ax² + bx + c = 0

They are:

- Factorise
- Complete the square
- Use the formula

I think I generally teach them in that order probably without much thought as to why. I guess the formula needs to be derived by using completing the square and factorising seems to follow on from multiplying out double brackets, which comes before all of this. The I question that I sometimes get from students is “what’s the point in learning factorising if the two other methods always work?”. Well, it’s quicker and you can do it on a non-calc exam is probably a standard response.

But have you tried using the formula without a calculator to solve a quadratic that you know will factorise? Have a go. Plug this:

x² - 3x - 28 = 0

into this:

…and solve without a calculator.

It’s a surprisingly satisfying experience, one that I would not want to deny my students.

You’ll need to know your square numbers because *b² – 4ac* will always give you a square number for quadratics that factorise. But the arithmetic is perfectly reasonable and is likely to be so for most quadratics that can be factorised.

When I did this recently, I had a great question from one of my students, as I was aching my brain trying to make up a quadratic that I knew would factorise. “If you just picked one randomly, what are the chances that you would be able to factorise it?”

I’ve since had chance to investigate this further. It’s a great question and there is a lesson in here, or at least an extension question to explore once the fundamentals of using the formula are secure.

I started approaching it by using the formula and focussing on *b² – 4ac* and what values of a, b and c would yield square numbers. To simplify the problem, I started with a=1, so I was looking for when *b² – 4c = 1,4,9,16,25, etc.*

I then looked at it from the other end, i.e. starting with e.g. (x+1)(x+n) what values of a, b, and c are yielded. Then vary further to look at (x+m)(x+n). I just started with a few values of n and m to see if I could spot patterns. I won’t spoil the fun by revealing those patterns, but this is very open-ended and could provide some intrigue for the right learners.

]]>

What is the ratio of the height of the red triangle to the height of the pink triangle?

Can you solve using trigonometry or only using Pythagoras?

The border is now 2cm, whilst the side length of the red triangle remains 4cm. What is the ratio of heights now?

Explore what happens for other border widths. Can you generalise for any border width w?

Geogebra file here.

Spoiler here.

I got a few solutions posted on Twitter for this, but the most elegant (so far…) has to be this from @mathforge. As he said: no Trig, no Pythagoras, just ratio.

]]>

This is what came to mind when joining Derek Ball’s session at the ATM conference this week, entitled “Recurring Decimals”.

In fact, we were going the other way: converting fractions to recurring decimals, specifically looking at fractions where the denominator is prime. It was a fascinating session, great for deepening subject knowledge. This blog post is my attempt to reflect on what I learned and my thoughts about how I might use this in the classroom, probably Year 10 or 11, but really any group that is confident with bus stop division could investigate this.

I was already aware of some pretty cool things that happen with sevenths, mainly from Don Steward’s blog:

Why should this happen? Why do we only see these six digits with sevenths? The process of bus stop division helps us see why and this is where I feel I would start with a class. This is good practice of a technique that should be secure but often isn’t in Year 9 / 10. It helps learn the seven times table and I don’t think it is too tedious to ask students to perform these six calculations manually:

Some learners might want to find 2/7 by doubling 1/7. And then maybe find 3/7 by adding 1/7 to 2/7 and so on. Even if they stick with the bus stop division, they will find that they are essentially doing the same six calculations:

10 ÷ 7 = 1 rem 3 20 ÷ 7 = 2 rem 6 30 ÷ 7 = 4 rem 2 40 ÷ 7 = 5 rem 5 50 ÷ 7 = 7 rem 1 60 ÷ 7 = 8 rem 4

The ones digit is always zero and the tens digit must be less than 7 so there are only these 6 options. Can we extend this rule to other fractions with denominator less than 10? Of course, all others except 3, 6 and 9 will terminate – why is this?

There are also some interesting things to notice about the 1/7 “wheel” before moving onto higher primes. I won’t spoil it for you, but suggest you include the fractions around the outside of the wheel to spot some of the patterns.

