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.

Larry Zimmerman strikes again

I wanted to share another few excerpts from Larry. He gave a talk last Tuesday at the NY Math Circle's PD for Middle School Teachers, and I was impressed again by how he asks questions to extend thinking about a problem, offer more entry points, and deepens my interest. He does it so quickly, I can't help but get interested and then my mind is thinking about 10 different but connected things at once, and whatever I get into relates to the bigger picture, and it's like magic differentiated instruction and guided inquiry. It's really awesome. We did a bunch of problems, and here are three that I wrote down his questions for:

Show that the sum of two odd numbers is an even number.

What is an odd number?

What is an even number?

Is 4 even?

Is 0 even?

Are there any even primes?

What’s the next even prime?

Are all prime #s greater than 2 odd?

Are all #s greater than 2 prime?

Is -4 even?

Is -6 even? Why?

So now, what is an odd number?

Is every even a multiple of 4?

Show that no perfect sq ends in 3.

Do you believe it?

Do any end in 6?

Did you know that every sq ends in 6 iff the number’s 10s digit must be odd?

Would anyone like to see a proof of that? Good! Go do it.

(I was curious to observe here that every sq is

- a multiple of 5

- one less than a multiple of 5

- one more than a multiple of 5)

Show that x² – y² = 2 has no solutions in integers

Translate this into words.

Say this in words: x² + y²

(x + y)²

x² + 5x + 6 = (x + 2)(x + 3) Find the value of x that makes this false.

Factor it using diff of squares: (x + y)(x – y) = 2

Note: remember given info

Integers factors must be 2 & 1

“I wanna savor that for a minute. To some people that sounds trivial, but to me it sounds profound. Never dismiss the obvious as being trivial. Even if it’s obvious it may be very very important.”

x + y = 2 and x – y = 1 (or other way around)

“Do something, do anything, and if it doesn’t work, SO WHAT?”

Larry Zimmerman Quotes & Notes

“It is perhaps more important to be able to compose problems than to solve them.”
“Problem creation is the essence of mathematics.”
“The first effort is rarely if ever the final product.” (we must emphasize editing)
“Creating problems is theme and variations.”
“Determine what you are looking for and WRITE IT DOWN.”
“I don’t know what I want them to realize.”
“Beware of reinforced suspicion,” which is not a substitute for problem solving.
“If I’m 100% I will not forget it, I write it down.”
“There is nothing wrong with clarity by redundancy.”

Types of Problems
• Surprising
• Novel
• Fruitful
• Charming, imaginative, alluring
• Historically significant
• Haunting, musical
• Beautiful, elegant, sublime

Elements of Problems
1. a goal (construct, prove, maximize, minimize, classify, compare, compute)
2. given information
3. special rules (sometimes)

Some Big Ideas
- Plant a seed and then walk away until someone says something. Students and teachers struggle with silence, but it’s important and it saves time.
- Keep a notebook (both teachers and students) of interesting questions.
- We are well versed at turning a lot of words into symbols. Do the reverse.
- There are 3 varieties of equations: identities (always true), conditionals (sometimes true), and contradictions (never true)

Mary O'Keeffe Best Games Ever

Number Bracelets Game (adapted from Marilyn Burns at
http://www.geom.uiuc.edu/~addingto/number_bracelets/number_bracelets.html)

Imagine that you have lots of beads, numbered from 0 through 9, as many as you want of each kind.

Here are the rules for making a number bracelet:

• Pick a first and a second bead. They can have the same number.
• To get the third bead, add the numbers on the first and second beads. If the sum is more than 9, just use the last (ones) digit of the sum.
  To get the next bead, add the numbers on the last two beads you used, and use only the ones digit. So to get the fourth bead, add the numbers on the second and third beads, and use the ones digit.
  Keep going until you get back to the first and second beads, in that order.
• How long (or short) a bracelet can you make?

Other Questions:
How long is the longest?
How short is the shortest?
How many number bracelets are there?
- How many different starting points?
- How many different bracelets?
Does it necessarily repeat?
Extensions:
Change the mod/cutting number
Change the rule (from Fibonacci to some other sequence)
Why no bracelet of 6?
Why no bracelet of 5, 10, other factors of 60?
What if only use even numbers between 0 and 9?

