From 3d07b05424b18b416dc84a1401dcc29e7efc7a1c Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Mon, 20 Jul 2026 16:59:01 +0200 Subject: [PATCH] =?UTF-8?q?=C3=A4nderungen=20mit=20Tobi?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- content/Anhang.tex | 2 +- content/Arbeitsmaterialien.tex | 22 +++++----------- content/Mathematischer_Hintergrund.tex | 2 +- content/Zeitplan.tex | 10 ++++++-- galton.pdf | Bin 871216 -> 979010 bytes galton.tex | 3 +++ material/ab1.pdf | Bin 107981 -> 110039 bytes material/ab1.tex | 2 +- material/ab2.pdf | Bin 82103 -> 65438 bytes material/ab2.tex | 34 +++++++++++++++---------- material/ab3.pdf | Bin 147605 -> 293717 bytes material/ab3.tex | 2 +- 12 files changed, 41 insertions(+), 36 deletions(-) diff --git a/content/Anhang.tex b/content/Anhang.tex index a7f9e6d..13c29ed 100644 --- a/content/Anhang.tex +++ b/content/Anhang.tex @@ -86,7 +86,7 @@ pagecommand={\thispagestyle{plain}} ]{material/ab2.pdf} \section{Arbeitsblatt 3} -Im dritten Arbeitsblatt lernen die Schüler*innen das Galtonbrett konkret kennen und beweisen semi-formal den zentralen Grenzwertsatz. +Im dritten Arbeitsblatt lernen die Schüler*innen das Galtonbrett konkret kennen und beweisen semi-formal den lokalen Grenzwertsatz. \includepdf[ pages=-, diff --git a/content/Arbeitsmaterialien.tex b/content/Arbeitsmaterialien.tex index ff9040e..9f8517b 100644 --- a/content/Arbeitsmaterialien.tex +++ b/content/Arbeitsmaterialien.tex @@ -1,19 +1,9 @@ -% 1. Einführung/ Motivation (lustig/ spielerisch) -% Spiel von Tobi (gibts nicht auch so nen Glücksspiel?) -% https://plinko-web-game.netlify.app/ : wieso macht man im Schnitt minus? -% Pachinko-Automat: Wo einwefen für max Gewinn - -% 2. Überleitung (interaktiv) -% Wabenaufgabe und Einführung über Pascallsches Dreieck, überführung Approx durch Normalverteilung: Tobi-AB +Der Einstieg erfolgt über die formale Definition des Binomialkoeffizienten, wobei der Fokus auf den einfachen Berechnungen und den beiden Basisfällen $k=0$ und $k=n$ liegt. Zur visuellen Unterstützung und zum Verständnis rekursiver Abläufe wird das Pascalsche Dreieck herangezogen. Ein kurzer in Form eines Unterrichtsgespräches geführter Exkurs zu den binomischen Formeln zeigt den Jugendlichen zudem praktisch, wie die BinomialkoeAizienten das Ausmultiplizieren von Klammern vereinfachen und das Auswendiglernen überflüssig machen.\\ -% 3. Arbeitsmaterial ZSG (mathematisch Verständnisorientiert) -% wollen über eine Diskrete Verteilung eine stetige Kurve legen! -% Die Verteilung ist durch die Fakultät diskret, da diese nur für n\in N definiert ist -% 1 -> n! -> gamma Funktion -> Sterling Approx geben (Visuell!) -% 2 -> Vorfaktor "abschätzen" -% 3 -> potenzen Killen um E funktion zu gewinnen! - % ln und umformen (Klassenstufe! -> Anschätzung und Ergebnis vorgeben, nicht rechnen) - % Rücksubstitution machen -Das dritte Arbeitsblatt, \emph{Galton x ZSG Movire-Laplace}, dient dem Beweis des zentralen Grenzwertsatzes von Movire-Laplace. Ziel ist es hierbei explizit nicht, dass die Schüler*innen den Beweis in jedem Detail, wie im Teil "Mathematischer Hintergrund", verstehen aber die Grundlegenden Argumentationen und mathematischen Tricks nachvollziehen. Außerdem ist vor allem das Beweisen etwas, dass im Rahmenlehrplan (\cite{lisum-2022} und \cite{lisum-2023}) nicht gefordert ist, wodurch es sich sehr gut für eine MatheAG oder Zusatzstunde handelt, in der Inhalte über den klassischen Unterricht hinaus behandelt werden.