# Rich Starting Points in A Level Mathematics – Core 3 and Core 4 Modules

**TES Secondary maths resource collections**

**Collection Author:** Craig Barton - Maths AST and creator of www.mrbartonmaths.com (TES Name: mrbartonmaths)

The RISP resources are some of my all time favorites. As we all know, there seems to be a relative scarcity of rich, challenging A Level material out there. All the emphasis seems to go into helping our students develop into independent thinkers and problems solvers at Key Stages 3 and 4, but then when they return after the 6 week summer holiday to begin their post 16 studies, quite often they witness a return to a more traditional style of teaching and learning with which many of them struggle to cope.

The RISP resources offer opportunities to really challenge and engage A Level students. They were created by the excellent Jonny Griffiths who explains his rationale for putting them together:

*“The idea that mathematics teaching should include, indeed, should be based upon, open-ended, investigatory, problem-solving activities is as old as the hills. But maybe the most obvious truths are sometimes the most easily forgotten. I hope I am not claiming to have re-invented the wheel here.*

*Perhaps some maths teachers have been less confident about using this type of work at A Level than at GCSE, falling back on more didactic methods for the ‘advanced’ material. My belief is that it is no harder to construct and use this kind of activity at A Level than at GCSE or Primary level, or indeed with University work”*

This Collection page provides links to all the RISPS activities related to content for the Core 3 and 4 mathematics modules.

**1. Functions**

- An exploration of the idea of a periodic function with range of activities. The tasks progress from simple to quite tough - they should help you to differentiate as you see fit.

**Risp 18: When does fg equal gf?**

- This activity offers a chance to practice the composition of functions, and produces some nice results along the way.

**2. Numerical Methods**

**Risp 39: Polynomial Equations with Unit Coefficients**

- Some easy-to-remember equations with which to begin numerical methods, ones that have straightforward rearrangements that work every time.

**3. Differentiation 2**

- A way to introduce implicit differentiation in a way that your students can picture. There are also some curious graphs to catch their attention!

- This activity heads towards using the Quotient Rule for differentiation, but it asks some interesting questions about functions in general along the way.

- An arithmogon is a simple mathematical diagram that emphasises the need to be able to both ‘do’ and ‘undo’ when it comes to mathematical processes.

**Risp 38: Differentiation Rules OK**

- A ‘game’ using cards to generate practice with using the Product, Chain, and Quotient Rules. The exponential and ln functions are differentiated too. An excellent poster activity.

**4. Integration 2**

**Risp 25: The answer’s 1: what’s the question?**

- Turning a question around sometimes makes the maths more interesting. Given the answer (x+1)/(x-1) what could the question be?

**5. Proof 2**

- We know the situation - there is an awkward lesson to be filled when most of the group are out on a trip. This task gives a thorough algebraic workout to those who remain.

**6. Algebra 2**

**Risp 19: Extending the Binomial Theorem**

- If your students need some practice at using the Binomial Theorem in the case where negative and fractional indices are involved, then this activity offers a mini-theorem that students can work towards proving whilst consolidating their Binomial Theorem knowledge along the way.

- An arithmogon is a simple mathematical diagram that emphasises the need to be able to both ‘do’ and ‘undo’ when it comes to mathematical processes.

**Risp 22: Doing and Undoing the Binomial Theorem**

- Given a rational function we can hopefully find the right hand side as a power series using the Binomial Theorem. What happens if we are given the right hand side - what are the constants on the left?

**Risp 32: Exploring Pascal’s Triangle**

- ‘Five choose two is ten - so that means that n choose r is n times r…’ Discuss!

**7. Trigonometry 2**

- Forgetting to change one’s calculator from degrees mode to radians mode is likely to be a familiar error for your students. Here this is turned into an activity that tests how to solve general trig equations.

**Risp 26: Generating the Compound Angle Formulae**

- Where do the compound angle formulae come from? This activity is a route in to this question that does not involve drawing triangles, but rather seeing what happens when we combine simpler functions.

**8. Parametric Equations**

**Risp 27: Playing with Parametric Equations**

- Given some clues about which parametric curve you are dealing wiith, can you find the missing constants?

**9. Integration 3: Differential Equations**

**Risp 28: Modelling the Spread of a Disease**

- Using dice to simulate the spread of a disease, given different infection rates and recovery rates. A good application of differential equation theory, especially the business of forming differential equations.

**Risp 30: How Many Differential Equations?**

- You are invited to build differential equations given some starting expressions (picked so that each DE requires a different method for solution.) Good consolidation for students who have already encountered simple DEs.

**10. Vectors**

- Given three expressions, find a way to make each one the odd one out. Designed here for revising C4, but the method can be used for any topic.