# The Setup

One time I made my students measure the heights of buildings around our school with an engineer’s transit. They had to use trigonometry. I was a pretty great teacher so I made them take all of their measurements multiple times. Man, they loved it. It was a great activity because I give out points for each successful thing they do, which really motivates them. Everything is on a scale of five to ten (I have a progressive attitude about grades), so when a team gives an answer I give them some points, and at the end of the week we average them to get final grades. (side note: this is extra great because I can give students feedback pretty quickly)

Anyway, when they were done we did some serious analysis on the board. One team measured the gym, and they took four separate measurements. They found that the gym building was 40, 39, 42, and 39 feet tall. To really make the significance of that stick I made a spark graph.

I gave that team three 9/10s, one for each incorrect measurement, and one 10/10, for the correct measurement of 40 feet.

The other team measured the science building, and I guess they were slow or something because they only measured it twice – 19 and 21 feet. In the end it worked out because we could still make a spark graph. Two 9/10s.

I was starting to think that the spark graphs weren’t that useful, but luckily this was on a smart board so I could have one of the students come up and drag them around.

We figured out that the science building was about 20 feet tall and the gym was about 40 feet tall.

Here’s what I couldn’t believe, though. I asked them what the average height was, and here’s what they did:

Whoa guys! I know the scale is supposed to start at 5 but I’ve got to give a zero for that. Very disappointing. Are you sure you’re in the right class?

# The Problem

The students couldn’t understand that the numbers they collected shouldn’t just be added up and averaged together. I mean, you can add up numbers and average them, but you have to understand what you’re doing. Averaging is an algorithm that really only works for equal, independent measurements of the same thing. Obviously the average height of the buildings is 30 feet. We have to make sure that the numbers we’re sticking into our averaging algorithm are actually compatible. Even though all six of our measurements were in feet… some are measurements of the science building and some are measurements of the gym. If you average them you get the average measurement, not the average height!

# The Punchline

Overall, I think the activity was a success. After I added up the points everyone got during the day and averaged them together, everyone had over 90%! Then I added up the points they got on homework and tests, and averaged that in too. Finally, I added up all the points that everyone had earned in the whole week and averaged those together, and I had a final grade of B+ for the class overall – pretty good!

Afterwards, someone asked me what they needed to work on to improve. I looked up their grade and saw they had an 85%, so I suggested they try to get more points the next time I asked a question. I love that self-motivation that points systems provide.

But I was the proudest when the director of maintenance heard that our class had been measuring the buildings. He actually came into the class to ask some advice! He needed to get new ladders so that he could easily repair the roofs of the building, and almost my entire class could easily answer that he should get – you guessed it – the thirty-foot model!