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 Five red-eye frogs and five orange frogs line up in a row with a space between them. They must change places. They can hop one frog over another frog or slide to an empty place next to them. Red-eye frogs can only move to the right Orange frogs can only move to the left How can they change places following these rules? Try the puzzle with 2 frogs on each side first, then with 3 frogs on each side … How many moves will it take? HELP Think about the order in which the frogs should move to avoid blocking in any of the frogs so that they can’t move. NEXT FROGS AND TOADS This is a competitive game for 2 players with almost the same rules as Frogs. Place some frogs at one end and some toads at the other end of a strip of cells with just one empty space between them. The rules for moving are the same as for frogs. The players take it in turn to move any one of the pieces. A player loses if he cannot move any of the pieces when it is his turn. When you have tried the puzzle with an equal number of frogs on each side, try it a number on one side different from the other side. Resources: Cut out paper frogs 2 colours.
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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.
GototheAIMSSEC AIMING HIGH websiteforlessonideas,solutionsandcurriculumlinks:http://aiminghigh.aimssec.ac.zaSubscribetotheMATHS TOYSYouTubeChannelhttps://www.youtube.com/c/mathstoysDownloadthewholeAIMSSECcollectionofresourcestouseofflinewith
theAIMSSECAppseehttps://aimssec.appFindtheApponGooglePlay.
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