qualifications.pearson.com Level... · Web viewSlide. Purpose of Slide. Additional Information....

Slide Purpose of Slide Additional Information

Slide 1 Introduction The other thing to be aware of if polls are anonymous. Other delegates can see how many people have voted for each option, but will not know who has voted for what. OK, that's all I need to go through for now. Your host today is Kielan McGill and your presenters are Kim Ogden and Narsh Srikanthrapala. Kim and Narsh are going to take you through the rest of the session, so it's with great pleasure that we hand you over to them now. Over to you Kim and Narsh.

OK, welcome everybody, my name's Kim Ogden. I will be as they just said presenting alongside Narsh today. For the first section I’m going to be doing the presenting and Narsh will be in the background answering any questions that you send in to him. About halfway through I will switch over and I will be answering questions and Narsh will be presenting. Just to let you know, both Narsh and I are school teachers ourselves, we both work in schools at the moment, and we freelance for Pearson as well, helping out with training and things like that for the new AS and A level specification. So if you do have questions that we can't answer, we may refer you to the Pearson maths team, but hopefully we'll be able to answer most of your questions that you might have. We'll also involve you in some polls and things today, so please do interact because that may influence some of what we do later in the session. The session as I said is being recorded and you will be able to access it later, along with other sessions that we've been doing on mechanics, statistics and further maths units as well, you can also access a delegate pack and download some content in that we've put the specification, sample assessment materials, formula booklet etcetera. And we've also got a version of this PowerPoint on pdf which you can revisit at your leisure later on, and you'll be able to present that if you wish to other members of your department.

OK, so we'll get started today then.

Slide 2 The A level reforms First of all, I’m just going to run very briefly through some information about the A level reforms. I’m sure you all know most of this already but so I will go through it quite quickly, but just in case you’re not too familiar. So new AS and A levels will be assessed at the same standard as they are currently. We know that they're fully linear now, so any AS level units will not contribute towards the full A level.

The AS is a standalone qualification, and in general, for all the new A levels, the AS content can be a subset of the A level content, and that is what is the case with the mathematics AS and A level.

Slide 3 A level mathematics So for mathematics, everything is now core content, and core here doesn’t mean pure maths any more, as we kind of got used to it meaning for the old spec. It just means compulsory, so everything is compulsory across all of the different exam boards, and the pure maths content is broadly the same as it used to be and the mechanics and the stats mostly comes from M1 and M2, S1 and S2. If you're looking through your specifications you'll notice the AS content is shown in a bold font.

The A level reforms - content

So requirement for assessment of problem solving, communication, proof, modelling and application of techniques, the pre-released large dataset obviously is for stats only so we won’t be talking through that today, and the calculator regulations have slightly changed in terms of what the calculators need to be able to do. So hopefully you're all familiar with that and have already got your students using new-style calculators with this new specification.

In terms of pure specific things for the calculators, all of the calculators that meet the requirements now can solve polynomials, obviously solve quadratics within that, solve simultaneous equations and do vector calculations. So those are some of the things that the calculators can do, which if you used old scientifics then you might not have had much familiarity with it.

Slide 5 Our design principles

For Edexcel they have separated the pure and the applied into completely distinct papers. There's a two to one ratio of pure to applied for the A level, slightly different at as but I’ll talk you through that slight difference a little bit later on. The large dataset is hopefully going to be for the lifetime of the qualification although it may be amended slightly later on, and further maths is designed to a parallel delivery with mathematics, as well as consecutive delivery obviously, if you have students doing maths in one year followed by further maths in the second year. None of the papers are non-calculator any more, everything is calculator assessment.

Slide 6 Today’s agenda So what are we going to cover today? Well we're going to look at the overarching themes of the new specification and particularly look at how that applies to the pure content. We're going to look at the specific content within pure and the assessment objectives, and we're then going to go onto for the most part of the session be looking at some specific areas of content. We're going to look at all of these four sections, but we may have to go through one or two of them a little bit quicker than the others, so we will do a poll on that later to see what your priorities are before we start that section. But you will be able to access resources and obviously the download for all four of those sections anyway, and then we're going to finish off by looking at some questions arising from the sample assessment materials that you may have come across.

Slide 7 Poll 1 OK, so we’re going to start with a poll, just so that we can get a sense of what everybody's experience level is, so if you can just vote on the poll now, if we can just bring that up for the delegates please, and just let us know how much experience you have, so are you new to A level completely, or what level of experience do you have if you have any experience.

OK, looks like we're almost there with the voting. If you just want to place your poll within the next few seconds.

OK, and could we publish the poll results now please for the delegates to see. So bit of a mixture, we’ve got quite a lot of you who've got more than ten years of A level teaching experience and a few people who are completely new to it, and a bit of a range in between, so that's good, that gives us a sense of what you’re already been used to over the last few years.

2

Poll 2 OK, great. Moving onto another poll actually before we go any further, so the second poll is just for us to get a sense of what you’re offering in your establishments that you’re currently working in. So are you offering an as opportunity at the end of year 12, is it potentially something that you might consider for some students at the end of the year, depending on how they’re getting on, or is it something that your schools or colleges are saying no to? So again if you can just finish off the voting on that please. Give you a few more seconds.

OK, lovely, can we publish those results as well now please? So we've got again a bit of a range but it looks like the majority there saying none of the students are going to be doing an AS exam. And then we've got a split there between some who are everybody's doing the AS and some who are potentially going to do it for a few students.

OK, that's great, that just gives us an idea of what we're working towards. OK. Let's take that off now and we'll start going through the rest of the content.

Slide 9 Overarching theme 1

So starting with the overarching themes of the qualification, so firstly here we’ve got the first overarching theme, which is to do with mathematical argument, language and proof, the key things here for the pure maths, I’ve highlighted here the set theory, so the set theory is more clearly specified within the content now and students may be required to use specific notation in terms of set theory, so that's one thing that is slightly different to the old spec. And also the proof element which is much more prevalent in the new specification than in the old specification.


The second overarching theme is about mathematical problem solving, so this is very much to do with taking problems that are in a particular context and interpreting the solutions. We'll talk a lot more about this one later when we look at the modelling. So also numerical methods here, so providing solutions to a required level of accuracy, and also evaluating accuracy or limitations of the solutions that students get as well.


And the final overarching theme here is the mathematical modelling, again we'll have a look at this in a whole lot more detail later on, we've got a whole section on modelling. So being able to translate the context that they're working with into a model, and then gauging and exploring the solutions that they come up with relating to that model. The interpretation of outputs in the context of the original situation, and I think there's actually more modelling within the pure than you might expect, though the modelling is more heavily weighted I think within the mechanics, you’ll notice when you look at the sample assessment materials that there is a lot of modelling within the pure SAMs as well.

