5 Maths Gems #87

Welcome to my 87^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Regular Recall
Starting a lesson with mixed topic questions seems to be fairly common practice these days. Many teachers use resources like Corbett Maths 5 a Day. Taking this idea a step further, some teachers on Twitter have recently shared examples of tasks where students are asked a question on what they did last lesson, last week, last term and last year. Here's a great example from @MissBanksMaths.

And here's an earlier example from @JonONeillMaths, who was inspired by the original idea in @87History's blog post about retrieval challenge practice grids.

This is one of those rare ideas that works well in most subjects. Here's an example for physics by @alexpboulton.

2. GCSE Papers
I've blogged about CrashMaths (@crashMATHS_CM) before. Their website is a good source of practice exam papers for both GCSE and the new A level. They've recently added a new set of Edexcel-style Higher GCSE papers (Set B), which includes this nice non-calculator question:

They've also recently added a couple of Higher AQA-style papers to Set A and some helpful GCSE worksheets.

Another useful set of resources on CrashMaths is for the Edexcel A level large data set. This includes an information and guidance document for students and six practice questions.

3. Goal Free Problems
If you've read Craig Barton's book 'How I Wish I'd Taught Maths', you'll already be familiar with Goal Free Problems. In Chapter 4, which is all about focusing thinking, Craig explains that while most exam questions are goal-specific, he now makes use of goal-free problems in the early knowledge acquisition phase. He also uses goal-free exam papers to kick-start the revision process. If you haven't already done so, read Craig's book to see some examples of these problems and to fully understand the goal-free effect and how it relates to Cognitive Load Theory.

Thanks to @MrMattock who has now created a free website - goalfreeproblems.blogspot.co.uk - which shares a large number of these problems for both Higher and Foundation tier. In each of these exam questions, the actual question has been removed and replaced with the words 'Work out what you can from this information'.

4. What Went Wrong
Thanks to Year 6 teacher @MrBoothY6 who has shared a large collection of common maths misconceptions on TES:

Over last couple of years I have used 'What Went Wrong?' questions in maths to encourage reasoning & address misconceptions. This year I collated them for every big maths objective except geometry. There's over 90 questions here for free if you want them: https://t.co/m2skd3Mo5x pic.twitter.com/tcljYQrRzJ

— Ashley Booth (@MrBoothY6) 9 April 2018

See my misconceptions page for more resources relating to common misconceptions.

5.  GCSE Revision
@AccessMaths has been busy making resources - his latest revision resource 'Progressive Overload' covers a number of key algebra skills and works well printed on A3. @podroberts helpfully worked out the answers too!

Also check out his new 'Fill in the Blanks' graph revision resources.

For more GCSE revision resources check out my GCSE 9-1 Revision post. I also have an A level revision post for the legacy specification.

Update
My 300th blog post was 'New GCSE: Bounds' - in this post I took a close look at the GCSE specification and resources for this topic. Before that I wrote 'BCME 9 Reflections' which included slides from my recent workshop 'Ideas that transformed my teaching'. Next week is the fourth anniversary of my blog, which means it's time for my annual gem awards!

Here's some other news that you might have missed:

• MEI have archived all of their Monthly Maths and M⁴ magazines and categorised their classroom resources by GCSE topic here.
• Tickets for the JustMaths conference are on sale. It takes place at Alton Towers on Monday 25th June (I'm looking forward to presenting at this one!).
• I've added a few new events to my conferences page - including Don Steward presenting at an MA/ATM event in London in June.
• I love MathsPad resources and this excellent new similarity proof resource is no exception! I've added it to my resource library.
• For legacy further maths revision materials, check out drolivermathematics.com. Thanks to @gismaths for sharing this website with me.
• If you teach A level maths in the London area, do take your students along to the IMA 16+ Lectures on 28th June at UCL - the programme is fantastic. 
• If you're an MA member and willing to help out on the MA bookstand at #mathsconf15 in Manchester on 23rd June, please get in touch.
• Thank you to Hannah Fry for sharing @oliviawalch's wonderfully illustrated "Some Myths about Math". 

