If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

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

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.

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

Knowing two of the equations find the equations of the 12 graphs of cubic functions making this pattern.

Which is bigger, n+10 or 2n+3? Can you find a good method of answering similar questions?

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.

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

Four vehicles travelled on a road with constant velocities. The car overtook the scooter at 12 o'clock, then met the bike at 14.00 and the motorcycle at 16.00. The motorcycle met the scooter at. . . .

The square ABCD is split into three triangles by the lines BP and CP. Find the radii of the three inscribed circles to these triangles as P moves on AD.

Can you adjust the curve so the bead drops with near constant vertical velocity?

Four vehicles travel along a road one afternoon. Can you make sense of the graphs showing their motion?

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

Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?

When I park my car in Mathstown, there are two car parks to choose from. Which car park should I use?

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

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

Find all the turning points of y=x^{1/x} for x>0 and decide whether each is a maximum or minimum. Give a sketch of the graph.

Sketch the graphs of y = sin x and y = tan x and some straight lines. Prove some inequalities.

Prove that the graph of f(x) = x^3 - 6x^2 +9x +1 has rotational symmetry. Do graphs of all cubics have rotational symmetry?

Looking at the graph - when was the person moving fastest? Slowest?

Two problems about infinite processes where smaller and smaller steps are taken and you have to discover what happens in the limit.

Here is a pattern for you to experiment with using graph drawing software. Find the equations of the graphs in the pattern.

Which curve is which, and how would you plan a route to pass between them?

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

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

Can you draw the height-time chart as this complicated vessel fills with water?

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

Explore the relationship between quadratic functions and their graphs.

This task depends on learners sharing reasoning, listening to opinions, reflecting and pulling ideas together.

Explore the relationship between resistance and temperature

Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

Various solids are lowered into a beaker of water. How does the water level rise in each case?

Draw a square and an arc of a circle and construct the Golden rectangle. Find the value of the Golden Ratio.

Explore the hyperbolic functions sinh and cosh using what you know about the exponential function.

This set of resources for teachers offers interactive environments to support work on graphical interpretation at Key Stage 4.

Can you work out which processes are represented by the graphs?

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

Two buses leave at the same time from two towns Shipton and Veston on the same long road, travelling towards each other. At each mile along the road are milestones. The buses' speeds are constant. . . .

Two cyclists, practising on a track, pass each other at the starting line and go at constant speeds... Can you find lap times that are such that the cyclists will meet exactly half way round the. . . .

Here are several equations from real life. Can you work out which measurements are possible from each equation?

Several graphs of the sort occurring commonly in biology are given. How many processes can you map to each graph?