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.

VolumeFrom 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.

For Valentine’s Day

An annual update, ideas and resources for Valentine’s Day …

It’s that time of year again – save your money and send your loved ones (or anybody!) a math-o-gram!

Desmos Valentine

Click on the image and move that slider!

Desmos have provided you with the means to send a math-o-gram to the mathematicians in your life!

Geeky people you could even use the Desmos API

NASA Valentines
For an alternative source of Valentine’s cards, we can turn to NASA! Take your pick from this post or these fabulous NASA images. (And don’t forget that NASA provides us with many Mathematics resources.)

From @OCR_Maths, we could try this puzzle.
OCR Valentine Puzzle
OCR often share some great maths puzzles, look out for them.

The excellent Maths Careers site is managed and maintained by the Institute of Mathematics and its Applications. If your students wonder where Mathematics is used they will find plenty of answers here. See for example Who employs mathematicians?  

cardioidFor Valentine’s day we have an appropriate article from Maths Careers, did you know that 6 is a kissing number?! And for your Further Mathematicians, What is the equation for a heart?

careers-heartsAlso from Maths Careers, see this post with instructions on how to make this wonderful pair of linked Möbius hearts.

If you wish to get creative and try this I advise watching the Numberphile video carefully; following the instructions worked as you can see from my creation here! I can verify that unless you follow the instruction to make sure the twist in each strip is in a different direction you will end up with a mess! Quite an interesting mess but certainly not two hearts!….
use-to-create-mobius-heartsNote the Desmos graphs on my strips. I created a file in Word valentine-mobius-hearts (or pdf: valentine-mobius-hearts) with Desmos images in a table. Adding dotted borders to the table gives guidelines for cutting. I began each cut by using the end of a paperclip to pierce the paper.

To create my strips I printed the document and then printed again on the reverse. I then cut out and trimmed the strips so there was no white space at the end – the picture here has been made using strips 10 cells long.

Transum Valentine Puzzle
Here’s a Valentine logic starter from Transum.

This Valentine Relay from Chris Smith is excellent as are all the other relays in this excellent set of resources. You can find more excellent resources from Chris on TES and follow him on Twitter here.valentine-relay

Valentine's Treasure HuntFrom Maths4Everyone, try this Valentine’s Day Maths Treasure Hunt. Students have to work out the costs of gifts, meals and other outings for Valentine’s Day.

You can find a whole collection of Treasure Hunts from Maths4Everyone here.

Also on TES resources, from a favourite TES author of mine, Andy Lutwyche, you will find this Valentine’s card Area and Perimeter problem.


Think Maths Parametric HeartFrom the MEI Archives, the February 2015 edition of the MEI Monthly Maths Magazine includes some connections between maths and Valentine’s Day. On page 7 note the article “A Happy Ending” which includes references to some Numberphile videos, Professor Ron Graham discusses the Happy Ending Problem and from Dr Emily Riehl, The Stable Marriage Problem. We also have a great Parametric Heart spreadsheet from Think Maths.

This edition of the magazine includes some lovely activities which link paper folding and proof.

Remaining with the loving theme you can express your feelings for WolframAlpha!

I Love YOU

and from the WolframAlpha archives, Computing Valentine’s Day.

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