Resources Anonymous

'Hello. My name is Jo and I'm a resourceaholic'. Yes, I confess, I'm utterly addicted to searching the internet for maths teaching resources.

It all started when I saw a PGCE student photocopying this Trigonometry Pile Up activity from greatmathsteachingideas.com. I work in a grammar school where pupils respond quite well to didactic teaching styles and textbooks exercises - arguably there's nothing wrong with this approach, but the lovely Trigonometry Pile Up worksheet reminded me that the internet is full of engaging resources and exciting teaching ideas. I started spending a lot of time looking for resources online and sharing these resources with colleagues. It became a bit of an obsession! Unfortunately my colleagues found it all rather overwhelming. And they're right - there's so many resources to choose from, no-one has the time to look at them all. So it's helpful when someone does the searching, filtering and classifying for us.

William Emeny recently posted a link to 'Mathematics 101: Leading Sites for Math Teachers' on his blog. Ooh, new websites! Heaven for a resourceaholic.

I haven't had a chance to look at all these websites yet, but here's some of my top resource recommendations from the websites I've looked at so far. I've focused on A level because that's where I think it's hardest to find interesting teaching ideas.

Statistics 1 
Illustrative Mathematics has some fantastic ideas for teaching S1. Here's a selection of examples:

• Correlation and Regression: a lovely activity on coffee shops and crime. Includes calculating a correlation coefficient, interpreting a regression equation and considering causality and outliers.
• Discrete Random Variables:  a short activity Sounds Really Good! (sort of) which features a real-life use of expectation. Bob's Bagel Shop is similar.
• Probability: the card activity Describing Events is an excellent introduction to probability and the activity Venn Diagrams and the Addition Rule is good too. I also like this short Titanic activity on independence.
• Data:  These activities on Haircut Costs and Speed Trap focus on comparing box plots. Describing Data Sets with Outliers and Identifying Outliers are about outliers and skewness in data (see my related post on teaching skewness).
• Normal Distribution: short contextual exercises Should We Send Out a Certificate? and Do You Fit In This Car?.

Core 1 and 2

• Logs: This booklet Logarithmic Functions from Mathematics Vision Project is packed full of excellent activities such as ordering logs, converting between exponential and logarithmic form, using log laws and solving equations with logs.  For related resources, see my earlier post on Logs. 
• Quadratics and Cubics: I like this short Cubic Graph task (answers here) from Mathematics Assessment Project. There's some great Forming Quadratics activities here too.
• Circles: Mathematics Assessment Project also has full lesson plans - I've successfully used the excellent materials from Equations of Circles 1 and Equations of Circles 2 with my Year 12s. 
• Surds: See resources listed in my earlier blog post, Surds. 

Finally, I thought it might be helpful to share this quadratic activity 'Find the coordinates' - my school uses this on Sixth Form Induction Day in our maths taster sessions (thanks to Susan Wall at Wilberforce College for this activity).

I'll continue to work my way through the website list and will feature recommended Key Stage 3 and GCSE resources in later posts.

All about gradient

When we teach linear graphs at Key Stage 3, we often miss the opportunity to explore useful real-life mathematics. In this post I share practical ideas and resources for teaching the concept of gradient.

A sense of steepness
At school we usually express gradients as integers or fractions but road signs in the UK show gradients as percentages.  Older signs showed ratios (rise:run) - arguably these were easier to interpret.

The road sign pictured indicates a gradient of 25% ie one quarter. We say the gradient of this road is 'one in four'. Think about this as 'rise over run' and therefore interpret this as 'we go up one unit for every 4 units we go across'. Wouldn't it be great if we could jump in the school minibus with our class and drive to a road with a 25% gradient to see how steep it is? Unfortunately this is probably impractical (though it could be set as a homework, depending on how hilly your local area is!).

There are some impressively steep roads in San Francisco but New Zealand boasts the 'world's steepest road' according to the Guiness Book of Records - Baldwin Street in Dunedin has a gradient of 35%. So how can we interpret this gradient? It means we go up 35 units for every 100 units we go across. We could write this as the ratio 35:100, simplified to 7:20. But we usually prefer to state gradient ratios in the form 1:n, so in this case the gradient is 1 in 2.86. Incidentally, 25,000 balls of chocolate are rolled down this 350m-long street in an annual charity Cadbury Jaffa Race.

Give your students a sense of 'steepness' by showing this short video of someone cycling up a road with 38% gradient. Looks like hard work! As a matter of interest, this rather technical article attempts to calculate the steepest gradient that one can cycle up.  This article says that anything over 16% is considered very challenging for cyclists of all abilities.

