Other blogs have included the cubes, but here’s a couple more examples.
First, Stacey demonstrated how to collect like terms, including different colours for negative and positive:
And second, Adam found the twelve pentominoes, and one of the two arrangements of a 2 x 30 rectangle. We then went on to identify the lines of symmetry of each, and the order of rotational symmetry where it exists.
All good stuff.
It started, as usual, with difficulties in the classroom. I have a set of quite simple equations on cards, to be matched to the appropriate solutions. The learners did fine apart from the ones in which the variable, call it x, appeared on both sides. And as usual for me I looked around for a suitable representation of x. As it happens I have a number of empty Oxo boxes in the cupboard I keep for work on volumes (an Oxo cube is 2 by 2 by 2 centimetres, and so are my plastic cubes). So I tried to display the equation using Oxo boxes to represent x.
It worked, after a fashion, and in the example above you can see we moved on to negative numbers. But how do I roll this out to a whole class, without needing an enormous number of Oxo boxes? I moved on to envelopes, with each envelop representing p, which was part of an exam question. I’m afraid that was too much for one group, who could see no connection between my envelopes and the question before them on the exam. Plus I don’t have that many envelopes to use as consumables.
So back to the cupboard and the tin of matches. I bought these first for the typical patterns questions on GCSE – a simple number pattern illustrated by a matchstick question. We’ve had some fun making our own.
We began with something simple, created by Shannon:
She set this for Salma, and then explained how to solve it, which was interesting in itself, demonstrating each move:
And finally she went to the board and gave us the symbolic version:
Then Salma made one for Shannon, which was not quite so complex:
And it was all so move involving and engaging than the usual book stuff. Equally, professionals seem stuck in the written mode and refusing to consider practical experiences. I’m not saying that my learners have had a better experience, but consider the version from the Standards Unit. I’ve tried this in the classroom, having lovingly produced laminated versions, and the whole experience was a failure.
From our positive matchsticks we progressed to negative matchsticks, and with numerous cubes of many colours we could use a range of letters to represent the variables – pink cubes for ‘p’ for example. I enjoyed it, and I hope the learners did, and gained a little more insight into the mystery that can be algebra.
And a final thanks to Shannon.
When Ofsted are on the way
Not long ago, and I’m sure it is a regular occurrence, there was a twitter post along the lines of ‘Ofsted in the morning – what can I do?!’. My reply was quite simple – it’s far too late to be worrying about that – inspectors need only speak to the learners to find out what normally happens, regardless of what you are doing on the day.
I have some simple criteria for the classroom – nothing to taxing and easy to measure:
Are the learners enjoying themselves?
Do they appear to be confident?
And that’s probably about it – all the rest is the fine detail.
Get these two sorted and the quality of the lesson will shine upon the faces of the learners.
Enjoyment is apparent to any visitor, and the elements are part of any lesson plan. Are there a variety of activities, is there hands-on, colourful and active things to do, are the learners engaged and challenged, are they co-operating with one another, and are the tasks designed to encourage co-operation? Do they take the initiative, or wait for guidance and direction at every move? Are they confident to make mistakes and be corrected by each other, rather than by the teacher? Do they encourage and reward each other, or is that the prerogative of the teacher? (In one never to be forgotten lesson in which I was being observed, one learner did something most noteworthy, and someone else shouted with obvious enthusiasm ‘She should get the prize!’ – I’m afraid I don’t run to prizes anymore, but I do keep a box full of stars.)
Confidence is built upon each lesson. Do the learners know how each lesson fits into the scheme of things, and do they have access to the scheme. Do you share the objectives, either in the lesson or in a plan that embraces the whole year? And is that pinned up or distributed at the start of the year? Do they have benchmarks to measure achievement, and are they clear and understandable? (My benchmarks are what would have been needed to get a desired grade on that particular past paper.) Is there a ‘virtual learning environment’, and do you remind the learners each lesson where they can find what we are doing, what we have done, and what we will be doing next? And do you post regular photographs so they can see themselves doing it? An alternative is to put the resources on a public site – I take great pleasure in showing them how to find my stuff, by putting my name and the topic into a Google search and seeing it as the top hit. Are the learners encouraged to seek other sources of learning outside of the classroom? Khan Academy has greatly improved, as has BBC Bitesize and a number of other free resources.
