Try to sort these graphs into families.

  • How will you choose to define the families?
  • Which features of the graphs will you use?
  • Are there some graphs which belong to more than one family?

Once you have sorted the graphs into families

  • Try to sketch a further example for each family.
  • Try to sketch a graph which would not belong to each family.

How else could you have sorted the graphs?

There are many ways to sort objects into families. It is the process of trying to do this that can be interesting (and often difficult).

You could start by looking at the graphs to see if they have any shared features. Trying to sort the graphs according to whether or not they have this feature may raise further questions. For example, you may need to be more precise about how you define features of the graphs, or you may have to think about different ways to describe these features. Sorting the graphs may also highlight other features that can be used to sort the graphs into further families.

Here are some calculus-related features of the graphs that you could use to sort the graphs into families.

  • Is the gradient always positive?
  • Does the gradient function change sign?
  • Is the second derivative positive?
  • Does the function have stationary points?
  • What type of stationary points does the function have?
  • Does the tangent to the graph ever cross the graph?

If you haven’t already tried Gradient match you could try to put the cards in pairs so that functions are paired with their gradient function.