## graphs and functions

December 15, 2009 Leave a comment

Although it is not a sin to confuse these two it’s good to know they are different so let’s get this straight. The graph is not the function. The function is the relationship between the varying quantities represented by the graph.

Problems about graphs and 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 graph used to be the more difficult task than interpreting the graph 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 tasks not only assesses students’ mathematical knowledge and skills but also the level of thinking and reasoning they are operating on. See posts on features of good problem solving tasks. Check this link for free graphing software and how to use it.