Det bedste sted - en undersøgelse med cirkler

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1 Det bedste sted - en undersøgelse med cirkler Ján 1 Šunderlík Introduction og Eva Barcíková In optics when it comes down to show path of rays of light through glass, lenses or systems of lenses, m physics teachers groan experiments are quite complex, and you need a lot of 1 Indledning equipment. It is difficult enough to show a ray of light in air you need smoke, dust or or way of making light visible. To show path of light in materials, you need special equipment smoke Bratislava, glass lenses Slovakiets etc. Now hovedstad, that snot always er en lille available, storby, and der adjustments ligger ved Donaufloden to system og can har usually mange only broer. be En done af turistattraktionerne by removing er piece også and udsigten putting fra anor disse piece broer. in. På To figur see 1 what ses tre happens af broerne if you ovenfra. make a De lens er ikke thicker, lige gamle. you have Til venstre to take out er den ældste current bro, lens som and også put in hedder new det: one. Den Students ældste Bro. can n observe I midten er den situation before change and after change but it is not exactly a gradual change that lets m nyeste bro, der hedder Apollobroen. Den sidste bro hedder Havnebroen. observe how path of light actually changes. We want to demonstrate how you can show path of light through a lens with help dynamic of geometry software (DGS). material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction not insteadof actual experiment (if one sees experiments only in simulation, pedagogicvalue is not quite same), but complementingit. It can as well be useful for mamatics teachers. Well, now where is mamatics? There is a lot of it in re! If a ray of light hits glass surface of an optical lens, a part of it gets reflec ted back in a certain angle, and anor part penetrates glass and continues re, in anor angle. The same happens when light reaches or surface of lens again mamatics is required to calculate angle in which light is reflected and refracted. For ideal lenses, re is an easy equation calculating se near centre of lens. With thicker lenses and light being more off-centre, calculations become more complex, and from equations alone it would be difficult to see what happens. With DGS it is possible to simulate properties of a lens without actually having to use a lens, laser light, etc. But even for DGS, we need mamatics to create simulation in first place. angle ofincidence (between ray of light and normal) is equal to angle of reflection: Fig.1: Tre broer i Bratislava set ovenfra Som helt almindelig turist kan man gå ad Donaupromenaden, som vi har farvet rød. Promenaden er ca. 455 m lang, og der er tre udsigtspunkter. Fig.1 Reflection Som of en light (ualmindelig) at a plane turist surface. kan man spørge sig selv, hvor er det "bedste udsigtspunkt", hvis man vil se Apollobroen (figur 2). project has been fund ed with support from European Commission in its Lifelong Learning Programme project has been funded with support from European Commission in its Lifelong Learning Programme only of and only of rein. and made of information rein.

2 In optics when it comes down to show path of rays of light through glass, lenses or systems of lenses, m physics teachers groan experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air you need smoke, dust or or way of making light visible. To show path of light in materials, you need special equipment smoke glass lenses etc. Now that snot always available, and adjustments to system can usually only be done by removing one piece and putting anor piece in. To see what happens if you make a lens thicker, you have to take out current lens and put in new one. Students can n observe situation before change and after change but it is not exactly a gradual change that lets m observe how path of light actually changes. We want to demonstrate how you can show path of light through a lens with help dynamic of geometry software (DGS). material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction not insteadof actual experiment (if one sees experiments only in simulation, pedagogicvalue is not quite same), but complementingit. It can as well be useful for mamatics teachers. Well, now where is mamatics? There is a lot of it in re! If a ray of light hits glass surface of an optical lens, a part of it gets reflec ted back in a certain angle, and anor part penetrates glass and continues re, in anor angle. The same happens when light reaches or surface of lens again mamatics is required to calculate angle in which light is reflected and refracted. For ideal lenses, re is an easy equation calculating se near centre of lens. With thicker lenses and light being more off-centre, calculations become more complex, and from equations alone it would be difficult to see what happens. With DGS it is possible to simulate properties of a lens without actually having to use a lens, laser light, etc. But even for DGS, we need mamatics to create simulation in first place. 2 Easy Fig. 2. beginnings Apollobroen (Wikimedia) light hits a plane surface Vi 2.1 kan Reflection godt blive enige om, at det "bedste sted" vil være stedet med den største vinkel, som Apollobroen When a ray kan of ses light fra. hits Der a plane er tre glass udsigtspunkter surface, a part (1, 2, of 3). it is Hvilket reflected. The er law ud fra of reflectionsaysthat vores antagelse bedst, hvis man angle vil se ofincidence Apollobroen? (between ray of light and normal) is equal to angle of reflection: Hvis nu du ser dig selv om en ualmindelig turist, dvs. du ligner en videnskabsmand mere, end du troede, eller måske snarere en problemløser. Den proces, vi nu skal igennem, er en spørgeteknik, som er en metode, der svarer til den måde, som matematikere arbejder på. Er du klar? Så lad os begynde. 2 Opgave 1 Hvordan kan man finde svaret på det foregående spørgsmål? Hvor på promenaden er det bedste udsigtspunkt, hvis man vil se Apollobroen? 2.1 Sådan kan det løses Vi kan forenkle spørgsmålet med en model. Den er lavet ud fra den omstændighed, at broen rører ved flodbredden. Vi forlænger derfor Fig.1 linjen Reflection langs promenaden of light at a plane og konstruerer surface. en linje, der ligger vinkelret project has been fund ed with support from European Commission in its Lifelong Learning Programme project has been funded with support from European Commission in its Lifelong Learning Programme only of and only of rein. and made of information rein.

