Higher GCSE Revision

Revision lessons can be hard to get right. It's much easier to engage students when you're introducing new material. Every year I try to deliver revision lessons that really add value, but it can be difficult to find inspiring resources.

This post features a selection of effective and engaging GCSE revision resources suitable for students working at grades A/A*. I'd be interested to see your recommendations too.

Picture Relay

My favourite activity for a GCSE maths revision lesson is this picture relay from alipon on TES.

I've delivered this lesson a few times and it's always gone really well. Each pair of students is given a picture and 20 questions. They have to bring me a correct answer to move on to the next question. The first few questions are fairly easy so it's fast-paced, but it quickly slows down as the challenge increases. And that's what I love about this resource - the level of challenge is spot on for students working at grade A/A*. There's plenty to get them thinking and the activity takes a good 40 minutes. Afterwards there are a number of discussion points (eg questions where students took different approaches), so it nicely fills a one hour lesson. My students really enjoy it.

The files provided on TES can be edited, meaning you can change or remove any questions you don't want to include.

Revision Races and Treasure Hunts
The writers of Number Loving provide some fantastic revision races. Here's an A/A* version. These activities make a nice change from past paper lessons. If you're planning to use a revision race then you might like this post about how to organise revision races from @Jeremy_Denton.

Higher Starter Questions
I found these Higher Tier: 25 Starter Questions which I used for a relay activity last year. Some of these questions are pretty challenging so I organised this one differently to my usual relays. I put piles of questions on a long table at the front of the room. I put an Excel spreadsheet up on the board - students names down the side (in pairs), question numbers at the top. They had to come to the front and choose a question. Once they'd solved it, they brought it to me and if they were correct I gave them a point in the appropriate cell on the spreadsheet. Students could see the spreadsheet updating throughout the lesson so they knew who was in the lead and they could also see which questions their peers had answered correctly (which helped them decide which question to pick next). It worked very well.

Another nice resource is this set of Higher Tier Prompts - students are given a picture and have to come up with a suitable exam question to go with it.

Revision Cards
Last year I did a circle theorems revision lesson in which students tested each other using these cards. Teachit Maths have cards for a range of topics here.

I've made my own A* revision cards that students can use to test each other. And for a more comprehensive selection, check out these brilliant flashcards from @tannermaths. The Mathematical Association sells these GCSE revision cards.

Of course there's also a lot of value in students creating their own revision cards. You could make each student a specialist for a particular topic - they spend half a lesson making quiz cards for their allocated topic, then in the second half of the lesson students rotate around the room speed-dating style, quizzing each other.

Tailored Practice Papers
Aside from past GCSE papers (most of which my students do at home rather than in class), there are three other categories of practice paper:

1. Targeting a grade. A* practice papers like this from danwalker on TES and this from John Corbett are very helpful if you want to challenge a high attaining class. Don't forget that high attainers need to practise the basics too! I sometimes give my A* students an old Intermediate Paper (from the days of three-tier GCSE) and tell them I expect full marks. I often find that students who can do topics like vectors and algebraic fractions have forgotten some Key Stage 3 topics like isometric drawing and pie charts, so the Intermediate Paper is a good wake-up call.

2. Practising a topic. Exam-style questions are really useful when you teach a revision lesson focussing on a specific topic. Sources include Corbett Maths, MathedUp!, Maths Genie and Bland.in.

3. Practising a skill. It's useful for students to practise a technique for 'Quality of Written Communication' questions. I particularly like this set of RULER questions, again from the excellent @tannermaths.

Read the full description of RULER resources on tannermaths.co.uk.

Revision Resources
There are thousands of maths GCSE revision resources available online so I can't list them all, but here's a few that you might find helpful.

• Did you know that Don Steward has a blog specifically for revision and practice resources? Here's an example of his A/A* revision questions.