Slide 12 Content changes - AS

Looking at key content changes in terms of the topics that you need to be teaching, starting with AS here, and the brand new content is in bold, so there's a significant element of proof now in the new specification. Graphical inequalities is something that is new.

3

Curve sketching on here, not that it's a new topic but that it has been clarified across boards what they’d be expected to do in terms of curve sketching, so if you’re familiar with Edexcel in the past, y = a over x squared for example is something that you wouldn’t have to teach in the old specification but is in the new spec.

Exponentials coming up in the AS, so some of this has moved down to AS from what used to be A2 level, and differentiation from first principles, again this is one of the areas that we will look at in more detail later, and there is now some vectors in AS, which obviously we all know used to be in the second year of our A level.

Slide 13 For the second year, A level content, again we've got more proof. Sequences has completely moved into the A level now, so some of that was previously in AS level. Radial measure as well used to be an as topic is now in the A level only.

Exact trig values and small angle approximations.

Again differentiation from first principles but in the A level we're looking at sine and cosine now.

Integrating the area between two curves, so in the old AS we had integrating the area between a line and a curve, and iterative methods, including the Newton-Raphson method, which if you're familiar with the old spec used to be in FP1. And trapezium rule is now exclusively in the A level, probably because it comes under the numerical methods section and there are no numerical methods in the AS, so that has been put with the rest of the numerical methods in the full A level.

Slide 14 Content changes – what’s gone completely

What's gone completely, volume of revolutions, vector equations of lines and the remainder theorem. And some of those things you may find in the further maths content, for example the volume of revolutions and the vector equations of lines.

Slide 15 Poll 3 OK, before we move on we're just going to do one more little poll here, so do you feel that you need additional support with familiarisation of the specific content changes. We're not going to cover any more of that within this section, but if you do feel that you need more support, there are, we have been doing some getting ready to teach events over the summer term, and a lot of that content is available online if you go to the website. So if you do feel, which most of you don’t, but for those that do, please do refer to the pre-recorded material from the getting ready to teach events.

OK, if you just publish the results for the delegates there so they can just see, most of you are obviously quite familiar with what the content is at this point. Great, OK so we'll move on then.

Slide 16 Assessment structure

And just look briefly at the assessment structure, so for the AS you’ve got two papers, one pure paper and one mechanics and stats mixed up together there. You will notice the percentages are slightly off the two thirds one

4

thirds weighting on the AS. The reason for that was at 50 marks the mechanics and stats paper was thought to not really be able to get into all of the content deeply enough to really assess what they know from those elements of the qualification. So that was brought up to 60 marks, and that therefore made the weighting slightly different for AS. But you’ll notice on the full A level that it's equally weighted with the two pure papers, and then you've got a one third weighting there for the mechanics and statistics.

And worth mentioning as well that any pure content can be assessed on any of the two papers, so it's not a case that paper one is the AS content and paper two is the A level content, any of the content can be assessed through either paper, so similar to the GCSE in that respect, there's no knowing what topics are going to come up on each paper.

Slide 17 Assessment objectives: AO1

And we're just going to look briefly at the assessment objectives. So AO1 is as it always has been really, standard techniques, routine procedures, facts terminology and definitions, so all the straightforward stuff I suppose really in there.


AO2 is about reasoning, interpretation and communicating mathematically. Some significant change that impact on the pure side of things here, so strand 3, which gives rise to spot the mistake style questions, so we're going to have a look at those later on. And you may have, you may have already seen some of those if you’ve taken a look at the SAMs. And strand 5, just to note that specific notation may be required, so definitely worth consulting the specification as to what that notation involves.


And moving onto AO3, which is solving problems within contexts. So here we've got all of the interpretation, evaluating the accuracy and the limitation of the solutions, and translating situations into contexts, so with the mathematical modelling.

I think for me when I looked at the sample assessment materials one of the things that struck me as a difference between the old and the new spec was the fact that they are in some cases required to come up with their own mathematical model, which tended to be in the past given to them and then potentially they had to interpret it. But I think now they, there's more of a focus on them being able to produce their own mathematical models.

Slide 20 Assessment objectives: breakdown

And then here's a breakdown of the assessment objectives, just to notice that AO1 is consistent throughout all three areas at A level, so the pure and the stats mechanics have a consistent percentage weighting there for AO1, but AO2 is much heavier on the pure and AO3 heavier on the stats and mechanics. So that's just something that might be worth noting.

Slide 21 Specific areas of assessment and content

OK, so we're going to move on now and look at our four specific areas of assessment and content. So when we were planning this session, we sat together, Narsh and I, and looked at what we had noticed when we were looking at the sample assessment materials and what we felt perhaps teachers would like a bit more support on. And we're actually going to get you guys to interact now, and just tell us which are your priorities, so we are going to look at all four areas, but we may need to go through one of them slightly quicker than the others.

5

Poll 4 So if we can bring up the poll now for the delegates, if you can just select your top two priorities out of the four areas that we've given you, and as I said we'll cover all of them but we may have to go through one or two of them in a slightly more detail and go through the others slightly quicker. So just so that you can let us know what your priorities are. I think we've still got a few votes coming in, so I’ll give you another few seconds there to make your decisions.

OK, can we publish the results now please?

OK, so you can see that the modelling seems like the biggest priority, and proof coming in second there. The spot the errors and the differentiation from first principles are slightly lower, so we may end up spending a little less time on those two.

OK, great. I think we shall move on then and start looking at modelling.

Slide 23 Modelling So what we're going to do here is have a look through some of the issues around modelling. We're going to have a look at some SAMs questions where the students would need to use mathematical modelling. And I think we'll have a look at where the increase in modelling and interpretation has come from, first off.

Slide 24 Modelling: AO3 So going back to the AO3 assessment objective about solving problems in contexts again just to reiterate that this is where it's coming from really, this third assessment objective. So the original problem can be in a mathematical or non-mathematical context, and there's that emphasis on them creating their own models and refining them, evaluating the accuracy and the limitation of their models.

Slide 25 Modelling: introducing the content

So how would we go about introducing modelling to the students as a concept? Well I think before you do this you probably want to speak with the teachers of the mechanics and statistics units if it's a shared group where you have perfect different teachers teaching different parts of the course. Because this is more heavily weighted in the stats and mechanics sections. So you might want to figure out who is going to be approaching this topic first out of you yourself and your colleagues, if you are sharing the different units. And you might want to discuss how you could share resources and techniques in terms of presenting to your students the ideas of modelling and interpretation.

It's worth mentioning modelling in general terms as well as in specific contexts to students. So I would start off probably by explaining what mathematical modelling is and the links to real life situations, and specifically from pure, how usually the model will take the form of an equation, which can be used to describe the variables that are related to the context.