Confidence in the learning process is built by the learners, not the teacher. Can they check their own understanding of learning, or is it done when the teacher marks the stuff and provides feedback. Do you give out answers and solutions or get learners to come to the board and share their methods and solutions? (This year I have been providing exam-board mark schemes with all class exercises and written tasks so that the learners can see if they are achieving the marks allocated.) And why wait until an inspector asks them if they know how they are getting on, or if they understand? Survey Monkey and Socrative are ideal tools for getting instant and regular feedback. I’ve posted feedback on the wall outside my classroom, which happens to be the corridor to the management suite, and done it in Wordle to make it even more satisfying!
When my boss asks if I have tracking sheets and records I’m often at a loss as to what to say. Is it about recording completion of homework, success on homework or class tests, or about learner satisfaction? I provide a handbook with the details and the criteria for the award, and a column for the learners to tick, date or whatever, but that is their responsibility. I keep records of past exam paper attempts, including modular ones with can be brief and easy to do in a short session, but are these really significant? More important is if I am genuinely interested in each and every one of them. Have I spoken to each person each lesson, either by asking a mathematical question or any other device?
Speaking to everyone is what I do, and over recent years I’ve found ways of engaging them that never occurred before. For example, when we need numbers at random for prime factor decomposition, I once thought it clever to use Excel to generate these. Then I bought a Bingo machine – turn the handle and out pops a ball, which a learner can do. Now I go round the room and collect house numbers which we duly factorise. For scale drawings I use local maps – learners stick a label on their home and measure the distance to college. For any statistical data we get it from them, including distance to college. (With this last we were able to utilise converting metric to imperial, all kinds of average, and calculating average speed.) Compare learner data to national statistics, and remember Census at School as a wonderful resource. I try to do something along these lines every lesson. This week we did ratio and proportion, and most classes came up with baking/cooking as something which uses proportion, and it just so happened that my son had sent by Snapchat a picture of his dumplings, which I kept for display, and which led to a discussion on taste, mixture, and what a dumpling equivalent would be in both India and Bangladesh – it all depends on who is in the room. 
So the question was ‘who makes the dumplings in your house?’ The trick is simple – put the learner at the centre of the experience, not the teacher. The learner appreciates this, and Ofsted can see it a mile off. But do it from the start, not the night before they are due to arrive!
Getting help from the learners
Too often we teachers think we know what is best, and if not, we know where to go to find out. We get professional development, observations and mentoring; we read books, blogs and tweets, and of course, we are the teachers, so we must know. But this week I had an opportunity to talk though an exam with a learner, after she had passed it, over a long cup of coffee far away from any learning environment, and she was able to give me a personal insight into how a different type of mind works, one that is not ‘naturally’ mathematical, which I guess mine is.
We’ve been working together to achieve the mathematical qualifications needed to become a teacher – first GCSE then the QTS test. The former was easy enough – there’s a wide enough spread on the curriculum for her visual and artistic talents to make up for some of her difficulty with number work. But the QTS test is mostly number, with a bit of statistics, and has constraints that a long GCSE paper doesn’t have. These are the mental test, with a few seconds to answer each question, and the second section which varies between the deceptively simple (‘can it really be that easy?’) type to the longer, wordy ones that may suggest a very clever question but can give any of us a feeling of ‘What!’ My approach was problem oriented, using the sample tests and similar questions I created, and we’d go through each type with regularity and diligence. Any real issues I’d return to with alternative approaches. For example, for averages I printed items of data on single cards – ‘Show me five cards with a mean of six; show me five cards with a medium of six and a range of eight’. (Averages) All ideas cheerfully copied from ideas in John Holt’s book ‘How Children Fail’, which should still be required reading for all teachers. When we struggled with fractional equivalents, including decimals and percentages, I produced visual images that included diagrams and numbers, which my learner stuck around her bedroom walls. PostersVisual Displays And I bought a set of Cuisenaire rods, which were well worth the small investment. But for things like currency conversion, and using speed/distance/time ratios, how does one do that practically in a classroom?