Ripley Street Ridge, San Francisco

Practical activities using trigonometry
In real life it is normally impossible to measure the rise and run of a slope so we use trigonometry to calculate these lengths. Picture a slope as a right-angled triangle - we can use a clinometer to measure the angle of elevation and perhaps a trundle wheel to measure the length of the hypotenuse. Trigonometry can then be used to calculate the rise and run in order to find the gradient. The table below shows gradients and their equivalent angles of elevation.  I think it's important that students understand the connection between a 100% gradient and a 45 degree angle.

Source: Wikipedia

Gradient can also be calculated using Pythagoras' Theorem. Say a man cycles 5km on a slope and knows (from his altimeter watch) that he has climbed 250m vertically.  Again, picture a right-angled triangle - the hypotenuse is 5000m and the height is 250m. Use Pythagoras to calculate the base (4994m).  Now use 'rise over run' to calculate the gradient - in this case we get 5%.

These real-life applications of mathematics can easily be made into practical class activities. For example you could use home-made clinometers and trigonometry to calculate the gradients of slopes in and around school, like staircases and ramps.

Resources for teaching gradient

• My worksheet on real-life gradients on TES
• A wide range of exercises in this pack. Note that this is a American resource so uses the terms slope and grade instead of gradient.
• A measure of slope - lesson and activities from Mathematics Assessment Project
• A worksheet of contextual gradient problems
• A gradient activity from Nrich
• Mixed gradient questions by linzbfc on TES
• PowerPoint on gradients by Andrea Frame on TES
• What is gradient? worksheet by MathsPad
• Alphabet slopes activity

Possible homework investigation questions
What is a funicular? Why can funiculars travel at steeper than 90 degrees?
What’s the maximum grade allowed for a railway without cable or cogs?
Why do you think road builders have to worry about the grade of the road?
Why do you think we have regulations for building wheelchair ramps?
At what steepness do avalanches occur?
Is it possible to ski at 90 degrees steepness?
(source)

I hope this post has inspired you to try out some new ideas when teaching gradient!  Let me know how you get on (@mathsjem).

Unusual gradient sign in Hertfordshire, UK

Maths Vocabulary

I found Cav's blog post 'Vocabulary and maths are not mutually exclusive' really interesting and it got me thinking about activities and teaching techniques related to maths vocabulary.

A teaching tool that can be used regularly throughout the year is a vocabulary knowledge survey. When you start teaching a topic, create a list of the new words that will be used and ask students to complete a chart showing their understanding of those words.  Repeat the activity later to see the improvement in understanding - I've not tried this myself yet but I think it would generate some useful questions as well as boosting pupil confidence.

In my post on Area and Volume I suggested these maths vocabulary exercises which make for nice 'end of term' lessons.  Here's an example of an activity from this pack:

Here's some more links that you might find useful:

• The Maths Magpie has created these great Maths Vocabulary Activities
• A list of key words for Key Stage 3
• Another vocabulary activity - perhaps an alternative to the knowledge survey described above
• Exercises with a Mathematics Dictionary 
• Background reading 'Building Mathematical Vocabulary' by Dr Madeline Kovarik

I was recently talking to a highly intelligent and articulate Year 13 pupil who said that she hated it when her teacher asked her to 'speak maths'. I asked her to elaborate. She said that when questioning the class, the teacher expected answers that used appropriate maths vocabulary - and even this pupil (who'll be off to Oxbridge in September) found this daunting.  We need to address this issue from a young age.  One tip is to always require pupils to answer questions in full sentences instead of just numbers or words - from primary school onwards. A very simple example - in response to the question 'what do we call a polygon with 8 sides?', don't accept the answer 'octagon' - ask for a full sentence such as 'An 8-sided polygon is called an octagon' to get pupils used to 'speaking maths' from an early age (source).

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Edit: 11/11/15

Since writing this post last year I've seen lots of great resources for developing vocabulary. I particularly like the Literacy Assessments featured in Gems 43. I also found the activity below worked well with my Year 7s.

Here's my guide to literacy in maths lessons:

Shortcuts vs Concept Development

I highly recommend this book Nix the Tricks - ‘a guide to avoiding shortcuts that cut out math concept development’. I felt uncomfortable reading it at times - my own teaching is full of these tricks and shortcuts because I’m strongly influenced by the time pressures and exam focus that the author references in her introduction. The book covers a great many teaching tricks and in each case makes excellent points about why these tricks should be avoided. Helpfully, alternative approaches are suggested. I’m not going to stop using crocodile references when introducing inequality symbols (I like the crocodiles!) and I'm not (yet) going to stop teaching gradient as 'rise over run', but I will certainly think twice about many of my methods now I’ve read this book.