As I’ve said we want to link it to other work that they may have done, but also maybe talking to them about careers where mathematical modelling is used regularly, to make it more relevant to the students.

Mention about how the whole point of mathematic modelling is to be able to draw conclusions and make interpretations about the scenario that they've been given. So again there is a lot of focus in this new specification

6

on them drawing out conclusions and making those interpretations.

Often now they'll be having to give a written conclusion or interpretation with reference to the context of the problem, and I think this something that was common in the old spec applied units but not so much in the pure, there was obviously modelling in the pure but it was more about them applying their mathematical skills to the model they were given rather than drawing out those conclusions and giving written interpretations.

Slide 26 Modelling: key points to embed throughout

So key points that you want to embed throughout your teaching of modelling. Obviously as we said before they may be asked to suggest their own model, so they're going to need to draw on their understanding from a variety of mathematical topics at once. It's not just a standalone topic that you can teach them, it's something that you would have to embed throughout your teaching, as and when it kind of becomes appropriate.

Models can be simple or complicated so it's worth discussing some basic models and then looking obviously later on at some much more complicated ones.

Students need to be aware that the results that they get can be approximate or they can be exact, depending on the type of model that they’re using.

Also being aware that sometimes a model is only valid for certain circumstances as well.

And mentioning to them about the links with stats and mechanics and where they'll look at the simplifications and assumptions they might make when doing mathematical models.

I think when you’re teaching your modelling in different contexts it's worth continually asking them about whether they can identify any limits or limitations of the model. So they get familiarised with having to do that.

Interpretation being a crucial skill to develop when using models. I think this is probably one of the aspects on the old spec, when I was teaching statistics for example, that students probably didn’t enjoy as much and a lot of students found the most difficult. A lot of maths A level students are not really into doing much writing, they prefer to just do the maths and get the right answer and move on and be happy about it, so having to interpret their results sometimes is something that they're not particularly keen on. It's also something that EAL students tend to find really tricky, particularly with the language skills, so it's something that you will need to work on there.

They may be asked to explain what the values within a model represent.

They may be asked to make estimates from the model, give reasons why the model may or may not be realistic.

Also their prior knowledge is really important, they might need to use some of that prior knowledge from GCSE when they're doing modelling, and sometimes they can be caught off guard by that I think, maybe when bearings comes up or area perimeters, surface area, volume, and they do, it does take them off guard because well, we haven’t been taught that in this A level so far so how come I’m having to use that area of my knowledge? So warning them about some of those things and covering some of those things if they need to go over them from GCSE.

7

And then finally the common sense element, this is something my students find difficult sometimes, I think maybe they don’t have much common sense, but knowing for example that £32 a year is not a realistic salary, or that a company designing a chair with a height of 2.5 metres might not be such a good idea. These are the kind of things that they might even be asked about within the exams, so they then need to be able to interpret and obviously look at the units within their model and make sure that they are applying those correctly to the context.

Slide 27 Modelling: topics at AS

So I’ve looked through for different topics where modelling is going to be an important element of your teaching, and if you look through the textbooks, most of the textbooks that you might be using, whether it's Pearson or a different company, will have some references to modelling within certain chapters. So these are the ones that have a modelling section within the chapters of the textbook, so at as we've got quadratics, straight line graphs, vectors, differentiation, so mostly the modelling there is to do with rates of change. Exponentials, obviously the modelling here is mainly to do with growth and decay, and logarithms, basically exploring trends in non-linear data, which links to their modelling with straight lines as well.

Slide 28 Modelling: topics at A level

Moving onto the A level, there's a series chapter, so the arithmetic and geometric series. Trigonometry, so most of the modelling within the trigonometry is using the r sign, theta plus minus alpha, or r cos theta minus alpha obviously, parametric equations as well, so most of that modelling is to do with using time as a parameter to model motion in 2d, so you'll find examples on using parametric equations to model the motion of an ice skater and their journey across an ice rink for example. The Newton-Raphson method, so mostly do with using that technique to find solutions to models that have been given to the students and then obviously differential equations as well.

So those are your topic areas. What we're going to do now is take a look at the sample assessment materials and see how that modelling aspect has manifested itself within the sample assessment materials. There are three questions in the AS SAMs out of the 17 questions on that paper, and then obviously we’ve got two papers at A level, and there are 7 questions out of the 31 in total across those two papers. We're going to have a quick look at all ten of those questions and just see how the modelling comes up, and hopefully then you can see the similarities and the differences between the old and the new specifications and the way these questions are going to come up.

Slide 29 Modelling: questions from SAMs – AS level 1

So starting off with the AS level, this is from question 8. Just to let you know at this point it might be a good idea for you to full screen your presentation which you do by just pressing, there's a button in the top right of the window where you can see the presentation, just because the writing here is coming out quite small, so you might find that it's going to help you if you have it on a full screen for this section.

So first question here is to do with finding the area of a triangle. Obviously we're talking about in the chapter where you’ve got the sine and cosine rule. And so the part b there, why is your answer unlikely to be accurate to the nearest square meter, we're looking here for something like that the information may not be accurate to four significant figures, so the information they’ve been given in the question may not have the accuracy

8

required. The other reason that they could use here is that the lawn may not necessarily be flat so modelling by a plane figure may not give us an accurate result for an area of the field. So that's the first one.


Second question, this one is obviously to do with modelling here with a straight line, so we’re looking really at part c, they’ve got to interpret the values of a and b. So they have to look at what the values of a and b are, and actually interpret that in terms of the model that they’re using. So a is the initial population, and b is the proportional increase in population each year. I think that one for b there might be quite tricky for students to spot, I thought maybe it might be worth when you’re teaching this topic linking to compound interest because that will be something that they're familiar with, using a percentage multiplier there for the compound interest, and then using the power obviously is the number of years which the interest is being applied. So that will be a familiar context for them to maybe link this one to.

And then they have to give reasons as well, in part c, reasons why this may not be as realistic, population model, so looking at the fact that 100 years is a long time, in the mark scheme I think it says something about that wars may occur or disease may affect the population over such a long period of time. Also the model predicts that there'll be unlimited growth which is obviously not very realistic in a real life context. So they really have to know not just about the mathematics here bit about how they interpret that model within the context that they've been given.


Third one here from the AS SAMs. I think this one is quite similar to, in the first part at least, with what you'll be familiar with from core two, where they're creating a model here based on the measurements that they're given and their links, obviously here would be with their shape, shape and space, topics from GCSE. But then for part b there they have to explain why x has to be between those particular values, so knowing things like the fact that distances are always positive values, that's part of the explanation that they have to give there, so again something that you'll want to embed while you’re teaching this topic to them.

