Visualizing Patterns: Fibonacci Numbers and 1,000 Pointed Stars. Abstract. Agenda. MAA sponsored since 1950

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1 /9/0 Visualizing Patterns: Fibonacci Numbers and,000 Pointed Stars National Council of Teachers of Mathematics 0 Annual Meeting Denver, CO (April 7 0, 0) Session 67 Scott Annin (California State University, Fullerton) Jairo Aguayo (Bonita High School, California State University, Fullerton) STARS Abstract Patterns in mathematics invite exploration and often arise in peculiar places. This talk will engage you in two such experiences appropriate for students in grades 9 : () Determining how many stars have a given number of points () Discovering new relations among the Fibonacci numbers via strategic visual arrangement of them Problem (00) How many,000 pointed stars are there? Define a regular n pointed star to be the union of n line segments P P, P P,, P n P such that *The points P, P,, P n are coplanar and no of them are collinear; *Each of the n segments intersects at least one of the other line segments at a point other than the endpoint; *All of the angles at P, P,, P n are congruent; *All of the n line segments P P, P P,, P n P are congruent; *The path P P, P P,, P n P turns counterclockwise at an angle less than 0 degrees at each vertex. Agenda Stars AMC and AIME problems Fibonacci Numbers Tiling k column arrangements of Fibonacci numbers Series of Middle/High School Mathematics Competitions MAA sponsored since 90 00,000 s of students participate annually from,000 s of schools nationwide

2 /9/0 SCHEDULE: AMC (November) AMC 0 and (February) AIME (March) USA Math Olympiad (April) Math Olympiad Summer Program (June) International Math Olympiad (July) Problem (00) How many,000 pointed stars are there? Activity Sheet # n = n = n = n = n = n = n = 6 n = 6 n = 7 n = 7 TOPICS: Algebra Number Theory Functions Counting and Probability Sequences Geometry Problem (00) How many,000 pointed stars are there? n Pairs of j values Number of producing stars n pointed stars (,) (,) and (,) (,) (,7) and (,) (,7) (,9) and (,) and (,7) and (,6) (,7) AIME questions: each answer is an integer in the range hours Calculators are not allowed Top 00 nationwide scorers are invited to take the United States Math Olympiad Problem (00) How many,000 pointed stars are there? Activity Sheet # #) Circle the numbers below that are relatively prime to n = 0 and determine the number of stars with 0 points. ANSWER:,,7,9,,,7,9 stars with 0 points #) Circle the numbers below that are relatively prime to n = and determine the number of stars with points. ANSWER:,,,7,,,,,6,7,9,,,6,,9,,,,7,,,, stars with points #) How many numbers from to 00 contain a factor of? 0 How many numbers from to 00 contain a factor of? 0 How many numbers from to 00 contain a factor of both and? 0 How many numbers from to 00 are relatively prime to 00? 0 How many 00 pointed stars are there? 9 #) How many numbers from to 000 contain a factor of? 00 How many numbers from to 000 contain a factor of? 00 How many numbers from to 000 contain a factor of both and? 00 How many numbers from to 000 are relatively prime to 000? 00 How many 000 pointed stars are there? 99

3 /9/0 FIBONACCI NUMBERS Your list of Numbers a b a + b a + b a + b a + b a + b a + b You create a list of numbers generated in the same way as the Fibonacci numbers (add the previous two numbers) The first two numbers are J(0) = a and J() = b Fibonacci Numbers The first two numbers are: f(0) = and f() = The sequence of numbers generated by adding the two previous numbers: f(n) = f(n ) + f(n ) TILING Lucas Numbers 7 9 The first two numbers are L() = and L() = The Lucas Numbers are a list of numbers generated in the same way as the Fibonacci numbers (add the previous two numbers): L(n) = L(n ) + L(n ) Directions: You are being asked to arrange tiles using only two types of tiles. One type of tile is of length by and the other type of tile is twice as long ( by tile). Objective: Before you begin to tile you must discover a way to look at all possible arrangements of length by n, where n is the length of the space available to tile.

4 /9/0 First, how many different ways can you arrange tiles if you are only required to arrange them to make by tile arrangement? Finally, how many ways to arrange tiles to make a by tile arrangement? Second, how many different ways can you arrange tiles if you are only required to arrange them to make by tile arrangement? What do you predict will be the number of ways to arrange tiles for a by? Or a by 6? Or a by n? Next, how many ways to arrange tiles to make a by tile arrangement? f(n) Number of Arrangements Number of different ways f(0) f() Picture of Arrangements f() f() f() f() f(n)

5 /9/0 Combinatorial Interpretation How many ways to arrange tiles of by n length with by tiles and by tiles? If the first tile is a by tile, then there are f(n ) ways to complete the arrangement. If the first tile is a by tile, then there are f(n ) ways to complete the arrangement. Hence, f(n) = f(n ) + f(n ). 9 What about using two columns? What is the linear combination for this two columns case? f(n) = f(n ) + f(n ) Continue through the three columns case. What is the linear combination for this three columns case? K COLUMN ARRANGEMENTS OF FIBONACCI NUMBERS f(n) = f(n ) + f(n 6) We want to arrange the Fibonacci numbers into various numbers of columns. We are familiar with the column case We know the linear combination to list these numbers in column is: f(n) = f(n ) + f(n ) Can we stop here and predict the formula for f(n) case when k=, k=, and so on? Is there a pattern? Is this enough to say there is a pattern?

6 /9/0 Results Results Number of Linear Combination Columns *f(n ) + *f(n ) *f(n ) + ( )*f(n ) *f(n ) + *f(n 6) _7_*f(n ) + ( )*f(n ) *f(n ) + *f(n 0) 6 *f(n 6) + ( )*f(n ) 7 9 *f(n 7) + *f(n ) Conclusion of our pattern We can now take our results from the table above and our description of the patterns that arise from the coefficients, to generalize all our linear combination to one recurrence relation for every n>k, where k : f(n) = _L(k)_ f(n k) + _( ) k+ _ f(n k) The pattern of the linear combination f(n k) in the k th row f(n k) in the k th row Study the linear combinations in the table as a function of k, the number of columns. We observe that a pattern arises among the coefficients. We describe this pattern: Thank you! Questions? The pattern of the linear combination Rate this presentation on the conference app. f(n k) in the k th row f(n k) in the k th row The coefficient is the Lucas number, L(k). + if k is odd. if k is even. In other words, ( ) k+ Download available presentation handouts from the Online Planner! Join the conversation! Tweet us using the hashtag #NCTMDenver 6