My blog focuses on sharing teaching ideas and some of my posts include exactly the kind of shortcuts that this book is so opposed to - like sohcahtoa triangles, or this trick for remembering exact values of trigonometric ratios (which doesn’t replace the triangle method - but it's a nice trick and worth sharing).

It was my excitement at discovering this method for multiplying matrices that inspired me to write this blog. I was astounded that I’d never seen this method before and immediately showed it to my FP1 students, who were equally impressed. I mentioned this method to a colleague, prompting a conversation about other 'alternative methods' we knew of. She told me about this method for finding Highest Common Factors. Again, I was astonished that I’d been happily teaching the Venn method - which I’d learnt from my mentor during a PGCE placement - when a quicker and easier method existed. I felt like I'd unnecessarily overcomplicated things for my students.

We all know that quicker methods aren’t always better methods. ‘Maths hacks’ such as these and shortcuts (such as those discussed in Tina Cardone's book) do nothing to help students’ underlying understanding of mathematics. But I still believe there's value in teachers sharing tips and alternative methods because sometimes we discover hidden gems - like the wonderful matrices method - that change the way we teach a topic for the better.

By the way, if you enjoy the book then there's more (including new sections in draft) on nixthetricks.com.

Area and Volume

Every maths department should own a giant tennis ball! Walk into the classroom with a large fluorescent ball under your arm and you've instantly got the attention of the class.

Students stand in a circle and are told they're only allowed to speak if they're holding the ball. Ask "how could we work out the surface area of this ball?" and throw it to a random student. Students throw the ball around the circle as they share their thoughts and suggest ideas. At some point a teacher prompt may be necessary, along the lines of, "why did I bring a tennis ball? Why not a football or netball or golf ball?". In my experience this prompt may lead some students to 'eureka' moments as they start thinking about the markings on the surface of the tennis ball. Eventually someone will spot that the surface is made up of 4 circles wrapped around each other, and they can then deduce that the formula for the surface area is 4πr². If they need convincing, show them this slide.

This is a very engaging activity. As pupils can't talk unless they're holding the ball, you'll see them wildly waving their arms about, silently begging their friends to pass them the ball so they can share their ideas!

Try it - I'd love to hear how it goes.

Area and volume is a wide ranging topic, from areas of triangles in Year 7 to volumes of hemispheres and frustums in Year 11. Here's a handful of my favourite resources:

• I hand out these rhymes for the area and circumference of a circle in Year 7, tell them to pick their favourite or come up with their own. Love this silly song too.
• Also for Year 7, these matching cards from the Nuffield Foundation involve calculating the area and perimeter of compound shapes.
• As always, plenty of great activities on teachitmaths.com. 
• At GCSE I use this problem solving challenge with questions from Median Don Steward. Pupils work on A3 paper in pairs. They need their brains switched on for this one. There's loads of brilliant activities on Don Steward's blog that could be used in a similar way.
• The Chalk Face has some lovely activities, including: cornflakes problem; Penny Farthing; fuel tank; golden balls and baby surface area. 
• There's a nice activity on finding the volume of different shaped glasses from Shell Centre. 
• Illustrativemathematics.org has fantastic geometry activities. I love the rich task Eight Circles which relates to finding the area of a circle.
• The Chalk Face also brings us this great interactive cuboid for teaching surface area. This could be followed up with harder surface area questions from Don Steward.
• Maths Sandpit has a fantastic resource for investigating the surface area of a cone.
• Challenging volume and area revision questions for Year 11. Some of these are quite tricky and demonstrate that this topic isn't just about substituting into formulae.

For related class discussion - why do we say hemisphere but semicircle? Why not semisphere or hemicircle? I believe it's because the word circle is from the Latin word circus (meaning ring), hence we use the Latin prefix semi, whereas sphere is from the Greek word sphaira (ball), hence we use the Greek prefix hemi. This article about semi/hemi/demi gives more examples. If you like this then you might also like these lovely maths vocabulary activities.

Finally, here's one for the teachers. It's not as obvious as it seems!

(answer)

Teaching with Desmos

I discovered Desmos last year through @ColleenYoung's fantastic blog Mathematics, Learning and Web 2.0. It quickly became my favourite piece of 'teachnology'. Before Desmos I occasionally used Omnigraph or Autograph, but I now use Desmos regularly because it's just so easy to use. It really couldn't be simpler. As it's web-based, I encourage my pupils (particularly at A level) to use Desmos at home to support their studies, for example to check graph sketches.