Prime Number Monopoly
Materials: 2 dice, 5 cards (one for each of the first 5 prime #s) with the following costs and deed contracts:

Prime Number Deed:
Cost:
2 – $250
3 – $167
5 – $100
7 – $71
11 – $45
The owner of this deed
splits the winnings
when another player gets a multiple of:
(prime number here)

Game: Roll two dice and can choose the amount of either number formed by those two digits. Can buy prime numbers. Then whenever another player chooses a number that is a multiple of your number, they split the $ with you.

Goal: Get as much money as possible (there will be times when it will be to your advantage to choose the smaller number). Play for a set amount of time, clearly stated in advance.

Questions:
How did I work out the pricing?
What’s the best prime real estate buy?

Mary O'Keeffe Quotes

“Math can and should be a social experience.”
“Get people comfortable with making mistakes and taking intellectual risks in public.”
“Give people powerfully compelling activities they will want to share with others, who will want to share them with others, etc.”
“Best way to learn is to share things with others.”
“I like to blurt out things when I’m not at all sure that I’m right.”
“We respect respect and we correct mistakes.”

Mary O'Keeffe Book Recommendations

About Teaching Mathematics by Marilyn Burns
Aha! Gotcha by Martin Gardner
Aha! Insight by Martin Gardner
The Annotated Alice by Martin Gardner
Feynman and Asimov
Melisande by E. Nesbit
Mathematical People (interviews)
More Mathematical People (interviews)
Raymond Smullyan logic stories
Calculus By and For Young People by Don Cohen http://www.mathman.biz/

David Hankin Questions, many of them good & big ones

• What is geometry?
• When we study geometric figures, what are we concerned with?
• When we study geometric figures, what are we not concerned with?
• What kinds of motion (transformation) keep a figure “the same”?
• What is the maximum number of these motions (isometries) are necessary to get from one figure to another if they have the same orientation? If they have different orientation?
• What minds of motion change a figure?
• What is an angle?
• Given a diagram, what do you think might be true? (rather than given a diagram and some true things, prove this other thing)
• What is symmetry?
• Why do we study almost exclusively symmetric figures in geometry?
• Is a line segment symmetric?
• Is an angle symmetric?
• Are triangles symmetric?
• What can we say about the points on a perpendicular bisector?
• What happens when we find the perpendicular bisectors of the sides of a triangle?
• When is the circum center ON (/in/out) a triangle?
• What happens when we find the angle bisectors of a triangle?
• How many lines of symmetry does a quadrilateral have?
• Is it possible for a figure to have more than four lines of symmetry?
• How many lines of symmetry does a circle have?
• How do I draw a symmetry line through a circle?
• What happens when we draw the diameter through a point on a tangent line?
• What happens when we draw perpendicular chords through a diameter?
• What happens when we draw progressively smaller and smaller perpendicular chords?
• Problem: Given a triangle and side lengths, find the lengths of the segments formed by the incircles.
• Problem: Given a quadrilateral and side lengths, find the lengths of the segments formed by the incircles.
• Must a quadrilateral have a circumcircle?
• Reflection Problem: Given two points A & B, and a line (below them), what is the shortest path from point A to the line and then to B?
• Reflection Problem: In pool/minigolf, where do you aim so that A bounces once (twice, 3 times, 4 times back to A!) before hitting B? When are these possible/not?
• What has translational symmetry?
• Translation Problem: Two towns on opposite sides of a 1 mile wide river. Where should we put the bridge (to minimize the distance)? What if there are two nonparallel rivers?
• What letters are preserved under 180 rotation (/half turn/point reflection)?
• What geometric figures are preserved under point reflection? (which have point symmetry?)
• What geometric property follows from the fact that the letter Z has point symmetry?
• Point Reflection Problem: two circles intersect. Draw through the point of intersection a line which creates congruent chords in each circle.

David Hankin Quotes

1. Majority rules in math right?
2. Did we prove it or did we just all agree?
3. Math is not about computation, but about proving things.
4. When something works well in math, you should milk it.
5. Geometry is something dynamic, not a bunch of stationary objects. Transformations help us see this. If you’re good at geometry, chances are you moved stuff around in your head.