Fun with the Fibonacci Sequence

1. Fun with the Fibonacci Sequence Alannah McGregor Gudrun Mackness Brittany Kozak

2. Background Information • Grades 6-7 • Mathematics: patterning and algebra • Time frame: 30 min • Lesson environment • Inquiry-based exploration • Exploration of a complex number pattern that results in a sequence that is found in nature and has been translated into art • Whole class and small group • 4-5 students per group

3. Materials & Prep work • iPad(s), loaded with 2 FREE apps • Camera Awesome • Doodle Buddy • T-chart (1 copy per student) • 1 column for flowers, fruit, and pinecone icons and names • 1 column to gather each object’s data • Pictures of the Mona Lisa with golden spiral focal point on her face; • 2 examples of a photo centered contrasted with a variation of the same photo using the spiral focal point • Cue cards • with “warm-up” number patterns • Each with picture of a flower and its name • Graph paper • Markers and pencils • Library picture book on Fibonacci sequence and golden spiral • Natural Objects showing Fibonacci numbers – e.g.: • Apple(s) cut horizontally through the centre, • Banana(s) cut into slices, • Pineapple(s) • Pinecone(s)

4. Lesson Overview • Pattern warm up and History of Fibonacci (5 min) • Find the numerical patterns ~ 3 min • Who was Fibonacci, and what is the Fibonacci sequence? ~ 2 min • Finding Fibonacci numbers (15 min) • Fruits, flowers, and pinecones ~10 min • Discovering the pattern ~5min • The Golden Spiral (10 min) • How to draw it using the sequence ~ 5 min • da Vinci’s Mona Lisa ~ 1 min • Camera awesome! ~4 min + extra time

5. Patterning Warm-Up 3 Minutes • Give students pre-made cards with simple patterns on them • Ask students to use what they know about patterns to figure out each pattern provided • Ask students to investigate the patterns independently or with a partner

6. Patterning Warm-Up (Continued) • Circulate to help or give students an added challenge (e.g. ask them to predict the 8th number in the pattern) • As needed: Look at patterns together as a group

7. History Of Fibonacci 2 Minutes • Provide students with background information on Fibonacci the mathematician • He found a number pattern in nature • He was credited for the ‘Fibonacci sequence’ even though it can be traced back to India

8. Finding Fibonacci Numbers 15 Minutes • Explain to the students that they will be looking at natural materials and images of flowers to see if they can find numbers in the Fibonacci sequence • Hand out the T-Chart and review with the students the meaning of the words “petal”, “section” and “spiral” (see next slide for example chart)

9. Example Observation Chart

10. Observation Chart Suggestions • Our example chart can be used with a few fruits we selected and set of cue cards made from our “Flower” pdf document (one flower per cue card) • If you wish to select your own Fibonacci objects, you can create your own chard (and cue cards if needed) • Include a visual representation of each object (either the real thing or a photo) • Include objects that show the whole range of Fibonacci numbers from 1 to at least 34 • An added challenge would be to have students draw the objects themselves

11. Finding Fibonacci Numbers • Provide students with: • Cue cards with pictured flowers • Banana sliced horizontally • Apple sliced in half horizontally • Pineapple • Pine Cone • Ask students to explore the number of petals on each flower has and number of sections in the pre-cut fruit • Tell students to document what they find on their observation chart

12. Fun with Fruit • Banana: tell students to use their hands to gently split the banana into its natural sections (there are 3) • Apple: Draw students’ attention to the sections in the core of the apple and ask them to count them (there are 5)

13. The Pesky Pineapple • The pineapple is an interesting object to have students examine because it provides a number of examples of Fibonacci numbers • However, accurately finding the number of spirals on a pineapple is difficult without the right materials • Have students use tape, or strings and pushpins to trace the spirals as they find them so they can keep track

14. Picture-Perfect Pinecone The pinecone also has examples of Fibonacci numbers in the spirals formed by its seed pods • Have students use the iPad to take a photo of the bottom of the pine cone • They can then use a doodle app (e.g. Doodle Buddy) to trace the line of each spiral directly on the image • Have students use the doodled image to count the number of spirals on the pine cone Alternatively, print a ‘birds eye view’ of the bottom of the pinecone so students can draw the spirals as they count them