Slide 32 Modelling: questions from SAMs – A level 1

Moving onto A level now, so we’ve got one here where they're given a whole bunch of worded information there at the beginning, something that my students tend not to really like very much. So making sure that they can read through the information, maybe highlight the key pieces of information that will be important to them. But they have been given two diagrams here as well, the graphs or two possible different models that could be used. So here they're having to look at different models and actually analyse for themselves which of the models might be better or what might be the limitation of those models. So for example for the first one, knowing that the volume cannot become negative, so that might be why that's not such a good model.

In part b though they then have to write their own model, so it’s not a show that question, as it may have been in the old spec, they actually have to write that model from scratch, using the information that they've been given, so they’ve been given enough there I think for them to be able to go on, but they haven’t been given any sort of equation to base their model on, so they have to recognise the shape of the graph, they've been told it's exponential but also then they have to figure out what that model would

9

have to look like. And then they have to go on and use their model in the second part there of part b.

So it's worth five marks there for both of those parts together, so they have to first of all figure out what their model would look like in general, and then insert some values that are given to them there on the graph to be able to get a model that's suitable. So that's quite an interesting one I think.


Next one here, so we've got a quadratic model this time. They have to understand for part a that they would need to make the value of h equal to zero in order to find that horizontal distance travelled by the arrow, so they have to interpret what they've been given there and figure out what to do initially, and then with reference to the model interpret the significance of constant 1.8 within the formula, so that's about linking with their knowledge of polynomials and the fact that the constant will always be the y intercept, but not just saying it’s the y intercept, actually translating it into the context of the problem and actually understanding what that y intercept actually represents in that context.


Next one, we've got another model here which is to do with logarithms and approximating it to a straight line graph here using logarithms. So this is a new topic, it's not something that you will have encountered in the old spec. So they've got to obviously use this graph here in part b to find the values of a and b for the exponential model. They've also got to know that they can't extrapolate from the graph and assume that the model will still hold. So extrapolation, something that probably only really came up in statistics in the old spec, they weren't really asked questions about that within pure contexts, and obviously then they have to be able to interpret the value of a as well later on in the question.


Moving on again, so we've now got, this one's pretty basic actually, I mean this is more like the standard old spec type modelling that we're familiar with. There's no real interpretation here, they don’t have to write their own model, they're simply given a model to use and they just have to substitute in appropriate values. So that I think is fairly straightforward. We're onto paper 2 now by the way of the two papers in the A level SAMs.


And also with this one, quite similar to what we've seen before in old spec, so for the most part it's very familiar. I think I’ve highlighted part d there as something that is worth drawing attention to. So part d is about how they would adapt the equation of the model to reflect an increase in speed. This I think requires some quite deep understanding of the structure of the model really. They have to identify that they would need to increase the value of the 80 within the formula, so I think for that they're going to need to know quite a bit about how the model, what's going on with that model and how the structure of the model itself actually works and what different numbers within that model are actually doing to the model itself. So that's quite an interesting one I think.


Number 14 here from paper 2, so here we've got the common sense coming into it. So again this is quite similar to the old core 2 differentiation maximum and minimum questions that students might have been familiar with. There's no diagram here with the cylinder so it might be that they would want to draw a little diagram of the cylinder and what's going on to help them. And in the first two parts it’s fairly standard what we're used to

10

again, but in part c they've got to give a reason why the company wouldn’t manufacture cans with the minimum surface area that they’ve found there in part b, so they’re looking for things like the fact that the radius is too big for a human hand to comfortably hold the can, or that in general drinks cans should be taller than they are wide, and this one wouldn’t be. Also that the can, the shape that it is there for the minimum surface area wouldn’t be very easy to stack on shelves necessarily or it wouldn’t fit in with the supermarkets' general shape of cans that they're used to. Things like vending machines etcetera might not be able to hold the cans. So all those kind of things, which is really a common sense element I think that perhaps students might struggle with.


And this is the last one now, so this one is a differential equations question from the second paper, second pure paper. And again this is pretty standard really, this is quite familiar I think in terms of old spec and how we are used to seeing a differential equations problem, so there’s nothing really standing out to me there that is very different from the old spec.

Slide 39 Modelling: teaching ideas

So let's take a look at some teaching strategies now, or ideas for teaching strategies. So I’m first of all going to look at a possible activity that you might look at with students when you first introduce modelling to them. It’s got quite a general feel to it in terms of the idea of modelling, so there isn't a specific topic area that you're looking at here, and depending on when you talk about modelling with the students, you may want to adopt the scenarios that I’ve given, depending on what they already know. It could also be used as a revision of modelling once you've covered all your topics, and maybe if you're looking at modelling throughout, this might be a good way of revisiting.

Slide 40 Modelling: sample task

So here it is, basically the students are going to do this through a discussion task. They have to look at the different scenario, ooh sorry, I’ve just skipped forward there, look at the different scenarios given, and the hints there tell them really what they should be doing. So thinking about what the variables should be and the relationship or pattern that they would expect those variables to follow. Thinking about as one variable increases what might happen to the other variable. Getting them to sketch out on a mini whiteboard what the graph might look like, and I think that's a good starting point when you're modelling, think about what the graph of the situation or the journey would look like, and then that will give them ideas as to what type of equation that might link with, so drawing on their knowledge of the graphs and the shape of the curves and what equation would match with that curve shape.

And you could extend this by getting them to look at what assumptions you might be making with the model, or even getting them to come up with some specific numbers that might go into the model and then actually plotting a graph and then coming up with an equation that would go with it. So I think that could take quite a long time within a lesson, give them plenty of time to discuss it, and I think that that's quite a nice way of introducing why modelling is quite important and how they might be able to apply it within their learning.

Slide 41 Modelling: teaching ideas

We'll look at a specific lesson now, possible lesson plan for modelling with exponentials. So the strategy that we're using here is embedding the teaching and modelling throughout teaching the particular topic that you're

11

talking about. This could be expanded afterwards, you could use exam wizard to generate some past exam questions on this topic to test their application of the skills that they're developing.

Slide 42 Modelling: starter So first of all probably get them to discuss why exponential models would be better than the new models in considering growth and decay, and there's lots and lots of discussion you could have with them there. I usually when I’m doing exponentials with them get them to have a class discussion about what situations in real life would be modelled using exponentials, so talking about growth and decay and getting them to come up with a few different areas that they might be able to model on those two areas of growth and also decay.

Slide 43 Modelling: task Once you’ve talked about all that general stuff, I would then go onto looking at an exponential model in general, and what that exponential model tends to look like. So this is one that I did with my students recently when I was teaching my year 13s exponentials, just before the summer holidays when we started core 3. Obviously the use of technology is important in the new spec, so this one uses graphical calculators but you could adapt it depending on what technology you have available, and in a few moments we'll look at a Geogebra task as well that is available for you that you could potentially use if you don’t have graphical calculators. The idea of this is to get them to plot this using the modify function on their calculators, so they can then change the values of a, b, c and d and look at how that affects the graph. And get them to really think about in a general exponential model, what the values of a, b, c and d actually represent in terms of the model itself.

