An interesting property of linear sequences - inspired by the 1% club.

The 1% club is one of my favourite quiz shows. It is the only quiz show I have actually applied to be on (no success unfortunately) but I play along on the app all the time, and also regularly complete the daily question that comes through the app. Yesterday (27th January 2026) had a very interesting question (from a maths point of view) that sparked a little dive into linear sequences. I resisted posting it yesterday as I didn't want to provide spoilers for any readers that also play along.

So, the 1% club daily question yesterday was this: 

What two digit number replaces the question marks in this sequence of numbers:

92, 23, 53, 83, 14, 44, ??

What made this interesting was the way I achieved the correct answer was very different to the way the app explained how to arrive at the answer (if you want to try and answer before I reveal the solution then don't scroll down too far!)

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The correct answer was 74. The reasoning the app gave was that if you reverse the digits of the list you get the sequence 29, 32, 35, 38, 41, and 44 and so the next value would be 47 which, when reversed gives 74. Which makes perfect sense. But it isn't how I arrived at 74.

I (as I am sure many other readers also) noticed that a lot of the jumps were +30 and that those that weren't were -69. There also seemed to be a regularity to when these jumps appeared; a jump of -69 followed by two jumps of +30. Given the jump of -69 from 83 to 14, I reasoned there would be a jump of +30 (although I was wrong about the regularity of the pattern of jumps as the next would actually be another -69).

Of course, once I realised that these two approaches both gave the same answer, I absolutely had to try and decide whether this was a property of this particular set of numbers, or whether it would be true for the reverse digits of all linear sequences made of two digit numbers.

Rather than diving in with the algebra straight the way (that is coming, don't worry), I decided to play with a few more sequences first to create further examples and see if this sequence was obviously a unique case (a very good problem solving strategy in general I find to allow for pattern spotting).

So I tried 30, 34, 38, 42, 46, 50 becoming 03, 43, 83, 24, 64, 05 - which quickly disabused me that there was any regularity to when a sequence went up or down, and then I tried 17, 24, 31, 38, 45, 52 becoming 71, 42, 13, 83, 54, 25.

It was at this point that I realised that the value of the differences were always 99 apart in the reversed sequences, in the first 30 and 69, in the second 40 and 59, in the third 70 and 29. It took me an embarrassingly long time to recognise that the subtractions were happening when the original linear sequence bridged a 10, or that if the linear sequence was going up in 3 (say) that the reversed sequence should be going up in 30.

I started to explore the algebra at this point a little, but quickly realised that I was getting confounded by the fact that I had only tried differences in the original linear sequences that were less than 10, so I tried 26, 39, 52, 65, 78, 91 becoming 62, 93, 25, 56, 87, 19 (which showed me it wasn't so simple as subtractions occurring when the original sequence bridged a 10, but was more about the units digit becoming smaller - which should have been obvious really) and also 12, 35, 58, 81 becoming 21, 53, 85, 18. This confirmed that the sum to 99 was still a thing - or more precisely that the subtractions were the positive differences subtract 99.

At this point I dived properly into the algebra, which I did as follows (again, if you want to try it first then don't scroll down):

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(I added some text to show clearly what the algebra implied that I didn't write in my own scribblings).

In terms of this as a task for pupils, I think there would be something interesting in offering KS3 pupils a chance to explore 'reverse linear' sequences - probably at a distance from linear sequences themselves. I think it might reinforce some properties of linear sequences and it would be very interesting to see if they spot the 99 link and how they try and justify it.

I definitely think there would be something about using the proof with a GCSE/Further GCSE/A-Level class, either as an example of constructing a logical proof or as an exercise for them as part of their practise in creating a deductive proof.

Of course, the question remains about what happens with linear sequences that stray into 3 digit numbers (single digits are trivial as we can just treat them as two digit numbers with first digit 0). I have answered this question to my own satisfaction and so will leave it as an exercise for the interested reader with one hint, which comes from when I shared the initial problem with other maths teachers at Twinkl and one of them came up with a third approach to the original problem (which is equivalent to what I have outlined and also leads to the correct answer):

"Add 30 each time but if the answer goes over 100 add the 100s digit to the ones digit".

