sign in sign up
<<
Home , Patterns in nature

’s Numbers Educator’s Guide

Nature’s Numbers Nature’s Numbers 1 Educator’s Guide

Dear Educator, Welcome to the Educator’s Guide for the traveling exhibit “Nature’s Numbers”. We hope you and your class will enjoy your trip. Our goal is to generate enthusiasm and a love of math by helping your students see that math is everywhere, even in nature. To that end, the activites in this guide are arranged to support the following ideas:

• Nature has some of the same found in math • Mathematical Inquiry • Exploring early math principles

The guide contains pre-trip and post-trip activities and a trip sheet for each grade level, but you can mix and match the activies to adapt the guide to your own classroom. We also have an annotated list of book and websites.

Have fun at “Nature’s Numbers”!

The Franklin Institute 2 Table of Contents

✸ Description of exhibit...... 3 ✸ K-4 Activities and trip sheet ...... 4 ✸ 5-8 Activities and trip sheet ...... 7 ✸ 9-12 Activities and trip sheet ...... 11 ✸ Additional resources ...... 14

Math in Nature

The of spots on your dog, the delicate of butterfly wings, the angular shapes of , and the rugged- ness of a coastline all involve math. “Nature’s Numbers” helps students understand the connections between math and nature with concrete, hands-on interactive exhibits.

The exhibit focuses on three main themes: Symmetry and patterns, Sectioning, and Mathematical Inquiry. Many things in nature are symmetrical, such as plants or snow-flakes, and follow a regular pattern. Other natural objects can be sectioned, or divided into different parts, much like a corncob into kernels. Mathematical inquiry forms the basis for our explorations, whether it’s investigating random patterns of chaos or finding different ways to solve a puzzle. 3 K-4 Activities

Pre-Trip Activity: Shape Claymation Objective: To discover how 2D and 3D shapes fit and relate to each other. Materials: Clay, plastic knives, pictures of shapes Discuss Before the Activity: Have students name different shapes, both 2D and 3D, around the classroom. Things to Do: 1. Try to find the pictured 3D solids around your classroom. 2. Have your students make solids out of clay. 3. Slice solids. What new 3D shapes do you create? What 2D shapes do you see in the face of the sliced parts? 4. Have students draw the new shapes. 5. Put the solids back together and try slicing in a different way.

cylinder cube rectangular solid cone

Extensions: ✸ Cut out pictures from magazines and make a collage of natural things and unnatural things. ✸ Cut various two-dimensional shapes from construction paper and have students use the shapes to make all the winter objects they can think of, such as from triangles, snowmen from circles, and houses from rectangles and triangles. After the Activity: Ask students what they discovered. What can they say about the way different shapes fit together to make a 3D shape? During-Trip Activity to Nature’s Numbers: (see attached trip sheet to guide your visit) Post-Trip Activity: Making Patterns Objective: Create patterns using squares and triangles. Materials: squares, triangles Discuss: Draw a pattern on the chalkboard and ask the children to describe the pattern using the names of the shapes. Repeat with another pattern. Things to do: 1. Distribute the triangle and square blocks. Have the students create a pattern using the triangles and squares, and draw and color their patterns on paper. Repeat the activity using different shapes. 2. Students can write about a pattern that they use every day.

Extensions: ✸ Grow rock candy crystals and examine the resulting shapes. http://www.exploratorium.edu/cooking/candy/recipe-rockcandy.html ✸ Examine patterns on objects in nature, by creating a pattern booklet. 4 K-4 Activities

