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Proportional relations

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

An exercise book costs 2 €. What do 3 books cost? Twice as many books cost twice as much. Such a relation is called proportional (proportional).
📄 Open worksheet (PDF, German)

The video

In the video we write a table of values (die Wertetabelle), draw the graph and use the rule of three (der Dreisatz).

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

Contents of the video

  1. 0:00 Introduction
  2. 0:18 The table of values
  3. 0:41 Proportional
  4. 1:23 The graph
  5. 1:45 The rule of three
  6. 2:32 Careful, mistake!
  7. 2:57 Try it yourself

The most important points

Proportional

Twice as many: twice as much. The quotient (der Quotient) is always the same: \(4 : 2 = 10 : 5 = 2\). The graph is a ray through the zero point.

Top ·\, 2, bottom ·\, 2. The quotient is always 2.
Top \(\cdot\, 2\), bottom \(\cdot\, 2\). The quotient is always 2.
All points lie on a ray through the zero point.
All points lie on a ray through the zero point.

The rule of three

4 kg of apples cost 10 €. First to 1 kg: both sides \(: 4\), 1 kg costs 2,50 €. Then to 6 kg: both sides \(\cdot\, 6\), 6 kg cost 15 €.

The rule of three in two steps.
The rule of three in two steps.

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

Try it yourself

Task

3 exercise books cost 4,50 €. What do 5 books cost?

Show solution

1 book: \(4{,}50 : 3 = 1{,}50\) €. 5 books: \(5 \cdot 1{,}50 = 7{,}50\) €.

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: proportionale Zuordnungen (Wertetabelle, Quotient, Graph durch den Nullpunkt, Dreisatz). 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.