    Ms. Cookie

  3. You might like the lesson "On Top of the World" at as well as the applet at

    I was always fascinated by the approximate formula d = 1.5*sqrt(h), where d is the distance to the horizon (in miles) and h is the height above sea level (in feet). I enjoy that the formula doesn't require a conversion from feet to miles or vice versa.