Problems about graphs of functions can be grouped into *interpretation* or *construction *tasks. The tasks may involve interpreting individual points, an interval, or the entire graph. The same may be said about construction tasks. It may involve point-plotting, a part of the graph, or constructing the whole graph.

Tasks involving constructing graphs are considered more difficult than interpreting graphs tasks but with the available graphing technology, constructing graphs is now easy. But not when you have to construct a relationship, not just graphs! In fact, I would consider it as an indicator of students deep understanding of graphs and functions when he or she can interpret and reason in terms of relationship shown in the graphs and from these be able to construct a new relationship, a new function. Here is a task you can use to assess this level of understanding. Note that in this task the graphs are not on grids to encourage holistic analysis of the graph rather than point-by-point. Interpreting graphs not on grids encourages algebraic thinking.

Below is a a sample a Year 8 student solution to the task above. This answer indicates that the student understands graphs and the function it is representing but he/she could still not reason in terms of relationship so resorted to interpreting individual points in *x* vs *y* and *y* vs *z* in order to relate *x* and *z*.

The figure below shows a solution of a Year 10 student who could reason in terms of the relationships of the variables represented by the graphs.

A similar solution to this would be “*x* is directly proportional *y* but *y* is inversely proportional to* z* hence *x* would also be inversely proportional to z”.

Both solutions are correct and both solved the problem completely. Note that initially students will use the first solution just like the Year 8 student. The Year 10 however should be expected and encouraged to reason in terms of relationship.

A good assessment task not only assesses students’ mathematical knowledge and skills but also assesses the level of thinking and reasoning students are operating on. See posts on features of good problem solving tasks.