Year 11+ Exploring and noticing

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

The items in the shopping basket add and multiply to give the same amount. What could their prices be?

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

Can you find the area of a parallelogram defined by two vectors?

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.

What is the same and what is different about these circle questions? What connections can you make?

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

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?

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