AQA 90 problems to solve- perfect for 9-1 Maths GCSE


AQA have recently released their ’90 problems’ to support teachers in delivering the problem-solving element of the new 9-1 maths GCSE. It’s a rebranding of the 90 problems they released to support the teaching of Further Maths GCSE in 2008. There a number of issues with many of the diagrams on the newer version, so I’ve attached the 2008 version for your interest, here:  AQA 90 Problem solving questionsAnswers

They are excellent resources and perfect for some little-and-often problem solving practice. Highly recommended.


Powerful, diagnostic, games-based AFL- Kahoot!!!


Kahoot is a tremendously useful, free AFL tool I recently came across after a Twitter recommendation. Students can use any web-enabled device (any OS platform) to take part in games-based quizzes. There are thousands of quizzes publicly available, or you can create your own in a simple drag-and-drop editing tool and add to the public pool. There are many maths-based quizzes already in the public pool suitable for KS3 and GCSE maths classes.

Students can access the quiz without needing logins and passwords, making it easy to use in class. They simply go to in their web-browser and type in a game-pin number. The game begins and students answer multiple-choice questions, scoring points for speedy correct answers. The leaderboard is updated after each question to build the tension!

After the quiz you can download a colour-coded spreadsheet of all your students’ responses. The students love it- I love the AFL!

See the video below of Kahoot in action to get a feel for it. To register for an account, go to

Kahoot in action

Bar modelling- a powerful visual approach for introducing number topics

Building on my recent post about a taxonomy for deep learning in maths, I have been trying to think a bit deeper myself about what each type of ‘deep learning link’ might look like. In particular, I have been researching and putting a lot of thought into what effective ‘visual models’ look like for the ‘key nodes’ I have previously identified as the most important foundation maths knowledge for students to master before starting their GCSE maths course. These are principally number topics.

Last year I became aware of the Singapore Maths Bar Modelling approached have recently found the time to research it further. I bought some Singapore textbooks and read about the work of Dr Yeap Ban Har. This video, featuring Dr Ban Har shows an exemplification of the approach for a typical functional maths problem:

Maths No Problem

In short, I really like the approach and am convinced it could enhance my own practice significantly by giving students powerful, but simple visual models they can draw upon and use to solve problems. I have been experimenting with some of the models in my lessons this year and have seen the positive effect they have had on student understanding of topics. What these visual models give you is an entry point when teaching a topic that all students seem able to grasp. It presents the concept in its rawest, simplest form without the distraction of lots of words or mathematical notation. The diagrams don’t replace the eventual algorithmic methods, but they provide an entry point where students seem to understand what it is they are trying to solve; something that often gets clouded when algorithms are presented to early on.

In primary education in Singapore, maths teachers follow a Concrete-Pictorial-Abstract (CPA) sequence when teaching maths topics. They start with real world, tangible representations, move onto showing the problem using a pictorial diagram before then introducing the abstract algorithms and notation.

The particular power of the bar modelling pictorial approach is that it is applicable across a large number of topics. Once students have the basics of the approach secured, they can easily extend it across many topics.

I have spent some time putting together some pictures showing how the approach can be used for different topics. They are not teaching slides, but rather ‘notes for teachers’ to demonstrate how this single model can be adapted to be the diagrammatic entry point for many topics.

To start with students are given blank (bar) rectangles (on plain paper) and then get used to dividing the bars into halves, thirds, quarters etc:


They can then calculate a fraction of a quantity by first drawing the fraction in the bar, showing the length of the bar to be the quantity and then calculating the length of the shaded part:


Again, you’ll end up at ‘divide by denominator, multiply by numerator’ eventually, but this does show the concept of what’s going on very nicely and is a good route into showing where the algorithm comes from.

Next up, equivalent fractions:


Then simplifying fractions:


A ‘fraction wall’ (as many teachers use traditionally in England) can be used for ordering fractions:


Adding fractions with the same denominators:


Adding fractions with different denominators:

bar modelling concept sketches


Multiplying fractions:


Dividing by fractions (works ok so long as you have integer answers). That’s enough to get across what is going on… Then you can lead into the method…

Slide10 Slide11


Converting mixed numbers and improper fractions:


Next up, understanding place value in decimal numbers. This approach lets you deal with lots of misconceptions like 0.62 not being larger than 0.7 etc. Importantly, this is now taking the bar model and putting a decimal number line onto it. This forms the basis for many topic models that follow:


bar modelling concept sketches

Now they have an understanding of how the decimal number line works, and they can draw bar models for fractions, they can combine the two on one diagram to convert between fractions and decimals:

Slide16 Slide17

Next they can learn that percentages are hundredths, and in doing so can put a percentage number line under the bar model:


They can now combine the fractional bar model with both the decimal and percentage number lines directly underneath it to convert between fractions, decimals and percentages. They draw the fraction bar first, then put on the decimal increments (by dividing 1 by the denominator) and finally put on the percentages (by dividing 100 by the denominator):


