Find a condition which determines whether the hyperbola y^2 - x^2 = k contains any points with integer coordinates.

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

Explore the relationship between quadratic functions and their graphs.

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.

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?

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

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

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

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.

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

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?

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

An inequality involving integrals of squares of functions.

In y = ax +b when are a, -b/a, b in arithmetic progression. The polynomial y = ax^2 + bx + c has roots r1 and r2. Can a, r1, b, r2 and c be in arithmetic progression?

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 ?

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