We then moved onto looking at 13ths. 11th are interesting too for different reasons, so I can see why we went onto 13th because there are some surprising relationships between 7th and 13ths. At this point, we started using calculators and I would do the same with a class. Or even better, open a spreadsheet, which I am always keen to do!

I’m not sure of the value of kids typing 12 calculations into their calculators, so I might give this image above as a print out for them to write on. Hopefully they will soon spot that there are two sets of recurring digits, i.e. 076923 in 1/13 + 5 others and 153846 in 2/13 + 5 others and which, this time can be written as two wheels:

Ideally at this point you’ll have the class hooked and they would be asking all sorts of questions. Well, maybe enough of them to get everyone else thinking. I would try really hard to encourage the students to come up with these questions. This is what I hope they will ask, but I hope they will also ask questions I hadn’t thought of, something which is a really special moment in any lesson.

- Do we see the same patterns in the sevenths wheel as we do in the thirteenths wheel?
- Is there something special about 6 points around the wheel?
- Why are there 2 wheels? Are there any patterns across the two wheels?
- What about other fractions, do they have multiple wheels?

On the image above, I have shown that as you move clockwise around the wheel, you are effectively multiplying by 10. This is obvious going from 1/13 to 10/13 but actually occurs for all other hops if we ignore the whole number part. i.e.

So if go 3 hops we have multiplied by 1000. I think this goes some way to explaining why the fractions that are opposite each other must sum to 1.

The next prime fraction: 17ths recur after its maximum of 16 digits, so effectively we have one, rather large wheel. Unfortunately, Excel gives up after 16 decimal places (please let me know if there is a way around this) but you can still see some patterns in here:

Beyond that, we found that the following fractions recur after the maximum number of digits so have only one wheel: 3,7,17,19,23,43,59,61 – someone had a much better calculator than me that showed 32 digits!

Other fractions worth exploring are:

- 31ths – contain 2 wheels of 15
- 37ths – contain 12 wheels of 3
- 41th – contain 8 wheels of 5

With the fractions that recur after a relatively few numbers of digits, we can find factors.

Because 1/41 is a decimal that recurs after only 5 digits, if follows that 41 must be a factor of 99999.

And in the larger wheels there are all sorts of patterns – pentagons and triangles in the wheel of 31ths apparently.

I’m not sure how far I’d go with a class, there could be several lessons worth in here. It would depend on how they responded, of course. Maybe 13ths would be sufficient unless…

This post captures only some of what we worked on in this session and highlights to me the depth of subject knowledge that can be gained by attending the ATM conference. Other sessions were more pedagogical in nature, but this one was pure fascinating mathematics and I was grateful to be surrounded by so many knowledgable and friendly people!

]]>

Sitting and watching (actually I rarely manage to sit still for long, the urge to get up and engage with students is too strong!) someone else’s lesson only gives half the picture, however. Going hand-in-hand with the lesson is the shared reflection on that lesson afterwards between teachers. And this is the point. It is not a lesson “observation” in the traditional, pre-2014 Ofsted sense. I am not there to evaluate the teacher in any way. I am a fellow professional who has another perspective on the learning happening in that room. Because I am not leading the lesson, I should be able to notice things, and I may notice different things than the teacher who is leading the lesson.

A lesson observation is traditionally is followed by “feedback” which is more often than not a one-way conversation between the observer and the observed. Usually it is a very polite affair which starts with a lot of “I liked…”, “I thought … was lovely” – the WWW. All nice to hear, but do you ever get that feeling that these are platitudes and really you are waiting for the EBI? The “I thought maybe you could…”, or “In the past, I’ve tried…” I’m not saying that this style of feedback is not useful, especially when the observer has many more years experience that the observed. But I would say that anyone with more than a few months’ experience in the classroom has something to offer and that the conversation should start off very differently.

Earlier this year I was invited by Danny Brown to join a lesson of his. It was last thing on a Friday and after the lesson I also joined his staff meeting. The lesson was fascinating, but it was the staff meeting that has really stuck in my mind since. It wasn’t a department meeting as such, but a voluntary gathering to reflect on a lesson that had been given by one of the department and observed by others. The focus was on something that was “noticed”. It was not an attempt to analyse everything that happened in the lesson, but a focussed discussion on something that was interesting for some reason and that we can all learn from. There was a high degree of respect and trust between these teachers and the discussion became deep, insightful and .

