From 906d343b82069aac07d4040e5df5dab799c6c782 Mon Sep 17 00:00:00 2001 From: Safak Date: Sat, 4 Jan 2025 05:10:53 +0100 Subject: [PATCH] =?UTF-8?q?Mathe=20Aufgaben=20alle=20eingetragen=20Erkl?= =?UTF-8?q?=C3=A4rtexte=20angepasst=20Verwandte=20Themen=20nach=20Vorgabe?= =?UTF-8?q?=20aus=20den=20Bsp=20Aufgaben=20verkn=C3=BCpft?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../mathe/topics/bruchrechnung/article.html | 26 ++++- .../topics/bruchrechnung/properties.json | 4 - .../mathe/topics/dezimalsystem/article.html | 11 ++ .../downloads/exercise1.pdf | Bin .../downloads/exercise2.pdf} | Bin .../topics/dezimalsystem/properties.json | 8 ++ .../mathe/topics/dezimalsystem/tasks.json | 26 +++++ .../mathe/topics/geometrie/article.html | 35 +++++-- .../mathe/topics/geometrie/properties.json | 9 +- .../mathe/topics/geometrie/tasks.json | 94 +++++++++++------- .../topics/graphen-diagramme/article.html | 7 ++ .../topics/graphen-diagramme/downloads/A1.png | Bin 0 -> 52169 bytes .../graphen-diagramme/downloads/exercise1.pdf | Bin 0 -> 62069 bytes .../graphen-diagramme/downloads/exercise2.pdf | Bin 0 -> 62069 bytes .../graphen-diagramme/images/diagramm.png | Bin 0 -> 62256 bytes .../topics/graphen-diagramme/properties.json | 8 ++ .../mathe/topics/graphen-diagramme/tasks.json | 8 ++ .../mathe/topics/prozentrechnung/article.html | 13 +++ .../prozentrechnung/downloads/exercise1.pdf | Bin 0 -> 62069 bytes .../prozentrechnung/downloads/exercise2.pdf | Bin 0 -> 62069 bytes .../topics/prozentrechnung/properties.json | 8 ++ .../mathe/topics/prozentrechnung/tasks.json | 26 +++++ .../punkt-vor-strichrechnung/article.html | 1 - 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Brüche werden verwendet, um Anteile an einem Ganzen darzustellen. So kann es vorkommen, dass nur ein Teil einer Pizza gegessen wird oder nur ein Teil einer Flasche getrunken wird. In der Mathematik wird ein solcher Anteil durch einen Bruch dargestellt. In der folgenden Grafik wird gezeigt, wie 3 von 7 Kästchen gelb markiert werden, und wie der entsprechende Bruch dargestellt wird.

Ein Bruch setzt sich aus einem Zähler, einem Bruchstrich und einem Nenner zusammen. Der Zähler wird über dem Bruchstrich geschrieben, der Nenner darunter.

$$\begin{array}{cl} 3 & \text{Zahler} \\ - & \text{Bruchstrich} \\ 7 & \text{Nenner} \\ \end{array}$$

Brüche addieren und subtrahieren


Brüche mit gleichen Nennern (= gleichnamige Brüche) werden addiert, indem die Zähler addiert und der Nenner beibehalten wird. So werden aus 2 von 6 Stücken plus 3 von 6 Stücken insgesamt 5 von 6 Stücken.


$$\frac{2}{6} + \frac{3}{6} = \frac{5}{6}$$


Zur Subtraktion gleichnamiger Brüche werden die Zähler subtrahiert und der Nenner beibehalten. Wenn von 5 von 6 Stücken 3 von 6 Stücke weggenommen werden, bleiben 2 der 6 Stücke übrig.


$$\frac{5}{6} - \frac{3}{6} = \frac{2}{6}$$


Brüche multiplizieren und dividieren


Brüche werden multipliziert, indem die Zähler und Nenner jeweils miteinander multipliziert werden: Zähler mal Zähler und Nenner mal Nenner. Beim Bruchrechnen mit der Grundrechenart Multiplikation spielt es keine Rolle, ob die Nenner gleich oder verschieden sind.


$$\frac{2}{9} \cdot \frac{5}{9} = \frac{10}{81}$$


Bei der Multiplikation von Brüchen mit unterschiedlichen Nennern (= ungleichnamige Brüche) werden ebenfalls die Zähler und Nenner jeweils miteinander multipliziert. Im Gegensatz zur Addition müssen die Brüche dabei nicht auf einen gemeinsamen Nenner gebracht werden.


$$\frac{4}{5} \cdot \frac{2}{3} = \frac{8}{15}$$


Die Division von Brüchen basiert auf der Multiplikation von Brüchen. Um zwei Brüche zu dividieren, wird aus der Division eine Multiplikation gemacht. Das Geteiltzeichen wird durch ein Malzeichen ersetzt. Dafür wird beim zweiten Bruch der Zähler und der Nenner vertauscht.


$$\frac{5}{8} : \frac{4}{7} = \frac{5}{8} \cdot \frac{7}{4} = \frac{35}{32}$$

+

Brüche werden verwendet, um Anteile an einem Ganzen darzustellen. So kann es vorkommen, dass nur ein Teil einer Pizza + gegessen wird oder nur ein Teil einer Flasche getrunken wird. In der Mathematik wird ein solcher Anteil durch einen + Bruch dargestellt. In der folgenden Grafik wird gezeigt, wie 3 von 7 Kästchen gelb markiert werden, und wie der + entsprechende Bruch dargestellt wird.