I'll leave you with this excellent maths joke, created by @treemaiden and illustrated by @aap03102. Have a great week!

New GCSE: Bounds

I like the bounds content in the new GCSE. I think that the introduction of error intervals has provided some clarity. Previously some students just couldn't get their head around why we use 3.65 as the upper bound for 3.6 ("but Miss, 3.65 rounds to 3.7. This doesn't make sense!"). The use of inequality symbols (in conjunction with instruction using number lines) really helps with conceptual understanding here.

3.55 ≤  x < 3.65

Of course error intervals aren't new - they just weren't previously assessed at GCSE. Upper bounds and error intervals are clearly explained in this extract from the CIMT MEP textbook chapter on Estimation and Approximation from the late 1990s:

Truncation

I also like the inclusion of truncation in the new GCSE specification, because it means that students need to think more carefully before answering a bounds question. Previously, bounds questions at GCSE were predictable, and students only really needed a superficial, procedural understanding of the topic. Things have changed.

Age is a good way to teach truncation, given that students will already be very used to truncating their own age, rather than rounding it to the nearest integer.

Ed is 36 years old. His age is represented by x. Give the error interval for x.

 36 ≤  x < 37

Jemma Sherwood has written a helpful post about truncation resources.

Discrete Bounds

My Year 11s really struggled with a bounds question in their mock exams this year. It's from a recent AQA paper so I can't share the actual question here, but this question is similar:

Two integers are rounded to the nearest 10. The rounded numbers are added to give to 40. What's the maximum total value of the orginal numbers?

Let's say that both numbers were 20 when rounded. The maximum each number could be is 24, so the highest possible total is 48. The common mistake here is to take the upper bound to be 25, giving an incorrect final answer of 50. So why was this such a common mistake? I expect it's because most of us teach bounds in a context of measurement, not counting. We spend a lot of time on continuous bounds, and very little (if any) time on discrete bounds.

Interestingly, my GCSE textbook doesn't contain a single question on discrete bounds. A bit of internet searching shows that BBC Bitesize has one question on discrete bounds:

The number of people on a bus is given as 50, correct to the nearest 10. What is the lowest and highest possible number of people on the bus?

The OCR Check-In Test on Approximation and Estimation has this question:

 Explain why the error interval of 400 cars to the nearest 50 cars could be written as  375 ≤  c ≤ 424 or 375 ≤  c < 425.

and the AQA Rounding Topic Test has a couple of discrete bounds questions, including this one:

Two performances of a show are each attended by 175 people, to the nearest 5. Work out the maximum possible difference between the numbers of people attending.

It's worth noting that the Government's GCSE subject content doesn't specifically refer to discrete bounds. This is the official content for bounds at GCSE:

"15. round numbers and measures to an appropriate degree of accuracy (e.g. to a specified number of decimal places or significant figures); use inequality notation to specify simple error intervals due to truncation or rounding

16. apply and interpret limits of accuracy, including upper and lower bounds"

OCR's specification does mention discrete bounds though, saying that students must "Understand the difference between bounds of discrete and continuous quantities".

AQA's excellent Teaching Guidance has this:

"Upper bounds do not necessarily require use of recurring decimals. For example, if the answer to the nearest integer is 7, the maximum could be given as 7.5, 7.49.... , or 7.49.

If this value of 7 represented £7, £7.49 would be expected for the maximum.

For continuous variables, students may be asked for the lower and upper limits rather than the minimum and maximum values."

The money example here is interesting - it's not something I have explicitly covered with my students. Don Steward has a good bounds exercise including money questions - I'll  make sure I use this next time I teach this topic.

Edexcel doesn't specifically refer to discrete bounds in either their specification or supporting material, but of course that doesn't necessarily mean they won't come up in an Edexcel exam.

