…LEARN ABOUT THEAPPEARANCE OF

DIFFERENT NUMERICALSEQUENCES AND

SYMMETRY IN NATURE,AND HOW THESE RELATE

TO THE WIDER WORLD

TODAY YOU

WILL...

Fibonacci, fractals, fraud prevention andmore – Dr Joanna Rhodes brings biology

and maths together with some trulyabsorbing activities…

52 www.teachsecondary.com

LESSONPLAN

NATURE BYNUMBERS

MAIN ACTIVITIES

It is Summer 2011 in Tennessee in theUnited States. A man plays his banjoon the veranda and crickets chirp inthe long grass. As the evening drawsnear millions and millions of cicadasin full chorus suddenly drown out themusic. The man puts his banjo down.There is little point trying to play overthem, he remembers from the lasttime this happened in 1998. Heknows this will happen again in 2024. What is so special about this pattern?The local cicadas have evolved tobreed to a 13-year cycle [AdditionalResource 1]. In other regions cicadasmass breed to 7 and 17-year cycles,

all of which are prime numbers.Numbers in nature are ubiquitousand fascinating. The same codes thathelp cicadas to avoid predators formthe basis of the encryption that keepsour credit cards and personal detailssafe from fraudsters.In this lesson, ideas about nature andnumbers are presented with strongcross-curricular links between biologyand mathematics at KS4. Studentsexplore How Science Works and howit relates to the wider world bringingin themes such as fraud prevention,data encryption, art, architecture and music.…

Use whole sunflower heads(fresh or dried), a cauliflower, arange of fresh flowers, pinecones and apples cut in halfacross the middle as physicalresources for students toexamine. Introduce the taskusing the video Nature byNumbers [AR3], which is adramatic introduction of theoccurrence of key numbers andangles in biological systems andis set with stunning imagery tomusic. A further video withnarrated explanation suitable forKS4 level students is calledNature and Numbers [AR4].Explain to students that theFibonacci series is created whenthe two previous numbers areadded together to give thesequence 1,1,2,3,5,8,13,21 and soon. Students should write outtheir own Fibonacci series anduse it to spot individual numbersfrom the series in nature. This is

LAY THEFOUNDATIONS

1It is helpful to begin by exploring theidea of simple symmetry withstudents. Students can discuss thehuman form and its symmetry. Mosttextbook illustrations show the bodyvery close to symmetrical so pupilscould take measurements fromimages to investigate this. However,a real human body is far fromsymmetrical. Students will find itintriguing to investigate how theywould appear if their face were trulysymmetrical. To help there is auseful internet tool called Symface[AR2]. This allow students to seehow some well-known celebritieswould look if they had symmetricalfacial features. They can upload apicture of themselves to generatetheir own results.

STARTER ACTIVITY

Lesson plan Science Qx_Layout 1 20/12/2012 17:26 Page 1

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FRACTALSFractals are one of the more recentdiscoveries in the field ofmathematics. Uncovered in the late1960s by Benoit Mandelbrot,fractals are geometric figures thatare made up of patterns and repeatthemselves at smaller scalesinfinitely. Fractals are also commonin everyday life, in nature, ourbodies, and even in popularculture. To help studentsunderstand the concept of a fractalit is useful to look at a fern. Eachfrond of the fern is a copy of theshape of the main plant. This was

+ STRETCH THEM FURTHER+ STUDENTS COULD PLOT THEIR OWNLOGARITHMIC SPIRAL. A WORK SHEETTO HELP CAN BE FOUND ON THEDISCOVERY EDUCATION WEBSITE [AR5].FOR FURTHER CROSS-CURRICULARLINKS YOU COULD DISCUSS HOWRECTANGLES WITH FIBONACCIDIMENSIONS ARE USED IN ART ANDARCHITECTURE. USE THE EXAMPLESSUCH AS LEONARDO DA VINCI’SPAINTINGS AND GRAHAMSUTHERLAND’S TAPESTRY IN COVENTRYCATHEDRAL. IN ARCHITECTURE THEEDEN PROJECT IN CORNWALL USEDFIBONACCI DIMENSIONS AS DID THEPARTHENON BUILT BY THE ANCIENTGREEKS AND GIZA GREAT PYRAMIDBUILT BY THE EGYPTIANS. IT IS EVENSAID THAT MOZART USED THESENUMBERS IN COMPOSING HIS MUSIC. ITALSO FEATURES IN POPULARLITERATURE AND CONSPIRACYTHEORIES SUCH AS THE DA VINCI CODEBY DAN BROWN. ASK STUDENTS WHATOTHER EXAMPLES THEY CAN FIND INNATURE AND THE ARTS [AR11].

