AFRICAN INSTITUTE FOR MATHEMATICAL SCIENCES SCHOOLS ENRICHMENT CENTRE (AIMSSEC)
AIMING HIGH
PATTERNS is the theme for this INCLUSION AND HOME LEARNING GUIDE
This Guide suggests related learning activities for all ages from 4 to 17+ Just choose whatever seems suitable for your group of learners
The FROGS PUZZLE was designed for a wide age range Primary and Secondary
FROGS PUZZLE
Fivered-eyefrogsandfiveorangefrogslineupinarowwithaspacebetweenthem. Theymustchangeplaces.
Theycanhoponefrogoveranotherfrogorslidetoanemptyplacenexttothem.
Red-eyefrogscanonlymovetotherightOrangefrogscanonlymovetotheleft
Howcantheychangeplacesfollowingtheserules?
Trythepuzzlewith2frogsoneachsidefirst,thenwith3frogsoneachside…
Howmanymoveswillittake?
HELP Thinkabouttheorderinwhichthefrogsshouldmovetoavoidblockinginanyofthefrogssothattheycan’tmove.
NEXT FROGS AND TOADS Thisisacompetitivegamefor2playerswithalmostthesamerulesasFrogs.Placesomefrogsatoneendandsometoadsattheotherendofastripofcellswithjustoneemptyspacebetweenthem.Therulesformovingarethesameasforfrogs.Theplayerstakeitinturntomoveanyoneofthepieces.Aplayerlosesifhecannotmoveanyofthepieceswhenitishisturn. Whenyouhavetriedthepuzzlewithanequalnumberoffrogsoneachside,tryitanumberononesidedifferentfromtheotherside.Resources:Cutoutpaperfrogs2colours.
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INCLUSION AND HOME LEARNING GUIDETHEME: PATTERNS
Early Years and Lower Primary TrytosolvetheFROGSpuzzlebypretendingtobefrogs.Startbychoosing2peopletobered-eyefrogsand2peopletobeorangefrogs.Youneed5chairs.The2red-eyefrogssitatoneendandthe2orangefrogsattheotherendwithanemptychairinthemiddle.
. .
And
.
.
Red-eyefrogsmovethiswayandorangefrogsthisway.
Thefrogshavetochangeplaces
Allthefrogscaneitherhoportheycanslideintothenextspaceifitisempty.
.
Canyouandyourfriendssolvethispuzzle?Canyoufindouthowtofollowtherulesandmaketherightmovessothatyouchangeplaces?
Hereare6frogsthatyoucancutout.Canyousolvethepuzzlewiththepaperfrogsinthesamewayaswhenyouactedasfrogsinthegame?
Canyousolvethepuzzlefor3frogsoneachside?
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Upper Primary Startbytryingtosolvethepuzzleandtofindtherightmovesfor5frogsononesideand5ontheotherasonpage1.Whenthisprovesdifficulttalkaboutproblemsolvingmethods.Inparticularworkonthesimplestversionoftheproblemwith2frogsoneachside.Thisisnotascriptforahomelearningsession,ratheritwouldbebesttoadaptthisplanaccordingtolearners’responses.• Ifyouhaveenoughpeopleactouttheproblemwith3girlfrogsand3boyfrogsand
7chairs,orfewerifyouhavefewerpeopleinyourgroup.• Ifthelearnersfinditdifficult,try2boyfrogsand2girlfrogsonthechairsand
makesureeveryoneunderstands therules.• Perhapstry2boyfrogsand3girlfrogsnext.• Thisiscollaborative,notcompetitive–everyoneshouldtrytohelptheplayersto
maketherightmovesoastosolvethepuzzle.• Whentheyknowwhattheyhavetodolearnerscantrythisinpairsusingcutout
paperfrogsorcounters.
Whenlearnershaveworkedoutthesequenceofmovessothattheycandothechange-overswithoutgettingstuck,suggestthattheyrecordtheirresultsinatableasbelow.Thinkaboutsimplecases:e.g.2boyand2girlfrogsFillinatableofresults.Howmanymoves?Howmanyhops?Howmanyslides?Isthereapattern?Canyouexplainthepattern?
Number of boy frogs
Number of girl frogs
Number of hops
Number of slides
Number of moves
1 2
2 2
2 3
3 3
3 4
4 4
4 5
5 5
Beforetheendofthesessionhaveawholegroupdiscussionaboutthestrategiesforsolvingthepuzzleandthenumberpatternsthatthelearnershavefound.
