# Proof

Index

Spoken language is so important in students’ development, the KS3 Programme of Study quite rightly stresses the importance of spoken and written language. Both spoken and written language and notation are key. This applies to all key stages from definitions and examples and non-examples at key stage 3 to proof by contradiction – writing negations of statements at KS5. For many students if their ability to write mathematics clearly and succinctly is poor this may well be a result of their not speaking about mathematics clearly. Conversations in class where students explain their thinking are so valuable.

“Most remarks made by children consist of correct ideas very badly expressed. A good teacher will be very wary of saying ‘No, that’s wrong.’ Rather, he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole of the art of teaching.”

W. W. Sawyer (2012). “Vision in Elementary Mathematics”, Courier Corporation

Dan Meyer really struck a chord with me at the MEI Conference in July 2021 – showing this quote from WW Sayer he said “There are no mistakes or misconceptions, just takes and conceptions.” That approach in our classes should certainly encourage an atmosphere where they are willing to share ideas and talk about mathematics.

Thinking about language and notation, and writing expressions from algebraic statements and vice versa there are some ideal activities for this, some suggestions….

This vocabulary page includes useful dictionaries and a glossary for teachers in KS1-KS3 as well as exam terminology.

In the slideshow you can see Jonathan Hall’s Worded Expressions, as always with MathsBot resources we have lots of choices – for example, hide either the sentences or expressions. With the ability to generate new expressions we have an endless supply; also from Jonathan Hall, see his Forming Expressions, these resources are ideal for self-study as well as for use in class.

From Don Steward, we have translating English to algebra, expressions, see also translating English to algebra, relationships. Also included here is an activity, A1 from the Standards Unit on Interpreting algebraic expressions. This includes 4 card sets to match, ideal for looking at multiple representations, students match algebraic expressions, explanations in words, tables of numbers and areas of shapes. One of the goals of the activity is to help learners to translate between words, symbols, tables, and area representations of algebraic shapes. The Standards Unit resources can all be accessed without a login from the very clear to navigate University of Nottingham site linked to in the Standards Unit post.

One of Chris McGrane’s Starting Points MathsCurriculum Booklets – Algebra 1 from Phase 3 features some great activities for writing algebraic statements, featured on the slides you can see a Smile activity, and Jo Morgan’s lovely Introduction to Writing Algebraically – this is such a good idea, as Jo says in the resource description if they know how to do it with numbers, then they just do the same thing with the algebra.

Further excellent resources on this skill are available on Maths4Everyone.

The last few slides here are from a Shell Centre resource, Evaluating Statements: consecutive sums. The resource is available on STEM Learning. Slides are available to show in class and also a very comprehensive Teachers’ guide which includes further suggested conjectures to discuss and common issues with suggested questions and prompts. For more on Consecutive Numbers see these Proof resources from nzmaths.

See also this search on STEM Learning for several other resources on evaluating statements. As noted earlier the Standards Unit resources can all be accessed from The University of Nottingham site. Try the following activities from the Standards Unit on evaluating statements: Number: N2, N9, Algebra A4, Geometry SS4, Statistics S2.

Excellent Statistics resource on evaluating statements for KS3 and KS4 comes from the Census at school site, no longer available in the UK (STEM Learning has a whole collection of the Census At School resources).

The following resources are all accessible to younger students. Once students have met some algebra it is possible to discuss how to express even and odd numbers, also consecutive numbers, consecutive even numbers and so on, and perhaps how to represent a general two-digit number xy as 10x+y.

From AQA comes an outstanding resource, GCSE Mathematics: 90 maths problem solving questions. Flow Chart provides an ideal activity to illustrate the steps to proof. Students can try numbers in the flow chart and see what happens – specialising, from this it should be possible to form conjectures, perhaps try more cases. Very usefully problems are indexed by strategy as well as by topic. The five strategies of the resource are 1 Set out cases, 2 Work back familiar; work back unfamiliar, 3 Find an example to fit, 4 Find key relationships and 5 Find mathematical features.

The finding mathematical features category has problems ideal for early journeys into proof, helpful also that the suggested tier Foundation, Higher or either is given.

