SCMP Summer 2009
Math in Nature Investigation
ENGAGE
Give each team of three a pineapple, several pinecones, and pictures of flower heads and a nautilus shell cross-section. Ask them to study the objects and look for something that would connect them to each other.
EXPLORE
As students are exploring the materials given them, move among the groups and listen to conversations they are having about the items. Ask probing questions to orient them to the task of finding a pattern or connection between the objects.
When most of the groups have found some information to share, ask the whole group to listen to the sharing while we record their findings on chart paper. Ask probing questions to extend the thinking that is shared.
EXPLAIN
Ask students to explain what they found related to the spirals and demonstrate the spirals on the first pineapple slide. Allow more time, if necessary, for teams to continue their investigating of the other items. Hopefully they will find more sets of spirals. Each time we will record the Fibonacci numbers they find on the chart paper in numerical order. When all teams have shared their findings and the numbers they found have been recorded, ask them to look for a pattern in the sequence of numbers and use their pattern to extend the sequence to lesser and greater numbers. Define the Fibonacci Sequence as a rule:
f1 = f2 = 1, f n+1 = fn + fn-1
Also define the rule in “everyday words.”
Plants do not know about this sequence – they just grow in the most efficient way. Many plants show the Fibonacci numbers in the arrangement of their leaves around the stem. Pineapples, pine cones, daisies and sunflowers show the number pattern in the number of spirals their seeds form around the buds on the ends of their stems. Palm trees show the pattern in the rings on their trunks.
Why do these arrangements occur? In the case of leaf arrangement, the growth pattern may be maximizing the space for each leaf or the average amount of light falling on each one. In the case of closely-packed seeds on a sunflower, the correct arrangement may be crucial for availability of space. The spirals are imperfect. The plant is responding to physical constraints, not a mathematical rule. The basic idea is that the position of each new growth is about 137.5 degrees away from the previous one, because it provides, on average, the maximum space for all shoots. This angle is called the golden angle and it divides the complete 360 degree circle in the golden section, 0.618033989…Take two consecutive Fibonacci Numbers, divide the smaller by the larger(for example, 55/89), multiple the result by 360 degrees and you get the measure of the external angle, 222.472 degrees. Subtract this from 360 degrees and you have the Golden Angle.
Show the slide of the three seed pods as this is being explained. The first pod is for a ¼ turn or 90 degrees. The second picture is with a 137.6 degree angle. The third pod has the golden mean angle and the two families of spirals are visible. Their numbers correspond to the numerator and denominator of one of the fractions which approximates the golden mean: 1/1, ½, 2/3, 3/5, 5/8, 8/13, 13/21, and so forth.
Fibonacci rabbits
The original problem was simulated to study how fast rabbits could breed in ideal circumstances. A pair of rabbits, one month old, is too young to reproduce but they can mate. When they are two months old, they reproduce and the cycle begins again. The rabbits reproduce every month after they first reproduce.
EXTEND
Fibonacci Rectangles
Each new square, added after the first two with side lengths of one unit, has side length equal to the sum of the previous two squares’ sides. Each of the side lengths will be Fibonacci numbers. By putting together quarter circles, one in each new square drawn, we create the Fibonacci Spiral. The spiral increases in size by a factor of Phi (1.618…) in a quarter of a turn (i.e., a point a further quarter of a turn around the curve is 1.618… times as far from the center. The Nautilus spiral turn takes a whole turn before points move a factor of 1.618… from the curve. These spiral shapes are called Equiangular or Logarithmic spirals.
Number of Petals on flowers
It is thought that flowers have Fibonacci numbers of petals because these petals grow off of the head which has spirals corresponding to Fibonacci numbers.
More flowers with Fibonacci number petals:
3 – lily, iris
5 – buttercup, wild rose, larkspur
8 – delphinium
13 – ragwort, corn marigold, cineraria
21 – aster, chicory
34 – plantain, pyrethrum
55 – michaelmas daisies
89 – the asteraceae family
EVALUATE
Show the remainder of slide show to check for understanding of several more Fibonacci number patterns in Nature with the plant and tree growth patterns.
References:
The Nature of Math by Michael Naylor at Departments: Integrating Math in Your Classroom. This web site describes grade-level appropriate activities for grades K through Algebra 1 which investigate mathematics in nature. Go to
The Fibonacci Series at
gives a nice background for the Fibonacci Math in Nature investigation.
Fibonacci Numbers in Nature at
provides extensive information about the Fibonacci series and the Golden Ratio with ideas to further your study of the “Mysteries of Mathematics.”
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