Collatz Sequence. Fibbonacci Sequence. n is even; Recurrence Relation: a n+1 = a n + a n 1.

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1 Fibonacci Roulette In this game you will be constructing a recurrence relation, that is, a sequence of numbers where you find the next number by looking at the previous numbers in the sequence. Your job will be to choose the first one or two numbers in the sequence (depending on which type of sequence you are creating), and your goal is to eventually reach a randomly chosen target number. There are two types of sequences you can construct: a Collatz sequence or a Fibbonacci sequence

2 Collatz Sequence a n { if a Recurrence Relation: a n+1 = 2 n is even; { 3 a n + 1 if a n is odd. Step 1: Choose whether you want to attempt an Easy, Hard, or Expert sequence. Step 2: Randomly select a number token from the green bag that goes with your choice. This is your target number! Step 3: Choose a green number (33, 34, 36, 38, 40, 42, 44, 46, or 48) as the first number in your Collatz sequence. Place your token on that number. Remember, your goal is to reach your target number by following the rules of Collatz sequences. Step 4: Move your token to the next number in your sequence by following the Collatz sequence rules: If your token is on an even number, move it to the number that is half of the number it s currently on. If your token is on an odd number, move it to the number that is equal to one more than three times your current number. Step 5: Repeat Step 4 until you move off that board, until your token reaches 1, or until you reach your target number. If you reached your target number, you win! Fibbonacci Sequence Recurrence Relation: a n+1 = a n + a n 1. Step 1: Choose whether you want to attempt an Easy, Hard, or Expert sequence. Step 2: Randomly select a number token from the red bag that goes with your choice. This is your target number! Step 3: Choose two red numbers (1-10) as the first and second numbers in your Fibbonacci sequence (they can be the same number). Place the white token on the first number and the red token on the second number. Remember, your goal is to reach your target number by following the rules of Fibbonacci sequences. Step 4: Move the white token to the number that is the sum of the two numbers your tokens are currently on. Step 5: Move the red token to the number that is the sum of the two numbers your tokens are currently on. Step 6: Repeat steps 4 and 5 until you move off that board or until you reach your target number. If you reached your target number, you win!

3 Further Study: Fibonacci Roulette A sequence of real numbers is a function from the natural numbers to the real numbers. It can be represented as a list of numbers, such as {1, 6, 7, 4, 3, 9, }. A recurrence relation is a type of sequence in which each entry (besides the first few) depends on the values of previous entries in the sequence. In a recurrence relation, if you know all the numbers in the sequence up to a certain point, you can use the recurrence rule to find the next one. Collatz Sequences and the Collatz Conjecture A Collatz sequence, named after the German mathematician Lothar Collatz, is a sequence of positive integers defined by the following recurrence relation: Choose any positive integer for the first entry in the sequence. Then for all positive integers n, if the nth number in the sequence is even, the n + 1st number is half that one; if the nth number in the sequence is odd, the n + 1st entry is three times that number, plus one. So for instance, if you start with the number 10, the Collatz sequence is {10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, }. Notice that at the end, the sequence just keeps repeating in the pattern {1, 4, 2, 1, 4, 2, 1, 4, 2, }. For every positive integer mathematicians have tried starting at, the sequence eventually arrives at this pattern. The Collatz Conjecture is that every Collatz sequence eventually arrives at this pattern, no matter where you start. However, mathematicians have been unable to prove this. Providing a correct proof that this is the case or giving a counterexample to the conjecture would guarantee you a place in mathematical history.

4 Fibonacci Sequences A Fibonacci sequence, named after the Italian mathematician Leonardo Fibonacci, is a sequence in which you start with any two integers, and every later term is the sum of the two previous terms. The original Fibonacci sequence begins with 1 and 1, so the original Fibbonacci sequence is {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, }. This sequence describes the number of pairs of rabbits you have each month under the following assumptions: 1. You begin with one immature pair of rabbits 2. A pair of rabbits takes one month to mature 3. Each month, each mature pair of rabbits produces another pair of rabbits 4. Rabbits are immortal The Fibonacci sequence has many interesting properties. Perhaps most spectacular is the fact that if you take the ratio of adjacent terms, you get a sequence that converges to the Golden Ratio: f lim n+1 = φ = = , n f n 2 which was a quantity known and used extensively by civilizations as old as ancient Greece. Recurrence relations are recursive definitions of mathematical functions or sequences. They are a fundamental mathematical tool. They can be used to represent functions and sequences that cannot be easily represented non-recursively. Both the Fibanocci sequence and the Collatz conjecture are examples of this. Fibonacci Sequence: a n+1 = a n + a n 1. a n2 if a Collatz Sequence: a n+1 = { n is even; 3 a n + 1 if a n is odd.

5 Common Core: Fibonacci Roulette Objective: This activity will show the students how to use patterns to understand mathematics and model situations. It will also show the student how we can communicate and generalize algebraic relationships. It will help students to understand how we use patterns and relationships of algebraic representations to generalize, communicate, and model situations in mathematics. Common Core Standards HS.F, IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x.the graph of f is the graph of the equation y = f(x). Common Core Standards HS.F, IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n + 1) = f(n) + f(n 1) for n 1. Enduring Understandings Students will be engaged in an activity that will introduce sequences, functions, recursive, subsets, domains and integers. Students will understand that sequences are formed in different ways by playing a game in which they will create their own sequences. Students will be able to recognize sequences as functions.