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!

Two ways to use comic strips in Maths teaching

I stumbled across a nice teaching idea in this blog post which is all about giving pupils time to summarise what they've learnt.  The author of the blog suggests that pupils draw comic strips in class or for homework. Here's some examples:

Example worksheet

Example of pupil's work

I once used comic strips for a different purpose in my teaching - I produced this comic strip for my GCSE class, which I showed in this Prezi slideshow to generate discussion about sampling methods.  The characters are all teachers at my school (with slightly amended names...).  It went down well with my class and I set them the task of creating their own versions. We used Bitstrips for Schools which is lovely but can be very time-consuming.

(that's meant to be me by the way!)

Skewness

Let’s talk S1, as this year’s exam is fast approaching.

I don’t like the way my S1 textbook deals with skewness. It basically just sets out methods for identifying skewness without really explaining what it is. In this post I discuss an alternative approach.

I normally use this starter question in my first S1 lesson of the year:

The point being that when there are outliers in a data set, the mean becomes rather useless (unless of course you wish to deceive people!).

Whilst teaching S1 I often find myself referring back to my Bibonacci starter (‘remember that CEO? He ruined the mean!’). So I decided to make a worksheet out of it - see below - which I used for the first time this year when teaching skewness (if you like this then please leave me a review on TES!). My pupils enjoyed the activity and it generated interesting discussions.

• Investigating skewness activity
• Investigating skewness teacher notes

There is a similar activity here - from the website Illustrative Mathematics.

The key teaching point is that when data is skewed, the median is more meaningful than the mean. We have various methods for identifying skewness in a data set (summarised in these slides – sorry, original source unknown) - we normally consider the shape of a frequency diagram or box plot. Sometimes skewness is apparent just from looking at the raw data (this sideways stem-and-leaf diagram is interesting). 

Pupils need to remember that positive skew means we have a few 'sparse' data points (possibly outliers) at the top end of the data set – and like in the Bibonacci example, this will pull the mean up. This will help them conceptually understand the various methods described in the textbook.

Finally, here's a neat trick that I learnt from my colleague Lizzie... Imagine a cute puppy sitting on the left of the graph. A positive skew is going to see the puppy (positive = yay, a puppy!) and a negative skew is going away from the puppy (negative = get away from this puppy!). It's a silly but simple and memorable way of ensuring that students don't get the graphs mixed up.

---

Similarity

I saw this great monster GCSE question on Twitter (thanks @ReviseJustMaths) - it inspired me to write a post about similarity!

Taken from Edexcel Higher Paper 1 November 2013

In this post I'll focus on resources and methods for teaching similar triangles.

Two triangles are similar if they're 'equiangular' (a bit of a mouthful so I teach 'AAA' instead). GCSE questions often require us to first match up equal angles before we start calculating unknown sides using ratio/proportional methods.  This 'matching up' normally involves using angle rules - see my post on FUN angles. The key point to emphasise in teaching is that it really really helps to draw the two triangles separately, and in the same orientation. Here's three examples to illustrate this:

Example 1 - Separate Triangles

Example 1 is straightforward - it's clear to see from the angle labels that the 8cm side in the first triangle is proportional to the side labelled x in the second triangle.  I'd still encourage my pupils to redraw the two triangles in the same orientation though.

Example 2 - Joined Triangles

Example 2 is less obvious.  My pupils often make the mistake of thinking that side BC is proportional to side CE.  I teach them to label all equal angles in the diagram, then draw triangles ABC and EDC separately. They'd then see that because <ABC = <CDE (alternate angles) and <ACB = <ECD (vertically opposite angles), the side BC is actually proportional to the side CD. 

Example 3 - Combined Triangles

In Example 3 we can use the same approach as in Example 2. Label all the equal angles in the diagram ie <ABD = <ECD (corresponding angles), then draw the triangles ABD and ECD separately. In this question it's likely we'll be asked to work out the length BC, which just requires a bit of thinking (ie calculate the length BD first, then subtract the length CD).  

