New GCSE: Sequences

I've been busy getting my head around the additional sequences content in the new 9 - 1 Maths GCSE and thought it would be helpful to share my findings. Quite a lot has been added to the higher tier. Previously students covered all the GCSE sequences content at Key Stage 3 so there was nothing new to teach at Key Stage 4. But now we need to include a few weeks' worth of sequences lessons in our GCSE schemes of work.

We've gone from this (from the current Pearson Edexcel Specification - GCSE 2012 Maths A):

...to this (from the new Pearson Edexcel GCSE (9-1) in Mathematics):

All the linear sequences content remains (see my previous post about methods for finding an nth term). The terms arithmetic progression and geometric progression are now used in both the specification and sample assessment materials - terms I normally don't introduce to students until Year 12.

Let's take a look at the new content in detail.

1. "Recognise and use sequences of triangular, square and cube numbers"
There's no specific mention of triangular numbers in the current Key Stage 3 and 4 curricula. I've always covered triangular numbers as an enrichment topic at Key Stage 3 anyway, but now students could be examined on this. It's likely that square, cube and triangular numbers will come up in Venn Diagram questions like in this activity from Flash Maths:

If you're looking for resources for square, cube and triangular number sequences then there's plenty available, like this worksheet from Channel 4 Learning and this lovely activity from Great Maths Teaching Ideas. Stacy Brookes (@Stacy_Maths) has written a very helpful post Special Numbers featuring many more resources.

2. "Recognise and use quadratic sequences... Deduce expressions to calculate the nth term of quadratic sequences."
It wasn't long ago that quadratic sequences were removed from GCSE - now they're back. I'm aware of three approaches for teaching students how to find the nth term of a quadratic sequence:

Pattern Spotting
In a fairly simple sequence, students might be able to make comparisons to known sequences (eg square numbers) to derive the nth term. This is a nice introduction to the topic - I have a worksheet on it here.

Finding the Linear Sequence in the Remainder
For more complicated quadratic sequences, I've always used the method outlined below (source: mathshelper.co.uk).

Finding the Coefficients
Huge thanks to Matt Dunbar (@MathsDunbar) of Trinity Maths for introducing me to a new method. In this method you find the value of the coefficients in the nth term an² + bn + c, where a is half the second difference, c is the zeroth term and b can be calculated using these values.

I've always wondered why we halve the second difference when finding the nth term of a quadratic sequence. Matt Dunbar kindly explained it to me (I feel like I should have worked this out myself!):

There's no shortage of resources for quadratic sequences - big thanks to Dawn (@mrsdenyer) for gathering a large collection here.

3. "Recognise and use geometric sequences"
Foundation and higher tier students will be required to recognise simple geometric progressions - for higher tier students these sequences may involve surds (for example 1, √2, 2, 2√2, …). The sample assessment materials provide two examples of geometric progression questions. This question appeared on AQA Paper 2 for both foundation and higher tier:

This question was from Edexcel Higher Paper 3:

We're fortunate that there's a decent amount of resources available for geometric sequences. Key Stage 5 textbooks and worksheets cover this topic, and a number of resource makers have already started producing activities including Don Steward and Ed Southall. Note that, unlike at A level, students won't have to find the nth term of geometric sequences.

4. "Recognise and use Fibonacci type sequences"
I'm pleased to see Fibonacci on the new GCSE specifications because, like triangular numbers, it should definitely feature in every child's mathematics education. Until now it's always been something we've taught for enrichment.

I was a bit confused by the meaning of 'Fibonacci type sequences' because I thought that there was only one Fibonacci sequence and by definition it is 1, 1, 2, 3, 5, 8, 13, ... . Thankfully Edexcel's sample assessment materials clarify this for us - this question appears on both their foundation and higher sample papers:

5. Recurrence Relations
The OCR specification mentions the use of subscript notation, and we are shown a recurrence relation. The AQA specification also mentions recursive sequences.

I'm a bit worried about how my GCSE students will cope with recurrence relations because my Year 12s often struggle with the notation.

