Compare & Discuss/Multiple Strategies

Have you seen…this important resource for our teaching?

From researchers at Harvard University and Vanderbilt University, the Compare and Discuss Multiple Strategies (CDMS) routine, and the curriculum materials for algebra courses?

Explore the curriculum materials and learn how to use the resources. See the About page for a summary of the Create & Discuss/Multiple Strategies routine which it is suggested should take about 20 minutes.

See also “One way is the wrong way to do math. Here’s the right way.” from The Harvard Gazette

Within each resource, you will find questions that could be used for discussion of the example, a “think-pair-share” routine and a “Big Idea” page that summarises the lesson objective for the example.

The teacher’s guide also explains each of the four strategies.

Compare & Discuss/Multiple Strategies – from researchers at Harvard University and Vanderbilt University

As you can see from the image here you can filter the resources by Type, Lesson Use and Topic, within each topic you will find several subtopics.

Mathematical Miscellany #57


Dr Austin Maths – Number, Ratio

My Ratio post features many activities and resources for teaching and learning ratio and proportion. This has been updated with some recent new resources including Amanda Austin’s always excellent resources and a really helpful Teacher Training pack from AQA. Also included in this post is a link to my post on bar modelling which is ideal for visualising ratio problems.

Don Steward – translating English to algebra, expressions

Spoken language is so important in students’ development, the KS3 Programme of Study quite righly stresses the importance of spoken and written language. Both spoken and written  language and notation are key; added to this Proof post, Don Steward’s translating English to algebra, expressions; see also translating English to algebra, relationships.

Added to Mathematics for Science is AQA’s teacher training pack, Maths in Science.

AQA Teacher Training Pack – Maths in Science

Included in the Activities booklet you will find examples for discussion, AQA suggest:

For each of the examples on pages 6–10 of the Activities booklet, discuss:

  • approaches you could use to help students to be able to access these types of questions
  • how this approach might be different in maths lessons
  • whether there any other differences between the two departments.
AQA Teacher Training – Maths in Science

Tim Brzezinski – Trig Reference Circle: Choose Your Own Radius

I have seen so many useful links for GeoGebra recently, including:

On Sarah Carter’s Math=Love site, you will find a wonderful collection of puzzles. The collection is helpfully indexed into several categories including Puzzles Related to Math Content and Number Related ChallengesSarah’s collection is included in my Puzzles and Games series of pages inder Collections.

For the Shared Factors puzzle illustrated here, John Golden created this GeoGebra version, Quadratic Puzzle.

John Golden – Quadratic Puzzle on GeoGebra

Sarah Carter has been inspiring some excellent GeoGebra resources! See also her Colour Square puzzle and GeoGebra resource from Duane Habecker.

Mathematical Miscellany #56


On Corbett Maths as well as the brilliant 5-a-day resources the videos and worksheets are also excellent. Do check the ‘More’ menu also, where you will find additional resources including a very comprehensive set of videos and worksheets for level 2 Further Maths.

Corbettmaths – Algebraic Proof, level 2 Further-Maths.

On the subject of proof, I will be publishing a very comprehensive post on teaching and learning proof next week.

I featured Corbett Maths earlier this year when he celebrated 200 million website views. Corbett Maths is such a go-to site – I use this a great deal and recommend it regularly to students. John Corbett has been busy – note his massive improvements to the brilliant 5-a-days.

Stoke Maths

Thanks to @MathsWithMsB I came across Stoke Maths MEP Starters.

These are very attractively presented high-quality resources. Looking at the Spot the Mistake PowerPoints for example, as you can see in the image below there are a great collection of questions that include full answers. It’s great to see Mechanics and Statistics collections.

The revision question starters provide very useful question sets.

I have added this to my Starters collection under Advanced Starters.

Stoke Maths – Spot the Mistake

Also from Ben Bently, a collection of low stakes quizzes using questions from Diagnostic questions.

Census at School – Ireland

Hunting for a resource I once used from Census at School (the UK site is no longer available), I found it on Census at School – Ireland. Cleaning Up Your Data. Many Statistics resources are available on the site – note the drop-down menu for the resources collection.

I found it again – on STEM Learning which happily has a whole collection of the Census At School resources.

From AQA Maths, check their excellent GCSE Maths Focus on success modular Teacher Training packs. Looking at the pack on Ratio for example you will find activities and discussion prompts on switching representations, combining ratios, problem-solving with ratio and dividing quantities in a given ratio.

AQA Teacher Training pack – Ratio

Mathematical Miscellany #55


In Mathematical Miscellany #54 I featured two excellent resources from Curriculum for Wales; a third is now available, “The Foundations of Algebra” is suitable for progression step 3 of the new #CurriculumForWales (age 11). The workbook contains chapters on patterns, commutativity, distributivity & associativity.