Don Steward 'A and A star mixture'

• MathsBox has an excellent 'Big Mistake' activity and two Higher GCSE Revision Treasure Hunts: here and here. These GCSE Revision Trail Cards from parabola on TES look good too.
• I like this 'One Question Per Topic' checklist from andrewshorty on TES. 
• There's loads of fantastic resources and ideas on MathedUp!. I love the 5 Minute Mock Paper analysis which is so much better than the boring mock analysis sheet I currently use. It's worth reading the recent posts on MathedUp! about GCSE revision. Check out the excellent resources including GCSE Revision Mats and Essential Skills Worksheets. 

  

  Source: mathedup.co.uk

I hope you've found some helpful resources in this post. Please comment below or tweet me to share your favourite GCSE revision resources.

Tricks and Tips 2: Sequences, Linear Graphs and Surds

I recently presented a workshop at the National Mathematics Teacher Conference (#mathsconf2015) entitled 'Tricks and Tips: Clever Methods for Explaining Mathematical Concepts'. This is the second in a series of posts summarising the content of that workshop for those who were unable to attend. The aim of my workshop was to encourage people to reflect on their subject knowledge and the effectiveness of their explanations. I also hoped that delegates would learn new methods that they might consider using at school. In today's post I'm covering sequences, linear graphs and surds. My previous post was on methods for finding a Highest Common Factor.

Linear Sequences

What's the n^th term of the sequence below?

5, 8, 11, 14, 17, ...

It might only take you a second to work it out - how did you do it? This is another one of those topics where there's lots of different approaches. Ask the same question to a random sample of teenagers from across the UK and you'll see a wide range of methods used.

Here I'm going to describe four different methods (there may be more!). As you read them, consider whether you're going to stick with the method you currently use, or try something new.

1. Zeroth term
Inspired by this post on Don Steward's blog, I taught the 0^th term method for the first time this year. To work out the value of q in the n^th term U_n = pn + q, we simply step back from the first term in the sequence (ie q = U₀).

In the example 5, 8, 11, 14, 17..., subtract 3 from the first term to get U₀ = 2. So the n^th term is 3n + 2. This method is so quick that it's now my preferred method for working out n^th terms.

When I taught this method it went well, although a misconception did surface later on. When students were asked to 'write down the first 3 terms of the sequence 4n + 3', some of them gave the answer 3, 7, 11 (ie they started their sequences from n = 0 instead of n = 1). This is something to be aware of next time I teach this method.

Jumping along a line - Median Don Steward

2. Shifting times tables
'Shifting times tables' is a popular method. The idea is that we compare our sequence to a times table.  In the example below, compare the sequence to the 3 times table. Ask your students how to shift the three times table to get the sequence and they'll spot that they need to add two, so the  n^th term is 3n + 2.

If you plan to use this method for the first time then I recommend this NRICH article Shifting Times Tables, which comes with an interactive tool. If you subscribe to MathsPad, they also have an interactive tool for identifying shifts.

This method works well for quadratic sequences too (a topic on the new GCSE syllabus). For example to find the n^th term of the sequence 4, 7, 12, 19, 28 we can compare it to n² and notice that it is shifted up by 3 (ie the n^th term is n² + 3).

3. Formula
It's fairly straightforward to derive the following formula for the n^th term:

Why do we restrict this formula to A level? I once took on a GCSE class in Year 11 and asked them to find some n^th terms in a revision lesson. I was surprised to see them all using 'the A level formula' - their previous teacher had taught them it, and why not? They seemed quite happy with it. I wouldn't use it with my Year 7s though - their algebra skills are very basic when they first meet sequences. In our example, this is what they'd have to do to find the n^th term:

Perhaps this is more effort than necessary. The 0^th term method is considerably quicker.

4. Substitution
This is the way I taught sequences for years. To work out the value of q in the n^th term U_n = pn + q, we substitute a value for U_n, p and n, then solve for q. For example in the sequence below, we know that U_n = 3n + q. We substitute the first term to get 5 = 3 + q. Therefore q = 2.

