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



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


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.


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