## Puzzle Of The Week

Puzzle of the Week is a free international puzzle competition for schools.

Students submit answers to a puzzle which is published weekly on Mondays. School and student performances are recorded and leaderboards are published on the results page. You can sign your school up for free on the ‘About Puzzle of the Week’ page.

http://www.puzzleoftheweek.com/

## Fraction Representation Makers

One of the things I always find difficult when teaching fractions is finding accurate ways to create the images I desire. I have found two sites which create the representations I want in my lessons.

The first is part of a large suite of online materials.

VisualFractions.com has a large range of tools linked to fractions. The page above is the basic fraction maker which I use a lot. The site has clear instructions for each area and works really well. The downsides, for me, are the lack of colour choice and no possibility to change the orientation of the final image. It can, however, create improper fractions.

The second site I use is TeachableMath.com‘s fraction shape maker.

This has the option to make shapes as polygons or circles, provides space to apply a custom colour and can rotate the images. Unlike my other option, this cannot make images for improper fractions.

As always, if you use these links in your lessons, I’d love to hear how you get on!

## MathsBot’s Manipulatives

MathsBot is a recent discovery of mine which has a few great tools aimed at older maths learners – on loading, it displays resources for GCSE pupils, for example.

However, it also has a section of online manipulatives which are great to use in the Primary classroom – particularly one which may be lacking somewhat in concrete resources.

Now, I know that online and virtual pictorial representations are no substitute for hands on experience, but they still have a place in the classroom.

Right now, there are 12 available:

• algebra tiles
• bar modelling
• counters
• counting stick
• Cuisenaire rods
• Dienes blocks
• fraction wall
• geoboard
• number frame
• pentominoes
• place value counters
• unit box

Of those, a handy way to create place value counters and Dienes blocks, will prove invaluable to me.

## ITPs for modern times

Old folk like me will probably remember the National Numeracy Strategy (NNS) Interactive Teaching Programmes (ITPs) – some of which were great and some of which were horrible.

It turns out, they were great tools to show loads of mathematical ideas on whiteboards – but because they were made in the late 90s, and in Flash, lots of them no longer work.

Ted Burch of mathsframe.co.uk has very kindly remade them all in html5 but retained their original look.

The reworked programmes should work on tablets and most modern devices.

There are currently 25 remade tools which are all available for free here:

## Nonexamples: Conceptual Variation

A great way to think about conceptual variation is to ask children when something doesn’t fit into a specific set of criteria – knowing not only the properties of a maths topic but being able to explain why particular objects don’t belong in that topic too is a feature of maths mastery.

Conceptual variation means the opportunity to work on different representations of the same mathematical idea. This might be, for instance, looking at multiple representations of a number with Diennes equipment, place value counters, Gattegno grids, place value grids, arrow cards etc. These numerous representations demonstrate to pupils the different conceptual ideas that underpin a mathematical concept.

So, in the context of place value, some will reveal the quantity or value of a digit, some will show the importance of the position of a digit, others will support the order of the number and some will reveal the additive or multiplicative nature of place value.

The use of conceptual variation can be applied to all areas of the mathematical curriculum.

One website I’ve found to aid in this aspect of conceptual variation is nonexamples.com

The site has a range of mathematical topics to choose from and will display a variety of appropriate images of both examples and non-examples relating to it. There are two modes currently: question and compare.

I feel the initial question mode would be useful for opening lessons, possibly mid-way through a unit of work. They would also be a wonderful resource to develop and assess children’s reasoning skills.

Compare mode, however, is something I would use to introduce a new concept – perhaps even just displaying examples and non-examples side by side as above and asking children what they notice.

## Numberblocks: NCETM Support Materials

Numberblocks is a pre-school BBC television series aimed at introducing children to early number.

These NCETM materials use each episode as a launch pad. They are designed to assist Early Years (and also Year 1) practitioners to move on from an episode, helping children to bring the numbers and ideas to life in the world around them.

You don’t even need to register with the NCETM to use them!

The materials are designed to be used in conjunction with the Numberblocks episodes. They highlight and develop the key mathematical ideas that are embedded in the programmes. Each set of materials comes in the form of a PowerPoint file, and includes the following features:

• The episode description summarises the story and the key things that happen
• The maths in the episode explains the key mathematical concepts that are featured in the episode
• Using mathematical language – because it is important that practitioners model precise and correct mathematical language, there are suggestions of key sentences that you might use and have repeated; they provide a language structure to connect each mathematical idea to different contexts. Children will initially use their own language to talk about the mathematics, and will develop correct and precise language if this is modelled by adults.

Overviews of each series, their storylines and the mathematics addressed, are available as downloadable PDFs from the link below.

More materials are available on the CBeebies’ Numberblocks page, also linked below.

## Statistics: The Birthday Paradox

This has always been one of those problems that I have loved exploring. I recently found a website that will model the outcome in a brilliantly beautiful way. Here’s a little more about it from the Birthday Problem Wikipedia page:

In probability theory, the birthday problem or birthday paradox concerns the probability that, in a set of n randomly chosen people, some pair of them will have the same birthday.

By the pigeonhole principle, the probability reaches 100% when the number of people reaches 367 (since there are only 366 possible birthdays, including February 29). However, 99.9% probability is reached with just 70 people, and 50% probability with 23 people. These conclusions are based on the assumption that each day of the year (excluding February 29) is equally probable for a birthday.

It may seem surprising that the probability is above 50% when there is a pair with the same birthday for a group as small as 23 individuals. This is made more plausible when considering that a comparison will actually be made between every possible pair of people rather than fixing one individual and comparing him or her solely to the rest of the group.

https://en.wikipedia.org/wiki/Birthday_problem

The Pudding’s Russell Goldenberg has created this website to demonstrate this paradox.

Have a visit, have a play and contribute to the data.

I would absolutely use this as a hook the next time I need to tackle statistics with an upper Key Stage 2 class!

## Opportunities to notice – developing number sense and reasoning

I have long been a fan of Steve Wyborney’s blog where he shares plenty of engaging and different ideas to develop mathematical thinking. Although based in Oregon, the ideas are still very relevant to the UK curriculum.

I recently came across his use of dot patterns to promote ‘noticing’ – not something we necessarily spend a lot of time on, but I now wonder why we don’t.

One big focus in mathematics teaching currently is the use of small step and space to let children think. This sort of activity undoubtedly lends itself towards these ideas.

Steve has created 10 pages of dotted patterns, the purpose of which is to promote a wide variety of partitioning, seeking different perspectives and recording observations in ways that can be very useful. I would guarantee there will be children in every class who see a different pattern or way of breaking things up to draw out different mathematics every time.

He recommends using 18 copies of the same pattern with the space underneath being vital to allow plenty of working room and many opportunities to explore the same thing in different ways.

The following image is from his explanation video of how to use the patterns (found here), demonstrating 4 ways to investigate one such pattern.

For a greater challenge, once children are used to working with these, you could develop more complex images with different coloured dots – each colour being worth a separate value – to explore larger numbers, perhaps. Each dot doesn’t need to necessarily only be worth 1, either.

The 10 patterns and some further information can be found at the link below.

## Prime Number GIF

This lovely animated GIF makes me feel like there is something worth getting children to investigate further – either as a lesson activity or as a project to create short films showing other numbers.

As always, I’d love to hear if you use the idea, or it has inspired you to do something different in a lesson.

## Venn Diagram in the wild!

I recently saw this on Twitter (via @MrATeachesMath) and just had to share it. I’ve copied the image below and embedded the original tweet too.

A nice reasoning question here would be: “What labels would this Venn diagram have?”