My normal approach with complex arithmetic problems is to work with simple cases. For average speed problems for example I encourage the learner to consider simple cases – if I drive at 40 miles per hour, how far will I go in three hours, or cycling at 8 mph, how long will it take me to travel 20 miles? For currency conversion, how many euros will I get for two pounds, three pounds, two hundred and fifty pounds? All slow and I assumed effective. But when the learner is faced with such problems in a test, does she have the time or skills to think ‘What did Colin do for that?’ or ‘Can I make it simpler?’. Certainly not in the QTS tests, when time is a premium, and I’m beginning to doubt if learners do in any exam. They have been encouraged to get down an answer, and too often this can be any answer. This week I have encountered a learner entirely happy with his own wrong methods, and almost every week I meet one who says, erroneously, ‘That’s what I’ve been taught!’ Not the mind-set that encourages thoughtful reflection in an exam, and there again, exams aren’t designed for reflection. So I keep making up questions, and saving them for future use. Conversions.
And now my aspiring teacher has given me some great tips, and ones well worth remembering. In all of the things for which instant recall of knowledge is required, the learner needs to have the ability to bring it straight to mind. Flash cards worked for us, providing drill and repetition. It hurts me to say it, but for some, if not most, learners it seems drill has an important role to play. It especially applies to number bonds. I really didn’t know my times-tables thoroughly until I became a teacher – I could go through the pattern of any table quickly enough, and correctly, to satisfy my needs. But I’m a mathematical person; is that any help to one who isn’t? Not just multiplication – learners need addition bonds, and secure ones on which they can build more complex additions, along with strategies to do that, and subtraction methods such as ‘shopkeeper’s addition’.
Finally, there is the issue of getting the brain stimulated and using as much of it as we can. Recent evidence shows the usefulness of playing a musical instrument to get those connections going. Another learner of mine is about to receive intensive help for dyslexia that includes playing with soft balls and using modelling clay. This learner practised tracing with her left hand what her right hand was doing. Don’t neglect the physical side of the brain; it all helps to get it going in top form. Perhaps a little session juggling three balls – an easy enough skill if taught properly. Use what the learner has, and encourage other things too. Often my individuals are artists, so encourage creativity and imagination. Don’t provide all the methods and solutions, let the learner come up with examples and ideas.
So I’ll be making some more flash cards, for use with individuals and whole groups. With a big group, they can play in pairs and groups – learners don’t need a teacher to direct every activity. But get things moving, and keep a kinaesthetic element in every lesson, if possible related to the topic in hand, but if not, remember that distributed practice is a hot topic of the moment, so use a topic from a different part of the syllabus if necessary. And keep testing the questions that are likely to come up, because that is the other method receiving current attention ‘Improving Students’ Learning.’
Actually, it was meant to start with a map. Give out a local map to each table, and ask the learners to stick a label on their home, a label on the college, and find the distance. If a learner lives ‘off the map’, then use Google Maps to find the distance, and convert to kilometres. A nice little exercise on scale, that provides suitable data for analysis – what is the average distance travelled, for example, and then possibly draw a network diagram and plan a route between the group of friends at the table. But we fell at the third hurdle: we found our homes, we measured the distance, but got stuck on the scale of 1:12000. What does that mean?
So out came Barbie, of which I keep a few in the cupboard. (I’m hoping one day to expand my collection to Kens and Barbies of obviously different ethnic origin, but at the moment I’m limited.) Is Barbie a scale model of a human? This always produces an interesting discussion, and we never fail to learn something new. For example, one’s feet are generally the length of one’s forearm, which is clearly not the case with Barbie. Much fun has been had by measuring ourselves and Barbie (not me – I keep out of that bit!), and a lot of mathematics covered. And we arrived at the conclusion, by measuring Barbie and measuring Charlotte, that the doll is on a scale of about 1:5.7.
We measured in centimetres, rather than ‘hands’.
And then, for some mysterious reason, possibly because we were playing with models, someone asked if we really did have a horse in the college. Do we have a horse? Yes, in the room next door! So in we went to measure the horse, and calculate the height of a suitable horse for Barbie, all still working to scale. Of course, not every institution is luck enough to have a life size horse.
And after that we went on to model cars and we did think of going out to the car-park to find a real Ford Focus, but it was raining so we did some internet research on dimensions. Eventually we were ready to go back to the maps, with a thorough and grounded understanding of what is meant by a scale of 1:12 000, or at least I hope so.
I did have a few other toys to work with, supposing we needed more practise, or should I say scale models. We have Dr Who, although a previous incarnation, a motorbike, and lots more. I have a big cupboard.
And there will always be one learner who does his own thing – or her own thing, and in this case it was Hollie, who declined to be photographed, but not a problem.
I ‘heart’ mathematics!