3 på broen. Det betyder, at alle steder langs med promenaden har en fælles stråle, og vi flytter bare toppunktet Andreas af Ulovec udsigtsvinklen tættere på broen. Det er således ikke svært at forstå, at jo tættere toppunktet er på broen, desto større er vinklen. Hvad så med de punkter, der ligger meget tæt på en af broens sider? Hvilken vinkel nærmer den sig (konvergerer)? Prøv på internettet at finde den bedste vinkel, hvis man skal se på en genstand. In optics when it comes down to show path of rays of light through glass, lenses or systems of lenses, m physics teachers groan experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air you need smoke, dust or or way of making light visible. To show path of light in materials, you need special equipment smoke glass lenses etc. Now that snot always available, and adjustments to system can usually only be done by removing one piece and putting anor piece in. To see what happens if you make a lens thicker, you have to take out current lens and put in new one. Students can n observe situation before change and after change but it is not exactly a gradual change that lets m observe how path of light actually changes. We want to demonstrate how you can show path of light through a lens with help dynamic of geometry software (DGS). material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction not insteadof actual experiment (if one sees experiments only in simulation, pedagogicvalue is not quite same), but complementingit. It can as well be useful for mamatics teachers. Well, now where is mamatics? There is a lot of it in re! If a ray of light hits glass surface of an optical lens, a part of it gets reflec ted back in a certain angle, and anor part penetrates glass and continues re, in anor angle. The same happens when light reaches or surface of lens again mamatics is required to calculate angle in which light is reflected and refracted. For ideal lenses, re is an easy equation calculating se near centre of lens. With thicker lenses and light being more off-centre, calculations become more complex, and from equations alone it would be difficult to see what happens. With DGS it is possible to simulate properties of a lens without actually having to use a lens, laser light, etc. But even for DGS, we need mamatics to create simulation in first place. Fig. 3 Udsigt til Apollobroen fra udsigtspunkter på Donaupromenaden (Foto fra Google Earth) angle ofincidence (between ray of light and normal) is equal to angle of reflection: Fig. 4 Udsigtspunkt C Fig. 5 Udsigtspunkt B Fig. 6 Udsigtspunkt A 2.2 Bemærkninger til læreren Vi kan nemt lade de studerende undersøge spørgsmålet og selv komme med løsninger. Det kan vil være en god ide at give dem en fotokopi af kortet eller lade dem udforske spørgsmålet vha. et geometriprogram. Det er nemt af finde et billede af det hele ovenfra med det gratis program Google Earth. Geometriprogrammet kunne f.eks. være GeoGebra. Det er vigtigt at opfordre de studerende til at formulere deres egne tanker Fig.1 og spørgsmål, Reflection of som light kan at a hjælpe plane surface. dem med at finde en løsning. Det er project has been fund ed with support from European Commission in its Lifelong Learning Programme project has been funded with support from European Commission in its Lifelong Learning Programme only of and only of rein. and made of information rein.

4 vigtigt, at de ikke får løsningen lige med det samme. Man kan også bede de studerende formulere deres Andreas forklaring Ulovec og lade dem skrive den ned på et stykke papir. In optics when it comes down to show path of rays of light through glass, lenses or systems of lenses, m physics teachers groan experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air you need smoke, dust or or way of making light visible. To show path of light in materials, you need special equipment smoke glass lenses etc. Now that snot always available, and adjustments to system can usually only be done by removing one piece and putting anor piece in. To see what happens if you make a lens thicker, you have to take out current lens and put in new one. Students can n observe situation before change and after change but it is not exactly a gradual change that lets m observe how path of light actually changes. We want to demonstrate how you can show path of light through a lens with help dynamic of geometry software (DGS). material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction not insteadof actual experiment (if one sees experiments only in simulation, pedagogicvalue is not quite same), but complementingit. It can as well be useful for mamatics teachers. Well, now where is mamatics? There is a lot of it in re! If a ray of light hits glass surface of an optical lens, a part of it gets reflec ted back in a certain angle, and anor part penetrates glass and continues re, in anor angle. The same happens when light reaches or surface of lens again mamatics is required to calculate angle in which light is reflected and refracted. For ideal lenses, re is an easy equation calculating se near centre of lens. With thicker lenses and light being more off-centre, calculations become more complex, and from equations alone it would be difficult to see what happens. With DGS it is possible to simulate properties of a lens without actually having to use a lens, laser light, etc. But even for DGS, we need mamatics to create simulation in first place. angle ofincidence (between ray of light and normal) is equal to angle of reflection: project has been fund ed with support from European Commission in its Lifelong Learning Programme project has been funded with support from European Commission in its Lifelong Learning Programme only of and only of rein. and made of information rein.

