Plot the graph of x^y = y^x in the first quadrant and explain its properties.

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

Draw the graph of a continuous increasing function in the first quadrant and horizontal and vertical lines through two points. The areas in your sketch lead to a useful formula for finding integrals.

Consider these questions concerning inverting rational functions

Make a functional window display which will both satisfy the manager and make sense to the shoppers

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

Decipher a simple code based on the rule C=7P+17 (mod 26) where C is the code for the letter P from the alphabet. Rearrange the formula and use the inverse to decipher automatically.

A function pyramid is a structure where each entry in the pyramid is determined by the two entries below it. Can you figure out how the pyramid is generated?

Crack this code which depends on taking pairs of letters and using two simultaneous relations and modulus arithmetic to encode the message.

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

We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.

Place a red counter in the top left corner of a 4x4 array, which is covered by 14 other smaller counters, leaving a gap in the bottom right hand corner (HOME). What is the smallest number of moves. . . .