Thinking about it, this is the same method I'd use to find the value of c in y = mx + c if I know the gradient and a point on the line. In fact given that linear graphs are simply graphical representations of linear sequences, any methods for finding the equation of a straight line work for finding the n^th term of a sequence. I don't do enough to make this connection with my students.

Linear Graphs

A straight line through the point (5, 7) has gradient 4. How would you find the equation of the line?

As I said in my post about linear graphs, I teach this differently at GCSE and A level. Wouldn't it be better to pick my preferred approach and stick with it? 

At GCSE my students write down y = mx + c and substitute values for y, m and x, then solve for c (let's call this Method 1). At A level my pupils use the formula y - y₁ = m(x - x₁) (let's call this Method 2). Look at the steps involved - for this question, the methods are equally efficient.

My current Year 10s had an American teacher last term - she covered linear graphs with them. When we came to revise this topic, they took a while to figure out where to start. Evetually I heard one say, "is this the point-slope thing? What was that formula again?". 

OK, so they'd been taught to call gradient 'slope', that's easily fixed. But they'd also been taught Method 2 and they couldn't remember the formula. This is a problem. Given that there's less to memorise, I think they'd be better off with Method 1. 

It's worth reading @srcav's post The Straight Lines Debate for more analysis and opinion on the two methods.

I find that all my students really struggle with linear graphs. It's one of those topics that frustrates maths teachers because it's hard to see exactly what it is that students find so difficult. I teach it over and over again from Year 8 to Year 12 and my students never seem to remember it from the previous year - I'm clearly doing something wrong!

One suggestion (thanks to @letsgetmathing for this) is to approach this topic from a less algebraic perspective - instead, focus on a table of coordinates. This is demonstrated in the example below. Students have to recognise that the y-intercept is at x = 0 and the gradient is the 'step' in the y values.

If your students struggle with the algebra-heavy Method 1 and Method 2, perhaps this table method might work.

Finally, I recommend that all teachers watch this video from James Tanton. He makes the concept of the equation of a line really clear.

3. Surds
Let's finish with three methods for simplifying surds.

In the book Nix the Tricks (essential reading for all maths teachers), we're shown a teaching trick called 'Jailbreak Radicals'. Thankfully I've never seen this 'zero conceptual understanding method' used in the UK.

Instead, most teachers tell students to identify a square factor and then split the surd accordingly eg √45 = √9√5 = 3√5.

Sometimes students struggle to spot square factors. In this case they might prefer to simplify using prime factors.

A geometric method is explained in the extract from Nix the Tricks below. Try to simplify a few surds like this yourself and see what you think. Notice that you still need to identify a square factor.

Source: Nix the Tricks

The feedback from my conference workshop suggests that some people find this unnecessarily complicated. But if you're going to try it, I recommend this post by @ChrisHunter36 and this resource (pages 5 - 6) by @314Piman.

That's it for today's post. I hope you've found it helpful. Please comment below or tweet me if you know of any alternative methods for teaching these topics. My next post will be all about quadratics - sketching, expanding and factorising.

5 Maths Gems #27

Hello and welcome to my 27^th gems post. This is where I share five of the best teaching ideas I've seen on Twitter recently. I'll be brief today because I feel like I'm drowning in marking, reports and lesson planning at the moment! Thankfully it's only one week until the Easter holidays, so there's light at the end of the tunnel.

1. Which One Doesn't Belong?
This website is pure genius from @MaryBourassa. 'Which One Doesn't Belong?' is 'a website dedicated to providing thought-provoking puzzles for math teachers and students alike. There are no answers provided as there are many different, correct ways of choosing which one doesn't belong'.  Here's a couple of examples:

Ask students to find a reason why each one doesn't belong. What a fantastic tool for generating interesting class discussion. Mary welcomes contributions to the website - details and templates are available here. This post gives some background on where the idea for the website came from.

2. C4 Integration

Some people find it easier to remember facts than others so bear with me on this one.

