Mathematical Miscellany #37

Screenshot_20200222-073804_MathFrom Microsoft, Math Solver, an app available for Android and iOS.
Either write a problem on the screen, type it or use the camera to scan a printed or handwritten maths photo and you will get a step by step explanation and any appropriate graphs. Additionally you get similar problems and online video lectures. Several languages are supported.

I like the choice of methods presented; here you can see that a quadratic is solved and you are given the choice of methods for a step by step solution – an easy way for students to compare methods.

Supported problems are as follows:

● Elementary: arithmetic, real, complex numbers, LCM, GCD (HCF), factors, roman numerals
● Pre-Algebra: radicals and exponents, fractions, matrices, determinants
● Algebra: quadratic equations, system of equations, inequalities, rational expressions, linear, quadratic and exponential graphs
● Word problems on maths concepts, number theory, probability, volume, surface area
● Basic Calculus: Summations, Limits, derivatives, integrals
● Statistics: Mean, Median, Mode, Standard Deviation, permutations, combinations

Select the icon in the top left for examples. I was curious about ‘word problems’; given examples are for example ‘Is 21 a prime number?’ and ‘Probability of rolling 1 dice and getting a 2?’

A little more experimenting needed I think, I tried an integration in this app (Android) and also using Photomath. The Microsoft app simply gave me a solution here with no step by step explanation, though it did reference similar problems from a web search; PhotoMath presented a complete explanation of the integration by parts.
Microsoft & PhotoMath

A reminder that for a really clear visual representation then WolframAlpha is excellent; the graph is returned with the query. Step by step solutions come with a subscription but the free option is so useful for checking answers and the visual representations mentioned.
WolframAlpha integral

I have mentioned the wonderful problem collection on Open Middle before, if you are not familiar with Open Middle do explore these excellent problems; you can read more about the type of problems you will find on the site on the About page. Note you can search by grade using the drop-down menus.

We could try Coordinate Parallelograms:Coordinate Parallelograms

Tim Brzezinski (@Brzezinski_Math) has started a brilliant collection of Open Middle themed problems on GeoGebra. Under Coordinate Geometry you will find his GeoGebra version of the problem.

GeoGebra problems

Brzezinski GeoGebra Open Middle

Algebra KO extract Nicola WhistonI wrote earlier on Knowledge Organisers, Nicola Whiston has started a collection of Knowledge Organisers which follow the White Rose Schemes of Learning, she is sharing the collection here, via Dropbox. These are really attractive and I think will appeal to students; they certainly appeal to me! I have updated that earlier post with Nicola’s resources.

Quanta Magazine have published this wonderful Map of Mathematics, to quote Quanta Magazine:

From simple starting points — Numbers, Shapes, Change — the map branches out into interwoven tendrils of thought. Follow it, and you’ll understand how prime numbers connect to geometry, how symmetries give a handle on questions of infinity.

For very young children, the Department for Education has announced 6 new apps available to improve reading, writing and speaking.

Following a competition to find the best educational apps for parents to engage young children in learning at home, the apps chosen cover activities ranging from interactive story books, handwriting exercises using Artificial Intelligence, and educational video games. The apps are published on the Hungry Little Minds website.

The expert panel who accredited the apps, chaired by Professor Jackie Marsh of the University of Sheffield and appointed by the Department for Education, included children’s digital media consultants, early learning charities and researchers at universities.


Mathematical Miscellany #36

Access Maths – Starters
Starters - Access MathsOn Access Maths you can now find a whole library of Starters covering Algebra, Sequences, Probability, Geometry and Number. The Starters have been designed to be used in the final term with a top set year 9 in the author’s school. These starters could be very useful for GCSE in Years 10 and 11.

Also on Access Maths, you can find all the Form Time Numeracy activities you ever wanted!

Though if you do want some more then look no further than Jonathan Hall’s (@StudyMaths) Form Time Ideas.

Topic Tests for Mathematics and Further Mathematics A Level
AQA Topic Tests
A growing set of really useful Topic Tests are available for both Maths and Further Maths A Levels from AQA. I like the index provided by the mapping documents, one for Maths and one for Further Maths. For each test there is a clear statement of what is assessed in the test which comprises two sections. The questions in section A test basics of the topic and those in section B require a bit more thinking. Mark schemes are provided for all tests.
(How to get access to AQA’s free portal – All About Maths).

AQA Algebra Topic Test example

AQA Topic Test – Series and Sums

Pyramid Puzzles – AndyLutwyche

Indices Pyramid Puzzles A Lutwyche

Indices Pyramid Puzzles – Andy Lutwyche

I frequently recommend Andy’s excellent resources, recent resources include his pyramid puzzles, currently available, he has puzzles on Indices, Expressions and Calculations (designed to be done without a calculator).

I recently recommended Andy’s Non Examples – Reasoning tasks, this collection has now grown.

AL Examples & non-examples

Non-examples Andy Lutwyche

Learning ScientistsI’ll end this collection with a little listening – literally a little listen from the Learning Scientists who include Bite-size research in their podcast collection. Their latest episode, Bite-Size Research on Seductive Details, looks at the details we include in our teaching that are not necessarily relevant for understanding the topic but make the lesson more interesting  – a good idea? Some further research into how you present new information to students.


Mathematical Miscellany #35

Non Examples – Reasoning tasks

AL Examples & non-examples

Non-Examples – Expressions and formulae, Andy Lutwyche

From Andy Lutwyche try Non-Examples – Expressions and Formulae – Reasoning Tasks, a resource with seven sets of five questions and solutions, some of which are correct and some of which are not. Students decide which are correct and explain how they have come to their decision. The topics covered are simplifying expressions, substitution, expanding and factorising expressions including quadratics, rearranging formulae and algebraic fractions. Resources like this can promote excellent class discussion.

