From aeec79dd48ed7810655a80dc88a5a0944ad9d9cf Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Wed, 15 Jul 2026 09:03:53 +0200 Subject: [PATCH] kleine Schmenkerlis --- bibliography.bib | 12 ++++++++++++ content/Anhang.tex | 13 ++++++++++++- content/Arbeitsmaterialien.tex | 24 ++---------------------- content/Mathematischer_Hintergrund.tex | 1 + content/Zeitplan.tex | 4 ++-- galton.pdf | Bin 808561 -> 1321702 bytes material/ab1.tex | 2 +- material/ab1_Lös.tex | 2 +- material/ab3.tex | 2 +- 9 files changed, 32 insertions(+), 28 deletions(-) diff --git a/bibliography.bib b/bibliography.bib index 75a820b..4e9f5e3 100644 --- a/bibliography.bib +++ b/bibliography.bib @@ -161,3 +161,15 @@ date = {2025}, } +@book{forster_analysis_2023, + location = {Wiesbaden}, + edition = {13., überarbeitete und ergänzte Auflage}, + title = {Analysis. 1: Differential- und Integralrechnung einer Veränderlichen / Otto Forster}, + isbn = {978-3-658-40130-6}, + series = {Lehrbuch}, + shorttitle = {Analysis. 1}, + publisher = {Springer Spektrum}, + author = {Forster, Otto}, + date = {2023}, + note = {Num Pages: 460}, +} \ No newline at end of file diff --git a/content/Anhang.tex b/content/Anhang.tex index 82fbbb3..8dab766 100644 --- a/content/Anhang.tex +++ b/content/Anhang.tex @@ -7,7 +7,12 @@ frame=true, scale=0.70, pagecommand={\thispagestyle{plain}} ]{material/ab1.pdf} - +\includepdf[ +pages=-, +frame=true, +scale=0.70, +pagecommand={\thispagestyle{plain}} +]{material/ab1_Lös.pdf} \section{Arbeitsblatt 2} @@ -29,3 +34,9 @@ frame=true, scale=0.70, pagecommand={\thispagestyle{plain}} ]{material/ab3.pdf} +\includepdf[ +pages=-, +frame=true, +scale=0.70, +pagecommand={\thispagestyle{plain}} +]{material/ab3_Lös.pdf} diff --git a/content/Arbeitsmaterialien.tex b/content/Arbeitsmaterialien.tex index 4b0aede..ff9040e 100644 --- a/content/Arbeitsmaterialien.tex +++ b/content/Arbeitsmaterialien.tex @@ -1,21 +1,3 @@ -Stellen Sie Ihre Arbeitsmaterialien hier dar. - -Dies ist der zweitwichtigste Teil des Dokuments und soll (inhaltlich) auch den zweitgrößten Teil des Dokuments ausmachen. Da Arbeitsmatierialien inhaltlich sehr unterschiedlich sind, ist es schwierig, hier genaue Vorgaben zu machen. Eine Bastelanleitung mit Bastelbögen hätte ja z.B. eine ganz andere Struktur als eine Sammlung von Aufgaben und kann schnell ein paar Seiten umfassen... - - Falls Sie Arbeitsmaterialien in separaten Dokumenten haben, erstellen Sie hier zumindest eine kurze Beschreibung der entsprechenden Materialien. - - -Ein paar Anregungen für die Arbeitsmaterialien (kommt natürlich auf das Thema an): -\begin{itemize} - \item Aufgaben - \item Musterlösungen bzw. Lösungsskizzen - \item Knobeleien - \item Basteleien -\end{itemize} - - -Bitte bedenken Sie, zumindest Lösungsskizzen zu den Aufgaben zu geben! Besser noch vollständige Musterlösungen. Als Lehrkraft ist es nicht immer einfach, die Lösungen schnell zu finden und das Dokument soll ja den Lehrkräften die Arbeit erleichtern. - % 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? @@ -32,8 +14,6 @@ Bitte bedenken Sie, zumindest Lösungsskizzen zu den Aufgaben zu geben! Besser n % 3 -> potenzen Killen um E funktion zu gewinnen! % ln und umformen (Klassenstufe! -> Anschätzung und Ergebnis vorgeben, nicht rechnen) % Rücksubstitution machen -Das dritte Arbeitsblatt 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.\\ +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.\\ 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. - -%Noch Checkfragen in den Beweis einbauen? \ No newline at end of file +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 12f5aa6..3d922ca 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -284,6 +284,7 @@ Die Gamma-Funktion berechnet wie folgt die Fakultät in $\mathbb{N}$: \Gamma(n+1)=n! \] \end{satz} +Ein Beweis hierzu befindet sich in \cite{forster_analysis_2023}. \subsection{Stirling-Approximation} Die Stirling-Approximation ist eine mathematische Näherung zur Berechnung der Fakultät einer Zahl. Die Approximation ist vorallem in der Stochastik und der statistischen Physik ein unverzichtbares Werkzeug. Wir schauen sie daher im Folgenden im Detail an. diff --git a/content/Zeitplan.tex b/content/Zeitplan.tex index 499904b..6fa7812 100644 --- a/content/Zeitplan.tex +++ b/content/Zeitplan.tex @@ -1,7 +1,7 @@ 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{Pascalsches Dreieck} (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{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.\\ Ab 12. Klasse:\\ -Um eine tiefgreifende Auseinandersetzung mit dem Galtonbrett zu ermöglichen empfehlen wir alle drei Arbeitsblätter anzubieten. Wir empfehlen mit dem Arbeitsblatt \emph{Pascalsches Dreieck} (c.a. 25 min) zu beginnen, daran das Arbeitsblatt \emph{Galton x Plinko} (c.a. 25 min) anzuschließen und das Arbeitsblatt \emph{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. 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. 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