Day 04: Heart of Mathematics

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Announcements:
- I understand that some of you were concerned enough about the Haitian situation to ask whether that had caused my absence. Thank you! It is a continuing concern, which may call me away ultimately.
- You owe me our first homework assignment....
- Assignment (hand this in Wednesday, 1/27): Estimate the following (remember our classroom list of the fundamental mathematical elements of an estimate):
  1. How much do you spend in groceries for a year?
  2. How much gasoline do you use in a year? (CNN website) What percentage of your budget is fuel?
  3. How many times will you breathe today?
  4. How tall is the Lucas Administration building?
  5. The southbound lanes of I-471 are backed up from the Big Mac bridge at the Ohio river, to Nunn drive. How many cars are stuck in traffic on that section?
  These should not be "one-liners": you should give reasons for things, justifications, draw pictures, use data, etc.
Review: A glance back at the games: how did they go?
- Cannibals and Missionaries
- Dot of fortune...
Now: we need to finalize the plan for the semester. I asked you to take a look at the book, and see if you have any topics you'd really like to cover. Do I have any candidates?
Today we start our unit on Estimation, with Section 2.1: Number Contemplation
- How did you like the reading of the section? Any new feelings about the text?
- Motivation: We have trouble estimating (I found this article a couple of years ago, which indicates that we have this problem generally, especially in financial situations).
- In this section we find an estimate of hairs on a human body (among other things).
  - How do our authors make their estimate?
  - Do our authors seek to estimate the number of hairs well?
  - What is the objective of the authors?
  - What is "the pigeonhole principle"?
    - Have you ever played a game, perhaps at a birthday party as a young child, that relies on the pigeonhole principle?
    - How do we know that there are two people on earth who have exactly the same number of hairs on their bodies?
      - Why do the authors exaggerate?
      - World Clock for Population, Illness, etc.
- What are some of the fundamental mathematical elements of the estimate?
  1. Collection of (quality) Data
  2. Approximation of the body by geometrical objects with known characteristics.
  3. Units and unit conversion
    - We need useful referents
    - Example: a dollar bill
    - Is one of your fingers three inches long?
    - Is a knuckle an inch long?
    - How many centimeters in an inch?
    - How many kilometers in a mile? (factors of 10 and English unit conversion)
  4. Computation (putting it all together)
  5. Bounding (that is, capturing the true value between two values)
  6. Sometimes it pays to be rough!
  7. Can we do a better job of estimating?
- Two things are important: an estimate and a bound (some idea of how bad your answer can be): e.g.
  A CNN-Opinion Research Corp. poll of 1,346 adults conducted July 31- August 3, 2009 found that 50 percent favored Obama's Health care plan, but 45 percent opposed it. The results had a margin of error of plus or minus 3 percentage points. (source)
- Problems to consider (exercises in estimation):
  1. How many ordinary dice would it take to fill this classroom?
  2. Do you have a temporal twin (someone who was born on the same day as you, and will die on the same day)?
    - Is there anyone who obviously does?
    - Is there anyone who obviously doesn't?

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