5 Maths Gems #160

Welcome to my 160^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Algebra Warm-Ups
@geoffkrall shared a year's worth of algebra warm-ups. These are designed to fit with the US curriculum where students study maths in discrete chunks, which is very different to the approach we take in the UK. In the US, students study a course in Algebra, then Geometry, then Algebra 2. Geoff explains in his blog post that he designed this set of warm ups to help students keep their algebra skills fresh throughout their geometry course. Thank you Geoff!

2. Level 2 Further Maths
Thank you to @tm_maths for sharing an AQA Level 2 Further Maths booklet. This free booklet is fully editable and features examples, questions and challenge tasks.

I've set up a new page for Certificate of Further Maths resources here. This is linked from the main menu at the top of my blog for easy access.

3. New Resources from Dr Austin

Prolific resource maker @draustinmaths has shared loads of fantastic new tasks on her website. This includes tasks on reverse percentages, constructions, set notation, quartiles, cumulative frequency, pie charts, bar charts and vectors.

4. Negative Number Hook

@tessmaths has been running a fantastic Padlet of mathematical hooks for years. One of my favourites is the classic Pythagoras Shortcut hook - I always show my students this picture!

Julia's most recent addition is this brilliant video about the depth of the ocean from MetaBallStudios. This might work well in a lesson on negative numbers.

5. Percentages

Thank you to @nathanday314 for sharing some excellent percentages resources. Check out this thread for more like this, and links to the tasks that inspired these resources. 

Back to School 

These pages are always popular in August and September:

• My Year 7 Maths Activities post which features ideas for first lessons with Year 7
• My Displays page which contains loads of fantastic maths displays for corridors and classrooms

Update

I've created a new page listing maths education conferences in 2022/23. There's not much on it yet, so please let me know of any upcoming conferences that I can add.

Did you see that Rob Eastaway has teamed up with Shakespeare improviser Rebecca MacMillan to run a series of talks for Key Stage 4 students in Birmingham in November? Much Ado About Numbers - Maths & Shakespeare presents a unique opportunity for a joint trip for Maths, English and History departments. I bet it will be brilliant.

I officially finish my Assistant Principal role and become Head of Maths on 1st September. I spent the first week of the holidays getting stuck into Head of Maths stuff, which I thoroughly enjoyed. I made a number of curriculum changes, rewrote our schemes of work, created assessments, planned first lessons, that kind of thing... I've also spent some time planning lessons for Certificate of Further Maths which I'll be teaching for the first time from September. I have a class of 32 excellent Year 11 mathematicians raring to go. After taking a four year break from teaching A level to set up a new school, it's lovely to have calculus back in my life. And trig identities! As much as I've enjoyed focusing on Key Stages 3 and 4 for the last few years, these topics are my happy place.

I'm off on holiday with my family for two weeks tomorrow - we're heading to a cottage in Cornwall. I can't wait. When I get back, I'm looking forward to a couple of chilled weeks at the end of August, including catching up with maths teachers at Dr Frost's triannual drinks.

I'll leave you with this picture tweeted by @p_millerd. I know a lot of people who'll relate to this!

New GCSE: Inequalities

The new GCSE specification has two additions under the heading Inequalities:

a) using set notation to represent solutions

b) solving quadratic inequalities.

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. 

Quadratic Inequalities

Here's a question from OCR's Sample Assessment Materials Higher Paper 5 (non-calculator):

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. 

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.