In a discussion about C4 integration on Twitter, @DrAliMaths told me how his students remember the integrals of sinx and cosx.

Seems silly, right? The very next day, I overheard my Year 13s getting in a muddle over signs when integrating sinx and cosx. I showed them the sic and cis mnemonic and they loved it! Instant recall of basic facts is very helpful when tackling complex integration problems. I even find myself using this mnemonic now. If I have to integrate sinx, I immediately think '(sic) sine integrates to cos - negative'. Previously I would have paused for a second to think about it, now it's automatic so I can get on with the trickier stuff.

And here's another tip for your A level students from @DrAliMaths:

Remember: 'D'ifferentiation 'D'ecreases the power.

'I'ntegration 'I'ncreases the power.

While I'm on the subject of integration, I must share this lovely idea from @sxpmaths. When his Year 13s do integration, they fold a piece of paper into 12 (see picture below) and fill it in as they learn more and more techniques. 

3. Hint Tokens

I read a nice post from @MrsOClee this week about a successful Pythagoras Lesson. In this lesson Pythagoras problems were put around the room and students worked in pairs to solve the problems - they had to present their teacher with as many correct answers as they could in an hour. They had 3 ‘hint’ tokens that they could trade in for help at any point. I like this idea - I have a habit of offering too much help, which can be counterproductive. The limited supply of hint tokens will encourage perseverance in students - they'll only ask for help when they really need it.

Spiderbox from illustrativemathematics.org

4. Surds
Two lovely surd resources were shared on Twitter this week. First up was @aap03102's excellent surds magic square.

And then @MathsPadJames (of awesome website mathspad.co.uk) shared this brilliant activity:

Followed by another set of examples:

5. Sticky Maths

John Smith (@HoDteacher) shared his post Sticky Maths which is full of fantastic ideas. Read it now! He's encouraging teachers to tweet their 'tricks of the trade' with the hashtag #stickymaths. An example is this clever way of setting out binomial expansions:

Binomial expansions can sometimes get a bit 'busy' and as a result mistakes can be made when simplifying. Setting out the workings vertically is a brilliant idea! It's so much tidier.

In this week's #mathsTLP, @El_Timbre suggested another vertical idea for introducing expansion at Key Stage 3. 

For example 3(2x+1) is 3 lots of (2x + 1) ie:

(2x + 1)

(2x + 1)

(2x + 1)

This makes the concept clear and simplification easy. And, as @mrallanmaths pointed out, when you move onto 20(2x + 1), students will be motivated to spot the pattern so they don't have to write out 20 terms!

I love these vertical stacking ideas and I look forward to all the #stickymaths ideas still to come.

What I've been up to
Did you read my post about methods for finding a Highest Common Factor? It's the first in a series of posts summarising the content of my recent conference workshop. The rest will follow over the next couple of weeks.

Ed Southall (@solvemymaths) and I have been running #mathsTLP on Sunday nights for three weeks now and it's going really well. We've been inundated with teachers asking for lesson ideas and thankfully there's been a fantastic response on Twitter, with loads of inspiring ideas and resources being shared. Join in on Sundays at 7pm to see for yourself!

Bonnie Attaway (aka Jo Morgan), Rock Legend

Today I trialled Times Tables Rockstars for the first time with my Year 7s. It was great fun. All students were totally engaged throughout the lesson - I played rock music while they excitedly competed against each other. I reminded them of the importance of instant recall of times tables for topics they will encounter over the next few years (like factorising quadratics). I'm really looking forward to using Times Tables Rockstars more in my new job - it's fantastic.

Speaking of fantastic resources, tomorrow I'll be using Chris Smith's Easter relay with my Year 8s. If you're looking for a nice Easter maths lesson then you can find it here. Enjoy!

