Engaging Explanations

I was delighted to present a session at #MathsConfOnline from Complete Mathematics, August 22. The resources discussed in the session are all provided here with some relevant additional content.

‘When presenting content and ideas to students we need ‘engaging explanations that are just right for the students: neither too short nor too long; neither too complex nor too simple.’

Coe, R., Rauch, C.J., Kime, S., & Singleton, D. (2020). Great Teaching Toolkit: Evidence Review. Evidence Based Education.

Great Teaching toolkit: Evidence review – complete report, note the individual elements on page 6, including 4 Activating hard thinking.

Rosenshine’s Principles of Instruction – Research-Based Strategies That all Teachers Should Know.
(Further commentary and resources can be found in my post, Rosenshine’s Principles in the Mathematics Classroom).

White Rose Maths – Small Steps-Functions

White Rose Secondary Resources including Complete Secondary Small Steps

AQA – 90 maths problem solving questions

AQA’s 90 maths problem solving questions supports the problem solving requirement (AO3) of GCSE Mathematics. Very useful summary lists link questions to process skills and to content areas. Answers are provided for all the problems and very helpful full commentaries are given on 30 of the problems.

AQA GCSE Examination Questions

AQA GCSE Assessment Resources, AQA legacy GCSE question papers


There is also a good deal of evidence that the use of worked examples can
be helpful in introducing new ideas (Booth et al., 2017; Sweller et al., 2019).
Particularly effective are ‘completion problems’ where students are given
partial solutions and required to complete them
. These can help students to
focus on the examples but also manage the difficulty level while retaining
authentic tasks

For a collection of such problems, see Fill in the blanks…

Open Middle problems in GeoGebra

Distributive Property: Open Middle Theme

Building Blocks – Andy Lutwyche

Andy Lutwyche-Building Blocks

Spot the mistake resources including Erica’s Errors and Clumsy Clive from Andy Lutwyche
Show that… and What was the question? from Andy Lutwyche

PhET Interactive Simulations, University of Colorado Boulder, https://phet.colorado.edu

Use of Technology including PhET Simulations

Colour in Mathematics

Colour in Mathematics – the use of colour to add clarity

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

Colin Foster – including Mathematical Etudes

Transum Mathematics

Transum – Venn Paint Transum – Topic Index
Transum  Maths Map (Students)

AQA – Bridging the gap

AQA – Bridging the gap

This set of AQA resources was originally designed to support teaching and learning for the cohorts of students who studied the 2007 Key Stage 3 Programme of Study and were preparing for the then new Mathematics GCSE (8300). The resources are still very useful for students in KS3-KS4.
KS3-4 Bridging the gap.

This series of termly tests covers Years 7, 8 and 9 and can be used to check understanding as well as showing younger students what GCSE papers look and feel like. Extension tests are included and analysers are available for all tests.
AQA Key Stage 3 tests for Years 7, 8 and 9.

AQA Key stage 3 tests

AQA KS3 Tests

Maths on Holiday

We have seen many great pictures on Twitter recently tagged #mathsonholiday.

Looking back over some previous posts, I thought I would check and update some holiday-themed posts.

An Italian holiday included a visit to the wonderful National Cinema Museum in Turin – see the Mathematics in the Movies post.

National Cinema Museum, Turin- Photograph by David Young
This image has an empty alt attribute; its file name is milan-cathedral.jpg
Milan Cathedral – photo by Sophie Young

Visiting Milan and wandering around the lovely Cathedral (one of the largest in Europe) I was struck by the geometry all around!

Some interesting sites on Mathematics and architecture:

From Maths Careers, see Interesting Buildings Based on Mathematics and from Tripbase, 9 Most Mathematically Interesting Buildings in the World.

The Tripbase article includes the cathedral in Barcelona Designed by Antoni Gaudi. 

Barcelona - Cathedral
Barcelona – Cathedral – photo by David Young

For some further reading Plus Magazine has an article on Perfect buildings: the maths of modern architecture by Marianne Freiberger.


A great resource on TES for when we get back to school is Laura Wilson’s Plans and Elevations, a PowerPoint of images showing plans and elevations of some famous buildings.

