AP 2nde, calcul mental 2nde
authorDenise sur Lya <sekhmet@lya>
Thu, 10 Mar 2016 21:44:20 +0000 (22:44 +0100)
committerDenise sur Lya <sekhmet@lya>
Thu, 10 Mar 2016 21:44:20 +0000 (22:44 +0100)
20 files changed:
2nde/AP/.~lock.liste_eleves_diagnostic.ods# [new file with mode: 0644]
2nde/AP/fiche_systemes.pdf [new file with mode: 0644]
2nde/AP/fiche_systemes.tex [new file with mode: 0644]
2nde/AP/liste_eleves_diagnostic.ods [new file with mode: 0644]
2nde/AP/systemes_1.pdf [new file with mode: 0644]
2nde/AP/systemes_1.tex [new file with mode: 0644]
2nde/DM_5/liste_eleves.ods [new file with mode: 0644]
2nde/calcul_mental/courbe1.pdf [new file with mode: 0644]
2nde/calcul_mental/courbe1.svg [new file with mode: 0644]
2nde/calcul_mental/courbe2.pdf [new file with mode: 0644]
2nde/calcul_mental/courbe2.svg [new file with mode: 0644]
2nde/calcul_mental/courbe3.pdf [new file with mode: 0644]
2nde/calcul_mental/courbe3.svg [new file with mode: 0644]
2nde/calcul_mental/courbe4.pdf [new file with mode: 0644]
2nde/calcul_mental/courbe4.svg [new file with mode: 0644]
2nde/calcul_mental/courbe5.pdf [new file with mode: 0644]
2nde/calcul_mental/courbe5.svg [new file with mode: 0644]
2nde/calcul_mental/courbes.pdf [new file with mode: 0644]
2nde/calcul_mental/courbes.tex [new file with mode: 0644]
header_a5.tex

diff --git a/2nde/AP/.~lock.liste_eleves_diagnostic.ods# b/2nde/AP/.~lock.liste_eleves_diagnostic.ods#
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+,sekhmet,Lya,10.03.2016 18:25,file:///home/sekhmet/.config/libreoffice/4;
\ No newline at end of file
diff --git a/2nde/AP/fiche_systemes.pdf b/2nde/AP/fiche_systemes.pdf
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diff --git a/2nde/AP/fiche_systemes.tex b/2nde/AP/fiche_systemes.tex
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+\input{../../header_a5.tex}
+
+\begin{document}
+\section*{Systèmes d'équations, méthodes}
+On souhaite résoudre le système suivant :
+\[ 
+\left\{\begin{array}{lll}
+2x-y = 11 & \hspace{1cm} & (1) \\
+3x+2y = -8 & & (2)
+\end{array}
+ \right.
+\]
+Résoudre un système d'équation, c'est trouver la (ou les) valeurs des variables (ici $x$ et $y$) qui vérifient \souligne{toutes} les équations.
+\section{Résolution par \emph{substitution}}
+\begin{enumerate}
+\item Dans l'équation $(1)$, exprimer $y$ en fonction de $x$. On obtient :
+\[ (1) \equi y =  \hspace{7cm} (1')
+\]
+\item Remplacer (substituer) l'expression de $y$ obtenue dans l'équation $(2)$ :
+\[
+3x + 2 \left( \hspace{5cm} \right) = -8 
+\]
+\item Résoudre alors l'équation obtenue, et trouver la valeur de $x$ :
+\vspace{4cm}
+\item Remplacer la valeur de $x$ trouvée ci-dessus dans l'équation $(1')$ :
+\[ y = \hspace{5cm} \]
+\end{enumerate}
+\section{Résolution par addition (ou pivot)}
+\begin{enumerate}
+\item Multiplier la première équation (c'est à dire le membre de gauche et de droite) par $2$ :
+\[ (1) \equi \hspace{8cm} (1')\]
+\item Effectuer la somme de l'équation $(1')$ et de l'équation $(2)$.
+\[ 
+\begin{tabular}{llll}
+ & \vspace{0.5cm} & \hspace{1cm} (1') \\
+\huge{+} \hspace{1cm} & 3x \hspace{0.5cm} + \hspace{0.5cm} 2y \hspace{0.5cm} = \hspace{0.5cm} -8  & \hspace{1cm} (2) \\
+\hline
+\vspace{1cm}
+\end{tabular}
+\]
+\item Résoudre l'équation obtenue, qui ne devrait contenir que $x$ comme variable.