Tricks and Tips 1: HCF

I recently presented a workshop at the National Mathematics Teacher Conference (#mathsconf2015) entitled 'Tricks and Tips: Clever Methods for Explaining Mathematical Concepts'. This post summarises the content of that workshop for those who were unable to attend. I have quite a lot to cover so I expect I'll need to write three or four blog posts. In this one I'm going to explain the rationale for the workshop and describe alternative methods for finding a Highest Common Factor. In subsequent posts I'll cover sequences, linear graphs, surds, quadratics, compound measures and a few more bits and pieces.

Workshop aim
How do you find the Highest Common Factor of two numbers? Do you use a Venn method? List the factors? Use the Euclidean Algorithm? Are there 'better' methods that you don't know about? (how do we define 'better'?). These are the sort of questions I want to explore. The aim of my workshop was to encourage people to reflect on their subject knowledge and the effectiveness of their explanations. I also hoped that everyone would learn new methods that they might consider using at school.

What determines the way we choose the explain things?

Most teachers establish teaching habits during their training and NQT year. The method they use the first time they teach a topic will probably stay with them throughout their career - unless they make a concious effort to try a new method. 

One of the few things I remember about GCSE maths was that I solved equations by 'moving terms over the equals sign' (the 'magic portal method'). This method is now considered to be a shortcut which stands in the way of conceptual understanding. These days it is more acceptable to teach students to solve equations using inverse operations (ie balance the equation by 'doing the same thing' to both sides). It's lucky that during my PGCE someone told me not to use the magic portal method because before then I was convinced that it was 'the' way to solve equations. 

Source: https://mrjasonto.wordpress.com

When we were student teachers, we drew our methods from a variety of sources. ITT courses don't cover much in the way of mathematical methods, so we were left to gather ideas from examples in textbooks, our memories from school, observations of teachers and conversations with colleagues. It's these conversations with colleagues that are vital, but we simply don't get enough time for them. Most of the new methods I've encountered over the years have not been through organised CPD (ie 'collaboration sessions') within my department, but instead through chance encounters. My first ever blog post was about an alternative method for matrix multiplication that I'd happened to stumble across online.

During my PGCE I was asked to teach a lesson on Highest Common Factor. I remember my mentor showing me the 'Venn Method'. As a result of that conversation I used the Venn Method for years, until by chance a colleague mentioned an alternative. Let's look at that alternative method now, and a number of others. Will you stick with the method you know and love, or will you try something new?

Highest Common Factor and Lowest Common Multiple

I've identified six methods for finding the HCF and LCM of two numbers. I'll explain each method here and identify any pros and cons.

1. Listing

Source: mathx.net

There's no harm in the listing method. It's brilliant in terms of underlying conceptual understanding - students can see exactly what they're trying to achieve here. List all the factors of the two numbers and find the biggest number that's in both lists - that's your HCF. List all the multiples of two numbers and find the smallest number that's in both lists - that's your LCM. Simple! Shame it's so time-consuming. And, in my experience, students sometimes miss factors from their list. One way to avoid this is by listing factors using a pairing method like this:

Factor rainbows are a pretty alternative (see this article from the NCTM).

2. The Venn Method

This is a popular method in the UK. First, we need to do a prime factor breakdown. By the way, if you're teaching prime factorisation then you might like these lovely factor tree activities from Don Steward.

Once you have the prime factors of each number, draw a Venn diagram and place the common factors in the intersection of two sets, as shown in the example below.

tutorvista.com

The HCF is the product of the elements in A intersection B (ie 2 x 2 x 2 x 2 in the example above) and the LCM is the the product of all the elements in A union B (ie 2 x 2 x 2 x 2 x 2 x 2 x 5). Note that UK GCSE students are not yet familiar with this terminology (ie intersection and union), but they will be under the new GCSE syllabus.

Even though I taught the Venn method for years, I'm not a huge fan of it. In my experience, students are ok with filling in the Venn diagram but then they often can't remember which 'bit' is the HCF. If they do remember the method then they probably don't have a clue why it works.

