You can download the slides here.

To see more free CPD from this series, view my Topics in Depth page here. And for more CPD from me, visit my YouTube channel.

I hope this is helpful.

Ideas and resources for teaching secondary school mathematics

Loughborough University Mathematics Education Network (LUMEM) offers free access to online CPD videos for maths teachers. I've recently produced another video for their collection. It's from my Topics in Depth series, and it's all about angles in polygons.

This CPD takes a detailed look at the Key Stage 3 topic 'angles in polygons', examining its place on the national curriculum and how it is assessed at GCSE. My presentation features a range of methods and discusses various pedagogical approaches. I share ideas for how to enrich the teaching of angles in polygons, adding depth and challenge that goes beyond the national curriculum.

You can access it here.

Like all my Topics in Depth CPD, this workshop aims to enhance teachers’ subject knowledge as well as providing teaching ideas and inspiration. You could watch the video directly before you teach the topic, to help you plan really good lessons.

If your whole department is going to be teaching angles in polygons, you could watch it together and discuss it in a department meeting.

You can download the slides here.

To see more free CPD from this series, view my Topics in Depth page here. And for more CPD from me, visit my YouTube channel.

I hope this is helpful.

Welcome to my 150^{th} gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

**1. ****Tasks**

@TickTockMaths shared an introduction to bounds calculations.

@canning_mrmaths shared a trigonometry task, based on an idea by @DanielPearcy.

**2. ****Mrdaymaths.com**@nathanday314 shares fantastic resources on Twitter so it's great to see that he's launched a website where we can find them all in one place. Check out mrdaymaths.com to download his displays, tasks, resources and presentations.

**3. Angles and Ratios Interweaved**

**5. QLA Workbook Generator**

@PiXLMattTheApp is always sharing free tools and resources for maths teachers. His latest is an online tool which allows you to create a student workbook from a QLA.

Update

Over half-term (which feels like months ago!) I wrote three blog posts:

It was a busy week for me at work last week. One of my big responsibilities is to run the Key Stage 4 Options process. I launched it last week, running events for both parents and students. At the same time, my school was treated to a MAT review (basically a Mocksted, though we're not meant to call it that...). This was stressful, mainly because we're all 100% sure that our school is outstanding - in every sense of the word - and we really wanted to make sure that the inspectors saw that. The entire maths department made me immensely proud, as did my Year 9 class (I was observed teaching them some experimental ideas that I picked up at the last mathsconf... It was a bit risky for me to go ahead with that lesson but thankfully it went well! Phew).

**Milestones**

This is a milestone blog post for me. When I wrote my 50th gems post I was presented with a special cake at a conference (thanks Julia et al!). When I wrote my 100th gems post I recorded a special podcast with Craig Barton, and Chris Smith sent me a trophy which I still proudly display on my bookcase. Today I've reached 150 gems posts. Have you read them all?! You should! There's a gems index here.

By the way, I know people love the gems posts, which are packed full of other people's great ideas, but I do also blog about my own ideas too! In fact I've written 274 posts which aren't gems posts, and the full archive is here.

Another milestone I recently passed (but failed to notice at the time!) was my ten millionth website visit. Thank you to all my readers for their support. I am immensely happy that my resource libraries save people time, and that my blog posts provide teachers with inspiration and ideas. Teachers who visit resourceaholic.com tend to do so on a regular basis, so I must be doing something right.

I'll leave you with this tweet which made me laugh. I wonder what my students would write down if I asked them the same question.

I've spotted several great tasks on Twitter recently. Here are three of them:

@TickTockMaths shared an introduction to bounds calculations.

@jontheteacher shared some tasks designed to introduce the concept of 'like' terms.

Nathan always has great ideas and presents them beautifully. For example, in this thread he explains how he designed a sequence of lessons that took Year 7 from substitution through to solving equations. The resources featured in this thread are available to download from his website.

