Good Mathematicians Can Go Backwards!

There is so much opportunity for thinking backwards when we teach – a great learning opportunity and also a problem-solving strategy.

I was delighted to present a session at IOE, UCL’s Faculty of Education and Society on Saturday 18th July 2022. We looked at lots of ideas and activities to get our students thinking backwards as well as forwards. The Slides from the session are available at the end of this post in PowerPoint or pdf format. The resources used are listed below.

Nrich Articles and Resources

Number

Manipulation

Questions such as this can make a great starter for a lesson and provide the chance to discuss number operations and the relationships between them. Manipulating numbers like this can also help with algebraic manipulation.

Looking for some more examples of this type, I came across a really useful resource on TES, “If I know this then I also know …” by Piers Butler. This would make an ideal lesson starter. As it is an Excel spreadsheet, I thought it would be simple to add another worksheet with the answers and created the Excel file CY If_I_know_this_then_I_also_know_ which is a copy of the original, but just adds another worksheet with the answers.

Number Operations

Algebra

Resource Collections

From AQA comes an outstanding resource, GCSE Mathematics: 90 maths problem solving questions. Strategies discussed include work back familiar and work back unfamiliar. Problems are indexed both by strategy and also by content.

Arithmagons

Transum – Arithmagons

This post on Arithmagons includes the resources discussed and many more.

See also Digitisers from Jonny Griffiths.

Standards Unit

A2 Creating and Solving Equations

This is an outstanding resource – many excellent activities here for the secondary classroom. Start by reading Improving Learning in Mathematics – Malcolm Swan

The resources are hosted by Nottingham University, including all the pdf files very clearly indexed. Note that this site includes the complete set of resources including the software; however, this software no longer works on modern browsers and mobile devices, but see note on my Standards Unit page for HTML5 versions.

Open Middle

Distributive Property, Open Middle

Mathsbot – Jonathan Hall
Variation Grids

Fill in the blanks

Nathan Day – Factor Trees

There are many suggestions for fill in the blanks resources in this post.

Coordinate Geometry Table, Transum, Adapted with permission from a learning resource created by Rob Southern @mrsouthernmaths.

The answer is – what was the question?

Andy Lutwyche – TES resources

Here’s the diagram – what’s the question?

Presentation Slides

GCSE Mathematics Paper 3

On Third Space Learning note their complete collection of free GCSE revision resources. See the GCSE Maths Checklist 2022 page which provides a GCSE maths checklist for all GCSE advance information (2022) topics listed by the exam boards Edexcel, OCR and AQA. Filters enable you to pick your exam board from AQA, Edexcel or OCR and your chosen paper.

Third Space Learning GCSE Checklist

A reminder of other useful resources for Paper 3 final revision we have:

From White Rose Maths (scroll down the page) their excellent pre-exam workshops for GCSE paper 3 for Edexcel, AQA and OCR exams boards. These workshops are suitable for foundation and higher students; the workshops are slideshows with answers included – ideal for final revision.

White Rose Maths

Corbett Maths

See GCSE 2022 on Corbett Maths, John Corbett has released lots of high-quality support material for Edexcel, AQA, OCR, and CCEA for both Foundation and Higher. You will find papers, revision checklists (hyperlinked to videos), also YouTube playlists, content is still being added. The papers are in the style of John Corbett’s “A Bit of Everything” papers – a resource I have always liked.


Mathsbox Skills Checks

On Mathsbox, we have some really useful skills checks – one or two-page documents with answer sheets aligned to the AQA, Edexcel and OCR advance information.


Craig Barton – Diagnostic Questions

Craig Barton has provided links for quizzes on Diagnostics Questions mapped to Edexcel’s and AQA’s Higher and Foundation advance information.


For Edexcel schools, a very useful read from Christine Norledge is GCSE Maths Paper 2 2022: Summary Of Topics, Questions & Planning For Paper 3. In this post, we have a very careful analysis of the papers and suggestions for selected revision topics for Edexcel’s GCSE Paper 3. You can find Practice Papers for AQA, Edexcel and OCR from Third Space Learning here.


PixiMaths – Edexcel Higher 2 & 3
Maths Genie

On Maths Genie, you will find videos, exam questions and solutions for AQA, Edexcel and OCR.

From PixiMaths, Best Guess Papers and Mark Schemes for Edexcel Paper 3.


