5 Maths Gems #28

Hello and welcome to my 28^th gems post. This is where I share five teaching ideas I've seen on Twitter.

1. Math Snacks
I spotted a tweet from @fawnpnguyen about the website mathsnacks.com. The website's tagline is 'Smart educational animations, mini-games, and interactive tools that help mid-school learners better understand math concepts'. Check out the animation Atlantean Dodgeball which is all about ratio. The video is clever and funny and the associated resources are very good.



The other animations are also worth a look. Number Rights, in which a passionate fractional activist rises up and demands equity for all numbers, is lovely (if a little bizarre).

2. Election Graphs
Cav (@srcav) wrote a post 'It's election time again' in which he presented some of the terrible graphs that have been distributed as part of the general election campaign. The example below is part of Liberal Democrat MP Greg Mulholland's (@gregmulholland1) campaign. I happened to be teaching graphs to Year 7 this week so I showed them this example. I asked them to identify the errors and discuss why someone would produce such a misleading graph. It was a really good discussion.

Adam Creen (@adamcreen) had a great idea for a related lesson - he produced a Mulholland Graphs activity in which students were asked to produce corrected graphs.

3. Two New Blogs
Stacy Brookes (@Stacy_Maths) has started a lovely new website www.missbrookesmaths.co.uk. She very helpfully writes blog posts featuring recommended resources. Stacy searches the internet so you don't have to! For example if you're planning a lesson on expanding single brackets, ratio or plans, elevations and isometric drawing then you're in luck - she has a post for each of these topics. There's plenty more on her website (and lots still to come!) so do explore.

Miss Norledge's (@MissNorledge) website www.norledgemaths.com is also excellent. Miss Norledge shares loads of great resources and teaching ideas - for example check out this post on Pythagoras' Theorem, this post on completing the square using algebra tiles and this post on multiplication methods. She also does a regular 'Pick of Twitter' post. 

Both websites are fantastic for a resourceaholic like me and I look forward to seeing them grow. Do make sure you're following @Stacy_Maths and @MissNorledge on Twitter.

4. Resources for the New GCSE
Here's an example of a topic that's new to GCSE Maths from September:

We now need to teach 'Fibonacci type sequences, quadratic sequences and simple geometric progressions... and other sequences'. I fear that we are under-resourced here. For some new topics, such as quadratic inequalities and compound and inverse functions, we can use existing iGCSE and A level resources. But some topics currently have very little available so we need to start creating resources. Ed Southall (@solvemymaths) has started the ball rolling by creating these lovely activities - Geometric Sequences Card Sort and Geometric Sequences Worksheet.

I want to start adding resources for new GCSE topics to my libraries so do let me know if you make or find something good.

5. Classroom Practice
I love it when teachers share photos of interesting work their students have done in class. This is what Twitter for teachers is all about - sharing and inspiring. Here's a small selection of student work that caught my eye in the last week.

3D enlargements from @ThetfordMaths

Interesting approach from @mburnsmath's student

Fractional representations from @surreallyno

Number bonds activity from @LttMaths...

...and solution from @BucksburnMaths

Update
I'm ridiculously busy at school at the moment. I teach four exam classes (Year 11, Year 13 and two Year 12 classes) so I'm doing a lot of final exam preparation. If you're in the same boat, you might be interested in my recent post about Higher GCSE Revision Resources. My resource libraries also contain revision resources for both GCSE and A level.

Last week my department had an Inset in which we looked at questions from the new GCSE and discussed our schemes of work. Looking at the Sample Assessment Materials made me realise how much work we all have ahead of us. Some of those new GCSE questions are very challenging. I hope to support teachers in delivering the new GCSE, starting with my recent post about quadratics.

#mathsTLP (Twitter Lesson Planning) continues to go really well (read my post about it here). Lots of teachers are enjoying finding ideas and resources through #mathsTLP - do join in at 7pm on Sundays, all welcome.

Last Friday I attended the UK Blog Awards 2015 (no, that's not my husband in the picture above! Just the Mad Hatter). I had a lovely evening and was incredibly pleased that my blog was Highly Commended in the Individual Education category. I haven't stopped smiling yet! Thank you all for your support.

I'll leave you with this puzzle, which was originally shared by @mathsExplorers back in October: "To solve this multiplication grid, place digits 1 to 9. You have to use each digit once and only once".

