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.


Pouring and Filling

Checking my blog statistics I noticed some visits to a 2011 post and realised that the links no longer worked, so some years later – an all-new version! Note the brilliant Painted Cube demonstration at the end of this post.

Some excellent watery demonstrations this week:
Depth Time graphs

Depth time graphs of filling different containers. (GeoGebra). Simply select the container you want and the speed of animation, select animate and watch the graph.

From the GeoGrebra resource team, a resource to interpret meanings of different rates of change across a graph that represents the water level in a hot tub.


From Michigan State University’s Connected Maths Project Student Activities, one of the Grade 7 Activities is Pouring and Filling which provides a great demonstration of relationships between the volumes of pyramids and prisms. We could tip the contents of a cone into the cylinder for example…

Each activity describes the purpose of the activity and has suggested uses, many activities as this does have helpful how to videos.

These activities are well worth exploring, for example in Grade 6 we have factor and product games and also the very satisfying locker problem. Note the TEDEd lesson video here.

Painted CubeOne of the grade 8 activities is the classic Painted Cube. This activity is so clear; it allows students to build a cube or cuboid out of unit cubes, colour the faces using a palette of colours, then rotate the object to paint the initially hidden sides. The expand option allows students to blow up the prism and inspect will count the number of cubes with 0, 1, 2 or 3 painted faces.

Bridging the Gap

AQA’s KS3-4 Bridging the Gap set of resources is excellent. Whilst 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 it remains just as useful now.

The teaching materials provided in 9 ‘Pockets’ focus on the areas new to this current GCSE specification; the use of these at KS3 helps ensure students have the relevant pre-requisite knowledge to progress to the GCSE specification. For example, Pocket 2 on function notation introduces the formal f(x) notation.

The nine resource sets cover Fractions and ratio problems, Function notation, Graphs in real-life contexts, Iterative methods for solving equations numerically, Set notation, number lines and Venn diagrams, Number sequences, Direct and inverse proportion, Growth and decay, and Vectors.

Each set of resources has three sections; the first, developing understanding, includes class-based, teacher-led questions, Skills Builders are standard progressive worksheets that can be used to drill core skills in a particular area and Problem Solving Activities are Extension activities for paired work or small group work to develop problem-solving skills.

Iterative techniques 1

AQA –  Bridging the Gap – Iterative methods

I have used many of these resources since they were first made available, for example, the resources introducing the concepts of using iterative methods to solve equations numerically in cases where an algebraic approach is not possible. I recommended these resources in my post on Iterative Methods originally written in May 2015, a post that remains popular.

The Teacher Guide includes in Section 3, five exemplar activities on mathematical proof and in Section 4 we have some suggested resources for embedding problem-solving activities throughout the mathematics curriculum. Included is a link to the classic Durham Maths Mysteries. AQA quite highlight the Directed Numbers mystery, where students must decide which number goes where on a 3×3 grid given thirteen cards each with a statement about directed numbers and how they lie on a 3 by 3 grid.

Carom Maths

Staying with the subject of Bridging the Gap, see this post, Transition Time which looks at some resources for transition from GCSE to A Level and also resources to help prepare for University.

Colour in Mathematics

Update July 21

The update for July 2021 includes many new examples and also a notes version to download.

I frequently use colour and highlighting in my explanations when doing worked examples with a class. It can be helpful in Mechanics, for example when taking moments.

I have added this image to my presentation, Colour in Mathematics or pdf version, Colour in Mathematics.

See also Colourful Chi Squared and Dijkstra’s Algorithm – a colourful version!

Chi Squared examplw