5 2.3 Matematiske begreber der udvikles Sammenligning af vinkler Matematisk modellering In optics when it comes down to show path of rays of light through glass, lenses or systems of lenses, m physics teachers groan experiments are quite complex, and you need a lot of 3 Opgave equipment. It is 2 difficult enough to show a ray of light in air you need smoke, dust or or way of making light visible. To show path of light in materials, you need special equipment smoke Det glass er meget lenses populært etc. Now blandt that snot turister always at available, tage en sejltur and adjustments på Donaufloden. to Så system lad os can også usually gøre only det. be På sejlturen done by har removing igen mulighed one piece for and at putting se Apollobroen anor piece fra forskellige in. To see what synsvinkler. happens Hvor if you er make det "bedste a lens thicker, you have to take out current lens and put in new one. Students can n observe sted" situation at se Apollobroen before change fra, hvis and vi kommer after fra change Den but gamle it is not Bro? exactly a gradual change that lets m observe how path of light actually changes. We want to demonstrate how you can show path of light through a lens with help dynamic of geometry software (DGS). material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction not insteadof actual experiment (if one sees experiments only in simulation, pedagogicvalue is not quite same), but complementingit. It can as well be useful for mamatics teachers. Well, now where is mamatics? There is a lot of it in re! If a ray of light hits glass surface of an optical lens, a part of it gets reflec ted back in a certain angle, and anor part penetrates glass and continues re, in anor angle. The same happens when light reaches or surface of lens again mamatics is required to calculate angle in which light is reflected and refracted. For ideal lenses, re is an easy equation calculating se near centre of lens. With thicker lenses and light being more off-centre, calculations become more complex, and from equations alone it would be difficult to see what happens. With DGS it is possible to simulate properties of a lens without actually having to use a lens, laser light, etc. But even for DGS, we need mamatics to create simulation in first place. angle ofincidence (between ray of light and normal) is equal to angle of reflection: 3.1 Sådan kan det løses Fig. 7. Den gamle Bro (i forgrunden) og Apollobroen (i baggrunden) Hovedspørgsmålet er, hvordan vi kan finde den største vinkel, som indeholder to punkter. Vi kan igen prøve at udarbejde (bruge) en matematisk model, som ikke tager højde for begrænsninger ved det menneskelige øje og andre faktorer. Derefter vil vi prøve at se mere realistisk på spørgsmålet. Udsigtsvinklen er det største vinkel, hvor en genstand kan ses med acceptabelt resultat. Størrelsen af vinklen øges, når toppunktet nærmer sig centrum af segmentet MN. Lad os kalde dette centrum for S. Fra punkt S er der samme afstand til punkt M og N. Men inden for den matematiske model kan vi fortsat overveje andre steder, der er lige langt fra begge slutpunkter. Vi taler om et uendeligt antal steder. Vi kunne betegne dem cirkler, der går igennem punkt A og B. project has been fund ed with support from European Commission in its Lifelong Learning Programme project has been funded with support from European Commission in its Lifelong Learning Programme only of and only of rein. and made of information rein.

6 In optics when it comes down to show path of rays of light through glass, lenses or systems of lenses, m physics teachers groan experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air you need smoke, dust or or way of making light visible. To show path of light in materials, you need special equipment smoke glass lenses etc. Now that snot always available, and adjustments to system can usually only be done by removing one piece and putting anor piece in. To see what happens if you make a lens thicker, you have to take out current lens and put in new one. Students can n observe situation before change and after change but it is not exactly a gradual change that lets m observe how path of light actually changes. We want to demonstrate how you can show path of light through a lens with help dynamic of geometry software (DGS). material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction not insteadof actual experiment (if one sees experiments only in simulation, pedagogicvalue is not quite same), but complementingit. It can as well be useful Fig. 8 Vinkelret halveringslinje, der er lavet i Geogebra Fig. 9. Vinkelret halveringslinje på et ark for mamatics teachers. Well, now where is mamatics? There is a lot of it in re! If a ray of papir light hits glass surface of an optical lens, a part of it gets reflec ted back in a certain angle, and anor part penetrates glass and continues re, in anor angle. The same happens when Det light andet reaches hovedpunkt or derfor surface at finde of det lens geometriske again mamatics sted, som is centrene required af to de calculate cirkler, der angle går in igennem which slutpunkterne light is reflected for Apollobroen and refracted. M og For N, ideal aftegner. lenses, re is an easy equation calculating se near centre of lens. With thicker lenses and light being more off-centre, calculations Undersøgelse become more complex, and from equations alone it would be difficult to see what happens. With Vi kan DGS bruge it is possible GeoGebra to simulate og vælge en properties passer (under of a værktøj lens without - cirkel actually og bue having værktøj) to use, hvor a lens, vi indstiller laser light, etc. But even for DGS, we need mamatics to create simulation in first place. radius hen ad vejen og derefter nærmer os cirklen på punkt M og N. Centrene af cirklerne danner et geometrisk sted, som vi kan forsøge at definere efter undersøgelsen. Det er bedst at have mindst tre cirkler. Vi kan 2.1 definere Reflection det via den induktive proces med at finde det geometrisk sted. Det geometriske sted af punkter, der er lige langt fra to punkter, er den vinkelrette halveringslinje. angle ofincidence (between ray of light and normal) is equal to angle of reflection: project has been fund ed with support from European Commission in its Lifelong Learning Programme project has been funded with support from European Commission in its Lifelong Learning Programme only of and only of rein. and made of information rein.

