# Proof

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!

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!

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.

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.

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.

# Have you seen…?#7

These wonderful, and reassuring A level notes for students from Mathsbox?

From the excellent MathsBox site, brilliant timing for this wonderful free resource: Pure Mathematics A Level Notes. The notes have a very comprehensive set of clear and well-explained examples which I am sure will appeal to students.

We now have Statistics and Mechanics notes as well.

Free A LEVEL MATHS – Statistics and Mechanic Notes

Statistics : https://t.co/umOaX5b9XJ
Mechanics : https://t.co/8vybq1RqMW

Thanks to @TheHBomb @MrsMathematica @JamesMaths1 @DrBennison for the suggestions/checking/corrections.

— SandraD (@mathsbox1) April 23, 2019

Mathsbox has many free samples to try, easily one of the best value subscription sites around in my opinion.

# Making Mathematics Stick

Following my presentation at the ATM-MA 2019 Conference, I have updated my Retrieval Practice page with many new resources. The complete (giant) set of slides is available, all links are provided separately on the Retrieval Practice page

Some highlights from the updates include:
Jonathan Hall’s wonderful NonExamples.com.

NonExamples.com – Jonathan Hall

All the excellent guides from Retrieval Practice.org

..and the brilliant Interactive Comic from Nicky Case – How to Remember Anything Forever-ish.

Interactive Comic by Nicky Case

Talking of making things stick, we are of course into the final run-up to public exams. For Further Maths, we have a lovely addition to Further Mathematics teaching resources with Mohammed Ladak’s ‘MathedUp’, see his A Level Further Maths Takeaway, a wonderful source of exam questions by topic with mark schemes. AQA questions have been used here. Also available on MathedUp are A Level Maths and GCSE Takeaways.

For GCSE, a recent addition on Corbett Maths is these Challenge Papers 2019. Note too, John Corbett’s “A Bit of Everything” papers (Edexcel or AQA). Brilliant resources for final revision.

# It’s Springtime…

Dandelions in the Breeze – with Desmos

We can celebrate Spring with Desmos.

For more flowers, perhaps a tulip, or we can, of course, turn to Polar Coordinates for some rose curves.

With my Further Mathematicians recently, we were looking at Cardioids
a+bsinθ and a + bcosθ. When do you get a dimple? When do you get an inner loop?
The University of Georgia – Polar Equations Alex Szatkowski investigating a + bcosθ.

Polar Equations – Alex Szatkowski

A little more exploring here I think! Through the above link, I chanced upon Jim Wilson’s resources for a Technology and Secondary School Mathematics course. The Polar Equations exploration is here.

It’s holiday time for schools – if you are still on the lookout for Easter resources, see Mathematics for Easter. For some lovely problems, check the Easter Challenges from Perton School Maths Department.

# Have you seen…#6

From the excellent MathsBox site, brilliant timing for this wonderful free resource: Pure Mathematics A Level Notes. The notes have a very comprehensive set of clear and well explained examples which I am sure will appeal to students.