Around this time of year, many UK schools will have taster sessions as part of a Sixth Form Welcome Event. What to do for such lessons?

I can never resist starting by asking students to do a graph sketch.

We’ll see who is immediately familiar with my 11th Commandment!

Graph plotters cannot usually cope with graphs with holes, ask WolframAlpha for a plot for example and we see

However, we can force WolframAlpha to show the point discontinuity…

See this cK-12 resource – Holes in Rational Functions for many examples.

Another possible graph sketch, y² = x²  which could lead to futher discussion on line pairs. Graph sketching is of course ideal for illustrating how technology can help, something we should be doing as a natural part of A Level courses.

A very useful source of ideas for such lessons is Underground Maths. In fact it was Straight Lines from Underground Maths which led to my investigation of line pairs with a group of Level 2 Further Mathematicians, perfect since they are required to be able to factorise expressions such as 12x² + xy − y² .

Underground Maths is very helpful for teachers of A Level, I particularly like the Resource suggestions (scroll down the page) for the A Level specification.

For each content statement, Underground Maths have suggested up to three rich resources and up to three Review questions. Each suggestion is hyperlinked to take you directly to the resource on the Underground Mathematics site. Resources that are particularly good at supporting the overarching theme of Mathematical modelling have been highlighted.

I find the Review Questions can be excellent for either introducing ideas or for review and have created a spreadsheet of my own for just Review Questions which lists the questions by title rather than by number. I keep the latest version of the spreadsheet on this page. Excel File: Underground Maths Mapping Review Questions August 19 or the same file in pdf format: Underground Maths Mapping Review Questions August 19

I have found the Review questions a valuable source of tasks for Year 11, 12 and 13. You can browse all the Review questions or narrow your search by question type; note the O/AO-level questions which are questions from old papers, definitely a useful source of resources for planning taster sessions for new Sixth Form Students. One can also search by line ( Number, Geometry, Algebra, Functions or Calcuus) and by Station. Try When is 6 × 7 = 42 a counter-example?

Working on these problems is ideal for students aiming for the highest grades and they are indeed appropriate for the more demanding A Level questions. Note the many Underground Mathematics Resource Types.

Review Questions in the words of the Underground Maths Team:

These are questions designed to test students’ understanding of one or more topics and to exercise their problem-solving skills. In many cases they can also be used as a classroom resource to help teach concepts and methods. They are mostly drawn from past examination questions and have been chosen as ones that are interesting in nature and require non-routine thinking. The hints and solutions are designed to explain the reasoning and highlight connections as well as giving the answer. In many cases, alternative methods or solutions are presented.

Read about the use of Review questions in the classroom on this Teacher Support page.

If you create an account you can easily save and organise your favourite resources. This list of favourites can be easily downloaded as a csv file.

I have so many favourites on Underground Maths, here’s one – Scary Sum!

To further organise your favourites you can create subcollections.

Note that following my session at Maths Conf Mini2 – August 2021, a new post on Proof will be published shortly.

There are some excellent resources available for teaching Proof. I have created a collection here.

Poof Worksheet – crashMATHS

From crashMATHS a Proof worksheet and solutions are available for A Level. There are proof questions on the A Level Maths Practice Papers available, including a rather nice question on the recently added Bronze set C for Edexcel.

A very valuable resource for A Level Maths is OCR’s section check in on Proof and for Further Maths: OCR Check in test-Proof

Maths Genie – Proof by Contradiction

For A Level and also GCSE questions by topic, Maths Genie is a go-to site, try Proof at AS Level or A Level – proof by Contradiction or GCSE.

Plymouth University – Proof by Induction

For Further Mathematicians, these very clear notes with exercises from Plymouth University include Proof by Induction.

Underground Maths has many Proof resources; perhaps try these proof sorts, Proving the laws of logarithms or Proving the quadratic formula. Or try this review question…

Building Blocks – Proof, Andy Lutwyche –

I have mentioned a favourite TES author, Andy Lutwyche in many posts; in his excellent Building Blocks series which have questions to take students through the various skills required for each topic, we have one for Proof.

