These are problems discussed in the “pre-STEP” sessions at the City of London Academy Southwark on 7, 14, and 21 November 2019.

Basic problems, with solutions

Extra problems, with solutions

The way we did each session was this:

• To start with, one student was the “model interviewee” and attempted Question 1 of the session the whiteboard.

• Then the students were divided into two groups. Group A had a solution to Question 2 explained to them, and Group B had a solution to Question 3 explained to them

• Then we brought back the students into pairs, each pair having one student from Group A and one from Group B. First, the student from Group A was “the professor”, and “interviewed” the student from Group B about Question 2. Then, the student from Group B was “the professor” and “interviewed” the student from Group A about Question 3.

• In the time left over, each pair of students cooperated in trying to tackle as many of the “extra” problems as they could manage.

They are problems of the sort that may be asked in interviews at those university maths departments which do ask questions in interviews.

They’re more similar to STEP exam questions than to A level. You don’t need to know more formulas or that sort of thing than for A level. But at A level you’re presented with problems where it is obvious what formulas and methods you should use for them. The only question is whether you can remember the method well and use it quickly and accurately.

In STEP, and in real life, you come across problems where the real issue is: what method should you use?

At university maths department interviews, they will *not* expect you to be able to answer all the questions completely and accurately. They’ll be looking to see if you can develop ideas to *try* to solve the questions, or at least to *start* solving them.

In the STEP exam, you only need to answer 4 out of 11 or 12 questions well in order to get an excellent grade. However, you do need to learn how to explain your reasoning clearly (which is not the same as “showing your working”).

Some ideas used in these problems which are very useful in maths generally but not much used in A level:

• Solving a simplified version of the problem first, then seeing if you can go from that solution to a solution of the original problem

• Proof by contradiction

• Sketching the *rough general shape* of graphs, rather than the exact curve (with intercepts and asymptotes and everything exactly worked out). Sometimes all you need is the rough general shape

• Doing diagrams wherever you can

• In mechanics (in the last “extra” question, for example), thinking not about the *exact* formula for the quantity you want to find, but the *shape of the formula* – which variables it includes. You can then calculate what power those variables appear with in the formula by checking that each side of the formula must be measured in the same units. A formula for escape velocity must have some expression on the right-hand side which works out as measured in metres per second.