I also love Nathan's 'No More Primes' game. Read his post to see how it works.

Thanks to @blatherwick_sam for sharing a lovely couple of tasks interweaving ratios and angles.

@PiXLMattTheApp is always sharing free tools and resources for maths teachers. His latest is an online tool which allows you to create a student workbook from a QLA.

I recommend that you watch Matt's video if you want to find out more about how this works.

Update

Over half-term (which feels like months ago!) I wrote three blog posts:

I also updated my conferences page which lists national maths education conferences. Please let me know of any additional events that I should add to this page.

Note that the MA Annual Conference is now open for booking. I'm one of the keynote speakers at this conference and am really looking forward to it.

Speaking of the MA, did you see that they have made the latest issue of their journal Mathematics in School freely available to read online? I'm a big fan of this journal and always look forward to receiving my copy in the post.

Speaking of the MA, did you see that they have made the latest issue of their journal Mathematics in School freely available to read online? I'm a big fan of this journal and always look forward to receiving my copy in the post.

It was a busy week for me at work last week. One of my big responsibilities is to run the Key Stage 4 Options process. I launched it last week, running events for both parents and students. At the same time, my school was treated to a MAT review (basically a Mocksted, though we're not meant to call it that...). This was stressful, mainly because we're all 100% sure that our school is outstanding - in every sense of the word - and we really wanted to make sure that the inspectors saw that. The entire maths department made me immensely proud, as did my Year 9 class (I was observed teaching them some experimental ideas that I picked up at the last mathsconf... It was a bit risky for me to go ahead with that lesson but thankfully it went well! Phew).

Finally, a personal milestone for me - my eldest daughter turned ten. A decade of parenting. 💖

I'll leave you with this tweet which made me laugh. I wonder what my students would write down if I asked them the same question.

On yesterday's quiz, just for fun, I ended with the request "in the spirit of Halloween, show the scariest math mistake you can." I wasn't disappointed by some of the student responses! pic.twitter.com/LV50BpE1us

— Chris Luzniak (@cluzniak) October 27, 2021

Labels:
Algebra,
Angles,
Bounds,
Fun,
GCSE,
Gems,
Misconceptions,
Quadratics,
Ratio,
Trigonometry

When I joined the teaching profession I was surprised by two things:

There have been a few occasions when I've tried to get conversations going on Twitter about misconceptions. Years ago, I set up a misconception sharing website but I didn't have time to maintain it. I also started a hashtag #misconceptionchat but I haven't used it often enough for it to gain traction. But I still believe that it's vital that this knowledge is shared widely. We all see misconceptions in the classroom every day. Some just need acknowledgment ('this is something students do, and our teaching should address this') whilst some are worthy of deep discussion.

There's a problem with sharing student misconceptions on social media though. Almost every time I've shared a misconception on Twitter, I've had replies from people criticising my teaching and suggesting that their approach is foolproof and would never lead to such misconceptions. Pfft. A more constructive dialogue (accompanied by a 'thanks for sharing this misconception, it's really interesting') would encourage teachers to share their observations more readily. Bear in mind that our students come to us with ingrained misconceptions that are not a reflection on our own teaching, but still need to be identified and addressed. We shouldn't hesitate to discuss these misconceptions with others - doing so enhances our pedagogical subject knowledge and makes us better teachers.

I think back to the time I tweeted a misconception that took me by surprise when I was marking a Year 7 fractions quiz.

I received a reply to my tweet from a teacher who told me I was making it up (unfortunately this teacher no longer has a Twitter account so his tweets have disappeared - I can't remember his exact words!). He said he'd give his students the same question the next day to see what happened, because he was adamant that this was not a mistake that any of his students would ever make. When he did so, he was absolutely shocked to find that they held the same misconception, and he later tweeted to apologise for not believing me. I guess it was a helpful misconception for me to share after all, as it alerted him to a gap in his students' understanding that he otherwise wouldn't have been aware of.