Tricks and Tips 3: Quadratics

Last month I presented a workshop at the National Mathematics Teacher Conference (#mathsconf2015) entitled 'Tricks and Tips: Clever Methods for Explaining Mathematical Concepts'. This is the third 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 quadratics, specifically methods for finding a vertex. My previous posts were on methods for finding a highest common factor and methods for sequences, linear graphs and surds.

The vertex of a quadratic graph
This comes up in the new maths GCSE, in questions like this which is taken from the Pearson Edexcel GCSE (9-1) Mathematics Sample Assessment Materials:

Pause for a minute and look at this question, because it's a great example of the change in difficulty in the new GCSE. Note the use of function notation and the term turning point. The equation doesn't factorise. The graph doesn't cross the x-axis. The coordinates of the turning point aren't integers. This is a notable step up from the type of questions asked in current GCSE exams.

For the purpose of this post, let's consider the function y = x² - 6x + 10. There are a number of ways to find the coordinates of the turning point - how would you do it?

1. Vertex Form
The term 'vertex form' is not commonly used in the UK. Vertex form is what you get when you complete the square. So we'd write the function as follows:

y = (x - 3)² + 1

Now we can identify the turning point straight away. I've always explained it a bit like this:

"The (x - 3)²  is squared so it can never be negative. The lowest it can be is zero. It's zero when x is 3. The lowest possible y value is 0 + 1. So we know that the minimum is at (3,1)..."

This explanation is in line with my thought process - it's the way I identify the turning point - but my students really struggle with it. Of all the things I teach, this is the explanation that gets the most blank looks! So this year I tried a different approach when revising this topic with my Year 11s. This time I relied on my students' knowledge of graph transformations. I told them to think of the graph y = (x - 3)² + 1 as a transformation of the graph y = x². It's been translated 3 units right and 1 unit up. The vertex moves from (0,0) to (3,1). They found this approach really easy - it made a lot more sense to them. Suddenly all my students were able to find the vertex of a quadratic function.

As long as students have studied graph transformations then this approach seems to work. This teaching order is worth bearing in mind when designing a Scheme of Work. 

From now on, I'm going to use the transformation method. But there are alternatives...

2. A formula
In some countries, students simply memorise a formula. They learn that the x coordinate of the vertex is -b/2a. They then find the y coordinate by substituting that value into the equation.

By memorising this formula, you can find the coordinates of the turning point of any quadratic function without completing the square. At my conference session I showed the video below - watch it to see how the method is explained. I'm not a fan of this approach. I don't want my students to simply memorise a formula - there's no conceptual understanding here.


3. Differentiation
Differentiation is always a pleasure. We don't do calculus at GCSE, but I thought it worth mentioning here that another method to find a turning point of a function is to set the derivative equal to zero. As you can see below, for a quadratic that will always give us x = -b/2a.

4. Symmetry
For a quadratic that intercepts the x axis, the vertex is the midpoint of the two roots. This works because parabolas are symmetrical. Up until recently I thought this approach wasn't possible for quadratics that don't intercept the x axis, but then I discovered James Tanton's method. It's described below for the function  y = x² + 4x + 5  - for more detail and examples, see this curriculum essay or this video.

If we apply this method to our original example, we rewrite y = x² - 6x + 10 as y = x(x - 6) + 10. We can see that two points on this curve are (0,10) and (6,10), so the vertex has x coordinate 3 by symmetry. Simple!

James Tanton has produced a brilliant pamphlet 'Guide to Everything Quadratic' which is helpful for any maths teacher preparing to teach quadratics for the first time. My Algebra and Core AS resource libraries are packed full of recommended resources for teaching this fantastic topic, such as this activity from Susan Wall.

Preparing for the new GCSE
As I was writing this post it occurred to me that there's a lot of really important things that maths departments need to do this term to prepare for the new GCSE. Writing new Schemes of Work is a huge job, as is finding suitable resources for teaching the new GCSE topics.

CPD for maths teachers is also really important. All maths teachers need to be familiar with the new GCSE content - they need to know what's been added and what's been removed. They need to look at lots of example questions.

The other thing that all teachers need to do now is a subject knowledge check - are there any topics on the new GCSE syllabus that you're not familiar with? This is particularly relevant for teachers who've never taught A level maths. Has everyone in your department thought about how to teach the new GCSE topics? It's time for some vital maths department CPD.

Gem Awards 2015

This is my 100^th blog post! It's also nearly a year since I launched resourceaholic.com. To celebrate this milestone I've been reading back through my gems posts. I've picked out some of my favourite gems, and those that were particularly popular with my readers. They deserve a special mention. So raise a glass of champagne with me, and toast the winners of the first annual Gem Awards!

