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Linear functions in everyday life

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

Taxi, mobile phone, bike rental: often there is a basic fee (die Grundgebühr) and a price per unit. That is a linear function.
📄 Open worksheet (PDF, German)

The video

In the video we calculate a taxi ride, forwards and backwards, with a table and a graph.

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

Contents of the video

  1. 0:00 Introduction
  2. 0:19 What do m and b mean?
  3. 0:59 The graph
  4. 1:20 Calculating backwards
  5. 1:42 Careful, mistake!
  6. 2:05 Try it yourself

The most important points

m and b in everyday life

b is the starting value, for example the basic fee. m is the price per unit. Taxi: \(y = 2 \cdot x + 3{,}5\).

With every kilometre, 2 € are added.
With every kilometre, 2 € are added.
5 km: 2 · 5 + 3{,}5 = 13{,}5. The ride costs 13,50 €.
5 km: \(2 \cdot 5 + 3{,}5 = 13{,}5\). The ride costs 13,50 €.

Backwards: a ride costs 17,50 €. \(2 \cdot x + 3{,}5 = 17{,}5\), so \(2 \cdot x = 14\) and \(x = 7\). The ride was 7 km long.

From the price to the distance: read across first, then down.
From the price to the distance: read across first, then down.

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

A mobile phone contract costs 5 € a month and 0,20 € per minute. What does a month with 100 minutes cost?

Show solution

\(y = 0{,}2 \cdot x + 5\); \(0{,}2 \cdot 100 + 5 = 25\). The month costs 25 €.

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: lineare Funktionen im Alltag (Grundgebühr als b, Preis pro Einheit als m, vorwärts und rückwärts rechnen). 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.