+ +

Ein Bruch setzt sich aus einem Zähler, einem Bruchstrich und einem Nenner zusammen. Der Zähler wird über dem Bruchstrich geschrieben, der Nenner darunter.

+ +

$$\begin{array}{cl} 3 & \text{Zahler} \\ - & \text{Bruchstrich} \\ 7 & \text{Nenner} \\ \end{array}$$

+ +

Brüche addieren und subtrahieren


+ +

Brüche mit gleichen Nennern (= gleichnamige Brüche) + werden addiert, indem die Zähler addiert und der Nenner beibehalten wird. So werden aus 2 von 6 Stücken plus 3 von 6 + Stücken insgesamt 5 von 6 Stücken.


+

$$\frac{2}{6} + \frac{3}{6} = \frac{5}{6}$$


+

Zur Subtraktion gleichnamiger Brüche werden die Zähler subtrahiert und der Nenner beibehalten. Wenn von 5 von 6 Stücken 3 von 6 Stücke weggenommen werden, bleiben 2 der 6 Stücke übrig.


+

$$\frac{5}{6} - \frac{3}{6} = \frac{2}{6}$$


+ +

Brüche multiplizieren und dividieren


+

Brüche werden multipliziert, indem die Zähler und Nenner jeweils miteinander multipliziert werden: Zähler mal Zähler und Nenner mal Nenner. Beim Bruchrechnen mit der Grundrechenart Multiplikation spielt es keine Rolle, ob die Nenner gleich oder verschieden sind.


+

$$\frac{2}{9} \cdot \frac{5}{9} = \frac{10}{81}$$


+

Bei der Multiplikation von Brüchen mit unterschiedlichen Nennern (= ungleichnamige Brüche) werden ebenfalls die Zähler und Nenner jeweils miteinander multipliziert. Im Gegensatz zur Addition müssen die Brüche dabei nicht auf einen gemeinsamen Nenner gebracht werden.


+

$$\frac{4}{5} \cdot \frac{2}{3} = \frac{8}{15}$$


+

Die Division von Brüchen basiert auf der Multiplikation von Brüchen. Um zwei Brüche zu dividieren, wird aus der Division eine Multiplikation gemacht. Das Geteiltzeichen wird durch ein Malzeichen ersetzt. Dafür wird beim zweiten Bruch der Zähler und der Nenner vertauscht.


+

$$\frac{5}{8} : \frac{4}{7} = \frac{5}{8} \cdot \frac{7}{4} = \frac{35}{32}$$

\ No newline at end of file diff --git a/webseite/config/subjects/mathe/topics/bruchrechnung/properties.json b/webseite/config/subjects/mathe/topics/bruchrechnung/properties.json index ead1b48..73c4446 100644 --- a/webseite/config/subjects/mathe/topics/bruchrechnung/properties.json +++ b/webseite/config/subjects/mathe/topics/bruchrechnung/properties.json @@ -3,9 +3,5 @@ "icon": "fa-chart-pie", "description": "Die Bruchrechnung ist ein Teil der Mathematik, der das Rechnen mit Brüchen beinhaltet, also das Teilen eines Ganzen in gleich große Teile, und umfasst Operationen wie Addition, Subtraktion, Multiplikation und Division von Brüchen.", "relatedTopics": [ - "schriftliches-multiplizieren", - "schriftliches-dividieren", - "punkt-vor-strichrechnung", - "rechnen-mit-klammern" ] } \ No newline at end of file diff --git a/webseite/config/subjects/mathe/topics/dezimalsystem/article.html b/webseite/config/subjects/mathe/topics/dezimalsystem/article.html new file mode 100644 index 0000000..911f94d --- /dev/null +++ b/webseite/config/subjects/mathe/topics/dezimalsystem/article.html @@ -0,0 +1,11 @@ +Grundbegriffe des Dezimalsystems
+Das Dezimalsystem verwendet die Basis 10. Jede Stelle repräsentiert eine Potenz von 10.

+Beispiel: + +$$435,12 = 4 \times 100 + 3 \times 10 + 5 \times 1 + 1 \times 0,1 + 2 \times 0,02$$ + +Dezimalzahlen auf eine bestimmte Stelle runden
+Runde immer zur nächsten Zahl auf oder ab, je nachdem, ob die Zahl über oder unter 5 liegt.

