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Inversely proportional relations

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

A cake has 24 pieces. The more children, the less each child gets. Such a relation is called inversely proportional (antiproportional).
📄 Open worksheet (PDF, German)

The video

In the video we share the cake, calculate the product and draw a hyperbola. Then comes the rule of three.

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

Contents of the video

  1. 0:00 Introduction
  2. 0:20 The table of values
  3. 0:40 Inversely proportional
  4. 1:16 The graph
  5. 1:33 The rule of three
  6. 2:24 Careful, mistake!
  7. 2:54 Try it yourself

The most important points

Inversely proportional

Twice as many children: half as many pieces. The product (das Produkt) is always the same: \(2 \cdot 12 = 4 \cdot 6 = 24\).

Top ·\, 2, bottom : 2. The product is always 24.
Top \(\cdot\, 2\), bottom \(: 2\). The product is always 24.
The points lie on a curve, the hyperbola (die Hyperbel).
The points lie on a curve, the hyperbola (die Hyperbel).

The rule of three

3 people need 8 hours. 1 person needs three times as long: 24 hours. 4 people need a quarter of that: 6 hours. At the bottom you do the opposite of the top.

Top : 3 and ·\, 4, bottom ·\, 3 and : 4.
Top \(: 3\) and \(\cdot\, 4\), bottom \(\cdot\, 3\) and \(: 4\).

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

5 people need 4 hours. How long does 1 person need? How long do 2 people need?

Show solution

1 person: \(4 \cdot 5 = 20\) hours. 2 people: \(20 : 2 = 10\) hours.

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: antiproportionale Zuordnungen (Produkt, Hyperbel, Dreisatz, Unterschied zu proportional). 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.