Sequences in square numbers

List the first 6 square numbers.

What do you notice about the difference between them?

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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.

How do we report and measure attainment in Maths? And Why?

An interesting discussion in the Maths office this week has led to some musings.  When the English education system moved away from Levels, my school took the new GCSE (9-1) grades and “translated” the old Teacher Assessed Levels (TALs) into new Teacher Assessed Numbers (TANs) as shown here on the school website:

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As I teacher I am required to provide a TAN on each student I teach at various points in the year. As I was doing this earlier this week I started thinking about what these are used for.  Here I am not talking about the actual process of assessment. I am just thinking about why we collect that information and what we do with it. It seems to me that there are 3 key recipients of this information each with their own agenda.

1, The Students.  This is a form of feedback, albeit a very blunt, summative piece of feedback that basically tells the student how good they are at Maths (in my case) summed up in a single score. The student may be aware of their previous score(s) so they may also get a measure of their progress.  They may also discuss it with their peers so get a sense of their relative position in their year group.  But mainly, it is a single measure that tells them where they are at today.  Students used to have a good understanding of what Level 4, 5, 6, etc. meant.  In fact I still hear of Year 8s and even 7s asking what level they are. It wasn’t that long ago.  Inevitably they will take time to get used to a new scoring system. In our case it is linked to 9-1 GCSE grades. But the key difference of course is that these TANs are their teacher’s opinion (based on summative end of unit assessments) on how they are doing, rather than the external impersonal authority of the exam board.

Due to the simplicity of the summative score this feedback doesn’t actually tell the student how to improve other than “work more” or maybe even “work less” depending on that student’s disposition and level of ambition towards that subject.

2, The Parents.  All parents want to know how their child is doing at school. Through parents’ evenings and other contacts, we can provide much more nuanced information on this progress.  But I believe most parents like the clarity of some sort of score. The score needs to be understood in context, e.g. in our case it looks like a GCSE grade but it is not any sort of prediction, well not until KS4 anyway.  The question for me is what do parents then do with this information? Obviously a full range of responses exist here from nothing at all, to deciding to get a tutor in and putting additional pressure on the student to work harder in whatever way they see fit. Even though the parents may do nothing, the mere fact that the TAN is shared with the parents is likely to have some impact on the students’ engagement with school, positive or negative.

3, The school leadership. By having a regular school-wide, “score” for each student per subject the school can do all sorts of analysis of the attainment and progress of their student base.  What the school then do with this information is myriad: e.g. decide on classes/sets, plan intervention including deployment of support staff, provide support to teachers, evaluate teachers as well as track overall school improvement.  The data may be shared with Ofsted although my understanding is that this is not statutory.  These are pretty wide-ranging but basically boil down to helping the school focus their efforts and resources in the right place.

It strikes me that these are 3 quite different and potentially conflicting sets of objectives. For example the school may wish to collect data that is useful for analysing whole school performance but is not relevant or motivating to individual students. (A Twitter conversation here with @LaSalleEd highlights how their MathsAge system shares specific content objectives with the student, but calculates an overall score solely for school use)

The dynamic between students and parents varies as children get older.  I believe there is a case for parents of primary children having information that their children don’t see, but as pupils approach GCSE they need a realistic view of what they are aiming for which can prove an incentive to work hard.

Lots of questions, not many answers, I’m afraid. I would like to understand more about what other schools do. I understand many have adapted the old levels system by basically changing the scale but didn’t see a need to make a broader change to reporting.

Please leave comments below or get in touch on Twitter, @mhorley.  Thanks!

 

Exterior Angles of a Polygon and other shapes

I find some students like to memorise this, or at least the first two or three rows and then remember that we keep adding 180 for extra size.

Reason being – an extra triangle.

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I feel that using exterior angles is more satisfying.  I find myself walking around an imaginary hexagon in my classroom with arm out stretched.