+Beispiel: +Rundung auf die zweite Nachkommastelle +$$3.146 \approx 3.15$$ \ No newline at end of file diff --git a/webseite/config/subjects/mathe/topics/punkt-vor-strichrechnung/downloads/exercise1.pdf b/webseite/config/subjects/mathe/topics/dezimalsystem/downloads/exercise1.pdf similarity index 100% rename from webseite/config/subjects/mathe/topics/punkt-vor-strichrechnung/downloads/exercise1.pdf rename to webseite/config/subjects/mathe/topics/dezimalsystem/downloads/exercise1.pdf diff --git a/webseite/config/subjects/mathe/topics/rechnen-mit-klammern/downloads/exercise1.pdf b/webseite/config/subjects/mathe/topics/dezimalsystem/downloads/exercise2.pdf similarity index 100% rename from webseite/config/subjects/mathe/topics/rechnen-mit-klammern/downloads/exercise1.pdf rename to webseite/config/subjects/mathe/topics/dezimalsystem/downloads/exercise2.pdf diff --git a/webseite/config/subjects/mathe/topics/dezimalsystem/properties.json b/webseite/config/subjects/mathe/topics/dezimalsystem/properties.json new file mode 100644 index 0000000..70df2a0 --- /dev/null +++ b/webseite/config/subjects/mathe/topics/dezimalsystem/properties.json @@ -0,0 +1,8 @@ +{ + "displayName": "Rechnen mit Dezimalzahlen", + "icon": "fa-table-columns", + "description": "Rechnen mit Dezimalzahlen bedeutet, dass wir Zahlen mit Komma, wie 3,5 oder 7,25, zusammenzählen, abziehen, multiplizieren oder teilen. Es ist wie normales Rechnen, nur dass wir besonders auf das Komma achten müssen.", + "relatedTopics": [ + "prozentrechnung" + ] +} \ No newline at end of file diff --git a/webseite/config/subjects/mathe/topics/dezimalsystem/tasks.json b/webseite/config/subjects/mathe/topics/dezimalsystem/tasks.json new file mode 100644 index 0000000..7074c5d --- /dev/null +++ b/webseite/config/subjects/mathe/topics/dezimalsystem/tasks.json @@ -0,0 +1,26 @@ +[ + { + "text": "Zerlege 732.45 in seine Bestandteile", + "vars": { + "?": "700 + 30 + 2 + 0,4 + 0,05" + } + }, + { + "text": "Was bedeutet die Zahl 56789 im Dezimalsystem?", + "vars": { + "?": "700 + 30 + 2 + 0,4 + 0,05" + } + }, + { + "text": "Runde 2,849 auf zwei Dezimalstellen.", + "vars": { + "?": "2,85" + } + }, + { + "text": "Runde 7,235 auf eine Dezimalstelle.", + "vars": { + "?": "7,2" + } + } +] \ No newline at end of file diff --git a/webseite/config/subjects/mathe/topics/geometrie/article.html b/webseite/config/subjects/mathe/topics/geometrie/article.html index 6dc42c3..7c6e11a 100644 --- a/webseite/config/subjects/mathe/topics/geometrie/article.html +++ b/webseite/config/subjects/mathe/topics/geometrie/article.html @@ -1,16 +1,29 @@ -Erklaerung: Die Flaeche eines Rechtecks oder Quadrats berechnet sich durch Laenge x Breite. -Formel: A = l x b -Beispiel: Rechteck mit l = 5 cm und b = 3 cm: A = 5 x 3 = 15 cm2 +Fläche eines Rechtecks
+Die Fläche eines Rechtecks oder Quadrats berechnet sich durch Laenge x Breite.
+Formel: $$A = l \times b$$ +Beispiel:
+$$l = 5 cm, b = 3 cm \\ A = 5 \times 3 = 15 cm^{2}$$

-Thema: Berechnung des Umfangs von Rechtecken und Quadraten -Formel: U = 2 x (l + b) -Beispiel: Rechteck mit l = 5 cm und b = 3 cm: U = 2 x (5 + 3) = 16 cm +Umfangs von Rechtecken und Quadraten
+Formel: $$U = 2 \times (l + b)$$ +Beispiel:
+$$l = 5 cm , b = 3 cm\\ U = 2 \times (5 + 3) = 16 cm$$