15. Interpreting the Pattern • Ask students to arrange the numbers they have found into order from smallest to largest (omitting repeated numbers). • Have students examine this number sequence to see if they can spot the pattern • If students need a hint, tell them to look at the two numbers before a given number (e.g. 3+5=8)

16. Completing the Sequence • Once students have interpreted the pattern, ask them to infer what numbers should precede 1 and 2 • If students need help, have them focus on the number 2. They will see that one of the numbers preceding it is 1, but another number is needed in order for 2 to make sense in this sequence. Students will need to determine that the existing 1 needs another 1 to make a sum of 2 (1+1=2) • They can use the same strategy to figure out that in order for the 1 preceding the 2 to make sense, a 0 should begin the sequence (0+1=1)

17. Golden Spiral 10 Minutes • Tell students that the Fibonacci sequence can be found all throughout nature because it is an adaptive way for plants to grow • Plants tend to grow in these ways to maximize the amount of sunlight they receive and the number of seeds they produce • Sometimes we see the patterns by counting parts of a living thing but sometimes we see it by measuring the way things grow • Show the students a photo of a nautilus shell (or the real thing if you can get one!)

18. Golden Spiral (Continued) • Tell students that the spiral growth pattern of the nautilus shell is called a golden spiral • A golden spiral has certain proportions and can be created using the numbers in the Fibonacci sequence • It can be created by drawing a series of connected squares wherein each square has the length/width of each number in the sequence. Start with 1x1, and then watch it grow!

19. Drawing the Golden Spiral • Provide markers and graph paper • Explain how to use the Fibonacci numbers to draw squares on graph paper: • For each number, draw a square box with that number of units as the side lengths • Zero produces no square, so the first three squareswill be (1x1, 1x1, 2x2). • Draw the squares in a counter clockwise spiral (1x1, then move left for another 1x1, then down for 2x2, and right for 3x3, then up, then left, then down, etc.) • Each new square fits along a line created by previous squares • Ask students to help you determine the dimensions of the next 2 squares. • Show them the trajectory of the spiral that follows the pattern of squares • Have students complete their own spiral independently, using Fibonacci numbers 1-21 (or more if materials & time allow) Check out this YouTube video for a demo: http://www.youtube.com/watch?v=ahXIMUkSXX0

20. The Golden Spiral in Art: the Mona Lisa • Give the students examples of how the golden spiral has been used in art • Show the students a photo of the Mona Lisa with the Golden Spiral superimposed over her face.

21. The Mona Lisa (Continued) 10 Minutes • Explain that Da Vinci used the proportions of the Golden Spiral when he drew Mona Lisa's face, which likely contributed to the fame of her beauty. • Point out to students (or let students discover) that her head is ever so slightly tilted, and that the slightly off-centre vertical line of the rectangle follows the centre line of her face. Also, the horizontal line of the rectangle follows the line of her eyes. The side of the spiral even follows closely to the curvature of her face on the right side.

22. iPad Photography • Explain how to use the Camera Awesome iPad app using the Fibonacci Spiral feature • Show students how the center of the spiral is the focal point of the picture (share some example photos taken using the app)

23. iPad Photography • Have students use the iPad to take two photos, one where the focal point is in the center of the spiral and one that does not use the spiral • Have students compare the photos and discuss which one is more aesthetically pleasing Caption for above and below

24. Closing/ Recap • The sequence is credited to Fibonacci but originally identified somewhere in India • These numbers can be found around the world in nature and even the human body but don’t apply to every single plant. • In addition to providing a demo for drawing the Fibonacci spiral, this YouTube video provides a great, kid-friendly overview of some of the many places where the sequence can be seen:http://www.youtube.com/watch/?v=ahXIMUkSXX0

25. References Campbell, S. C. (2010). Growing patterns: Fibonacci numbers in nature. Honesdale, PA: Boyds Mill Press. Posamentier, A. S. (2007). Fabulous fibonacci numbers. Amherst, NY: Prometheus Books. Van de Walle, J. A., Folk, S., Karp, K. S., & Bay-Williams, J. M. (2011). Elementary and middle school mathematics: Teaching developmentally. (2nd Canadian ed., pp. 278-279). Toronto, Ontario: Pearson Canada.