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.

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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.