In addition to the detailed content for Proof, for A Level Mathematics, note that Mathematical argument, language and proof, is one of the overarching themes and so must be applied, along with associated mathematical thinking and understanding, across the whole of the detailed content of the specification.
The following resource compilation provides a variety of resources for teaching Proof at this level.
Proof sorters can be ideal for showing students clear language for proof; they can just be concerned with the logic of the proof while seeing examples of the language used.
From Nrich, try this Interactive Proof Sorter example which works on a phone or tablet as well as on a 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 onto an A4 page!
Proof Sorters – Further examples
- Nrich – The Quadratic Formula
- Nrich – Sum of an Arithmetic Sequence
- Nrich – Geometric Sequence
- Nrich – the Square Root of 2 Is Irrational (electronic version)
- Nrich – Curve Fitter a proof by contradiction, related problem Curve Fitter
- TeachitMaths card sort for root 2 is irrational
- Underground Maths Proving the laws of logarithms
- TeachitMaths – Integration by Parts – Card Sort
- Euclid’s proof of infinitely many primes
RISP 1 from Jonny Griffiths’s collection is Triangle Number Differences – A lesson on proof using some light number theory. RISP 12 – Two Repeats provides excellent algebra practice; given a starting premise, students have to solve the puzzle using algebra, working logically and systematically to arrive at the solution.
The Proving Ground, by Jonny Griffiths is an introduction to mathematical proof. This complete e-book, available free from ATM, offers forty easy to understand problems classified by one of three levels, level 3 has problems that have not been resolved. The book is very clearly structured, a notes page is provided for each problem, best accessed after trying the problem first. There are also Learning Pages that introduce different proof techniques. The home page is the contents page with links to all parts of the book.
On TL Maths the play list for Proof is here.
Before we worry about the logic of proof by contradiction and proof that root 2 is irrational, perhaps an ideal activity is the Shell Centre’s Evaluating statements about rational and irrational numbers (scroll down for the lesson resources.)
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 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 notes on solutions are provided for teachers. In the task illustrated here, a full proof is given, and students are asked to explain each step.
From Loughborough University, see Self-Explanation Training.
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 proof by contradiction on the Bronze set C for Edexcel, see question 12 on A level paper 1 in this set.
A useful resource for A Level Maths is OCR’s section check in on Proof.
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…
More from Nrich
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.
Proof by contradiction requires the skill of negating a statement. AMSP’s Topic Plans include Proof in the Year 2 section. Check the Reasoning and proof activity pack, also this set of five videos on Reasoning and proof, lesson plans and presentations are also included including proof by contradiction.
From the University of Toronto, Logic and Mathematical Statements – worked examples of writing the negation of a statement with very useful generalisations. This resource is one of the resources from a series of resources on Logic and Mathematical statements designed to prepare students for first-year calculus. Nine high school topics which students are expected to know are available.
Physics and Maths Tutor, Proof Year 1, cheat sheet and exam questions by topic.
For Year 2, Proof by Contradiction, there is a brief summary and two examples (one of which is the proof by contradiction that there are infinitely many prime numbers) on the Algebraic Methods cheat sheet.
For Further Mathematicians, these very clear notes with exercises from Plymouth University include Proof by Induction.
On Physics and Maths Tutor, Proof by Induction resources are available for AQA, Edexcel, and OCR.
From Nrich, these STEP resources include Proof by Induction.
Excellent videos are available for Further Maths, including Proof by Induction.