Rationale
Enrico Fermi was an Italian physicist who, among other things, worked on the Manhattan project. As legend goes, during the Trinity test, where the first nuclear bomb was detonated, Fermi took a handful of strips of paper and dropped them into the oncoming blast wave. By pacing off how far backwards the strips were blown, he made an estimate of the energy released in the explosion. Using these seemingly crude methods, he estimated it was equivalent to 10 kilotons of TNT, within a factor of two of the true value.
A little bit below the surface, however, you realize his methods weren’t crude at all. Fermi was meticulous in everything he did, and his estimates were no different. His assumptions that formed the foundation of the estimates were clearly cataloged and defined. He exploited specific symmetries inherent to problems and drew upon his deep physical intuition to decompose the complex into the manageable.
Fermi’s ability to get very good results through a series of estimates is now the stuff of science education legend. That’s why Fermi questions now bear his name! He would give these questions to his students in an attempt to train them to think about problems that seemed beyond calculation by breaking them into pieces that can be reasonably estimated and then combined into an answer. His students drew on observations, experiences and logic to answer questions that, at first, they thought they couldn’t even scratch the surface of.
My students love to ask "when will I need this in life" and this unit attempts to give them something they can use every single day. The true value of a physics education comes in the ability to ask questions, make observations and solve problems. Not just math problems or physics problems, but real life problems. Should I rent or buy? What's this noise my car is making? Having the ability to step back and decompose questions into their parts puts answers and understanding within reach. This ability isn't intrinsic to certain persons, either. It can be taught explicitly and developed through experience.
Good number sense is fast becoming just as important in our data-driven society as basic literacy. Numbers are cited constantly in media and literature, but how deeply do our students really understand them? What's the difference between a million dollars and a billion dollars? Numbers so huge may seem out of reach to our kids. How much smaller is a bacterium than a grain of salt? Again, not a hard question, but the small size of both masks the vast comparative difference. Consider this: In our own bodies, bacterial cells outnumber “our” cells – the eukaryotic cells that carry our genes – by about 100 to 1, but their total weight is only a couple of pounds out of our 100 – 200. So while bacterial cells are orders of magnitude more abundant, this implies that they must be many more orders of magnitude smaller than our own cells. Number sense, especially when orders of magnitude are concerned, is a tough skill to develop and internalize. Physics provides the perfect platform for developing an initial number sense and cultivating a fundamental understanding of the powers of ten in students over the course of a year.
Once an understanding of orders of magnitude and relative size is established, the importance of precision becomes paramount. Precision and relative accuracy are concepts that are baked into the nature of science, but students rarely develop real competency in. How precise does an answer need to be? Is writing down more digits better? If the calculator gives me all these digits, I’d better write them down, right? To students, the answers to these questions seem obvious. To science teachers, these student intuitions are obviously wrong.
Fermi became known not just for his accomplishments as a physicist, but also for asking and answering big questions in a way that required a set of justified assumptions in order to arrive at an answer. Students can learn to take a question that, on the surface, seems impenetrable and break it into smaller pieces (or constituent quantities). These pieces can be related to each other by their units and arranged in an order that produces the desired result. Then students have to make a judgment and decide if each quantity should be looked up or estimated. When they estimate, students have to realize that they've thrown any hope of a precise answer out the window. Since different people may estimate differently, there needs to be a rationale behind the choice of numbers. Students have to be able to communicate the assumptions they've made to arrive at a solution, and to justify them. To answer Fermi Problems requires students to not only develop skills that aren't actively taught in other classes, but also shake free of most of what they've learned about solving problems. There isn't going to be an exact right answer and that causes students a great deal of stress, but the payoff is a robustness in problem solving and thought that isn't developed at any other point in secondary education.
These skills are all highly transferable. Estimation and number sense can partner to help students be more critical of statistics and more data literate. The understanding of quantities as they relate to the real world can contextualize them in a way that by-the-book pure math classes often fail to do. Routinely deconstructing problems can help students to be more independent problem solvers, by developing their ability to take the first steps toward finding their own answers to questions. Finally, all of these skills are moot if the results can't be effectively communicated and supported. This unit seeks to set students up with foundational skills that will lead to increased success in the physics classroom and beyond.
From the scenarios presented in the problems in this unit, students should get a better sense of what sorts of questions they can answer with the math they already know. The surprising result is that they have access to way more computational power than they’d ever expect with just unit sense, order of magnitude sense and some multiplication. Students are also going to need to develop a systematic approach to problems where the components they need to solve it aren’t immediately provided. Hard problems are separable into easier problems. Students can apply an iterative process of problem solving steps until they develop an intuition about answering questions.
When students invest time in thinking about the problems, they should obtain a sense of how situation and purpose inform an appropriate level of precision in the numbers they use. Significant figures are more than just a set of rules that artificially constrict the answers in a science classroom. As students look numbers up, measure quantities and estimate, they will be challenged to develop an understanding of where the idea of significant figures actually comes from and a sense of how to use them in a meaningful and intuitive way. Students will also need to examine the quantities they need and sort them into categories based on what qualifies as “encyclopedia knowledge” and can be looked up, what needs to be estimated, and what needs to be further broken down and calculated. As students deal with these different numbers of different orders of magnitude, they’ll also develop a sense of how they relate to each other.
Finally, from solving the problems, students will learn how quantities of different orders of magnitude interact with each other. Operationally, addition and subtraction will behave extremely differently than multiplication and division. How do numbers get bigger or smaller when combined? When do numbers have little to no effect when combined? Students will also need to understand why estimates are acceptable in some situations and not others. For some reason, students are irrationally tied to all of the digits that pop up in a calculator window, even when infinite precision is completely inappropriate. The unit will explore how to break them of this habit and also get students discussion the rationale and process used in solving problems, instead of just supplying a functionally meaningless string of digits.
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