7 Fig. 10. Vinkelret halveringslinje lavet med værktøjet "passer" og "Midtnormal" halveringslinje på kortet i GeoGebra (Foto fra Google Earth) Vi kan In optics også when fortsætte it comes med punkter, down to der show er forskellige path of fra rays den of vinkelrette light through halveringslinje. glass, lenses or systems of lenses, m physics teachers groan experiments are quite complex, and you need a lot of Vi equipment. kan udvikle It ideen is difficult med midtpunkt enough to af show segmentet. a ray of Lad light de in air studerende you need smoke, undersøge dust vinklerne or or i en cirkel, way der of har making centrum light i midtpunktet visible. To show af segmentet path of MN. light Lad in materials, os kalde you cirklen need k 1 special. Hvad equipment er vinklen smoke MSN? Vi kan glass nu vælge lenses et etc. hvilket Now som that snot helst always punkt, available, f.eks. X and på cirklen. adjustments Hvor stor er system vinklen? can Hvad usually med only andre be punkter? done by Hvordan removing kan one vi definere piece and den putting centrale anor vinkel piece og forholdet in. To see til what den perifere happens vinkel? if you make Størrelsen a lens af thicker, you have to take out current lens and put in new one. Students can n observe den perifere vinkel er halvdelen af den centrale vinkel i cirklen. situation before change and after change but it is not exactly a gradual change that lets m Ud observe fra denne how viden path og informationerne of light actually changes. om en passende We want vinkel to demonstrate fra opgave how 1 kan you vi can finde show de steder, path of hvorfra light through vi kan se a broen lens with i denne help vinkel. dynamic of I forbindelse geometry med software undersøgelsen (DGS). vil vi gerne finde ud af, hvor stor vinkel material β can sammenlignet be useful for med science den centervinkel. teachers, who Størrelsen can use it af to model vinklen experiments øges, når toppunktet with lenses, S nærmer reflection sig langs and refraction aksen af segmentet not insteadof MN til centrum. actual experiment Derefter (if skal one cirklen sees experiments k mindst have only et in punkt simulation, pedagogicvalue is not quite same), but complementingit. It can as well be useful fælles for mamatics med linje c. teachers. Den maksimalt Well, now tilladte where is tilnærmelse mamatics? af punkt There S til is linjen a lot of MN it in forekommer, re! If a ray når of cirklen light kun hits rører glass ved linjen surface c i of et an punkt optical X. lens, Dette a vil part være of it vinklen gets reflec MSN, ted back og derfor in a certain vinklen angle, MXN and den maksimale. anor part penetrates glass and continues re, in anor angle. The same happens when light reaches or surface of lens again mamatics is required to calculate angle in which light is reflected and refracted. For ideal lenses, re is an easy equation calculating se near centre of lens. With thicker lenses and light being more off-centre, calculations become more complex, and from equations alone it would be difficult to see what happens. With DGS it is possible to simulate properties of a lens without actually having to use a lens, laser light, etc. But even for DGS, we need mamatics to create simulation in first place. angle ofincidence (between ray of light and normal) is equal to angle of reflection: 3.2 Bemærkninger til læreren Fig. 11. Center- og periferivinkel I forbindelse med undersøgelser kan arbejdet nogle gange føre ind i en blindgyde. Det er nødvendigt at være klar til at understøtte de studerendes undersøgelsesarbejde med andre startpunkter eller inden for nøglepunkter, som kan bringe dem videre i arbejdet. 3.3 Matematiske begreber der udvikles Matematisk modellering Vinkelret halveringslinje - det geometriske sted Forholdet mellem central og perifer vinkel project has been fund ed with support from European Commission in its Lifelong Learning Programme project has been funded with support from European Commission in its Lifelong Learning Programme only of and only of rein. and made of information rein.

8 Cirkel tegnet rundt om trekant 4 Opgave 3 Havnebroen In optics when er ikke it comes nær så down spændende to show som Apollobroen. path of rays of Men light det through er den glass, bro, lenses der leder or systems den hurtige of trafik lenses, hen over m Donaufloden. physics teachers Det er groan det første experiments adgangssted are for quite rejsende, complex der, and kommer you need med a bil lot eller of bus fra equipment. motorvej D1 It is for difficult at se enough den gamle to show by. a ray Og of endnu light in en air you gang: need hvor smoke, er det dust bedste or sted or til way at se Apollobroen of making fra light Havnebroen? visible. To show path of light in materials, you need special equipment smoke glass lenses etc. Now that snot always available, and adjustments to system can usually only be done by removing one piece and putting anor piece in. To see what happens if you make a lens 4.1 thicker, Sådan you have kan to det take løses out current lens and put in new one. Students can n observe situation before change and after change but it is not exactly a gradual change that lets m En observe mulighed how kunne være path of at bruge light actually konceptet changes. om det We geometriske want to demonstrate sted. En anden how er you at can bruge show teorien om path of "power light point" through på a cirklen. lens with Lad os help koncentrere dynamic of geometry os den software første. (DGS). material can be useful for science teachers, who can use it to model experiments with lenses, Vi kan foretage nogle indstillinger og bruge GeoGebra til at lave en model over situationen. Vi lægger reflection and refraction not insteadof actual experiment (if one sees experiments only in kortet simulation, over broerne pedagogicvalue fra Google Maps is not ind quite i GeoGebra same), og undersøger but complementingit. spørgsmålet It can her. as well Vi starter be useful med at konstruere for mamatics en linje teachers. c på den Well, ene now side where af broen is mamatics? og et punkt X There på denne is a lot linje. of it in Derefter re! If bruger a ray of vi værktøjet light hits "Vinkel" glass i GeoGebra surface of og an måler optical vinklen lens, a MXN. part of Hvad it gets nu, reflec hvis ted punktet back in X a (der certain ligger angle, på linjen and s) kan anor flytte sig part langt penetrates linjen c? Flyt glass derefter and continues med punktet re, in X anor langs linjen, angle. The same og se, happens hvad der when sker med vinklen. light reaches Kan du forklare or hvorfor? surface of lens again mamatics is required to calculate angle in which light is reflected and refracted. For ideal lenses, re is an easy equation calculating se Hvordan effects kan but vi this finde is just ud a af, model, hvor and vinklen it does er størst? work well only with thin lenses and with light falling in Hvordan near hænger centre udsigtsvinklen of lens. With sammen thicker med lenses afstanden and light fra genstanden? being more off-centre, Hvor er det punkt calculations på Havnebroen, become more hvorfra complex, der er and den from største udsigtsvinkel equations alone alt efter it would afstanden be difficult fra genstanden to see what - Apollobroen? happens. With DGS it is possible to simulate properties of a lens without actually having to use a lens, laser light, Forestil etc. But dig even f.eks. for et punkt DGS, i we floden. need Hvordan mamatics er det to create forbundet simulation med cirklerne, in hvis first centre place. er på den vinkelrette halveringslinje? Vi undersøgte det i spørgsmål 2. Vi ved fra spørgsmål 1, at centrum af cirklen 2 Easy ligger beginnings på den vinkelrette light halveringslinje, hits a plane men vi surface ved ikke præcist hvor. Vi leder efter centrum S x for cirklen k x, der vil have berøring med linjen s og gå gennem punkt M og N. Hvordan kan vi finde ud af, hvor vinklen er størst? Vi When kan lade a ray de of studerende light hits a komme plane glass med surface, forslag, a eller part of vi it kan is reflected. bruge The tidligere law of reflectionsaysthat forslag. Lad f.eks. de angle ofincidence (between ray of light and normal) is equal to angle of reflection: studerende undersøge spørgsmålet vha. cirkler i GeoGebra, forsøge at få cirklen til at nærme sig punkterne MN og linjen c. Vi kan derefter tale med de studerende om, at det formentlig opfylder vores formål, men at det ikke er matematisk nøjagtigt. Lad os gå baglæns - flytte vores tanker fra slutresultatet og gå bagud. Den proces vil føre os til en løsning. Vi har brug for at undersøge den endelige løsning. Vi skal se efter et geometrisk sted med samme afstand fra linjen og punkterne M eller N. Hvilket geometrisk sted ligger lige langt fra punkt M og fra linjen c? Metatænkning og metakommentarer er meget væsentlige element som en del af undersøgelsesprocessen. Vi kan dele spørgsmålet op i to. A) Hvilket geometrisk sted beskriver centrum af de cirkler, der rører ved to faste punkter? B) Hvilket geometrisk sted beskriver centrum af de cirkler, der rører ved en fast linje, og som går igennem et fast punkt? project has been fund ed with support from European Commission in its Lifelong Learning Programme project has been funded with support from European Commission in its Lifelong Learning Programme only of and only of rein. and made of information rein.