Desmos on the Interactive Whiteboard
When teaching exponential functions to Year 12, I used Desmos on my IWB to show the graphs of functions such as y = 2^x and y = 3^x. We discussed our expectations for graphs such as y = 2^-x and used Desmos to check.  When we plotted y = (-2)^x we got an interesting result - which had me stumped for a minute - can you explain it?

Desmos is particularly helpful for demonstrating concepts in Further Maths (such as the directrix and focus of a parabola) - there's some good examples in this blog.

Desmos in the IT Room

My school doesn’t own tablets and IT rooms are rarely available, so my pupils don't often get the chance to use technology during their Maths lessons. But I was so keen to share Desmos with my Year 9 class, I made a big effort to get organised. I produced this worksheet for them to use in an IT room, with the aim of introducing different types of graph (quadratic, cubic, reciprocal) and simple transformations. After a very short demonstration of Desmos, the class spent two whole lessons working independently. I was pleased with their progress and I hope that they'll remember Desmos and continue to use it as a learning tool both at home and at school.

I loaded my worksheet onto TES today - along with a graph transformation activity that I use at Key Stage 4 - and it got me thinking about whether any other resources are available for teaching with Desmos.  I found a few examples on TES, such as Investigating straight lines with Desmos by Tristan Jones and Using Desmos to teach Coordinate Geometry (KS3) by phildate. I expect that the range of teaching resources available will grow as increasing numbers of teachers discover and embrace this fantastic tool.  

Venn Activities

This post is about using Venn Diagrams as a teaching tool, rather than teaching the concept of Venn Diagrams themselves.  To quote the excellent Johnny Griffiths, 'The Venn Diagram with three bubbles is the maths teacher's friend! ... To be able to fill in an eight-region diagram shows a really deep understanding of the distinctions that a good student has to be able to make, while the weaker student finds being faced with a picture such as this much less threatening than a page of exercises'.

Before we start using Venn activities in Maths lessons, let's think about whether pupils are familiar with how Venn Diagrams work. I admit to being a bit clueless about exactly what children study at primary school. The primary Maths Curriculum doesn't specifically mention Venn Diagrams, but I believe they are often used as a teaching and learning tool at Key Stage 2. Pupils are also likely to come across Venn Diagrams in other subjects, such as science, at both primary and secondary level (Physics teachers might be interested in this paper). In my experience most Year 7s have seen them before, and if not then it doesn't take long to explain the concept.

Here's some examples of how we can use Venn Diagrams in teaching secondary Maths.

Grouping Activities

I love these activities because they provide great opportunities for pair, group or class discussion. They're also quick and easy to prepare and explain (print the empty Venns onto A3 paper for pairwork, or just draw an empty Venn on the board for a whole class activity).

• My Year 10s were fully engaged in Don Steward's fantastic linear equations activity. It helped consolidate their knowledge of linear graphs and allowed me to check their understanding.
• Johnny Griffith's excellent RISPs (Rich Starting Points for A Level Core Mathematics) includes the Venn activities Risp 10 (coordinate geometry) and Risp 4 (periodic functions).  In the teacher notes for Risp 10 Johnny shares his views on the usefulness of Venns.
• A Venn activity for Solving Quadratics by Factorising by aka nka on TES. 
• A GCSE extension activity on quadratic graphs from MEI.
• I haven't tried this yet but Don Steward also gives us this lovely number activity.

Interactive Venns

Venns work well on the Interactive Whiteboard - here's some examples.

• Nrich has some interactive Venn diagrams on types of number.
• MathsPad has some lovely Venn activities (most require a subscription). I've successfully used this Factors, Primes and Multiples Activity with my Year 8s.   
• Teachitmaths.co.uk has interactive Venn activities - for example for shapes and prime factors, or you can create your own (subscription required).

Other Uses

• When I introduce complex numbers in FP1 we create a Venn Diagram during class discussion about types of numbers (real, rational, integers etc) and their Greek symbol notation.  The NCETM has a similar Venn activity in this resource sheet.
• Up until recently I used Venn Diagrams to teach Highest Common Factor and Lowest Common Multiple (method explained here), but I now prefer the simpler method described in this post.
• Quadrilaterals can be classified using a Venn Diagram like this one.  We can then answer questions such as 'Is every square a rectangle? Is every rhombus a square?' A 'family tree' also works well.

In conclusion - Venns are everywhere! But in secondary Maths they are only formally taught to those studying iGCSE (set theory) and Statistics 1 at A level (probability). I'm pleased to see that Venn Diagram methods in probability are on the new Maths GCSE Syllabus so will be more widely studied in future. But in addition to being taught as a topic in their own right, Venn Diagrams are a versatile teaching tool with numerous applications in the classroom. And they have the added bonus of being created by a British Mathematican! Well done Mr Venn.