Pre-Trip Activity: Nature Walk Objective: Discover the many ways math is present in nature. Materials: Optional: magnifying glass Things to Discuss After the Activity: What can students say about math in nature? Do they recognize anything in nature seen in the exhibit? Things to Do: 1. Pick a nature spot. It can be anywhere—the schoolyard, a neighborhood park, or the beach. 2. Look carefully at plants and animals. Examine their shapes, count the , petals, and legs present. Look for symmetry. During-Trip Activity to Nature’s Numbers: (see attached trip sheet to guide your visit) Post-Trip Activity: Symmetry Objective: Cut with varying numbers of line of symmetry. Materials: Paper, scissors, pencils Things to Discuss: Review the definition of symmetry. State that symmetry is when you have some-thing that is the same on both sides. Then ask how can we create line symmetry. Demonstrate by drawing a shape with line down the middle of it on the board. This shows that each half is a mirror image of the other. Things to Do: 1. Fold a square piece of paper in half lengthwise, widthwise or diagonally. 2. Trace a pattern on one half of the paper. Cut it out. Or, just start cutting. You can cut on the fold, but make sure you leave some of the fold left. 3. Open it up! How many lines of symmetry does your snowflake have (1)? 4. Can you make a snowflake with two lines of symmetry? Three? How can you make a six-sided snowflake.

© 1994 Encyclopedia Britannica

Extensions: ✸ Cut into different shapes, press into paint and make patterns. ✸ Make a kaleidoscope. ✸ Explore symmetry by manipulating pictures of students’ own faces. http://regentsprep.org/regents/math/symmetry/photos.htm 5 K-4 Activities

“Nature’s Numbers” K-4 Trip Sheet

1. Find and draw something that reminds you of a butterfly. Why does it remind you of a butterfly?

2. Find one of the puzzles in the exhibit and try to solve it. Draw the solution. How did you figure out the answer? 6 5-8 Activities

Pre-Trip Activity: Symmetry Objective: Explore the concept of bilateral symmetry and discover its relevance in nature. Materials: hand mirrors, objects from nature with symmetry (shells, , , ), geometrical shapes with 3,4,5, and 6 sides. If you divide a symmetrical object, the line Things to Discuss: down which you divide that object is called a 1. Pose the question, “What do you know about symmetry and how we can tell if line of symmetry an object has symmetry?” Discuss a definition. 2. Use a volunteer as a visual example on in the human body. Ask the students, “Which way could we divide him so that the two halves would be the same? Lead a discussion on how our bodies are not perfectly symmetrical due to uneven facial features, and asymmetrical haircuts. Things to Do: 1. Divide the class into teams. Distribute mirrors. Ask each group to investigate if the shape has a line of symmetry by using their mirrors. When the shape is (If don’t have natural objects use copies of placed in front of mirror, does its reflection complete the whole shape? Does it pictures. Pictures can be folded to test for symmetry) have more than one line of symmetry? Are there pattern from your findings? (The number of sides and number of lines of symmetry will be the same.) 2. Distribute natural objects and look for the lines of symmetry. Extensions: ✸ While on a walk, have students find natural and man-made symmetry, like traffic signs. ✸ Download SymmeToy, a Windows program for creating paint patterns, symmetry roses, tessellating , and so forth: http://www.hufsoft.com/software/page4.html During-Trip Activity to Nature’s Numbers: (see attached trip sheet to guide your visit) Post-Trip Activity: Kaleidoscope Objective: See symmetry in the familiar form of a kaleidoscope. Materials: 35mm black plastic film canister, cardboard, overhead projector trans- parency black, construction paper, matte board, margarine tub lid, pin, markers Things to Do: 1. Cut three 7/8" x 1 7/8" rectangles out of the cardboard, transparency, and construction paper. Cut one rectangle of the same size out of the matte board. 2. Puncture a 1/4" diameter eyehole in the center bottom of the film canister. 3. Slide the three rectangles of cardboard into the canister so they form a triangle. Do the same with the construction paper, then the transparency. Set aside. 4. Use the markers to make a colorful design on the outside of the margarine lid. 5. Push the straight pin through the center of the margarine lid then stick it into the edge of the 7/8" side of one of the pieces of matte board. Slide the matte board into the film canister along side one of the cardboard pieces. 6. Hold the kaleidoscope up to the light and slowly turn the wheel. The three mirrors symmetrically reflect the pattern on the lid, which produces the kaleidoscope effect. Extensions: ✸ Build a Butterfly Kaleidoscope. http://www.carolina.com/elementary/activities/kaleidoscope/kaleidoscope.asp ✸ Have students' describe in a journal what they are seeing (lines of symmetry) in the kaleidoscope designs. 7 5-8 Activities

