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Rules of arithmetic with decimals

This page was translated from German with AI. It has not been checked by a native speaker. Original page in German

With the rules of arithmetic (die Rechengesetze) you calculate faster, often in your head. And you learn what comes first.
📄 Open worksheet (PDF, German)

The video

In the video numbers are swapped and grouped until it gets easy.

Subtitles are available in ten languages. Choose them at the bottom right of the video.

Contents of the video

  1. 0:00 Introduction
  2. 0:17 Swapping
  3. 0:48 Grouping
  4. 1:18 Multiplication before addition
  5. 1:57 The distributive law
  6. 2:36 Careful, mistake!
  7. 2:53 Try it yourself

The most important points

Swapping and grouping

With plus and times you may swap (vertauschen) and group (zusammenfassen) the numbers: \(2{,}7 + 4{,}5 + 1{,}3 = 2{,}7 + 1{,}3 + 4{,}5 = 8{,}5\), \(0{,}5 \cdot 7{,}3 \cdot 2 = 1 \cdot 7{,}3 = 7{,}3\).

First the pairs that make a whole number.
First the pairs that make a whole number.

Brackets first, then times and divided before plus and minus (Punkt vor Strich)

\(2 + 0{,}5 \cdot 4 = 2 + 2 = 4\), but \((2 + 0{,}5) \cdot 4 = 2{,}5 \cdot 4 = 10\).

With and without brackets.
With and without brackets.

The distributive law (das Verteilungsgesetz)

\(4 \cdot 2{,}25 = 4 \cdot 2 + 4 \cdot 0{,}25 = 8 + 1 = 9\). Also backwards: \(3{,}5 \cdot 6 + 3{,}5 \cdot 4 = 3{,}5 \cdot 10 = 35\).

Distributing and grouping.
Distributing and grouping.

The worksheet

Here you can see the worksheet for the video (in German). First it explains everything again, then there are practice tasks at three levels. You can also open it as a PDF and print it.

📄 Open worksheet (PDF, German)

The worksheet – Page 1
Page 1/4
The worksheet – Page 2
Page 2/4
The worksheet – Page 3
Page 3/4
Show the worksheet solutions
The worksheet – Page 4
Page 4/4

Try it yourself

Task

Calculate cleverly: a) \(1{,}6 + 3{,}9 + 0{,}4\) b) \(5 - 0{,}5 \cdot 4\)

Show solution

a) \((1{,}6 + 0{,}4) + 3{,}9 = 5{,}9\) b) \(5 - 2 = 3\)

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Rechengesetze mit Dezimalbrüchen (vertauschen, zusammenfassen, Klammer zuerst, Punkt vor Strich, Verteilungsgesetz). Stelle mir nacheinander fünf kurze Aufgaben. Warte immer auf meine Antwort. Sag mir dann, ob sie richtig ist, und erkläre kurz, warum. Benutze einfache deutsche Sätze.

How this was made: The idea, structure and examples of the series are mine. I made them with the help of AI: a language model (LLM) built the slides of the video following my instructions. The voice was generated with Qwen3-TTS, modelled on my own voice.

The subtitles, the translations, the worksheet and this page were also made with the help of AI. The translations have not been checked by native speakers.

The videos, worksheets, texts and prompts in this series are licensed under CC BY 4.0: you may use, change and share them if you name the source.