Also recommended from AQA Maths, check their excellent GCSE Maths Focus on success modular Teacher Training packs including Problem Solving, A02.

Magic or Number tricks can be an ideal way into algebraic proof, try MathsPad – Number Tricks or be dramatic with Number Jumbler from Nrich! I have used this successfully with students from Year 7 to Year 12. Also try the suggested task, Double-Digit.

Nrich has this collection Reasoning, Justifying, Convincing and Proof for Lower Secondary; see also – Thinking Mathematically. A search on Nrich on Proof returns a wonderful selection for all ages and stages. The article Reasoning: the Journey from Novice to Expert from the Nrich Primary Team describes what NRICH see as a five-step progression in reasoning. relevant for secondary students also. 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.

For also the Nrich article Proof for all (st)ages.

Progression Maps in Reasoning Skills from NCETM are a valuable resource for Primary Students, also valuable for KS3 teachers.

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!

Diffy is an activity that encourages students with exactly what they need in learning about proof, to play around with different cases, spot patterns and make predictions. In a post by Don Steward on Diffy he has numerous excellent questions for students to explore, he suggests trying constant gap diffys for example, younger students who can manipulate like terms could try an algebraic proof here.

Stella’s Stunners is a library of more than 700 Non-Routine Mathematics Problems for Middle and High School Students; these problems certainly encourage mathematical thinking. You can filter the problems to narrow your search and usefully create a pdf document of your chosen problems. Perhaps try Symbol Pushing or Logic. Try the Introductory Problem Set which consists of twenty problems designed to show you how to use the problem library. A comprehensive user guide is available.

Building Blocks – Proof, Andy Lutwyche –

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

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. Also, check the Key Skills for Algebraic proof. 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.

The legacy coursework tasks, provide some very useful activities for algebraic proof, a search for these old coursework tasks returned this Edexel document which has numerous investigations with mark schemes.

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). In the GCSE questions by topic section, under Algebra you will find Algebraic Proof. A new navigation system is currently under development on the site.

On Corbett Maths, check Algebraic Proof:    Video    Practice Questions     Answers. These resources are for level 2 Further Maths but would also be useful for high ability higher level GCSE students.

For Geometry – much early exploration with Dynamic Geometry such as GeoGebra or Autograph is ideal. Key Stage 3 students can explore such activities and get a real feel for the simple theorems such as the angle sum of a triangle or this demonstration (choose Theorem 06) in the first slide where they can move points but see that the two interior angles always add to the opposite exterior angle.

Also shown here: Tim Brzezinski – Triangle Exterior Angle

In slide 3 we could treat this as a goal free problem. What is the diagram showing you? This idea came from Colin Foster’s article – Trapezium Artist: Some thoughts on the formula for the area of a trapezium where he discusses a Year 8 lesson on area. Colin Foster states “The formula for the trapezium stood out as being the only one that wasn’t immediately ‘see-able’. With thought, those for the triangle, rectangle, parallelogram and kite could all be seen to be correct at a glance. That got us thinking about different ways of proving the formula. We were seeking something not only believable but striking enough as an image to stick in our minds. All our methods involved converting to simpler shapes”

“I have ended up quite keen on Method 6”

Thinking about Pythagoras see slides 4, 5 and 6. Slide 6 shows Nrich Proving Pythagoras.

For US President James Garfield’s proof of Pythagoras – see this article from MAA, Mathematical Treasure: James A. Garfield’s Proof of the Pythagorean Theorem.

Robert Smith has written many great Autograph resources, see his triangle puzzle on slide 6 and circle theorem on slide 8. Features of Autograph I really like include the ability to have text boxes on the page and the excellent styling options including colour choices – all can make explanations clear for students,

Proof Sorters can be ideal to show students clear language for proof, they can just be concerned with the logic of the proof whilst seeing examples of the language used.

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!

Proof Sorters – Further examples

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.

RISP 1 from Jonny Griffiths 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.

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 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 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.

From Nrich, see preparation for STEP examinations. The STEP resources 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

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.

More from Nrich

Plymouth University – Proof by Induction

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