In the OCR sample assessment materials we have this question:

There's not many recurrence relation resources suitable for GCSE students, but A level textbooks may be useful and there's some great stuff from CIMT MEP. In fact the CIMT chapter on sequences covers pretty much everything we need for sequences in the new GCSE. Also, do have a look at AQA's resources for this topic - their recurrence relations activities are great.

6. "And other sequences...!"
I don't like this vagueness in the specification but I believe this refers to, for example, sequences like this:

where the numerator and denominator are separate linear sequences.

---

So, as you can see, sequences has suddenly become a much bigger topic at GCSE. We already have some resources, but I'd love to share more so please let me know about anything you create. I'm starting to build a collection in my algebra library.

I hope this post has helped you figure out how this topic has changed under the new GCSE specifications. Let me know what you think.

5 Maths Gems #34

Hello and welcome to my 34^th gems post. This is where I share five teaching ideas I've seen on Twitter. The summer holidays are finally here! Let's be honest though - although it's lovely to have the opportunity to rest and play, the majority of us do a fair amount of school work over summer - creating resources, organising and tidying, preparing for September, catching up on reading... Have you read all 34 of my gems posts? Now's a good time to start!

1. Magnitude
Standard Form is all about really big numbers and really small numbers. It's nice to put the topic into context - I've written a few times about The Scale of the Universe which is wonderful. If you're looking for a way for your students to interact with this then check out the post 'The beauty of the very very big and the very very small – Standard form' by @mhorley. Thanks also to the lovely @MissBLilley who shared this video which puts size into context - she used it to introduce Standard Form and her students were amazed!



2. Prime Time
@MathedUp shared a nice activity on prime factor decomposition. I really like this because it makes it so clear that every number is either prime or composed of primes.

I love this prime clock from the brilliant Minimal Math Concepts. A few tweeters were unhappy to see the number one on it though!

Whilst on the subject of primes, check out this excellent primes and factors puzzle from @LearningMaths:

@LearningMaths always produces excellent resources. Here's another couple of examples:

Fractions starter by @LearningMaths

Number machines by @LearningMaths

3. Alphabet Lesson
In a Twitter discussion about first lessons of the year, the ever creative @missradders shared an idea for a Maths Alphabet lesson. In this lesson, students come up with the best word they can for each letter. They need to be able to tell you what it means - for example for the letter A they might define add, angle, area, average, acute or adjacent... As Hannah says, "this is a good gauge of prior knowledge/ability ... and gives you time to chat to them". What a lovely idea for a first lesson with Year 7.

If you want to take this idea further, your class could design and produce a maths alphabet poster like the examples shown below. 

Source: http://www.vernalpool.org/treefrog/pop/09F-MPS-MATH.htm

Source: http://www.vernalpool.org/treefrog/pop/04R-B-M.htm

Last year I wrote a post 'Your First Maths Lesson with Year 7' which has lots more ideas for September.

4. A Level Prompts
@dannytybrown has been busy developing a series of pre-teaching prompts for A level topics. These are designed to be given to students a few days before a topic is taught. They are intended to prompt class discussion and potentially reduce the variation in prior knowledge amongst students. A few examples are shown below. Do keep an eye on Danny's blog squeaktime.com for more on this.

 5. Maths with Pizzazz
Thank you to @japleen_kaur1 for sharing a set of resources I've never seen before - Maths with Pizzazz worksheets. Hundreds of topics are covered. Many of the activities involve practising skills and using answers to work out riddles and jokes. Here's a couple of extracts:

Extract from Page 211 - Algebra with Pizzazz!

Extract from Page 159 - Pre-Algebra with Pizzazz!

I could spend days looking through these resources! Thankfully there's an index. What I really need to do is split the PDFs into separate worksheets, labelled and grouped by topic. But that would take weeks. Any volunteers?!

Recommended Reading
This summer @DrBennison and @MissNorledge are undertaking an impressive #summerblogchallenge. They intend to write a post every day, so keep an eye on their blogs this summer - blog.ifem.co.uk and storeroom.norledgemaths.com.

