Higher level problem solving (HLPS)

Thursday 12th November 2020

Here is a problem that can be solved using only AS level Mathematics concepts. How would you approach it?


I like this problem, as it rewards students for using good problem-solving strategies, such as drawing a diagram and introducing suitable variables1. It also requires students to show some rigour in their solution; many students attempting this problem determined the correct minimum for A but did not prove that their result was the minimum.

Would you be surprised to learn that this is a STEP question?

STEP questions have a reputation for being hard, as STEP – the Sixth Term Examination Paper – forms a compulsory part of any application to study maths at the University of Cambridge. However, what makes STEP challenging is not the difficulty of the content, but the need to really understand the subject matter. Even though STEP 1 is now discontinued, carefully chosen STEP questions make ideal practice for any A level Mathematics students aiming for high grades.

A particularly useful resource is the searchable STEP databaseOpens a new window. A search for 'differentiation' will find the question given above, which is STEP 1, 2006, Q2. You can also find the same question by searching for 'optimisation'… or 'goat'. You can also view MEIOpens a new window's free worked solutionsOpens a new window.

If you want prechosen problems to solve, there is a structured courseOpens a new window available, with short modules that make excellent tasks for independent study.

Questions like this form the basis of our regular online problem-solving classes for students. We also explore these with staff in our HLPS professional development. Current events with a focus on problem-solving, including on-demand professional development, can be found on the AMSP website. For more details about any these resources or about any of our problem-solving opportunities in the South, please email me or contact your local Area Coordinator.

1 Lots of standard problem-solving strategies can be found in Pólya’s classic work How to Solve ItOpens a new window. This excellent book about teaching problem solving is still relevant today. Pólya’s other works, Mathematics and Plausible Reasoning and Mathematical Discovery are less well-known, but are also fascinating.

By Phil Marshall

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