Pre-Trip Activity: Generalized Sequences Objective: To understand Fibonacci sequence and how it is expressed in nature. Materials: pictures of flower petals, cauliflower florets, pinecones, and heads Things to Do: 1. Write this sequence 1 1 2 3 5 8 on the board. Ask the students to find the Good site for all things Fibonacci; good illustrations. pattern. (Add the previous two numbers to get the next number in the sequence). http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fi Have students calculate 12 more terms of the sequence. Tell students that bonacci/fibnat.htm this sequence has intrigued mathematicians for centuries and it has appeared in different patterns in nature. 2. Divide students into groups. Distribute illustrations and ask them to answer these questions: (The numbers vary, but they are all Fibonacci numbers.) a. Flower petals: Count the number of petals on these flowers. Are they Fibonacci numbers? (Lilies and irises have 3 petals, buttercups have 5 petals and asters have 21 petals; all are Fibonacci numbers.) b. Seed heads: Each circle represents a seed head. Do the circles form ? Start from the center; find a going towards the left. How many seed heads can you count? c. Cauliflower florets: Locate the center of the cauliflower. Count the number of florets that make up a spiral. Are they Fibonacci numbers? d. Pinecones: Look carefully; do the seed cases make spiral shapes? Find as many spirals as possible. How many seed cases make up each spiral? Are they Fibonacci numbers? Things to Discuss: 1. Ask which shape emerges the most often from the clusters of (the spirals). Discuss if there are advantages to this shape. (Seeds may form spirals because this is an efficient way of packing the most number of seeds into an area). 2. Ask where else they see the spiral shape in nature ( shell). Are those spirals also formed from Fibonacci numbers? Are these shapes pleasing? To conclude, discuss other pleasing patterns in nature, such as leaves, branches, etc. Do they have mathematical basis? Extensions: ✸ Video “Mathematical Eye: Fibonacci and Prime Numbers” Available from TRS ✸ Video On Demand at http://vod.opb.org/ ✸ Further explore Fibonacci numbers in nature. Challenge students to find other patterns of numbers in crystals and rocks, in the distance of from the sun and so on.

During-Trip Activity to Nature’s Numbers: (see attached trip sheet to guide your visit) 8 5-8 Activities

Post-Trip Activity: Fibonacci Art Objective: To use the Fibonacci numbers to create unique artwork. Materials: Paper, 8 1/2” x 8 1/2” white paper, rulers, pencils, crayons Things to Discuss: In nature, plants with spiral -growth patterns have spiral ratios that are adjacent terms in the Fibonacci sequence. In a daisy, the ratio is 21 to 34, while in a pine cone, the ratio is 5 to 8. In many wallpapers and floor designs, Fibonacci ratios are used because they are pleasing to the eye. Things to Do: 1. To create a pattern, students will take the paper and fold it in half, then unfold. Fold it in half another way, then unfold it. Do this two more times in different directions. There will be four creases intersecting the middle. 2. Use a ruler to trace a straight line between the halfway points (creases) on the edges of your square. Repeat this with each new square until you are in the center of the paper. 3. Shade in one of the smallest triangles in the center. Then shade in one of the large triangles touching it. Continue in the same directions, shading larger triangles until you reach the outside of the paper. Extensions: ✸ Walk around the school’s neighborhood look for Fibonacci patterns in archi- tecture and buildings.

example of Fibonacci Art 9 5-8 Activities

“Nature’s Numbers” 5-8 Trip Sheet

1. Logarithmic Spirals Can you replicate the pattern (nautilus shell)? Draw it below. Then using Fibonacci sequence, divide the spiral into a set of rectangles. List the sequence you used below.