Foundations of Algebra – Curriculum for Wales

I do like the above exercise which as the Teacher’s Guide acknowledges is based on Don Steward’s work, directed number arithmetic speed up and Chris McGrane’s Alternative representation of integers. A further useful resource for such an exercise is Jonathan Hall’s Directed Number MCQ Generator on MathsBot with which you can generate all the addition and subtraction multiple-choice questions you want; choose between Counters on or off.

Foundations of Algebra – Teacher’s Guide

As with the other two resources, a very comprehensive teacher’s guide is also available. You can see the contents here, this resource with its carefully chosen and varied activities and exercises will help students with the foundations of Algebra.

PhET Sims

On the subject of negative numbers, from PhET simulations we have another excellent resource in their latest addition to the Mathematics collection. To use the number line as a model for ordering real numbers and also to illustrate operations with negative numbers we can use the excellent, Number Line: Distance. Also available are Number Line: Integers, and Number Line: Operations. All are excellent for students to explore.

This resource has been added to my post on Negative Numbers which looks at some resources to develop understanding of operations with positive and negative integers and exercises for practice.

A popular post on this blog is on Venn Diagrams, first written in 2016 this has recently been checked and updated with some new resources including always excellent resources from Amanda Austin on Dr Austin Maths. Included in her Probability resources you will find an excellent section on Set Notation and Venns.

Dr Austin Maths

GeoGebra retweeted this from Javier Cayetano Rod

…the translation:

“Adding some leaves to the stem of a flower can be the perfect excuse to talk about translations and turns in space @debora_pereiro . Added to the flower generator in @geogebra

Which in turn led me to this amazing GeoGebra collection, Flowers 3D, Author: Deborah Pereiro Carbajo.

Have look at Flowers from Curves, simply brilliant!

Flowers from curves – Deborah Pereiro Carbajo

Explore the collection – you could be a while!

Complete Maths has made Robert Smith’s session “Web Autograph, a First Look” freely available

Highlight equivalence – Math Whiteboard

I wrote earlier this year on the excellent Math Whiteboard. This is completely free to use; If you create a whiteboard you can then get a link for that whiteboard which you can share. When I have created a whiteboard I then save a second copy so I can always return to the original.

With an individual subscription ( currently $15 for a year) it is possible to access all the features of Fluid Math including as an authoring tool for creating Math Whiteboard activities. It is also possible to save your activities.

A new feature is available – the ability to create answer boxes to check for correct answers. You can see examples using this feature here.

Mathematical Miscellany #54

Mathematical Alphabet – Sarah Carter

If you are looking for posters, there are so many excellent free resources for Maths from puzzles to information, careers and many more.

This post, Posters has been updated and all links checked. Check two of the new additions to the post – a brilliant Mathematical alphabet from Sarah Carter on

Casio font for ClassWiz series

Or if you want any calculator symbols – try Casio’s collection of fonts.

Diving Deeper – Curriculum for Wales

Superb resources from Curriculum for Wales:
Diving Deeper with chapters on place value, negative numbers and fractions. The Learner Guide includes many activities, it’s great to see bases other than 10 as well! Scroll down the page for English versions, there are several resources available including answers, a teacher’s guide and additional tasks.

Calculation Guide-pdf or note the Word version also.

Dr Frost Maths – KS3 White Rose Maths

The KS3 (UK ages 11-14) page has had various updates recently including several new sources of resources. From Dr Frost Maths choose KS2/3/4 from the resources view or select the White Rose Maths course. For further resources linked to the White Rose Schemes of Learning, check this very useful feature on Transum Mathematics, where you can find resources linked to the White Rose Schemes of Learning as well as to areas of the National Curriculum. The page includes many valuable transition resources.

Transum Mathematics

From AMSP for students at the other end of the school, check their very helpful new topic plans for AS/A Level. With common content for all boards these are useful whichever board you are teaching. Have a look at this example on Integration:

AMSP Topic Plans AS A Level Year 1 (AS) Pure

The Mathigon page has been updated, note Mathigon’s professional development section that includes details of upcoming free events which are held regularly and recordings of all the past event recordings.

Follow Mathigon for latest news.

News on Mathigon seems to come thick and fast! Their Mobile Apps have recently been updated and on Monday, 8/9 August check the Charts, Statistics and other Data Science tools coming to Mathigon.

Two new additions for this Core Maths page: from the University of Plymouth, CIMT check Core Maths Subject Support which includes notes, examples and exercises.

From @KerryDunton and published on Dave Gale‘s blog, reflectivemath’s blog check this Core Maths Further Reading List.