FAIR SHARES AND GEOMETRIC SPIRALS
Grades 4-8

The lesson opens with the problem of how to divide a triangular piece of land fairly among four people. After a fair method is established, one person leaves town and relinquishes his rights to the property. Dividing his piece of property leads to the notion of recursion and geometric spirals. The follow-up activities with centimeter rulers and a variety of regular polygons will let the students decide whether this method of fair division always works or if it only works in special cases. In the extension activities, the students will have an opportunity to look for other examples of spirals and exponential growth in nature and explore why nature favors a spiral growth pattern.

Math Talk #118: Getting into Shapes: Playing with Polygons

Students will be able to:

• identify when a polygon has been divided fairly;
• describe what is meant by recursion;
• find the midpoint of a segment accurately with a centimeter ruler;
• draw geometric spirals in a variety of regular polygons;
• identify spiral growth as it occurs in nature;
• explain why nature favors a spiral growth pattern.

• large rectangles of paper meant to simulate candy bars
• centimeter rulers, one per student
• worksheets with regular polygons drawn on them
• colored pencils or markers
• (optional) chambered nautilus - real shell or drawing of one
• (optional) a collection of other examples of spirals in nature

Ask the students, "What do we mean by 'fair' when we divide a candy bar?" The teacher should have some pre-cut rectangles of paper available to simulate a candy bar. When students give vague answers such as, "cut it in two," the teacher can cut the paper rectangle into two unequal parts. The teacher should lead the students to state their definition of "fair share" as precisely as possible.

Say, "Besides dividing candy bars and other kinds of food, what are some other situations in life that require that we be 'fair' with how we divide something?" The student responses might include ideas such as sharing the space in a bedroom with a sibling, sharing decisions about what TV show to watch with the rest of the family, splitting up chores around the house with other family members, taking turns playing with toys, inventing rules for games that are 'fair' and make the games more fun to play.

"One situation in life that might use the notion of 'fair share' is when a family meets to divide the estate left after a relative dies. We are going to watch a video segment about a family faced with just such a dilemma and see one way of resolving this problem."

To give students a specific responsibility while viewing, ask them, "As you watch the video about the brothers dividing up their land, what do you think would be a fair division of the ranch?"

BEGIN the video Math Talk #118: Getting into Shapes: Playing with Polygons when you see the word "Verboserosa" in a triangle. PAUSE after the sheriff says, "How do you divide up a three sided figure among four still-growing boys?" Allow time for the students to share their opinions by coming up to the chalkboard and illustrating how they would divide up the triangular piece of land. Keep these illustrations on the chalkboard so they may be referred to later in the discussion.

To refocus the students on the video say, "We are going to see the boys' suggestions for dividing the estate. Be prepared to explain why you think their suggestions are or are not fair." RESUME the video. PAUSE after the three solutions are shown and the script says, "...and Parnell gets this." Let the students draw the three solutions shown in the video on the chalkboard and discuss why each solution is or is not fair.

To refocus the students on the video ask, "How is Dirk's hint about the solution the same or different from the solutions we have already seen displayed by the class?" RESUME the video.

PAUSE when "Use Your Noodle" comes on the screen. Ask students if they think Dirk's solution is going to follow the same line of thinking as any of the solutions proposed by their classmates.

FAST FORWARD to just after the part where Dirk asks, "Have you figured it out?" Say, "Let's see how Dirk's solution compares to ours." RESUME the video. PAUSE after "It's for you, Parnell" in order to validate the solutions that the students gave that agreed with Dirk's.

FAST FORWARD to the scene that shows the Cartwrong boys gathered around the map of the divided land. Say, "You are going to hear a new word: recursion. It won't be defined for you but see if you can figure out what it means from the context." RESUME the video. STOP after the sheriff says, "Looks like a good example of recursion to me." Ask the students if they figured out what recursion means? Accept any comments that include the idea of repetition or doing the same thing over and over again.

Note to the teacher: In mathematics, an important part of the recursion process is that the place where we end a process becomes the place where we begin the process again. For example, if we start with 7, add 2, and end with 9, when we do a recursion we begin with 9, add 2, end with 11. Then we begin with 11, add 2, and end with 13. Recursive growth (or decay) occurs because we keep changing the beginning place but repeat the same process. It is not important that the students be able to articulate these finer points of recursion. As they practice recursion, the idea of beginning where they ended will appear intuitively.

