Fibonacci pattern
Give each student several pattern blocks and call on volunteers to make pattern cores with blocks in at least two different shapes (for example, square, square, trapezoid, hexagon). Then ask them to read the pattern core. Ask each student to make a copy of the pattern core. Now place the students in groups of six and ask them to make the pattern using all six repeats. Then record on a table the number of shapes that they used in all. (An example of how four repeats of the sample pattern would be recorded is shown on the table below.)Then ask what the entries would be if there were eight or ten people. Encourage the students to skip count to find the answers.
Then ask what the entries would be if there were eight or ten people. Encourage the students to skip count to find the answers.
Because children often have difficulty memorizing the multiplication tables for 6, 7, and 8, you may wish to encourage them to look for patterns as they skip count these tables. Because memory is enhanced if the skip counting is done to music, you may find singing the skip counting both fun and effective. Tunes that work well for the 6 multiplication table are “Frère Jacques” and “Mary Had a Little Lamb.” The 7 multiplication table fits well to the tune of “London Bridges” or “Kumbaya.” “This Old Man” is a good melody for the 8 multiplication table, as is “The Bear Went over the Mountain.” You can add a visual dimension by asking the students to circle on a hundred chart the numbers they say while skip counting. After they have recorded all three skip-counting patterns, call on a volunteer to describe each pattern in words. As a study guide, you may wish to have the students construct a table to record the skip counting patterns.
Now display the pattern 1, 1, 2, 3, 5, 8, 13, 21. Tell the students that this special pattern is named after a man who lived in Italy many years ago and that the Fibonacci pattern is different from those they have studied before. Ask the children to speculate on the rule for this pattern. [Each pair of numbers is added to get the next number in the series.] and what the next 3 numbers in it will be [34, 55, 89]. Ask the students to make a list of the first ten Fibonacci numbers, and then record the difference between each pair of numbers. [The differences will be 0, 1, 2, 3, 5, 8, 13, and so on.]
If it is appropriate for your class, you may wish to give the students calculators and have them find the quotient of each pair of numbers, dividing the smaller number by the larger in each pair. [The quotients will be 1, .5, .667, .6, .625, .615, .619, .617, .618] Ask the students if they notice anything about the quotients. [After the first 2, they are all close to .62, and they follow a pattern of over, then under, .62.] Encourage the students to record and describe this pattern in their portfolios.
Why will a Fibonacci representation never contain two consecutive 1’s?
[The nature of the Fibonacci pattern is that each number is equal to the sum of the two previous numbers. Consequently, two consecutive 1’s should be replaced by 0’s, and the next larger place value should increase by 1. For instance, the incorrect Fibonacci representation 110 can be interpreted as the decimal sum 3 + 2 + 0 = 5; but since 5 is a Fibonacci number, this should instead be represented as 1000.]
Discovering the Fibonacci pattern through spiral counts
If you have a pinecone readily available, count the spirals (in each direction) along the pinecone�s outer surface. If you have a fresh pineapple available, count the spirals on the outer surface of the pineapple. (Then cut and eat the pineapple!) Using the picture of the daisy, count the spirals (each direction) in the daisy center. (It may help to print the picture and draw the spirals!) Using the picture of the sunflower, count the spirals (each direction) in the sunflower. (It will be even better if you have an actual sunflower seed head.)
Discovering the Fibonacci pattern through mathematical explorations
A mathematical bee, starting in cell one of the simple honeycomb design shown in the figure wishes to stroll to another higher number cell via a path of connected cells not necessarily in numerical order, but always increasing. That is, each step of the journey must head to the right, so the bee can only move from one cell to a neighboring cell of a higher number. In how many different ways can the bee travel from cell 1 to cell 4? To cell 5? To cell 6? Can you see a pattern developing?
The mirror idea is very good, but visual patterns will already exist. standing in the center, you'll see a condenced fibonacci pattern all around you in the "aisle tabs". this is because the tabs at the front will be smaller and will raise of the top of the aisles less. As the Aisle cubes move outwards, their cooresponding tabs will become larger, and will be raised higher in the air. This way, you see all the tabs when standing in the center. The tabs will be color coded based on the leading lane that passes them, so there will be a rainbow fibonacci sequence of increasingly larger tabs as you look upwards off the tops of the aisles form the center. N.B. this beauty will, for the most part, only work when standing in the center... Capice? I'd draw it, but it's actually harder to draw than describe. Anyone up for it?
Of the 13 "note events", 8 are unstressed (di's - single notes) and 5 are stressed (dum's - notes with octave doubling), and they break into a Fibonacci-style 5 + (5 + 3) grouping. At the same time the left hand plays a regular 8 note or chordal um-pah um-pah accompaniment. The 13 right-hand and 8 left-hand note-events together produce the satisfying Fibonacci total of 21. The melody here can be parsed for nested relationships in a similar (but not quite identical) way to the limerick pattern.
Another piece of jazz-inspired music which has an identical metrical pattern to the limerick is the song It ain't necessarily so from George Gershwin's Porgy and Bess(1935). Whereas there is no suggestion that the metrical rhythms of limerick and ragtime were deliberately worked out, but rather that they emerge as an inevitable reflection of nature - with Gershwin there may be an element of deliberate shaping involved. Along with a number of popular American composers from the first half of the twentieth century he was taught by the remarkable engineer and musician Joseph Schillinger. His massive tomes, The Mathematical Basis of the Arts and The Schillinger System of Musical Composition, suggest many different systems for generating rhythm patterns and other forms of musical material which Schillinger explicitly relates to patterns in the physical world. He mentions the Fibonacci sequence as being rather special, but its potential is only hinted at and not developed. Gershwin was diligently supervised by Schillinger during the composition of Porgy and Bess, but a detailed study of the extent of this influence has yet to be completed.
Musicologists have uncovered Fibonacci patterns and related Golden Section proportions in the work of many other composers, particularly figures such as Debussy and Bartok whose work looked to folk and other influences outside of the European mainstream.