2. Rosetta Kaleidoscope Using the mirrors, how can you make one of the images whole? At what degree does the mirrors have to be at to make it whole? (120 degrees,180 or 360) Explain your answer. 10 9-12 Activities

Pre-Trip Activity: Making Geometric Puzzles Objective: Create and solve puzzles using geometric shapes Materials: rulers, thick paper (card stock), scissors Things to Discuss: Give students the T-puzzle and have them try to solve it. Discuss what makes the puzzle difficult. What edges do we assume must be next to another piece? Why? Things to Do: 1. Have students choose a letter or number. 2. Give paper rulers and pencils to students to make their own puzzle. 3. Have students draw the outline of their letter or number, then divide the character into sections, and finally cut along the puzzle lines 4. Have students share their puzzles with others.

T puzzle

Extensions: ✸ The Japanese would challenge other to solve geometric puzzles, they called it sahgaku. During-Trip Activity to Nature’s Numbers: Try some out here (see attached trip sheet to guide your visit) http//matcmadison.edu/is/as/math/kmirus/reference/San Gaku.html Post-Trip Activity: How Many Steps Does it Take Objective: Use the towers of Hanoi puzzle as an introduction to series Materials: dime, penny, nickel, quarter and paper with three circles on it. Things to Discuss: Discuss the puzzle Tower of Hanoi and its rules. The puzzle consists of different sized disks (in this case the coins) and 3 poles (the circles on the paper). Stack the coins form largest to smallest and put them on one circle. Try to move the stack to a new pole but only move one disk at a time and do not put a larger disk on a smaller disk. See solution in the solutions packet. Things to Do: 1. Break the students into teams. Starting with 1 disk and working their way up to 4 discs, each group should find the least number of moves needed to move a pile of disks to a new pole. 2. Have students report back the results and find the lowest number of moves for each number of disks. Have them search for the pattern and calculate the next 3 numbers in the series. 3. Allow students some time to try to create an equation that calculates the least number of moves needed. It might help to begin by naming the variables like X1 = 1(number of moves for 1 disk), X2 = 3(number of moves for 2 disks) The equation should be Xn=2Xn-1+ 1 where n = number of disks. You move the old pile to the middle pole (Xn-1), the new bottom disk (+1)to the far pole and then move the old pile back onto the bottom disk(Xn-1)in it’s new location. Extensions: ✸ Make a computer model that predicts the number of moves for large numbers of disks. http://www.math.toronto.edu/mathnet/games/towersmath.html 11 9-12 Activities

Pre-Trip Activity: Make a Objective: Make your own fractal tree. Materials: graph paper, ruler, colored pencils, pictures of real trees and fractal trees Things to Discuss: Begin by introducing the definition of . Then discuss how a tree can be seen as a fractal because each branch has smaller branches “branching” off of it. Have the group come up with other things in nature that are fractal. Show students examples of real trees and fractal trees. Things to Do: 1. Have students come up with the “line” they will repeat. It should look like a trunk or branch and be very simple to repeat at a small scale. Students should draw their pattern so it extends across half the graph paper. This will be the trunk. 2. Students should then draw copies of the branch at 1⁄3 the size to the top of their trunk. (Students can decide how many branches they want to add but they should always add the same number) 3. Students should continue to draw branches to the top of each old branches at 1/3 the size of the previous branch. 4. Students can change the colors of the branches to look like flowers or leaves in fall. Extensions: ✸ Check out these fractal trees at: http://www.math.union.edu/research/fractaltrees/FractalTreesDefs.html ✸ Make a computer model that makes fractal trees. Try out this one on-line: http://id.mind.net/~zona/mmts/geometrySection/fractals/tree/treeFractal.html Post-Trip Activity: Koch Snowflake During-Trip Activity to Nature’s Numbers: (see attached trip sheet to guide your visit) Materials: graph paper, pen Things to Discuss: Talk about what fractals the group saw in the exhibit. Discuss the difference between area and perimeter. What would the area and perimeter of a figure that contains small replicas of the whole be like? Things to Do: 1 Begin by drawing an equilateral triangle in the middle of the graph paper. 2. Divide each side of the triangle into equal thirds. 3. Draw three more triangles with a base as the middle third of each side of the original triangle. 4. Repeat this step with each triangle 5. Once the snowflake is created have students determine the area and the perimeter. Next have the students determine the perimeter and area when the triangles repeat infinitely. While there is a limit to the area, the perimeter will be infinitely long! Extensions: ✸ Try making another fractal in nature like a or just a beautiful shape. ✸ Try making a computer program to create a Koch snowflake. Try this one on-line: http://math.rice.edu/~lanius/frac/koch/koch.html