Here is a very nice Geogebra visualisation of this:

And here’s an activity which reinforces this idea.  Use a pencil as an arrow.  Does it always rotate through 360?  Does it always turn the same direction?  Use algebra, maybe?irreghex_42603_mdimages

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Mathematical Timeline

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I made this years ago, but just realised it is not on this blog.

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It contains snippets of bits of maths from 500BC up to the present day to adorn your classroom or corridor. The focus is on interesting, but accessible maths rather than historical completeness! Mostly inspired by the excellent Alex’s Adventures in Numberland (Alex Bellos) as well as Number Freak (Derrick Niederman)

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Best printed A3 in colour. Here is the whole thing as a pdf, and here as a word doc

 

Circle Theorems with Geogebra – Part 2

This is the second post in this series.  Part 1 is here.

 

1, Tangents

What is a tangent?  If you look at a line and a circle, it can either not touch the circle at all, cut it in two places, or if we are really careful, just touch it in exactly one place.

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The special thing about a tangent is that there is only one line that can form a tangent at any point on the circle. If you move the line then the point of contact also moves.  Geogebra works by drawing both tangents from a point to the circle.  By putting points at the intersect you can see how they move.

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As you connect the points on the circumference to the centre, stop to ask what we know about the lines that have been drawn here.  Then what do we notice about the two tangent lines and radius as we move them?

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To prove the point we can measure the angle between the radius and the tangent:

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And measure some lines too, whilst moving things around a bit:

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We can deduce that DE = FD, by drawing in one more line and proving that we have congruent triangles AED and AFD.

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2, Cyclic Quadrilaterals

What is a quadrilateral? How many lines, how many vertices? So what is a cyclic quadrilateral? Once you’ve established this, draw one.

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Then go round measuring the angles. Before you move one of the points, ask students to predict what will change and what will stay the same.

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Equation of Straight Line

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EQUATION OF STRAIGHT LINES
These slides are in groups of 4 designed to be used as a mini-whiteboard exercise for groups that need reinforcement on equations of the more basic straight-line graphs that can be determined by looking at patterns in the co-ordinates of points on the line (i.e. all points have a y-coordinate of -4 means the line is y=-4)

The idea is that you show the line itself first and ask students to show on their mini-whiteboards the equation of the line if they know it. If they don’t, then showing them the next slide with a bunch of co-ordinates provides the scaffolding. The next slide then reveals the equation and the final slide is to make the point that the coordinates themselves are arbitrary.

Collaboration

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I’ve had a couple of experiences of collaborative lesson planning at different schools. What we are doing at the moment with our mixed attainment Year 7 classes feels like the best I have experienced. I’ve been pondering why this is.

Collaborative planning can be a hard sell .  It can feel inefficient.  Some may be thinking “I could have planned this lesson in the time we’ve been sitting here discussing it”.  It’s sometimes difficult to get everyone involved, especially if the less experienced teachers perceive that the more experienced teacher will know the “best” way to teach a topic and therefore don’t feel they have much to offer.  It’s taken me time to appreciate that there is no “best” way to teach anything – it’s a complex interplay of the teacher’s experience and preferences, the relationship with the class and what’s gone before in the sequence of learning.

However, it is also patently inefficient for 6 teachers teaching the same topic to the same age children to be creating 6 sets of powerpoint slides and worksheets.  This struck me when I started teaching. How much time do teachers spend creating their own resources and lessons as if they were the first person to have ever taught it!?  At the time I thought why doesn’t just one person in the department to it, or, better still, one person in the country.

If classes are in sets then there is a justification that each teacher needs to “tweak” things for that particular class and the “level” they are working at. I would speculate that this is usually down to the teacher’s preconception of what the class will or will not make a decent stab at rather than a clear understanding of the individuals’ prior attainment with that topic.  However even if tweaking is required (we still tweak for our mixed attainment classes) it is still a whole lot quicker than starting from scratch.