From CIMT who are one of my favourite sites for a reason – see this GCSE additional unit on Proof. A favourite site because if you are ever short of examples it is highly likely you will find something on CIMT who have everything from resources for little people to Advanced Level and everything in between!

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Nrich has this collection Reasoning, Justifying, Convincing and Proof for Lower Secondary. A search on Nrich on Proof returns a wonderful selection for all ages and stages. We have tasks to introduce ideas of proof to younger children (see also Mastering Mathematics: Developing Generalising and Proof) to preparation for STEP examinations. The STEP resources include Proof by Induction, useful for Further Mathematics Students.

Also from Nrich, try this Interactive Proof Sorter example which works on my phone as well as on my laptop. This would make a good starter – if you want to give out paper copies for students to work on as they come in, you can easily fit 4 copies to an A4 page!

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Teachit Maths, though a subscription site offers an extensive collection of activities as free pdf files. A search on Proof returns some great resources. I do like this Worksheet on Proof which has 20 varied tasks aimed at older students 16-18, though some would be accessible to younger students. Full teachers notes on solutions are provided. In the task illustrated here, a full proof is given and students asked to explain each step.

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From Don Steward on Median, we have many wonderful proof resources. Try always and never or multiple proofs. Why just multiply out brackets when we can do a little proof?

On Dr Frost’s site, it is possible to browse all his excellent resources by topic so if, for example, we search on KS2/3/4 then Algebra, we see Algebraic Proofs. Under Proof the Year 9 file PowerPoint file is excellent for high ability students, you will also see a very useful worksheet on counter-examples. I do like Dr Frost’ Full Coverage resources which are compilations of GCSE questions (GCSE – UK qualification taken at age 15-16). Answers are provided at the end of the document. Explore this outstanding site full of very high-quality resources, all Dr Frost’s clear indexing make the resources simple to find.

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MathsBot is another superb site and very easy to find questions by topic, the GCSE Exam Style Questions are a good example. Select any filters and note the many question topics.

This search of TES resources returns several highly rated free resources on proof. Try Algebraic Proof – Expressions and Proofs from James Clegg, the worksheet “teases out expressions to show certain situations (e.g. the sum of 2 consecutive odd numbers) and features options on an “answer grid” at the bottom of the page.” There are also some questions to try.

Maths4Everyone – Algebraic Proof Workbook

From the excellent Maths4Everyone this Algebraic Proof (Workbook with Solutions) has numerous problems to try as well as very clear examples. Answers are provided – highly recommended (as are all the resources on Maths4Everyone by David Morse). A new navigation system is currently under development on the site, we see that many proof resources are on the site.

Looking at the Subject content for A Level Mathematics we see that Appendix A, pages 16-22 describes the mathematical notation for AS and A level qualifications in mathematics and further mathematics. Appendix B, pages 23-26 is on mathematical formulae and identities.

Checking individual examination board specifications shows us the formulae which will be provided in the examination; each looks very similar.

• AQA Mathematics
• Edexcel Mathematics and Further Mathematics
• OCR and OCR (MEI) formulae sheets will be provided with each examination paper; details are provided at the end of each specification.

I think it is useful for students to be aware as they study the course which formulae they must know and which will be provided; though they should be very familiar with any provided formulae.

Teaching Calculus from the new specifications I see that the formula for Differentiation from first principles is provided which seems fair. Looking at MEI’s very helpful advice on Integrating Technology into your scheme of work we see some suggested resources for teaching differentiation, including this GeoGebra resource on First Principes. I like the way one can choose between numeric and algebraic.

MEI – Use of Technology, Differentiation

Staying with Calculus and technology, note that Desmos allows you to very easily see a function and its gradient function; note the requirement of the subject content that students should be able to sketch the gradient function for a given curve.


A resource I found very useful for the matching a functions with their gradient functions comes from Underground Maths. I included Gradient Match which can be used interactively online in this post on introducing gradients at GCSE. Note that you can simplify the task by giving students the set of six functions and the six gradient functions separately.

 

 

 

 

 

 

 

A new page in the series of pages on Underground Mathematics …

If you create an account (all free) on Underground Mathematics you can easily save and organise your favourite resources.