Just because we haven't personally seen a misconception, it doesn't mean it doesn't exist. Sometimes we just haven't asked the right questions.

**A flaw in my explanation**

I take the blame for this one, even though I don't teach the popular 'sausage method' which could directly lead to this misconception. In case you don't know what the 'sausage method' is, check out the video here to here to see it in action (there are dozens more videos like this). Basically it involves drawing a ring around the two diagonal terms that can be simplified (the ring is sometimes referred to as a sausage, balloon or peanut).

Of course, the circled terms can only be simplified in certain expansions, so we should be careful to avoid implying that this ring can always be drawn.

So what did I do wrong in my teaching? Well in my initial modelling I always completed the grid then wrote out the four terms underneath the grid before simplifying them. I did that every time, until my students had built up some fluency in using the grids. Then I got a bit lazy and rushed my modelling, to the point where sometimes I skipped writing out the four terms underneath the grid, and took to drawing a ring around the like terms. I distinctly remember doing this when demonstrating a triple bracket expansion. So although I never said that 'drawing a sausage' is part of the procedure, and I certainly never said 'these two terms can*always* be simplified', I may have accidentally implied this in my modelling. Lesson learnt! Not every misconception is a direct result of a flaw in my explanation, but I think this one probably was.

**Misconception baggage**

**Pre-empting misconceptions**

We can also address misconceptions through task design and curriculum sequencing. I recently had a Year 8 student who was factorising this expression:

*4abc*^{2} + 10a^{3}b^{2}c

- no book existed about the common misconceptions in each topic.
- no book existed about the various methods that can be taught in each topic.

I felt that both of these should be standard reference books for trainee maths teachers. But they didn't exist. I addressed the latter (see my book 'A Compendium of Mathematical Methods') but no one has fully addressed the former. We still don't have an easy-reference book or website featuring all the common misconceptions in one place. Instead, we repeat the mantra 'inexperienced teachers should learn about common misconceptions from their experienced colleagues'. That's a great idea, unless you find yourself working in dysfunctional maths department where no one talks maths. Then you just have to work it all out for yourself.

I think back to the time I tweeted a misconception that took me by surprise when I was marking a Year 7 fractions quiz.

Just because we haven't personally seen a misconception, it doesn't mean it doesn't exist. Sometimes we just haven't asked the right questions.

Occasionally I see a student doing something in a lesson and realise it has resulted directly from my own flawed explanation. So I reflect on this, and I adjust my explanation the next time I teach it. I experienced this recently when I gave my Year 8 students a test on expanding double brackets. One student did well on every question, until he got to this one:

Expand and simplify (𝑥^{-2} + 4𝑥^{10})(𝑥^{2} + 5𝑥^{3})

I was delighted that this student knew that 𝑥^{2} ✕ 𝑥^{-2} = 1, demonstrating his knowledge of index laws (other students wrote 𝑥^{0} which was also acceptable).

However, a misconception relating to expanding arose that I wouldn't have spotted if I hadn't used this particular question.

Can you see what he's done? He seems to have assumed that the two diagonal cells can always be simplified. So there's a misconception in his underlying algebraic knowledge that needs to be addressed (i.e. that 4𝑥^{12} + 5𝑥 can be simplified). There's also a step in the expansion procedure that he has fundamentally misunderstood. A couple of other students did something similar.

I take the blame for this one, even though I don't teach the popular 'sausage method' which could directly lead to this misconception. In case you don't know what the 'sausage method' is, check out the video here to here to see it in action (there are dozens more videos like this). Basically it involves drawing a ring around the two diagonal terms that can be simplified (the ring is sometimes referred to as a sausage, balloon or peanut).

Of course, the circled terms can only be simplified in certain expansions, so we should be careful to avoid implying that this ring can always be drawn.