Similarly to fraction of an amount earlier, you can use this approach to introduce percentage of an amount. Starting with showing how to find 10% of a number (and thus why you divide by 10), it serves as a nice way into multiplicative reasoning approaches to ‘build off the 10%’ to find other percentages:


Other percentage topics then follow, such as percentage increase and decrease:


By putting both the decimal and percentage number lines on the bar model for this, you can clearly show where percentage multipliers come from, including the ones less than unity for percentage decreases:

bar modelling concept sketches

You can introduce calculating a percentage change using the bar model approach:


It’s a particularly nice diagrammatic way in to teaching reverse percentages:


Once the above techniques have been mastered, it can be used for showing how compound interest works as follows. Particularly nice is if you turn the bars so they are vertical, it shows the exponentially increasing relationship:


It works for ratio too:

bar modelling concept sketches


Slide27 Slide28

I’ve used fraction walls for years, but it is the inclusion of decimal and percentage number lines built onto the fraction bars that is new for me. It opens up diagrammatic routes into so many topics and in such a coherent, simplistic way. The universality of the approach is what particularly impresses me; from humble beginnings of shading rectangles, the same model leads all the way up to reverse percentages and compound interest. If done in the correct order, there is a beautiful journey of progression all using one simple model. Each topic builds off the last logically as the model is manipulated in different ways.

The visual models won’t ever replace the algorithmic approaches, but instead will I hope provide my students with a better understanding of ‘what is going on’ when we are at the early stages of learning a topic. I hope their conceptual understanding is improved and this in turn enhances their procedural understanding through it giving it a purpose and something visual to hook onto. If they can ‘see the bar picture’ for problems with simple numbers it is my hope that the algorithmic approaches that follow that will enable them to solve problems with more challenging numbers with stick better. If they can represent a problem by drawing one of these models, they may have a better understanding of what the problem wants them to do.

I plan to develop resources to support teacher explanations and student activities in these topics in the coming months with the support of keen beans in my department and will share them with you when they have been tried and tested. There are no silver bullets in education, but this does represent a decent step forward for my teaching. Much to learn still!

Brainfeed- fantastic educational videos app for kids


I highly recommend you check out the Brainfeed app in the iOS appstore. It is essentially a collection high-quality educational videos about a wide-variety of topics. For example: How does the brain work? Does space go on forever? How can you outrun a cheetah? How big is the ocean? There are mathematically themed videos too.

The strength of the app is the quality of the videos and that you know they will be appropriate for your young learners. Rather than letting your children search on Youtube and potentially come across inappropriate material, you know with the Brainfeed videos that they’ll be entertaining, engaging, educational and appropriate.

They remind me a lot of the TEDEd project; videos I have regularly featured on this blog. Similar to these, the Brainfeed videos are top class!

Here are some sample screenshots:



You’ve never seen the GCSE Maths curriculum like this before…

Whole network no labels low res

What you’re looking at is the GCSE Mathematics curriculum. Each node represents a topic, e.g. transformations, ordering decimal numbers, frequency polygons etc. There are 164 nodes in the diagram representing all topics on both the foundation and higher tier curriculum. The nodes are connected by 935 links. Each link represents a connection between two topics whereby one is the prior learning required to be able to access the other. For example, equivalent fractions is linked to adding fractions because you need to be able to do the former before you can learn how to do the later. Continue reading brilliant maths enrichment videos

Numberphile_-_Videos_about_Numbers_and_Stuff is a website devoted to enrichment maths videos. The quirky, fast-paced style of the presenters is engaging and fun. Whether finding maths deliberately hidden in episodes of The Simpsons or finding the flaw in the Enigma code, Numberphile videos never fail to inspire.

Here’s the latest one about shapes/ solids of constant width:

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G&T website shout out: IB Maths, ToK, IGCSE and IB Resources

Lyapunov exponents of the Mandelbrot set (Stee...

Lyapunov exponents of the Mandelbrot set (Steel Beach) (Photo credit: Arenamontanus)

I got an email from a reader, Andrew Chambers drawing my attention to a website he has built for IB students studying in Thailand. The content they study is generally equivalent to A-level. His site IB Maths, ToK, IGCSE and IB Resources features lots of resources to enrich maths learning of gifted and talented pupils at  KS3 and GCSE. In his own words, here are the highlights:  (links for everything from using ESP tests to look at probability models, to using a mobuis strip to help understand extra dimensions to chaos theory or fractals….. – a large number of embedded Youtube videos looking at speed of light travel, Mandelbrot sets, synathesia, the golden ratio in nature and more…..

and the homepage:  – has a number of blog posts on everything that I think of that could be useful to teaching – from correlations on the latest premier league wages to league position analysis, to maths podcasts to sequence puzzles….

There’s also a load of maths dingbats that I’ve created – which work really well as lesson starters and in quizzes…

I have taken a look at think it is just the thing my super-keen year 8s would love to explore. Enjoy!

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The Magic of Pineapples goes live in paperback via Amazon!