Confusingly, it seems that some people use a different Venn method which involves putting all factors (not prime factors) into a Venn diagram and identifying the highest factor in the intersection (see example below). This is another form of the listing method described above - it's just a different way of organising the list. Let's call this Lenn Method - it's a hybrid of Venn and Listing.

http://edtech2.boisestate.edu/brianroska/506/finalproject/gcf.html

3. Prime Factor Pairing

An alternative to the Venn Method is to do the prime factorisation but then skip the Venn. Write the prime factors of each number out as shown in the example below so it's easy to see which factors appear in both number - the product of these is the HCF. This method is featured in this post by Don Steward.

4. Euclidean Algorithm

I love the subtraction-based Euclidean Algorithm. It sounds complicated but it's incredibly easy. Try a few examples yourself to see how straightforward it is.

The method (including why it works) is explained in James Tanton's video below. I really like this method but for some reason I'm hesitant to use it with students... Would you?

Note that this method doesn't give you the Lowest Common Multiple, but it's easily found once you've got the Highest Common Factor. 

This looks like a pain, but cancelling helps - in the example above I found the HCF of 60 and 84, so to find the LCM I multiply 60 by 84 and divide by the HCF.

5. The Indian Method

I've stopped using the Venn Method at school and now use this instead. I'm not sure it's really called the Indian Method, I'm only calling it that because of this video. At my school we call it the Korean Method because a Korean student introduced it to us! I've found that my students really like it. It's hard to go wrong. It's easy to explore why it works too.

Say we want to find the Highest Common Factor and Lowest Common Multiple of 315 and 420. Write down the two numbers, then (to the left, as in my example above) write down any common factor. I've chosen 5. Now divide 315 and 420 by 5 and write the answers underneath (63 and 84 in this case). Keep repeating this process until the two numbers have no common factors (ie 3 and 4 above). 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 (to find the LCM, look for the L shape).

6. The 'Upside Down Birthday Cake Method'

I should mention this method because I keep spotting it online (video here). The only difference between this and the Indian Method is that here we can only remove prime factors. This seems unnecessary - the Indian Method is quicker. In the example below, why divide by 2 if you spot larger common factors? Why not start by dividing by 4, 6 or 12?

Source: https://uk.pinterest.com/pin/13229392628819982/

So that's it - six methods for finding a HCF and LCM.

Integers
I really like this set of questions from Don Steward. The following question confused one of my brightest Year 10s:

"A person has a rectangular plot of land measuring 8.4m by 5.6m. To survey the number of dandelions they want to divide it equally into the minimum number of square plots. What is the size of each square plot and how many such squares will there be?"

My student's approach to this question was to attempt to find the HCF of 8.4 and 5.6 using the Indian Method. This is what she did:

But she realised that her answer made no sense. 28 can't be the HCF of 8.4 and 5.6. Can you see where she went wrong? Although you can divide 8.4 by 7 (indeed, you can divide 8.4 by anything), it doesn't mean than 7 is a factor of 8.4. The numbers on the left must be factors of the numbers at the top. Non-integers don't have factors. A better approach would have been to convert the measurements to centimetres as shown below. 

Factors vs Multiples

If your students struggle to remember the difference between factors and multiples then they might find this helpful: for factors, think of a factory (where separate parts are put together). For multiples, think of multi-packs eg if cokes are sold in multi-packs of 6 then I can buy 6, 12, 18 etc. These ideas are taken from this resource from the thechalkface.net.

Conclusion

Did you know all of these methods? Will you try something new? Please let me know of any good methods that I've missed.

Even if you decide to stick with the method you're currently using, at least you've now reflected on how you teach this topic. Teachers rarely have the opportunity to pause and reflect.

This post should be read alongside Ed Southall's post Complements #9 LCM and HCF which explains the underlying concepts.

The whole presentation from my workshop is here. In my next post (give me a few days to write it!) I'll cover methods for teaching sequences, linear graphs and surds.