nestedshapes.html

NESTED SHAPES LESSON

In the "Finding Patterns" section of the Beyond Numbers exhibit, visitors explore ratios of spiraling squares that directly relate to the Fibonacci series of numbers. The Fibonacci series is composed of numbers, the next of which is equal to the sum of the two preceding numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 and so on. Visitors also explore natural spirals and ratios with a nautilus puzzle.

NCTM Standards and/or AAAS Benchmarks addressed:

Mathematics as Corrections:
Students use mathematics in other curriculum areas. Students use mathematics in their daily lives.

Patterns and Function:
Students describe, extend, analyze, and create a wide variety of patterns.

Geometry and Spatial Sense:
Students develop spatial sense. Students recognize, describe, extend and create a wide variety of patterns.

The Mathematical World, Symbolic Relationship:
Students learn that similar patterns may show up in many places in nature and in the things people make.

VOCABULARY

Fibonacci series - the list of numbers beginning with 0 and 1, the next of which is equal to the sum of the two preceding numbers, 0, 1, 2, 3, 4, 5, 8, 13, 21, 34, are the first several numbers.

score - lightly cut a line in thick paper so that it bends easily.

INTRODUCTION

In three of the following four activities students will explore curving patterns. In the second activity students will use addition to write the Fibonacci series that describes generations of honeybees. Each activity stands alone or can be done in sequence with the others. In the first activity, students draw a simple spiraling pattern on paper. In activity three, they construct a series of curves that builds on the Fibonacci series. In the last activity they will construct and connect polyhedrons to see the geometry of the growth patterns that cause some natural objects, such as a ram's horn, to spiral.

"The key to maintaining the spiral growth of the shell is to allow the outer surface, the surface farthest from the axis around which the coiling takes place, to grow more than the inner surface."
-Peter S. Stevens [Patterns in Nature, page 88]

OBJECTIVES

Students will construct two-dimensional and three-dimensional nested shapes to simulate spiraling patterns of growth.

MATERIALS

Natural objects or pictures of natural objects showing spirals -- sea shells, pineapples, sunflower head, eddies, narwhal's tooth, curling palm fronds, hurricanes, plant tendrils (morning glory, peas, honey suckle, grape), DNA, etc.
Activity 1 materials: crayons or markers, square pieces of paper -- origami paper is excellent for this activity. Older children may construct their own squares.
Activity 2 materials: pencils, graph paper, the larger the better.
Activity 3 materials: scissors, cutouts reproduced on card stock paper, transparent tape.

PROCEDURE

Activity 1- Spiral Pattern

Show students examples of spirals in real and/or pictured natural objects. Point out that mathematicians study patterns and that mathematics can describe the patterns of a spiral. Tell students that they will be making spiraling shapes and give these directions (for older students, you may wish to distribute copies of printed directions for them to follow.)

Spiral Pattern DIRECTIONS

Take a square piece of paper and fold it in half, then unfold it. Fold it in half another way, then unfold it. Can you find two other ways to fold it in half? Do so. Your paper should have four creases intersecting in the middle.

Use a ruler to trace a straight line between the half-way points on the edges of your square (where the crease meets the edge of the paper). Now you have a smaller square. Trace straight lines between the halfway points on the edge of your new square (where the creases meet the edge of the first square). Now you have a smaller square. Repeat this as far as you can into the center of the square.

Shade in one of the smallest triangles in the center. Then shade in one of the large triangles touching it. Continue in the same direction, shading larger and larger triangles until you reach an outside triangle.

BULLETIN BOARD

Put a 4 x 4 completed pattern in the middle of a black-papered bulletin board. Provide meter sticks and pencils. Allow students to extend the pattern outward.

Activity 2 - Fibonacci series

Have students continue the series that starts 1,1 by adding the two numbers to find the next. Then have them add the last two numbers to find the next, and so on. If you like to sing with students, teach them the Fibonacci Fractal Fugue that is written in the On-the-Bus Activity section of this manual.

Point out that mathematicians have noticed Fibonacci patterns in many mathematical relationships and in many features of nature.

Post this family tree for a honeybee on the chalk board. With honeybees, the male bee has only one parent, but females have two. (Male bees have half the number of chromosomes that females have) Point out the number pattern shown on the right.

Activity 3 - Fibonacci Squares
In the middle of a large piece of graph paper, outline a 1 x 1 square (a). Above it, outline another 1 x 1 square (b). Use these two squares as one side of a new square (c - which will be 2 x 2). Use the three squares as one side of a new square (d - which will be 3 x 3). Repeat the process and you will be making a spiraling set of squares that represent a Fibonacci series.
In each square, connect diagonal corners with arcs as shown.
Activity 4 - Rams horn spirals
Distribute one card stock copy of each of the three cut-outs (at the end of this lesson) to each student. Have students use ball point pens or sharp pencils to score on the fold lines of the cut-outs before they're cut out. Then have them cut on the outlines of the figures. Crease them with the scored lines inside. Have them put white glue on the shaded tabs to connect the figures. Have the students make several of each shape, then use transparent tape to connect the shapes, having them meet at the arrows.
Note that shape 1 forms a straight line, shape 2 forms a curve, and shape 3 forms a spiraling shape. Have students explain why they think shape three forms its shape. Point out that on a ram's horn, the outside grows more quickly than the inside, so the length of the outside of one month's growth is longer than the inside.

Class Management Tip:

Glue may be dispensed in plastic soda bottle caps and spread with toothpicks.

Have students locate pictures of other growing things that make a spiral.

ASSESSMENT
Have students describe how they constructed their shapes.