9 A) Vi undersøgte det i spørgsmål 2. B) Til at begynde med bruger vi en induktiv tilgang for bedre at kunne forstå konceptet. Først skal vi overveje, hvilken udsigtsvinkel er størst? Vi kan In optics tegne when cirkler it comes på planet, down der to rører show ved linjen path og of rays et punkt. of light through glass, lenses or systems of lenses, m physics teachers groan experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air you need smoke, dust or or way of making light visible. To show path of light in materials, you need special equipment smoke glass lenses etc. Now that snot always available, and adjustments to system can usually only be done by removing one piece and putting anor piece in. To see what happens if you make a lens thicker, you have to take out current lens and put in new one. Students can n observe situation before change and after change but it is not exactly a gradual change that lets m observe how path of light actually changes. We want to demonstrate how you can show path of light through a lens with help dynamic of geometry software (DGS). material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction not insteadof actual experiment (if one sees experiments only in simulation, pedagogicvalue is not quite same), but complementingit. It can as well be useful for mamatics teachers. Well, now where is mamatics? There is a lot of it in re! If a ray of light hits glass surface of an optical lens, a part of it gets reflec ted back in a certain angle, and anor part penetrates glass and continues re, in anor angle. The same happens when light reaches or surface of lens again mamatics is required to calculate angle in which light is reflected and refracted. For ideal lenses, re is an easy equation calculating se Fig. effects 12 Undersøgelse but this is af just parabler a model, i Geogebra and it does work Fig. well 13 only Undersøgelse with thin lenses af paraboler and with på light papir falling in near centre of lens. With thicker lenses and light being more off-centre, calculations For become at definere more det complex, ønskede and geometriske from equations sted vil vi alone gerne it would bruge den be difficult viden, to vi see har what fået om happens. cirkler With og DGS it is possible to simulate properties of a lens without actually having to use a lens, laser light, geometrisk konstruktion. (Denne tilgang er for erfarne matematikstuderende, der kan bruge deres etc. But even for DGS, we need mamatics to create simulation in first place. viden i nye situationer til at undersøge nye koncepter.) Vi kan 2 Easy nu gå beginnings fra kortet over til light den hits universelle a plane model, surface fordi vi ikke ved præcist, hvor punktet S (centrum af den ønskede cirkel) ligger. 1: Vi kan vælge et punkt og kalde det punkt M, der er givet på planet. (Vi kan ikke flytte det. Det er i virkeligheden When a ray end of light af broens hits a ender.) plane glass Hvad surface, er det punktsæt, a part of der it is er reflected. lige The langt law fra of et reflectionsaysthat givet punkt? angle ofincidence (between ray of light and normal) is equal to angle of reflection: Det er en cirkel med centrum i punkt M og en konstant afstand r - radius. Fig. 14 Undersøgelse af paraboler - trin 1 Fig. 15Undersøgelse af paraboler - trin 2 project has been fund ed with support from European Commission in its Lifelong Learning Programme project has been funded with support from European Commission in its Lifelong Learning Programme only of and only of rein. and made of information rein.

