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## Patterns and Growth

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**Patterns and Growth**John Hutchinson TM MATH: Patterns & Growth**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**Is there a pattern?**TM MATH: Patterns & Growth**Here’s one.**TM MATH: Patterns & Growth**Here’s another.**TM MATH: Patterns & Growth**What is:**1 + 2 + 3 + 4 + …..+ 98 + 99 + 100? TM MATH: Patterns & Growth**Look at:**There are 100 different 101s. Each number is counted twice. The sum is (100*101)/2 = 5050. TM MATH: Patterns & Growth**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**If there are n people in a room the number of handshakes is**n(n-1)/2. TM MATH: Patterns & Growth**Problem 2: How many intersections?**Given several straight lines. In how many ways can they intersect? TM MATH: Patterns & Growth**2 Lines**1 0 TM MATH: Patterns & Growth**3 Lines**0 intersections 1 intersection 2 intersections 3 intersections TM MATH: Patterns & Growth**Problem 2A**Given several different straight lines. What is the maximum number of intersections? TM MATH: Patterns & Growth**Is the pattern familiar?**TM MATH: Patterns & Growth**Problem 2B**Up to the maximum, are all intersections possible? TM MATH: Patterns & Growth**What about four lines?**TM MATH: Patterns & Growth**What about two intersections?**TM MATH: Patterns & Growth**What about two intersections?**Need three dimensions. TM MATH: Patterns & Growth**Problem 3**What is the pattern? 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,… TM MATH: Patterns & Growth**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**This is the Fibonacci Sequence.**Fn+2 = Fn+1 + Fn TM MATH: Patterns & Growth**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**Sums of squares**TM MATH: Patterns & Growth**Pascal’s Triangle**TM MATH: Patterns & Growth**=32**TM MATH: Patterns & Growth**Note**1 1 2 3 5 8 TM MATH: Patterns & Growth**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**Let’s count them.**TM MATH: Patterns & Growth**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**Lets count them.**Q = quarter, H = half-dollar TM MATH: Patterns & Growth**Observe**5 2 3 C D E F G A B C 8 13 TM MATH: Patterns & Growth**C 264**A 440 E 330 C 528 264/440 = 3/5 330/528 = 5/8 Observe TM MATH: Patterns & Growth**Note**89 55 144 89 TM MATH: Patterns & Growth**Flowers**TM MATH: Patterns & Growth**References**TM MATH: Patterns & Growth