Einfache Umrechnung von Flächen- und Volumeneinheiten

In meiner Tätigkeit als Nachhilfelehrer, immerhin sind es schon etwa 4 Jahre, habe ich immer wieder fest gestellt, dass eine Großzahl der Schüler Schwierigkeiten mit dem Umrechnen von Flächen- und Volumeneinheiten hat und sich kaum ein Schüler die korrekte Umrechnungszahl merken kann. Kein Wunder, denn vorallem Flächeneinheiten und Volumenangaben bzw. deren Umrechnung gehören nur selten zu den Alltagserfahrungen der Schüler. Dabei kann man mit einfachen Grundlagen den Schüler zur Umrechung befähigen.

Reihenfolge der Längeneinheiten
Für die Schüler ist es nie ein Problem die Einheiten der Größe nach zu ordnen: 1 mm, 1 cm, 1 dm, 1 m, 1 km
Diese Größen sind für sie greifbar und im Schulunterricht aber auch im außerschulischen Alltag allgegenwärtig. Außerdem kennen sie auch meist die Umrechnungszahlen der Längeneinheiten:

1 mm – 10 – 1 cm – 10 – 1 dm – 10 – 1 m – 1000 – 1km

Umrechungszahl – was nun?
Dabei wissen Schüler auch, dass von einer kleineren zur größeren Einheit dividiert und umgekehrt mit der Umrechnungszahl multiplizieren müssen. Spätestens bei der Einführung der indirekten Proportionalität ist es den Schülern verständlich. Um nun direkt von mm in dm umzurechnen erhält man die Umrechnungszahl durch Multiplikation auf dem Weg von mm zu dm, also 10×10=100.

Multiplizieren und dividieren mit 10er-Potenzen
Ist die Bruchrechnung eingeführt, dann ist es einfach zu zeigen: Das dividieren und multiplizieren mit 10er-Potenzen ist Kommaverschiebung. Dabei ist die Regel einfach: die Anzahl der Nullen in 10, 100, 1000 oder der Exponent 10^1, 10^2, 10^3 entspricht der Anzahl der Kommaverschiebungen. Die Richtung der Verschiebung ist nicht schwer, wenn man sich bereits überlegt hat, ob man dividieren und multiplizieren muss. Einziges Problem ist noch das Komma bei 356 cm zu suchen – man kann ja auch einfach 356,00000… cm schreiben und dadurch ein Komma “hinzufügen”. Außerdem lässt sich die Zahl vorn und hinten mit beliebig vielen Nullen versehen – die Ziffern schreibt man sonst üblicherweise einfach nicht mit.

Der Schritt zu den Längen- und Volumeneinheiten
Die Schüler kennen meist schon die Operation des quadrierens bzw. des potenzierens allgemein. Damit lassen sich die Umrechungszahlen der Volumen- und Flächeneinheiten einfach bestimmen. Motivation hierfür sind die Einheiten selbst: cm² oder m³. Damit sind für Flächeneinheiten intuitiv die Umrechnungszahlen zu quadrieren, für Volumeneinheiten mit 3 zu potenzieren. Es ergeben sich damit also folgende Umrechnungszahlen:

1 mm² – 10²=100 – 1 cm² – 10²=100 – 1 dm² – 10²=100 – 1m² – 1000²=1000000 – km²

1 mm³ – 10³=1000 – 1 cm³ – 10³=1000 – 1 dm³ – 10³=1000 – 1m³ – km³ wird so gut wie nie verwendet

Was ist mit Hektar, Ar, Liter und Milliliter?
Anhand der großen Umrechnungszahl von m² zu km² erkennt man, dass sich beispielsweise eine sehr große Zahl wie 23900 m² nur sehr schwer in km² ausdrücken lässt, da diese Größe mit  0,0239 km² auch noch sehr klein ist. Damit lässt sich der Einschub von Ar und Hektar erklären. Die Umrechnungszahl ist dabei die gleiche, wie von dm² zu m², also 100:

1 m² – 100 – 1 a – 100 – 1 ha – 100 – 1 km²

Die “Größe” Liter ist meist durch Getränkeverpackungen bekannt. Indem man nun überlegt, wie viel ein dm² ist, merkt man, dass dieses Volumen einer Liter-Verpackung entspricht. Damit weiß man: 1 dm³ = 1 l. Nun kennen Schüler meist die Umrechnungszahl bspw. 1 m – 1000 – 1 mm oder g – 1000 – mg, so dass sie wissen, dass 1l – 1000 – 1ml ist.

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