Alf Coles writes about how he tries to create 'spaces for exploration' for the students in his classrooms.

The illustration shows the graphs of twelve functions. Three of them have equations y=x^2, x=y^2 and x=-y^2+2. Find the equations of all the other graphs.

Here is a pattern composed of the graphs of 14 parabolas. Can you find their equations?

The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.

Explore the two quadratic functions and find out how their graphs are related.

Here are some more quadratic functions to explore. How are their graphs related?

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?

This task develops knowledge of transformation of graphs. By framing and asking questions a member of the team has to find out which mathematical function they have chosen.

Substitute -1, -2 or -3, into an algebraic expression and you'll get three results. Is it possible to tell in advance which of those three will be the largest ?

Explore the relationship between quadratic functions and their graphs.

Alison has created two mappings. Can you figure out what they do? What questions do they prompt you to ask?

Clearly if a, b and c are the lengths of the sides of a triangle and the triangle is equilateral then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true, and if so can you prove it? That is if. . . .

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?