So if I can ask my host to take us to the Geogebra task now, and I’ll just show you how this might work or what you might want the students to be looking at.

Slide 44 Geogebra task Just while that's loading I ought to let you know that this Geogebra task is available in your delegate pack, I believe it should be in there as a Geogebra file, but the one here is part of the online Pearson maths textbook activities, so there are Geogebra activities linking to certain examples within the textbooks, and this is one of them.

So what the model starts with there is the value of a is zero, and also c as zero, and b and d are both equal to 1. So if you can just take the slider for b please and just adapt that for me, so I’d probably get the students to look at this first, what's happening as you adapt the value of b and get them thinking about the y intercept there and thinking about what that would mean in terms of a context. If you can take it negative as well for me please, the b value, looking at how when b is negative it changes the shape of the curve. If you can just put b back to 1 now, and then if you can drag d, positive first and then negative as well for d, so looking at how that is affecting the speed of growth of the speed of decay. And obviously, you know, you might want to talk to them about how usually with exponential models we're using time along the x axis and so most of the negative values might not be appropriate in a lot of cases. And obviously if they take the value of a and move that perhaps if we can take a upwards and you can talk about how a is linked to the location of the asymptote. So

12

there are various different things that you can explore there with that activity.

OK, I think we'll go back to the PowerPoint now at this point if that's OK please.

Slide 45 Modelling: example So getting them to kind of figure out what the model in general will look like, and how the different values are going to affect the model and the shape of the curve is really important. And then I would probably give them an example that they could work through. I would ask them to sketch the graph first based on what they've just done and be thinking about these key questions when they're thinking about that, so what is the meaning of that 150 and how does that impact on the shape of the curve, what's the meaning of the 80 and what's that doing to the curve. And the significance of the negatives, the fact that the 80 is negative and the fact that the t over 40 is also negative and how that affects what the shape of the curve looks like and they could even go back to their calculators or the Geogebra and actually have a look at maybe plotting this on there and having a look at what the curve is doing.

And obviously if you don’t have access to technology you could just use their knowledge of curve sketching here, you wouldn’t have to use technology, but I think it's a nice time to use it and obviously with it being more important in this spec I think it's a good opportunity there.

Slide 46 Modelling: apply your understanding

And then I’d probably get them to look at an exam question and apply their understanding to it and then talk through what they’ve done with that.

Slide 47 Modelling: common errors and misconceptions

So that's just one example of a potential lesson on modelling and how you can kind of embed it within the specific topic itself.

Some general points again, so units in modelling problems, don’t forget that they'll need to check carefully what units they're going to be dealing with, so some of the most common errors and misconceptions come up with units. If they can try and remember to give the units with their final answer when they’re modelling, and if they're giving written conclusions or interpretations, make sure they're referring to the units in the context of the problem as well.

One in particular that is really problematic is if they have for example some question where it is mentioned in days but later on in the question it asks for an estimate of what’s happening after 6 hours. That tends to trip them up, so really important that they are constantly reminded of that as a potential place for error.

Slide 48 Modelling: questions from SAMs A level 1

So just really quickly going back to the SAMs, here's an example where it is given in years but then they're asked for the mass after six months. So what was a fairly basic modelling question does still incorporate that potential tripping point.

Slide 49 Modelling: questions from SAMs 2 A level 2

And another one here with the meerkats again, these two were the ones that I said were the most similar to the old spec I think if you remember, and both of these are ones where you would potentially have problems, just from misreading or not paying attention to the units. So here p is in thousands and then they’re asked to deal with the fact that there's a

13

thousand meerkats, and making sure that they realise that they would have to use 1 in that instance and not 1000 when they're substituting.

Slide 50 Modelling: common errors and misconceptions

And other common things that might go wrong, so use of significant figures, so being really careful that if their answer requires a particular degree of accuracy make sure that they do their calculations to a greater degree of accuracy than that, so if their final answer needs to be correct to three significant figures making sure that they process using four, just to be certain that they're going to be OK with their final answer there to the correct accuracy.

Slide 51 Modelling: questions from SAMs – AS level

And again just referring back to that SAMs question there, that's a particular question where that could have been an issue or where that was, their knowledge of that was necessary to be able to answer the final part of that question.

So that's it for modelling, hopefully it's given you some ideas of how you're going to embed your teaching of modelling. We're going to now move onto spot the errors. And as it was less popular I am going to go through this one nice and quickly for you, and after this section I’ll then be handing over to Narsh.

Slide 52 Spot the errors So here we go with spot the errors.

First of all just to remind you where this comes from, it's because of this new strand within the assessment objective AO2 about assessing the validity of mathematical arguments. So that strand in itself has given rise to these questions coming about within the sample assessment materials.

Slide 53 Spot the errors: questions from SAMs AS level

There's only one question on each of the SAMs, so one in the AS level, and one across both papers in the full A level. So we're not looking at them quite so much for impactful area as we were with modelling, and quite simply this one is literally spot the errors, identify the errors made by the student.

Slide 54 Spot the errors: questions from SAMs A level

And the paper 2 one that you're looking at here is giving them two different students' responses, again part a there is simply find the error that the student has made. Part b is slightly more detailed I suppose because it's asking them to explain why it's wrong and then how the error might have occurred. So they don’t just need to say what the error is here but why is that error wrong. So for why it's wrong, it's almost like a disprove by counterexample I suppose, because they just have to show that minus 26.6 is not a solution of the equation. How the error occurred is slightly more difficult for students I think. The wrong answer is introduced by the squaring within, they squared both sides at the beginning of the solution. And perhaps given an example that if x is equal to five when you square that you get x squared is 25, but x squared is 25 has two separate solutions of plus or minus five. So talking them through examples like that is going to be quite useful I think.

Slide 55 Spot the errors: prior knowledge

Prior knowledge, well this could be assessed through any topic area through the specification, there is no one topic where it's going to come up, so it literally could be anywhere I think. So we need to make sure that

14

we're embedding it as a skill through our normal teaching, and it could form the basis of an independent revision session, if you have covered everything in order to address and look at those sample assessment questions.

Slide 56 Spot the errors: ways in and starters

Ways in and starters, well obviously mistakes in general, so you might have classic mistakes posters up in your classroom, I think they're quite common in maths classrooms. Some quotes here, I quite like the one that says the person who doesn’t make mistakes is unlikely to make anything. I find often that A level students tend not to like admitting that they make mistakes, they like to think they’re amazing at maths and therefore they wouldn’t make mistakes, but obviously they all do, so talking about how that's OK and not to be afraid of mistakes within their learning.

