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

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?

April 2002TM MATH: Patterns & Growth3 PeopleHandshakes

April 2002TM MATH: Patterns & Growth4 PeopleHandshakes

April 2002TM MATH: Patterns & Growth5 PeopleHandshakes

April 2002TM MATH: Patterns & Growth6 PeopleHandshakes

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

April 2002TM MATH: Patterns & Growth8 Here’s one. PeopleHandshakes

April 2002TM MATH: Patterns & Growth9 Here’s another. PeopleHandshakes

April 2002TM MATH: Patterns & Growth10 What is: … ?

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

April 2002TM MATH: Patterns & Growth12 Look at: = 3  7 = = 4  7 = 28

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

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

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

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

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

April 2002TM MATH: Patterns & Growth18 Is the pattern familiar? LinesIntersections

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

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

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

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

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

April 2002TM MATH: Patterns & Growth24 Note = = = = = = = 43

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

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 Every 8th Fibonacci number is divisible by 21.

April 2002TM MATH: Patterns & Growth27 Sums of squares      13

April 2002TM MATH: Patterns & Growth28 Pascal’s Triangle

April 2002TM MATH: Patterns & Growth29 1= 1 11= 2 121= = = =32

April 2002TM MATH: Patterns & Growth30 Note

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?

April 2002TM MATH: Patterns & Growth32 Let’s count them. MonthAdultsBabiesTotal

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?

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

April 2002TM MATH: Patterns & Growth C DEFGABC Observe

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

April 2002TM MATH: Patterns & Growth37 Note

April 2002TM MATH: Patterns & Growth38

April 2002TM MATH: Patterns & Growth39

April 2002TM MATH: Patterns & Growth40

April 2002TM MATH: Patterns & Growth41

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

April 2002TM MATH: Patterns & Growth43 References