I have been practising this ever since in discussions following lessons, be they informal “feedback” with colleagues (I dislike this term because it implies a one-way flow of traffic) or more formalised TRGs as part of my Teaching for Mastery work. It takes some practise. Commenting on something without evaluating it can be tricky. You sometimes feel like you aren’t really making a point. But actually just clarifying what happened at a particular point can then open into useful conjectures as to *why* that happened. This is where different insights from different people in the room can become really powerful and is the essence of a fruitful TRG discussion.

We need to see a major culture shift in our schools. For too many years, lesson observations have been about scrutiny and accountability and not about close collaboration of a team of professionals seeking to improve their practice. This has led to a culture of fear in schools where many teachers still would rather not have someone “observe” them because it causes anxiety as they feel they are being judged. I would warmly welcome anyone into my classroom at any time and would always want to know what they noticed, but I recognise that is not a common attitude amongst teachers. We need to practice how we share these noticings with each other so that they are truly supportive, non-judgemental and lead to fruitful discussions. And we need to be open and receptive to these discussions and realise that they are about mathematics and learning.

The approach can be very time effective. We don’t need to sit through an entire lesson to notice something interesting, 10 minutes might be enough. One noticing might spark a couple of useful insights on a short post-lesson conversation. I might call this “noticings-lite”. It’s not a huge investment of time, the bigger challenge is the shift in culture.

If we are serious, though, we do need to organise this and having more than one adult seeing the same lesson can generate the range of perspectives. This is what the Teaching for Mastery programme is achieving this year and for once it is coming with funding to enable teachers to be out of class. In my experience watching each other and carefully analysing lessons is simply the most powerful form of CPD there is, far more beneficial than most whole-school INSET. I hope it continues to grow in a funded sustainable manner to increase the skill levels of all teachers.

]]>

Until recently, this has seemed to me like a fairly reasonable thing to do. Sometimes more useful than others depending on the nature of the test and the actual results produced by the students, but generally OK… Takes a bit of time to enter that data but, hey, I need to add up the scores anyway and I love a nice spreadsheet…

But I wanted to know what others thought. So I posted the following poll on Twitter:

Maths tchrs: Are you marking mocks at the moment?

What do you think of QLAs? (spreadsheets where you enter marks for each Q)

Vote, then RT!— Mark Horley Maths (@mhorley) March 19, 2017

As Dave Gale (@reflectivemaths) pointed out, I might have added a category…

@mhorley I’d like a vote option of

Has some use but not convinced it’s worth the time.I do mail merge the results for parents evening.

— Dave Gale (@reflectivemaths) March 19, 2017

I agree, that would have been good, but actually what is more interesting than the results is the discussion that this generated. It seems that there is a quite a wide range of opinion on the humble QLA.

So, let’s start with the positives. Some people really value them:

@mhorley Essential in my view. One of the most useful ways of identifying focus areas (whole class and individual).

— Maths Mr Cox (@MathsMrCox) March 19, 2017

@mhorley Individual feedback for student revision (I provide @hegartymaths clip numbers) and focuses my planning. Absolutely essential.

— Mr. Taylor (@taylorda01) March 19, 2017

@mhorley @solvemymaths @Just_Maths @mathsjem I’m a huge fan of QLA myself, I feel the time it takes is well worth it.

— Jonathan Hall (@StudyMaths) March 19, 2017

@taylorda01 @mhorley I’ve always found they accelerate progress. They often expose gaps I thought were closed.

— Kim McKee (@MrsKimMcKee) March 19, 2017

@mhorley I create the spreadsheets for our dept but have a student slip. Used for independent learning pic.twitter.com/XYf3FUinRM

— Terri Ridings (@TerriRidings) March 19, 2017

Others felt that the workload requirement was unrealistic so maybe we should get someone else to do it but that the end result may have some uses:

@mathsjem @mhorley have used google docs and kids to enter own data with a good group before. Can have issues with accuracy tho.