A mathematical curiosity?

 In writing my new book 'Practising Maths' I referenced a lovely result (you will have to buy it to see how) that sums such as 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1, etc. all produce square numbers.

If you haven't come across this result before then feel free to have a look at it for a minute (even try and prove it) - if you are familiar with consecutive triangular numbers summing to square numbers, it is closely related.

The curiosity I noticed was that I knew 121 was also square. So I became interested in the fact that 1 + 2 + 1 is square, and 121 is square. I decided to look into the others, and it turns out they are also square! Well, the ones up to 12345678987654321 are square anyway.

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This of course raised a question - is this a reflection of something deeper? You might like to spend some time exploring and coming to your own conclusion before you read on.

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I guess the truth is a little of both.

If we consider squaring polynomials of increasing order with unit coefficients we get the following:

These are, of course, the same expressions as above, but in base x rather than base 10. So, if we substitute x = 1 into the expressions we get the sums on the left of the above table. However, if we substitute x = 10 into the same expressions, we get the numbers on the right of the table.

In terms of a task, we could offer the first few rows of the table to pupils and ask them what they notice/wonder. They might explore when the pattern breaks and why. We might encourage them to write out the numbers using explicit base 10 notation, such as 1 × 100 + 2 × 10 + 1 and see what insights this brings out. Pupils with the necessary algebra skills might even explore the expansions given above. Or we might just show it to pupils as an example of a mathematical curiosity.

New Maths and Dyscalculia Assessment!

A new assessment for identifying and supporting difficulties in mathematics learning was launched in late July that has the potential to be of significant help in strengthening school’s and parent’s ability to appropriately plan for learners struggling with key aspects of maths study.

The assessment has been designed by the co-founder of the Dyscalculia Network and experienced specialist teacher, Rob Jennings, alongside Jane Emerson, the Director of Emerson House – a centre for dyscalculia, dyslexia and dyspraxia.

Rob Jennings and Jane Emerson (used with permission)

Split into 19 sections, the assessment provides for a comprehensive examination of learner’s abilities regarding early number concepts such as number sense and counting, different mental and written calculation strategies, interpreting word problems, working with and converting between fractions and decimals, as well as basic length measure. The sections have a mix of verbal and written questions, with some of the earlier ones requiring the use of counters. The authors suggest that the assessment should take roughly one hour (based on trials that have taken place to help refine the test questions and assessment approach), however are at pains to point out that the assessment should not be limited by time, either overall or for any one of the sections, as this could lead to anxiety for the pupil that could skew the results.

What sets this assessment apart, for me, from other assessments and on-line screeners for dyscalculia and/or other maths difficulties, is the level of detail that the assessor (which could be a parent, teacher, or TA – not necessarily a qualified assessor) is encouraged to capture about the pupil. As well as simply getting an answer correct or incorrect, the assessor is encouraged to note down (through the use of a provided assessor’s booklet) how long each section took, the strategies that pupils used (to help capture whether these are efficient or immature strategies), and any comments or questions that the pupil makes – either to themselves or to the assessor. This sort of information is potentially crucial in formulating a proper plan for addressing the difficulties that the assessed pupil is facing with mathematics. In addition, there are actually two assessments, an A assessment and a B assessment, which means that they could be used as a pre- and post-test for an intervention specifically designed to support the pupil.

Access to the assessment, assessor’s booklet, and answers comes through the purchase of the companion guide called (straightforwardly), “The Maths and Dyscalculia Assessment”, with a link and redemption code for the online materials included in the guide.

The companion guide comes with much more than simply a step-by-step guide on administering the test itself. Included is also guidance for the assessor in preparing for the test, including how to make sure the necessary things are organised in advance, how to create a good environment for conducting the test, what to be on the lookout for and to record during the assessment, and even making sure that the learner is at ease during the assessment. There is also a chapter on interpreting  the results of the assessment, including what issues may have been highlighted by the assessment, what might then be included in a teaching plan if these issues have arisen, and suitable specific interventions that might be required, which I would definitely recommend reading this before  administering the assessment – I feel like it would sharpen my focus on certain aspects of the test and approaches that a child takes to the test beyond what is given in the step-by-step guide.

In addition to the guidance provided for before, during and after the assessment, the companion guide also contains a host of other information and support for working with pupils that have difficulties with mathematics, including different checklists or screeners that could be used in advance of the assessment or to support its findings, a template summary report along with a completed exemplar to help capture the results of the assessment and plan for future teaching with the pupil, and an in-depth family questionnaire that can be used to provide extra detail and context to be used in the summary report. Both the templates for the summery report and the family questionnaire are both included in the online materials accessed through the guide, which means that the complete assessment package can be used with as many pupils as is needed. The guide also contains lots of information about how dyscalculia and other maths specific and non-specific learning difficulties (such as maths anxiety, dyslexia and the like) might impact maths learning and attainment, as well as some interesting statistics about occurrence of maths difficulties and co-occurrence with other difficulties, and a host of sources of further information and resources that could be useful for parents and educators.

As I went through the book and the online assessments, I reflected heavily about the numerous pupils I had encountered that displayed some or many of these difficulties. In the last department that I led we benefited from the assistance of a part-time numeracy intervention tutor and I can see how this sort of assessment would be invaluable in supporting her work alongside those of the main class teachers, as well as contributing hugely to the work of our SENDCo and inclusion team in pinpointing the difficulties that pupils were having and initiating conversations with parents and other agencies about the diagnoses and support that these pupils might benefit from. Despite my limited experience in the field, I have never seen anything that is designed to capture how pupils approach maths problems alongside their accuracy and time taken in completing them, or the depth of insight to guide future planning that this test provides and for that reason alone I think this book/assessment is well worth a look for any school or parent that wants to get a real handle on the maths difficulties that their children are facing.

From GCSE Sequences to Calculus

As part of the recent mathsconf39 I ran a session on using structure to promote algebraic thinking, and as part of that I looked at using a couple of different structures that can be used to help with the teaching of sequences. This got me reflecting on a link that I taught and highlighted to my Level 2 Further Maths and A-Level students about the links between the sequences they learn at GCSE/pre-GCSE and the calculus with polynomial functions that they first learned at Level 2 Further Maths or A-Level.

Part of the first work that pupils do with sequences is learn about linear sequences, and their defining characteristic being that of constant difference. Later they learn about quadratic sequences, and their defining characteristic being constant second difference. I always highlight that this means that the differences in the terms for a quadratic sequence form their own linear sequence. This establishes a progression from quadratic to linear to constant, or, in reverse, constant to linear to quadratic. I then ask them to predict what type of sequence would have a quadratic difference – which pupils quickly identify as cubic and can then extrapolate further from there.

This of course perfectly mirrors the progression that different polynomials go through when they are differentiated or integrated. Cubics differentiate to quadratics, which differentiate to linears, which differentiate to constants. So, when I first introduce differentiation to pupils, I ask them about where they have seen this progression before. Someone almost invariably mentions sequences, and if they don’t, I might start by writing out a quadratic sequence to prompt them.

A great question to ask students then is, of course, why? Why this connection between sequences and differentiation? This allows me to reinforce the idea of differentiation as about the rate of change, as when we are identifying the type of sequence we are examining the change between the terms.

It is worth then examining a particular sequence and its nth term, and the links to differentiation. I will typically pick one that has a coefficient of n², such as the sequence 3n² + 4n – 1:

n = 1		= 2		= 3		= 4
6		19		38		63
	+13		+19		+25
		+6		+6

 

If we differentiate the nth term of the sequence, we of course get 6n + 4, whereas the nth term of the linear 1^st differences is 6n + 7. At first this appears to be a discrepancy, until you see that the 1^st differences are in the 1.5^th, 2.5^th and 3.5^th position. We can account for this by shifting each value back by 3 (half of 6):

n = 1		= 2		= 3		= 4
6		19		38		63
10		16		22		28
	+6		+6		+6

 

Or by substituting n = 1.5 and T_n = 13 into 6n + a = T_n:

6 × 1.5 + a = 13

9 + a = 13

a = 4

Either way the nth term of the first differences can be shown to actually be 6n + 4, which is precisely what we get when we differentiate the nth term.

This provides a nice way of interleaving other algebraic manipulation and structure into the study of calculus, as well as reinforcing the idea that differentiation and sequences are two sides of the same mathematical idea – sequences being an introduction the study of changes and growth of discrete functions whilst differentiation (and integration) being the study of changes and growth of continuous functions. It also just adds to the impression for pupils that the landscape of mathematics is connected in unexpected but beautiful ways.

ResearchEd National Conference 2025

The first Saturday in September has become a somewhat annual tradition for me to take down to Parliament Hill School in London for the ResearchEd National Conference. This year I was joined by two of my fellow National Education Leads; Ash Morris, who leads secondary science; and Sarah Hutson-Dean, who leads primary science, who I am delighted have contributed their experience to this blog as well.

Me, Ash and Sarah meet in London

Like many others, we started the day in the session with Professor Becky Francis CBE, hoping to maybe get a few advance insights into what might be reported when the Curriculum and Assessment review lands at some point before Christmas. I don’t think there was anything that isn’t already in the public domain, but there were some interesting points made, including the results of the 2024 parents survey indicating that the majority of parents think that the attention given to both core and “non-traditional” subjects is about right in schools, and that academic subjects featured in the top five list of things that parents and pupils at KS4 would like to spend more time on (alongside employment and interview skills, creative thinking and problem solving projects, digital skills, and finance and budgeting). Professor Francis did confirm that the review will examine GCSE volume and time spent in examinations, look at how to facilitate greater choice of subjects whilst maintaining the breadth of curriculum offer that is a key strength of our curriculum. Another key issue the review is looking at is disadvantage, where gaps are still significant, with a commitment to ensuring that “every young person can see themselves in the curriculum, and that it challenges discrimination and extends horizons”. Climate science, digital literacy and financial education were also mentioned as being needed to help young people navigate the opportunities and challenges they would face in the future. Professor Francis echoed the detail of the interim report when she highlighted there was a lot to celebrate about our education system, and many things that we do very well, and so “evolution not revolution” was what the review was looking to achieve.

After this it was my time for my own session on front-loaded feedback in maths, which I gave to a surprisingly large audience (given the other speakers on at the same time). The session seemed to go down well (I got top ratings from everyone that filled in the feedback form) and people coming to see me later to say that they had enjoyed it. 

Me in presenter mode!

Of course, the test for any session is whether people make use of it afterwards, so if you were there and you do incorporate the ideas then please let me know!

Oracy and literacy have been big on the agenda in recent months, so I decided to go and listen to Sarah Davies’ talk on using oracy to support literacy fluency in the next session. Hearing about Sarah’s journey after taking responsibility for literacy in her deputy head role was fascinating, and her commitment to ensuring that each child was treated as an individual and received the right pathway of intervention for them was inspiring. She rightly pointed out that if kids are not able to access exam papers, then all the content knowledge in the world won’t help them – and I was surprised to see maths exams being quite high up in the literacy demands (third only to English and Geography). One thing that Sarah shared that really resonated with me was a conversation she had with one of her staff members about why their top performing pupils were not able to secure places in top universities despite their excellent academic record, because they did not possess the oracy skills necessary to do well at the interview stage – something that would clearly hamper them well beyond applying for universities.

From Sarah Davies’ presentation

During this session, Sarah and Ash were back in the sports hall for the session by Peps McCrea and Dr Jen Barker. Here are Sarah’s thoughts on the session:

Peps McCrea and Dr. Jen Barker shared a powerful message: inclusive teaching is effective teaching. With educational outcomes declining and many teachers feeling underprepared to support children with SEND, they unpacked five drivers of system failure. Yet their talk was full of hope. They reminded us that how we learn is far more similar than different. Our shared cognitive architecture means we all benefit from strategies that secure attention, provide high-quality feedback, and establish strong routines. By designing lessons to be as accessible as possible for the widest range of pupils, and keeping adaptations minimal so every child engages with the same learning, we can lift outcomes without sidelining teachers. Practical strategies like highlighting key sections of text, chunking information with guiding questions, using choral responses, encouraging pair talk, and making the most of mini-whiteboards showed how small, thoughtful choices can have a big impact!

Sarah (with Ash) followed this up by listening to Professor Rob Coe talking about teacher expertise, and said:

Dr. Rob Coe’s session was a brilliant reminder of just how much teachers matter. He showed us that the impact of great teaching goes way beyond test scores, shaping students’ chances in life and even their wellbeing. So what sets great teachers apart? It’s a mix of deep subject knowledge, clear explanations, strong routines, supportive relationships, and lessons that really make students think. Rob shared the ‘Great Teaching Toolkit,’ which breaks this down into four areas: knowing your content, creating a positive and motivating environment, managing the classroom so learning time is maximised, and designing tasks that stretch thinking. One point that really stuck was that just clocking up years in the classroom does not guarantee expertise. Without the right feedback and focus, we can plateau. His challenge to schools was clear: if we want teachers and pupils to keep thriving, professional growth has to be a priority.

After a gorgeous lunch in the speakers lounge and a trip round to the farmer’s market, it was time for the afternoon sessions. The buzz recently with my fellow Twinkl National Education Leads has been around the recently released writing framework so I was very interested to go along to Clare Sealey’s session talking about it. Clare shared some great advice from her work on the sector panel and what the writing framework does and does not mean schools need to consider or do.

 

Whilst I was hearing all about the writing framework, Sarah and Ash were delving into the world of adaptive teaching and its links to cognitive science. Here is what Ash had to say about the session:

The standout session of the day for me was delivered by Jade Pearce, Trust Head of Education for Affinity Learning Partnership, on ‘Adaptive Teaching x Cognitive Science.’ Having always been passionate about the science of learning and committed to teaching in an evidence-informed way, I expected to leave with reassurance that my practice was on the right track. Instead, I came away with fresh ideas and practical adaptations, particularly around the ‘I do, we do, you do’ model for introducing new content, as well as approaches to questioning. Rather than relying solely on questions to check for listening or recall, the session highlighted different styles of questioning and their purposes, which added real depth to my thinking.

The final session I attended was led by Adam Robbins, unpacking the overly simplistic ideas that seem to exist in areas of the profession about motivation, particularly the idea that extrinsic motivation is a “bad thing”, and autonomy being absolutely necessary for motivation. Adam pointed out that there are plenty of times that people are motivated by extrinsic factors over which they have no control, but that these don’t necessarily lead to lack of intrinsic motivation, before unpacking the research that shows that there are levels of extrinsic motivation that tend to come with maturity as well as how extrinsic factors such as competence and relatedness can “fill the gaps” when intrinsic motivation might be lacking. A simple example is marking pupil, which has to be done (low autonomy) but that teachers might be motivated to do because they think it will make them more able to support pupils (competence) or because it feeds into a wider department plan that they are committed to (relatedness).

Outside of the sessions, it was great to catch up with old friends and colleagues like Kat Howard, Kris Boulton, Mark Lehain and David Faram. This is part of what keeps me coming back to ResearchEd; the combination of hearing about things that are really working for people in schools, insight into the latest thinking that might impact education, and the chance to touch base and interact with people who care about education as much as I do just cannot be beaten!

Ash agreed that it was a great day, saying:

Aside from the sessions themselves, it was inspiring to be surrounded by so many dedicated and passionate people, all giving up their Saturday in the name of research! A personal highlight was spending time with my colleagues, Pete and Sarah, and enjoying some excellent Greek food from the nearby farmers’ market at lunchtime, with Pete leading the way.