Autograph – Free

A major announcement at La Salle Education’s Maths Conference #21 today –

Autograph for graphing and geometry is now part of the La Salle Education family and with the backing of La Salle Complete Maths, Autograph (version 5) is now available free to everyone forever. Teachers and students can download the software.

The dynamic graphing software has 4 modes, Statistics, 2D Graphing, 3D Graphing, and, introduced in version 4, Complex Numbers.

Perfect timing for myself and my Further Maths students as we are just about to study Complex Numbers. It is very simple to enter complex numbers and menu options mean that various operations are easily and quickly demonstrated, including displaying the nth roots of a complex number. A comprehensive manual is available – see the Help menu. Manuals can also be found on the Autograph site including useful summaries.
Autograph Complex Numbers

More for the Further Mathematicians, we can easily show transformations in 2D and 3D, use pre-sets or enter a matrix for a transformation .

Autograph - shear

Or very simply show your chosen number of terms of the Maclaurin series for sin x (select the graph, right click, choose create and select Maclaurin series).
Maclaurin series

Newton RaphsonAutograph is also excellent for illustrating the Newton Raphson method – something which will be very useful in teaching Numerical Methods for A Level.

So many excellent features are available – this is a powerful and sophisticated program. Note that a web application version is coming too.

Time to get studying the manuals and tutorials!
If you are unfamiliar with Autograph, you may like this training material by Simon Woodhead, Alan Catley, Craig Barton and Douglas Butler to get started.



Mathematical Miscellany #33

PhET and OneNote Integration

PhET to OneNoteI have often mentioned the excellent PhET simulations; it is now possible to embed these in OneNote. Simply paste any public PhET link, such as that for Balancing Act ( on to a OneNote page and it will render it as a live interactive embed.

Reading the Microsoft blog on this integration, I noted with which one can create mind maps; the free model allows 3 Private Diagrams and Unlimited Public Diagrams with PDF & Image Download. This looks like a simple way to produce such diagrams and I lke the fact that they can be integrated with OneNote.

MEI – Ritangle.
Registration for MEI’s Ritangle opens on October 7th. Ritangle is a competition for teams of students of A level Mathematics, the International Baccalaureate and Scottish Highers requiring no knowledge of mathematics beyond A level Mathematics. Students can use Technology to help with some of the questions.

There have been some great questions to challenge your older stdents in these Ritangle competitions. note the links to the questions and answers to the 2016-2018 competitions.

MEI Ritangle 2018 q1

MEI Ritangle – 2018, Question 1

This competition has been added to the updated Competitions page. This page includes Puzzle of the Week, a free international puzzle competition for schools.

Long Division & Multiplication – Formal methods

Mathisfun Long DivisionNoting the popularity of Long Division & Multiplication – Formal Methods I have checked and updated this page. This includes Algebraic long division.

Mathisfun Algebraic Long Division

Mathisfun – Algebraic Long Division

I’ll end with some highly recommended reading, Tom Sherrington on The #1 problem/weakness in teaching and how to address it.

For more reading, note the reading series of pages, including  Research – Learning and Teaching and Research – Mathematics Learning and Teaching.



Knowedge Organisers – Mathematics

In a post on Cognitive Science in the Classroom, I mentioned Knowledge Organisers, or to be more precise I mentioned Kris Boulton’s  “When shouldn’t I use knowledge organisers?”. Kris has written on why they are less applicable to maths. Certainly, I have not used knowledge oranisers for Mathematics myself with one exception, I have used  William Emeny’s Angle Facts; as Kris Boulton says in his article, “Maths is super-dense with concepts, and processes, but really only very few facts.” Noting topics where students do need to know more facts, he includes angle facts.

When I have used Will’s angle facts, I have adapted it so some content is missing, particularly the section on basic angle facts, students can be given just the diagrams for example and asked to recall the basic angle facts. I have also asked students to recall as many basic angle facts before they see the list as in the organiser, so using it following retrieval practice or as a retrieval exercise.

As noted in this post on Knowledge Organisers from Durrington Research School, it is not about the knowledge organisers  themselves but how they are used for planning, teaching and testing. Note the example shared of a Maths Knowledge Organiser, Year 10 Expressions, Foundation. I like the fact that key vocabulary is included here, the few key facts needed and of course by far the largest section – key concepts with examples.

To quote the Durrington blog, “Maths are using their range of knowledge organisers to support homework tasks. Firstly, the students can access their maths knowledge organisers are any time using our online system Connect. This means that students have scaffolding in place for when they are working outside of the classroom. Furthermore, every fortnight the maths team set a homework that is based on retrieval quizzing. The students are required to use the knowledge organisers to find the answers to upcoming quizzes and then actually sit the quiz in class on the due date for the homework. Students who score less than 12 out of 15 are then supported in making flashcards on the questions, again gaining the information from the knowledge organiser, and use these to retest until they are successful. This strategy demonstrates how knowledge organisers can be used to support learning through the testing effect.

non examples – Jonathan Hall

If I want definitions, characteristics and examples (clarified with the use of non examples), then I could return to the Frayer model. (See Frayer Models.)