12 9-12 Activities

“Nature’s Numbers” 9-12 Trip Sheet

1. Try the T-puzzle Draw the solution

2. Go to Wild Weather: This exhibit shows something called Sensitive Dependency on Initial Conditions. This means if you vary where the pendulum starts initially, even the smallest amount, it can change the path of the pendulum swings a great deal.

Pull the 2 pendulums from the same point. How many swings does it take for the two pendulums to be going in different paths? Attempts # of swings 1st attempt 2nd attempt 3rd attempt

Start the two pendulums from different points. How many swings does it take for the two pendulums to be going in different paths? Attempts # of swings 1st attempt 2nd attempt 3rd attempt

3. Towers of Hanoi What is the least number of moves you can make to solve the puzzle? Attempts # of moves 1st attempt 2nd attempt 3rd attempt Additional Resources 13

Books: Beyer, Jinny. Designing Tessellation: The Secrets of Interlocking Patterns. Contemporary Books, 1999. Grades 5-8 The author shows how the combination of pattern and symmetry can result in stunning geometric designs.

Devlin, Keith. Life By the Numbers. John Wiley & Sons, 1998 Grades 5-8 This book focuses on the role plays in everyday life.

Devlin, Keith. Mathematics: The Science of Patterns. Owl Books, 1996. Grades 6-12 To most people, mathematics means working with numbers. But as Keith Devlin shows in Mathematics: The Science of Patterns, this definition has been out of date for nearly 2,500 years. Mathematicians now see their work as the study of patterns real or imagined, visual or mental, arising from the natural world or from within the human mind.

Garland, Trudi Hammel. Fascinating Fibonaccis. Dale Seymour Publications, 1987 Grades 6-9 This is a really excellent book - suitable for all, and especially good for teachers seeking more material to use in class.

Sitomer, Mindel and Harry. Spirals. (1974). New York: Thomas Y. Crowell Company. Grades K-4 Explore spirals in nature and in the manmade world with this fabulous book.

Time Life Books. Right In Your Own Backyard: Nature Math. (1992). USA: Time Life Inc. Grades 2-6 Fun stories and activities, good information, and great pictures help students see math in nature.

Tompert, Ann. Grandfather Tang’s Story. (1990). New York: Crown Publishers. Grades K-4 Grandfather Tang tells a story about fox fairies using Chinese tangrams, or seven shapes that can be flipped to form different pictures. Students will enjoy making their own tangrams.

Wyatt, Valerie. The Math Book For Girls and Other Beings Who Count. (2000). Toronto: Kids Can Press Ltd. Grades 2-6 Check out easy activities and connections to real life applications of math in this fun story featuring mini-mathematical loving Nora. Additional Resources 14

Websites:

http://www.snowcrystals.com Check out beautiful pictures and great information about snowflakes.

http://www.montessoriworld.org/Handwork/foldingp/snowflak.html This website has great instructions to cut different types of snowflakes.

http://plus.maths.org/issue3/fibonacci This website has brief biography Fibonacci with pictures.

http://www.branta.connectfree.co.uk/fibonacci.htm This British site by Brantacan explains Fibonacci numbers and how they are related to flowers, pinecones, , palm trees, suspension bridges, spider webs, dripping taps, CDs, your savings account and quite a few other things.