There are, of course, plenty of lessons resources available either for free or by buying into schemes such as Boardworks, which could mean that nobody has to create any PowerPoint slides ever again. Or your department could follow a text book scheme which often have on-line resources to support them. But picking up someone else’s slides and just stepping through them is unlikely to result in a good lesson, teaching is not as simple as that.  It’s not really about the PowerPoint.  There are many subtle nuances within the flow of a lesson that are nothing to do with what is on the screen, the worksheet or the text book. And the temptation to spend to glance briefly at the resources just before the lesson without really thinking properly about the sequence of learning is always there.

This year, our Year 7 teachers have been meeting once a week for 55 mins in a timetabled PPA slot we call a TRG (Teacher Research Group).  We are doing the same for Year 8 teaching. It is voluntary for teachers to attend these TRG sessions as it is not additional PPA time.  So far, all teachers do regularly join and generally feel that they get back the time they give up.  There is one designated lead teacher for each year group who “chairs” the TRGs and is responsible for keeping the files and folders organised.  Being the lead shouldn’t be a big burden but it is important to have someone clearly identified in this role.

The TRGs vary, but generally will start with looking at a very simple overview of the next 8 lessons or so with names and dates alongside which we might spend a few minutes discussing – effectively the medium term plan.  We try to keep a week ahead of the lessons being taught.  Too far in advance and we are likely to forget what was discussed.  Going the other way, we risk the situation (which we have sometimes slipped into) where lessons are shared the night before and we miss the opportunity to discuss them.

It is this discussion of the individual lessons that is the crucial part of co-planning. The teacher that has planned the lesson will run through it, with the other teachers thinking about their own class and picturing how things will run. As we go, someone makes changes on the slides or writes brief notes on them to follow up later.  The discussion focuses on specific explanations and questions, but can also cover the flow of the lesson, the practicalities of classroom management or verbal questioning.  Often there is not a clearly defined end point for a lesson, it is up to teachers to manage the pace with the class and draw the lesson to a close at an appropriate point. This can mean that we sometimes get out of sync by a lesson or two but this is manageable.  We go at a decent pace in the TRG which enables us to get through a week’s worth (4 hours) of lessons.  We have 6 classes in each year group, so that means that each teacher is creating a lesson just under once a week.  The starting point for this teacher is sometimes last year’s lesson, or sometimes we will briefly discuss the key learning points and key questions for the lesson in the TRG to give the teacher planning it enough to go away with to start planning.  Homeworks are also created collaboratively and are linked closely to the lessons aiming to include some more challenging open-ended or problem-solving questions alongside practice of core concepts.

In addition, we do “learning walks” once per week where one class will be covered, usually by the Head of Department.  The teacher will go round the other classes to see the lessons in action.  They are looking to see how the pupils are responding to lessons we have planned and are not commenting on individual teachers. Every few weeks we will devote some of the TRG time to learning walk reflections where we discuss more generally what we have noticed and what we need to change.

Personally I have learned a lot from our TRGs this year.  As well as an efficient way to plan high quality lessons, it is also a very effective form of ongoing CPD. I feel more confident actually teaching the lessons having had the chance to discuss them with colleagues beforehand. It has been a fundamental part of our transition to teaching in mixed attainment groups, which I don’t think would have worked without collaboration.  I also notice that we have more informal reflective discussions between teachers about how the lesson went as we all have the common ground of having taught the same lesson.

The key challenge of course, is time. It would be great to continue this model right up to Year 13 but timetabling all these TRG sessions will be difficult. My message to anyone in a position to make a decision on this would be to have a go, make a start.  If TRGs are effective then they deliver high quality lesson planning and ongoing CPD for all staff involved.  These are two pretty fundamental aspects of teaching, aren’t they?  So let’s dedicate the time to them.   Marking policies might be a place to look at for clawing back some of that time.  But that’s a topic for another post…