Exploring and noticing structure - advanced

Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?

In this problem, we define complex numbers and invite you to explore what happens when you add and multiply them.

Complex numbers can be represented graphically using an Argand diagram. This problem explains more...

How can you decide if a graph is traversable?

Where do people fly to from London? What is good and bad about these representations?

Investigate the transformations of the plane given by the 2 by 2 matrices with entries taking all combinations of values 0, -1 and +1.

Explore a new way of multiplying with matrices.

Change one equation in this pair of simultaneous equations very slightly and there is a big change in the solution. Why?

What happens when you find the powers of this matrix?

Can you work out which spinners were used to generate the frequency charts?

Starting with two basic vector steps, which destinations can you reach on a vector walk?

This problem challenges you to sketch curves with different properties.

Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

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

10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?

Explore the shape of a square after it is transformed by the action of a matrix.

Here you have an opportunity to explore the proofs of the laws of logarithms.

There are many different methods to solve this geometrical problem - how many can you find?

Which of the statements must be true?

Explore the strange geometrical properties of the Koch Snowflake.

The equation a^x + b^x = 1 can be solved algebraically in special cases but in general it can only be solved by numerical methods.

In this short problem, try to find the location of the roots of some unusual functions by finding where they change sign.

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

What can you deduce about the gradients of curves linking (0,0), (8,8) and (4,6)?

Find the location of the point of inflection of this cubic.

Get started with calculus by exploring the connections between the sign of a curve and the sign of its gradient.

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?

Looking at small values of functions. Motivating the existence of the Maclaurin expansion.

Solve these differential equations to see how a minus sign can change the answer

How would you sort out these integrals?

Investigate the normal distribution

Farey sequences are lists of fractions in ascending order of magnitude. Can you prove that in every Farey sequence there is a special relationship between Farey neighbours?

Explore how matrices can fix vectors and vector directions.

Consider these questions concerning inverting rational functions

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

Investigate the family of graphs given by the equation x^3+y^3=3axy for different values of the constant a.

Exponential functions grow pretty quickly...