— Nicke Jones (@NEdge9) March 19, 2017

@mhorley @MathsMrCox get the students to input there scores as homework and then send you the spreadsheet. Copy paste into big file

— Yuvraj Singh (@YuviLite) March 19, 2017

@mhorley @Just_Maths @mathsjem If someone handed me the analysis I’d probably use it to inform some queries, but if I have to create it…

— solve my maths (@solvemymaths) March 19, 2017

And then others who question their value and have stopped doing them:

@mathsjem @mhorley I found making notes on the markscheme as I went along effective. It made a note of why students were making mistakes.

— Richard Deakin (@RichardDeakin) March 19, 2017

@mhorley @MrBlachford The QLA model works well for short predictable 1question = 1 concept tests, but otherwise lots of false positives and negatives…

— Requires Improvement (@RequireImprove) March 19, 2017

This last point is crucial to me. There is a problem if we are doing something that is a waste of time. Time is a finite resource and every additional task we do drains our energy for the important part of the job – being in a classroom full of kids. But actually the problem is greater than it being a waste of time. There is a danger that we read TOO much into our wonderful spreadsheets. The practice of labelling a question with a single topic is highly dubious. Even if we got more sophisticated and labelled each question with the multiple topics that are embedded within it, something which is more prevalent with the new GCSE vs. the old, this still struggles to capture what the question is about and how difficult it is. There is a wide range of difficulty and sub-topics in say, a question on standard form. Answering a simple conversion question accurately could give false confidence that this topic was sorted.

With key skills tests lower down the school where one question maps directly onto one topic, there could be value in charting this progress over time. But for GCSE papers?

There are some elaborate tools available such as Pinpoint learning which produce reports and customised question booklets based on the results. However they still rest on the premise that we can get a reliable read on a students’ proficiency, confidence and competence in a topic by answering one question on it.

Today, as I was marking the latest set of year 11 mocks (the 5th they have done since December), I thought I would do a little experiment. I generally mark the papers first, then enter the data into the spreadsheet. The alternative is to enter the data as you go along. This may seem like a trivial difference, but when we are spending several hours doing this work, it’s worth analysing the most efficient way to do it. I’m not the fastest marker, but I calculated that I averaged 7 minutes per paper when entering the data at the same time and 4.7 minutes per paper when not going near the computer. I was surprised the difference was as much as this. I then timed how long it took me to enter the data, which averaged 1.7 mins per paper. So yes, about 10% quicker to enter the data at the end.

Apart from being mind-numbingly boring (which some teachers get their very understanding partners to help with!), this data-entry task takes 30-50 mins per paper for a class of 30. So at least 1.5 hours work for a full 3-paper set of mocks.

I could spend 1.5 hours entering all that data into a spreadsheet that probably won’t get used, and if it does get used may lead to false conclusions. Or I could use that time for something else, like planning carefully how I want to hand back their test papers ensuring that I am using the lesson time most effectively, maybe selecting or writing new questions that I have a pretty good hunch everyone needs to work some more on. That hunch comes from doing the marking in the first place and doesn’t need a spreadsheet to back it up.

]]>

These are not intended to be fraction of an amount questions. An approach could be to decide upon an amount, but the intention is to direct students to drawing a representation of each question.

]]>

*The power of collaboration and Twitter has led me to Bruce Gray (@bucksburnMaths) who teaches in Bucksburn Academy in Aberdeen. I am a recent convert to teaching mathematics in mixed attainment groups rather than sets but I am a pragmatist at heart and fully understand the reservations and difficulties with this approach. It is not a small decision for a maths department to switch from teaching in sets to mixed attainment groups, especially if other subjects retain sets as my school does. My summary of how to do it which I have written about before:*

*Collaboration and co-planning must be part of it**Do it one year at a time, i.e. start in Year 7, and review each year*

*To find out more, check out www.mixedattainmentmaths.com. The first #mixedattainmentmaths conference was held in January and plans are being put into place for another one in June. If you are interested, leave a comment on this post or get in touch via Twitter or email.*