Onlygoasfarasthelearnersarecomfortablewith.Donottellthemtheruleorformula.Youcansuggestthattheymightliketoplaythegameagainjustforfunandtrytofindmorepatternsandanswers,andmaybearule.The‘rule’canbegiveninwords,itdoesnothavetobeinaformula.
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Lower and Upper Secondary Forsecondarylearnersthisexerciseleadstowardsalgebra.Agoodanswerfromthelearnerswouldbeaformulainwordslike:
”‘youmultiplythenumberoffrogsofeachtypetogetthenumberofhops andyouaddthenumberoffrogsofeachtypetogetthenumberofslides”.
Someyounger‘highflyers’, andtheyear9learners,maybeabletoexpresstheformulaalgebraicallyandevengiveaproof.
• Witholderplayersintroducetheactivityby5boysand5girlsactingitout,withhelpfromthe class.
• Whentheyfindittoodifficultaskforsuggestionsforworkingontheproblem.
• Iftheplayershaveexperienceofproblemsolvingtheyshouldsuggesttryingsimplecases,ifnot theteacherwillneedtosuggestit(andpointoutthatthisisageneralstrategyforproblem solving).
Key questions • Canyoufindaformula(rule)inwordsforthenumberofhops?• Canyoufindaformula(rule)inwordsforthenumberofplacesmoved?• Canyoufindaformula(rule)inwordsforthenumberofslides?• Canyoufindaformula(rule)forthenumberofmoves?• Canyouprovetheformulaalwaysworks?• Canyoufindaformulainsymbols?EdouardLucas(1842-1891)inventedthispuzzlewhichisalsoknownastheLucaspuzzle.HewasaFrenchmathematician,wellknownforhisinventionsofmathematicalpuzzlesandrecreations,andhealsoinventedtheTowerofHanoipuzzle.https://mathshistory.st-andrews.ac.uk/Biographies/Lucas/
SOLUTION TheformulaforthenumberofmovesforRred-eyefrogsononesideandGorangefrogsontheotherisR+G+RG.
ThisisR+GslidesandRGhops.
Proof
Eachfrogmusthopexactlyonceoverallthefrogsoftheothertype(orbehoppedover)so thereareRGhops.
TheRfrogsononesidemustmove(G+1)placesinonedirectionandGfrogsontheothersidemustmove(R+1)placesintheotherdirection.SothetotalnumberofplacesmovedisR(G+1)+G(R+1)=R+G+2RG.
Witheachhopafrogmoves2placessowithRGhopsthefrogsmoveatotalof2RGplaces.ThenumberofplacesmovedisR+G+2RGsothereare(R+G)slides.
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Why do this activity?Patternsareattheheartofmathematicsand,inthisactivity,learnersdeveloptheirproblemsolvingskills.Theproblemstosolvearefirsttofindapatternofmovesthatwillsucceedinthe2setsoffrogsswappingplaces,thensecondlytofindnumberpatternsinthenumbersofmovesandthirdlytofindaformula(orrule)thatgivesthenumberofmovesasafunctionofthenumbersofeachtypeoffrog.Thephysicalengagement(peoplemathsactivity)promoteskinaestheticlearning.Iflearnersworkinpairsandsmallgroupstheydeveloptheircommunicationandreasoningskillsthroughdiscussion.Usingatabletorecordresultshelpslearnerstoworksystematically.
Learning objectives Indoingthisactivitystudentswillhaveanopportunityto:• developlogicalreasoningandproblemsolvingskills;• developcommunicationskillsthroughtalkingroupsandasaclass;• (forolderlearners)practisedescribinganumberpatterninanalgebraicformula.
Generic competences Indoingthisactivitystudentswillhaveanopportunityto:• thinkmathematically,reasonlogicallyandgiveexplanations;• thinkflexiblyandapplyknowledgeandskills;• interpretandsolveproblems;• collaborateandworkwithapartnerorgroup.
Follow up FROGS AND TOADS
Thisisacompetitivegamefor2playerswithalmostthesamerulesasFrogs.Placesomefrogsatoneendandsometoadsattheotherendastripofcellswithjustoneemptyspacebetweenthem.Therulesformovingarethesameasforfrogs.Theplayerstakeitinturntomoveanyoneofthepieces.Aplayerlosesifhecannotmoveanyofthepieceswhenitishisturn.
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