Resources

• I love these resources from the Mathematics Assessment Project. In particular, the sorting cards on pages 15 - 18 provide a challenging and engaging activity.  I asked my pupils to sort the cards into three categories (similar/not similar/can't tell), stick them onto A3 paper and annotate with reasons.  
• Similar pairs is a nice matching activity from Don Steward.
• This activity from Teach Mathematics has been tried by Mr Collins and reviewed in this blog post. 
• This is a great booklet of practice questions. There's also some challenging questions in this worksheet from mathsmalakiss.com.
• CIMT has lots of excellent similarity resources.
• This is a lovely rich task from Shell Centre

For resources for area and volume of similar figures, see my resource library.

Revision with Year 11

As this year's Maths GCSE exams are fast approaching, I thought it might be helpful to share a few of my revision resources.

My Year 11s expressed their frustration in trying to remember the differences between statistical graphs – particularly histograms, frequency polygons and cumulative frequency graphs. A key area of confusion is when to use midpoints, endpoints, class widths etc. I created this for them to help clarify these points:  GCSE Graphs Revision Summary

They also fretted about which formulae they had to remember for volume and area. In class we went through this sheet - I asked them to label each formula according to whether they needed to learn it or whether it was provided. I then gave them this formula summary to take away.

In one revision lesson my Year 11s tested each other using these circle theorem revision cards from teachitmaths.co.uk. They loved this lesson and demanded cards for every topic! I supplied them with the other cards from teachitmaths.co.uk, as well as my own mixed A* revision cards. I also encouraged them to make their own cards at home.

There are many good revision checklists and progress trackers available online.  I've just discovered this great progress booklet for Higher GCSE, created by Emily Hughes who writes the blog http://ilovemathsgames.wordpress.com/.

Finally, in my last lesson with my Year 11 class I went through these ‘hints and tips’ slides (while they ate cake, of course!). I pulled these slides together over time from a variety of sources so sorry for not crediting the originators of this material. It includes the rather tasteless but memorable picture below, which got a huge reaction from my class! ‘If you don’t like it, don’t do it!’ I told them. :)

Source: @mathequalslove

A Level Revision

I created these revision cards for a classroom activity, but they're equally helpful for pupils studying at home. These cards cover all the facts and definitions that pupils need to learn for each module - I haven't included anything that is given in the formula booklet.

These cards are based on the Edexcel specification.

Core 1 and Core 2 revision cards
Core 3 revision cards
Statistics 1 revision cards
Statistics 2 revision cards
Decision 1 revision cards
Mechanics 1 formulae revision  (thanks to my boss for creating the M1 cards!)

I hope to do C4 and FP1 next year!

Hope they're helpful!  If you want to leave me a review or make suggestions, please do through TES or Twitter.

Trial and Improvement

In this post I want to address a common misconception that comes up when teaching trial and improvement. Here's an example which illustrates the problem:

These trials tell us that the solution is between 2.1 and 2.2. The question requires an answer to one decimal place, so our final solution will be either 2.1 or 2.2.

At this point it looks like our solution will be 2.1, because x = 2.1 gives us the closest answer to five. But this is a cubic function, not a linear function, so this is an incorrect assumption (and a common misconception amongst students).

We must now check the midpoint of 2.1 and 2.2:

Let’s draw our last three trials on a number line:

We can see that the solution must lie between 2.15 and 2.2.  All values between 2.15 and 2.2 round to 2.2, therefore our final answer must be 2.2. 

You can see here that it was absolutely necessary to check x = 2.15. And not just because you'd lose a mark in a GCSE for failing to check it! You could plot the function, as shown below (use Desmos - it's fantastic!) and zoom in on the solution to help your pupils understand this point.

I got this example from a discussion on TES.  There is a similar example in this blog post.

Resources
While I'm on the subject of trial and improvement, let's take a quick look at a few teaching resources:

• Teachitmaths has a great PowerPoint explaining the method.
• I particularly like the way this worksheet by jhofmannmaths is set out.
• Justmaths.co.uk has a Whodunnit activity and exam questions.
• Mr Barton Maths has a 'Win a Million' game.
• Teachmathematics.net has a good Excel activity.
• Mr Collins writes about this animal zoo resource in his blog.
• Finally, an extension worksheet on linear interpolation.

Multiplication and Colour

This simple idea shows how important it is to use different colours when writing on the board - no matter what topic you're teaching.  It really helps pupils keep track of what you're doing.

