1. Math Journal 9-4 3. Is this graph a Function? Yes or No Domain: ______________________ Range: _______________________

2. Unit 2 Day 3: Types of Sequences Essential Questions: How can we describe two types of sequences? How can the Fibonacci sequence be used to explain a recursive pattern?

3. Vocabulary Sequence: an ordered list of numbers that often forms a pattern. Term: each number in a sequence (synonyms – member, element) Arithmetic Sequence: a sequence formed by adding a constant number to each previous term. This constant number is also called the common difference. Recursive Sequence: a sequence which uses the previous term to determine the next term.

4. +3 The pattern is “add 3 to the previous term”. To find the next two numbers, you add 3 to each previous term. 11 +3 = 14, and 14 + 3 = 17. Look at your vocab from today! What type of sequence would this be? Arithmetic!! We are adding the same number, 3, each time. The “Common Difference” is 3. Describe the pattern, then find the next two numbers. 2, 5, 8, 11

5. Example 1: Find the common difference for the arithmetic sequence: -7, -11, -15, -19 d = -4 -11 - (-7) -11 + 7

6. +? What is the total difference between 2 and 30? If we have to split that total difference between all four arrows, what would the “common difference” be? + 7 +7 +7 +7 Example 2 2, ___, ___, ___, 30 These numbers form an arithmetic sequence. What is the common difference? What are the three missing numbers? 30 - 2 = 28 28/4 = 7

7. Term #Term Common Difference Answer 12+79 29 16 3 +723 4 +730 5 +737 Term #Term Common Difference Answer 12+79 29 16 3 +723 4 +730 5 Term #Term Common Difference Answer 12+79 29 16 3 +723 4 5 Term #Term Common Difference Answer 12+79 29 16 3 4 5 Term #Term Common Difference Answer 12+79 2 3 4 5 Complete the table. What is the second term in the sequence? Analyzing Example 2:

8. Bus Stop #Arriving TimeTraveling TimeNext Stop 16:30 AM7 Min6:37 AM Example 3 Buses on your route run every 7 minutes between 6:30 A.M. and 9:00 A.M. You get to the bus stop at 7:07 A.M. Use the information to complete the table, then determine how long will you have to wait for a bus.

9. Bus #Arriving Time Traveling Time Next Stop 16:30 AM7 Min6:37 AM 2 7 Min6:44 AM 3 7 Min6:51 AM 4 7 Min6:58 AM 5 7 Min7:05 AM 6 7 Min7:12 AM 7 7 Min7:19 AM How long do you have to wait for the next bus if you arrive at 7:07 AM? Analyzing Example 3:

10. Quick Write: Look at the two tables you have in your notes… by definition of a function, do you believe a sequence is a function? Why or why not?

11. Not all patterns are arithmetic sequences. Some sequences are recursive (requires a previous terms to continue). For example, if 1 and 1 were the first two terms in a sequence and the function rule was to add two previous terms to find the next, what would be the next number? 1 }+ 2 1 1 2 }+ 3 Recursive Sequences

12. 1 1 2 3 }+ 5813213455 This specific sequence is called the Fibonacci Sequence. It is a natural pattern that exists and shows how a recursive pattern may appear in numbers. The Fibonacci Numbers

13. Term NumberTerm 13 27 3 4 5 6 7 10 17 27 44 71 What is the 7th term in the sequence? Example 4: Complete the recursive sequence table using the same rule as the Fibonacci sequence.

14. The Fibonacci numbers are found many places in the natural world, including: The number of flower petals. The branching behavior of plants. The growth patterns of sunflowers and pinecones, …… It is believed that the spiral nature of plant growth accounts for this phenomenon. Fibonacci Numbers In Nature

15. The lengths of bones in a hand are Fibonacci numbers. Fibonacci Numbers In Anatomy

16. http://dsc.discovery.com/tv-shows/other- shows/videos/assignment-discovery- fibonacci-sequence.htm

17. Summary Essential Questions: How can we describe two types of sequences? How can the Fibonacci sequence be used to explain a recursive pattern? Take 1 minute to write 2 sentences answering the essential questions.