So what did I do wrong in my teaching? Well in my initial modelling I always completed the grid then wrote out the four terms underneath the grid before simplifying them. I did that every time, until my students had built up some fluency in using the grids. Then I got a bit lazy and rushed my modelling, to the point where sometimes I skipped writing out the four terms underneath the grid, and took to drawing a ring around the like terms. I distinctly remember doing this when demonstrating a triple bracket expansion. So although I never said that 'drawing a sausage' is part of the procedure, and I certainly never said 'these two terms can

As I mentioned earlier, our students come to us with ingrained misconceptions that need addressing - sometimes the student has been carrying that misconception around for years. On a daily basis I'm furious at whoever was responsible for inventing the acronym BIDMAS because it leads directly to a totally avoidable misconception. I'm not opposed to mnemonics at all, but they should have chosen a better one. Even BIDMSA would have preferable. Anyway, I'm not here to rant about BIDMAS (I have a video about it here) but this is something that we know is a big problem. I shouldn't have been surprised when I saw the classic mistake recently, but what surprised me was the context:

I have a class of very strong mathematicians in Year 10, and one of the best students in the class asked me for help on this question:

'Write as a surd in simplest form: 10sin60 - 12tan30 + 8cos30'.

This was in a task on simplifying expressions involving exact trigonometric ratios, which is basically an exercise in manipulating surds.

"Miss, can I just check something with this question. I have to add 12tan30 and 8cos30 first, right? Because of BIDMAS?"

Who knew that the order of operations would be a problem in my lesson on exact trig values...

There's some debate about whether we should show our students common misconceptions up front. Some people think it might confuse them, but I'm all for it. In my teaching I directly show the mistakes people make and we discuss the reasons, and I make it clear what's gone wrong. That's not to say it always works though.

When teaching angles on a straight line, I'm careful about the wording (I always very specifically say '*adjacent* angles on a straight line sum to 180') and I talk up front about how the angles need to be next to each other - they need to form a half-turn - in order to add up to 180. As a class we explore and acknowledge the common misconception. Yet still I have one or two students who do things like this:

Even though I don't have 100% success rate at eliminating all misconceptions, I still think addressing them head-on is worthwhile.

We can also address misconceptions through task design and curriculum sequencing. I recently had a Year 8 student who was factorising this expression:

He put up his hand to ask me whether the index in the first term only related to the c. A good question! It then occurred to me that I hadn't explicitly addressed this in my teaching, but I had previously designed a resource exactly for this purpose. I have an exercise on substitution that aims to draw out misconceptions around algebraic notation - see extract below.

**A misconceptions journal**

I made a note of a few misconceptions this term because I thought they might be interesting to blog about. So I ended up with a notepad on my classroom desk where I jotted down some things students said in lessons. This made me reflect on whether I should make a habit of doing that - keeping a log or journal of misconceptions that I could later reflect on and discuss with colleagues. Perhaps all teachers should do this. It's probably totally impractical though - over the course of a day's teaching we process dozens of misconceptions. And this isn't a bad thing - good teaching should be designed to draw them out and address them! If we're not seeing misconceptions, it doesn't mean they don't exist. They may just be hiding.

I used this in a subsequent lesson with the same class. When circulating during the lesson, I spotted that a number of students had incorrectly evaluated Questions 10 to 15 by multiplying first. I know this is a common misconception so I should have probably done some more work on mini-whiteboards at the start of the lesson to draw it out and address it.

I made a note of a few misconceptions this term because I thought they might be interesting to blog about. So I ended up with a notepad on my classroom desk where I jotted down some things students said in lessons. This made me reflect on whether I should make a habit of doing that - keeping a log or journal of misconceptions that I could later reflect on and discuss with colleagues. Perhaps all teachers should do this. It's probably totally impractical though - over the course of a day's teaching we process dozens of misconceptions. And this isn't a bad thing - good teaching should be designed to draw them out and address them! If we're not seeing misconceptions, it doesn't mean they don't exist. They may just be hiding.