\\ +Im weiteren Verlauf wird eine Brücke zu den aus der Grundschule bekannten Zahlenpyramiden geschlagen. Im Gegensatz zum klassischen Aufbau von unten nach oben wird hier eine Pyramide mithilfe von Rekursion von oben nach unten aufgebaut. Dieser Vergleich veranschaulicht, dass dasselbe mathematische Konstrukt (das Pascalsche Dreieck) auf verschiedenen Wegen berechnet werden kann. Eine Zusatzaufgabe zur Rekursion stellt zudem eine Verbindung zur Informatik und der Simulation von Zufallsexperimenten durch Computeralgorithmen her. \\ + +Der nächste Schritt widmet sich der Modellierung des Galtonbretts. Nach der Demonstration an einem realen Tischmodell (oder digitaler \href{https://www.arndt-bruenner.de/mathe/scripts/galtonbrett.htm}{Simmulation}) sowie dem Besprechen der Modellannahmen zeichnen die Schülerinnen und Schüler Pfade ein und zählen diese ab. Dies hilft ihnen, ein mentales Modell zu entwickeln und die Verbindung zwischen der Anzahl der Pfade und dem Pascalschen Dreieck zu verstehen. Die letzte Aufgabe führt die Gruppe schließlich kollaborativ an die Binomialverteilung heran, deren Formel gemeinsam im Kurs erarbeitet werden kann.\\ + +Das dritte Arbeitsblatt, \emph{Galton x LGS Movire-Laplace}, dient dem Beweis des lokalen Grenzwertsatzes von Movire-Laplace. Ziel ist es hierbei explizit nicht, dass die Schüler*innen den Beweis in jedem Detail, wie im Teil "Mathematischer Hintergrund", verstehen aber die Grundlegenden Argumentationen und mathematischen Tricks nachvollziehen. Außerdem ist vor allem das Beweisen etwas, dass im Rahmenlehrplan (\cite{lisum-2022} und \cite{lisum-2023}) nicht gefordert ist, wodurch es sich sehr gut für eine MatheAG oder Zusatzstunde handelt, in der Inhalte über den klassischen Unterricht hinaus behandelt werden.\\ Für das Arbeitsblatt sollten die Schüler*innen bereits mit der Binomialverteilung, Zufallsvariablen, Logarithmen sowie der e-Funktion vertraut sein und mit etwas Formalismus umgehen können. Tiefgreifende Kenntnisse zur Normalverteilung sind nicht vorausgesetzt und werden auf dem Arbeitsblatt in ausreichender Form wiederholt bzw. können selbständig aufbereitet werden. Es ist empfohlen mindestens das Arbeitsblätt \emph{Pascalsches Dreieck} und optimaler Weise auch noch das Arbeitsblatt \emph{Galton x Plinko} vorher bearbeitet zu haben, um das Konzept des Galtonbrettes und dessen Zusammenhang mit der Binomialverteilung verstanden zu haben.\\ Auf dem Arbeitsblatt wird die formale Modellierung des Galtonbrettes vorgestellt und soll durch eine Zuordnungsaufgabe durch die Schüler*innen vertieft werden. Im Anschluss soll der Bezug zwischen Binomial- und Normalverteilung wiederholt bzw. erarbeitet werden. Das Kernstück des Arbeitsblattes stellt der Lückentext-Ähnliche Beweis dar. Hier sollen die Schüler einige Mathematische Argumentationen nachvollziehen und einige Umformungen selbst durchführen, um den Beweis nachvollziehen zu können. Sollten die Schüler leistungsschwächer sein oder an einem Punkt fest Stecken gibt es zur Differenzierung Lösungskarten für die Lücken, welche den Schülern in nicht sortierter Reihenfolge zur Verfügung gestellt werden können. \ No newline at end of file diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index ba0f5bf..f1622fa 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -1,7 +1,7 @@ %"ltex.language": "de-DE" \subsection{Beweis des zentralen Grenzwertsatzes} -Das Galtonbrett (nach Francis Galton) dient der Veranschaulichung der Binomialverteilung und der experimentellen Bestätigung vom zentralen Grenzwertsatz im Spezialfall der Binomialverteilung. Im Folgenden formalisieren wir den Weg einer Kugel durch das Brett