# Rich Starting Points in A Level Mathematics – Core 1 and Core 2 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 resources related to content for the Core 1 and 2 mathematics modules.

**1. Basic Algebra**

**Risp 3: Brackets Out, Brackets In**

- A chance to practice expansion of brackets and factorisation within the context of a simple algebraic result that can be spotted and then proved.

**Risp 8: Arithmetic Simultaneous Equations**

- You need to revise simultaneous equations at the start of the A Level course, but you wish to take account of your students’ prior learning and differing abilities.

- 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.

**2. Coordinate Geometry**

**Risp 5: Tangent through the Origin**

- An investigation using a computer or graphics calculators that could lead towards simple differentiation, or simultaneous equations or equal roots of a quadratic.

- Given two points at the end of a diameter, what is special about the circle equation we can form from these clues?

- The Venn Diagram with three bubbles is the maths teacher’s friend! This pdf suggests ways in which this idea can be used in a range of settings.

- When does x^2 + y^2 + ax + by + c = 0 give you a circle? Sometimes a, b and c are such that nothing appears on your screen when you try to graph the curve.

- Using a graphing program together with its constant controller facility, one can produce six parabolas simultaneously - what happens if we ask them to behave in certain ways?

- 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.

- You are given four clues - but they can’t all be true together. What combinations of the clues define a parabola? A good test of how to sketch a quadratic curve.

- A slightly off-the-syllabus activity that would be useful for an end-of term lesson. It rehearses the distance formula, and encourages ideas of curve-sketching and plotting. It also introduces the conics.

**3. Polynomials**

**Risp 6: The Gold and Silver Cuboid**

- The edge-length, the surface area and the volume of a cuboid are examined here, and it is a chance to look at the solution to polynomial equations with the help of a computer.

- The Venn Diagram with three bubbles is the maths teacher’s friend! This pdf suggests ways in which this idea can be used in a range of settings.

- A chance to use the Remainder Theorem and Factor Theorem in the context of a simple result from number theory that provides a motivation for the work.

**4. Curve Sketching**

**Risp 6: The Gold and Silver Cuboid**

- The edge-length, the surface area and the volume of a cuboid are examined here, and it is a chance to look at the solution to polynomial equations with the help of a computer.

- A chain hangs in a catenary - but how close is this to being a parabola? A chance to bring in a discussion of percentage error and some related graphs…

- The initial challenge is to see how many curves students can build from a given set of cards. Then questions of increasing difficulty are asked about these curves; what questions would you ask?

- We know what a^b means, but what does a^b^c mean? An activity that extends understanding of powers, while introducing some nice graphical work too.

- You are given four clues - but they can’t all be true together. What combinations of the clues define a parabola? A good test of how to sketch a quadratic curve.

**5. Uncertainty and Inequalities**

- A chain hangs in a catenary - but how close is this to being a parabola? A chance to bring in a discussion of percentage error and some related graphs…

**6. Indices and Surds**

- We know what a^b means, but what does a^b^c mean? An activity that extends understanding of powers, while introducing some nice graphical work too.

**7. The Language of Mathematics and Proof**

**Risp 1: Triangle Number Differences**

- A chance for students to explore ideas of proof while focussing on some simple ideas from number theory. The only prior knowledge needed is ‘prime’ and ‘triangle’ as applied to numbers.

**8. Sequences and Series**

- How many types of sequence behaviour can your students create using just this given set of cards? The range of possibilities is surprising!

**Risp 14: Geoarithmetic Sequences**

- When you start with sequences, there are a number of different behaviours that you want your student to encounter. This activity gives rise to most of the kinds of sequence you want to examine, and their classification makes for good practice.

- When does the nth term of a sequence equal the sum of the first n terms? A chance to consolidate the formulae for summing arithmetic and geometric sequences.

**9. Differentiation**

**Risp 36: First Steps into Differentiation**

- How to start on the calculus? A key question - one answer is here. An approach through pattern- spotting that should bring all your students with you.

**10. Integration**

- e plays such a fundamental role in A Level maths that it is good to introduce the number early. Here there are three quick starter activities that introduce e, with one main task on integration that leads to e.

**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?

**11. Trigonometry**

- Given four facts about a triangle, only three of which are true at any one time, how many triangles can you make?

**12. Logarithms and Exponentials**

**Risp 31: Building Log Equations**

- You are given a set of cards, including three ‘log’ cards, and you have to build sensible equations using them. Are these equations always, sometimes, or never true?

- A chain hangs in a catenary - but how close is this to being a parabola? A chance to bring in a discussion of percentage error and some related graphs…