Finally, I will end on a positive, with a lovely quote from the late Malcolm Swan:

"Frequently, a ‘misconception’ is not wrong thinking but is a concept in embryo or a local generalisation that the pupil has made. It may in fact be a natural stage of development."

Misconceptions posters from classicmistake.nhost.uk |

Welcome to my 149^{th} gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

**1. Entry Level Maths**Entry Level Maths is a qualification designed to provide a progression route to GCSE Maths for the lowest attaining students. To support schools using this qualification, @MarsMaths has shared sets of questions for each topic at each of the different stages. A lot of time and effort has gone into creating the resources at marsmaths.com and I'm sure this will be helpful for SEND departments and Entry Level teachers.

**2. Make Your Own Codebreaker**

**3. Oops, I Forgot!**

**4. Linked Maths**Thank you to @l88belle for sharing a task on expanding double brackets that interweaves fractions, surds, area, volume and solving equations. It's helpful to think about the ways we can make links between different topics. I look forward to seeing more from Belle on her website linkedmaths.weebly.com.

**5. Resources**

Thanks to @jshmtn for sharing a lesson on index laws with lots of good ideas to borrow. I never thought of using pi as a base!

@draustinmaths continues to add new resources to her website. Latest additions include some excellent surds tasks.

**Update**

Speaking of conferences, you only have a couple more weeks to sign up to speak at the MA Conference which is taking place between 12th and 14th April 2022. I really encourage teachers to propose a workshop, even if you've not delivered at a conference before. The MA's 2021 online conference was a huge success. The beauty of online conferences is that they are widely accessible - people who are unable to attend in-person events can easily participate. In 2022, two days of the MA Conference will take place online and the final day will take place in person in Stratford-Upon-Avon. I plan to attend both the online and in-person days. In fact, I'm delivering one of the keynotes.Finally, I'll leave you with this tweet from @mansbridgemaths which made me laugh! It's a great task, similar to 'What's z?' from MathsPad.

If you're on half term this week, have a good one!

@PiXLMattTheApp shared an activity generator which gives teachers the opportunity to create their own codebreaker tasks. Select a topic and write a joke or sentence to encode. It's very easy to use. This is one of a range of tools on mathswhiteboard.com that are worth checking out.

I like this idea shared by @fawnpnguyen.

Been doing "Oops, I forgot" (OIF) routines. I haven't seen any T do this other than @NatBanting at his session.

— Fawn Nguyen (@fawnpnguyen) October 13, 2021

Write 2 fractions that =1. Then, OIF, the denominators need to be diff...OIF, 1 denominator is 2-digit, etc.

It has layers [literally] that I really like.

Read the thread for more information.

For an idea of how it works: you read a series of instructions to students while they use mini-whiteboards and adapt their answer each time. For example:

"Sketch a quadrilateral"

"Oops, I forgot - it should have four right angles"

"Oops, I forgot - it should have an area of 24".

"Oops, I forgot - it should have a perimeter greater than 24".

Thanks to @jshmtn for sharing a lesson on index laws with lots of good ideas to borrow. I never thought of using pi as a base!

Half term has finally arrived, much to everyone's relief. My school gets a two week October half term (I know, we're very lucky) but my daughters only get one week off. So I've had a week to myself - it's the only week in the entire year that I get a little bit of 'me time' (in between dropping them off and picking them up from school), so it does wonders for my mental health. As well as taking a bit of time to chill, I also used this week to contribute to some discussions about maths education. On Monday I attended an MEI Curriculum Committee meeting, and on Tuesday I participated in a Sheffield Hallam Uni/Royal Society roundtable on textbooks and curriculum materials. On Wednesday morning I chatted to Julia Smith about methods, and in the afternoon I recorded a new Topics in Depth CPD workshop on angles in polygons for Lumen (watch this space!). On Thursday evening my husband and I went to see Harry Baker perform, which was absolutely lovely. Harry is a mathematician and poet - I must book him to come and speak at my school.