Also available is Non-Examples – Shapes (and Angles) – Reasoning Tasks.
For more on Non-Examples – see this post on Frayer Models.

To log or not to log – Underground Maths

To log or not to log

To log or not to log – Underground Maths

I used a favourite Underground Maths resource this week – To log or not to log? This has worked really well every time I have used it. The activity requires students to think about the methods which could be used to solve the various equations. I have always found that in addition to working on indices and logarithms this task has exposed some misconceptions, with students trying to invent some new and invalid laws of logarithms!

This problem is classified as a Problem Requiring Decisions.

Students are often used to problems being posed in such a way that they have all the information that they require in order to start, and no more. Problems (especially from the real world) are very often not like this, and so resources of this type will give students the opportunity to develop the skills needed to deal with this. Some problems might not contain enough information, so students may need to decide on classifications, make assumptions or approximations, or do some research in order to move forward. Some problems might contain too much data, so that part of the challenge is to identify the useful information.

UM Powerful QuadraticsHere’s another Underground Maths task, Powerful quadratics, which will certainly give your students food for thought, as the authors state:

When students are familiar with concepts and ideas they often benefit from exploring them further to improve their understanding. These problems aim to allow this further exploration, and for example, might bring different techniques together, highlight interesting or unusual cases, or probe the definition of mathematical terms.

From Matthew Scroggs have a look at for a wonderful collection of puzzles including Advent and Christmas themed puzzles. Try Matthew’s Christmas Card 2019 with its nine puzzles – a great card for your mathematical friends!

Year 13 will get that card this week, thank you Matthew!

TeachIt Maths Advent Calendar

Teachit Maths – Advent Calendar

On the subject of Christmas, there are still loads of doors to open on those Advent Calendars and if you are looking for activities for the last week of term, try the Christmas 2019 collection.






AQA – Maths Digest
From AQA, have a look at their Maths Digest written to support Mathematics Teaching and learning. Useful whichever examination board you use, the digest offers tips and resources. I do like “Small things make a big difference” on avoiding common exam mistakes. This PowerPoint highlights where marks are often unnecessarily lost.

“Top tips for perfecting exam techniques” by Julia Smith provides her top tips to help students perfect their exam technique and to help gain crucial marks. In the article, Julia refers to AQA’s list of command words, so useful to show your students.

See also Reference for further resources on mathematical vocabulary.


In the Recommended resources section, I have reminded readers about AQA’s brilliant GCSE Mathematics: 90 maths problem  solving questions.AQA Problem Solving Questions

AQA 90 ProblemsThere is a helpful introductory section for teachers and note also the helpful Classification Tables by Strategy and by Content Area.

Em,  has a brilliant PowerPoint with all the questions and answers – see it here.


As part of the Maths Digest, you can also find information on AQA’s GCSE papers.

Working on Bivariate Data this week with my Further Maths students we were certainly able to use technology …

This GeoGebra applet allows students to move points and watch the effect on the line of best.
GeoGebra Correlation

This can be used in class by asking students to plot the points, draw their lines of best fit and then comparing with the computer. This worked really well on my phone, I simply sent myself an email with the link and was able to move points easily. This could also be used with younger classes when talking about lines of best fit.

We can also demonstrate correlation coefficients and lines of best fit with this PhET simulation on Least Squares Regression.

Correlation & Line of best fit

Choose from a range of examples or choose Custom to add your own points and guess then check the correlation coefficient. You can also draw your own line of best fit and compare it to the theoretical line of best fit. Note the option to include residuals for both your own attempt and the line of best fit.

For more on resources for Regression see this post; you could also use Desmos, GeoGebra, Excel or WolframAlpha.



Mathematical Miscellany #34

A Level Maths and Further Maths Resources
Jack Brown has over 990 teaching videos for the A Level Mathematics specification on his TLMaths site. There are now more videos for the core content for Further Maths, to date, videos for Proof and Matrices are available. These are included on this page for Further Maths teaching resources.

Desmos New Features
Keep an eye on new features from Desmos, we now have for example, Spearman’s rank coefficient. The data in this example comes from the exam paper, 
MEI S2 January 2006, question 3. Remember that Desmos has many useful and easy to use Statistical functions.
Normal Distribution

MEI Calculator Tasks
For resources using technology for A Level Maths and Further Maths, MEI’s series of tasks are excellent; all the student tasks, whether for Autograph, Casio, Desmos, GeoGebra, or TI-Nspire are structured in the same way, beginning with instructions on the use of the technology, also given are questions, a problem and further tasks. Integrating Technology Into Your Scheme of Work suggests activities aligned to the specification.

Note we now have a recently added collection of Autograph Tasks, currently for A Level Mathematics.  Remember that Autograph is now free, these tasks are helpful for teachers wishing to learn more about Autograph in that clear instructions are given on how to construct the objects in Autograph.

November 5th – Bonfire Night

OCR Bonfire Night

OCR Puzzle

Last week I mentioned Halloween resources from Perton Maths Department and OCR Maths, this week we have Bonfire Night. Check the Perton Maths bar model bonfire challenges and from OCR, check their latest Maths Puzzles including a challenging number grid for Bonfire Night.

Maths Week England – 11th-16th November 2019
Maths Week England aims to “Raise the profile of Mathematics throughout England, you can see the full aims here. Note the various competitions running including a Secondary and Post 16 quiz featuring questions for every age group from Y7 to Y13 and also the resources available. MangaHigh, Sumdog and TimeTables Rockstars are running competitions using their software for all schools whether or not they subscribe and we also have MEI’s Ritangle competition and Desmos Art Competition. All the details are on the Maths Week England competition page.

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.