The other day, I wanted to start easing my sixth graders into estimation (before diving into Andrew's estimation180.com), so I put this up as a warm-up:

After we talked about some estimation strategies such as using your hand as a guide (see picture below) and identifying lower and upper limits of reasonable answers, many were eager to try another problem. As each of my classes is only 37 minutes in length (crazy, right?), I told them that we could try another one the next day.

"I'm Crushing Your Head!"

So, here's the problem I gave them the next day...

And sure enough, their guesses were much more informed. As with yesterday's estimation, I made line plots for each class's data and we could see that many more students were closer to the right answer. As a class, we felt that progress was made.

And then came the beauty of the line plot itself. For every class, I asked: what do you notice? In one particular class, we noticed that the data points were spread out. In another class, we saw that we had outliers. In another class, we saw that somebody guessed 18 inches, so they really must have been thinking that the rectangle was a square. In another class, we noticed that the data was skewed to the left or closer to a bell curve. In many of the classes, we noticed that students typically underestimate (which I'm very interested in understanding why, but I'm not going to delve into that here).

Later in the day, I noticed that the data from one class was very similar to a previous class. So I put both data sets up, and all of a sudden, we weren't just evaluating different students' guesses, but two different data sets. Finally, I added a third set, and we started having discussions about which class guessed the best. And the kids were really into it and coming up with some interesting ideas about how to determine the best class.

And I thought, this is awesome. Not only are my students driven to become better at estimating, but now they're looking at using math to help figure out if they're getting better at it and if they're better than somebody else. (They're downright vicious when you throw a little competition their way.)

By the way, the answers to the two estimation challenges are: The first rectangle is 12 inches by 32 inches. The second rectangle is 18 inches by 26 inches.