I want to offer an example of a mathematical exploration which is likely to enrich learners’ appreciation of an apparently unrelated aspect of graphs of functions.

Imagine the graph of . Is there a point on the curve at which the tangent passes through the origin?

One thing that emerged when I used this with sixth-formers some years ago now is that, although their reasoning was excellent, their choice of notation left something to be desired. It had to do with what was being changed currently, and what was a standard variable. Thus using [x, y] to denote the particular point on the curve at which the tangent passes through the origin gets you into a notational mess which is at best unconvincing. Learning to use a distinct letter like t, acting as a parameter, a quantity fixed for the moment but likely to be changed, is a really useful practice!

Now repeat this for the graphs of  ,  , … and for , , … . At some point, ask yourself what is the same and what is different about these points. Then ask yourself why these points of tangency lie where they do (I am being careful not to shortcut surprises).

Now try the same task on equations of the form  for various values of λ.

Also try the same task on equations of the form  for various values of μ.

Is there anything special about the exponential or the quadratic?

Might there be value in developing the task so as to make use of Taylor approximations? For example, where do the points on  lie at which the parabola tangent to the curve at that point also passes through the origin and the point [1, 0] as λ varies, or as μ varies, or as λ and μ vary together in some way?

If learners have a coherent story about why the points of tangency where the tangent passes through the origin lie where they do, they will have made a connection with transformations of graphs.

This is intended to illustrate how mathematical explorations can be used to enrich learners’ appreciation and comprehension of mathematical topics, through offering a collection of instances of some more general phenomenon that links objects with concepts.

John Mason

John Mason worked for the Open University Mathematics Department, where he designed two of the mathematics summer schools, contributed to numerous courses, and then helped form and run the Centre for Mathematics Education. He has written numerous books and booklets, as well as research articles and book chapters, the best known being ‘Thinking Mathematically’ (1982, 2010).