Wednesday, February 10, 2010

Slide Rules with John Ewing

The Math for America President John Ewing lead a workshop on slide rules last week, and I was really excited to have the opportunity to get to know him better and learn about slide rules. At 29, I had literally never seen a slide rule. I had no idea how they worked. I will be forever grateful to John for changing this.

Caution: I was confused off and on throughout this workshop because, never having seen a slide rule before, I had no idea which indices to look at when, or how to keep track of decimals when I chained operations. If you are like me you will need to play around with these a bit to figure them out.

Classroom benefits of integrated slide rule use:
- Estimation
- Scientific Notation
- Complex arithmetic
- Number sense
- Meta understanding of accuracy (how close to actual), precision (how reproducible), significance (related to precision…how many digits?) and difference b/w those.
- What makes an answer foolish. (Don’t need 100 digits to decide how much paint to use.)
- How functions work: increasing, decreasing, even concave up and down all were natural notions

John's questions:
How did the advent of technology and the handheld calculator affect the way that people approached and thought about calculation? How does it affect us? John is interested to think about why some things are harder or just different to teach today without slide rules.

Interesting History:
First handheld calculator, the HP-35, cost \$395 in 1972 (~\$2000 today)
People used to carry around book of log and trig tables…had interpolation tips as well to increase accuracy by 1 digit or so.
The Regents exam provided log tables

Gauss complained about the time consuming annoyance of calculation, and he’s famously good at it.

John Napier’s 1614 invention of logs was revolutionary. Before every calculation was done by hand. Napier got a really good feel for what the log function looked like (do high school students know what it looks like?)
Log xy = log x + log y (converts mult into add)
Log x/y = log x – log y (converts div into sub)
This is what was revolutionary, since addition & subtraction are way easier than multiplying and dividing.

Reverend William Oughtred (www.oughtred.org) used this trick to manipulate logs:
- Label rulers with numbers 1-10 but at log distances
- Developed within a decade of Napier’s work, but took two centuries to catch on
- Need increased in 19th century with engineering and war (cannon aiming) and so got popular in 1850, with the addition of the middle slider.

Tricks with slide rules:
- easy to chain calculations
- squaring things just means double log
- geometrically, just double the scale
- reverse to find square roots! (sq on A scale, root on D scale)
- cube the number on the K scale (and cube root backwards!)

Problems:
1. Want to paint a large sphere with radius 12.5. Paint label says 1 gallon covers 450 sq ft. How many gallons do you need?
2. Cylindrical tank has radius .82’. How tall should it be to hold 65 gallons? (NB: 1 cubic ft = 7.48 gal)
3. Have a tank 5.2’ high. Want it to hold 63 gallons. What is the radius? (uses sq root)

My inspirations after this:
- Teach decimal addition with just regular sliding rulers, play with measurement and perimeter. Use as a way of building meta-cognition about the sense of answers.
- Do a calculator correction activity, e.g.: Fix the problem: 42/85 = 2.0238
- Adapt what John did and run a math circle style workshop in which we re-discover the concept of sliding rulers to do arithmetic, the amazing transformation that happens when the rulers have logs on them instead of our regular rulers (I’d be interested in constructing a log table for that matter), how to chain operations, etc. Apparently real engineering slide rules have log log scales, so play with those too.

This pdf is what John gave us (kindly already cut out, just not folded) to make our very own slide rules.