INFO BAR

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Cicadas have evolved to breed in a prime number cycle and the relationship between prime numbers is asmathematically elusive to us as it is to the predators that the cicadas manage to outwit with their unusualchoice of cycle. This year Shinichi Mochizuki of Kyoto University in Japan may have come up with amathematical proof, which allows us to work with prime numbers in a way that nature has evolved to donaturally. This could revolutionise the security of encrypted data such as bank and personal details as well asimproving national security and communications between nations. Ask students to come up with their ownway to encrypt a message. They will need to provide a key so that the intended recipient can decode themessage. As an additional challenge ask them to use principles from the natural world that they haveinvestigated during the lesson in order to do this.

SUMMARY

especially rewarding if physicalresources have been provided. Auseful worksheet is available fromDiscovery Education (linked to theDiscovery Channel) and can bedownloaded along with fullinstructions for use [AR5]. Imagesof flowers and seed heads arehelpfully reproduced for studentsto refer to if completing questionsas homework. Finally, brainstormsome animals with spiral features(the shape is similar to the spiralof the snail, the nautilus, andother seashells). Ask students toresearch what function the spiralcould serve?

generated mathematically byMichael Barnsley and zooming inand out is demonstrated well in anumber of videos available online[AR6]. Romanesco broccoli is alsoa useful resource where it is clearto see that each of the smallersprouts around the vegetableresembles the broccoli as a whole[AR7]. The Fractal Foundationrecommend a number of different“Fractivities” that provide anengaging introduction to fractals innature [AR8]. One of theseinvolves drawing a fractal tree,which generates a formremarkably similar to a real tree.Students practise measuring thedistance between branches, andthen determine the ratio. Theyalso measure the angles of thebranches, and identify as obtuseor acute. Another engagingactivity is a fractal scavenger hunt[AR9]. Encourage students tophotograph examples that theyfind always looking for self-similarity as with a fern (whereeach leaf also looks like the wholefern) and trees (where eachbranch looks like a smaller versionof the whole tree). Although theseactivities are mathematical innature they should be accessibleto all pupils. A specific task whichmay be suitable to engage lowerability pupils is to use toothpicksand mini marshmallows to create afractal tetrahedron [AR10].

HOME LEARNING+ Ask students to solve thefollowing problem [AR5]: Femalehoneybees have two parents, amale and a female, but malehoneybees have just one parent,a female. Can you draw a familytree for a male and a femalehoneybee? What patternemerges? Are they Fibonaccinumbers? (The male bee has 1parent, and the female bee has 2parents. The male bee has 2grandparents, and the femalebee has 3 grandparents. Themale bee has 3 great-grandparents, and the femalebee has 5 great-grandparents.The male bee has 5 great-great-grandparents, and the femalebee has 8 great-great-grandparents. The male bee has8 great-great-great-grandparents,and the female bee has 13 great-great-great-grandparents.)

+ Imagine you have arrived as ascientist to an alien planet that isinhabited by extra-terrestrial floraand fauna. Where might you lookfor the Fibonacci series? Designyour own imaginary beings thathave features from the Fibonacciseries. Do you think thesecreatures look more realistic thanwhere numbers not found inFibonacci are used?

+SCIENCE | KS4

Dr Joanna L. RhodesM.Chem, D.Phil, MRSC teaches atShelley College,Huddersfield.

+ ADDITIONALRESOURCESAR1: PRIME NUMBER CICADASTINYURL.COM/3CFKK3SAR2: SYMFACE WEBSITETINYURL.COM/N5UFJLAR3: NATURE BY NUBMERS MUSICALCLIP TINYURL.COM/YBC7JXYAR4: NATURE AND NUMBERSNARRATED CLIPTINYURL.COM/CRFCGRG AR5: DISCOVERY EDUCATION –FIBONACCI NUMBERS IN NATURETINYURL.COM/D9A7769AR6: MICHAEL BARNSELY FERNTINYURL.COM/CHVG8HOAR7: FRACTALS IN NATURETINYURL.COM/D5L2QM4AR8: THE FRACTAL FOUNDATION –FRACTIVITIESTINYURL.COM/C9WX5CNAR9: FRACTAL SCAVENGER HUNTTINYURL.COM/YBKAWKXAR10: FRACTAL TETRAHEDROMSTINYURL.COM/C2OJWV5AR11: FIBONACCI NUMBERS AND THEGOLDEN SECTION IN ART,ARCHITECTURE AND MUSICTINYURL.COM/7A5HU4H

+ ABOUT THE EXPERT

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