Mathematical Gems


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Math at FlexTech is dominated by students burying their noses in ALEKS. I prefer to be a guide on the side teacher, but once students get into ALEKS, most become fixated on the process of completing topics and completing their pie. ALEKS provides no motivation behind the mathematics. This leaves me frustrated and the students bored.

I've started adding a short vignette to each math class, and it is remarkably effective in getting their attention. It adds some balance to their education, and they are ready for someone to introduce an idea about math that they can then think about.

  1. Given a square with side length and area defined to be Show that this implies that the area of a rectangle with side lengths and is
  2. Archimedes' demonstration that given the definition (circumference of a circle), then using a clever argument that transforms the area of the circle into an area of a rectangle. This presaged integral calculus.
  3. Magic squares. I showed them a typical then gave them a partially filled square to see if they could recreate the square. On the next day...
  4. Game day! Tic-tac-toe, followed by the Sum-to-15 game: players take turns choosing from the numbers 1-9. Numbers can only be chosen once. Objective is to have 3 of your numbers sum to 15 before you run out of numbers. See if they connect this with magic squares.
  5. Constructive procedures for magic squares. Dazzle your friends with a magic square, for example.
  6. Wisdom of crowds problem. Choose a topic the students have some knowledge about (e.g. the number of downloads of a particular well-known video). Collect observations, take the median. Yes, this might not work. But odds are good that the median is a good estimate.
  7. Prove that the diagonal of a square has length One method: use two unit squares cut along the diagonal, and rearrange the four pieces to make a new square with diagonal of length 2.
  8. Show that is irrational using a numeric argument. Show again via reductio ad absurdum via geometric argument using a right triangle with integer side lengths and such that
  9. Show the primes are infinite.
  10. Ask how we know a set has the same number of members as another set (e.g. 1-1 correspondence).
  11. Show there are just as many even natural numbers as there are natural numbers.
  12. Show there are just as many integers as there are natural numbers.
  13. Show there are just as many rationals as there are natural numbers.
  14. Show there are more reals than there are rationals.
  15. Can the Hilbert Hotel accommodate more guests?
  16. Review rules for identifying whether a number is divisible by 2, 3, 5, 11. As for 7...
  17. Casting out 9s to check addition and multiplication.
  18. Short division. For example, show the students how they can calculate without the usual sequence trailing down their page. This prepares them for factoring...
  19. How many factors does 1260 have? Connect this to combinations.
  20. What is the sum of all the factors of 1260?
  21. What fraction corresponds to How about And
  22. Monty Hall problem.
  23. Bits (binary digits), bytes, kilobytes, megabytes, terabytes, petabytes.
  24. Looking the other way, ask the students to pick two numbers and write them on the board (for your sake, you might limit them to integers with fewer than 4 digits). Then have them create a Fibonacci sequence of 10 numbers (i.e. compute the next 8), listing the numbers in order. Have the rest of the class help with the addition. Then ask the students to sum the 10 numbers, but not to write the sum on the board. When they are done, glance at the board and tell them the sum of the 10 numbers. Or glance at the first two numbers, write the sum of the 10 on a piece of paper, and reveal that paper when the students are finished. If you can't figure out the trick, e-mail me.
  25. More Fibonacci sequence. Explore, then use spreadsheet to compute ratios of consecutive terms, while changing starting values.
  26. Continued fractions. . Compute with spreadsheet. Then compute with algebra.
  27. The sum of geometric series uses a similar trick to the continued fractions and repeating decimals problems. Illustrate the infinite series with a video of highly elastic LiquidMetal.
  28. Gauss and
  29. If every student in the room gives a high five to every other student in the room, how many high fives are there?
  30. Pigeonhole principle. How many socks from a drawer with sock colors are need to guarantee a match?
  31. If there are people in a room high-fiving each other, is it true that at least two people have high-fived the same number of people?
  32. Simpson's paradox.
  33. Russell's paradox.
  34. Find shortest/cheapest path through directed graph having edge costs.
  35. I love it when students say imaginary numbers are useless because they are "imaginary." Aren't all numbers imaginary? If 3 is real, how much does it weigh? Where does it live? Maybe 3 is merely a useful idea—just like imaginary numbers, which are pervasive in physics, probability theory, and computer-generated art. This is a good chance to discuss the Mandelbrot set, and how it is generated.

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