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Powerful Quadratics

This comes in two parts, with the first being less fiendish than the second. It’s great for practising both quadratics and laws of indices, and you can get a lot from making sure that you find all the solutions. For a real challenge (requiring a bit more knowledge), you could consider finding the complex solutions.


You're invited to decide whether statements about the number of solutions of a quadratic equation are always, sometimes or never true.

Factorisable Quadratics

This will encourage you to think about whether all quadratics can be factorised and to develop a better understanding of the effect that changing the coefficients has on the factorised form.

Picture the Process I

Age 16 to 18

This resource is from Underground Mathematics.

Here are descriptions of eight real-world processes.  For each, try to sketch a suitable graph.  If you are not familiar with the background science, then try to use what you do know to reason through what a graph might look like.

You do not need to calculate or plot exact points, just sketch graphs that give the general shape.

As you produce your sketches, make a list of the features that you are considering.

Printable versions of these cards can be downloaded here


Temperature of a cup of tea over time. Height of the valve on a bicycle tyre as the bicycle moves forwards.
Height of a tennis ball thrown straight up and then caught. Distance fallen by a parachutist jumping out of a plane.
Reading on the odometer (mile counter) of a car driving on a motorway. Radius of a spherical balloon as it is inflated.
Volume of water remaining in a cup as water is sucked out through a straw. Distance along a tape measure measured in inches compared with distance measured in metres.

Once you have sketched graphs for some of the eight processes described, click on the following link to read a further question and move on further.



This is an Underground Mathematics resource.

Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.

Visit the site at to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.