10 Forstil dig nu et punkt på cirkelperiferien, der kan flyttes frit rundt. Lad os vælge et punkt og kalde det punkt Andreas S. Dette Ulovec punkt ligger på cirklen l. Tilsvarende vælger vi punktet M også på cirklen k, der har samme centrum som S. Tilføj nu linjen p til dine overvejelser. Denne linje er tangent til cirklen i punkt T, som er forskelligt fra punkt In optics M og when ligger it på comes cirklen down k. Fordi to show p tangerer path cirklen of rays k of ved light vi, through at ST er glass, vinkelret lenses på linjen or systems p. Ud of fra definitionen af den cirkel, vi talte om i begyndelsen af vores overvejelser, ved vi også, at SM = ST. lenses, m physics teachers groan experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air you need smoke, dust or or way of making light visible. To show path of light in materials, you need special equipment smoke glass lenses etc. Now that snot always available, and adjustments to system can usually only be done by removing one piece and putting anor piece in. To see what happens if you make a lens thicker, you have to take out current lens and put in new one. Students can n observe situation before change and after change but it is not exactly a gradual change that lets m observe how path of light actually changes. We want to demonstrate how you can show path of light through a lens with help dynamic of geometry software (DGS). material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction not insteadof actual experiment (if one sees experiments only in simulation, pedagogicvalue is not quite same), but complementingit. It can as well be useful for mamatics teachers. Well, now where is mamatics? There is a lot of it in re! If a ray of light hits glass surface of an optical lens, a part of it gets reflec ted back in a certain angle, and anor part penetrates glass and continues re, in anor angle. The same happens when light reaches or surface Fig. of 16 Undersøgelse lens again mamatics af paraboler is required - trin 3 to calculate angle in which light is reflected and refracted. For ideal lenses, re is an easy equation calculating se Det væsentlige er, at punkt T ikke er fast, så vi kan frit flytte det hen ad linjen p. Til gengæld er punkt near centre of lens. With thicker lenses and light being more off-centre, calculations M fast. become more complex, and from equations alone it would be difficult to see what happens. With Hvad DGS kan it is vi possible sige om to afstanden simulate fra punkt properties S til of M a og lens afstanden without actually fra punkt having S til to linje use p, a lens, når vi laser flytter light, cirklen etc. But (vi flytter even for den ved DGS, punkt we need T)? Flyt mamatics punkt T, og to create prøv at forudsige, simulation hvad in der first sker. place. Hold øje med, hvad der sker. M_S_p.html angle ofincidence (between ray of light and normal) is equal to angle of reflection: Fig. 17 Formodning om linje og punkt for det geometriske sted Forestil dig, at punkt S trækker røde punkter efter sig. Igen - hvad er navnet på dette geometriske sted? project has been fund ed with support from European Commission in its Lifelong Learning Programme project has been funded with support from European Commission in its Lifelong Learning Programme only of and only of rein. and made of information rein.

11 Kurven er en parabel. : den er sættet af alle de punkter på planet, hvis afstand fra et fast Andreas punkt Ulovec M er lig med deres afstand fra den faste linje p. Hvor mange cirkler er der, som rører ved punkt M og linje p, og hvis radius er r? Hvor punkter er der på linjen? Vi In har optics hidtil when antaget, it comes at punkt down M er to et show fast punkt path på of planet. rays of Hvad light sker through der, glass, hvis vi lenses lader or punkt systems M flytte of sig lenses, langs cirklen, m f.eks. physics cirklen teachers m med groan centrum experiments J? are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air you need smoke, dust or or way Lad of os making have en light linje visible. p og To en show cirkel m path med of centrum light in materials, J. Forestil you dig need et punkt special med equipment samme smoke afstand til denne glass linje lenses og cirklen etc. Now m. that snot Hvor mange always af available, den slags and punkter adjustments er der? to Hvilken system form can for geometrisk usually only sted be definerer done by de? removing Lad os igen one gøre piece det and trin putting for trin: anor piece in. To see what happens if you make a lens thicker, you have to take out current lens and put in new one. Students can n observe Forestil situation dig nu, before at punktet change M kan and flyttes after frit gennem change but den it is givne not exactly cirkel a m gradual med centrum change Sthat 1. lets m observe how path of light actually changes. We want to demonstrate how you can show path of light through Hvad er a det lens punktsæt, with der help er dynamic of lige langt geometry fra en given software cirkel? (DGS). Det material en koncentrisk can be useful cirkel. for science teachers, who can use it to model experiments with lenses, reflection and refraction not insteadof actual experiment (if one sees experiments only in simulation, pedagogicvalue is not quite same), but complementingit. It can as well be useful for mamatics teachers. Well, now where is mamatics? There is a lot of it in re! If a ray of light hits glass surface of an optical lens, a part of it gets reflec ted back in a certain angle, and anor part penetrates glass and continues re, in anor angle. The same happens when light reaches or surface of lens again mamatics is required to calculate angle in which light is reflected and refracted. For ideal lenses, re is an easy equation calculating se near centre of lens. With thicker lenses and light being more off-centre, calculations become more complex, and from equations alone it would be difficult to see what happens. With DGS it is possible to simulate properties of a lens without actually having to use a lens, laser light, etc. But even for DGS, we need mamatics to create simulation in first place. Fig. When 18 Undersøgelse a ray of light af hits parabler a plane - trin glass 1 med surface, cirkel a part Fig. of 19 it is Undersøgelse reflected. The law af of parabler reflectionsaysthat - trin 2 med cirkel angle ofincidence (between ray of light and normal) is equal to angle of reflection: Denne koncentriske cirkel er også et sæt af alle centre for cirkler, der rører en given cirkel m. Alle disse centre er lige langt fra det givne centrum S 1. Opgaven med at finde sættet af alle centre for de cirkler, der rører en fast linje og en fast cirkel, er således den samme som at skulle finde det sæt af alle centre for de cirkler, der rører ved en fast linje, og som går igennem et fast punkt. parabola_aplet_k_p.html project has been fund ed with support from European Commission in its Lifelong Learning Programme project has been funded with support from European Commission in its Lifelong Learning Programme only of and only of rein. and made of information rein.