Here's a few recent posts that I recommend:

• @solvemymaths wrote a wonderful, inspiring post '10 reasons why it's a great time to be a maths teacher'. Read it, you'll love it.
• @mrlyonsmaths wrote an excellent post 'The pain of problem solving...' in which he shared examples of students' work using his problem solving template. It's well worth a read.
• @jennypeek wrote a lovely post "Well whistle me Dixie, the answer is Pi!" about maths in the movies.
• @christianp published an article via @aperiodical about the 30 second arithmetic challenge. He's built a brilliant online challenge - see how many you can do! I find it really hard doing mental arithmetic against the clock, but the more I do the better I get. I'm already addicted, which is good - it will keep my brain active over summer!
• I enjoyed @ColleenYoung's post about Arithmagons. They're such a versatile activity. I recently spotted this great spin on Arithmagons from MEI:

Update

I have a long list of things I want to add to resourceaholic.com this summer, but in the last couple of weeks I've been preoccupied. I'll be running a workshop on A level resources at #mathsconf5 in Sheffield in September - I've been doing a lot of research in preparation for this. Along the way I've added lots of excellent new resources to my A level resource libraries. My presentation is nearly finished and I'm really pleased with it - if you teach A level and you're coming to the conference then do come to my workshop to see what I've discovered.

My two most recent blog posts both feature in my five most popular posts ever - one was about resources for end of term lessons, the other shared advice for teaching low attaining students.

My youngest daughter Hettie and I had the pleasure of meeting Rob Smith (@RJS2212) yesterday. Rob has spent the first week of the summer holidays travelling the length of the country to visit as many tweeting maths teachers as he can - check out #48hourmathstour to see what he's been up to.

Finally, look what I've got - Mathematical Mr Men stickers! If you've not seen the 50 Mathematical Mr Men before then do check out them out on solvemymaths.com. They're awesome.

Words of Wisdom: Teaching Foundation GCSE

As expected, the last #mathsTLP of the school year was a bit quieter than usual. As host, one of my jobs is to keep the chat flowing. So I filled a gap in conversation with a question:

I hadn't been planning to ask this question - it was just the first thing that came to mind at a moment when I wanted to get people talking. But then something quite extraordinary happened. Over the few days that followed, I was inundated with helpful advice - I received over a hundred replies to my question. These words of wisdom come from experienced teachers. Teachers who've taught low attainers, cared about them, believed in them. Teachers who've done everything they can to give these students opportunities in life, who've put huge amounts of effort into supporting them. I admire these teachers. I want to be one of these teachers.

"Throw away the handbook" ~ @solvemymaths

I received so much advice that it was all a bit overwhelming. I went back through every notification and made notes. It would be wrong not to share some of the ideas here. After all, I'm not the only teacher who'll be teaching low attainers for the first time next year. I hope that others will also benefit from all this good advice.

1. Building relationships

"Make them believe in themselves" ~ @FKRitson

There were lots of replies about how to build self-esteem, confidence and trust. Advice included:

• "Friday biscuits or a cup of tea works wonders. Instill confidence & let them know you believe in them. With one of my groups we took a lesson off to design and decorate our own mugs. Best quote 'Miss, you've got a teapot! That's like a dead posh restaurant'".  ~ ‏@Ms_Kmp. "I was amazed at the impact a cup of tea had for one young man. Came for help & a cuppa every Wed!" ~ ‏@takepi21
• "Instilling confidence is the key. Make the lessons non-threatening (they are in the set because they "fear" maths).... Make it different to the diet they've been served thus far; try stuff, it will breed confidence." ~ ‏@andylutwyche 
• "Your sales pitch is that it's 'us against the exam', 'we can beat this'. Their self belief in maths will be low, they will be v sensitive to feeling patronised". ~ ‏@BodilUK 
• "Give them lots of evidence to show they're getting C grade work right. You're not promising it as a grade, but they need possibility." ~ @adamcreen 
• "Lots of repetition! They'll prob be low in confidence so build this up. Word things in ways that don't belittle them. ... If they don't get something, then "maybe I didn't explain it very well..." It helps their confidence." ~ @WelshReady 

2. Pick your battles

"As Napoleon said 'If you can't be strong everywhere, pick a place and be strong there.'"  ~ ‏@MrMattock

A number of replies were about choosing which topics to teach. I won't be able to cover the whole syllabus - far from it - so I should focus on a limited number of key topics.

