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Patterns and Growth

Patterns and Growth. John Hutchinson. Problem 1: How many handshakes?. Several people are in a room. Each person in the room shakes hands with every other person in the room. How many handshakes take place?. Is there a pattern?. Here’s one. Here’s another. What is:.

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Patterns and Growth

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  1. Patterns and Growth John Hutchinson TM MATH: Patterns & Growth

  2. Problem 1: How many handshakes? Several people are in a room. Each person in the room shakes hands with every other person in the room. How many handshakes take place? TM MATH: Patterns & Growth

  3. TM MATH: Patterns & Growth

  4. TM MATH: Patterns & Growth

  5. TM MATH: Patterns & Growth

  6. TM MATH: Patterns & Growth

  7. Is there a pattern? TM MATH: Patterns & Growth

  8. Here’s one. TM MATH: Patterns & Growth

  9. Here’s another. TM MATH: Patterns & Growth

  10. What is: 1 + 2 + 3 + 4 + …..+ 98 + 99 + 100? TM MATH: Patterns & Growth

  11. Look at: There are 100 different 101s. Each number is counted twice. The sum is (100*101)/2 = 5050. TM MATH: Patterns & Growth

  12. Look at: 1 + 2 + 3 + 4 + 5 + 6 = 3  7 = 21 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 = 4  7 = 28 TM MATH: Patterns & Growth

  13. If there are n people in a room the number of handshakes is n(n-1)/2. TM MATH: Patterns & Growth

  14. Problem 2: How many intersections? Given several straight lines. In how many ways can they intersect? TM MATH: Patterns & Growth

  15. 2 Lines 1 0 TM MATH: Patterns & Growth

  16. 3 Lines 0 intersections 1 intersection 2 intersections 3 intersections TM MATH: Patterns & Growth

  17. Problem 2A Given several different straight lines. What is the maximum number of intersections? TM MATH: Patterns & Growth

  18. Is the pattern familiar? TM MATH: Patterns & Growth

  19. Problem 2B Up to the maximum, are all intersections possible? TM MATH: Patterns & Growth

  20. What about four lines? TM MATH: Patterns & Growth

  21. What about two intersections? TM MATH: Patterns & Growth

  22. What about two intersections? Need three dimensions. TM MATH: Patterns & Growth

  23. Problem 3 What is the pattern? 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,… TM MATH: Patterns & Growth

  24. Note • 1 + 1 = 2 • 1 + 2 = 3 • 2 + 3 = 5 • 3 + 5 = 8 • 5 + 8 = 13 • 8 + 13 = 21 • 13 + 21 = 43 TM MATH: Patterns & Growth

  25. This is the Fibonacci Sequence. Fn+2 = Fn+1 + Fn TM MATH: Patterns & Growth

  26. Divisibility • Every 3rd Fibonacci number is divisible by 2. • Every 4th Fibonacci number is divisible by 3. • Every 5th Fibonacci number is divisible by 5. • Every 6th Fibonacci number is divisible by 8. • Every 7th Fibonacci number is divisible by 13. • Every 8th Fibonacci number is divisible by 21. TM MATH: Patterns & Growth

  27. Sums of squares TM MATH: Patterns & Growth

  28. Pascal’s Triangle TM MATH: Patterns & Growth

  29. =32 TM MATH: Patterns & Growth

  30. Note 1 1 2 3 5 8 TM MATH: Patterns & Growth

  31. Problem 3A: How many rabbits? Suppose that each pair of rabbits produces a new pair of rabbits each month. Suppose each new pair of rabbits begins to reproduce two months after its birth. If you start with one adult pair of rabbits at month one how many pairs do you have in month 2, month 3, month 4? TM MATH: Patterns & Growth

  32. Let’s count them. TM MATH: Patterns & Growth

  33. Problem 3B: How many ways? A token machine dispenses 25-cent tokens. The machine only accepts quarters and half-dollars. How many ways can a person purchase 1 token, 2 tokens, 3 tokens? TM MATH: Patterns & Growth

  34. Lets count them. Q = quarter, H = half-dollar TM MATH: Patterns & Growth

  35. Observe 5 2 3 C D E F G A B C 8 13 TM MATH: Patterns & Growth

  36. C  264 A  440 E  330 C  528 264/440 = 3/5 330/528 = 5/8 Observe TM MATH: Patterns & Growth

  37. Note 89 55 144 89 TM MATH: Patterns & Growth

  38. TM MATH: Patterns & Growth

  39. TM MATH: Patterns & Growth

  40. TM MATH: Patterns & Growth

  41. TM MATH: Patterns & Growth

  42. Flowers TM MATH: Patterns & Growth

  43. References TM MATH: Patterns & Growth

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