Holiday Pictures

All photographs by David Young, taken in the attic space of the Casa Milà Barcelona a building designed by Antoni Gaudí, constructed between 1906 and 1910.

cosh x Desmos image

Imaginary Exhibition


I do like to keep my eye open for mathematical pictures hence: (see Tetrahedral Numbers on Mathisfun)

Neuwied Schloss cannons Photograph by David Young
Neuwied Schloss Cannons –  Photograph by David Young
Bottle sequence
Barrel holding equivalent of 200,000 bottles


The Mercier champagne barrel which contained 200,000 bottles of Mercier Champagne, Mercier Champagne House, Epernay France

Algebra Resources

Back in March, I wrote, Number Resources, still very much a beginning and a work in progress, I have been reorganising pages. Making things easy to find is a never-ending task! Having written this blog since 2009, many older posts are still very useful – I want to make those easier to find.

Hence some new pages, the first created was Number under GCSE (14-16). This shows GCSE content and links to some favourite posts, also to sites that are particularly easy to search by topic. Further posts and resources and sites will be added, this is quite a project, but watch this space!

I have now added Algebra which brings some posts together such as:

Algebraic Notation

Factorisation of Quadratic Expressions

Lyszkowski’s method

Simultaneous Equations

White Rose Maths – Exemplar Questions


Transum Mathematics – Iteration


Activities to help learn reasoning and proof

Links to sites that are particularly easy to search by topic for Algebra are included, including some brilliant Underground Mathematics questions to challenge your GCSE students.

Underground Mathematics Algebra Review Questions (old O/AO level).
Some favourite tasks to challenge your GCSE students:

Resource typeTitle
Review questionTwo values of x that differ by 5 satisfy x2 −12x + k = 0, what is k?
Review questionHow small can this triangle be?
Fluency exerciseQuadratic solving sorter
Review questionCan we solve these simultaneous equations of degree 1 and 2?
Review questionCan we simplify these algebraic fractions?
Review questionIf we know two values satisfying a quadratic, can we find the quadratic?
Rich exampleQuadratic grids
Building blocksWhich quadratic?
Review questionCan we find the three inequalities that define this region?
Review questionCan we solve these simultaneous equations that involve reciprocals?
Package of problemsName that graph
Building blocksGradient match
Fluency exerciseMultiple manipulations
Review questionWhen are these quadratic inequalities true together?
Building blocksA tangent is …
Review questionWhen are the coefficients of a quadratic equal to its roots?
Many ways problemTwo-way algebra
PhET Interactive Simulations, University of Colorado Boulder, https://phet.colorado.edu

If you like the old exemplification examples – there are still some great questions, these have been added to the following pages:

Algebra Exemplification Examples

Algebra – Exemplification Examples

Watch this space for further additions to all these pages.

Mathematical Miscellany #69

A compilation this week:

From Julia Smith (@TessMaths), have a look at this wonderful collection of Mathematical Hooks.

In Mathematical Miscellany #68 I included From Complete Maths (@LaSalleEd) this brilliant 128-page Task Booklet from Jonathan Hall (@StudyMaths). The tasks which aim to promote mathematical thinking and behaviour in the classroom have been designed to be used with an appropriate model or manipulative. (See also NCETM’s Using Mathematical Representations at KS3.)

Note the playlist, Tasks on the Complete Maths YouTube CPD channel where Jonathan Hall is sharing his thoughts on some of these tasks. He recently shared this on Negative Combinations:

From Sparx Maths, have a look at their Transition Booklets, one for Year 6 to 7 and one for GCSE to A Level; these booklets are free whether you have Sparx Maths or not. The Year 6 to 7 booklet with over 130 questions is based on the Year 6 DFE/NCETM ‘ready to progress’ guidance from June 2020. The GCSE to A Level booklet uses over 120 questions taken from 11 key topics on Sparx Maths. Answers can be obtained by completing a form with your school details.

In my post on Proof I included some Geometrical proofs. We could treat this diagram 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”

For an animation to illustrate this, see Trapezoid Area (2) from Tim Brzezinski.

…and in case you are wondering about Trapezid Area (1)

George Stone (@DrStoneMaths) is producing one slide summaries of interesting educational research papers.

For research summaries – see also the reading pages Research – Mathematics Learning & Teaching including easy to digest research on Mathematics education from Cambridge Mathematics (see their Espresso page), and Research – Learning & Teaching which includes Research in 100 Words from Chris Moyse and summaries of educational research from Tom Sherrington.