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

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

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

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

What biological growth processes can you fit to these graphs?

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.

Four vehicles travelled on a road. What can you deduce from the times that they met?

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

Quadratic graphs are very familiar, but what patterns can you explore with cubics?

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

Which line graph, equations and physical processes go together?

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

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.

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.

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?

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

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

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

I took the graph y=4x+7 and performed four transformations. Can you find the order in which I could have carried out the transformations?

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

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

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

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.

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 ?

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

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

Explore the relationship between resistance and temperature

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

Can you find the lap times of the two cyclists travelling at constant speeds?

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

This is a beautiful result involving a parabola and parallels.

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

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

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

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

Position the lines so that they are perpendicular to each other. What can you say about the equations of perpendicular lines?

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

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

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.

When I park my car in Mathstown, there are two car parks to choose from. Can you help me to decide which one to use?

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. . . .