Slide 57 Spot the errors: teaching ideas 1

Obviously the best source of the material that you're going to get for this topic is really the students that are in front of you, so they're an endless source of mistakes really if you’re looking through their work. But also yourself, I tend to make silly mistakes all the time in my teaching, and I like to think it's because I'm so smart that my brain’s working at a speed so fast that my body can’t keep up, so I’ll say five and I’ll write three for example, and students love to point that out, but embrace that and make sure that students know that that's OK, making mistakes. And then get them comfortable with sharing their mistakes with each other.

So some quick teaching ideas, use mistakes that happen in class as opportunities, so keeping a misconceptions board that you can discuss at a later date, so as errors occur or discussions that you overhear, have a little space on your whiteboard where you write down those mistakes as they happen or those discussions as you hear, and then get students to try and come up with reasons why those mistakes have been made. So focus in on how the misconception arose and not just correcting it I think is really important. That's just an example there from a piece when I had key stage three and four students in my classroom, but obviously it can apply to A level as well. We have, in my college we have whiteboards in the corridor, so this is quite a nice thing to have in the corridor, write some misconceptions up and get students to correct them or come to you and tell you what happened with that misconception.

Referring to examiners' reports, so if you look in the scheme of work for the news specs there are examiners' report quotes for all topics, so they’re a really good source of possible errors. Photographing and sharing student work when you’re marking, using a visualiser during lessons so again getting them comfortable with sharing their mistakes and not get upset with you when you put their mistakes up on the board and discuss them with the rest of the group.


Playing maths consequences, so this is a really quick and easy one to prepare, works really well for example once you’ve done differentiation and integration, so you can get them to, give them pieces of paper with a function on it, get them to differentiate it and then fold the piece of paper over, and then integrate, pass it on and then the next person integrates and folds it over, and the next person differentiates and so on and so forth and then at the end opening those up and seeing if the top is the same as the bottom. And if it’s not where has it gone wrong, what was the error that happened and why did that happen, and having those discussions. When I

15

do this I often then scan in the responses and email them to the students and as a homework get them to identify all of the errors that they can find in their own work.

Pass it on tasks as well, give them each in pairs or groups an exam question to work on and then give them a time limit, so put a timer on the board and after two minutes they have to pass it onto the next group, and that gets them then looking back through each other's work, checking for mistakes, looking for errors, before they then continue on with the question, so that's quite a nice one as well.


In the homework get them to choose errors they made, write an explanation of what went wrong.

Embedding self and peer marking and correction of work. My students, they mark all their own homework, I don’t mark any of their homework for them, they take the work and also I give them the mark scheme. They mark it before submitting and then when I get it it comes to me completely marked and corrected where they can, so they just use a different colour pen for this. And so they also leave me comments, so where they can't correct something themself they would then leave me comments and then that's what I then spend my marking time doing is just responding to their comments and just giving them hints to what they can do to improve and then I give them reflection time in the next lesson and get them to actually correct their mistakes themselves.

Also recalling scripts of their precious exams and using them with obviously the students' permission in future lessons. In case you're not aware you can now recall your scripts from Edexcel for free, so any summer exams from last year as long as you get students to sign a form to say that they're happy for you to recall their scripts, you can now do that completely for free, and you'll get pdf files I think they are of your students' work. So they're really great for using in your lessons, as long as the students are OK for you to do that.

Slide 60 Spot the errors: strategy in exams

And strategy for actually answering these questions in the exams, well I would be getting them to first of all check through the steps of the solution and see if they can spot the error straight away, but if they’re struggling with that, getting them to just answer the question themselves and then go back to the given solution and make a comparison and then they may be able to see the errors much better if they’ve actually got what the correct solution should be in mind.

Slide 61 Spot the errors: questions from SAMs

So again just going back to this one from the SAMs that I shared with you, just so that you can then have a look at the mark scheme for this one and just see the level of detail that's required.

Slide 62 Spot the errors: mark scheme

So if you take a look here at the bottom there, you can see the kind of level of detail that they would be looking for, according to the sample assessment materials. So make sure just that not only they can identify the mistakes and why they’ve happened but also that they can communicate that within their exam, and give enough detail really, to make sure that they're saying enough to get those full marks.

And that brings me to the end of my section, so I’m going to now hand over to Narsh, who’s going to move on and take you through some proof

16

and some differentiation from first principles. Thank you very much, over to you Narsh.

Slide 63 Proof Thank you very much Kim. Hi everybody, hope you’re well this evening. Just before I get started, just going to pop this quote up while I say this, I may get you to do some maths later or obviously only if you wish, so feel free to grab some paper and a pen, and obviously if you’re in a room with some other colleagues then you could discuss some of these problems too while we go through them. I’m going to spend a bit more time on the proof section than I will on the differentiation from first principles section and then we'll hopefully have some time at the end to have a go at a few questions.

Slide 64 Proof: spec So I’m going to start off looking at the spec, which Kim’s already shown you. So in the overarching themes it says that students ought to be able to comprehend and critique mathematical arguments, which she was just talking about, proofs and justifications of methods and formulae, including those relating to applications of mathematics.

And there's an entire proofs section within the specification document, and the specification document I’m sure most of you have seen and is downloadable from the Pearson website. And I zoom in on that, the first section here is on proof by deduction, and we're going to be looking at proof by deduction, and if anywhere in the spec, I think Kim mentioned this earlier, anything in the spec which is in bold type is assessable both at AS and A level, whereas if it's not in bold type, such as here, then it's only assessable in the A level exam.

Slide 65 Proof by exhaustion We've got proof by exhaustion, so trying every possibility and showing that it works in every possible case, and disproving by counterexample, so showing that something isn’t true by finding an example where it doesn’t work.

Slide 66 Proof by contradiction

And then proof by contradiction, which is only in the A level and not in the AS. There are a couple of technically relatively high order mathematic skills here, so showing the proof that the square root of 2 is irrational, or that the list of prime numbers does not stop, is infinite. And so I’m going to be doing the proof of the second of those later on, that will probably be where I spend most of the time.

Slide 67 Proof by deduction So to start with, this is again straight from the specification document, so this is a snapshot from the document, from Edexcel, and they suggest as an example, prove that the expression n squares subtract 6n add 10 is positive for all values of n. And questions won't always be worded with such guidance, but here in the spec they've said using completion of the square, and so hopefully if a student were to have that, they would go ahead and complete the square nice and rapidly, and they may not know exactly how this proof is going to pan out, but hopefully by completing the square as you've seen that I’ve done there in my best handwriting, they will hopefully have at this point some idea of how they're going to use that to make the proof that's been asked for in the question. So they've got this statement and they're now looking at this and hopefully they're thinking to themselves right, they said use completion to square, I’ve expressed it in completed square form, how can I use this to prove that n squared subtract

17

6n, add 10 is positive for all values of n.