Did you see my latest blog post? I wrote about curriculum sequencing and shared the slides from my recent conference session.

Speaking of conferences, you only have a couple more weeks to sign up to speak at the MA Conference which is taking place between 12th and 14th April 2022. I really encourage teachers to propose a workshop, even if you've not delivered at a conference before. The MA's 2021 online conference was a huge success. The beauty of online conferences is that they are widely accessible - people who are unable to attend in-person events can easily participate. In 2022, two days of the MA Conference will take place online and the final day will take place in person in Stratford-Upon-Avon. I plan to attend both the online and in-person days. In fact, I'm delivering one of the keynotes.

This is the task that caused a Year 8 to ask me, "are you a psychopath?" last week.

— Mr Mansbridge (@mansbridgemaths) October 18, 2021

There's no better endorsement than that. pic.twitter.com/Y0xEK7v6Ej

If you're on half term this week, have a good one!

Yesterday I presented a session on curriculum sequencing at #mathsconf27 in Ashford. Thank you to everyone who attended. In this post I will summarise the main points and provide a link to the slides.

I started by talking about the national curriculum. It's statutory for all local-authority-maintained schools in England to teach the Department for Education's programmes of study. In reality the vast majority of non-local-authority-maintained schools (e.g. academies) teach them too. I talked about how heavily prescribed the content is for maths. With the exception of science (which is also heavily prescribed), all other subjects have a fair amount of creative scope over the content they teach. If you're a Head of English you might choose to teach Macbeth, and if you're a Head of History you might choose to teach the Black Death. If you're a Head of Maths you have pretty much no say over what you put on your school's curriculum (in terms of the content), other than perhaps a bit of enrichment that goes beyond the national curriculum.

But that's not to say we have no control over curriculum at all. We can control the sequencing of the curriculum and we can control our pedagogy, resources, methods and approaches. Basically we can control the implementation of the curriculum. This implementation varies widely by school.

I talked about how odd it is that we don't all follow the same sequence, and that there is no agreed 'best order' for the topics we teach. In some countries (I talked about a specific example from Shanghai) a great deal of research is done on getting the order right, down to the minutia of 'is it more effective to introduce area first, or perimeter first?'. Yet in England our decisions about curriculum sequencing (which are typically made in isolation by each Head of Maths in over three thousand different secondary schools) are normally not research-informed. I also talked about the massively different approaches to curriculum sequencing seen in the United States, where students spend an entire year on algebra and an entire year on geometry. To us this seems unusual, and to them our approach seems unusual. I shared some images from Ben Orlin's very funny post about this. And the question I asked is, 'Has anyone actually researched which of these approaches is more effective? Which approach optimises student experience and progress?'.

I talked about why Heads of Maths and teachers should be interested in curriculum sequencing. It's not just because Ofsted might ask them about it! I also talked about an interesting point made by White Rose in their post Order, Order! The Importance of Sequencing - although our sequencing decisions are mainly determined by prerequisites (i.e. you can't study this topic until you've done that one), there are also considerations relating to student experience.

After this long introduction about why we should think about curriculum sequencing, the rest of my presentation was broken down into three parts: prerequisites, interweaving and common practice.

Teachers should think about the prerequisites for every topic they teach. That goes without saying. When I start teaching Pythagoras, the first thing I should do is ask myself the question 'what maths do my students need to know to access this topic?'. This is done by Heads of Maths when creating their scheme of work, and it's also done by teachers when planning their own lesson sequences. Prerequisites have a large part to play in the order of our curriculum. For example I wouldn't put area of a circle as the first topic in Year 7, because in order to access that topic I need to first ensure that my students are fluent in rounding, calculator use and squaring.

I reflected on the fact that students at my school do rounding in January of Year 7 but then don't actually use rounding in any topic until January of Year 8. So perhaps it makes sense to move rounding to Year 8, when it can be taught and then immediately used in topics like Pythagoras, circles and volume.