-Thema: Flaechenberechnung bei Dreiecken -Formel: A = (1/2) x Grundlinie x Hoehe -Beispiel: Ein Dreieck mit einer Grundlinie von 6 cm und einer Hoehe von 4 cm: A = (1/2) x 6 x 4 = 12 -cm2 -Uebungsa \ No newline at end of file +Flaechenberechnung bei Dreiecken
+Formel: $$A = \frac{1}{2} \times G \times h$$ +Beispiel:
+Ein Dreieck mit einer Grundlinie von 6 cm und einer Hoehe von 4 cm:
+$$A = \frac{1}{2} \times 6 \times 4 = 12 cm^{2}$$ + +Umfang und Fläche eines Kreises
+Formel: $$U = 2 \times \pi \times r \\ A = \pi x r^{2}$$ +Beispiel:
+Kreis mit $$r = 3 cm \\ U = 2 \times 3,14 \times 3 = 18,84 cm \\ A = 3.14 \times 9 = 28,26 cm^{2}$$ + + +Volumenberechnung bei Quadern
+Formel: $$V = l \times b \times h$$ +Beispiel: Ein Quader mit $$l = 5 cm, b = 3 cm, h = 2 cm\\ V = 5 \times 3 \times 2 = 30 cm{3}$$ \ No newline at end of file diff --git a/webseite/config/subjects/mathe/topics/geometrie/properties.json b/webseite/config/subjects/mathe/topics/geometrie/properties.json index 6228864..98aa2d7 100644 --- a/webseite/config/subjects/mathe/topics/geometrie/properties.json +++ b/webseite/config/subjects/mathe/topics/geometrie/properties.json @@ -1,11 +1,8 @@ { "displayName": "Geometrie", - "icon": "fa-chart-pie", - "description": "", + "icon": "fa-compass-drafting", + "description": "Geometrie geht es um Formen, Linien, Flächen und Körpern. Man lernt dabei, wie man Figuren wie Kreise, Dreiecke oder Würfel misst, zeichnet und versteht.", "relatedTopics": [ - "schriftliches-multiplizieren", - "schriftliches-dividieren", - "punkt-vor-strichrechnung", - "rechnen-mit-klammern" + "dezimalsystem" ] } \ No newline at end of file diff --git a/webseite/config/subjects/mathe/topics/geometrie/tasks.json b/webseite/config/subjects/mathe/topics/geometrie/tasks.json index 01f087e..67ed857 100644 --- a/webseite/config/subjects/mathe/topics/geometrie/tasks.json +++ b/webseite/config/subjects/mathe/topics/geometrie/tasks.json @@ -1,38 +1,62 @@ [ - { - "text": "Ein Quadrat mit Seitenlänge $$a = 4 cm$$ berechne die Fläche", - "vars": { - "Fläche in cm^2": "16" - } - }, - { - "text": "Ein Rechteck mit $$l = 7 cm$$ und $$b = 2 cm$$ berechne die Fläche", - "vars": { - "Fläche in cm^2": "14" - } - }, - { - "text": "Berechne den Umfang $$U$$ eines Quadrats mit $$a = 6 cm$$", - "vars": { - "U in cm": "24" - } - }, - { - "text": "Berechne den Umfang $$U$$ eines Rechteck mit $$l = 8 cm$$ und $$b = 4 cm$$", - "vars": { - "?": "24" - } - }, - { - "text": "Ein Dreieck mit der Grundlinie $$G = 8cm$$ und einer Höhe $$h= 5cm$$ berechne die Fläche", - "vars": { - "Fläche in cm^2": "20" - } - }, - { - "text": "Ein Dreieck mit der Grundlinie $$G = 10cm$$ und einer Höhe $$h= 7cm$$ berechne die Fläche", - "vars": { - "Fläche in cm^2": "35" - } + { + "text": "Ein Quadrat mit Seitenlänge $$a = 4 cm$$ berechne die Fläche", + "vars": { + "Fläche in cm^2": "16" } + }, + { + "text": "Ein Rechteck mit $$l = 7 cm$$ und $$b = 2 cm$$ berechne die Fläche", + "vars": { + "Fläche in cm^2": "14" + } + }, + { + "text": "Berechne den Umfang $$U$$ eines Quadrats mit $$a = 6 cm$$", + "vars": { + "Umfang in cm": "24" + } + }, + { + "text": "Berechne den Umfang $$U$$ eines Rechteck mit $$l = 8 cm$$ und $$b = 4 cm$$", + "vars": { + "U in cm": "5/15" + } + }, + { + "text": "Ein Dreieck mit der Grundlinie $$G = 8cm$$ und einer Höhe $$h= 5cm$$ berechne die Fläche", + "vars": { + "Fläche in cm^2": "20" + } + }, + { + "text": "Ein Dreieck mit der Grundlinie $$G = 10cm$$ und einer Höhe $$h= 7cm$$ berechne die Fläche", + "vars": { + "Fläche in cm^2": "35" + } + }, + { + "text": "Ein Kreis hat den Radius $$r = 4cm$$ Berechne den Umfang. Runde auf eine NkSt", + "vars": { + "Umfang in cm": "25,1" + } + }, + { + "text": "Ein Kreis hat den Radius $$r = 5cm$$ Berechne den Umfang. Runde auf eine NkSt", + "vars": { + "Umfang in cm": "31,4" + } + }, + { + "text": "Ein Quader hat den die Länge $$l = 4cm$$ Breite $$b = 3cm$$ und Höhe $$h=2cm$$ Berechne das Volumen $$V$$", + "vars": { + "Volumen in cm^3": "24" + } + }, + { + "text": "Ein Quader hat den die Länge $$l = 6cm$$ Breite $$b = 4cm$$ und Höhe $$h=3cm$$ Berechne das Volumen $$V$$", + "vars": { + "Volumen in cm^3": "72" + } + } ] \ No newline at end of file diff --git a/webseite/config/subjects/mathe/topics/graphen-diagramme/article.html b/webseite/config/subjects/mathe/topics/graphen-diagramme/article.html new file mode 100644 index 0000000..704b81a --- /dev/null +++ b/webseite/config/subjects/mathe/topics/graphen-diagramme/article.html @@ -0,0 +1,7 @@ +Einfache Darstellung von Daten
+ +Graphen und Diagramme stellen Daten visuell dar, z.B. als Balkendiagramm oder Kreisdiagramm.