• "If you can give them a quick win … that is always good. Index laws are nice as they are legitimately C-Grade but easy as hell! ... They won't get good at it all, so decide what you want them to understand and reinforce that." ~ ‏@MrMattock 
• "Don't be picky either, they might get Pythag but not directed numbers." ~ ‏@DJUdall 
• "Don't be bound by the grade of a topic - expanding brackets is easier than many E grade topics! … Focus on topics with visual cues and predictable processes, such as transformations or scatter graphs." ~  ‏@danicquinn 
• "When gearing up for the exam, you may want to condense your curriculum into 'hot topics'. Keep it focussed. ... We picked out the favourite 30 or so question types from exams over the last 5 years." ~ @jase_wanner 
• "Doubt you'll be able to get through everything. Pick most important bits and teach them well" ~ @Teachanana

Many teachers told me that numeracy is key. I will need to have "a relentless focus on core numeracy skills" (@ThainMike).

"Numeracy skills and times tables. Again, again and then some more!" ~ @RandomNumeracy

3. Memorisation

"Go for memorisation through every memorisation technique known to man" ~ @solvemymaths

For the vast majority of our students, conceptual understanding is vital. Teaching maths 'properly' is a maths teacher's raison d'être. But these are exceptional circumstances. Give a low attaining student the opportunity to get a C in GCSE Maths - by any means necessary - and you've opened a window of opportunity in a young person's life.

• "I find repetition helps. Do things quickly but repeat often". ~ ‏@MissMathsDla
• "Bring variety to memorisation: flash cards, memory king (one stands up and gets drum roll before reciting first 10 primes, etc)... " ~ @danicquinn
• "Most of their problems are issues with fluency so they can't *get* C/D grade topics when modelled. Remedial work on number is essential for quite a while at the start of the year. ... I always start with first 15 squares, first 5 cubes, and roots/ cube roots. Make it compulsory to be able to recite them in 10 seconds within a week & time/ game them. ~ @StuartLock
• "Find interesting ways to reinforce key number concepts; that will be their most serious need" ~ @MrMattock
• "You want silly dances, phrases, gesticulations, daft images to associate...everything". ~ @solvemymaths
• "YMCA becomes y = mx + c... angle sizes memorised by arm actions... and the circle song is a must. Diddle, the median' s the middle" ~ @7wittermiss.

A number of people recommended Sticky Studying for memorisation techniques.

4. Resources

"Never let them let go of the basics. Drill them in every lesson". ~ @DJUdall

Many teachers recommended resources. The mostly common recommendation was Numeracy 5-a-day from Corbettmaths.

Times Table Rockstars was also recommended. @MrReddyMaths suggested adapting it by teaching the tables upfront each week and cutting out the music - "Emphasise 'competition' with themselves by highlighting how much they're improving over the week".

Mathsbox was recommended, as were Andy Lutwyche's resources on TES, Bread and Butter resources from Just Maths and Essential Skills from MathedUp!.

Thanks to @tessmaths for sending me a set of colourful Foundation Revision Cards from the MA - these will be very helpful.

@DIRT_expert suggested using Cumulative Starters to build up skills and tackle fear. Students are asked the same questions at the start of each lesson but more questions are added as they learn new content.

5. More ideas

"You mightn't enjoy lessons at times; you'll enjoy the intellectual challenge it'll give you" ~ @siobhanorb

Here are some more things I need to think about in my planning:

• I need to find out about the literacy levels of my students and their attendance. 
• A few people suggested entering them for Higher GCSE rather than Foundation (there are mixed views on this). 
• @ColleenYoung suggested I do lots of short mini-tests with immediate feedback. 
• ‏@danicquinn suggested mini mocks, where you first work together on a "mock mock" (same questions, different numbers) before the real one.
• @NEdge9 talked about the importance of students knowing the meaning of key words. She said that sometimes they can do the maths but can't remember what the words mean. 
• @missdenzil said that lessons should comprise small tasks, lots of different activities, rewards, competition, whiteboard work, and mastering basics.