Exam Questions

Bounds questions involving calculations can be fairly challenging for students, particularly as sometimes it's not immediately obvious that the question involves bounds. Here's an example from an old Edexcel Linked Pair paper:

"Sian is driving on a motorway.
Sian drives for 2.8 miles, correct to the nearest tenth of a mile.
It takes her 200 seconds, correct to the nearest 5 seconds.
The average speed limit on this part of the motorway is 50 miles per hour.
Did Sian drive at a speed within the average speed limit? You must explain your answer."

Another type of challenging GCSE question is one that says "by considering bounds, work out the value of x to a suitable degree of accuracy, justifying your answer". For examples of questions like this, see Dr Frost's 'Full Coverage: Bounds' resource which has an example of every different question type from past Edexcel papers.

Resources

I've already mentioned Don's excellent resource, the CIMT resources and Dr Frost's full coverage GCSE questions. For a comprehensive list of bounds resources, see my resources library.

It's worth noting that John Corbett has helpful videos, textbook exercises and practice questions on limits of accuracy and limits of accuracy: applying. And Edexcel's new content resources include a helpful worksheet on bounds.

I'm doing a revision lesson on bounds with my Year 11s on Monday and will be using this excellent bounds GCSE revision resource from Maths4Everyone. 

For Interest

Finally, in my research into how mathematics was taught in the 18th, 19th and 20th centuries, I've found few references to rounding. However, in 'Practical Mathematics for All' (McKay, 1942) I found this nice explanation of 'limits of error':

If anyone knows of any earlier references to upper and lower bounds, I'd love to see them.

Thanks for reading! This was my 300th blog post, so a bit of a milestone for me. To read other posts about teaching specific GCSE topics, you can view my topic collection here.

Five things you might not know about the new GCSE content #2

I recently wrote a post about five changes to the content of maths GCSE. In today's post I list five more changes that may not be widely known.

1. Invariance

When teaching shape transformations to Higher Tier students, you'll now need to ensure that students are be able to identify invariant points. This is described in the specification as follows: "Describe the changes and invariance achieved by combinations of rotations, reflections and translations".

We are given this example question from AQA:

"Write down the coordinates of a point that is invariant when it is reflected in the line y = x".

Here, students will need to know that the points (1,1), (2,2) etc lie on the line y = x and would therefore be unchanged if reflected in this line.

There's a set of resources for this topic in the Mind the Gap Maths Toolbox and Peter Mattock has shared a resource 'Invariance activity sheet'.

2. Geometric Sequences
When I saw that geometric sequences now appear on the GCSE specification I assumed I'd be making use of my AS level (C2) resources. This is not the case.

In C2 we find the nth term of a geometric sequence using the formula U_n = ar^n-1.  This gives us a 'position-to-term' rule. For example if we have the sequence

3,  6,  12,  24,  48, ... 

then we have a = 3 (the first term) and r = 2 (the common ratio), so the nth term is U_n = 3(2^n-1).

Although geometric sequences do come up at GCSE, there's two points from the specification that are worth noting when teaching this content:

1. Geometric sequences will be in the form rⁿ , for example:

2,  4,  8,  16,  ...  

3,  9,  27,  81, ...

√5,  5,  5√5,  25,  ...  (Higher Tier only)

2. Students are not required to find expressions for the nth term of these sequences. They only have to do that for linear and quadratic sequences.

Examples of new GCSE questions include:

The nth term of a sequence is (√3)ⁿ. What is the 5^th term of this sequence?

and...

U_n = 2U_n-1     U₁ = 2     Write down the first four terms of this sequence.

In both examples students are using an nth term formula but not deriving the formula themselves. So when you teach this topic I suggest that you focus on:

• recognising geometric sequences
• finding a common ratio and using this to continue a sequence
• substituting into both position-to-term and term-to-term rules (including those using subscript notation). 

If you've spotted this question in OCR's Practice Paper 5 you may think that it contradicts what I've written here about geometric sequences:

Here is a sequence.