1. Best Website - Mathigon
There was a buzz about Mathigon when I featured it in Gems 13. It's full of beautiful animations and captivating explanations. The page Polygons and Polyhedra is a good place to start if you want to explore Mathigon. The page on sequences is excellent too. Mathigon also has an amazing mathematical origami section and the 'Alice in Fractalland' slide show is really good. The website is still under construction and I can't wait to see what's coming next.

2. Best Activity - Desmos Polygraph

Desmos' brilliant online graphing calculator has been a staple in my mathematical toolkit for a couple of years now. In Gems 19 I wrote about the launch of Desmos Polygraph. I later tried Polygraph in a lesson with my Year 10s and absolutely loved it - you can read about my experience here. It was one of the best lessons I've taught this year and I can't recommend it enough. Desmos' activities are all very cleverly designed and easy to use. I've been waiting for an opportunity to try out some more Desmos activities - particularly Water Line, which I featured in Gems 13. 

Other activities that have appeared in my gems posts and were shortlisted for this award include the 'blanks' activity from Gems 17, the scavenger hunt from Gems 3 and Dan Meyer's excellent 'Fry's Bank' from Gems 7. They're all worth a look.

3. Creative Lesson Award - jemmapduck.com
@JemmaPDuck wins this award for her end of unit review ideas. In Gems 5, I wrote about her suggestion to show students a page from a boring maths textbook and challenge them to come up with their own better version. I love the idea of students designing their own textbook page complete with explanations, examples and exercises. In Gems 7, I featured another of Jemma's ideas - at the end of a topic, her students create a 'cheat sheet' summarising the topic. She then gives them a test and lets them refer to a cheat sheet during the test, but gives each student someone else’s cheat sheet. After the test, students give each other feedback on how useful their cheat sheet was. I love both of these ideas from Jemma - she's a reflective teacher with loads of creative ideas and her blog is well worth following.

4. Best Display - The Maths Magpie
This award goes to the 'I don't know what to do next' poster from the fabulous @TheMathsMagpie. I featured it in Gems 4 and you can download it from TES.

Honourable mention goes to display expert Clarissa Grandi (@c0mplexnumber) for her amazing set of displays and Ed Southall (@solvemymaths) for his wonderful Mr Men Poster Pack.

5. Best Classroom Equipment - Vertical Whiteboards
If I had unlimited budget to furnish my maths classroom then the walls would be covered in vertical whiteboards. I was hugely inspired by @nathankraft1's post 'Every Math Teacher in the World Should Do This...Right Now!' (featured in Gems 7).

I've written about related ideas including acrylic sheets and large whiteboards for collaborative problem solving.

6. Subject Knowledge Award - Etymology

I wrote about the importance of talking to students about the etymology of maths words in Gems 10. I now spend time doing this in most lessons. It's just a small change to my practice, but I think it's an important one. For example, instead of just telling my Year 12s to write down the title 'Binomial Expansion', I now discuss the meaning and origin of the word Binomial. In doing this I often have to do a bit of research ahead of the lesson, so my subject knowledge is improving all the time. In Gems 23 I shared news of a new Twitter account @etymathology which tweets really interesting maths etymology facts.

7. Best Gadget - The Scale of the Universe
Because I don't have access to a class set of tablets, I don't feature apps for students in my gems posts (with the exception of the great apps I featured in Gems 13 - I often play Make 10 on my phone when I'm on the tube now. It's addictive!). But I do occasionally feature apps for teachers (like Plickers and Quick Key) and animations to show on the Interactive Whiteboard. The winner in this category, featured in Gems 12, is this fantastic tool 'The Scale of the Universe 2'. It's an interactive resource that you could use when teaching standard index form. It allows you to zoom in and out from the tiniest things (quantum foam!) to the biggest things (the Universe) - all measurements are given in standard form.

8. Bright Ideas - Find the Factors
Some things are hard to categorise, but I spot them on Twitter and they capture my imagination. The winner of this category is @IvaSallay's Find the Factors resources. I featured these in Gems 11 - they're 'an excellent way for children and adults to review multiplication facts, use logic, and strengthen brain power'. They are lovely puzzles.

Other bright ideas worth a special mention are Kathryn Forster's Pret Homeworks, which I first featured in Gems 3 and Calligrams, which I featured in Gems 6.

