3 1 Recursive Sequences Patterns are everywhere you
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3. 1 – Recursive Sequences “Patterns are everywhere you look”
Learning Target � By the end of section 3. 1, I will be able to recognize a recursive pattern and find out the pattern, either increasing or decreasing.
Vocabulary � Recursive sequence: An ordered list of numbers defined by a starting value and a rule. › You generate the sequence by applying the rule to the starting value, then applying it to the resulting value, and repeating this process.
Example 1: The Empire State Building � The Empire State building in NYC has 102 floors and is 1, 250 ft high. How high up are you at the 4 th and 25 th floor? What floor is 217 ft above ground? Lets answer this using a recursive sequence Floor Number 0 1 2 3 Height (ft) -4 9 22 35 4 … … 10 … … … 217 25 … Answer: Starting value = -4 Rule = Add 13 to the current floor height to get the new floor height. Graphing Calculator: 1. Press -4 “enter” 2. Press +13 “enter” 3. Keep repeating until you have reached the desired floor 4. Count how many numbers and that is the floor number
Investigation: Recursive Toothpick Patterns In this investigation, you will learn to create and apply recursive sequences by modeling them with puzzle pieces made from toothpicks. Figure 1 Figure 2 Figure 3 • Make Figures 1 – 3 of the pattern using as few toothpicks as possible. How many toothpicks does it take to reproduce each figure? How many toothpicks lie on the perimeter of each figure? • Copy the table with enough rows for six figures of the pattern. Make Figures 4 – 6 from toothpicks by adding triangles in a row and complete the table. Number of toothpicks Perimeter Figure 1 Figure 4 Figure 2 Figure 5 Figure 3 Figure 6
Investigation: Recursive Toothpick Patterns • What is the rule for finding the number of toothpicks in each figure? What is the rule for finding the perimeter? Use your calculator to create recursive routines for these rules. Check that these routines generate the numbers in your table. • Now Make Figure 10 from toothpicks. Count the number of toothpicks and find the perimeter. Does your calculator routine give the same answers? Find the number of toothpicks and the perimeter for Figure 25.
Mini Project: Recursive Toothpick Patterns You’ll create your own puzzle piece from toothpicks. Add identical pieces in one direction to make the succeeding figures of your design. • Choose a starting figure that has more than 3 toothpicks. • Paste figures 1 – 3 on your paper. Write recursive routines to generate number sequences for the number of toothpicks and perimeter (fill in a table for your figure like in the investigation). • Find the values for ten puzzle pieces. • Your project should look like slide 4 but a different starting figure.
Examples Find the missing values in each sequence. 3, 7, __, 15, __, 27, … 2. 7, 1, -5, __, -17, __, … 3. 5, 13, 21, __, 45, … 1. Homework = pg. 161 (1 – 4)