2,  2√7,  14,  14√7,  ... 

a) Work out the next term   (1)

b) Find the nth term   (3)

c) Find the value of the 21st term divided by the 17th term.   (2)

I asked OCR for clarification and they sent a helpful reply:

"...the Assessment Objectives around making deductions, inferences and problem solving mean that some questions may involve taking known elements of content and taking them that bit further, as seen in the question. Given that students at Higher tier have to know how to find the formula for the nth term of a quadratic sequence and also to 'recognise and use sequences of... geometric progressions (r^n where n is a... surd) and other sequences', the question is an example of how higher ability students might be expected to make a deduction from known content. I don't expect this to be a question we'd regularly examine in live assessment, but is the sort of thing we'd want to include in sample assessment to give an indication of how the content and Assessment Objectives are brought together in writing questions."

3. Scatter Graphs

There's a bit more to scatter graphs now, though some teachers may have already been covering these things. Students will need to know:

• that correlation does not necessarily indicate causation
• the difference between interpolation and extrapolation, and the dangers of extrapolation.

Although I will use the words interpolate and extrapolate in class, I doubt these words will be used in exams. Questions are likely to be in the form of this example from Edexcel:

You should not use a line of best fit to predict the number of units of electricity used for heating when the outside temperature is 30°C. Give one reason why. 

Students may also have to identify an outlier from a scatter graph (note that this is informal identification of outliers - ie by eye) and decide whether to ignore it when drawing a line of best fit.

The Scatter Graphs: True or False activity from MathsPad covers all of this content and is absolutely excellent.

Don't forget that some statistics content has been removed from the new GCSE, such as questionnaires and stratified sampling. It's worth looking at these exam questions and this OCR Check-In Test to get an idea of the new sampling content.

4. SUVAT

The early versions of the DfE’s new GCSE maths specification contained content relating to the 'suvat equations', but this content was removed from the final published draft. However the formulae were retained in the appendix (in the section entitled 'Formulae that candidates should be able to use, but need not memorise. These can be given in the exam, either in the relevant question, or in a list from which candidates select and apply as appropriate').

This means that the exam board might include these formulae in a question, but this will be no different to how a student would be expected to work with any formula or equation provided (for example, students may have to substitute into or rearrange a suvat formula).

In short: you don't need to teach this topic at GCSE in the way you would teach it in M1 at A level, but you might find it useful to use these formulae in your teaching of algebra - for example using this great resource from Christine Norledge.

5. Don't follow the textbooks... yet!
I've said before that if you plan to invest in textbooks for the new GCSE then it's best to wait for the second editions. I've already found a few inconsistencies between the specifications and first edition textbooks. For example a helpful conversation with @STABMaths confirmed that an Edexcel textbook contains graph stretches, as does a sample paper, but these won't be examined (as stated in my previous post).

Also, in my post about real life graphs, I said that I was surprised to see sketching cubics in my new GCSE textbook - including identifying roots from factorised expressions.

Cubic graphs from Edexcel GCSE (9 - 1) Mathematics: Higher Student Book 

This is what the specification says about cubic graphs:

recognise, sketch and interpret graphs of ... simple cubic functions ...

This is on both the Foundation and Higher Tier. The AQA Teaching Guidance provides further details, stating that students should be able to:

draw, sketch, recognise and interpret graphs of the form y = x³ + k where k is an integer

There's certainly no mention of the type of cubic graphs that feature in my textbook, and I think this is one of the things that will probably be removed from subsequent editions.

A few more clarifications
I asked OCR what they are frequently asked about the new GCSE. In addition to questions regarding grading and teaching time, they are often asked to clarify the following:

• whether students will be expected to differentiate to find the gradient of a curve (no, they will have to estimate gradient from a tangent at a point, or potentially a chord between two suitable points) - see my blog post about this. 
• whether Foundation Tier students are required to calculate turning points of quadratic graphs (no, at Foundation they’ll just need to read from the graph. At Higher they will find turning points by completing the square).
• if Higher Tier students have to know about the equations for any circle (no, only those centred around the origin).

This is really helpful information from OCR.