Binary

Maths teaching is often rushed. We cover topics too quickly, we test regularly, we repeat the same thing every year. By Year 10 our pupils are bored. It's so unnecessary and frustrating. If only we spent more time on rich Mathematics. I'd love to slow down, go off on tangents, let my pupils investigate and discover... The only time I get the chance to digress from the Scheme of Work is at the end of the summer term.  So I seize the opportunity to engage my pupils in interesting concepts and ideas - like the binary system.

It's easy to get pupils interested in this topic. Here's some suggestions:

• Show them a picture of a binary watch. I once knew a Year 13 who actually wore one of these to school every day! My pupils were intrigued by this and wanted to know how to read it.
• You could lead into this topic through other number systems (Roman Numerals etc).
• Perform this fantastic binary magic trick.  My pupils loved it! They were desperate to know how it worked.
• Tell them the joke "There are only 10 types of people in the world: those who understand binary and those who don't". 

Here's a few examples of resources you could use in a binary lesson:

• Nice PowerPoint 'Binary Bot' from MauveStorm on TES (designed for Key Stage 2 but could be used for Year 7/8)
• The slides that I use when I teach binary.  Includes how to count binary on your fingers! 
• Binary Window Frames from mathsticks.com (free registration to access)
• A worksheet to practise converting to and from binary
• Binary Crossums
• PowerPoint from teachitmaths.co.uk on binary, denary and hexadecimal numbers
• Teaching binary using cups
• Lots more resources listed in this blog.

Last year I went to a fantastic lecture by Simon Singh (@SLSingh) about Maths in the Simpsons.  His book 'The Simpsons and their Mathematical Secrets' is absolutely brilliant, a must-read for all Maths teachers.  Singh writes about a scene in the Futurama episode 'The Honking' in which Bender is alarmed when he reads the binary code 0101100101. When read in a mirror, the code reads '666'.  I told my Year 13s about this - they loved it. I wanted to show them the clip from Futurama but all I found was this YouTube video in Spanish (?) which they thought was hilarious. I also explained the reference to the 'Redrum' scene in The Shining - most of them had never seen the film so I showed them this clip and managed to terrify half the class! Myself included. :)

Student Communication using Blogs

I've taken a few days away from posting on here because my baby was born on Monday... But even though I've been a bit tied up, I've still been able to share last minute motivational messages with my exam classes using two student blogs - one for my GCSE class and one for the whole Sixth Form. In this post I discuss my experience of communicating in this way.

GCSE Class
I started this blog for my Year 10 class two years ago. Initially I posted after each lesson. I'd post a summary of what we'd covered in the lesson, including links to further reading in case they hadn't understood something or wanted to extend their learning (here's an example post).  I also posted classwork answers and links to worksheets and homework sheets in case they missed a lesson or lost a homework. I found this was the most useful aspect of the blog - my pupils never had an excuse for missing a homework!

I found it a bit time-consuming to write a blog post three times a week and by the time this class were in Year 11 I was updating the blog less often, and my posts got shorter. To be honest there were times when I felt far too busy to post, but I did persevere.

Since this class have been on study leave, I've been posting motivational messages and last minute revision resources. I can see from the hit tracker on Blogger that my pupils have been receiving these messages.

Sixth Form
When I took on the role of Key Stage 5 Coordinator I decided to set up this blog as the primary tool for communicating with A level pupils.  My school has 200 pupils studying AS and A2 Maths and Further Maths.  Around 20 pupils subscribe to my blog by email and I hope that number will increase greatly next year (if I successfully promote it to the incoming Year 12s in September). It's been a very useful way of sharing resources and news - for example about trips, courses, open days and competitions.  Over the course of the year I've done an average of 3 - 4 posts per month so it's been less time-consuming than my other blog.

Summary
Although I've continued to use 'old-fashioned' forms of communication (noticeboards, handouts etc), I feel that blogging has been an excellent method for communicating with pupils, especially during study leave. Blogger - and similar tools - are very quick and easy to use.

My blogs have been primarily for communicating with pupils, unlike other blogs run by my school (like this one) which give pupils the opportunity to share blog posts themselves.

I've seen Twitter used by other Maths Departments to communicate with pupils - I think this is an excellent idea and my next goal is to encourage my Head of Department to set this up for my school.

Maths Movie Quiz

Spiked Math has some excellent Maths Movie Quizzes.  I chose these ones to entertain my Year 13s at Christmas.  Great fun for Maths teachers too!