Overassuming Linearity

Sue reports:

One student wanted to average the two angles at 16 degrees each. Another said the observer could stand on a stool to be a little higher, so the angles would be 16 degrees each. Their answers were very close. There were other good (but wrong) methods that all came down to assuming this relationship was linear in a way that it’s not. Since their answers were very close, it was hard to help them see what was wrong with their reasoning.

She ends with “Can anyone help me here?” And she’s got some good comments. Here’s one from Kate Nowak:

This example you’ve picked is a toughie, for two reasons. As you point out, 18 and 14 degrees are both darn close to 16 degrees, so if you make that simplifying assumption 1) the problem is much easier and 2) the answer is pretty close to what it should be. But, you have to realize you are making a simplifying assumption that will throw your answer off a little bit, which I take it was not what your students were doing. They thought the result should lead to the exact, correct answer.

Anyone here have any ideas to send her way?

6 replies on “Overassuming Linearity”

I will confess I haven’t done the math on this, but I think if we make the height bigger, then the off-ness becomes a very big deal. So… what if you presented your method using the 18 and 14 and had students compare with their 16 and 16 method… and then asked “What if the flagpole were the Empire State Building? What if it were Mt. Everest?” and found out how much of a difference the 16-degree assumption made?

Another extreme to go to, I guess, would be to change how high up she was and thus the angles of to the top and bottom of the triangle.

Finally, making a dynamic sketch and changing the angles and watching what happens to the distance to the flagpole would probably be most illustrative. Do you have access to iPads with Sketch Explorer or computers with GeoGebra or Sketchpad?

That is a good idea! I think I’ll play with this on Geogebra this weekend.

I’d also like them to get practice distinguishing when we can use proportional reasoning and when we can’t. I want to compile a collection of tempting situations.

My idea is basically the same as Max’s but make the distances MUCH bigger, like on the order of light years. “Imagine we are trying to figure out how far apart these two stars are…” or something similar.

As regards the stand-on-a-stool idea, I think the key fact in their reasonging is that their assuming that the total angle (14º+18º) is constant regardless of the height of the line of sight, and that’s not true. This is easily seen in the extreme case when the height of the line of sight is really above the line of the flag, for in this case the total angle is very small. In fact, the total angle is a maximum when the line of sight lies right in the middle of the flagpole, so that the angles of depression and elevation are equal.

Yes! I only went to the top or bottom of the pole, so the decrease in angle size wasn’t yet obvious. I don’t know what stopped me from going higher (or lower). I think this is the easiest way to show students that the angle sum will change.

If you change the angle to 16 degress, then you are actually calculating the length s = x*sec(pi/90). Since sec(pi/90) = 0.99939, their answer will be close, but not exact. The percent relative error being 0.061%. If this is a tolerable amount of relative error, then their answer is acceptable, but if they are suppose to have an exact answer, they are obviously wrong. Hope this helps

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