Reading further on Knowledge Organisers, I do recommend this collection of blog posts and articles from St Mary’s Catholic Academy, on Retrieval Practice which includes the use of Knowledge Organisers. Once again, the emphasis is, quite rightly, on how they can be used fro rerieval practice.

Searching for Mathematic Knowledge Organisers, I have come across some resources I wish to explore further, such as the Henry Box School on Knowledge Organisers where the school are sharing Knowledge Organisers for each subject, recognising the support parents can offer. On TES, GCSE Maths 1-9 Knowledge Organisers is a (free) set of 50 files, described as “A full set of Knowledge Organisers containing the facts, definitions, formulae etc. that students need to know for the new GCSE Maths specification, broken down into individual units. On each Knowledge Organiser, content shaded in grey is for Higher Tier only.

Can be used for homeworks, revision, starters, plenaries or any other ways you might find useful.”

The files have been created in Word, I like the 3 column format which includes, Topic/Skill, Definitions/Tips and importantly, Examples.

I will return to this topic as I have further resources I wish to explore….

…and I want to ask my students what they think.



Mathematical Miscellany #32

Small SatisfactionA nice little starter from Transum, Small Satisfaction. I do like the related activities, I know some of my students will do this first level quickly, the second and third levels will provide them with something to think about, or perhaps the alternative, Satisfaction.

Remember that Transum provides starters for students all the way through to Avanced level.

A divisibility test for 7…
Staying with Transum, Divisibility Tests 2-12 provides a handy summary of divisibility tests. What really caught my eye here is the divisibility test for 7. Scroll down the page for the comment from Transum which tells the story of the origins of the test – a 12-year-old student. And note the proof – something for our A Level students perhaps!

Note the exercises to practise using the divisibility tests. Delightfully Divisible will keep everybody busy! See also, from Plus magazine – Finding the nine. (There is a link to a very clear solution in the video).



Mathisfun is an attractively presented site, the divisibility rules are clearly explained and if you scroll to the end of the page you will see multiple-choice questions to try.

Alternative tests for 7 exist, but I like Chika’s Test!

Mudd Math Fun FactsThe page on the divisibility tests for 7 is a reminder of the site – Mudd Maths Fun Facts. This searchable collection of Mathematics fun facts from Harvey Mudd College Math Department make ideal lesson starters or perhaps useful for those odd moments. Note the search on the left, it is possible to search by topic, difficulty level and keywords.

For example, try:

ITP - Number LineA blast from the past for many teachers perhaps, from MathsFrame, the ITPs have been made available for modern browsers – no Flash needed. Developed in 2002 to help with the teaching of maths in UK Primary Schools, the National Numeracy Strategy Interactive Teaching Programs cover many areas of numeracy including number, shape, measure and statistics.

To finish this collection, some Number Puzzles from Mathisfun.

Mathisfun puzzle

Mathisfun number puzzle

Note the complete Mathisfun Puzzle Index.


Cognitive Science in the Classroom

To learn about Cognitive Science, a good place to start is with The Learning Scientists, who are cognitive psychological scientists whose main research focus is on the science of learning. Note the Frequently Asked Questions and Downloadable materials for teachers and students.

The Learning Scientists have a very comprehensive collection of podcasts, many of these are bite-size, short but really clear – ideal for busy people!

For many examples of Mathematics resources we can use in the classroom and reading, see my page on Retrieval Practice and other learning strategies.

Retrieval Practice Guides

Retrieval  – Library

For, I think one of the most comprehensive and well-organised reading lists I have come across see The go-bag on the CogSciSci blog. Blogs, articles and research have been included with a real focus on the application of cognitive science to the classroom. the list is in 23 sections with a clear summary of each item.

The CogSciSci blog concerns cognitive science in the science classroom, but this reading list is relevant for teachers of any subject. Note also the study modules available for teachers, including Retrieval Practice. In section 7 of the Retrieval Practice module I was interested to see the Retrieval Roulettes and like the KS4 Chemistry triple new Excel resource linked to in the third paragraph. Questions and answers on the first sheet can be for any subject, this particular resource has all of GCSE chemistry (AQA) but you could just copy and paste questions and answers. For Mathematics, we do have Jonathan Hall’s Retrieval Facts on MathsBot or his Recap and Recall; to choose questions by topic, use his GCSE Revision Grid. Other resources can be found on my Retrieval Practice page, including this Custom Starter from Transum which allows teachers to select the number of questions and the topics to include; scroll down the page and choose the topics you want from the Concept Selection.
Transum Revision

It is possible to save a particular selection of topics as the URL for your selection will be generated. It is also possible to drag the panels so your questions are displayed in the desired order. The beginning of a lesson can be an ideal time to review previous learning, starters like this can be ideal.

To return to the reading list, I was particularly interested to read Kris Boulton’s “When shouldn’t I use knowledge organisers?”. Kris has written on why they are less applicable to maths. The one knowledge organiser I have used myself is William Emeny’s one on Angle Facts. In Mathematics, rather than knowledge organisers, we can turn to Frayer Models.

non examples – Jonathan Hall