+Beispiel:
+Erstelle ein Balkendiagramm fuer die Anzahl der Schueler in jeder Klasse (Klasse A: 20, Klasse B: 18, Klasse C: 22). +
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+ +Formel: $$Prozentsatz = \frac{Teil}{Ganzes}\times 100$$ +Beispiel:
+20 von 100 Schuelern haben eine Eins: $$\frac{20}{100}\times 100 = 20\%$$ +
+Einfache Zinsrechnung
+ +Formel: $$Zinsen = \frac{(Kapital \times Zinssatz \times Zeit)}{100}$$ +Beispiel:
+Ein Kapital von 1000 Euro bei 5 % Zinsen für 1 Jahr: +$$Zinsen = \frac{(1000 \times 5 \times 1)}{100} = 50 Euro$$ + diff --git a/webseite/config/subjects/mathe/topics/prozentrechnung/downloads/exercise1.pdf b/webseite/config/subjects/mathe/topics/prozentrechnung/downloads/exercise1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..047d232d1a883b5ee9ddd2945de84dd9ea4ac0d3 GIT binary patch literal 62069 zcmdSBWmsLyvM!7U3+_JA;O_431lI{n+}+(ZxH|-b2AALx+yViD69}%s{hP@;Ywxwr z-us?=pZoo}JcBuV^r-5ts;=?&TVwD*r6M84$jZcxj7Zf4{y|1$1+V}dOstU+1q7Ji zdO3oa#euFsTL%kf6`%#k1;7Trtj4SWw6~x!1JMGQ)m@$4OkLkPgFtEy4z2*UKdvhR zoo!_8%^kpF{Az;C0AMkYkPx#X$Q5V?9uUCF{)eQd703hRtl|tZ2RVc6O~Es;va|o1 zLr4f25oB-n+mv6Tf3xYgEzIH$ZuYJKR%UrCGZ#I;Z!H7BU-f=<{fi#h5%5f4Q-6D+ z26Az5a|X`~HdV^O-u2fPD}eL&H7Pa#>mO25>;P7dKl(TTth~QkU}q#fU8U7sfvzC1 zlC(N23jloliVnEV3IHE}NU1nGn5u(Z^_antW_}CubY+&c1BX)Vk1z2*zGQzH4vvnQ z6;RBH4@;z-9o!tjGyh(s-zUuK%x|56 z_AZXU95nS}7FTDM0J&S4g4CqNz@xtfnz(>P{#da;g5UyR`D?s>4g6=kq^xXRLC(xl 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B. 50 % bedeutet die Hälfte. Bei der Zinsrechnung geht es darum, wie Geld wächst, wenn man es verleiht oder spart, z. B. bekommst du Zinsen, wenn du Geld auf ein Sparkonto legst.", + "relatedTopics": [ + "wahrscheinlichkeitsrechnung" + ] +} \ No newline at end of file diff --git a/webseite/config/subjects/mathe/topics/prozentrechnung/tasks.json b/webseite/config/subjects/mathe/topics/prozentrechnung/tasks.json new file mode 100644 index 0000000..f1940e1 --- /dev/null +++ b/webseite/config/subjects/mathe/topics/prozentrechnung/tasks.json @@ -0,0 +1,26 @@ +[ + { + "text": "Berechne, wie viel Prozent 25 von 200 sind.", + "vars": { + "%": "12,5" + } + }, + { + "text": "Wenn 15 von 60 Schülern Sport machen, wie viel Prozent sind das?", + "vars": { + "%": "25" + } + }, + { + "text": "Kapital $$K = 2000 €$$ Zinssatz $$z = 4 \\%$$ Zeit $$t = 2 J$$ Berechne den Zins $$Z$$", + "vars": { + "Z": "160" + } + }, + { + "text": "Kapital $$K = 1500 €$$ Zinssatz $$z = 3 \\%$$ Zeit $$t = 1 J$$ Berechne den Zins $$Z$$", + "vars": { + "Z": "45" + } + } +] \ No newline at end of file diff --git a/webseite/config/subjects/mathe/topics/punkt-vor-strichrechnung/article.html b/webseite/config/subjects/mathe/topics/punkt-vor-strichrechnung/article.html deleted file mode 100644 index 790d2fb..0000000 --- a/webseite/config/subjects/mathe/topics/punkt-vor-strichrechnung/article.html +++ /dev/null @@ -1 +0,0 @@ -

Diese Regel besagt, dass zuerst Multiplikationen und Divisionen und danach Additionen oder Subtraktionen durchgeführt werden. Anhand eines Beispiels wird nun gezeigt, wie diese Regel angewendet wird (und wie Fehler dabei passieren können).

a) 5 + 2 · 4 = 5 + 8 = 13 (richtige Lösung)

b) 5 + 2 · 4 = 7 · 4 = 28 (falsche Lösung)


Erklärung:

In beiden Fällen wird das Ergebnis der Aufgabe 5 + 2 · 4 gesucht. Im Fall a) wurde die Aufgabe richtig gelöst, indem zuerst die Multiplikation berechnet wurde. Das ergibt 2 · 4 = 8. Anschließend wurde 5 + 8 = 13 gerechnet. Wie zu sehen ist, wurde erst die Multiplikation ausgeführt, bevor die beiden Zahlen addiert wurden. Im Fall b) wurde jedoch ein Fehler gemacht, da hier erst addiert und dann multipliziert wurde. Das Ergebnis wurde dadurch falsch berechnet


Noch eine kleine Anmerkung:

Links des "=" muss immer dasselbe stehen wie rechts des "=". Wenn so gerechnet wird, wie es in den Beispielen gezeigt wird, passiert das automatisch. Dies wird hier erwähnt, weil es in einem späteren Kapitel – in den sogenannten Gleichungen mit Unbekannten – noch genauer besprochen wird. Zu diesem Zeitpunkt sollte jedoch einfach versucht werden, den Beispielen zu folgen und das Konzept in den Übungsaufgaben selbst anzuwenden.