I showed examples of topics that can be interwoven and how the order of our curriculum can create opportunities for interweaving. For example, I think it's really important that angles should follow equations in Year 7. If angles is done first, it can end up being a repeat of primary school angles. But if angles is taught second then teachers can make the most of opportunities to interweave equations and angles. This adds depth and challenge to the topic of angles, as well as giving students the opportunity to make use of their newly acquired equation solving skills.

I gave the example of how I recently taught index laws to Year 8 and then expanding brackets with the same class, so was able to make use of index laws within my lessons on expanding brackets. It's important that students understand that each bit of maths they learn takes them to the next step, and allows them to access more complex maths. We don't need to answer 'Why are we doing this?' questions from students with tenuous real-life application nonsense. "All roads lead to calculus", and they are on a journey heading in that direction.

In the last part of my workshop I shared examples of schemes of work. Even though we all teach the same curriculum, there is very little consistency in curriculum sequencing across schools. And even though Heads of Maths put a lot of time into designing their schemes of work, when you look at all the different orders side-by-side, it almost looks random.

In my recent survey of almost 800 Key Stage Three maths teachers, I found huge inconsistency in when topics are taught. Look for example at the topics with few prerequisites (shape transformations and constructions) which are spread across Year 7, 8 and 9.

I talked about the idea behind spiralling in order to build depth of knowledge. I feel like spiral curriculums are sometimes seen as the opposite of mastery curriculums, but that's not necessarily the case. The worst kind of spiral curriculum touches on topics at a surface level and returns to them every year. Things are rushed rather than mastered, meaning each year it's like starting from scratch again. A better form of spiral curriculum takes a strand of maths in its entirety (e.g. angle geometry) and each topic within that strand adds a piece to the jigsaw, building on prior knowledge and deepening understanding of the strand as a whole. Angles could be taken as five topics: 'basics', parallel lines, polygons, bearings and circle theorems. Each year we teach one of these topics, and each time we do so we not only add a new piece to the jigsaw but also build depth of understanding of the strand as a whole.

The approach taken in the new DfE/NCETM curriculum sequence seems to differ to this. This has challenged my thinking, and I would love to hear the NCETM team speak about the rationale for their proposed order. I know the team behind this are very experienced and knowledgeable and will have given it a lot of thought, so this deserves our attention.

In the last part of my presentation I shared my own five year curriculum sequence. I made it as an example for the purpose of this presentation, really just to see how difficult it was to do, and it's therefore simply based on my own thoughts and experience. The main challenge was time: we just don't have enough of it. It's frustrating. I've said it for years and I'll say it again: we simply have too much prescribed content in our national curriculum. If we are going to have any hope of teaching this content properly - actually in depth, without just skimming the surface of it all - then the content needs to be reduced. They could start by removing constructions... 🙃

I ended with the message that curriculum sequencing is fascinating, and it's something maths teachers should think about and talk about.

So that's a brief summary of my talk. There was a lot more in it, but that's the gist. I hope that people who attended found it helpful.

I spoke more about all this in my recent podcast with Craig Barton.

You can download the slides from my workshop for use with your own department.

***

Finally, I just want to say a huge thank you to La Salle for organising this conference. As much as I like the virtual conferences, they are nothing compared to the in-person ones. Being in a room full of maths teachers and having impromptu conversations about schools and teaching is just so powerful. I enjoyed all the sessions I attended, and I think Kris Boulton's workshop was one of the best I have ever been to. And given that I have been going to La Salle's conferences since 2014, that's saying something.

I also want to say thank you to those who joined me at my drinks on Friday night. At work on Friday I was stressing that I'd booked a table for ten people but might end up sitting at it all by myself, but in the end I was joined by Chris, David, Sudeep, Nathan, Nathalie and Rachel, and then later in the evening by many more conference-goers. I ended up dancing in a club until the early hours. I can count on one hand the number of times I've been out dancing like that since I became a mum a decade ago! It was so much fun. After my first big night out in my forties, I was a bit worse for wear at the conference - but it was worth it.