"Let's be sure we understand how this recursive process worked." Give each student a centimeter ruler and a worksheet with a large equilateral triangle drawn on it. Have each student use the centimeter ruler to find the midpoint of each side of the triangle. Connect the three midpoints to form the four congruent triangles. Suggest that the students color the three outer triangles with three different colors so they can keep track of the spiraling inward. (See figure #1.) Have the students find the midpoints of each of the sides of the innermost triangle, connect these three points to form four smaller triangles, and color in the three outer triangles of this section. (See figure #2.) Show the students how the colored triangles are forming congruent areas so the triangle is being "fairly" divided. Continue this process until the sides become too small to measure.

Do students think that this method of dividing regular polygons will always create spirals that have the same area? Allow some time for sharing conjectures. Suggest that members of the class work with a variety of different regular polygons: squares, pentagons, hexagons, octagons to find the answer to this question. The teacher may wish to have these regular polygons pre-drawn on worksheets if the students have not already learned how to quickly draw these regular shapes.

Allow time for students to share their work with the class and conclude that this method of dividing a regular polygon always works. Ask the students if they can think of anything in nature that reminds them of these spirals.

All of the shapes we have worked with so far have featured the spiral moving in towards the center of the figure. Many shapes in nature follow the same recursive pattern of growth but spiral outward. One example of this is the chambered nautilus. The teacher can show the class an actual shell or a picture of one. When the chambered nautilus outgrows its shell, it builds on "an addition" and moves into the addition. The process is repeated again and again as the chambered nautilus repeatedly outgrows it shell. A typical nautilus moves through about 36 chambers during its lifetime.

A few other examples of geometric spirals in nature include the arrangement of seeds in the heads of sunflowers and daisies, the webs of certain spiders, and the Milky Way galaxy. There are many, many more examples which the students are encouraged to find on their own.

Have a banker or financial advisor visit the class and show how compound interest is a recursive process.

Have students research the interest rates that area banks are offering on savings accounts. Using a calculator, have them figure out what $1000 would be worth after three years using the various interest rates they found.

Ask a zoologist to share information about a collection of spiral shells.

A springtime field trip into the woods will uncover many examples of spirals, such as fiddlehead ferns, spider webs and some flowers. Students could make a log of their findings and accompany their descriptions with sketches.

Using the Internet, have students contact classes in other parts of the country or world to find out what kinds of plants or animals in their part of the world exhibit spiral growth patterns.

Math: Find the actual area of one of the spirals. If the original triangle has an area of one, then the area of the spiral equals f(1,4) + f(1,4) ¥f(1,4) + f(1,4) ¥f(1,4) ¥f(1,4) + f(1,4) ¥f(1,4) ¥ f(1,4) ¥ f(1,4) +. The students could find the areas of the spirals in other polygons. Instead of starting with an area of one, the students should find the actual area of the polygon they are working with in square centimeters.

Math: Students could investigate whether the pattern of equal areas holds in non-regular polygons.

Math: Students could investigate the difference between arithmetic spirals and growth patterns vs. geometric spirals and growth patterns.

Math: The solution to the problem in this video resulted because a three-sided polygon needed to be divided by four people. The problem became much simpler when the triangle only needed to be divided by three people. Challenge the students to find a simpler (more realistic) solution than making spirals.

Science: Students could look for more examples of plants, animals, and fossils that exhibit spiral growth, and discuss why growth so frequently follows a spiral pattern.

Science: Students could find more information about the chambered nautilus.

Language Arts/History: The geometric spiral approaches but does not touch the center of the polygon. The ancient Greeks were fascinated by this idea and created paradoxes to illustrate this concept of limit. You could read a paradox to the class.

Language Arts/multicultural: Another example of geometric growth is exponents. Students could investigate the idea of doubling and compare a similar tale as told in different cultures by these three books: A Grain of Rice by Helena Clare Pittman (New York: Bantam Skylark Books, 1992), The King's Chessboard by David Birch (New York: Dial Books for Young Readers, 1988) or The Rajah's Rice: A Mathematical Folktale from India adapted by David Barry (New York: W. H. Freeman & Company, 1994)

Master Teacher: Linda Dodge

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