The Magic of Pineapples Kindle 1563x2500 (1)The Magic of Pineapples, my latest book has just gone live in paperback format on Amazon. I wrote the book to inspire kids and ‘curious-minded’ adults into appreciating the beauty of maths. It has just been reviewed by TES Maths Advisor, Craig Barton who says:

“It is impossible to claim maths is boring after reading this wonderful book. I will also never look at a pineapple in the same way again! A must read for the curious minds of students and adults alike.”

The book is available in paperback via Amazon at these links:



It is also available in the Kindle Store. Just search for The Magic of Pineapples.

Here are some video trailers for the book that I’d really appreciate if you could share via your social networks:

Pattern in the Primes- The Magic of Pineapples:

Cherry in Matchstick Glass- The Magic of Pineapples:

Addition and Handshakes- The Magic of Pineapples:

Think of a Number- The Magic of Pineapples:

I hope you enjoy the book and a big thanks for your support :-)

Ideas for helping students tackle ‘wordy’ functional questions

Since the introduction of the ‘functional’ questions on the maths GCSE last year it has become important for pupils to improve the skill of taking a ‘wordy’ question and interpret what it is requiring them to do. Many pupils find this difficult and seem to give up before they have even read the question. Through a couple of strategies obtained from an excellent INSET I attended recently, I have had some success in getting pupils to improve their interpretation of functional questions. Surprisingly, these strategies were presented by our Head of English under the umbrella of a training session based on literacy, but I have found them to work well in the functional maths part of our subject.

Two colour highlighting

After reading the question once, get pupils to read it again twice more. On the second time they should highlight all the numbers in the question (both those in digits and in words). The third time they read it they should highlight in a different colour all the ‘key maths vocabulary’ words that are important to the context of the question. For example words like: more, each, difference, total, profit etc. Through reading three times, each with a different focus it seems many pupils improve their interpretation and understanding of the questions. It is a strategy for breaking down the process of interpreting a question into a series of smaller tasks.

Highlighting numbers in one colour and key maths vocabulary in another

Highlighting numbers in one colour and key maths vocabulary in another

Cartoon story boards

Another strategy that seems to work well with some pupils is to get them to create a ‘cartoon picture’ for each sentence of the question. For example, if the question begins ‘Sue buys 24 books for £2 each’ pupils could draw a picture of a book with a £2 sign on it and a ‘X 24’ beside it. They work through the question creating a cartoon picture for each sentence. They then look at the whole cartoon story board they have drawn and it is a pictorial representation of the problem. I have found that many pupils understand the question better looking at their story board, than looking at the text. I think this may be due to them creating a mental picture of the problem in their imagination, something that is essential for solving functional problems. Here is an example of a story board one of my pupils drew today for the above question and then their solution:

Cartoon story board for the 'Sue buys 24 books for £2 each' question

Cartoon story board for the ‘Sue buys 24 books for £2 each’ question

Cartoon story board of the above problem and then the student's solution

Cartoon story board of the above problem and then the student’s solution

Another cartoon story board and solution to a similar problem by a different pupil

Another cartoon story board and solution to a similar problem by a different pupil

Do you have any other strategies that you use when teaching pupils how to tackle functional questions? If so share them with us in the comments section!

The Magic of Pineapples- A Brain Tingling Adventure Through Amazing Mathematics

The Magic of Pineapples Kindle 1563x2500 (1)

How are children’s puzzles and patterns based on infinity related? What are the similarities in the maths behind the shape of tropical storms and how pineapples grow? Why is much of internet security built on one of the great unsolved problems in mathematics?

A year ago I decided to write a book that answers these questions. The Magic of Pineapples was born. A year later, it’s ready! The aim of the book is to inspire maths-curious teenagers and adults into a life-long love of the subject. I want the readers to realise that maths is not just a set of routine steps that you blindly learn to enable you to live in our society, but a portal into a whole new way of seeing and understanding the world around you. I want them to see the beauty of the subject.

The content of The Magic of Pineapples is accessible to anyone proficient in secondary school maths. I teach 11-16 year olds and I wanted to write a book they could understand. Readers are not just spoon-fed facts however, with the book setting numerous challenges for them to tackle before the interesting results are discussed. The Magic of Pineapples is a hands-on, book that leads the reader into making some of the most famous mathematical discoveries themselves.

Combining interesting historical events with contemporary applications, the book makes links between many different real world phenomena, showing the reader how often the maths underlying the behaviours is the same. For example, the maths used by Carl Friedrich Gauss to quickly add all the numbers between 1 and 100 (1 + 2 + 3 … + 100 ) is the same as that used to calculate the number of handshakes that take place in business meetings!

The Magic of Pineapples discusses the big ideas in maths in a way 11+ year olds can understand and relate to. If it inspires some people into a love of the subject I’ll be very happy.

Available in the Kindle Store now:

The Magic of Pineapples in the UK Kindle Store

The Magic of Pineapples in the US Kindle Store

Coming also in paperback via Amazon soon!

Here is a video trailer about the book: