5 Maths Gems #162

Welcome to my 162^nd gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Scaffolded Tasks
Thank you to @JaggersMaths for sharing a highly accessible Pythagoras task with lots of scaffolding.

And here's an extract from another task which is structured to make it both accessible and challenging. Thanks to @scott_math83 for this set of probability tree questions.

2. Number Etymology Video

@berniewestacott shared a lovely video 'Why is Eleven not called Oneteeen?'. This is a really interesting summary of some of the etymology of numbers. I love this kind of thing.

3. Angles
Teacher and task designer @SegarRogers shared a set of clever angles tasks designed to give students a deep understanding of complementary angles.

He tweeted his tasks here, here and here.

4. Indices

@giftedHKO shared an indices task that she used with her Year 10s last week. I'm also teaching this topic to Year 10 at the moment, so this was a well-timed tweet for me! She had them complete a grid with all the powers of two through to ten up to the fifth power. This supported them in completing fractional indices questions with larger numbers.

5. Fractions
@robertkaplinsky shared a challenge from openmiddle.com which might be suitable for Year 7s doing fractions. This lovely task is written by @IanKerr_math.

Update
I love September - teaching new classes is so exciting - but it's always manic and exhausting isn't it? I've been struggling with mixed feelings in my new role. I'm loving being Head of Maths again - I've done this job before so I feel very confident in the role, plus I get to spend all my time focusing on the thing I'm most passionate about (maths!). But at the same time it's frustrating for me to step down from the senior leadership team after three years of putting my heart and soul into building a new school. I know I'll get used to it. I must keep reminding myself of all the benefits of changing role - I no longer spend two hours a day on systems and spreadsheet work, because they've now hired a data officer to do these things. Instead, I've been swamped with Year 11s asking for help with maths at every opportunity (we've not had Year 11 in this school before, so this is a novelty for us!). So I'm still just as busy as I used to be, but I think my expertise is being put to better use now. I might blog separately about some of the things I've already done in my capacity of Head of Maths, and some of the things I've planned, as this might help support new Heads of Maths. Watch this space.

Another thing I'm massively enjoying this year is teaching AQA's Certificate in Further Maths GCSE. Six years ago, my timetable was mainly A level. I now haven't taught A level for so long, I'd forgotten how much I enjoy it. Certificate in Further Maths isn't A level, but I'm teaching stuff like Binomial Expansion and Trig Identities, which I love. I have a class of 31 keen mathematicians who meet after school every Wednesday. It's going to be a rush to get through the content - this week we were doing some work on exact trig values and I discovered that their fluency in working with surds isn't quite where I expected it to be - but I think I am going to absolutely love teaching this course.

Here are some things from Twitter that you might have missed in the last couple of weeks:

• @mathsaurus shared his free online course to help students prepare for the UKMT Senior Maths Challenge which take places on 4th October 2022
• @Ridermeister shared his problem solving booklet containing 151 problems aimed at Year 13s who want to prepare for a university interview in Mathematics.
• @PiXLMattTheApp shared a tool to create a bespoke maths newsletter in thirty seconds! With this tool you can create a newsletter that could be sent home to parents or used as a form time activity.

In other news, I loved wearing my new mathsy Popsy dress 'Marva' at school last week. I will wear it again on our Open Day in a couple of weeks. Colleagues keep asking me how many maths dresses I have. Five! One for every day of the week.

I look forward to seeing you all at some of the conferences coming up in the Autumn term. I'll be doing a new 'Topics in Depth' presentation on area at #mathsconf30 in Manchester on 15th October (I hope! I've booked a train to Manchester, but due to issues with the train operator, there are currently no trains home on Saturday or Sunday. This is stressful!). I'm also speaking at the Pixl Maths Conference at The Oval on 12th October. And if you work for Harris, I'm speaking about Key Stage 3 challenge at the Harris Federation October Conference at the ExCeL Centre on 14th October. While writing this I've realised that I've unintentionally agreed to do three conference presentations in one week! Oops - sounds intense! Luckily I will be on half-term the following week to recover.

I'll leave you with this surdic expression to simplify -  your GCSE students might enjoy this. This was shared by @mathisstillfun, a great account to follow for regular maths problems.

5 Maths Gems #161

Welcome to my 161^st gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Mathematicians Display

@nathanday314 has updated his excellent Famous Mathematicians From Around the World Display. This features 82 mathematicians from 53 countries. 

The original inspiration came from a display by @DrStoneMaths and @MrYoungMaths which I featured in Gems 132. 

For more classroom and corridor displays for maths departments, see my displays page. 

Do make sure you follow Nathan on Twitter if you don't already. He has been sharing loads of great resources over summer including tasks on simultaneous equations, bounds and indices. For example I love the task below, which is an adaptation of @karenshancock's indices task.

Speaking of @karenshancock's resources, I also love this sequences fill in the gaps task. I've added it to my resource library.

2. Vectors Task

It's always great to see new content on my favourite website MathsPad. They have now completed their curriculum booklets for all of Year 7, 8 and 9. These booklets are full of brilliant tasks - I use them all the time.

I also love their new free Vectors & Quadrilaterals Worksheets. The idea is to work out which type of quadrilateral will be formed if you follow a sequence of vectors. This provides practice working with vectors whilst prompting thinking about whether pairs of vectors are equal in length, parallel or perpendicular, and encouraging mental visualisation. There are two levels of difficulty.

I've added this to my resource library.

3. Certificate in Further Maths Resources

In my last post I mentioned that I've set up a new page linking to Certificate in Further Maths resources. This can be accessed from the main menu at the top of my blog. Since then, I've added three additional resources to this page:

• Content list for pupils from @mathshysteria. This handy student-friendly checklist may also be helpful for teachers' planning as it shows which topics are covered at GCSE and which are specific to Certificate in Further Maths.
• Certificate in Further Maths questions from A Level papers from @tm_maths. This will be a helpful source of challenging questions.
• Videos and exam questions by topic on an excellent new website from @1stclassmaths.

4. GCSE Revision 

In the Autumn term @MrsEVCartwright gives her Year 11 students a revision sheet per week on a topic covered so far in the course. Each sheet contains a recap of the main ideas and techniques, followed by some questions to complete for homework that week. You can download this resource from TES.

Back in Gems 155 I shared another GCSE revision resource - PowerPoint files containing GCSE maths questions by topic for OCR, Edexcel and AQA. I subsequently awarded it a Gem Award for 'Most Useful Resource' because I use it all the time in my GCSE lesson planning. @gcse_math has now shared this resource on its own website - gcsemathsquestions.co.uk. If you missed this earlier in the year, do check it out.

5. Place Value

Thank you to @Scott_Math83 for sharing some new resources. I like these place value tasks using fractions and decimals.

 

And this rounding problem is great.

Resource Libraries

If you're new to teaching then you might not know about my resource libraries which contain listings of resources for Key Stages 3 to 5, organised by topic. I use these libraries every day when planning lessons as they provide a quick way to browse quality free resources. 

I try to keep my Key Stage 3 and 4 libraries updated on an ongoing basis, and when I start teaching A level again next year I'll do some work on my A level libraries too. Links sometimes get broken when people move or delete their resources - if you find a broken link then do let me know! 

I also add new resources fairly regularly. For example I've recently added more links to resources from Dr Austin Maths including tasks for indices, standard form, factorising quadratics, inequalities and graphs.

The quickest way to access my resource libraries when you're planning lessons is using the menu across the top of my blog.

Update

I've mainly been doing DIY over summer and am now a bit fed up with painting. My house is in disarray (I currently don't have a bedroom!) so I'm looking forward to returning to school on Thursday. I had a really lovely break though, which included a very sunny family holiday in a cottage in Cornwall, and a great night out with maths teachers at Jamie Frost's house (thanks @nkl_17 for the photo!).

I'm looking forward to meeting my new classes next Monday. I'll be teaching Year 7, 8, 10 and 11, plus I have a Certificate in Further Maths class who will have lessons after school. My school first opened four years ago, so Summer 2023 will be our first ever GCSE results. It's a big year for us! This year we'll also be setting up our new A level courses ready for next September. In a couple of months we'll be advertising for a Key Stage 5 Coordinator to join our maths department - watch this space if you're interested in applying!

Good luck to everyone starting in new schools this week, particularly ECTs, and good luck to those of you embarking on teacher training. September is such an exciting month.

5 Maths Gems #160

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

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

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

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

3. New Resources from Dr Austin

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

4. Negative Number Hook

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

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

5. Percentages

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

Back to School 

These pages are always popular in August and September:

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

Update

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

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

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

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

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

Year 9 Assessment Questions

Last week I blogged about our Year 8 assessments. I set out the principles we follow when we write our Key Stage 3 assessments, and I shared examples of the questions that challenged our highest attaining students. 

Today's post is about Year 9. My school's Year 9 cohort has a very wide range of maths attainment, which was further exacerbated by the lockdowns in Year 7 and 8. One notable feature of this year group is how incredibly good the high attaining students are. The top set teacher finds it hard work to sufficiently challenge them in lessons. So when I made the end of year assessment, I had to ensure there were plenty of questions in there to make them think. I don't want anyone coming out of a maths assessment bragging that they found it really easy.

Here are some of the more challenging questions from our end of Year 9 assessment.

Ratio

At a concert the ratio of men to women is 5 : 3.

The ratio of women to children is 7 : 4.

Show that more than half of the people at the concert are men.

This was a middle-of-the-paper question. It's not massively challenging - I taught set four out of five and a few of them managed to get full marks on it. But I wanted to include it here because it's a nice question requiring a bit of reasoning. It was originally from an AQA specimen GCSE paper.

A more difficult ratio question was this one from Edexcel, which also tested another Year 9 topic: changing the subject.

The ratio (y + x) : (y - x) is equivalent to k : 1.

Find a formula for y in terms of k and x.

Surface Area

Surface area is a great topic because it presents many opportunities for problem solving and reasoning. For example, this classic question may have been fairly straightforward for a high attainer, but I like the way it's also accessible to anyone who does a bit of thinking. I was delighted that a few students in my class worked this out. We put it in our non-calculator paper so it also tested their arithmetic.

The total surface area of a cube is 294cm².

Work out the volume of the cube.

A slightly more difficult question was this one from Edexcel. It doesn't have any particularly challenging reasoning in it, but has got a multiple steps to work through, including unit conversion which might be missed. I like questions where students have to identify that they need to work with surface area rather than volume.

The most challenging surface area question I included was this one from AQA. Very few students could do this, even our highest attainers. We will revisit questions like this in Year 10.

Bounds

I like this coal question from WJEC. In Year 9 we teach bounds and error intervals for the first time. We introduce some basic bounds calculations, but go into greater depth on this at GCSE. This question fitted perfectly. However, I thought our students would find it easier than they did. I had it near the start of the paper, but most the students in my class only picked up one mark on it. My students are all working at a Grade 4 level though. Our higher attaining students had no problem with this one.

Probability

We taught Venn Diagrams to Year 9 this year. There were some fairly straightforward Venn diagram questions nearer the start of our end of year assessment, where students basically just had to complete various Venn Diagrams. But for challenge I wanted to test understanding of notation as well as probability, so I adapted an OCR AS level question. Only our very best mathematicians answered this correctly.

Algebraic Proportion

Algebraic proportion can be incredibly procedural. As long as students read the question carefully, once they know how to do it, it's almost a guaranteed five marks. So I included a non-calculator proportion question that was a bit different to questions they'd seen before. What I liked about this was the accessibility: no one from my class got all the marks, but they did manage to pick up one or two.

Right-Angled Trigonometry
This question was my pièce de résistance. I figured that if our super clever Year 9s breezed through all the other challenging questions I threw at them, they would surely have to stop and think at this point. This SQA question is designed to be solved using the Sine Rule. But our students don't do the Sine Rule until Year 10. This question can be done with right-angled trigonometry. The way I did it was by splitting the base into x and 350 - x, then forming two equations for the height and equating them. Even if our students managed to get this far, solving the equation would be fairly challenging for them because they haven't seen anything like this before.

As it happened, none of our Year 9s managed to solve it in the way I envisaged. But one very smart student came up with a genius (albeit inefficient!) method of trial and improvement. I yelped with joy when I realised what he'd done:

It's such a delight to see students using creative approaches like this.

I also found some great challenging questions involving standard form and percentages, but I will stop at this point otherwise this blog post will go on forever! Like I said in my last post, there are numerous places we can find good assessment questions for Key Stage 3. It's a shame they aren't centrally produced anymore - the old KS3 SATs contained great questions, but at least we can still draw on those to make our own assessments.

If you have any good Year 9 assessment questions you'd like to share, please tweet me. 

Thanks for reading!

Year 8 Assessment Questions

This year I produced our end of year assessments for Year 8 and 9. We have a few basic principles that we follow when we create our Key Stage 3 assessments: 

• Single Tier Key Stage 3 Assessments. I know this one is controversial, but I want every student in the year group to take the same assessment. I've never been happy with prematurely allocating students to tiers in Key Stage 3. When we've done that in the past, I've found that students end up getting 'stuck' in certain sets. So now, although our regular end-of-topic tests are tailored to each class, our big assessments (twice a year) are one-size-fits-all. This makes the assessments more difficult to write, but helps us get a better sense of relative attainment across the whole year group.
• Testing All Taught Content. We know it's frustrating for students to spend ages learning and revising topics that are then not tested. So we try to include questions from every topic taught during the year. And we would never test on topics not yet taught - this is counterproductive (it damages confidence and tells us nothing). This is why we don't give full higher tier GCSE papers at the end of Year 10 - I'm strongly opposed to it. 
• Testing Knowledge, Skill and Reasoning. We want to test procedural fluency and reasoning skills, but we don't believe there's a strict separation between these things. Well-written questions test both, so we try to find as many of those as we can.
• Challenge. It's so important to include a good level of challenge. I can't stand it when students finish a 60 minute assessment in 20 minutes and then sit there looking bored because they found it easy. There should be plenty of questions that make them think. And given that the outcomes of these assessments help us determine the groups we'll put students in next year, we need our assessments to provide us with enough information to get these decisions right. If half the year group get 90%+, it's not helping us identify our strongest mathematicians.

Question Style

I don't have time to write many of the questions myself. Thankfully there are plenty of good questions around, and tools like Dr Frost mean they are easily accessible. Here's an example of a question that is well suited to a single tier Key Stage 3 assessment:

This is from a WJEC Foundation paper. What I like about this is that Part a is very accessible to all students with a calculator. So the vast majority of students can get a mark here. Part b requires an understanding of inverse operations. Some of our lower attaining students found that more difficult than we expected them to.

Here's a similar one. I adapted this from an old AQA GCSE question. The majority of students got Part a correct. But Part b required a bit more thinking, and it surprised me that many students got it wrong, suggesting we need to work on their proportional reasoning.

Questions with multiple parts where the first part is highly accessible and the parts get increasingly difficult are perfect for single tier assessments. Ideally every paper would full of these, then you don't end up with the situation where some students can only do the first four questions and can't access the rest of the paper. Here's another example of a question that has both accessible and challenging parts:

Perhaps this ramps up in difficulty too quickly. There could be an additional part between b and c with moderate difficultly.

If a paper is full of questions like this then it's both accessible and challenging. Next year I want to include more of this. This year my paper ended up with the classic format 'easier questions at the start, harder questions at the end', which doesn't create a nice experience for the lower attaining students.

In the remainder of this post I want to share some of the more difficult questions I included in my Year 8 assessments. Some of our high attainers said they found the mid-year assessment too easy, and I didn't want a repeat of that. 

Circles

In Year 8 our students are introduced to pi, and they learn how to find the circumference and area of a circle. We go into depth by exploring questions involving semi circles and quarter circles, which takes a lot of reasoning. MathsPad has some great tasks for this:

And for even more challenge, there are problems like this:

There's absolutely no need to accelerate onto the formulaic ^θ/₃₆₀ πr² when there's so much reasoning to be done in Year 8 with semi circles and quarter circles.

To test these reasoning skills I included this question in the non-calculator assessment (taken from OCR's sample GCSE questions):

And this question in the calculator assessment (I can't remember the source of this one - I think it might be Edexcel?):

Even our very best Year 8 mathematicians found these questions challenging. In my very bright Set 2, no student got more than one mark on either question. This suggests we still have work to do on developing their reasoning. 

Fractions

Although this wasn't one of the most challenging questions on my Year 8 assessment, I decided to give this question a mention here because it surprised me. Fractions questions are often so procedural, I wanted to include a fractions question that took a bit more thinking. Even though the vast majority of students in my class have no trouble adding, subtracting, multiplying and dividing fractions, this question was answered very badly. Even halving five eighths proved conceptually challenging. 

Volume

There are lots of 'high tariff' volume questions to choose from. These are good for a single-tier assessments because most students should be able to pick up some of the four or five marks available even if they can't complete the whole question. 

This cylinder question (adapted from AQA) stumped most of my students. They all correctly worked out the volume of water in the Cylinder A, but then found the volume of the whole of Cylinder B and divided one volume by the other. This is a classic example of students following a formulaic approach to answering a question, rather than stopping to think about what the question actually says.

Interweaving

As much as possible I tried to find questions which combined more than one topic they'd studied this year. This is particularly easy to do for Pythagoras and ratio, which both come up all over the place. For example, this question combined ratio and percentages:

Joseph’s flock has 55% more sheep than goats.

What is the ratio of goats to sheep in the flock? Leave your ratio in its simplest form.

This was originally an IMC question. Can you guess what went wrong? Most students in my class simplified 45:55 instead of simplifying 100:155 (or equivalent). They have the knowledge they need to answer this question correctly. They just didn't think.

They had much more success on this 'problem solving' style question on the non-calculator paper which also involved ratio and percentages.

Andrew is paid £250 a week.

Each week, he:

- shares his pay with his sister in the ratio 3 : 2

- saves 12% of his share.

How many weeks will it take Andrew to save £360?

This is an AQA GCSE question from 2014. It was done pretty well.

And even though we'd spent time on questions just like this back in December, not a single student in my class picked up any marks on this non-calculator question:

This is a SQA specimen question but I added the requirement to answer without indices to add extra challenge. It interweaves index laws, fractions and expanding brackets. All three topics were on our Year 8 curriculum.

Although we do a lot of work on fractional and negative indices in Year 10, we do look briefly at negative indices in Year 8. So the higher attaining students should know that x^-1 = ¹/_x  and x⁰ = 1. They didn't get this far though. A few students realised that they should be adding the indices, but they seemed to panic at the thought of adding 1/2 to -3/2. Now I think about it, I remember A level students finding this kind of thing difficult when we used to teach C1 too. Perhaps this is something to spend more time on when we do fractions, instead of skipping over fractions with common denominators.

*****

I've featured a number of GCSE questions here, but I should emphasise that our Key Stage 3 assessments draw on numerous sources, including old Key Stage 2 SATs questions and old Key Stage 3 questions. Anything that's right for our students goes in, we're not precious about what it was originally written for. I wish I had more time to write my own questions.

I'd love to see some of the questions that most challenged your Key Stage 3 students in their assessments this year, so do tweet me some examples. 

I'll write a similar post soon about some of the questions I used to challenge Year 9.

*****

For context, I work at a comprehensive school in the London Borough of Sutton. There are five grammar schools in our borough, which slightly reduces our intake of high attainers. We are proudly non-selective. It's a fairly affluent area so only 14% of our students are pupil premium. We are quite ethnically diverse (41% White British compared to 66% nationally). We have 22 students with EHCPs. We are a new school with around 200 students per year group and we currently only have Years 7 - 10 (we'll have Year 11 next year). In general, most of our students have excellent attitudes to learning and want to do well. The average score on our end of Year 8 assessment was 57% and the highest score was 95%.