See you all at the next one.

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**

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.

**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**

**4. Factorising Task**

**5. Probability Spinners**

Here are a few other things you might have missed:

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.

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!

@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.

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

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

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!

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...

Labels:
Factors,
Gems,
Multiple Choice,
Percentages,
Primes,
Probability,
Quadratics,
Speed,
Units

I was going to call this blog post ‘pimp my worksheet’ but I realise then unless you’re familiar with MTV programmes from the mid-noughties, you won’t get the reference and you’ll think I’m just weird.

The point I want to make in this post is a simple one:

Take this example, created last week by my excellent colleague Sarah. She made this for her top set Year 10 class. I borrowed it. I like that it starts with scaffolding and, with a simple tweak to the question style, ends with challenge.

I will be using these with Year 8 in a few weeks! Their first topic of the year is index laws, and then they move onto expanding double brackets. This task neatly combines the two topics. A lovely bit of interweaving.

Instead of spending ages hunting for challenging activities, you may be able to quickly and easily increase the challenge level of an existing exercise.

Take this example, created last week by my excellent colleague Sarah. She made this for her top set Year 10 class. I borrowed it. I like that it starts with scaffolding and, with a simple tweak to the question style, ends with challenge.

At both the White Rose Secondary Maths Brunch in January and the MA Conference in April, I talked about the use of ‘working backwards’ in tasks to get students thinking. I shared examples of tasks which feature elements of working backwards, such as 'fill in the gaps’ activities. Here's an example from Kyle Gillies.

And here's an example from Andy Lutwyche.

You can read more about tasks like this in Paul Rowlandson's excellent blog post "This Way, That Way, Forwards and Backwards".

After Sarah shared her resource with my department last week, I realised that it’s pretty simple to convert a standard drill exercise into a ‘working backwards’ task that requires more thinking. It’s just a case of editing a few questions.

Here’s an example that I used yesterday in Year 10's first lesson on surds. Notice how they’re not just simplifying, they’re also ‘unsimplifying’ which I think might help build depth of knowledge.

Here’s an example that I used yesterday in Year 10's first lesson on surds. Notice how they’re not just simplifying, they’re also ‘unsimplifying’ which I think might help build depth of knowledge.

It's a simple idea to combine scaffolding and challenge in a single task. Many resources like this already exist, but it was probably quicker for me to make my own rather than search for one. It took only a few minutes to create.

Note to new teachers: once you become a pro at using equation editor shortcuts, it's super quick and easy. Press thealtbutton andequalsat the same time in Word or PowerPoint to insert an equation, then follow this guide from Jamie Frost. You'll get the hang of it quickly and won't have to keep referring to the guide.

My lessons often combine short 'drill' exercises (i.e. practise a procedure) and longer 'thinking' tasks (i.e. use reasoning). The former normally come from screenshotting from Corbett Maths or CIMT. The latter are typically MathsPad or Don Steward. I draw on many other sources, normally going via my own resource libraries to save time. For example, in my next surds lesson I've taken an extract from an old Solomon worksheet that I found in my surds listings.

Note to new teachers: screenshot using thesnipping toolto easily insert tasks into your lessons. You don't need to write exercises from scratch when they already exist.

Incidentally, in the same Solomon worksheet I spotted these questions:

Anyway, I've digressed. Back to my original point:

You can add more challenge to a straightforward exercise by tweaking it slightly. Blank out some of the questions and provide the answers instead. Getting students to work backwards makes a task less procedural and gives opportunities for reasoning ("*if this is how it worked going forward, how must it work going backwards?*"). Even if it's just a tiny tweak it may be a worthwhile one, to avoid mechanical repetition and make students pause and think.