__Introduction__

I like coincidences. Particularly when they provide learning opportunities for my students. We had just spent time learning about percent and percent increase and decrease. The problems in the text were good but not really challenging. The coincidence was that I was reading John McPhee’s *The Curve of Binding Energy *(1974) and I’ll start the story with what he said on page 18.

__The Story__

“Thousands of miles of tubes, pipes, and other conduits were needed to create a network of flow wherein the gas could now go through a membrane, now return to try again, now go on to a new membrane, gradually advancing, in a process of separation and elimination, until what had begun as seven-tenths of one percent U-235 was more than ninety percent U-235 – fully enriched, weapons-grade uranium.”

I’d never heard this detailed an explanation of how weapons-grade uranium was made. But what really got my attention was that his statement could be a percent increase problem. I worked it out before I gave it to the class, rounding off the initial “more than ninety percent” to a manageable 90%.

Further, I decided that it would be an in-class extra credit exercise and allowed that the students had to first work within their assigned group, but once they had an answer they could discuss it with other groups.

I did this because the percent increase is 12,757% and this size percent increase would cause the groups to question what they did, even if they got that number. There were occasional answers to the problems in the text that resulted in percent increases of more than 100% but nothing quite like this. Once I gave them the problem and answered any preliminary questions and they got to work, I roamed the room listening to the strategies they came up with to do the problem.

The first issue was how to numerically express seven-tenths of one percent. One group asked if they could talk to another group to get help expressing it. So, I stopped the class and said that if they are willing to accept the following condition, they can work as a class to get the answer. The condition was that everyone in the class would get the same grade. They accepted. There was an eruption of conversation and as I roamed around, I was asked if what they got was right. I just referred them to other members of the class.

Once there was consensus on how to represent the initial percent, they simply continued with what they had learned about setting up percent increase problems. By the way, I taught a somewhat non-traditional method that doesn’t use a formula, rather it uses a somewhat modified percent proportion approach. You can look at it in this blog at *Percent Proportion*.

Several groups quietly called me over to show me their result, asking if they were right. Some were and some weren’t but I wouldn’t say yes or no, reminding them of the condition under which they were working. So, more talk, discussion and exchange of how to set up the problem.

It was interesting to watch the evolution of the shared work – people got up and moved around the room; some asked to and did use the white board; I heard a lot of “show me” and “why did you do that?” and “that doesn’t seem right”. But, ultimately there was class consensus on the right answer.

They did, however, insist that I walk through how I thought about it even though they got it. So, I put on what I call my “slow-motion-math” hat and gave them the following:

Ninety percent is 90/100, so the amount of increase is 90 – .7, or 89.3%. Seven-tenth of one percent is (7/10)(1/100) or 7/1000 (I did this because I saw a lot of questioning on how to express it). This in percent is .7/100 (or if you were sure, you could have just written .7 over 100). So the question can be put in a percent increase frame. First, the amount of increase is 89.3 and since it started at .7, the amount of increase relative to the beginning point can be expressed as 89.3/.7, or 893/7 (they questioned if doing this would give the same answer and we discussed this). Using the proportion statement 893/7 = x/100 gives a percent increase of 12,757%, rounded. So, doing the original problem led to some other related talk about fractions, decimals and rounding. Neat.

After all was said and done, I got questioned about this exercise because there was a sense that it was a trick question. I have noticed that when students feel uncertain about a math problem, the frequently asked question is just that. I then heard stories from the class about their prior math experiences where trick questions unfortunately were used to presumably teach them something about math, but the only learning was frustration because a lot of the tricks were beyond the bounds of what had been taught and in essence they quit. Given what I heard, I may have quit too. Somehow they concluded that math is just knowing the right tricks.

But once they were accepting that it was an interesting problem, I noted to them that as they read books, magazines, watch TV or come across “mathy” stuff, they might play with it as we did with this problem. And of course they noted to me to record the “A” for all of them.