*Anyway, enough from me. Grab a cuppa and read Mr Gray’s story. Just, please – make sure you get to the end!*

Mixed set maths classes. It’s a phrase in the past that has made my brain shriek in horror and caused me to physically shudder with trepidation. Throughout my time teaching (coming up to a full decade this year!!!) I’ve often wondered “Where would you even begin?” whenever the topic has been brought up in meetings and CPD events and pretty much every time I’ve came to the same conclusion: “Not a damned clue! I’m glad we don’t do it in our school!” My PT (Principal Teacher, i.e. Head of Department) had been talking about it for a while after seeing it at a Stirling Maths Conference (the highlight of any Scottish Maths teacher’s year!) but, much to my relief, nothing had actually came of it.

Then, at the start of the 2015/2016 session everything changed. I actually missed the first three weeks of the new school year. My wife (also a teacher) and I had our daughter on the first teaching day of the school year and I missed the same thing we did every year with our next S1 classes (equivalent to Year 7) – get them in, do some introductory lessons, slap a diagnostic assessment to see what they know, and then split them into the traditional “top”, “middle” and “bottom” sets before getting stuck into delivering content. I didn’t think anything would be different when I returned as “That’s the way we’ve always done it.” Little did I know that I was going to go back and everything had changed! Much to my dismay I returned from my three weeks of paternity leave to discover the department had decided to bite the bullet and take the plunge into the world of mixed set teaching.

I can’t remember exactly what my reaction was to this news but it was not wholly positive – I’m imagining a combination of an ashen faced “You did *what*?” and a sinking feeling in the pit of my stomach. The way the S1 classes had been timetabled to come to Maths meant it wasn’t really possible to set them all in an effective way and would have resulted in a couple mixed ability classes so, in order to give all pupils the same experience, we were just going to keep them all in their register classes rather than in ability groups. A sound reason. The fact my colleagues had discussed it at length and believed that it result in a better standard of learning experience for all pupils also reassured me so I decided to stop moaning, roll my sleeves up, and get on with it.

I hated it.

Serious, genuine, unbridled hatred for the whole damned thing.

I tried my best to be positive, I tried my best to make it work, I tried my best to deliver lessons that meet the high standards I have always set for myself and it felt like every single time I tried during those first few months I failed. Completely, utterly and hopelessly. I remember my misery coming to a head one rainy Winter morning. I was running over my latest set of failed lessons in my head during a 6 a.m. session of making baby’s bottles before going to work when something inside snapped. “I don’t want to go to work. I hate my job” I muttered to myself through a choked back sob with slumped shoulders and eyes reddening with tears. The whole thing had made me totally and utterly miserable. I had no idea what to do. For the first time in my career I had serious doubts about my future. I’d always said that if the job ever became a chore, if I ever fell out of love with it, then I would throw in the towel and find something else. Rather than doing anything silly – like quit my job there and then or hide in the stationary cupboard until the whole thing went away – I had a talk with my PT about it. I am really lucky and have always appreciated his approachableness and the support he’s offered over the years. I was glad I was able to go through the cathartic process of getting it all off my chest and it helped me break the problem down and see what the actual issues were. As far as I can remember there were three points that made those mixed classes the bane of my existence.

1) Organisation

Organisation has never come easy to me. It’s something that I’ve had to work hard at. In recent years I think I’ve been a bit guilty of getting complacent in terms of my organisation and fallen into a routine. I’d gotten to know what each level of class should be working on, how I’d go about delivering the content and what resources I’d use as part of the learning and teaching. Building this bank of knowledge up has really helped me keep on top of my day-to-day and long term planning but it also meant that when faced with something new (like this new system of mixed ability classes) I was lost. The rug was well and truly pulled out from underneath me. My safety blanket of experience was gone and it meant most lessons had to be planned from scratch – something I really wasn’t organised enough to do.