I also liked the Constant Character featured in Gems 22, the vertical Binomial Expansion featured in Gems 27 and Slice the Pie, a fraction activity that I featured in Gems 15.

9. Best Problems - Brilliant
I featured Brilliant.org in Gems 21 and now use it all the time to find interesting problems for my lessons. It find it particularly useful for sourcing starters and extension questions.

Special mention also goes to Ed Southall's set of puzzles on solvemymaths.com. His puzzles are excellent, and often very challenging.

10. Lifetime Achievement - Don Steward
A lifetime achievement award for Don Steward's Median blog is well deserved. I rarely write a post where I don't recommend some of Don's fantastic resources. Don's blog is always my first port of call when I'm planning lessons. He has done amazing things for maths education and his work is appreciated all over the world.

Contributors
I was going to have a 'best tweeters and bloggers' award category but the list quickly became far too long! But I do want to mention that Chris Smith and Ed Southall have probably been the most frequent contributors to my gems posts, so I thank them for continuing to share a huge number of creative ideas and resources. I'd also like to thank all my fellow bloggers and tweeters for keeping me inspired and excited about teaching maths.

Speaking of awards, I'm off to the UK Blog Awards event next week (I'm a finalist in the Individual Education blog category). The frontrunner to win this category is the fantastic ictevangelist.com. I'm looking forward to the quirky Alice in Wonderland themed celebrations. I love the invitation that arrived in the post yesterday, complete with a tiny key for a tiny door!

CPD - Maths teachers lead the way

Maths is a really hard subject to teach. Not only do we have to tackle negative perceptions - maths is difficult, maths is boring, maths is pointless - but we also have to explain some really tricky concepts. We need to develop skills in students such as problem solving, visualisation and reasoning. These skills don't come easily. Teaching maths is a huge challenge, and it's really important to get it right.

Sometimes it's helpful to bounce ideas around with our fellow maths teachers. We all benefit from sharing good practice. Sometimes we need inspiration. Up until recently, there were few opportunities to get this kind of support. But there's a revolution happening in maths education. It's called #mathsTLP.

What is #mathsTLP?

TLP is Twitter Lesson Planning. This was Ed Southall's clever idea - you can read his original post about it here. It takes place every Sunday evening between 7pm and 8pm. Let me explain how it works. If you're looking for an idea or a resource, tweet your request and include the hashtag #mathsTLP. Something like this:

People who are monitoring the #mathsTLP hashtag will see your request and respond to it - like this:

At the end of the chat, Ed (@solvemymaths) collates the resources on his blog.

#mathsTLP isn't just about sharing resources and lesson ideas, it's a forum in which maths teachers talk to each other about teaching maths. What could be better than that?

Is it working?

It's bloody brilliant. We're working together for the sake of our students. I cannot think of a more effective use of social media.

#mathsTLP has been running for 4 weeks now and we've covered a huge range of topics including sequences, inequalities, circle theorems, simultaneous equations and probability. Through #mathsTLP I've connected with lots of new people and have discovered many fantastic new resources. 

For a brand new hashtag, it's getting a lot of tweets (the orange line below represents #mathsTLP, blue and green are the well-established hashtags #mathschat and #mathscpdchat respectively).

Beyond maths?

Subject-specific CPD is considerably more helpful than the sort of general 'teaching and learning' CPD that is often delivered at school Insets. At some schools - mine included - no time or money is allocated to subject-specific learning. Since my NQT year, I haven't received any training that's actually had an impact on my teaching. Twitter is my sole source of development - without it I'd be working in a bubble.

Twitter's maths teacher community is brilliant - its members have really mastered the effective use of Twitter as a platform for collaboration. I wonder whether other subjects are heading the same way. Do other subjects have an equivalent to #mathsTLP, a chat specifically for sharing lesson ideas and resources?

Outside of Twitter, there are initiatives which offer hands-on help, advice, knowledge and inspiration. A fantastic example is The Physics Factory, a 'grassroots network of teachers working to put the fizz back into physics'. Much like those who participate in #mathsTLP, the people behind The Physics Factory recognise that teachers need to work together. They share ideas and resources, they develop subject knowledge. They don't tell teachers how to teach, but they inspire teachers to love what they do.

Subject-specific initiatives are essential. To have an impact in the classroom, teachers need to be in love with their subject, and derive authority from it. Immersion in their subject is vital.

I'm feeling very positive about all this collaboration. If teachers continue to work together like this, we're heading in the right direction. I'm proud to be part of #mathsTLP and I hope other subjects are inspired to follow our lead.