Das vorherige Beispiel sollte noch einmal durchgegangen werden. Anschließend folgen eine Reihe weiterer Beispiele. Jedes sollte genau angesehen werden, um die Berechnung zu verfolgen.


Auch hier gilt: Zuerst Multiplikation oder Division, danach Addition oder Subtraktion.


Aufgabe: 8 · 3 + 2 = ?

Lösung: 8 · 3 + 2 = 24 + 2 = 26


Aufgabe: 2 + 3 · 4 = ?

Lösung: 2 + 3 · 4 = 2 + 12 = 14


Aufgabe: 6 : 3 + 2 = ?

Lösung: 6 : 3 + 2 = 2 + 2 = 4


Aufgabe: 5 - 8 : 4 = ?

Lösung: 5 - 8 : 4 = 5 - 2 = 3


Aufgabe: 5 · 3 + 8 : 4 = ?

Lösung: 5 · 3 + 8 : 4 = 15 + 2 = 17


Erläuterungen:

In den ersten vier Beispielen wurde konsequent die Punkt-vor-Strich-Regel beachtet. Es wurde zuerst multipliziert oder dividiert und danach addiert oder subtrahiert. Beim letzten Beispiel mussten sowohl eine Multiplikation als auch eine Division durchgeführt werden. Es spielt dabei keine Rolle, in welcher Reihenfolge diese beiden Operationen ausgeführt werden – wichtig ist nur, dass die Addition aufgrund der Punkt-vor-Strich-Regel am Ende berechnet wird. Das bedeutet also, dass erst 5 · 3 sowie 8 : 4 berechnet werden müssen und danach die beiden Ergebnisse addiert werden.

diff --git a/webseite/config/subjects/mathe/topics/punkt-vor-strichrechnung/properties.json b/webseite/config/subjects/mathe/topics/punkt-vor-strichrechnung/properties.json deleted file mode 100644 index 56eca32..0000000 --- a/webseite/config/subjects/mathe/topics/punkt-vor-strichrechnung/properties.json +++ /dev/null @@ -1,9 +0,0 @@ -{ - "displayName": "Punkt- vor Strichrechnung", - "icon": "fa-plus-minus", - "description": "Die Regel \"Punkt vor Strichrechnung\" besagt, dass bei mathematischen Berechnungen Multiplikation und Division immer vor Addition und Subtraktion ausgeführt werden müssen, um das richtige Ergebnis zu erhalten.", - "relatedTopics": [ - "rechnen-mit-klammern", - "bruchrechnung" - ] -} \ No newline at end of file diff --git a/webseite/config/subjects/mathe/topics/punkt-vor-strichrechnung/tasks.json b/webseite/config/subjects/mathe/topics/punkt-vor-strichrechnung/tasks.json deleted file mode 100644 index 0637a08..0000000 --- a/webseite/config/subjects/mathe/topics/punkt-vor-strichrechnung/tasks.json +++ /dev/null @@ -1 +0,0 @@ -[] \ No newline at end of file diff --git a/webseite/config/subjects/mathe/topics/rechnen-mit-klammern/article.html b/webseite/config/subjects/mathe/topics/rechnen-mit-klammern/article.html deleted file mode 100644 index c55b313..0000000 --- a/webseite/config/subjects/mathe/topics/rechnen-mit-klammern/article.html +++ /dev/null @@ -1 +0,0 @@ -

Wie wird es gemacht, wenn bei einer Aufgabe zunächst eine Addition ausgeführt werden soll und danach eine Multiplikation? Die Antwort lautet: Es wird eine Klammer gesetzt. Die mathematische Regel lautet dann ganz einfach "Eine Klammer wird zuerst berechnet". Um bei den ganzen Klammern, Punkt vor Strich usw. nicht durcheinander zu kommen, folgt hier noch einmal ein kurzer Leitfaden:

1. Wenn eine Klammer in der Aufgabe vorhanden ist, wird diese zuerst berechnet

2. Danach werden Multiplikation und Division ausgeführt

3. Als letztes folgen Addition und Subtraktion


Der Leitfaden von oben sollte gemerkt werden, um dann die folgenden Beispiele zu verstehen:


Aufgabe: (2 + 3) · 5 = ?

Lösung: (2 + 3) · 5 = 5 · 5 = 25


Aufgabe: 8 · (3 + 4) = ?

Lösung: 8 · (3 + 4) = 8 · 7 = 56


Aufgabe: 2 · (1 + 2) + 5 = ?

Lösung: 2 · (1 + 2) + 5 = 2 · 3 + 5 = 6 + 5 = 11


Aufgabe: 3 · (8 + 2) : 10 + 1 = ?

