The Nature of Math, by Michael Naylor

Your students will be astounded by just how often math appears in nature

Can your students find Fibonacci numbers when looking at a pineapple?

The wonders of mathematics in nature will surely amaze your students. These fun math activities are only the beginning!

Leaf sort (Grades K-1)
If you're lucky enough to have fall leaves on your school grounds, go on a leaf hunt and have your students find three of their favorite leaves. You can also assign students to bring in three leaves from around their home. Place them in a pile and decide as a class how to sort the leaves – by size? By color? By number of holes? Students can sort them in a number of different ways, focusing on the attributes of the leaves.

Next, ask your students to decide on one or two attributes they like in a leaf, like "small and red." Have them draw a picture showing many different kinds of leaves, and circle the ones that match their attributes.

Symmetry in nature (Grades K-3)
Leaves have nice bilateral symmetry, meaning there's a line of symmetry right down the middle. Ask your students to find pictures in magazines or on the Internet that show reflective symmetry in nature. Examples include people, plants, animals and astronomical objects. Students can make a collage of their pictures with the lines of symmetry drawn.

Fibonacci hunt (Grades 3-6)
Fibonacci numbers show up often in nature. The Fibonacci Sequence is named for Leonardo Fibonacci, an Italian mathematician who lived from 1170-1250. The sequence begins 1, 1, 2, 3, 5, 8, 13, 21... with each term being the sum of the previous two terms (8 = 3+5, 13 = 5+8, 21 = 8+13, etc.).

Give your students the first eight terms of the sequence and challenge them to find the rule and extend the pattern.These numbers show up often in plants, pinecones, sunflowers, artichokes, celery, lilies and daisies. Ask your students to find examples of these numbers in natural objects outside or at home.

It's fun to bring in a pineapple from the grocery store and demonstrate the Fibonacci numbers. Looking at the bottom of the fruit, you can easily see spirals in the patterns. A pineapple makes for an excellent demonstration – the pattern of polygons on the husk forms several spirals that can be followed in different directions. Count the number of spirals in different directions (masking tape helps), and you'll find they are Fibonacci numbers. You'll probably find 13 spirals in one direction, eight in another and five in a third direction.

Golden ratio sequences (Grades 6-8)
Ask your students to find the ratios between consecutive Fibonacci numbers:

1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.666...
8/5 = 1.6
13/8 = 1.615384...
... etc.

Point out that the ratios are going up and down but getting closer and closer to some value. That value is about 1.61803... an irrational number that is called "the golden ratio." The number has fascinated mathematicians and artists for millennia.

Now have your students create their own "Fibonacci-like" pattern by choosing two starting numbers and applying the same rule to build a sequence. For example, if one chooses 3 and 7 as the starting numbers, the sequence is 3, 7, 10, 17, 27, 44, 71, 115, etc.

Ask your students to find the ratios between the terms of their own sequence. They will be surprised to find that their sequence, and everyone else's in the room, also converges to the golden ratio! Wow!

Phi = the golden ratio (Grades 7-8)
The precise value of this ratio is, and is often abbreviated as the Greek letter phi, which looks like this: Ø. Have your students punch up this value on their calculators.

• Have them find out what happens if they square this number. 1.61803 x 1.61803... = 2.61803...; or "Ø squared = Ø + 1." Weird!

• Ask them to figure out what happens if they take the inverse of this number. 1/1.61803... = 0.61803...; or 1/Ø = Ø - 1. Even weirder!

As you can see, these are two

Golden rectangle (Grades 4-8)
A rectangle with side lengths in the ratio 1:Ø is said to be the most beautiful rectangle of all and has inspired painters and sculptors throughout the ages. There are many ways to construct golden rectangles; here's a fun way that uses Fibonacci numbers.

Supply your students with graph paper. Have them outline a 1 x 1 square near the center of the paper. Add a 1 x 1 square adjacent to the first, then a 2 x 2 square so that it touches both of the 1 x 1 squares. Continue to add squares, circling around the outside of the figure. In each square, students write in the side length – the lengths form the Fibonacci sequence.

Golden spiral (Grades 4-8)
When the page is filled, your students can draw a quarter circle in each square in order to form an approximate "golden spiral" as shown below. This spiral is related to the shape of ram's horns, sea- shells and other spirals found in nature.

This is just a small taste of the wonders of mathematics in nature. Try searching the Internet for "the golden ratio" – you'll find pages and pages of ideas.

Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.