Top Tips for STEP 3

5 min read

UPDATE: From 2024, OCR will be replacing CAAT as the official testing body for STEP. The STEP 2 and 3 test papers will not be changing dramatically. For more information on scoring and important dates, please see the OCR website.

The STEP 3 syllabus is based on all of A-level Maths and A-level Further Maths. Common topics in STEP 3 include:

  • Elementary number theory
  • Vieta’s Formulas and equating coefficients
  • Complex numbers
  • Proof by induction
  • Coordinate Geometry using Cartesian and Polar coordinates
  • Sequences defined iteratively
  • Integration by parts and by substitution
  • Differential equations
  • Curve sketching
  • Trigonometry
  • Complex numbers
  • Inequality manipulation
  • Vector geometry and matrices in 3D
  • Pulleys
  • Collisions
  • Simple Harmonic Motion
  • Circular Motion
  • Hooke’s Law
  • Equilibrium and friction
  • Poisson distribution
  • Elementary Probability
  • Continuous Random Variables

Five STEP 3 Tips

Study the problem progression

There’s no multiple choice in STEP. All the problems consist of multiple parts, often related. Results and techniques are usually developed throughout the question, and a key skill for both STEP papers is reflecting on the methods you’ve used and identifying how you can adapt them to solve later parts. Here’s a STEP 3 example question from last year:

The first part introduces a technique called ‘proof by infinite descent’. Rearranging the first equation to make  the subject shows that  is even. Hence  is divisible by 8 so can be rewritten as . Dividing the whole equation by 2 then gives the required equation. We can then repeat this two more times (with  and ) to obtain the final required equation. Since we can then repeat this whole argument with this equation, we can indefinitely find smaller and smaller triples of integers which solve the original equation. However the absolute values of these decrease at each step, so this process cannot go on forever. This yields a contradiction.


After the first part is answered, you should take time to reflect on the components of the proof. For the second part we should consider how the proof can be adapted. The coefficients are different this time: . In the first part we used divisibility by 2, but since two of the coefficients are multiples of 3, this suggests we should use divisibility by 3 instead. This ends up working very well and the proof method is very similar to that in the first part. Finally, just start by making the b term the subject of the equation.

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Take time to reflect after you solve a problem

Here’s another STEP tip for you: The most important part of the problem-solving process which students often miss out on is reflection on the problem once it’s solved. Here are some key questions you should ask yourself:  

  • What are the question components and their corresponding solution components?
  • What new techniques are developed throughout the question?
  • How are earlier parts connected to later parts?
  • What existing methods that I already knew about were used in unconventional ways?
  • Can I apply one of these techniques to a slight variation of the problem?
  • Is this question similar to other questions (possibly non-STEP) that I’ve seen before?
  • Are there alternative solution methods? (You may want to check the mark schemes for this)
  • Are there parts of the STEP 3 syllabus that I should look over again to ensure I understand them fully?

Work through lots of STEP past papers to figure out what questions you are best at

You get free choice of the 6 questions you want to answer. You can choose to do more than 6 but it’s highly recommended that you focus on just 6 questions and do those as best as you can. By working through lots of STEP past papers, you’ll get a feel for what your strengths and weaknesses are. For example if you’re good with parametric equation questions, which tend to come up early on the paper, then you can plan to do that question first if it comes up.

Or perhaps you find that you’re struggling with Statistics and Mechanics a lot more than with Pure Maths questions. Then you might want to consider focusing solely on Pure Maths questions and doing 6 of those in the exam. At the start of the exam, take a look through the paper and pick out some preferred questions that look like they’re more in your comfort zone. This will allow you to secure some easier marks towards the start.

STEP 3 Timing tips and how to get ‘unstuck’

You have 3 hours to do 6 questions, which means on average you should be spending 30 minutes per question. You may even want to spend around 25 minutes per question to give yourself time at the end to go over your work and reattempt any tricky parts.

What’s even more important is how not to lose excessive amounts of time on difficult question parts. When working on a hard problem it can be easy to think yourself into a rut. If you notice that you’re going in circles or not really getting anywhere, try to remind yourself of the different methods that are on the table, e.g. algebraic, geometric, calculus, as well as results and techniques that have been developed in earlier parts. Think about what those techniques and results are really saying and what they can enable you to do. Sometimes a small shift in perspective is all that’s needed to find a path to a solution.

If you’re still struggling to come up with ideas then don’t dwell on the question for too long. Move to the next part and give your subconscious mathematician some time to process the problem. Then later when you revisit the problem you’ll be able to see it with fresh eyes, which will often help you find a new approach.

Use available STEP resources

There are lots of great resources for STEP 2 preparation that can help you if you’re stuck. These include worked solutions on the STEP and MEI websites, and a guide dedicated to STEP preparation. Read our article on STEP 2 Top Tips. You can also read our article ‘What are the best STEP Resources?’ to boost your STEP preparation.

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