Lösung: 3 · (8 + 2) : 10 + 1 = 3 · 10 : 10 + 1 = 30 : 10 + 1 = 3 + 1 = 4


Erklärungen:

Gerade die beiden letzten Aufgaben könnten abschreckend wirken. Es wird jedoch mit den ersten beiden Aufgaben begonnen: In beiden Fällen wurde zunächst die Klammer berechnet. Das Ergebnis wurde dann mit der dahinter oder davor stehenden Zahl multipliziert. Bei der dritten Aufgabe wurde zuerst die Klammer berechnet, dann die Multiplikation und schließlich die Addition ausgeführt.


Kommen wir zur letzten Aufgabe: Auch hier wurde zunächst die Klammer berechnet. Mit dem Ergebnis der Klammer konnte dann entweder als erstes multipliziert oder dividiert werden, das ist mathematisch gleichgültig. In meinem Rechenweg wurde zunächst multipliziert, dann dividiert, und schließlich addiert. Die Aufgabe sollte noch einmal Stück für Stück durchgegangen werden, um Klarheit zu erhalten.


Priorität bei Klammern / mehrere Klammern

Eine kleine Schwierigkeit muss zu guter Letzt noch angesprochen werden: Was passiert bei mehreren Klammern und verschachtelten Klammern? Dazu folgen zunächst zwei kleine Beispiele samt Lösungen, danach die Erklärung.


Aufgabe: (5 + 3) + (1 + 2) = ?

Lösung: (5 + 3) + (1 + 2) = 8 + 3 = 11


Aufgabe: ((3 + 4) + 2) + 1 = ?

Lösung: [( 3 + 4 ) + 2 ] + 1 = (7 + 2) + 1 = 10


Erklärung:

Im ersten Beispiel gibt es zwei Klammern. Welche davon zuerst berechnet wird, ist vollkommen egal. Beide Klammern werden einzeln ausgerechnet und die Ergebnisse anschließend addiert. Kommen wir zum zweiten Beispiel: Hier gibt es zwei ineinander verschachtelte Klammern. Die Rechenregel dabei lautet ganz einfach: Immer zuerst die innere Klammer berechnen.


Zur besseren Übersicht bei verschachtelten Klammern sehen diese meist etwas unterschiedlich aus: Die innere Klammer wird üblicherweise mit "(" und ")" gekennzeichnet, während die äußere Klammer mit "[" und "]" versehen wird. Natürlich gilt wie immer: Um die Rechenregeln wirklich zu beherrschen, sollten die Übungsaufgaben bearbeitet werden.