2) Workload

I direct fallout from the above point. Having to plan a new S1 course as it was being delivered required a lot of work all the time. I felt that as soon as I had something planned that I was happy with I had to get on with the next topic – there was no time to stop, collect my thoughts and reflect on what worked as we had to hammer on with the next set of lessons. The big thing that is needed to make mixed ability lessons work is differentiation and while this often takes place in my non mixed set lessons it needs to be done in a much more meaningful and effective way. A greater range of activities needs to be planned which leads to a much heavier workload. Matters were not helped by the fact that I was usually reluctant to use resources I’d found online and instead preferred to use resources of my own. This really was unsustainable.

3) “Won’t somebody think of the children!?!”

I may make light of it with that heading but this was a real sticking point for me. At every meeting we had as a department to discuss how we and the classes were getting on with this new set up I would constantly be bringing up the fact that I was hating it. Everything I said would usually start along the lines of “Look, I want this mixed ability set up to work BUT…” I meant the first bit, I really did want it to work, but I knew I was not meeting all the needs of every child in my class. This was something that really hurt to admit to. I didn’t feel I was doing a good enough job with the children at the extreme ends of the ability scale by aiming “right down the middle” with my teaching. The higher attaining pupils weren’t being stretched at all and the less able pupils found the work too challenging. This was the aspect that made me feel the worst – the fact that I perceived myself to be failing in terms of delivering an education and experience that each and every child has the right to. If you’re not doing that then you quite frankly aren’t doing your job.

Unloading and actually putting into words what was making me miserable was a good starting point to actually addressing the issue however it wasn’t something that was really possible to fix on my own. The department I work in is one I am passionately proud of and a huge part of that pride is down to the people I work with. I may have more in terms of “years served” in the profession than most of my colleagues but whenever I watch any of them teach I almost always come away going “I wish I could do *that* as well as they do.” and I have nothing but the utmost of respect for them. They are the reason I didn’t jack it in and quit when I had my crisis of faith. There is no way I’d willingly leave a team like the one we have. Things didn’t suddenly get better overnight and it took time me to be able to even say “mixed ability classes” without wanting to spit but I personally, did get there. The three things that helped me get there:

1) Divide and conquer

This is key in combating the workload. We had five mixed first year classes all following the same course and everyone planning different things for them. It was a colossal waste of teacher time with all the development work being needlessly repeated. The way around this grew quite organically. I found myself talking to my colleagues more and asking things like “What are you planning?” and “How are you going to deliver this part of the course?” Then I found I was given a well needed boost in confidence by being able to say “Oh, I’ve got something like that already.” and firing off an email with the activities attached. It actually got to be quite exciting opening emails and finding a bundle of emails with new activities ready to use. What ultimately helped me get over my reluctance of using other people’s resources is the fact that I have a deep set respect for my department as both teachers, work colleagues and friends and I trust them wholeheartedly with each and every aspect of the job. Each and every one of them passes the test of “Would you want them to teach your child?” with flying colours and it’s an absolute privilege to work with them. This job really is much, much easier if you are surrounded by good people! Our planning system developed beyond the scattershot of emails being sent back-and-forth and got more organised with everyone taking a topic, collating resources and developing a differentiated plan for the department to deliver. I’ve really grown to enjoy this. The fact that the resources for the current block of work has been prepared in advance frees up time for you to really focus on the next block and really look at what you want to teach and how you are going to differentiate the resources. In terms of what the pupils are doing it really gives consistency across the department. It means all the pupils are covering the same content and getting the same experience.

2) Everyone needs to buy in

This is vital for mixed ability classes to work. Every member of the department needs to be willing to do their share of the prep and to give the teaching of mixed classes a go. I found it much easier to deal with and was willing to give it a go (despite the fact I absolutely loathed the system) because everyone else was in the same boat and me refusing to even try to work as part of the team would just let everyone else down. I find the pressure of “Oh no! Everyone is going to see what I’ve prepared!” a bit of positive pressure. It means I have upped my game and made sure I’ve put a variety of different activities in my blocks of work. It also means I’m doing things that I would never have done with my classes before. I would have balked at the idea of something like a “treasure hunt“ or something similar but, after being taken out of my comfort zone with these classes, I now I find myself willing to give these a go with my set classes.