Jess Prior Breakfast Warm-Up, Edexcel

Edexcel Breakfast Warm Ups from Jess Prior. Jess has provided warm-ups with four different levels of difficulty – two Higher, and two Foundation for Edexcel paper 3.



From Prime Maths, Foundation and Higher Edexcel GCSE revision papers, Paper 3.


If you are looking for a set of short questions, remember Wayne Chadburn’s little bit of Maths each day calendars, see May (paper 3). These calendars have been written to focus on each of the papers and the topic lists announced by Edexcel. Solutions are provided.

Wayne Chadburn

From 1st Class Maths A full set of prediction papers for AQA GCSE Maths including video solutions is available free.


See also – A Level Summer 2022 and Examinations – Mathematics Summer 2022.

A Level Summer 2022

Naiker Maths

From Naiker Maths, A level Predicted papers based on advance information from Edexcel.

Mark schemes are provided.


Dr Tom Bennison – AQA Practice Questions

For AQA A Level Dr Tom Bennison is adding revision resources to this page. As you can see from Tom’s post he is planning to provide a set of “practice” papers for AQA A-Level Maths and AQA A-Level Further Maths, a paper for OCR FSMQ and then some “Exam warmup” grids for use before the exams.


ExamQ – Mr Watts

On ExamQ from Mr Watts you can quickly search for Edexcel Maths GCSE and A-Level exam questions. You can choose from GCSE Higher, AS or A level, you can also select by exam series, paper, area and topic. Selecting Summer 2022 Advanced Information displays a menu of questions by paper based on the Advance Information.

The interface is very clear indeed, easily display the questions or mark schemes. You can also see the original paper and mark scheme.


From Pearson/Edexcel, comes a very valuable resource, under Summer 2022, Support, A Level Mathematics Topic Tests, is a new set of topic tests (no login required) for Pure Mathematics (10 tests), Mechanics (5 tests) and Statistics (5 tests) are provided. These are very comprehensive, not only do we have the questions and mark schemes but further notes. Looking at an example, the first test of the Pure Mathematics tests is on Proof, 5 varied questions cover the knowledge, skills and understanding relevant to Pearson’s qualification. The tests are designed to be used with the advance information for the subject as well as general marking guidance for the qualification.

Also available in a similar format from Pearson Edexcel are A Level Further Mathematics Topic Tests. Tests are available in both Word and pdf format for Papers 1 and 2 (9 tests), and Papers 3A (7 tests), 3B (8 tests), 3C (5 tests), and 3D )5 tests.

Pearson Edexcel Further Maths Topic Tests

The first test in the Paper 1 and 2 pack for example is on Proof, 7 questions provide comprehensive coverage of Proof.


Whilst not written for the advance information, the following resources may be useful if you are searching by topic for A level Maths and Further Maths:


Core Maths

For Core Maths, note the very helpful section, Pre-release materials in Getting Started with Core Maths from Catherine Van Saarloos.


GCSE Revision Papers 2 & 3

Many resources including practice papers are available for all the GCSE papers and are detailed in the post Examinations Mathematics Summer 2022.

The entries here are recent additions, I will keep this updated as any more resources become available for papers 2 and 3.

From White Rose Maths (scroll down the page) their excellent pre-exam workshops for GCSE paper 2 for Edexcel, AQA and OCR exams boards. These workshops are suitable for foundation and higher students; the workshops are slideshows with answers included – ideal for final revision.

White Rose Maths
PixiMaths – Edexcel Higher 2 & 3

From PixiMaths, Best Guess Papers and Mark Schemes for Edexcel Papers 2 and 3.


Jess Prior Breakfast Warm-Up, Edexcel

Edexcel Breakfast Warm Ups from Jess Prior. Jess has provided warm-ups with four different levels of difficulty – two Higher, and two Foundation for Edexcel paper 2.


From Maria Howard – an excellent PowerPoint – revision for Edexcel Paper 2F. This is great – choose the topic you want – answers included.

Maria Howard – revision for Edexcel paper 2F

From Prime Maths, Foundation and Higher Edexcel GCSE revision papers, paper 2 and Paper 3.


1st Class Maths

From 1st Class Maths a new GCSE Edexcel paper 2H.


