For this lesson you will need:

1. For this lesson you will need: • A pencil and a Highlighter  • A calculator • Your thinking caps!

2. Trends, Patterns, Predictions…Oh My!!!

3. LEARNING GOALS: • You will all be able to: • Describe what a pattern is • Discover and analyze patterns found within different sets of numbers • Describe what the “Fibonacci Sequence” is and where it can be found outside of the classroom.

4. Agenda for today • Define what a pattern is • A pattern warm-up • Talk about the numbers that make up Pascal’s Triangle and look for patterns with those numbers to predict future rows in the triangle • Learn about the Fibonacci Sequence and its applications in the real world.

5. Patterns • What is a pattern ? • Where do we see patterns ? ♦ Something that repeats ♦ Forming a consistent or characteristic arrangement ♦ In fabrics or clothing ♦ The days of the week Mon-Sun ♦ Architecture ♦ Work schedules or Class schedules follow a set pattern ♦ Tire treads

6. Patterns in Mathematics • Math can be thought of as the “science of patterns”. • There are 2 basic types of patterns used in mathematics: • Logic Patterns – categorizing objects based on characteristics like shape, color, texture, etc… • Number Patterns – relationships among different numerical values/quantities.

7. Patterns • Discovering patterns can help us predict what will happen next. • Let’s give it a go in the warm-up!!!

8. WARM - UP FINDING PATTERNS Find the missing terms in the patterns below. (Be able to explain the “rule” you used to find the next term in each sequence.) • 1 , 3 , 5 , 7 , ____ , ____ , ____ • A1 , B1 , A2 , B2 , A3 , B3 , A4 , _____ , _____ , _____ • 2 , 4 , 8 , 16 , 32 , _____ , _____ , _____ •  ,  , , ______________ , ______________ , _____________________ • A 1 C , E 2 G , I 3 K , M 4 O , ______ , ______ , • 1, 4, 7, ____ , 13 , _____ , 19 , _____ , _____ • 160 , 80 , 40 , _____ , 10 , _____ • 17 , 15 , ____ , ____ , ____ , 7 • 09 , 18 , 27, 36, 45, 54, ____ , _____ , 81 , 90 • 9 , 98 , 987 , 9876 , _________ , __________ , _____________

9. WARM – UP (answers) • 1) 1 , 3 , 5 , 7 , 9 , 11 , 13 • 2) A1 , B1 , A2 , B2 , A3 , B3 , A4 , B4 , A5 , B5 • 3) 2 , 4 , 8 , 16 , 32 , 64 , 128 , 256 •  ,  , ,  ,  ,  •  • 5) A 1 C , E 2 G , I 3 K , M 4 O , Q 5 S ,U 6 W • 6) 1, 4, 7, 10 , 13 , 16 , 19 , 22 , 25 • 7) 160 , 80 , 40 , 20 , 10 , 5 • 8) 17 , 15 , 13 , 11 , 9 , 7 • 9) 09 , 18 , 27, 36, 45, 54, 63 , 72 , 81 , 90 10) 9 , 98 , 987 , 9876 , 98765 , 987654 , 9876543

10. Pascal’s Triangle … A brief history • Pascal's triangle is a triangular array of numbers. It is named after the French mathematician Blaise Pascal, but other mathematicians studied it centuries before him in India, Greece, Iran, China, Germany, and Italy. • The triangle contains many different hidden number patterns; many of which we will talk about later on. • The numbers in each row of the triangle are precisely the same numbers that are the coefficients of binomial expansions. [ex: (x + y)³ = 1x3 + 3x2y + 3xy2 + 1y3 ] • Its known applications in mathematics extend to calculus, trigonometry, plane geometry, and solid geometry.

11. HANDOUT: Fill out as many boxes as you can (pg2)

12. Pascal’s Triangle …of AWESOMENESS!(page 2) * What patterns do you notice within the triangle? * What method(s) can you use to find the numbers in the next row? * Do you notice any patterns that go across the triangle or diagonally through the triangle? * Fill in as much of Pascal’s Triangle as you can!

13. Patterns in Pascal’s Triangle

14. Diagonal Patterns in Pascal’s Triangle • The diagonals going along the left and right edges contain only 1's • The diagonals next to the edge of the “1’s” diagonals contain the natural numbers, or counting numbers. • The 3rd inner set of diagonals are the “triangular numbers” ; number amounts that make equilateral triangles (all sides are the same length)

15. Pascal’s Triangle (page 1) Highlightall of the hexagons that contain odd numbers. Odd numbers: 1 , 3 , 5 , 7 , 9 , 11 , etc.

16. It should look a little something like this 

17. Pascal’s Triangle & the Sierpinski Triangle • The pattern obtained by coloring only the odd numbers in Pascal's triangle closely resembles the fractal called the Sierpinski triangle. This resemblance becomes more and more accurate the more rows you add to Pascal’s triangle and the farther you zoom out.

18. Fibonacci Sequence 0,1, 1, 2 , 3 , 5 , 8, 13, 21, 34 , 55 , 89, 144, 233, 377, 610 , 987, 1597, 2584 , 4181, 6765, 10946, 17711, 28657, 46368 , 75025, 121393 , 196418… 1 2 3 5 8

19. Fibonacci Sequence • The next number in the sequence is found by adding up the two numbers before it. • It’s that simple!

20. Fibonacci Sequence (Spirals) • When you make squares with the widths 1, 2 , 3 , 5 , 8 and so on you get a nice spiral: • The squares fit neatly together!!! For example 1 and 1 make 2, 2 and 3 make 5, etc....

21. Fibonacci Spirals found in Nature

23. You can create your own Fibonacci-like sequence! • Start with any two numbers you like • Then add the two previous numbers to generate the next term. My example: 1 , 4 , 5 , 9 , 14 , 23 , 37 , 60 , 97 , 157 , 254 …

24. Fibonacci Trick! (Does it work?) • Trick : The sum of the first ten numbers in your sequence will automatically be 11 times the amount of the 7th term in your sequence. • Let’s see if it works with my sequence My example: 1 , 4 , 5 , 9 , 14 , 23 , 37 , 60 , 97 , 157 , 254 … Sum = 407 11 (37) = 407  IT WORKS!!! COOL 

25. Tonight’s Homework: • Create your own Fibonacci-like sequence. • Find a general formula for the “Fibonacci # Trick”. Prove why the trick works from a mathematical standpoint. (*Hint: call the first term of your sequence “a” and the second term of your sequence “b”)

26. Now you are all expert mathematical pattern investigators!!!

27. Are you ready to learn one last awesome fact about mathematical patterns? YES!!!!!  Adding the diagonal rows of Pascal’s triangle create the Fibonacci Sequence!!!! How crazy cool is that?