12 In optics when it comes down to show path of rays of light through glass, lenses or systems of lenses, m physics teachers groan experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air you need smoke, dust or or way of making light visible. To show path of light in materials, you need special equipment smoke glass lenses etc. Now that snot always available, and adjustments to system can usually only be done by removing one piece and putting anor piece in. To see what happens if you make a lens thicker, you have to take out current lens and put in new one. Students can n observe situation before change and after change but it is not exactly a gradual change that lets m observe how path of light actually changes. We want to demonstrate how you can show path of light through a lens with help dynamic of geometry software (DGS). material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction not insteadof actual experiment (if one sees experiments only in simulation, pedagogicvalue is not quite same), but complementingit. It can as well be useful for mamatics teachers. Well, now where is mamatics? There is a lot of it in re! If a ray of light hits glass surface of an optical lens, a part of it gets reflec ted back in a certain angle, and anor part penetrates glass and continues re, in anor angle. The same happens when light reaches or surface of lens again mamatics is required to calculate angle in which Fig. 20 light Formodning is reflected om and det refracted. geometriske For ideal sted lenses, og cirklen re is an easy equation calculating se Vi near burde nu centre være of klar til lens. at With gå tilbage thicker til lenses kortet and og light finde being det more bedste off-centre, sted. Vi vil calculations konstruere to geometriske become more steder complex, for den vinkelrette and from halveringslinje equations alone til segment it would MN be difficult og parabolen to see what til linje happens. s og punkt With M. DGS Skæringspunktet it is possible for to simulate disse kurver properties giver os centrum of a lens af without cirklen actually k, der er having tegnet to rundt use om a lens, punkt laser M, light, N og etc. linje But s. even Skæringspunkterne for DGS, we for need denne mamatics cirkel og linje to create s giver os simulation det "bedste in sted", first vi place. ønskede, så vi kunne se på Apollobroen fra Havnebroen. angle ofincidence (between ray of light and normal) is equal to angle of reflection: project has been fund ed with support from European Commission in its Lifelong Learning Programme project has been funded with support from European Commission in its Lifelong Learning Programme only of and only of rein. and made of information rein.

13 In optics when it comes down to show path of rays of light through glass, lenses or systems of lenses, m physics teachers groan experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air you need smoke, dust or or way of making light visible. To show path of light in materials, you need special equipment smoke glass lenses etc. Now that snot always available, and adjustments to system can usually only be done by removing one piece and putting anor piece in. To see what happens if you make a lens thicker, you have to take out current lens and put in new one. Students can n observe situation before change and after change but it is not exactly a gradual change that lets m observe how path of light actually changes. We want to demonstrate how you can show path of light through a lens with help dynamic of geometry software (DGS). material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction not insteadof actual experiment (if one sees experiments only in simulation, pedagogicvalue is not quite same), but complementingit. It can as well be useful for mamatics teachers. Well, now where is mamatics? There is a lot of it in re! If a ray of light hits glass surface of an optical lens, a part of it gets reflec ted back in a certain angle, and anor part penetrates glass and continues re, in anor angle. The same happens when light reaches or surface of lens again mamatics is required to calculate angle in which light is reflected and refracted. For ideal lenses, re is an easy equation calculating se near centre of lens. With thicker lenses and light being more off-centre, calculations become more complex, Fig. and 21 from Anvendelse equations af tillært alone viden it would til løsning be difficult af opgave to see 3 what happens. With DGS it is possible to simulate properties of a lens without actually having to use a lens, laser light, etc. But even for DGS, we need mamatics to create simulation in first place. 4.2 Bemærkninger til læreren Denne 2 Easy opgave beginnings er velegnet til 10. light og 11. hits kl. a(1. plane og 2. år surface af gymnasiet). Enkelte studerende kan have svært ved selv at finde løsningen. Der kan være flere, hvis de ikke er vant til at arbejde på denne måde. Selvom der kan være vanskeligheder, bør vi lade de studerende prøve at komme med forslag. Der burde When også a sættes ray of light tid af hits til, a at plane de selv glass kan surface, forsøge at a part finde of deres it is reflected. egen The tilgang law til of det. reflectionsaysthat Efter nogle angle forgæves ofincidence forsøg (between kan man ray skabe of light et undersøgelsesforum and normal) is equal med to en angle mindre of reflection: grad af åbenhed, som kan hjælpe de studerende med at bryde spørgsmålet op i mindre dele og således forstå hovedideen. Ved at arbejde på denne måde kan man hjælpe de studerende med bedre at kunne forstå brugen af koniske kurver. Det kan også være med til at få dem til at arbejde på måder, der ikke kan lade sig gøre uden at bruge dynamiske geometriprogrammer. Til slut beder vi de studerende samle deres resultater sammen og formulere en model (på en deduktiv måde), som kan fungere under forskellige omstændigheder og med forskellige genstande. Arbejdes der på denne måde, kan det måske være med til at få dem til at forstå, hvordan og hvorfra matematiske læresætninger kommer. 4.3 Matematiske begreber der udvikles Parabol - geometrisk sted Matematisk modellering project has been fund ed with support from European Commission in its Lifelong Learning Programme project has been funded with support from European Commission in its Lifelong Learning Programme only of and only of rein. and made of information rein.

