1. April 2002TM MATH: Patterns & Growth1 Patterns and Growth John Hutchinson

2. April 2002TM MATH: Patterns & Growth2 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?

3. April 2002TM MATH: Patterns & Growth3 PeopleHandshakes 10 21 3 4 5 6 7

4. April 2002TM MATH: Patterns & Growth4 PeopleHandshakes 10 21 33 4 5 6 7

5. April 2002TM MATH: Patterns & Growth5 PeopleHandshakes 10 21 33 46 5 6 7

6. April 2002TM MATH: Patterns & Growth6 PeopleHandshakes 10 21 33 46 510 615 721

7. April 2002TM MATH: Patterns & Growth7 Is there a pattern?

8. April 2002TM MATH: Patterns & Growth8 Here’s one. PeopleHandshakes 100 211 331 + 2 461 + 2 + 3 5101 + 2 + 3 + 4 6151 + 2 + 3 + 4 + 5 7211 + 2 + 3 + 4 + 5 + 6

9. April 2002TM MATH: Patterns & Growth9 Here’s another. PeopleHandshakes 100 211 + 0 332 + 1 463 + 3 5104 + 6 6155 + 10 7216 + 15

10. April 2002TM MATH: Patterns & Growth10 What is: 1 + 2 + 3 + 4 + …..+ 98 + 99 + 100?

11. April 2002TM MATH: Patterns & Growth11 Look at: 1234…9899100 999897…321 101 … There are 100 different 101s. Each number is counted twice. The sum is (100*101)/2 = 5050.

12. April 2002TM MATH: Patterns & Growth12 Look at: 1 + 2 + 3 + 4 + 5 + 6 = 3  7 = 21 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 = 4  7 = 28

13. April 2002TM MATH: Patterns & Growth13 If there are n people in a room the number of handshakes is n(n-1)/2.

14. April 2002TM MATH: Patterns & Growth14 Problem 2: How many intersections? Given several straight lines. In how many ways can they intersect?

15. April 2002TM MATH: Patterns & Growth15 2 Lines 1 0

16. April 2002TM MATH: Patterns & Growth16 3 Lines 0 intersections 1 intersection 2 intersections3 intersections

17. April 2002TM MATH: Patterns & Growth17 Problem 2A Given several different straight lines. What is the maximum number of intersections?

18. April 2002TM MATH: Patterns & Growth18 Is the pattern familiar? LinesIntersections 10 21 33 46 510

19. April 2002TM MATH: Patterns & Growth19 Problem 2B Up to the maximum, are all intersections possible?

20. April 2002TM MATH: Patterns & Growth20 What about four lines?

21. April 2002TM MATH: Patterns & Growth21 What about two intersections?

22. April 2002TM MATH: Patterns & Growth22 What about two intersections? Need three dimensions.

23. April 2002TM MATH: Patterns & Growth23 Problem 3 What is the pattern? 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,…

24. April 2002TM MATH: Patterns & Growth24 Note 1 + 1 = 2 1 + 2 = 3 2 + 3 = 5 3 + 5 = 8 5 + 8 = 13 8 + 13 = 21 13 + 21 = 43

25. April 2002TM MATH: Patterns & Growth25 This is the Fibonacci Sequence. F n+2 = F n+1 + F n

26. April 2002TM MATH: Patterns & Growth26 Divisibility 1.Every 3 rd Fibonacci number is divisible by 2. 2.Every 4th Fibonacci number is divisible by 3. 3.Every 5th Fibonacci number is divisible by 5. 4.Every 6th Fibonacci number is divisible by 8. 5.Every 7th Fibonacci number is divisible by 13. 6.Every 8th Fibonacci number is divisible by 21.

27. April 2002TM MATH: Patterns & Growth27 Sums of squares 1 2 + 1 2 1  2 1 2 + 1 2 + 2 2 2  3 1 2 + 1 2 + 2 2 + 3 2 3  5 1 2 + 1 2 + 2 2 + 3 2 + 5 2 5  8 1 2 + 1 2 + 2 2 + 3 2 + 5 2 + 8 2 8  13

28. April 2002TM MATH: Patterns & Growth28 Pascal’s Triangle 1 11 121 1331 14641 1510 51

29. April 2002TM MATH: Patterns & Growth29 1= 1 11= 2 121= 4 1331= 8 14641= 16 1510 51 =32

30. April 2002TM MATH: Patterns & Growth30 Note 1 11 121 1331 14641 1510 51 1 1 2 3 5 8

31. April 2002TM MATH: Patterns & Growth31 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?

32. April 2002TM MATH: Patterns & Growth32 Let’s count them. MonthAdultsBabiesTotal 1101 2112 3213 4325 5538 68513

33. April 2002TM MATH: Patterns & Growth33 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?

34. April 2002TM MATH: Patterns & Growth34 Lets count them. Q = quarter, H = half-dollar 1 tokenQ1 2 tokensQQ-H2 3 tokensQQQ-HQ-QH3 4 tokensQQQQ-QQH-QHQ-HQQ-HH5 5 tokensQQQQQ-QQQH-QQHQ-QHQQ HQQQ-HHQ-HQH-QHH 8

35. April 2002TM MATH: Patterns & Growth35 23 5 8 13 C DEFGABC Observe

36. April 2002TM MATH: Patterns & Growth36 Observe C  264 A  440 E  330 C  528 264/440 = 3/5 330/528 = 5/8

37. April 2002TM MATH: Patterns & Growth37 Note 144 89 55

38. April 2002TM MATH: Patterns & Growth38

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42. April 2002TM MATH: Patterns & Growth42 Flowers # PetalsFlower 1 White Calla Lily 2 Euphorbia 3 Lily Iris 5 ColumbineButtercupLarkspur 8 BloodrootDelphiniumCoreopsi 13 Black-eyed SusanDaisyMarigold 21 DaisyBlack-eyed SusanAster 34 DaisySunflowerPlantain

43. April 2002TM MATH: Patterns & Growth43 References