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Why do this problem?

This problem encourages students to get into the real meaning of graphical representation without getting bogged down in algebraic calculations or falling back into blind computation. It will also encourage them to think about error in measurement.

Possible approach

There are two levels at which the graphical data can be interpreted. At a basic level, the students can easily see if a measurement increases or decreases from point to point or whether the measurement is positive or negative. At a more advanced level, they can suggest some rate of change of the measurements from point to point: although there are no units on the charts, there are certain key points (the grid lines) which allow some 'indirect' by-eye measurement. Students will need to realise this more subtle point to make full progress. (Note that the points have been carefully placed by the question setter!)

The question of the accuracy of the 'by-eye' measurements can raise interesting discussion about the accuracy of the measurements. Since no context is given in the question it is natural to assume total accuracy, but would this be the case in practice?

Key questions

  • We only have two measurements, but what information can we deduce from these?
  • How can we relate this information to the equations?

Possible extension

Think of other equations which might match the points on the graphs. How many could you think of?

Possible support

Let students leaf through a science textbook searching for graphs and charts. Do they notice that the same shapes of charts appear frequently? For each chart they find, which measurements would be a possibility?

You might also first try Real-life equations.