Friday, October 29, 2010

Scientific Notation: Big Numbers and Small Sizes

Next year I'll do scientific notation a few days earlier, right after we start working on exponents. This semester I followed the book's strange order just so that there'd be more time between mastery tests - students needed a break.

On Wednesday, I introduced scientific notation. We practiced converting between standard notation (writing numbers the usual way) and scientific notation. Most textbooks I've seen give rules that involve the words left and right. Being a bit dyslexic myself, I don't find those very helpful. I have my students tell me: big numbers have ten to a ... positive power, and small numbers have ten to a ... negative power. They were relieved to have a slightly easier topic, and enjoyed our introduction.

On Thursday, I wanted to work with them on multiplication and division, and on recognizing what to do in story problems. The size of the numbers makes using common sense hard, so I emphasize making up their own parallel problem, with the same structure but easier numbers. (Thanks, George Polya, for all your good ideas.) How many of this tiny thing in this biggish space? Let's think about how many 2 inch things in an 8 inch thing - oh yeah, divide biggish space by tiny size to get how many. (We got lots of practice on unit conversions in these problems.)

As I prepared for that class, I lamented my lack of internet in the classroom. I wanted to show students a number of sites. To help them understand big numbers, I started with the National Debt, which is currently around $13 trillion. I don't know about you, but I think my brain loses it somewhere between a million and a billion. ($1 million = a house in the hills, $1 billion = 1000 of those houses?) I can sort of see what a million dollars is, but a billion is just huge, and so is a trillion. So how do we get a feel for the difference between one huge number and another? This site shows hundred dollar bills.  A million dollars fits in a briefcase, a billion takes ten warehouse pallets, and a trillion ... the picture reminds me of the photo of the Better World Books warehouse.

(Terrible resolution. It looks better when I see it in my email. I got this copy from Google images. Did I mess it up somehow?)

I showed them a million plastic cups by getting it on my screen before class and then walking around the class, showing them my laptop screen. With proper internet capabilities, I also would have shown my class the Powers of Ten film (9 minutes long) and the Universcale.

Since I can't easily have the internet tell stories for me, I mostly had to do it myself.  I just read The Ghost Map, by Steven Johnson, and folks in the math department were dressing up as detectives for Halloween. So I told the story from this book of the detective work John Snow (a medical doctor and researcher) did in 1854, as he gathered evidence for his theory that cholera was spread through drinking water. Most scientists and doctors at the time thought cholera was spread through bad air (miasma), but cholera causes severe diarrhea, so Snow suspected drinking water. A severe outbreak of cholera hit the Broad Street neighborhood of London in late August, 1854, and over the course of just a few days hundreds of people died. Snow figured out that the Broad Street pump (no drinking water in the homes) was the cause of the contagion. I told this story in class, and we looked up the size of the cholera bacteria (a student's cell phone got us that much internet at least): 1.5 microns, which is 1.5x10^-6 meters.

I asked how many it would take to make a line of them across the room. We figured that out, and then found how long a line a trillion of them would make. We also figured out how long a line the burgers sold at McDonald's would make (over 100 billion sold).

Two out of three classes really got into it. (The afternoon class is a tough sell.) I had a great time with a topic that's usually been much less fun.


  1. For next year, here's another site that's similar to the pallets of $100 bills site: My students like the final page that shows various names for the stuff bigger than trillions, etc. (and how 1 Googol is still not the largest number we can write out).

  2. Thanks! The cows were hilarious. I'll point my students there (this year).

  3. Douglas Hofstadter as an article (which appears in Metamagical Themas) called "Number Numbness", in which he talks a lot about having intuition for large numbers, and related issues.


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