als stochastischen Prozess mithilfe einer Binomialverteilung und beweisen anschließend den zentralen Grenzwertsatz von Moivre-Laplace \cite{nahrstedt_monte-carlo-methode_2015}. +Das Galtonbrett (nach Francis Galton) dient der Veranschaulichung der Binomialverteilung und der experimentellen Bestätigung vom zentralen Grenzwertsatz im Spezialfall der Binomialverteilung. Im Folgenden formalisieren wir den Weg einer Kugel durch das Brett als stochastischen Prozess mithilfe einer Binomialverteilung und beweisen anschließend den (lokalen) zentralen Grenzwertsatz von Moivre-Laplace \cite{nahrstedt_monte-carlo-methode_2015}. \begin{definition}[Modell des Galtonbretts]{galton}{nach \cite[S. 252-253]{buchter_elementare_2005}} Sei $(\Omega, \mathcal{F}, P)$ ein Wahrscheinlichkeitsraum. Der Fall einer Kugel durch ein Galtonbrett mit $n \in \mathbb{N}$ Reihen wird modelliert durch eine Folge von stochastisch unabhängigen und identisch verteilten Zufallsvariablen $X_1, X_2, \dots, X_n$, wobei $X_i \in \{0, 1\}$. diff --git a/content/Zeitplan.tex b/content/Zeitplan.tex index 6fa7812..a4cb5c8 100644 --- a/content/Zeitplan.tex +++ b/content/Zeitplan.tex @@ -1,7 +1,13 @@ Da sich das Themengebiet über einige Niveau- und Klassenstufen des Brandenburger Rahmenlehrplanes streckt \cite{lisum-2023}, und das Galtonbrett sich auch sehr gut eignet, auf verschiedenen Komplexitätsstuufen zu unterrichten, schlagen wir folgende Unterteilung vor:\\ Ab 10. Klasse:\\ -Eine Grundlegende Auseinandersetzung mit dem Galtonbrett und den Zusammenhang mit der Binomialverteilung und dem Pascallschem Dreieck, lässt sich wunderbar ohne höhere mathematische Inhalte erarbeiten. Hierzu eignen sich insbesondere das Arbeitsblatt \emph{Galton x Pascal} (c.a. 25 min) und \emph{Galton x Plinko} (c.a. 25 min). Wir würden auch empfehlen beide zu bearbeiten. Wenn eine ausblicksartige Auseinandersetzung im Zusammenhang mit der Normalverteilung gewünscht ist (ohne in die mathematischen Details dieser einzutauchen) kann hierzu auch noch Aufgabe 1 bis Aufgabe 3 des Arbeitsblattes \emph{Galton x ZSG Movire-Laplace} (c.a. 15min) mit eingebunden werden.\\ +Eine Grundlegende Auseinandersetzung mit dem Galtonbrett und den Zusammenhang mit der Binomialverteilung und dem Pascallschem Dreieck, lässt sich wunderbar ohne höhere mathematische Inhalte erarbeiten. Hierzu eignen sich insbesondere das Arbeitsblatt \emph{Galton x Pascal} (c.a. 25 min) und \emph{Galton x Plinko} (c.a. 25 min). Wir würden auch empfehlen beide zu bearbeiten. Wenn eine ausblicksartige Auseinandersetzung im Zusammenhang mit der Normalverteilung gewünscht ist (ohne in die mathematischen Details dieser einzutauchen) kann hierzu auch noch Aufgabe 1 bis Aufgabe 3 des Arbeitsblattes \emph{Galton x ZSG Movire-Laplace} (c.a. 15min) mit eingebunden werden. Als Ablauf empfehlen wir folgendes Vorgehen:\\ + +\emph{Galton x Plinko }(Seite 1) $\rightarrow$ \emph{Galton x Pascal} $\rightarrow$ \emph{Galton x Plinko} (Seite 2)\\ +$\overset{optional}{\rightarrow}$ \emph{Galton x LGS} (Aufgabe 1 bis 3) Ab 12. Klasse:\\ -Um eine tiefgreifende Auseinandersetzung mit dem Galtonbrett zu ermöglichen empfehlen wir alle drei Arbeitsblätter anzubieten. Wir empfehlen mit dem Arbeitsblatt \emph{Galton x Pascal} (c.a. 25 min) zu beginnen, daran das Arbeitsblatt \emph{Galton x Plinko} (c.a. 25 min) anzuschließen und das Arbeitsblatt \emph{Galton x ZSG Movire-Laplace} (c.a. 40min) zum Abschluss zu benutzen. Das Arbeitsblatt \emph{Galton x Plinko} ist hierbei als optional zu betrachten und dient lediglich der vertiefenden Auseinandersetzung mit dem Galtonbrett und der Motivation. \ No newline at end of file +Um eine tiefgreifende Auseinandersetzung mit dem Galtonbrett zu ermöglichen empfehlen wir alle drei Arbeitsblätter anzubieten. Als verkürzte Variante ließe sich hier auch das Arbeitsblatt \emph{Galton x Plinko} auslassen. 