So that's it - I hope you're now feeling more informed about the new GCSE content. Do let me know your thoughts and questions.

Multiple Choice Questions

Multiple choice questions (MCQs) have always been popular in maths in some parts of the world. In recent years they have gained popularity in England and Wales, and are now commonly used as a diagnostic tool in the classroom. MCQs are also now a key feature of AQA's GCSE papers.

Some teachers have been asking where they can find MCQs that are aligned with our curriculum. I've listed some sources below - if you have anything to add, please email me.

1. Foundation and Higher GCSE Multiple Choice Questions by Topic
Suitable for Key Stage 3 and 4.
I found these multiple choice quizzes at work - I believe they were written by Julie Bolter, probably to accompany a textbook. There are quizzes for a number of topics including sequences, standard form and coordinates.

2. Foundation and Higher GCSE Multiple Choice Revision Questions
Suitable for Key Stage 3 and 4.
There's a large selection of multiple choice questions here, both in PowerPoint and Word. The topics are mixed so these would be useful for exam revision or retrieval practice. Thanks to Mitesh Patel for sending me these. We don't know the original source so please tweet or email me if you know.

3. Diagnostic Questions
Suitable for Key Stages 1 - 5.
This fantastic website from Craig Barton has a huge range of multiple choice questions, contributed by users. You can quickly create quizzes for use with mini-whiteboards, voting cards or Plickers. The most powerful aspect of this website is the opportunity to set students quizzes online - this allows teachers to read explanations and therefore analyse students' thinking.

It's worth checking out the ready made quiz collections including quizzes from White Rose, UKMT, Edexcel, AQA and OCR.

Grace Horne, who is Head of Maths at Kettering Science Academy, has sent me her amazing set of multiple choice quizzes. She's made 85 consistently formatted quizzes, each containing five questions drawn from diagnosticquestions.com. Quizzes are categorised by topic and GCSE tier. Answers are provided. This is a very helpful resource! Huge thanks to Grace for sharing this.

4. SQA
Suitable for Key Stage 5.

The Scottish Qualifications Authority used to set Objective Tests for Higher Mathematics. These contain lovely questions for Year 12 and 13, covering topics such as calculus, trigonometry, sequences, vectors and the equation of a circle. Thanks to Helen Dey for telling me about these. Thanks also to @mrallanmaths for sharing this editable bank of Objective Test questions.

5. Integral

Suitable for Key Stage 5.
MEI provides a great selection of multiple choice tests for A level Maths and Further Maths on Integral (school login required). The example below is from a Normal Distribution assessment for Edexcel S1.

6. 30 Number Starters
Suitable for Key Stage 3 and 4.
Ben Cooper (@bcoops_online) on TES provides a PowerPoint of 30 multiple choice quizzes for various number topics including fractions, decimals and percentages. 

7. Functional MCQs
Suitable for Key Stage 2 and 3.
Thanks to Jon Treby for this set of Functional Multiple Choice Questions. They are split by topic and all questions are contextual.

8. Foundation of Advanced Mathematics 

Suitable for Key Stage 4.

Stuart Price (@sxpmaths) wrote a post about using multiple choice questions from Foundation of Advanced Mathematics exams for Year 10 revision competitions. There's a number of past papers to draw questions from - most of the topics are from GCSE and the questions are suitable for extension in Year 10 and 11.

There's lots to explore here! If I've missed anything, please email it to resourceaholic@gmail.com. Thank you!

New GCSE: Tangents and Areas

At first glance it appears that calculus features in the new GCSE specification. On closer inspection it turns out that our students will find the gradient of a curve by drawing a suitable tangent rather than by differentiating. And instead of integrating, students will use the trapezium rule (or similar) to find the area under a curve. So calculus remains reserved for Key Stage 5, but our students will now be better prepared for calculus when they first meet it. GCSE will have given them a conceptual understanding of rate of change and an ability to interpret this contextually.

This post talks you through this new GCSE topic - it tells you what you need to teach and provides links to resources.