3) Be willing to try new things

You can’t be precious about the way things have been done in past. This is something I’ve been guilty of before. I’ve often uttered the phrase “Well, I’ve never done it like that before.” when faced with an alternative way of teaching something before trotting out exactly the same lesson as I’d delivered the same time the previous year. I feel (actually that’s not strong enough – I know for a fact) teaching mixed ability has made me a better teacher. I’ve had to think more carefully about what I teach, how I teach and the kind of activities. I also feel I had become a bit complacent in my teaching. I didn’t feel like I was an expert but I definitely was a bit arrogant in terms of “My way is pretty damned good and there isn’t really much point in changing things.” I’ve now realised that it can be a very freeing experience to burn everything down and start building from scratch (ought it’s not something I’d advocate doing too often!). Another thing that has helped is relinquish control (to a degree). Letting others have control of your resources is part of this but giving pupils more freedom over the tasks they are doing has been a very enlightening experience. For mixed ability teaching to work there needs to be a range of differentiated materials provided for most lessons. I assumed there would be a large group of pupils who would always go for the easy route and would pick the straightforward tasks every time. The truth of the matter is I am continually genuinely surprised at the kinds of comments my pupils make. “I’m going to try the easier sheet but if it’s too easy I’ll come back and get this tricky one”, “I think it’s going to be tough going but I’m going to give this challenging option a go” and similar such comments are regular occurrences and while there is the odd pupil who will go for the easier activity rather than pushing themselves I am often surprised by the maturity and insight the kids show when given this responsibility for their own learning.

While I feel I am a convert to the church of mixed ability teaching I would not say I’m a believer who follows its teachings without questioning. Maybe it’s just my suspicious nature but I view each new topic warily and mutter “How this going to work then?” to myself before diving in. I’m only half-way through my second year of teaching this way and I can still see potential dark clouds looming. Whether or not they actually come to anything remains to be seen but it would be foolish to ignore any of these potential problems. Again, I could break these issues down into a trinity.

1) Resources and budget

It turns my stomach whenever I hear of cuts to education be at a national, local authority or individual school level however, despite the fact that education should be viewed as an investment rather than a black hole money is just poured in to, these cuts are a very real and current issue which means we are expected to do exactly the same job with less funds to go towards day-to-day items such as basic stationary, photocopying and printing budget and the purchase of resources. A major factor that has made our mixed ability teaching achievable is that we have moved away from textbooks (for the most part, there are still times where a good, well written textbook is the best resource for the job in hand) to differentiated worksheets, card matching activities and active learning tasks all of which rely heavily of having a healthy photocopying budget. We are trying to make savings by reusing resources but that is not always possible and it is inevitable that new copies of resources will have to be made. The fact our class sizes are small has also made progress with this system much more manageable. It goes without saying it’s far easier to get around a class of 20 individuals than a class of 30+. I know we’d cope with bigger class sizes but I don’t think he pupils would get as rich an experience and from a classroom management side it would be that bit trickier to administer. Our teacher numbers and class sizes look safe for the future but it is still a very realistic concern to have.

2) The extreme ends of the ability

I think we are doing a pretty good job of catering for a majority of the pupils in the classes as the lessons are well differentiated or the content is delivered in such a way to make it accessible. There is still a nagging doubt though that some pupils don’t have all their needs being met. It’s easy to worry about the low end pupils (and when I think about differentiation my mind automatically goes to “Right, I must plan something for the less able pupils”) and it’s easy to forget the high end pupils or to just give them more work rather than something truly challenging. I don’t think that this is a problem inherent to mixed ability class. Even with classes set by ability you will have individuals that fall out with the main group. Now I sit down and organise and type out my thoughts it seems less of a problem and more something to keep in mind and not let slip.

