I haven't written a gems post in a long time! I've been too busy at work. So I've got a big list of resources and ideas from Twitter, enough to fill three or four posts. Let's make a start...
1. New GCSE Resources
Teaching new GCSE? Edexcel have come to our rescue by launching a set of resources for new GCSE topics. There's loads of useful stuff here including worked examples, exercises with answers, and extension material. Sixteen topics are covered include functions, iteration, Venn diagrams and geometric progressions.
It's been a long time since I last taught Year 7. Now I've got a low attaining Year 7 class I'm going to start adding lots of new resources to my resource libraries (for example I've recently added place value and addition). With my Year 7 class in mind, resources from mathsframe.co.uk (@Mathsframe) have caught my eye. There's over 200 free resources here - they're aimed at primary schools but secondary teachers may find them useful too.
Craig Barton has been writing a great set of 'Inspect the Spec' posts on TES. If you're teaching new GCSE, you'll find these posts helpful. They contain information about what's in the GCSE specification (what's changed and what's the same) along with recommendations of TES resources.
5. Number Line
Megan Guinan (@MeganGuinan1) first tweeted about this tool ages ago, but it's taken me a while to find a use for it. I recently used the zoomable number line from mathsisfun.com to help explain decimal place value to my Year 7s. Now I realise it will be useful for quite a few topics. My Year 7s were 'wowed' by it and asked me to keep zooming in (forever!). Then they realised it could also zoom out to show really big numbers. It's always great to see kids getting excited by numbers!
Update
In case you missed them, here are my most recent posts:
I've now sold half the tickets for #christmaths16. There's still 60 tickets left but bear in mind that the first 60 tickets sold in only 9 days so I recommend that you book now so you don't miss out! Visit christmaths.co.uk for more information.
I'll leave you with this lovely picture from an old maths textbook. Thanks to @john_overholt for sharing this.
This post explains these changes and provides teaching support and resources.
Set Notation
Students will now be required to represent solutions to inequalities using set notation. This is in addition to representing solutions on number lines and graphs. The OCR specification gives us two examples of set notation:
If you're not familiar with set notation, it's explained here (it's commonly referred to as 'set-builder notation'). Note that either a vertical line or a colon can be used to represent 'such that'.
Image source: coolmath.com
The examples from the OCR specification imply that GCSE students will not need to know symbols representing number types (eg ℤ for integers), and therefore will not be required to express their answers like this:
Image source: mathsinsfun.com
Set notation comes up elsewhere in the new GCSE specification - under the title 'Venn Diagrams and Sets' we have this:
Source: OCR specification
So do we have any resources to practise this? There's plenty of resources relating to set notation and probability in my Data library. I've also made this simple worksheet so students can practise using set notation to represent inequalities.
Find the range of values of x for which x2 - 3x - 10 ≤ 0
If you haven't taught AS level maths or iGCSE then you might not have taught this topic before. There's a few different methods for solving quadratic inequalities. A common method involves sketching a graph of the quadratic function then identifying the required region (see example below). To use this method, our GCSE students will need to be confident in sketching quadratics (this is covered in the new specification). I often find that my Year 12 students make mistakes in their final answer because they don't bother sketching the graph. To me, sketching the graph is essential.
Desmos is a fantastic tool for exploring quadratic inequalities - thanks to Cathal (@CGA_PGS) for sharing this example.
An alternative method for solving quadratic inequalities involves using a number line and test points. In this method students still have to find the critical values, but instead of sketching the graph they check the value of a test point in each region.
Given that this topic has been taught in iGCSE and A level courses for some time, there's surprisingly few resources available. There's a few resources in my Algebra library, including a 'Spot the Mistake' activity I made, but I need more! If you know of anything else then please share it.
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.
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)2is 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.
Hello and welcome to my 26th maths gems - this is where I share teaching ideas and resources I've seen on Twitter. There's a lot going on this week - exciting times for maths teachers! Pi Day is fast approaching and it's a big one this year - 3.14.15. Many of us are busy planning Pi Day activities at school - I've included a few ideas in this post. Speaking of Pi Day, excitement is building for La Salle Education's National Mathematics Teacher Conference next weekend. I have loads of great stuff to share in my workshop - I'm nervous but really looking forward to it! The other big event this week is tonight's inaugural #mathsTLP. This is a collaborative lesson planning session at 7pm on Twitter hosted by @solvemymaths and I. Please join in to share ideas - for details see my previous post.
Things people do on Pi Day - Spiked Math
1. Helpful Exercise Books
I spotted foldable vertical number lines when reading Sarah Hagan's excellent blog Math = Love. I'm definitely going to use these next year - I'll try it out with my new Year 7s in September. They'll stick a vertical number line in the back of their exercise book. The number line can then be unfolded and referred to whenever it's helpful to do so. It's good to get students in the habit of using number lines, and I think that vertical number lines are more helpful than horizontal ones.
Speaking of number lines, this post about Open Number Lines by @mburnsmath is worth a read. I use number lines like this on the board all the time.
Back to exercise books, I was really interested in @Ms_Kmp's post 'Indexed Learning' which is about students numbering every page in their exercise books. The post describes the numerous advantages of page numbers - I particularly like the idea of students creating a topic index at the back of their book.
I particularly like this question involving algebra - find the value of f.
3. Reverse Questioning
I often write about Don Steward's resources - his website is wonderful. Ed (@solvemymaths) arranged for Don Steward to run a workshop at Hudderfield University a couple of weeks ago and I was gutted I couldn't go. Thankfully Ed wrote a very helpful post which included Don's presentations and activities. Hannah (@missradders) also wrote a lovely post about the key things she took away from the workshop.
The 'reversing the question' activity below caused quite a stir on Twitter - it's a fantastic idea. Thanks to @PardoeMary we've managed to find the original source of the problem - the 'Bag of Flour' task was described by Alan Bell as an example of a ‘making up questions’ task in Mathematics Teaching Issue 118.
Fawn Nguyen's post about her experience using this activity with her 6th graders (equivalent to Year 7) is worth reading.
I like all the 'making up questions' material - here's another set of examples from Don's presentation:
Do read Hannah and Ed's posts about the other amazing stuff that Don Steward shared in his workshop.
4. Communication and Vocabulary
Are you following @ExplainingMaths on Twitter? He shares great teaching ideas. I like his sentence stems display which helps students to develop their communication skills.
I also like his idea to get students to create and regularly update their own mathematics dictionaries.
Finally, I loved what he did with the World's Hardest Easy Geometry Problem which I wrote about in Gems 23 - he made an interactive display on the wall for students to have a go at.
5. Pi Day
My school doesn't normally do anything for Pi Day but this year we have a few things planned, including a pie baking competition, a treasure hunt and a Poetry Pi-Cital. The latter involves students writing and reciting a short poem about Pi, inspired by the poem at the start of this video:
Here's a few more Pi Day ideas that you might not have thought of:
Pi recital competition. Prize for the student who recites the most digits of Pi.
‘Tell a maths teacher an interesting fact about Pi’
That's it for today's gems. If you're coming to the conference next week then I look forward to meeting you, either at my workshop or during the Tweet Up (information about both sessions can be found here). Do come and say hi.
Finally, here's some pictures of the fantastic work done by my Year 10s - I love my new circle theorems wall. It was an easy and enjoyable lesson to run - I just gave them paper plates, straws and pins and they got on with it!
I haven't written a gems post in a long time! I've been too busy at work. So I've got a big list of resources and ideas from Twitter, enough to fill three or four posts. Let's make a start...
3. MathsFrame