If you are looking for a set of short questions, remember Wayne Chadburn’s little bit of Maths each day calendars, see these for April (paper 2) and May (paper 3). These calendars have been written to focus on each of the papers and the topic lists announced by Edexcel. Solutions are provided.

Wayne Chadburn

Many further resources for all the exam boards are detailed in the post Examinations Mathematics Summer 2022.

Simultaneous Equations

A collection of resources on solving simultaneous equations…

White Rose Maths – Exemplar questions

We could start with the White Rose Year 10 materials. An Autumn term block is on simultaneous equations; excellent free publications from White Rose include the scheme of learning with small steps, exemplar questions & answers, and Foundation and Higher End of Block assessments with answers.


Building Blocks – Andy Lutwyche

Staying with the idea of small steps, check Building Blocks on Simultaneous Equations, a collection of 14 problems from Andy Lutwyche; I do like this Building Blocks series – a useful reminder of the skills students need and the question types they can encounter. As Andy states – “This takes students through all the skills required to solve simultaneous equations graphically (only linear graphs), by elimination and by substitution including one linear and one non-linear up to GCSE level. Work from the bottom building the skills up to the most complex style of question.”

I have often used the Building Blocks resources as revision resources with GCSE students.

Andy Lutwyche

From Andy, we have several resources on simultaneous equations including Clumsy Clive who is making mistakes on his homework again!


Dr Austin Maths

From Dr Austin Maths we have a lovely collection including some very recent additions on Simultaneous Equations. The collection includes a favourite format of mine – fill in the blanks.

Dr Austin Maths

Note also a new type of activity, sort it out – see Simultaneous Equations sort it out where students must decide which of 4 categories the pair of equations belongs to, which is the most efficient method for solving the equations by elimination?


On Transum Mathematics, several resources on Simultaneous Equations are available. This really is a lovely collection with a great variety of resources. See also the topic list for teachers.

Transum Mathematics – Simultaneous Shopping

For a great introduction to simultaneous equations, try Simultaneous Shopping, a reminder of just how helpful diagrams can be. Level 1 is illustrated.

Simultaneous Shopping – Level 5

Level 5 uses coupons to introduce the idea of negative coefficients.


Don Steward

From Don Steward, see his many (34) Simultaneous Equations posts including an article with some suggested steps for teaching the topic and a series of 6 on generalising which you will see were written in February 2019, a PowerPoint is available for this series.


PixiMaths

From PixiMaths, her Simultaneous Equations collection starts with a lovely introduction using bar models. This collection includes four lessons, including equations where one is linear and the other quadratic.


Matt Woodfine – Maths White Board

On Matt Woodfine’s Maths White Board, search for Simultaneous Equations, illustrated above is a retrieval board on linear simultaneous equations.


Colin Foster’s wonderful Mathematical Etudes Project includes simultaneous equations; the article describes a complete lesson, to quote Colin Foster:

In this lesson, students explore a set of two-variable linear equations, constructed so that when you solve any pair of them simultaneously you get solutions which are all nonnegative integers.


From Craig Barton – see Simultaneous Equations Staircase, rather like Colin Foster’s lesson this will provide practise of solving simultaneous equations, whilst at the same time sparking students’ curiosity. Starting with linear sequences, students form a pair of simultaneous equations and solve them. What happens if they start with 2 different sequences?


We could use the excellent Mobile Puzzles from The Transition to Algebra (TTA) project, an initiative of the Learning and Teaching Division at Education Development Center, Inc. (EDC).

Mobile Puzzles

Open Middle

Here’s a lovely problem from Open Middle to really get your students thinking, this post includes a Desmos page and a Graspable Maths canvas for the problem.


From Teachit Maths, we have a collection on Simultaneous equations.

For practising problems with words try Simultaneous scenarios, the free pdf includes the questions and solutions. Subscribers can access a PowerPoint version.

We should of course illustrate solutions graphically, when working on simultaneous equations using algebra, it is so easy to show a graphical illustration. For exploring graphical solutions, we could use David Smith’s Simultaneous Equations Graphical Solution on Geogebra which includes questions for students to try.

From Ishak Hassan, this Simultaneous Equations GeoGebra resource allows you to create pairs of simultaneous equations.

You can change 3 points on the screen including the intersection of the two lines.

We are so fortunate in Maths to have so many resources available to us, just a sample of some further resources: