5 Maths Gems #148

Welcome to my 148^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers. 

1. Distance Time Game

Thank you to @MrChapmanMaths for sharing this Graph Game from @davidwees. Best played on a computer (not a phone), this game is absolutely brilliant for developing understanding of distance-time graphs. It's really fun.

This reminds me of an activity I did on my PGCE where we walked/ran across a room trying to track distance time graphs. That was fun too, but required specialist equipment that I haven't seen since. 

The Graph Game led me to more content from David (listed here) including this excellent Factors Game. I played this with my daughter and we were quickly deep in discussion about factors and primes, and practising division. What a great game!

2. MCQ Generator
@mrshawthorne7 has created a dynamic multiple choice question generator. Select a topic and it generates random questions that can be used to identify and diagnose misconceptions. You can choose to hide the choices initially, to encourage students to do some thinking before they see the possible answers.

3. Percentages Task

I like this task shared by @MrsEVCartwright. Students are shown the working, and have to work out what the question might have been. 

4. Factorising Task

Here's another nice task, this time from @canning_mrmaths. It's an Open Middle task on factorising quadratics.

5. Probability Spinners

I recently presented a CPD session on probability where I talked about how spinners make an excellent fuss-free teaching tool. Following on from this, the team at MathsPad have created a Probability Spinners Interactive Tool which is free to use. It has sections on finding probabilities as fractions, decimals or percentages, expectations of frequency of outcomes after a given number of spins, and the results of repeated trials, including a bar chart representation and a relative frequency table. 

Subscribers can also access a Probability Trees - Spinners Interactive Tool. MathsPad's interactive tools are always excellent - you can see their full collection here. 

It's also great to see that MathsPad's new range of curriculum booklets is expanding, with the recent addition of an Expressions booklet for Year 8.

Update
My last blog post about resource design went down well. If you missed it, you can catch up here.

I recently recorded a podcast with Craig Barton where we chatted about teaching for depth and curriculum design. You can listen here.

I've spent the last week dealing with hundreds of access requests for the resources I store on Google Drive. This is because Google did a security update which broke all my links (thanks Google!). Links have to be fixed individually unfortunately - every evening I come home from work to find dozens of emails from people trying to access resources, at which point I fix those particular links. I have a feeling this is going to continue for months until they are all fixed! Apologies if you click on a broken link on my blog at any point. I'm working on it!

Are you going to the upcoming maths conference? I'm really looking forward to an in-person conference. It will be good to catch up with everyone. It's happening on 16th October in Kent and you can get a ticket here. I'll be speaking about curriculum sequencing in Period 4. 

Here are a few other things you might have missed:

• @jemmaths published a video 'Line Managing the Mathematics Department' which is highly recommended if you're a Head of Maths.
• @andyparkinson31 has indexed five hundred Numberphile videos by curriculum area and shared this resource on TES.
• The NCETM published an excellent set of subject knowledge audits which will be incredibly helpful for ITE providers.

Last week saw the launch of the DfE's Key Stage 3 Maths Guidance, which was written by the secondary team at the NCETM. This guidance suggests an ordering of the Key Stage 3 curriculum. I was surprised to see this! It links closely with the workshop on curriculum sequencing I'm running at mathsconf in two weeks.

The guidance provides valuable material which can be used in department meetings to help teachers prepare to teach each topic on the curriculum. For each area of maths, there are sections exemplifying the key mathematical ideas. These sections feature information on the common difficulties and misconceptions and suggest questioning and other strategies for teachers to use.

My school held Open Morning yesterday. Limited tickets plus a very clever one-way self-guided tour worked really well to eliminate any crowding. I am fortunate to work with a brilliant team of mathematicians - they are pictured below (shout out to Mariam who was off sick so missed the photo!). We will have Year 11 for the first time next year and we'll be recruiting. If you want to join our team, look out for a job advert in the Spring Term. 

I'll leave you with this units meme from Reddit which I first saw shared by @MrYoungMaths. I showed my Year 9s, who looked at me with blank faces while I chuckled away...

5 Maths Gems #139

Welcome to my 139^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers. 

Well here we are, at the end of the most gruelling term of our lives. I'm sure we are all incredibly relieved to have made it to end of term in one piece. We should all feel proud that we've continued to fill our students' lives with the joy of maths, and we will continue to do so, no matter what gets thrown at us. Thankfully we now we have a short respite - from the physical exhaustion of pandemic teaching, if not from the anxiety. Throughout the Christmas break many of us will be steeling ourselves for January, when the level of challenge in schools and colleges will be ramped up even further. I wish I could offer words of encouragement and solace, but to be honest I am totally lost for words. I just hope that you all manage to have a safe and peaceful break over the festive period.

Perhaps on the first day of the holidays we all need to switch off from maths teaching, but I know some of you take comfort in browsing through teaching ideas and resources. It can be a cheerful distraction from the sadness that surrounds us. So, on with the gems.

1. Tasks

Thank you to @SegarRogers for sharing this brilliant speed, distance, time task. I love this. I've added it to my resource library so I remember to use it next time I teach speed. 

Thanks also to @ashtonC94 for continuing to share his expertly written tasks, including these on standard form and this interesting take on recurring decimals.

Here's another excellent set of tasks from @giftedHKO. The focus here is pie charts - a great topic which links nicely to proportional reasoning. 

2. Teaching Tools 

Thank you to @123lots for sharing @mathforge's handy Make Your Own Axes tool.

Also, I'm not sure whether I've featured this site before - thank you to @jemmaths for the reminder about @tesseralis’s website tessera.li. This website features a beautiful polyhedra viewer which teachers might find useful.

3. Website Updates

@MathsDunbar has completed Version 3 of his Trinity Maths Programme for the new National Curriculum. Matt has made ten of his best interactive topic files available as free downloads, which are macro-enabled Excel spreadsheets.

@MrMorleyMaths has completed and published Phase 2 of his website mrmorleymaths.co.uk. Every topic now has a starter activity of questions, with answers, testing prerequisite knowledge.

4. GCSE Revision Resource

Thank you to @Billyads_47 for sharing a GCSE revision resource.

Produced a revision booklet to use with yr11 during a revision day. I included @corbettmaths found & higher checklist to make it a 12 page stunner. Many thanks to @username_FINAL for providing the inspiration for a revision formula sheet. @suffolkmaths https://t.co/XjkxJjOqF6 pic.twitter.com/PBnpyLDenO

— Billy Adamson (@Billyads_47) November 23, 2020

Don't forget that my GCSE Revision Resources page is absolutely packed with excellent revision resources for your Year 11s.

5. Multiplication Tool

Thank you to @mathforlove for sharing these visual flash cards for multiplication that he has produced in conjunction with @MathigonOrg. This is a brilliant free online tool for children learning their times tables.

Christmas

I know it's too late for me to share Christmas resources, but in case you want to use them next year...

@Ayliean shared some lovely maths-themed Christmas cards.

An early ChristMaths present for you

✨printable mini Christmaths cards✨https://t.co/kvAPiHMuPS

4 per page, 8 pages to a class 💗❄️
Feel free to put your own message inside and remember print flip on short edge 🤓🎄 pic.twitter.com/N8u2hrU6Ns

— Ayliean (@Ayliean) December 13, 2020

This snowflake dot-to-dot activity shared by @giftedKHO is really nice. 

And @DrBennison shared his annual A Level Calculated Colouring, which is always fantastic.

I do have a page of Seasonal Resources. I didn't even get time to look at it myself this year, but it is there if you are ever after a themed maths activity. 

Update
I recently published another 'Dose of Don' post from Anne Watson - do check it out if you haven't already.

I have also created another set of warm-up booklets for my classes. These are specifically designed for the curriculum my school - each week we cover three different retrieval topics - but feel free to borrow and adapt if you like. In the most recent set, most questions were taken from CIMT resources. 

There are a number of online maths conferences coming up in the Spring term. See my conference listings for details. I will be speaking at two of them - #MathsConfMini on Friday 22nd January and the WRM Secondary Maths Brunch on Saturday 30th January. 

Don't forget to check out Marvellous Maths 2 - an entire online training course for maths teachers.

Congratulations to Craig Barton on the fifth anniversary of his incredibly successful education podcast. You can read his reflections on five years of podcasting here.

Finally, I should mention that as my own school continues to grow (we will have Year 10 for the first time next year) we will be recruiting a new maths teacher in the Spring term, to start in September 2021. If you are interested in coming to work with me and want to get a flavour of what the school is like, check out our Open Day video and our Christmas video, and get in touch if you want to have a chat about it.

I'll leave you with this video, shared by @berniewestacott, which features Douglas Clements speaking at the White House on early childhood maths education. It's under five minutes, and well worth a watch.

Have a very merry Christmas, maths teachers. Time to get some rest. x

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