Source: http://www.jandmranch.com/2013/02/09/teaching-three-digit-multiplication-to-my-aspergers-daughter/ 

Radians

Radians scare my Year 12s. They often choose to 'think' in degrees and just convert their answers into radians. I wonder if I could do a better job of introducing the concept.  I've just discovered this animation that is so absolutely fantastic, I had to share it.

I found it in this blog post but it originated here.  Both are worth a read.  

I love this post 'Degrees vs Radians' which presents an open letter to students wondering, “Why Do We Use Radians Instead of Degrees?”.  

Colin Beveridge has strong feelings on the subject - both teachers and students would benefit from reading his post 'Why Radians Rock (and Degrees Don't)', in which he convinces the reader of the unquestionable benefits of using radians.

Next Level Maths has a nice explanation of radians, including an interactive activity.

Incidentally, if you're excited by animations then you might like the other animations in this article '7 Animated GIFs That Will Make You Instantly Understand Trigonometry'.

A Percentages Trick

Just a quick post today to share a great tip:

You can swap the percentage and the amount:  16% of £25 is difficult to calculate in your head, but 25% of £16 is precisely the same amount.

So if you want to quickly work out 16% of £25 without a calculator, find 25% of £16 instead (ie divide £16 by 4). Simple!

Think about why this works.

Source: www.aperiodical.com

Linear Simultaneous Equations

Solving linear simultaneous equations is rather satisfying. In this post I suggest some techniques and resources for teaching this topic.

Equality and Lincoln
When I saw the film Lincoln I was struck by Abraham Lincoln’s reference to Euclid’s First Axiom ‘Things which are equal to the same thing are also equal to one another’. A well-made point in a discussion about racial equality, but also an interesting hook into a method for solving simultaneous equations such as

y = 4x + 6

y = 2x + 3

Both 4x + 6 and 2x + 3 are equal to y, therefore logically we have 4x + 6 = 2x + 3 (this is essentially just a substitution method). Why not show this clip from Lincoln in a lesson - it's a nice cross-curricular link between History and Mathematics and could even lead to a worthwhile ethical discussion (this blog gives a good explanation of Lincoln's point)

Elimination and 'SSS'
When solving simultaneous equations by elimination, my pupils find it very easy to remember the ‘SSS’ rule ie ‘Same Sign Subtract’. For example, in the equations 3x + 5y = 59; 2x + 5y = 56, this rule says that because the 5y in each equation is positive (‘same sign’), we subtract one equation from the other.

However, I noticed that when solving simultaneous equations such as -3x + 5y = 41; 2x + 5y = 56, some pupils saw that the -3x and the 2x were different signs and incorrectly decided that this meant they should add the two equations. So it’s worth clarifying which signs we are referring to in the ‘Same Sign Subtract’/’Different Sign Add’ approach.

Graphical Solutions
It’s so important for pupils understand that whenever they solve linear simultaneous equations, they are in fact finding the point of intersection of two straight lines. It’s essential to pose a question involving parallel lines (eg solve 3x + 4y = 10; 6x + 8y = 15) and get them thinking about why there are no solutions.  I like to show them this cute picture too.

Wordy Equations
This is one of my favourite simultaneous equation questions: “A zoo has several ostriches and several giraffes. In total they have 30 eyes and 44 legs. How many ostriches and how many giraffes are in the zoo?”. Make sure you discourage pupils from using the letter O for ostrich, because it looks like a zero.

Resources
There's lots of resources online – here’s a few I’ve picked out:

• Don Steward’s Median blog has a huge range of fantastic questions and activities.
• TeachitMaths has a great ‘wordy question’ worksheet and clear slides on elimination and substitution.
• UEA has a worksheet on linear simultaneous equations with a nice progression of questions.
• There is a useful worksheet on Solving Simultaneous Equations Graphically by Tristan Jones on TES. 
• This Geogebra worksheet by themathsteacher.com is a good exercise in solving simultaneous equations graphically with ICT.
• Peter Bland and JustMaths both provide exam questions on this topic.
• MEI has a nice extension activity. 

Surds

Surds - what a fabulous topic to teach! I love the abundance of resources available and I’ve listed some of my favourites below.

