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The Heart of Mathematics

The Heart of Mathematics. An invitation to effective thinking Edward B. Burger and Michael Starbird. Chapter 1 Fun and Games An introduction to rigorous thought. Make an earnest attempt to solve each puzzle. Be creative. Don’t give up: If you get stuck, look at the story in a different way.

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The Heart of Mathematics

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  1. The Heart of Mathematics An invitation to effective thinking Edward B. Burger and Michael Starbird

  2. Chapter 1Fun and GamesAn introduction to rigorous thought • Make an earnest attempt to solve each puzzle. • Be creative. • Don’t give up: If you get stuck, look at the story in a different way. • If you become frustrated, stop working, move on, and then return to the story later. • Share these stories with your family and friends. • HAVE FUN!

  3. Lessons for Life • Just do it. • Make mistakes and fail, but never give up. • Keep an open mind. • Explore the consequences of new ideas. • Seek the essential. • Understand the issue. • Understand simple things deeply. • Break a difficult problem into easier ones. • Examine issues from several points of view. • Look for patterns and similarities.

  4. Story 1.That’s a Meanie Genie

  5. Story 2. Damsel in Distress

  6. Story 3.The Fountain of Knowledge

  7. Story 4. Dropping Trou

  8. Story 5. Dodgeball

  9. Story 6. A Tight Weave

  10. Story 7. Let’s Make A Deal

  11. Story 8. Rolling Around in Vegas

  12. Story 9. Watsamattawith U?

  13. Chapter 2Number Contemplation Arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number… PLATO

  14. Section 2.1: CountingHow the Pigeonhole Principle Leads to Precision Through Estimation Understand simple thing deeply.

  15. Question of the day How many Ping-Pong balls are needed to fill up the classroom?

  16. The Hairy Body Question Are there two non-bald people on the Earth who have the exact same number of hairs on their bodies?

  17. Johnny Carson Johnny Carson was the most watched person in human history. Estimate the total number of viewers who watched Carson over his 30 year reign on the Tonight Show.

  18. Pigeonhole Principle Why are there two trees with leaves on the earth with the exact same number of leaves? Why does every person have many temporal twins on earth, that is, people who were born on the same day and will die on the same day?

  19. Pigeonhole Principle State the Pigeonhole Principle in your own words.

  20. Section 2.2: Numerical Patterns in NatureDiscovering the Beauty and Nature of Fibonacci Numbers There can be great value in looking at simple things deeply, finding a pattern, and using the pattern to gain new insights.

  21. Question of the day What is the next number in the sequence? 1, 1, 2, 3, 5, 8, 13, 21, ___

  22. Pineapples List as many observations about the pineapple as you can.

  23. The Daisy Count the spirals in a daisy.

  24. Comparing Numbers The pineapple has two sets of spirals: 8, 13 The daisy has two sets of spirals: 21, 34 Compare these numbers: 8, 13, 21, 34 Do you notice a pattern?

  25. Noticing a pattern Find the next two numbers in the sequence: 8, 13, 21, 34, ___, ___

  26. More of the pattern… What numbers must have come before 8, and how many numbers before 8 exist? __?__, 8, 13, 21, 34, 55, 89, …

  27. Fibonacci Numbers The following sequence of numbers are called the Fibonacci Numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

  28. Comparing Fibonacci Numbers Compare the size of adjacent Fibonacci Numbers. What do you notice? Compare 1 to 1 Compare 1 to 2 Compare 2 to 3 Compare 3 to 5 Compare 5 to 8… and so on.

  29. Fibonacci QuotientsFind each quotient. What do you notice?

  30. What number do we get? As the Fibonacci Numbers in the previous quotients get larger and larger, what number are we approaching?

  31. Express each non-Fibonacci Number as a sum of non-adjacent Fibonacci Numbers 1 = Fibonacci Number 2 = Fibonacci Number 3 = Fibonacci Number 4 = 1 + 3 5 = Fibonacci Number 6 = 1 + 5 7 = 2 + 5 9 = 1 + 8

  32. Express each non-Fibonacci Number as a sum of non-adjacent Fibonacci Numbers

  33. Unending 1’s

  34. The Golden Ratio

  35. The Golden Ratio

  36. The Golden Ratio Solve this equation for phi!

  37. Fibonacci Nim Rules: • Start with a pile of sticks. • Person one removes any number of sticks (at least one but not all) away from the pile. • Person two removes as many as they wish with the restriction that they must take at least one stick but no more than two times the number of sticks the previous person took. • The player who takes the last stick wins.

  38. Section 2.3: Prime Cuts of NumbersHow the Prime Numbers are the Building Blocks of All Natural Numbers Are there infinitely many primes, why or why not?

  39. Question of the day Can you write 71 as a product of two smaller numbers?

  40. Write the following numbers as products of smaller numbers other than one. 12 21 36 108

  41. Prime Numbers A natural number greater than 1 is a prime number if it cannot be expressed as a product of two smaller natural numbers.

  42. The Prime Factorization of Natural Numbers Every natural number greater than 1 is either a prime number or it can be expressed as a product of prime numbers.

  43. The Infinitude of Primes There are infinitely many prime numbers.

  44. Fermat’s Last Theorem It is impossible to write a cube as a sum of two cubes, a fourth power as a sum of two fourth powers, and, in general, any power beyond the second as a sum of two similar powers.

  45. The Twin Prime Question Are there infinitely many pairs of prime numbers that differ from one another by two? Examples: 11 and 13, 29 and 31, 41 and 43 are twin primes.

  46. The Goldbach Question Can every positive, even number greater than 2 be written as the sum of two primes? Examples: 4 = 2 + 2 6 = 3 + 3 8 = 3 + 5 10 = 5 + 5 12 = 5 + 7 14 = ? 16 = ?

  47. Section 2.4: Crazy Clocks and Checking Out BarsCyclical Clock Arithmetic and Bar codes Identifying similarities among different objects is often the key to understanding a deeper idea.

  48. Question of the day Today is, Monday, March 10. On what day of the week will the Fouth of July fall this year?

  49. Mod Clock Arithmetic Devise a method for figuring out the day of the week for any day next year. How many years pass before the days of the week are back to the same cycle?

  50. More Mod Clock Arithmetic… Formulate a numerical statement about when x = y mod 12.

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