Other

This page contains other aspects of high school math, such as math reform, book reviews, magic, and so on.

Math Magic

Math magic is a good way to motivate divisibility tests and illustrate the power of algebra in explaining the magic.

The interesting number 1089, part 1: Magic1089part1  (tex)    Magic1089part1  (PDF)

The interesting number 1089, part 2: Magic1089part2 (tex) Magic1089part2  (PDF)

Mind reader (Is it magic, or just math?) how

Number trick demonstrated and explained

More mind reading: Fido Puzzle

Divisibility Tests: Divisibility .tex code   Divisibility  PDF version

Motivating FOIL: MotFOIL (.tex)  MotFOIL (PDF)

Richard Wiseman's "The Prediction"

Understanding the Prediction:  PredictionExp  (PDF)

Understanding "The Test": TheTestRW  (PDF)

Coin Flipping

The probability of flipping heads on a coin is not 1/2. The assumption that flipping heads on a coin is 1/2 is a mathematical model and not reality which is akin to using 3.14 for pi. Coin tossing is a deterministic process in physics as demonstrated by a coin tossing machine, "To make his point, Diaconis commissioned a team of Harvard technicians to build a mechanical coin tosser -- a 3-pound, 15-inch-wide contraption that, when bolted to a table, launches a coin into the air such that it lands the same way every single time. Diaconis himself has trained his thumb to flip a coin and make it come up heads 10 out of 10 times. But what he really wanted to know was whether unrehearsed tosses -- by ordinary folk who flip coins with unpredictable speeds and heights and catch them at different angles -- would show that the outcome of the act was, in fact, random." Persi Diaconis, Susan Holmes, and Richard Montgomery are authors of the article "Dyanmical Bias in the Coin Toss" (.pdf). There is a Numberphile video with Diaconis (about 8 minutes) that gives a brief overview and there is a YouTube lecture by Diaconis (about 55 minutes) with more detail. One of the main assumptions is that you start the coiin with the heads side up is

Mathematician William Feller was a well known expert in probability who wrote a classic book An Introduction to Probability Theory and Its Applications in which you can find (by click on "Look Inside") the following quote on page 19: "As a matter of fact, whenever refined statistical measures have been used to check on actual coin tossing, the result has invariably been that head and tail are not equally likely. And yet we stick to our model of an "ideal" coin even no good coins exist. We preserve the model not merely for its logical simplicity, but essentially for its usefulness and its applicability.".

The coin flipping model has two assumptions built into it:

  1. There are two outcomes (heads and tails)
  2. The two outcomes are equally likely.

The first assumption isn't always true. The Abstract of paper (by Murray and Teare) mentions the odds of an American nickel landing on its edge is about 1/6000.

The deterministic nature of coin flipping can be found in the Phys.org article "Heads or tails? It all depends on some key variables" which says:

"But first, here's what the researchers concluded: Using a high-speed camera that photographed people flipping coins, the three researchers determined that a coin is more likely to land facing the same side on which it started. If tails is facing up when the coin is perched on your thumb, it is more likely to land tails up. How much more likely? At least 51 percent of the time, the researchers claim, and possibly as much as 55 percent to 60 percent -- depending on the flipping motion of the individual.In other words, more than random luck is at work."

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