3) What effect will it have long term

In the short term the mixed ability classes have been a hugely positive experience. There is a genuine buzz in the department when we have these classes in and I think it’s making me get better at differentiating with classes across the board. It’s got rid of the classic problem of kids being turned off from Maths because “I’m stupid – I’m in a bottom set, I’ve always been in a bottom se and always will be so there’s no point in trying to better myself.” Sure, there’s kids that still don’t like the subject – you can’t please all of the people all of the time – but comments of “I can’t do it.” are gradually being replaced “I’ll give this a go” and “This looks tricky. I’m not ready for it… YET!” It’s really encouraging to hear little comments like that being made. It’s part of the reason I get up in the morning! The long term effects remain to be seen. My gut feeling is that it will help attainment. If pupils are engaged and motivated it feels like that has to be the inevitable outcome but we can’t say that for certain without cold hard facts so we’ll just have to wait for the dreaded benchmark of “What did they get in their exams” a couple years down the line before labelling our journey into mixed ability teaching a success.

In conclusion… Well, I’m not sure I have one. Do I still hate mixed ability teaching? Is there still the “Serious, genuine, unbridled hatred for the whole damned thing” I mentioned at the start of this ramble?

No, not at all!

Hate has given way to something else. Love? No, not that but thoughts along the lines “I’m can’t do it” have definitely been replaced with a gleeful “I’ll give this a go.”

]]>

- 3.4 – 3.04
- 5.2 – 5.02
- 7.8 – 7.08
- 8.2 – 8.02

Find other questions like this. (The “weakest” student in the class told me the pattern before I’d even finished writing the fourth question on the board.)

What do you notice? Why is the answer to Q2 the same as the answer to Q4?

Can you create a question with 0.54 as an answer? How many different answers are there to these types of questions?

Then:

- 5.7 – 5.007
- 8.3 – 8.003
- 6.4 – 6.004
- etc.

These are more tricky and test the skills of column subtraction, something that should be secure by Year 7 but may not be. Maybe an opportunity for collaboration amongst students to show how.

And then finally, try these two calculations. Which is easier and why?

7 – 1.392

6.999 – 1.391

Show on a number line why this works and then try some more. I think these questions are interesting to explore. But I would hesitate to recommend it as a must-do method to solve e.g 8-2.5687. Whether or not it is easier to turn it into 7.999-2.5686 or not is an interesting discussion and one which I would want my students to form their own opinion on, not be too swayed by mine.

]]>

As with all things Geogebra, I always try to start with a blank sheet (see other posts on this here and here). This time, I’m not using the Geogebra app itself but just launching it from within a Chrome browser window which works pretty well.

Once it is launched, I right click in the middle to remove the axes, but I am going to leave the grid on.

Then I create the triangle by constructing a line, a perpendicular line…

…and a third point which I then join to create a triangle using the polygon tool.

Next, measure the base angle of the triangle remembering the convention that angles are measured in an anti-clockwise direction.

The next bit is a tad fiddly. You need to right click on the line segment to change the label to “value”. Then do the same for the other two sides of the triangle so that you now have one angle and all three side labelled.

So far, this has taken about 2 minutes to create from a blank screen. You could do it in advance of the lesson, but I think it is worth doing it in front of the class, maybe having practiced it a couple of times. Using “something I created earlier” is less powerful – it looks like some sort of trick, somehow.

Now you have everything set up you can start moving the points as shown here.

I start by moving point B, thus keeping the angle fixed. I would ensure students have calculators in front of them and ask them to calculate opposite divided by adjacent. Then move the triangle to get different values for side lengths. Then do the calculation again. The answer is the same, of course. I might ask them how they could get that directly from the angle (tan angle). Depending on where the discussion goes with that, I might then move on to look at sin and cos.

Finally, I always like to talk about how things were done in the old days, being careful to point out that I’m not that old and that I *didn’t* actually use these…

I explain that the sin button on your calculator is basically just looking up the values in the sin column of a table like this – not actually true, I know, but it helps understand what’s going on so that’s OK for me!

]]>

What do you notice about the difference between them?

Express the difference between the square numbers as a sequence in terms of n?

Express algebraically the difference between the nth square number and the (n+1)th square number?

Use you algebra skills to show that they are the same thing.

]]>