I’m still hoping one day that I’ll have learners designing a scale model of the Tardis, although I appreciate they aren’t easy to find outside of the capital.
Sam and the Tardis – not many about these days.
And after that? We collected the data on distances, with a little help from one another we converted to kilometres, and each learner wrote their measurement on the board, along with those who had used Google Maps. From there the learners were invited to analyse the data, and we readily found the range and the mean. The mean gave us 5.9 km, which was suitably suspect since nine learners lived closer than this and only two lived a greater distance than this. By then it was time to round off, so for a final activity the learners lined up in order of distance travelled, furthest on the right, nearest on the left. (They would not submit to a group shot I’m afraid.) It keeps them talking as they have to ask one another the detail. Then I invited the person who thought he or she was ‘the’ average to take one step forward, and the girl in the middle duly obliged without the slightest prompting. Median sorted, and all done!
Each time we cover probability in the classroom I warm more to the topic. Not just because it offers an opportunity for the learners to get hands-on experience with simple equipment and addresses a variety of learning styles, but also because it forces discussion, and we can bring in so much simple mathematics that would otherwise be rather dry and dull.
My first resource then is a large tin of equipment – dice of every shape and size (all the regular Platonic solids, to be precise), playing cards, lots of coins of different value, spinners, a Bingo machine with ninety numbered balls, coloured counters and cubes, and so on – and a healthy imagination.
My next resources (all available on the TES resources site) consist of lots of sets of cards for problems, discussion and ordering. The first set I made were True/False cards, with some obvious answers and some not so obvious – results of tossing one coin, two coins (are two heads, two tails and one of each equally likely?), lottery results, outcomes of dice, weather events, football results, and so forth. We’ve enjoyed discussing these over the years, but I often feel my input has been greater than that of the learners. So I introduced matching cards – the event on one card and the probability, as a percentage, on another card. Ordering these were simple once the probability was determined, and big displays with number lines were produced for classroom decoration. Then I introduced some experiment cards – the results from two coins, adding two dice, looking at football results, etc., so that the learners have to actually do it, rather than guess at the answer, which is often the case. Finally I developed the A4 cards, duly laminated, with fifteen scenarios that covered possible natural events like snow in June (I can add snow in October to that this year) and experimental events, like scoring a double six with two dice, picking the ace of Spades from a pack, any ace, any Heart, and so on, ranging from impossible to certain, with some unlikely, some very unlikely, and some very likely, along with those for which it is possible to calculate a value. These are my favourites, since it involves fifteen people trying to arrange themselves in order of likelihood, with any spare learners helping the proceedings. There is nothing I like better than taking them into a big space and watching as they do the work and I sit back and take a photograph, for display and checking back in the classroom. (It always impresses management too – seeing the learners taking responsibility in what one described last week as ‘warm and supportive atmosphere.’)
It is so easy with probability to think the learner has grasped a concept, but the teacher can easily set up scenarios in the classroom and ask the learners to decide if it is fair, or true, and make sure they have the resources to answer it. Are more boys than girls born each year in the UK? Is a draw in football less likely than a win/lose result? Is seven the most likely score when adding two dice? Teachers too often think because they have said something the learner has grasped it – the first fallacy of teaching.
And finally it can bring in so much. The obvious things are being able to convert between common fractions, decimals and percentages, and between equivalent fractions if one fiddles the answers a bit. Being able to compare and order fractions is something the learners have to do to align themselves on an imaginary number line. And then the whole area of combinations – the possible results from two successive events, for example. For years I would entreat my learners to be systematic, until the day I asked if anyone knew what ‘systematic’ meant – not a single response. All skills required by the National Curriculum and GCSE, and useful skills for life, being able to consider and enumerate the whole set of possibilities. And of course, and understanding of likelihood really is a skill for life – how many times do we make stupid mistakes by simply not considering the likely outcomes?
Do try some of these things yourself, and keep it active and meaningful. It might also be rather fun.
I’ve been reading the reports of the summer GCSE exams – not the press reports on the issues surrounding GCSE English, but the reports that matter to me, teaching GCSE maths. Thanks to technological innovations, and a certain forward-looking exam board, we get two types. The first is the overall report of the exam itself, with grade boundaries, pass rates, and the chief examiner’s comments on the papers and the individual questions. The second is the local analysis that provides information on the institution, classes, and individuals down to the level of each question. So for example I know that we performed better than the national average, we exceeded the average on almost every topic, and that candidate W got question one wrong. And what strikes me about the reports is not the self congratulation of being better than average, but topics on which nationally so many learners perform badly, and the examiners comments on these issues. And I find it rather disconcerting.