14 Anvendelse af modeller, udvikling af viden til at finde ud af løsningen på et problem fra den virkelige verden Induktiv tankegang Lave formodninger In optics when it comes down to show path of rays of light through glass, lenses or systems of lenses, m physics teachers groan experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air you need smoke, dust or or way 5 Afsluttende opgave of making light visible. To show path of light in materials, you need special equipment smoke glass lenses etc. Now that snot always available, and adjustments to system can usually only be Hvis vi tænker på, at en bekvem udsigtsvinkel ville være 36 o, så kan vi generalisere vores løsning fra done by removing one piece and putting anor piece in. To see what happens if you make a lens opgave 2 til et geometrisk sted, hvorfra vi kan se Apollobroen ved denne vinkel. thicker, you have to take out current lens and put in new one. Students can n observe Vi kan situation igen lade before de studerende change and komme after med deres change but egne it is ideer. not exactly a gradual change that lets m observe how path of light actually changes. We want to demonstrate how you can show path of Vi light vælger through konstruktionen a lens with af det help geometriske dynamic of geometry sted, hvorfra software vi kan (DGS). se broen ved en 36 o vinkel, ved at udregne vinklen i en ligesidet trekant. På denne måde kan vi udregne vinklen MSN, som skulle være material can be useful for science teachers, who can use it to model experiments with lenses, 72 o. Fra dette punkt kan vi lave en cirkel. Vi gør det samme med den anden side. Vi har derefter et sæt reflection and refraction not insteadof actual experiment (if one sees experiments only in punkter simulation, med den bedste pedagogicvalue udsigtsvinkel is not til quite at se Apollobroen same), but complementingit. i Bratislava. It can as well be useful for mamatics teachers. Well, now where is mamatics? There is a lot of it in re! If a ray of light hits glass surface of an optical lens, a part of it gets reflec ted back in a certain angle, and anor part penetrates glass and continues re, in anor angle. The same happens when light reaches or surface of lens again mamatics is required to calculate angle in which light is reflected and refracted. For ideal lenses, re is an easy equation calculating se near centre of lens. With thicker lenses and light being more off-centre, calculations become more complex, and from equations alone it would be difficult to see what happens. With DGS it is possible to simulate properties of a lens without actually having to use a lens, laser light, etc. But even for DGS, we need mamatics to create simulation in first place. angle ofincidence (between ray of light and normal) is equal to angle of reflection: Fig. 22 Løsning på indledende opgave (Google Maps) project has been fund ed with support from European Commission in its Lifelong Learning Programme project has been funded with support from European Commission in its Lifelong Learning Programme only of and only of rein. and made of information rein.

15 In optics when it comes down to show path of rays of light through glass, lenses or systems of lenses, m physics teachers groan experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air you need smoke, dust or or way of making light visible. To show path of light in materials, you need special equipment smoke glass lenses etc. Now that snot always available, and adjustments to system can usually only be done by removing one piece and putting anor piece in. To see what happens if you make a lens thicker, you have to take out current lens and put in new one. Students can n observe situation before change and after change but it is not exactly a gradual change that lets m observe how path of light actually changes. We want to demonstrate how you can show path of light through a lens with help dynamic of geometry software (DGS). material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction not insteadof actual experiment (if one sees experiments only in simulation, pedagogicvalue is not quite same), but complementingit. It can as well be useful for mamatics teachers. Well, now Fig. where 23 Apollobroen is mamatics? (Wikimedia) There is a lot of it in re! If a ray of light hits glass surface of an optical lens, a part of it gets reflec ted back in a certain angle, and anor part penetrates glass and continues re, in anor angle. The same happens when light reaches or surface of lens again mamatics is required to calculate angle in which light is reflected and refracted. For ideal lenses, re is an easy equation calculating se near centre of lens. With thicker lenses and light being more off-centre, calculations become more complex, and from equations alone it would be difficult to see what happens. With 6 Yderligere DGS it is possible undersøgelse to simulate properties of a lens without actually having to use a lens, laser light, etc. But even for DGS, we need mamatics to create simulation in first place. Hvordan ændres situationen, hvis vi i stedet bruger et kamera med et andet fokus? Hvordan 2 Easy kan beginnings vi finde et sted på light promenaden, hits a plane hvorfra surface vi kan se to broer ud fra den bedste vinkel? Hvordan ændres situationen, hvis promenaden ikke er vinkelret på broen i opgave 2? Hvad ændres, hvis promenaden buer, som den gør mellem Apollobroen og Havnebroen? Hvor er så helt præcist det bedste sted at se Apollobroen fra? angle ofincidence (between ray of light and normal) is equal to angle of reflection: Disse spørgsmål kan føre til opgaver med cirkler ud fra Apollonius' tanker. Det er måske bare et tilfælde, at Apollobroen lader os få øje på et cirkelproblem, hvor vi kan bruge Apollonius. Der er dog i virkeligheden mange flere eksempler, der ikke bare kan bruges i matematiktimerne. Det er også vigtigt at lade de studerende bearbejde det, de har fundet ud af, på ensartet økonomisk vis, så de kan bruges til yderligere udforskning og opgaveløsning. 7 Diskussion I vores undersøgelse benyttede vi os meget af tilnærmelse ved at lave skitser på et kort fra Google Maps i GeoGebra. Disse resultater var kun vejledende, men de kan opfylde vores formål med at udvikle de matematisk begreber. Det vil være fint at diskutere dette med de studerende. For at forbedre nøjagtigheden kan vi bruge GPS og tilføje koordinater, der vil hjælpe os med at finde det "bedste sted". Cirkler kan studeres og bruges på mange måder. I vores artikel brugte vi cirkler som et begreb, de studerende kendte. I forbindelse med undersøgelsen hjalp cirklerne dem med at forstå det nye begreb ved at løse en opgave. Vi vil Fig.1 gerne Reflection endnu of en light gang at a understrege, plane surface. hvor vigtigt det er at foretage project has been fund ed with support from European Commission in its Lifelong Learning Programme project has been funded with support from European Commission in its Lifelong Learning Programme only of and only of rein. and made of information rein.