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 June and later in July for the NANAMIC Summer Conference 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

• Liz Woodham – Using NRICH Tasks to Develop Key Problem-solving Skills
• Working Backwards at KS1
• Working Backwards at KS2
• Working Backwards to Move Forwards (Age 11 – 18)

Number

• Tables – Transum Mathematics Fast Factors
• Nrich – Missing Multipliers
• Nrich – Number Pyramids
• Open Middle – Order of Operations
• Mathsbot Directed Number Variation Grid
• Starting Points Maths – Prime Factorisation
• Underground Maths – Ab-surd

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.

Algebra

• Nrich – Missing Multipliers ( factorisation of quadratic expressions)
• Factorisation of Quadratic Expressions
• Mathsbot – Worded Expressions
• Mathsbot – Forming Expressions
• Distributive Property Open Middle Theme (GeoGebra)
• Open Middle – Create a System of Two Equations
• Underground Maths – Two-way algebra
• Underground Maths – Divide it up
• Underground Maths – Can you find… cubic edition
• Underground Maths – Can we find the three inequalities that define this region?

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

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

See also Digitisers from Jonny Griffiths.

Maths White Board

Standards Unit

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

• Open Middle
• From the home page – see Get Our Favorite problems
• Depth of Knowledge

Mathsbot – Jonathan Hall
Variation Grids

Fill in the blanks

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

The answer is – what was the question?

Andy Lutwyche – TES resources

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

Presentation Slides – PowerPoint, followed by pdf