And equally worrying is the fact that the chief examiner appears to be using ‘cut and paste’ over successive years. The following appeared in 2009 and again in 2011 in identical form: ‘All candidates should have a calculator for the calculator paper. Examiners become very concerned when it is clear that candidates are desperately trying to solve complex problems by hand, suggesting a calculator is not available.’ Do we learn nothing, us teachers?
A few years ago coursework was abolished for GCSE mathematics, with the comment from the examiner that we could ‘now concentrate on teaching’. I was worried then that someone within the examination system was dismissing the benefit of investigative learning. So many teachers see our job as one of teaching, training and drilling, with little focus on learning. And things haven’t improved much – a recent comment on a badly answered question demanded that ‘This needs to be taught more rigorously than before.’ And when a question has been answered well, credit is given to the teacher: ‘Many candidates had clearly been drilled into the correct form of words, and for them this was an easy mark.’
But it is clear some topics have not been sufficiently drilled. In June 2011 ‘The difference between perimeter, area and volume remains a mystery to many candidates.’ And asked to draw a rectangle of a given area in 2012, incorrect answers showed ‘…confusion between area and perimeter. A very small minority drew a triangle instead.’ There remains then an issue with simple shapes: ‘Two third of candidates could not write down the mathematical name for the quadrilateral.’ It was actually a trapezium, and a drawing of one is on the inside cover of the paper, with the formula for finding its area. Other technical terms confounded the candidates, many of whom could not differentiate between parallel and perpendicular. And three dimensional shapes fare worse. ‘Not many grasped what the question was asking. It was clear that many candidates struggled to visualise what shape would need to be added to make a cube.’
Shape and space questions may throw up some serious misconceptions, but other topics are equally susceptible to a lack of understanding of the technical terms. In 2012 ‘Many confused median with mean and mode.’ And I rather like the uncertainty about certainty – I’m quite happy to believe the sun will come up tomorrow, but apparently we need to make this more clear to the learners: ‘Candidates should be advised about the practical interpretation of likelihood, e.g. that although nothing can be considered to be truly certain, like the sun rising tomorrow, that for all intents and purposes the probability that the sun will rise tomorrow is as close to certainty (i.e. unity) as makes no difference.’
It seems to me that it is an issue of communication, but I’d welcome discussion. If learners can’t distinguish between words, isn’t it because they have not had sufficient opportunity to use those words in the appropriate context, rather than not having been ‘drilled’? (And it’s so long since I used the word ‘drill’ I’m not sure how to do it, not in a learning environment.) What do I do to be better than average? Sit back and let the learners do the talking – let them make and then name the shapes, let them count the vertices, edges and faces, and report their results. I set investigations involving area and perimeter, which they share, using our special mathematical language. I bring in the learners own personal data, which we share and analyse, and I hope to give as an example when we do it again this year. And we talk about events and likelihood, so that I’m quite confident that my learners appreciate that statistically speaking, the sun will come up tomorrow.
What I don’t do is use classroom time for repetitive examples from books and worksheets, or actually anything that asks the learners to work alone and not communicate with one another. And on what did we do worse than average? We dipped below on a handful of questions on manipulating simple algebraic terms. I’ve hopefully addressed this with some kinaesthetic card matching activities that combine shape and algebra questions, which incidentally was a question on one of our summer papers. It proved difficult for almost everyone. ‘The last and most challenging question on the paper had only had 4% of correct responses with many not attempting the question at all.’ So I’m learning from my mistakes, as hopefully the learners do, and plan to do better next time, which for many learners is not an opportunity that comes their way.
‘Teach me to divide’ she said
One wet afternoon when rain had stopped all play.
‘I can share OK to ten, but I need to learn the long way.’
Whatever for, my innocent young friend, for don’t we have a calculator as our foil,
And did not Ada Lovelace work such miracles, to take away such toil?
Learn algebra by all means
Seek patterns and relationships
For life is never really what it seems.
But better still, learn a poem, or a few
So that on some moonlit night
You might recite a verse or two,
To some young lover, arm in arm with you.