Two things I always do when teaching surds:

• Pupils list the first twelve square numbers in their books for reference throughout the lesson. They need to readily recognise square numbers in order to simplify surds so the more they practise listing them, the better.
• When manipulating expressions containing surds (eg expanding brackets) I make comparisons to what they already know about algebra. For example √2 and √3 can be thought of in the same way as x and y (ie not ‘like terms’) whereas 2√3 and 5√3 can be thought of in the same way as 2x and 5x (ie we can add them to get 7√3).

The NCETM suggests a couple of nice ‘hooks’ for getting started teaching surds:

• Ask pupils to find a way of drawing a line with a length of exactly √5 units (the hypotenuse of a right angled triangle with sides 2cm and 1cm) 
• Ask pupils to divide the length of an A4 piece of paper by its width. Repeat for A3 and A5. What do they notice? (The answer is always √2)

This blog post has some more ideas for introducing the concept of surds. And I like this on the history of surds.

Resources
Here’s some great teaching resources (there's hundreds more online). Most of these work equally well at GCSE and A level:

• Is it rational?
• Two interactive tasks from Mathspad
• Pairs activity 
• Introducing surds
• True or false activity
• Surds -Applying and problem solving
• Multiplication squares
• Mr Barton’s activity
• Standards Unit N11
• Surds Connect 4
• Surds resources - leannegadsby on TES
• Manipulating Radicals
• Tripods
• And here’s some resources recommended by the TES.

For a lovely set of surds problems, see my Problem Sets page.

Finally, if you're looking for something really creative, check out this Wheel of Theodorus Art Project.

Edit 29/12/14: Thanks to @runningtstitch for suggesting an alternative method for simplifying surds:

Large Square Roots

In C1, the non-calculator A level module, my pupils sometimes get stuck solving quadratic equations that are hard to factorise. This might be because the leading coefficient is greater than one (see my earlier post) or it may just be because the numbers in the quadratic are large.  I normally suggest that they persevere with factorising, but if they have to resort to using the quadratic formula then they may need to square root a large square number. Here's some simple ideas for how to do so - in this example I'm finding the square root of 324, though these methods work equally well for larger square numbers.

Note that if the quadratic equation can't be factorised in the first place then we can either use the formula or complete the square to solve it, and we will end up with a surd in our final answer.  My next post is all about teaching surds!

Coded Averages

Teaching coding in Statistics 1 is a bit of a pain.  My pupils just don't like it, and it's hard for me to get enthusiastic about something that didn't really exist when I studied Statistics (perhaps because it's not entirely necessary?). When I was revising this topic with my Year 12s this year, I used a really simple real-life example:

Imagine I'm queuing up to buy four items of clothing, which cost £29.95, £49.95, £59.95 and £99.95. To pass time I decide to work out the mean price of these 4 items. To make the numbers easier to deal with, I add 5p to each amount and then divide by 10, giving me £3, £5, £6 and £10. It's very easy to work out the mean of these 'coded' prices in my head - it's £6. And now I can uncode that mean by multiplying by 10 then subtracting 5p (the inverse of the original coding).  That quickly gives me the actual mean price of the clothes, which is £59.95.  

Resources

Here's a few resources I've found for teaching this topic:

• Teachitmaths.co.uk has a lovely worksheet for helping pupils understand the affect of coding. Just beware of the error in the extension questions - both data sets should have six values, not five.
• Schoolworkout.co.uk has this sorting activity and this worksheet.
• Phildb on tes.co.uk has written this worksheet.

There's a limited selection here so perhaps I should write something myself - I'll add it to my list!

Angles Facts

I've always taught angles in parallel lines as Fs, Cs and Zs, but I recently discovered that pupils prefer to think of Fs, Us and Ns, which work in exactly the same way but may be easier to remember because they spell the word FUN.

(image from greatmathsteachingideas.com)

Of course there's still the challenge of remembering the more formal descriptions that they will need to use in their GCSE exam (for example they are required to use the term 'corresponding angle' in their answers, instead of F-angle).  I suggest that they think of 'FC' (football club - as in Tottenham Hotspur FC) for F angles = corresponding angles.  I also suggest they could remember that Z angles = alternate angles by thinking of the first and last letters of the alphabet. But this doesn't work if I change Z to N!  I'll have to think of something else (or even better, get them to come up with something).