So they would hopefully then realise that this section here would necessarily be greater than or equal to zero, because the lowest it can possibly be is when n is 3, giving a bracket of zero, and obviously if you square zero you still get zero. So obviously this is going to be at least zero for every value of n, and this symbol here, for all values of n. You don’t need to know that symbol but there's no problem using it.

And if you add 1 to something that's definitely at least zero then you'll get something that's at least 1, and you'll certainly get something that's bigger than zero, so now we've got rid of this equality part, and turned it into a strict inequality. So now everything that it could possibly be is initially at least zero and then we add one, so we finally get a value that is definitely definitely positive.

And so there's a nice simple proof by deduction and not much different to the sort of questions that could have been asked in the old spec, in terms of completing the square, but just taking on that extra level and using it for a proof.

I’ve added in one extra line here which I don’t think would be required for any marks in an exam, I think you’ve shown it in the previous line, but for me, I like to go back to what the initial question asked me to do, and the initial question asked me to show that this expression was greater than zero, or positive for all values of n, so I've gone back to show that final line.

Slide 68 Proof by… OK, so here's something else that I’ve snapshotted directly out of the specification. Suppose x and y are odd integers, less than 7, prove that their sum is divisible by 2. Now this time they haven't said what type of proof they're asking for, they haven’t said show by completing the square, so they haven’t really given an awful lot of guidance, and so the student might well be sat there thinking right, OK, I can see that it says prove so I’m being asked for some sort of proof. Is it deduction, is it exhaustion, is it contradiction, what is it? And obviously that’s not trivial, but there are a few clues in the question of how they might realise what type of proof this is.

Slide 69 Proof by exhaustion Now I’ve done a model solution here. Just because I’ve tried to squeeze it onto one page I’ve gone in a funny order. So I’ve started off with this and I’ve said right, I’ve got x and y integers less than 7, so x and 1 could be 1, 3 or 5. And at that point, even if I haven't decided if it's a deduction or a contradiction or an exhaustion question, I’m looking at this and I’m saying OK, I’ve only got three possible values for each of my two variables, so now exhaustion is on the cards, because there’s only a handful of things I could be looking for. And that's something I’d want my students to be thinking as they go through a question, is OK, the fact that there's only three possibilities for x and y here makes exhaustion much more likely than if there were for example 20 or 30 different things to look at.

I’ve then done a table of what happens when I add x and y because they were asking about the sum, so every time I add my two integers I get an even number, and we've got a few options for how to complete this question, because we could have written in here if x was 5 and y was 3, which I haven’t bothered with, because I’m arguing that I know that here,

18

now, I’ve said I’ve dealt with this in my model solution that I know that addition's commutative, so swapping the order around isn’t going to matter. Again I don’t think it'd be necessary for a student to use the word commutative, so long as they justified why they hadn’t reversed the values, or they could alternatively just do their table to have all the extra options, you wouldn’t need to swap them where the values are the same, so you're only looking at three more options, so they could just put those three extra rows into the table. And they can see that every single time you get a result that is divisible by 2, i.e. an even number, and so it's been proved.

Slide 70 Disproof by counter example

OK. Disproof by counterexample. Now I think for me, if it's worded in the same way that this specification suggested question is worded, then it's very obviously a disprove question because you’re showing it's untrue, and if you're going to show that it's untrue then all you need to do is find one value of, sorry, I’ve just noticed a question come in. One value of a number which doesn’t work for the statement.

So hopefully students will identify it very quickly and then they can just try a few values, I would always try a few low values to start with, and this one works, so when I try n as 1, the value, the statement, and I get n squared take away n add 1 being not a prime cause obviously 1 is not a prime, and so I would then just make a comment about that, and there it is.

Slide 71 Proof by contradiction: infinity of primes 1

OK. Proof by contradiction, so proof by contradiction only comes up in the a level, it doesn’t come up in the AS, and I think the irrationality of root 2 is probably a little bit more trivial, I would t call either of these proofs trivial but I think the irrationality of root 2 is slightly more trivial than the infinity of primes, so I’m going to go through the infinity of primes, I’m going to take a little bit of time to go through this nice and slowly so that those of you that are new to this can follow it, and you may want to have a go at doing this alongside me if you want to.

So we're going to look at the infinity of primes, and to start with we're going to, sorry, to do a contradiction we're going to start by assuming that the claim is false. So if we talk about the number of primes being infinite, so we've got a whole list of primes and there is no limit to how many primes there are, then that means that they're infinite. And we’re being asked to do this as a proof by contradiction. So if we're going to start by assuming that the claim is false, we're going to start by assuming that the list is finite and not infinite. And that means that there are n primes. So if we've got a finite list we've got n primes and we can list them, p1, p2, p3, p4, p5, p6 etcetera.

So I would suggest if you are writing along then make a little list of primes, p1, p2, p3, you don’t need to write the actual prime numbers, just the order of them, so the first prime p1, the second prime p2 etcetera, and that means that you can write the list up to pn. If I were you I wouldn’t write the list up to pn, you’d need quite a lot of paper for that, I would probably just have a little dot dot dot in the middle there, p1, p2, p3, p4 dot dot dot pn.

And then this proof essentially requires a little trick effectively and you need to know the little trick. So this is probably a proof that you'd probably get your students to learn. We can construct an integer n, and we're going to construct it by multiplying all those primes that we’ve just listed

19

initially, so we’ll multiply p1 by p2 by p3 by p4 by p5 etcetera, up to pn, multiply them all together and add 1. Now there's no reason I don’t believe for students to have to write this formula, but there's no reason why they couldn’t, and the notation shouldn’t be too much of a challenge for them, because this symbol pi for product is not very different to the sigma they use for summation, which I’m sure they'll see a lot in statistics.

So we're multiplying all the p values, prime values, from 1 up to n, and then add 1. And that will give us a number and we're going to call that number n, and on my next page we're going to start off at this point, and we're going to start off by considering what happens when we think of its factors. So we want to know whether all the prime numbers below it are factors of this number. If they are all factors of it then obviously it's not a prime. If none of them are factors of it apart from 1 which isn’t prime and itself, then this thing must be a prime, or, yeah, sorry, that must be a prime.

So have a think about that for a moment, and have a think about whether, which of the primes below this number are going to be factors of this number n, and I’ll move the slide on in about 20 seconds.

Slide 72 Proof by contradiction: infinity of primes 2

OK, so we start off with this value, n equals the sum of 1 and the product of all of the other prime numbers.