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Untersuche nun die Anzahl der Pfade zu jedem der Stäbchen und vergleiche diese mit dem Pascalschen Dreieck. 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Vermutung auf, wieso in der Mitte aller Fächer die kleinsten Faktoren liegen. - \answer{3} + \item Stell eine Vermutung auf, wieso in der Mitte aller Fächer die kleinsten Multiplikatoren liegen. + \answer{4} - \item Stell eine Vermutung auf, wieso die Faktoren an den Randfächern am größten sind. - \answer{3} + \item Stelle eine Vermutung auf, wieso in der Mitte aller Fächer die kleinsten Multiplikatoren liegen. + \answer{4} - \item Argumentiere, wieso man auf lange Sicht verliert oder gewinnt. - \answer{3} + \item Stelle eine Vermutung auf, wieso an den Rändern aller Fächer die größten Multiplikatoren liegen. + \answer{4} + + \item Stelle Vermutungen auf, ob man auf lange Sicht gewinnt oder verliert. + \answer{4} \end{enumerate} + \newpage - Mit dem Wissen über das Galtonbrett und seine Verteilung können wir nun fundierte Schlüsse über Plinko ziehen: + \section{Galton x Plinko Teil 2} + Mit dem Wissen über das Galtonbrett und seine Verteilung können wir nun fundierte Schlüsse über Plinko ziehen. Website: \url{https://plinko-web-game.netlify.app/} \begin{enumerate}[itemsep=11pt] - \setcounter{enumi}{4} - \item Die Wahrscheinlichkeit mit der eine Kugel in Plinko nach links bzw. rechts hüpft ist \hrulefill - \item Nenne die Anzahl der Ebenen in Plinko. \hrulefill + \item Bestimme die Wahrscheinlichkeit, die eine Kugel in Plinko hat, dass Sie nach links oder Rechts hüpft? \hrulefill + \item Nenne die Anzahl der Ebenen in Plinko. \small(Hinweis: beginne bei 0 an zu zählen) \normalsize + \hrulefill \item Gib die Anzahl an Faktoren (Fächern) an. \hrulefill - \item Gib nun eine Zufallsvariable $X\sim \text{Bin}_{n,p}$ an. \hrulefill + \item Gib nun eine Formel an, um die Wahrscheinlichkeit zu berechnen, dass eine Kugel im Fach $k$ landet. \small (Hinweis: Galton x Pascal) \normalsize \hrulefill \item Berechne die Wahrscheinlichkeit folgender Ereignisse: \begin{enumerate}[label=\alph*),itemsep=11pt] - \item Die Kugel landet beim Multiplikator $0,3x$. \hrulefill + \item Die Kugel landet beim Multiplikator $0,3 \times$. \hrulefill \item Eine Kugel holt ihren gesetzten Betrag exakt wieder rein. \hrulefill - \item Eine Kugel landet in einem der beiden höchsten Faktoren. \hrulefill -% \item Zwei Kugeln holen mit Einsatz von je 1\$ einen Gewinn von gesamt 4\$. \hrulefill + \item Eine Kugel landet in einem der beiden höchsten Multiplikatoren. \hrulefill \end{enumerate} \item Bestimme den Erwartungswert, wenn der Einsatz immer 1\$ ist. + \answer{3} \item Überprüfe die These: "\textit{Je höher mein Einsatz ist, desto mehr bekomme ich raus.}". + \answer{3} \end{enumerate} diff --git a/material/ab3.pdf b/material/ab3.pdf index 9cc05a81de30bfb344bf650cf13fecef001994ec..1252bccbb83b43ab1d830c1d211b53073feffa2c 100644 GIT binary patch literal 293717 zcmZ_01z418*EYOuKo~@nkUG~ox^chn_x;E5|ND6MY{6@;`--*Bb*^)*`{7W#rXV16NsNM{@pt8~>aN!B zH58((7*+?9`xMgBtb!V>f{G5B4y=OLSh=0eO@SJ%u*%9(nA@8ngCzbN%xxed zArAja7+^3MMdSmAeApr%F35)i@?npB7+^%r;73#BM{^#E|5+;b|9fBhtdj5->a2na)-J4u 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