Day 03: Heart of Mathematics

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Announcements:
- Your first homework assignment (fun and games) is due Monday - any questions?
- Assignment (hand this in Monday, 2/2) : Estimate the following:
  1. How much do you spend in groceries for a year?
  2. How much gasoline do you use in a year (gallons)? How much do you spend on fuel each year? (This CNN website may prove useful.)
  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?
  Your answers should be typed (although figures and diagrams may be hand-drawn).
A glance back at the other games: any questions?
- Dodge Ball
- Dot of Fortune
- Sir Pinsky
- Birthdays example
- Others?
Today we start our unit on Estimation, with Section 2.1: Counting (part of the chapter on Number Contemplation)
- Motivation: We have trouble estimating
  In this article on personal finance, the author asserts that
  "What's really surprising is that people are almost always wrong in the same direction. They underestimate the benefits of saving and they underestimate the costs of borrowing."
  (We'll talk more about personal finance in a unit later in the semester.)
- So let's see how well you do at estimating:
  I'm going to start off with a challenge, which you should undertake with a partner or two: problem 6 from Test 1 of the fall semester, 2008
- In section 2.1 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 are some of the fundamental mathematical elements of the estimate?
  - Collection of (quality) Data
  - Approximation of the body by geometrical objects with known characteristics.
  - Units and unit conversion
    - We need useful referents.
    - 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)
  - Computation (putting it all together)
  - Bounding (that is, capturing the true value between two values)
  - Sometimes it pays to be rough!
  - 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 1,041 adults conducted July 27-29 found that 62 percent favor a timetable for withdrawing U.S. troops from Iraq, but 37 percent oppose setting deadlines. The results had a margin of error of plus or minus 3 percentage points.
- What is "the pidgeonhole principle"?
  - Have you ever played a game, perhaps at a birthday party as a young child, that relies on the pidgeonhole 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.
- 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? That is, is there someone who was born on the same date as you (i.e. within 24 hours of you), and will die on the same date (within 24 hours of you)?
    - Is there anyone who obviously does?
    - Is there anyone who obviously doesn't?

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