Resources

There are loads of good resources for teaching this topic. Here are a few examples:

• I love the 'Spot the Angle' activity from mathspad.co.uk. This website also has a great interactive Parallel Lines Tool.
• Teachitmaths.co.uk has a great angle card sort activity and useful revision cards.
• Mark Horley has made a Basic Angle Properties activity.
• Missbrookesmaths.com has a nice angles in parallel lines tick or trash activity.
• There are a number of angle chases here - thanks to @MathedUp.
• Greatmathsteachingideas.com has a number of lovely worksheets, activities and applets including Angles in Parallel Lines Colouring Fun and First Letter Prompts which gets pupils thinking about the language of their angle fact answers.
• This comprehensive booklet 'Introducing Angles' by Gareth Evans is helpful. 

There are loads more resources for this topic in my shape resources library. 

More Angle Facts

On a related note, here's some tips for teaching more basic angle facts:

• Marie Darwin's blog suggests that we remember the sizes of acute, obtuse and reflex angles by noting that the sizes from smallest to largest are in alphabetical order.  She also points out that the A in the word Acute makes an angle less than 90 degrees (see picture).
• For remembering that complementary angles add up to 90 degrees, you could think of a compliment as being the right thing to do.  
• mathsisfun.com has a couple more ideas - I like their suggestion that the C of the word complementary stands for 'corner' (ie a right angle) and the S in supplementary stands for 'straight' (ie 180 degrees on a straight line).

Interestingly, the word complementary comes from Latin completum (meaning completed) because the right angle is thought of as a complete angle.

Factorising Harder Quadratics

My pupils panic at the sight of a quadratic with a leading coefficient greater than one.  I factorise these quadratics by inspection (the 'guess and test' method) but my pupils aren't satisfied with this suggestion - they want a more structured approach.

A commonly taught method in the UK involves splitting the middle term in two (sometimes called the 'Grouping Method'). This is explained very clearly here (thanks to SRWhitehouse for this resource). Teachitmaths.co.uk has a PowerPoint explaining this method. It's worth watching James Tanton's video 'Splitting the Middle Term' too. He's not a fan!

'Grouping'. Source: Flat World Education

An alternative, which seems easy at first but paves the way for a large number of misconceptions, is the 'slide and divide' method. The method, and its associated problems, are nicely described in Nix the Tricks.

Nix the Tricks offers an interesting alternative - I've provided two examples here but it's worth reading the book for the full explanation.

It's also worth looking at this post by Don Steward to see his tap top method for finding factors and for lots of helpful practice questions.

Also worth a mention: when I first start teaching quadratic factorisation, I like to use this well-designed sum-product worksheet from greatmathsteachingideas.com as a starter. It's good practice of an essential skill.

Highest Common Factor and Lowest Common Multiple

There are many methods for finding the Highest Common Factor and Lowest Common Multiple of two or more numbers.  For years I've been teaching pupils to put the prime factors into a Venn Diagram, as described here. 

I recently discovered an alternative method that is impressively quick and simple.  It is described in this video as the 'Indian Method'. It's similar to the 'upside down birthday cake method' but it's much quicker because there is no requirement to use primes.

Say we want to find the Highest Common Factor and Lowest Common Multiple of 24 and 36.
Write down the two numbers, then (to the left, as in my example below) write down any common factor (ie 2, 3, 4, 6 or 12).  I've chosen 6.  Now divide 24 and 36 by 6 and write the answers underneath (4 and 6 in this case).  Keep repeating this process until the two numbers have no common factors (ie 2 and 3 below).  Now, your Highest Common Factor is simply the product of numbers on the left. And for the Lowest Common Multiple, find the product of the numbers on the left and the numbers in the bottom row. It's easy to remember which is which - to find the LCM, look for the L shape.

It's so quick!  And simple!  Try it.

Don Steward features an alternative method in this blog post.  He mentions that you can find a LCM by dividing the product of the two numbers by their HCF ie in this example, (24 x 36)/12 = 72.

See my Resources Library for resources for teaching HCF and LCM.