Specification and Exam Questions
Here's the relevant extracts from the new GCSE specification:

This all becomes clearer when we look at example exam questions. Huge thanks to Tom Bennison (@DrBennison) for doing the hard work for me here - he's been through the sample assessment materials for all exam boards and collated the relevant questions in this Subject Knowledge Check. In a typical question, students are given a velocity-time graph and asked to find the total distance travelled and/or the acceleration at a specific time.

AQA helpfully provides additional clarification about the specification in their teacher guide. The extract below is from GCSE Mathematics (8300) Teaching Guidance (available to All About Maths members) which provides a number of additional example questions.

Methods
The diagrams below are taken from this extract from a new GCSE textbook which sets out the standard methods that our students will use.

Students will already know how to find the gradient of a straight line (ie 'rise over run' or equivalent). To estimate the gradient of a curve they will have to draw a tangent, as shown here:

They'll also need to determine whether the gradient is positive or negative.

To estimate the area under a graph, students will have to split the area into sections. The AQA Teaching Guidance says 'the trapezium rule need not be known but it is recommended as the most efficient means of calculating the area under a curve'. Unlike at A level, they won't be given the formula in the exam.

The alternative to using the trapezium rule is to split the area into a number of triangles and rectangles.

Motion Graphs

The methods described above are fairly straightforward. I think interpretation might prove trickier (eg understanding what motion graphs are showing). Students will need to know that speed, acceleration and deceleration are rates of change, and that the area under a velocity-time graph represents distance. There's a lot of new concepts and vocabulary here.

Bear in mind that motion graphs come up in Physics GCSE too. As shown in this extract from the AQA Physics GCSE Specification, it's exactly the same content:

Other Contexts
I was initially confused by the mention of 'graphs in financial contexts' in this section of the specification - I've seen questions in which students have to interpret financial graphs but nothing involving tangents or areas. Thanks to @DJUdall for sharing the picture below (taken from this new GCSE textbook) which shows an example of estimating a rate of change in a financial context.

Could exam questions cover other contexts, besides motion and finance? It's possible. In the example below from CIMT students are asked to find the volume of water represented by the shaded area. To understand why the area under the graph represents volume we can consider the units - the units on the horizontal axis are seconds, and the units on the vertical axis are m³/s, so when we multiply the two together we get m³.

Instantaneous Rate of Change

Students will need to understand the difference between an average rate of change over a period of time and an instantaneous rate of change. In my post 'Introducing Differentiation' I talked about how to give students an intuitive understanding of the gradient of a curve at a point. It's worth reading the section entitled 'An Instantaneous Rate of Change' for ideas, videos and resources (including my worksheet 'Thinking About Gradient' which was designed for A level but is now suitable for GCSE).

Resources
Resources for this topic are listed below. I'll add these to my library and will continue to add new resources as I find them.

• @DJUdall has produced this excellent graphing activity which covers both tangents and areas under graphs. 
• @jase_wanner has written this super hero activity on distance-time and speed-time graphs to develop students' understanding of how to interpret these graphs.
• Gradient on a curved graph by Owen134866 on TES gives students the opportunity to practise drawing graphs and tangents. My worksheet Finding the gradient of a curve using a tangent is similar, but students aren't required to draw the graphs themselves.
• Using Graphs from CIMT covers a wide range of graph topics. Section 17.2 covers areas under graphs using the trapezium rule and Section 17.3 covers tangents to curves. The full range of resources for this module are available on TES.
• Nuffield Mathematics provides a Free-Standing Mathematics Activity 'Speed and Distance' which is about finding the area under a speed-distance graph. Resources include slides, a student sheet and teacher notes.

'Graphing' by @DJUdall

I hope this post has been useful in helping you prepare to teach this new GCSE topic. Please let me know if you have any resources to share.

You might also find my other posts about new GCSE topics helpful: Sequences, Inequalities and Quadratic Graphs.

See my New GCSE Support Page for resources and links for all new GCSE content. 

'Drawing a Tangent Line' by NCTM

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. 

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".