+\vspace{4cm}
+\item Remplacer la valeur de $x$ trouvée dans une des équations (au choix !), et en déduire la valeur de $y$.
+\vspace{4cm}
+\end{enumerate}
+
+\section{Vérifier à la fin (facultatif mais utile quand même !)}
+Remplacer les valeurs de $x$ et $y$ trouvées dans les équations de départ, et vérifier qu'elles sont vérifiées :
+\[ 2x - y = \hspace{8cm}\]
+\[ 3x+2y = \hspace{8cm} \]
+
+On peut aussi utiliser la calculatrice : \bouton{résol}, \bouton{2} (PlySmlt2), \bouton{2} (SOLVEUR SYST D'ÉQUATIONS).
+
+\section{Récapitulatif}
+Les malins/malignes trouveront comment adapter la méthode à 3 équations et 3 inconnues, 4 équations et 4 inconnues, etc !
+\begin{enumerate}[A)]
+\item {\bf Méthode par substitution}
+\begin{enumerate}[1)]
+  \item Choisir une équation, et exprimer une des variables en fonction de l'autre.
+  \item Remplacer l'expression trouvée dans \souligne{l'autre} équation.
+%  \item \emph{(Refaire les étapes 1 et 2 avec les deux équations restantes, de façon à "éliminer" encore une inconnue)}.
+  \item Résoudre l'équation "finale", n'ayant qu'une inconnue.
+  \item Remplacer la valeur trouvée dans la première équation, pour obtenir la seconde inconnue. %\emph{(Remplacer la valeur de la variable dans une des équations, pour obtenir une deuxième variable, etc.)}
+\end{enumerate}
+\item {\bf Méthode par addition, ou pivot}
+\begin{enumerate}[1)]
+  \item Trouver une "combinaison" des deux équations qui "supprime" une des inconnues (par exemple, $3\times(1) - 2\times (2)$). Souvent, une simple addition ou soustraction fonctionne.% \emph{(Choisir une équation qu'on va appeler "pivot", et trouver une telle combinaison entre le pivot et une autre, puis entre le pivot et la troisième.)}
+  \item Effectuer la combinaison.% \emph{(Effectuer toutes les combinaisons : on obtient deux équations à deux inconnues. Refaire alors les étapes 1 et 2 avec ces deux équations)}.
+  \item Résoudre l'équation obtenue, qui n'a plus qu'une seule inconnue.
+  \item Remplacer la valeur trouvée dans la première équation, et en déduire la première inconnue.% \emph{(remplacer la valeur dans l'équation qui ne contenait qu'une autre variable, et en déduire une inconnue. Puis remplacer les deux inconnues trouvées dans la première...)}
+\end{enumerate}
+\end{enumerate}
+
+\section{Un peu d'entraînement}
+
+\begin{enumerate}
+\item Résoudre le système suivant par substitution :
+\[
+\systeme{3x + y = -14\\x = 2y}
+\]
+\item Résoudre le système suivant par pivot :
+\[
+\systeme{x+ y = 3\\
+5x - y = 3
+}\]
+\item Résoudre, de la manière que vous voulez, les systèmes suivants :
+\[
+\systeme{3x+7y=35\\x+y=29}
+\qquad
+\systeme{8x + 2y=-1\\-10x + 5y = 5}\]
+\item Petit problème : \og Deux avocats et trois poireaux coûtent \ueur{4,85}. Mais un avocat et quatre poireaux coûtent \ueur{5,30}. Quel est le coût d'un avocat et d'un poireau ? \fg
+\end{enumerate}
+\end{document}
diff --git a/2nde/AP/liste_eleves_diagnostic.ods b/2nde/AP/liste_eleves_diagnostic.ods
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diff --git a/2nde/AP/systemes_1.pdf b/2nde/AP/systemes_1.pdf
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diff --git a/2nde/AP/systemes_1.tex b/2nde/AP/systemes_1.tex
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+\input{../../header_a5.tex}
+
+\begin{document}
+\section*{Systèmes}
+\exo{Équations à une seule inconnue}
+Résoudre les équations suivantes
+\begin{enumerate}
+\item $3x + 6 = 0$
+\item $-2x + 7 = 1$
+\item $6x - 2 = 4$
+\item $2x + 8 = 3x - 1$
+\item $5x - 1 = 3 -3x$
+\end{enumerate}
+
+\exo{Substitution}
+Dans les systèmes suivants, une des inconnues est déjà exprimée en fonction de l'autre. Effectuer la substitition pour résoudre le système.
