Post by Maura, who taught for Ethan today.

We turned on the computers today and checked out Ethan’s website and this blog. I asked the class what they had done on Tuesday (some of them reviewed the blog – good for them). The students told me about the three different problems they had worked on. When I asked them what those problems had in common, they had some good answers: the examples were connected to the real world, to solve them we use estimation and count zeroes, we can solve a problem by breaking it in to steps, etc. Great.

We then put this into practice by working on problem 1.8.42 from the book asking about the average size of an e-book using information about the Amazon Kindle. Someone read the problem for us and then the students told me what the important information was. We had two numbers to work with: 1,500 e-books and 1.4 GB of storage. What does GB mean? Someone knew it stood for Gigabyte but they weren’t sure how big Giga was. I diverted into a brief discussion about metric prefixes, in particular kilo, Mega and Giga. We did a couple of examples of those and then returned to the problem, along the way reinforcing the idea that sometimes to solve a problem you need to look something up or figure out a definition. Another tool in the toolbox.

The question was to find the average size of an e-book. The students suggested two ways to solve the problem. One was to “cross-multiply”, setting up the equation *1/1,500 e-books = x/1.4 GB*.

They identified *x *as what we didn’t know – the size of an average e-book. Someone else suggested that we solve the problem by dividing: *1.4 GB/1,500 e-books.*

We discussed this and agreed that it worked out be the same result. I asked them how they knew to set it up this way and one student talked about the answer we would get in terms of the size of an e-book. That is, we expect e-books to be somewhat large (maybe a megabyte) and that setting up the division this way gets us an answer that makes sense. In other words, we need to think about the problem as we are solving it.

We did the division two ways. Some of them estimated the number of e-books to 1,400 to be able to divide easily: in that case we get the average size to be 1,000,000 bytes or 1 Megabyte. Others used the Google calculator and got 933,333.333333. Then we talked about these answers. Does it make sense to use all the 3s in the Google answer? Not really, they said, since the original numbers were estimates and because we are estimating. We agreed that we could report it as 930,000 or even 900,000 bytes (or 0.9 MB). I think some of them were a bit concerned that there were two “different” answers. More on that later.

We spent the rest of the class time on the carbon footprint exercise. I showed them the chart from *The Boston Globe* about the carbon footprints of different activities (cheeseburgers, running a load of dishes, etc.) and then asked each row of students to take an activity and estimate the total carbon footprint of that activity in the U.S. in one day, then move on to the next activity. They needed consistent guidance from me and the tutor but overall I was impressed with how well the students worked. Each group made it through at least two of the activities. It was interesting to watch them work. At first several groups wanted to use the 5% of the population figure that was in the book. I explained that this was for the orange juice example and that they needed to make their own estimate for their activity. Quite a few of them also turned to the internet for solutions. We talked a bit about how useful or distracting that can be and how the exercise really is looking for them to *think* about the question and make a reasonable estimate. After a good 15 minutes of chatter they put their answers up on the board and we talked through them. For the cheeseburger activity one group had 180 billion grams while the other group had 125 billion grams. I asked each group to explain their reasoning and they both had estimated the percentage of the population who eats a cheeseburger in a day, based on their experiences and what they knew about other people. It took some discussion, but the class did seem to get the idea that these two estimates were in the same ballpark and so were consistent. The other activities were also surprisingly similar and for the most part were the same order of magnitude. Even the people who started with a number from the web then did an estimate based on that number. Again, the idea that the answers can both be correct, even though they aren’t equal, is a challenging one. But the exercise went every well and I was pleased both with the effort they put into it and the good thinking that they did.

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