I have had great difficulty in embarking on this little piece; how can I write about criticism without being critical, when I find being critical so difficult? Of course, I come home from work and complain, and from lots of other places and complain, but in the safety of my own dinner table and in the knowledge that my grumblings will not leave the room. What is it that prevents me from sharing my expert knowledge with colleagues and other professionals – is it simply my own diffidence, or do we lack the culture that might encourage robust criticism of one another? As teachers we have ingrained habits and attitudes, the way we always do things, and often assume other methods are either inferior or simply wrong. (Actually, they may be better ways of doing things, but we lack the skills of doing it that way.) And to compound this we may be doing things in the way we were taught ourselves – we who succeeded in the education system now relying on the methods that brought us success.
By way of example I will take one of my pet hates – the taking of the mouse. As a learner myself at a local university I was following the teacher’s instructions to load a programme, but getting somewhat lost I asked for assistance. The teacher came over, took hold of the mouse, and proceeded to do the thing for me. ‘Get off my mouse!’ I shouted, and explained that I was doing the learning, not her, and it required me do to the thing. More recently a variation on the situation happened. I was in the process of interviewing a prospective learner. I had given her a set of simple instructions to follow to undertake an online diagnostic test and asked a colleague the password to get on the local system. My colleague immediately took the learner’s mouse, logged her in, selected the options, and set her off on the test. Inside I’m shouting ‘get off!’ but I can’t say it, then or later, for whatever reasons I still can’t fathom. From the first meeting my aim is to empower my learning, and when I stand back to do so, and someone else jumps in, I get rather irate inside, but keep it inside.
Going back to the same university teacher, on a course for teachers, she had us cut out a triangle, tear off the corners, and arrange them to make a ‘straight line’, thus proving, she said, that the angles of a triangle add up to 180 degrees. ‘No we haven’t!’ I cried ‘you have demonstrated, rather roughly, that it works for this specific triangle, and in no way proved the general rule.’ I’m not confident that she grasped the difference, but why is it we find it so easy in a hierarchical order? I’m always happy to criticise someone teaching me, and someone I’m teaching, but never fellow teachers.
Online forums and resources seem to exasperate my feelings. One of my managers often suggests that I look at something that has been recommended to her at some training session, or that she has read about, and I’m quite happy to write a page of thoughtful criticism. When I see things that I think are a waste of money, considering the huge amount of free resources available, then I simply ignore them. But sometimes I see free things that so offend my pedagogical beliefs I want to offer that criticism. And often those things have been willingly shared by other teachers who, like me, spend hours preparing resources they are glad to share. If it is something that is based on a sound idea I’m happy to steal the idea and build upon it in my own way, and I hope people do the same with me. But what if the principles are at odds with what I passionately believe in?
Should we keep it bottled up, being the best place for it, or should we share our thoughts in the hope that all of us can learn from one another, and that the listener or the reader will hopefully appreciate the place that criticism is coming from – a deeply held love of learning?
Rich Questions
Many years ago, in the early days of Inset and ‘Baker Days’, when people still knew who Baker was, a colleague gave a demonstration of a simple resource, an apple, and how it could be used to enhance his teaching of modern languages. I’ve never forgotten it, as the epitome of a crass and thoughtless teacher showing off his knowledge without a care for our bored minds and beleaguered bums. He spoke without pause, and we probably forgot every word he said. I’m entirely in favour of simplicity, and love a simple question to start things off, and watch the threads take off in a variety of directions; in fact, the simpler the better. My group on Friday afternoon are returning learners of assorted ages, nationalities, and existing skills. I’m confident all of them have a wide range of mathematical knowledge, and my job is to get them back into critical thinking mode before term starts in September. We’ve covered some algebra and some ‘shape & space’, and lots of incidental number stuff in the first couple of weeks, and this week returned to shape.
Me – ‘Draw something with an area of 18 square centimetres.’
I’d supplied centimetre square paper, rulers and pencils, and the rest was up to them.
After a few moments:
A – ‘I’ve got them all – rectangles 1 x 18, 2 x 9, and 3 x 6 – that’s all there is.’
C- ‘What about 4 x 5? 4 + 5 + 4 + 5 = 18?’
A – ‘No, you are confusing area with perimeter – area is the inside, perimeter is round the outside.’
(A demonstrates to C on C’s drawings the difference between the two concepts.)
Me – ‘Are you sure this is all?’
Another pause…
C – ‘Can we use halves?’