diff --git a/webseite/config/subjects/mathe/topics/rechnen-mit-klammern/properties.json b/webseite/config/subjects/mathe/topics/rechnen-mit-klammern/properties.json deleted file mode 100644 index 1f87512..0000000 --- a/webseite/config/subjects/mathe/topics/rechnen-mit-klammern/properties.json +++ /dev/null @@ -1,9 +0,0 @@ -{ - "displayName": "Rechnen mit Klammern", - "icon": "fa-code", - "description": "Beim Rechnen mit Klammern werden die Rechenoperationen innerhalb der Klammern zuerst ausgeführt, bevor die restlichen Berechnungen im Ausdruck vorgenommen werden, um die korrekte Reihenfolge der Rechenschritte einzuhalten.", - "relatedTopics": [ - "punkt-vor-strichrechnung", - "bruchrechnung" - ] -} \ No newline at end of file diff --git a/webseite/config/subjects/mathe/topics/rechnen-mit-klammern/tasks.json b/webseite/config/subjects/mathe/topics/rechnen-mit-klammern/tasks.json deleted file mode 100644 index 0637a08..0000000 --- a/webseite/config/subjects/mathe/topics/rechnen-mit-klammern/tasks.json +++ /dev/null @@ -1 +0,0 @@ -[] \ No newline at end of file diff --git a/webseite/config/subjects/mathe/topics/schriftliches-addieren-subtrahieren/article.html b/webseite/config/subjects/mathe/topics/schriftliches-addieren-subtrahieren/article.html index fc1c3cf..471856e 100644 --- a/webseite/config/subjects/mathe/topics/schriftliches-addieren-subtrahieren/article.html +++ b/webseite/config/subjects/mathe/topics/schriftliches-addieren-subtrahieren/article.html @@ -1 +1,20 @@ -Lorem Ipsum \ No newline at end of file +Bei der Addition und Subtraktion grosser Zahlen hilft es, die Zahlen stellengerecht untereinander zu schreiben. +
+Beispiel:
+$$ +\begin{align*} +7421&\\ ++5634&\\ +\hline +13055& +\end{align*} +$$ + +$$ +\begin{align*} +9876&\\ +-1234&\\ +\hline +8642& +\end{align*} +$$ \ No newline at end of file diff --git a/webseite/config/subjects/mathe/topics/schriftliches-addieren-subtrahieren/properties.json b/webseite/config/subjects/mathe/topics/schriftliches-addieren-subtrahieren/properties.json index c78182c..0e517f7 100644 --- a/webseite/config/subjects/mathe/topics/schriftliches-addieren-subtrahieren/properties.json +++ b/webseite/config/subjects/mathe/topics/schriftliches-addieren-subtrahieren/properties.json @@ -1,7 +1,7 @@ { - "displayName": "schriftliche Addition und Subtraktion", - "icon": "fa-chart-pie", - "description": "Die Bruchrechnung ist ein Teil der Mathematik, der das Rechnen mit Brüchen beinhaltet, also das Teilen eines Ganzen in gleich große Teile, und umfasst Operationen wie Addition, Subtraktion, Multiplikation und Division von Brüchen.", + "displayName": "Schriftliches Addieren und Subtrahieren", + "icon": "fa-plus-minus", + "description": "Bei der schriftlichen Addition und Subtraktion rechnest du größere Zahlen Schritt für Schritt untereinander. Du fängst rechts bei den Einerstellen an und arbeitest dich nach links, wobei du manchmal Zahlen \"übergibst\" oder \"ausleihst\".", "relatedTopics": [ "schriftliches-multiplizieren", "schriftliches-dividieren", diff --git a/webseite/config/subjects/mathe/topics/schriftliches-dividieren/properties.json b/webseite/config/subjects/mathe/topics/schriftliches-dividieren/properties.json index bd3d157..997f565 100644 --- a/webseite/config/subjects/mathe/topics/schriftliches-dividieren/properties.json +++ b/webseite/config/subjects/mathe/topics/schriftliches-dividieren/properties.json @@ -3,6 +3,7 @@ "icon": "fa-divide", "description": "Schriftliches Dividieren ist eine Methode zur schrittweisen Aufteilung einer Zahl durch eine andere, wobei man die Teilschritte nacheinander schriftlich notiert, um das Ergebnis systematisch zu berechnen.", "relatedTopics": [ - "schriftliches-multiplizieren" + "schriftliches-multiplizieren", + "schriftliches-addieren-subtrahieren" ] } \ No newline at end of file diff --git a/webseite/config/subjects/mathe/topics/schriftliches-multiplizieren/properties.json b/webseite/config/subjects/mathe/topics/schriftliches-multiplizieren/properties.json index 5187937..662786e 100644 --- a/webseite/config/subjects/mathe/topics/schriftliches-multiplizieren/properties.json +++ b/webseite/config/subjects/mathe/topics/schriftliches-multiplizieren/properties.json @@ -3,6 +3,7 @@ "icon": "fa-x", "description": "Schriftliches Multiplizieren ist eine Rechenmethode, bei der zwei Zahlen schrittweise multipliziert werden, indem man die einzelnen Stellen der Zahlen nacheinander verrechnet, die Teilergebnisse notiert und am Ende addiert, um das Gesamtergebnis zu erhalten.", "relatedTopics": [ - "schriftliches-dividieren" + "schriftliches-dividieren", + "schriftliches-addieren-subtrahieren" ] } \ No newline at end of file diff --git a/webseite/config/subjects/mathe/topics/wahrscheinlichkeitsrechnung/article.html b/webseite/config/subjects/mathe/topics/wahrscheinlichkeitsrechnung/article.html new file mode 100644 index 0000000..1ea92d1 --- /dev/null +++ b/webseite/config/subjects/mathe/topics/wahrscheinlichkeitsrechnung/article.html @@ -0,0 +1,4 @@ +Grundbegriffe der Wahrscheinlichkeitsrechnung
+Formel: $$ Wahrscheinlichkeit = \frac{Gewünschte Ereignisse}{Mögliche Ereignisse}$$ +Beispiel:
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B. beim Würfeln eine 6 zu bekommen. Dabei rechnet man, wie viele Möglichkeiten es gibt und wie oft ein bestimmtes Ergebnis vorkommen kann.", + "relatedTopics": [ + "graphen-diagramme" + ] +} \ No newline at end of file diff --git a/webseite/config/subjects/mathe/topics/wahrscheinlichkeitsrechnung/tasks.json b/webseite/config/subjects/mathe/topics/wahrscheinlichkeitsrechnung/tasks.json new file mode 100644 index 0000000..dff688b --- /dev/null +++ b/webseite/config/subjects/mathe/topics/wahrscheinlichkeitsrechnung/tasks.json @@ -0,0 +1,14 @@ +[ + { + "text": "Was ist die Wahrscheinlichkeit $$W$$, bei einem Würfel eine gerade Zahl zu würfeln?", + "vars": { + "W": "1/2" + } + }, + { + "text": "Was ist die Wahrscheinlichkeit $$W$$ für eine rote Karte aus einem Kartenspiel (52 Karten, 26 rot)", + "vars": { + "W": "26/52" + } + } +] \ No newline at end of file diff --git a/webseite/topic.php b/webseite/topic.php index b645856..872474f 100644 --- a/webseite/topic.php +++ b/webseite/topic.php @@ -166,7 +166,7 @@ if (!isset($topicData)) {

- Aufgabe ' . ($taskIndex + 1) . ': ' . htmlspecialchars($task->getText(), ENT_QUOTES, 'UTF-8'); ?> + Aufgabe ' . ($taskIndex + 1) . ':
' . htmlspecialchars($task->getText(), ENT_QUOTES, 'UTF-8') . '

'; ?>