Summary

@MrBenWard summarised the key messages nicely in this tweet: 

When I first meet my new class in September, no doubt some things will take me by surprise. But the advice I received this week has got my head buzzing with ideas and I feel ready - dare I say excited - to get started with this class.

Huge thanks to all who tweeted and emailed me ideas and resources. In order to keep this post a reasonable length, I haven't been able to include all the replies I received, but every single piece of advice was valuable and will be factored into my planning for next year. Thanks also to @ColinTheMathmo for helping me collate and organise the tweets. 

I'm so grateful that I can rely on the maths teacher Twitter community to support me throughout the year. A challenge lies ahead, but I've got expert advisers at hand to see me through it. 

End of term resources

It can be hard to get on with teaching new topics when there are lots of end of term events disrupting the timetable. This post lists resources you might find useful.

Vocabulary Activities
I love these mathematical crosswords for developing students' knowledge of mathematical vocabulary. There's a wide range of crosswords to try (maths teachers will enjoy the 'branches of mathematics' crossword on on page 34).

Mathematics Instruments
and Equipment Crossword

Teacher Resources Online has loads of great activities including these vocabulary exercises, some of which would work well in an IT Room or with iPads. 

I like this polygon word search too.

There are more word searches here.

Crossums and Number Searches

Sticking with puzzles, there's loads of crossums and number searches here, including the lovely Number Sets Search pictured below. I gave this to my Year 8 class this week (we've just done some work on basic sets and Venns). I also gave them this excellent Number Types activity from Churchill Maths and we did this interactive whiteboard Drag and Drop Venn diagram activity from flashmaths.co.uk.

Cryptography

There are loads of great cryptography resources at Teacher Resources Online and CIMT. The Imitation Game is now available on DVD if you're looking to make cryptography a theme in the last week of term.

Quizzical
There's a nice set of maths quizzes from Pinnacle here, including three 'silly quizzes' at the end.

Magic Squares
Don Steward has a magic squares blog with all the magic squares you could ever need.

Logic and Lateral Thinking
CIMT has lots of logic resources. Maths lessons are probably the only place where students have an opportunity to develop the art of logical thinking - it's a shame we don't do more of this.

Extract from Plane Passengers (CIMT)

I've also put a load of brainteasers in this dropbox folder.

Investigations
Now is a good time of year for students to get stuck into an investigation. This dropbox folder has a large selection of investigations, including those from Nuffield and Bowland.

Problem Solving
On my problem solving page I list links to lots of problem solving resources.

Research
Teachit Maths has a lovely selection of end of term resources here. I particularly like the Random Mathematics Research Quiz and the Greatest Mathematics Country Debate, both of which require students to do internet research.


Durham Maths Mysteries
Thanks to @pwp1303 for pointing me in the direction of these lovely resources. If you want to keep doing topic-based maths in the last week of term then try these activities.

Relay Races and Team Challenges
Chris Smith (@aap03102) has created some excellent resources that would work well at the end of term. Check out his Maths Relay Races and Maths Group Challenges.

More ideas

• If you're looking to get outside the classroom on a sunny day,  try the Maths Scavenger Hunt from Transum. There's loads more lovely activities on the Transum website to explore.
• My students love these riddles! 
• Find the Factors puzzles are excellent (it's always worthwhile practising times tables). 
• A game of Countdown always goes down well. Put a twist on it with Algebra Countdown from @stephenbodman.
• Memory Maths is one of my favourites - get students to draw the grids on mini-whiteboards.
• My colleagues have enjoyed playing the Lobster Game and the Pirate Game with their classes this week.
• Quadratic colouring is a great activity for students who know how to plot quadratic graphs.
• Why not teach your students Binary? See my post about teaching this topic. 
• Don't forget the videos on Numberphile. It's only two minutes long, but one of my favourites is the delightful video '1 - 200 on Google Image Search'.
• I shared some ideas for lessons on Fermat and Pascal in my post 'End of term!' last year. 

I hope this has been helpful. There's lots to choose from. Enjoy the last week of term!