Me – ‘Did I say we couldn’t?’
C – ‘In which case we could have 4 x 4.5, 1.5 x 12…’ and goes on to list numerous other cases. By this time, C has moved entirely from the concrete to the abstract, and has ceased to draw rectangles.
Meanwhile, B is thinking…
B – ‘Can we have a square?’ And answering herself: ‘No, because the side would be the square root of 18, and that isn’t a ‘real number’’. She might not have said real, but something like it, indicating it wasn’t something we could measure with a ruler. I actually take part for a moment, stop asking questions and open a calculator on the interactive board to show that root 18 is a damned tricky number, if you want to catch hold of it with a calculator.
Me – ‘You might like to come back to that one.’
B- ‘What about triangles? Can we have an equilateral triangle?’
Me – ‘That might require some serious mathematics, but there are lots of other types of triangles that you might look at.’
C then produces a number of right-angled triangles, and demonstrates to B how she has halved assorted rectangles of 36 square centimetres to produce triangles of 18 square centimetres.
Meanwhile, A is ordering her thoughts around the number of variations on the original question.
Me – ‘Are there an infinite number of shapes with that area?’
A – ‘Yes. I’m thinking of a splodge, and that splodge can be any shape at all that covers 18 squares.’
Me – ‘A splodge – I like that. Sticking with rectangles for a moment, are there an infinite number of rectangles?’
A – ‘Yes there are. If I have an area of 18, and a given length, I can divide to find the width.’ We are now into re-arranging formulas, by intuition.
Me – ‘What did we refer to when we were looking at patterns last week, and general terms?’
C – ‘A variable.’
So now I’m inclined to open a spreadsheet on the screen, and A demonstrates confidence and fluency with Excel. I put in some headings, and she explains the formula for the ‘Area’ column.
A – ‘It is the contents of the cell A2 multiplied by the contents of the cell B2.’ Spoken like a pro, and delivered with more clarity than a good many IT teachers could muster. We pause whilst A explains the concept of spreadsheets to B and C. She then does all the clever bits, like copying down formula, and creating a new sheet for doing it backwards, with area fixed and length as the variable. After which A and I get into a technical discussion where she initially sees ‘infinity’ as going on forever, and I’m hoping for her to see the infinite number of cases bounded by the ones she has. In other words, she can see rectangles stretching forever and getting thinner, and I’m seeing an infinite set of widths, say, between 4cm and 5cm. But time is pressing.
Back to D, who by now has moved from right-angled triangles to isosceles and scalene triangles, all the while rediscovering her knowledge and naming those shapes.
M- ‘What about the square – is there any way to construct it, without having to measure root 18, and thinking of what you have done to find the triangles?’
D – after a very short reflection: ‘Yes, if you draw a square of 36, then cut off the corners to create… What’s it called?’
A – ‘A rhombus?’
D – ‘Yes, a rhombus, a square that’s half of the square, another square inside it, tilted…’
Me – ‘Come up and draw it!’
She does, and we have to break off shortly, with the briefest recap of what has ensued, what has been revisited, learnt, expanded upon. It’s show time, and C and B have a suitable array of rectangles and rectilinear shapes to share.
And an hour was far too short. What could we have done with two, and what might we do next term when they start for real:
Do all the shapes have the same perimeter, and if not, which has the least? Is there a maximum?
Will we find that parallelograms are easy to compute, and are trapeziums just as easy?
Are there any simple regular shapes that have an area of 18 sq cm, apart from a square (which wasn’t that simple at all), or do they all behave like equilateral triangles and need skills we might not learn in a few months?
Will we get to circles, and rearrange that formula that Maggie Thatcher said every pupil should know, to find a circle of 18 sq cm, and has it the smallest perimeter? (Cherry pie delicious, apple pies are too.)
And can we generate the formulas for the ones we’ve found, producing simple elegant statements with symbols and operators?
And could we use those formulas to generate variations, as we rearrange them according to known variables?
I’m not entirely pleased with myself, although I’m pleased that everyone has shown learning, and that both A and D have demonstrated true mathematical thinking and insight. B was not confident that he’d contributed, but the others reassured him that he’d be supported in future. I think I was indulging, and could have possibly tried to keep things simple at this early stage. And next term I will. But there again, challenging is good. And I had fun, and remember, that’s the important bit, I’m the teacher!
colinbillett 