\mnb150ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fmodern\fprq1 Courier New;}{\f4\froman\fcharset1 Times New Roman;}{\f5\fswiss\fprq2 Arial;}{\f6\fswiss\fprq2 Helvetica;}{\f7\froman\fprq2 Times New Roman;}} {\colortbl\red0\green0\blue0;\red255\green0\blue0;\red0\green0\blue128;\red0\green0\blue255;\red0\green128\blue0;\red128\green0\blue0;} \deflang1031\pard\ri4\plain\f3\fs20\cf0\b ________________________________________________________________________________ \par \par \plain\f3\fs20\cf0 Inhalt....: Gleichungen schrittweise loesen \par Kategorie.: Unterrichtsmaterial \par Mathematik: Analysis, Geometrie R^2, Sonstiges \par MuPAD.....: 3.0.0 \par Datum.....: 2004-08-24 \par Autoren...: Thomas Himmelbauer \par Funktionen: plot, ViewingBox, Color, plot::Point2d, XAxisTitle, YAxisTitle, \par Funktionen: XTicksDistance, YTicksDistance, PointStyle, \par \plain\f3\fs20\cf0\b ________________________________________________________________________________ \par \plain\f4\fs32\cf0 \par \plain\f7\fs24\cf5\b Vorbemerkung: \par \plain\f7\fs24\cf5 \par Die Materialien stammen aus dem Unterricht einer 3. Klasse Gymnasium (7. Schulstufe, Sekundarstufe \par 1) in Mathematik. Die Klasse hatte pro Woche 4 Stunden Mathematik. Der Lehrstoff bestand in etwa \par aus dem Rechnen mit Ganzen Zahlen, dem Arbeiten mit Termen und Variablen und ebener Geometrie). \par Die Sch\'fcler haben in der Schule im PC-Raum zu zweit an einem Ger\'e4t gearbeitet. Jeder Sch\'fcler hatte \par auch zu Hause einen PC mit einer MuPAD-Lizenz. \par Au\'dferdem wurde die Lernplattform Elsitos verwendet. Dadurch konnten Lehrer und Sch\'fcler Dokumente \par ins Internet stellen und austauschen, z.B. Haus\'fcbungen. \par Diese Klasse soll aber der 9. Schulstufe als Notebookklasse gef\'fchrt werden. Die Besch\'e4ftigung mit \par MuPAD war als erste Vorstufe dazu gedacht. MuPAD wurde in der Regel nur eingesetzt, um bereits \par gelernte Zusammenh\'e4nge an das CAS zu \'fcbergeben oder Fertigkeiten zu vertiefen. So wurde das L\'f6sen \par der Gleichungen von Textgleichungen an MuPAD ausgelagert, um sich ganz der Aufstellung der Gleichung \par aus dem Text widmen zu k\'f6nnen. \par Das Erkennen von Termstrukturen konnten durch das Zeichnen von Baumstrukturen verbessert werden. \par Auch beim L\'f6sen von Gleichung durch \'c4quivalenzumformungen konnten die Berechnungen an MuPAD \par \'fcbertragen werden. Dadurch konnten die gesamte Konzentration auf die Umformung gelenkt werden. \par Au\'dferdem f\'fchrt MuPAD immer die angegebenen Umformung durch. Was nicht immer den Zielvorstellungen \par von Sch\'fclen entspricht. z B. f\'fchrt die Subtraktion von 3 von der Gleichung 3x=7 nicht zu x=4) \par Die MuPAD-Graphik wurde zum Zeichnen von Polygonen verwendet. Einerseits um durch die h\'fcbsche \par Graphik die Freude an der Sache zu heben, andererseits um objektorientiertes Denken, exakte Eingaben \par und den Umgang mit Koordinaten zu schulen. \par \par Um die Lernmotivation f\'fcr MuPAD hoch zu halten, wurde eine Schularbeit von den 5 Schularbeiten ganz \par mit MuPAD geschrieben. Dabei wurde die Klasse geteilt, so dass jeder Sch\'fcler einen eigenen PC zur \par Verf\'fcgung hatte. \par \plain\f4\fs32\cf0 \par 10. Schul\'fcbung: \par \par Schrittweises L\'f6sen von Gleichungen: \par \par \plain\f4\fs24\cf0\b Beispiel 1: \par \par \plain\f4\fs24\cf0 Zun\'e4chst speichern wir die Gleichung unter dem Namen gl ab.\plain\f4\fs24\cf0\b \par \par \plain\f4\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}gl:=11*x-54=13*x-64 \par \pard\ri4\plain\f4\fs22\cf0 Nun subtrahieren wir von beiden Seiten der Gleichung -11x und nennen \par die neue Gleichung gl1. \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}gl1:=gl-11*x \par \pard\ri4\plain\f4\fs22\cf0 Hernach addieren wir 64 und nennen die neue Gleichung gl2. \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}gl2:=gl1+64 \par \pard\ri4\plain\f4\fs22\cf0 Zuletzt dividieren wir durch 2 und erhalten die L\'f6sung. \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}gl3:=gl2/2 \par \pard\ri4\plain\f4\fs22\cf0 Mit dem Substitutionsbefehl k\'f6nnen wir die Probe machen. indem wir \par in der Gleichung x durch 5 ersetzen. \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}subs(gl, x = 5) \par \pard\ri4\plain\f4\fs22\cf0\b Beispiel 2: \par \plain\f4\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}gl:=(6*x-1)^2-(4*x+3)^2-(3*x+2)^2=(2*x+1)^2+(7*x-9)*(x+28) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}gl1:=expand(gl) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}gl2:=gl1-11*x^2 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}gl3:=gl2+48*x \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}gl4:=gl3+251 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}gl5:=gl4/239 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}subs(gl, x = 1) \par \pard\ri4\plain\f4\fs22\cf0\b L\'f6se die folgenden Gleichungen schrittweise und mache die Probe \par durch Einsetzen: \par \par \'dcbungsbeispiel 1: \par \par {\pict\wmetafile8\picw3708\pich720\picscalex99\picscaley98\picwgoal2118\pichgoal413 0100090000030F04000007001C0000000000050000000B0200000000050000000C02D0027C0E03 0000001E00050000000C02D902980E050000000B0200000000030000001E00050000000C02E102 A30E050000000B0200000000050000000B0200000000030000001E00050000000C02EA02C00E05 0000000B0200000000050000000B0200000000050000000B0200000000030000001E0005000000 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Weitere Anregungen finden Sie in der Buchreihe \plain\f5\fs20\cf1 Mathematik 1 x anders\plain\f5\fs20\cf4 . In dieser Reihe \par wird eine Vielzahl unterschiedlichster mathematischer Probleme mit MuPAD gel\'f6st. Die \par B\'fccher k\'f6nnen unter \plain\f6\fs20\cf3 www.schule.mupad.de/literatur\plain\f5\fs20\cf4 kostenfrei kopiert werden. \par \plain\f5\fs20\cf3 \par \plain\f3\fs20\cf0\b _______________________________________________________________________________ \par \plain\f3\fs28\cf1 \par }