+\begin{enumerate}
+\item $
+\systeme{y = 2x\\ 3x + y = 5}
+$
+\item $\systeme{y = -x + 1\\ 2x + y = 17}$
+\item $\systeme{x = 2y - 1\\3x + y = -10}$
+\item $\systeme{y = \frac{x}{2} + 1\\ x + y = 5}$
+\end{enumerate}
+
+\exo{Résolution par addition/pivot}
+Résoudre les systèmes suivants par pivot. La combinaison à effectuer est indiquée.
+\begin{enumerate}
+\item $\systeme{x+y = 6\\x-y = -4}$ \emph{Effectuer (1) $+$ (2)}
+\item $\systeme{x + 2y = 2\\x -3y = 7}$ \emph{Effectuer (1) $-$ (2)}
+\item $\systeme{2x + y = 1\\3x+2y = 4}$ \emph{Effectuer $-2\times$ (1) $+$ (2)}
+\item $\systeme{2x -3y = -71\\5x+2y = -16}$ \emph{Effectuer $2\times$(1) $+ 3\times$(2)}
+\end{enumerate}
+
+\exo{En roue libre}
+Résoudre les systèmes suivants, en choisissant la méthode.
+\begin{enumerate}
+\item $\systeme{3x + y = 2\\x-y = 2}$
+\item $\systeme{2x + 3y = -76\\3x+2y = -9}$
+\item $\systeme{5x +6y = 1\\ -2x+3y=1}$
+\end{enumerate}
+
+\exo{Petits problèmes}
+Résoudre les problèmes suivants. Indication : exprimer les données sous forme de système d'équations...
+
+\begin{enumerate}
+\item Si un stylo et un cahier coûtent \ueur{1,40}, et 5 stylos avec 3 cahiers coûtent \ueur{5}, quel est le prix d'un stylo et d'un cahier ?
+\item J'ai ramassé 56 trèfles dans le jardin. Il y a des trèfles à trois feuilles, et des trèfles à quatre feuilles. La somme de toutes ces feuilles est de 172. Combien y a-t-il de trèfles à trois feuilles et à quatre feuilles ?
+\item Une fille dit à sa mère \og Tu as 25 ans de plus que moi. Il y a 7 ans, tu avais 6 fois mon âge\fg. Quels sont les âges de la mère et de la fille ?
+\end{enumerate}
+
+\end{document}
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+
+\newcommand\image[1]{\includegraphics[width=\textwidth]{#1}}
+\newcommand\ptimage[1]{\includegraphics[width=0.45\textwidth]{#1}}
+
+
+\begin{document}
+
+\slide{Courbe représentative d'une fonction
+\bigskip
+
+La courbe ci-dessous peut-elle être une courbe représentative de fonction ? Si oui, préciser son ensemble de définition.}
+
+\slide{1)
+
+\image{courbe1.pdf}
+}
+
+\slide{2)
+
+\image{courbe2.pdf}
+}
+
+\slide{3)
+
+\image{courbe3.pdf}
+}
+\slide{4)
+
+\image{courbe4.pdf}
+}
+\slide{5)
+
+\image{courbe5.pdf}
+}
+
+\slide{
+\ptimage{courbe1.pdf}\ptimage{courbe2.pdf}\\
+\ptimage{courbe3.pdf}\ptimage{courbe4.pdf}\\
+\ptimage{courbe5.pdf}
+}
+\end{document}
index eed42ea..4e3adc1 100644 (file)
 \smallskip
 
 }}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%% classe sésamath
+
+\newcommand\ucm[1]{\ensuremath{#1~\text{cm}}}
+\newcommand\upc[1]{\ensuremath{#1~\text{\%}}}
+\newcommand\ueur[1]{\ensuremath{#1~\text{\eur}}}
+
+%%%%%%%%%%%