So we've got two possibilities. Either n is composite, i.e. n is not a prime number. If n's not a prime number then either n has a prime factor, in which case one of the primes used in that list must have been a factor. So we constructed this number by multiplying all of the values of the primes below it and adding 1. So for one of those prime factors, sorry, prime numbers to now be a factor of this doesn’t really make any sense, because we constructed it in such a way that we multiplied that prime that we’re now claiming as a factor by a load of other numbers, and then we added 1. So we know that when we divide it by that prime factor, it must give a remainder of 1. Which means that there must have been another number, another prime within the list from p1 up to pn, but we said that we'd listed every single prime from p1 up to pn, which means that our list was not exhaustive, and so we've got a contradiction of our initial claim. And if we've got a contradiction of the initial claim it can't be true.

But there’s another option. It could be that our number, this new number here, is prime. And if it's prime, we know that it's definitely bigger than the biggest prime that we wrote down before, because we wrote down all of the primes, the finite list, and our finite list ended at pn. If pn's our biggest prime and we multiplied a whole bunch if other primes with it and added 1 to our answer, n's definitely bigger than pn. If n's bigger than pn and it's also prime, then pn can’t possibly be the last prime, because we've just invented one that's bigger. So therefore we've again got a contradiction of our initial claim, that p1 through to pn is our finite list of prime numbers.

So either way, whichever of these two options we've got, we have a contradiction of our initial claim that we had a list of prime numbers, and so therefore we've proved that the list of primes is infinite, because our initial assumption, go back to that slide, the initial assumption, at the top there, was a bad assumption. So this claim that there are a finite number of primes can’t have been true.

Slide 73 Differentiation: first OK. We're going to now look at differentiation from first principles, and so 20

principles there's an argument, when I introduce differentiation from first principles I usually have a little conversation about whether Newton or Leibnitz did this first. I’m going to leave this url on the screen, you might want to make a note of it and either share it with students or you may wish to look at it yourselves. There's a bit of a debate about Newton and Leibnitz and who discovered this first, and generally most books attribute both of them to having done it around the same time.

Slide 74 Differentiation: first principles 2

So we know that this is all assessable at either level, and we've got in here on that slide that I showed you earlier, the differentiation from first principles for small positive integer powers of x can be asked so first of all, in the formula book they give you this statement about the derivative of a function. So the derivative is a limiting process and it's the limit of a small increment in the y direction, and we can find the gradient using this and a small increment in the x direction, our increment in the x direction is x, sorry, is h, and if we increase by the same amount in the y direction and then find the gradient of that tangent then we can find the derivative.

Slide 75 Differentiation: question 1

There are some other formulae that are in the same place and we're going to be using those later on. And so here's a question I’ve written myself, show that the derivative of 5x cubed is 15x squared. So if I were a student, if I had a student who was asked this question I would initially want them to do a sketch. Now they don’t have to do a sketch, I don’t think they'd get any marks for their sketch, but I think it's worthwhile. So I’ve done a sketch of 5x cubed, I’ve only done it in the positive positive quadrants, so just up here, and I want to know about the gradient at this point which I’m calling x, so I want the gradient there. And in order to find the gradient there I’ve given myself another point a bit further along where the x coordinate here is x at h, and so this distance here is h. And that gives me a y value of x add h cubed and then multiply it by 5.

Now I know that really if I’m finding the gradient at this point I’m finding the gradient of this tangent, which touches that point. And I’m going to approximate the tangent, and I’m going to approximate the tangent by using this chord. And this chord is joined to the second point that I mentioned. And so differentiation from first principles works by now finding the gradient of that chord, so I’m going to use that chord as an approximation for the gradient of the tangent, and I’m going to say that the tangent would have the same gradient as this point here.

And the gradient of the chord is going to be the difference in the y values, divided by the difference in the x values, and so to get the difference in the y values I need to subtract this height and this height, so I need to take this height away from this height. This height is 5x cubed, subtracting that away, this height is 5h cubed so I take it away from that. And I need to divide by the change in x, and the change in x is this coordinate subtract this coordinate, so x add h subtract x. And as Kim said earlier, these might appear quite small, so you might want to make sure that you're full screen on this PowerPoint right now, and I’ve gone through the lines of working.

So you can see here that I’m first going to expand and if they're confident with binomial they'll be able to expand that nice and quickly, and then expand some more, multiply the 5 into the bracket, and then the first term here is going to be added to the negative 5x cubed, so it will disappear and we'll be left with 15x squared h and 15 x at h squared add 5h cubed. We

21

can factorise out one of the hs, and that h can then cancel with the h on the denominator. When I do cancel those I get 15x squared add 15xh add 5h squared, and there's the gradient of this chord.

And now we can say right, but what happens if I make this distance from here to here smaller, and I bring this point closer and closer and closer to this point here. If I do that and keep on doing it and I get closer and closer and closer, eventually in the limit I’ll get to that point. And so when that's happening I’m making this h value smaller, I’m making this distance cross here smaller.

And you may notice that I’ve used delta y delta x here, and I’ve also used h notation, and there's no problem with doing that. If you want to just stick to using delta notation then that's not a problem, and if you just want to stick to using h notation that's not a problem, I’m sure there are many of you who have done these sort of briefs with either of the notations. Both will be accepted in the exam.

And so if we carry on, we get down to the final limits, the limiting process, and in the limiting process we can see that the final two terms that I wrote down earlier, the final two terms here are both going to disappear. And so we get our final derivative, 15x squared, so we've proved what was asked. OK.

Slide 76 Differentiation: first principles – SAMs A level Paper 1 Q10

Right, we're going to look at a trig proof now, this is a bit less trivial, so I’ll do this a bit slower. This is a question from the SAMs, I’m not going to do this question just yet, this is asking to show that the derivative of sine theta is cos theta, but what I want to draw your attention to, and I’ve checked this with some of the maths team, is this bit that they give us here as an assumption, so we can assume these formulae, so this bit we know anyway from the formula book, if they give us that, and these parts, they’re giving us as something that we can assume.

So I’ve written my own question, given that theta is measured in radians prove from first principles that the derivative of cos theta is negative sine theta. And so very similar to what they've asked in the SAM, just here, but I’ve done it the other way around, I’m asking to show that the derivative of cos is negative sine. And in order to do that, again I would want my students to do a sketch, and this time when they do their sketch they're going to have, so I’ve done it again in the positive positive quadrant just to start with, I’m going to find the gradient here, I’ve drawn my tangent, I'm going to use that chord as my approximation, and I’m going to need to use the difference in the y values divided by the difference in the x values to get my approximate gradients.

So I’m just going to leave that on the screen for a moment, and if you’re confident with trigonometry or you’ve got the formula book to hand, it is downloadable from the website and I think off the top of my head it's also in the delegate pack that you can download, then you should hopefully be able to use the formula that's given for cos a plus b, so expand this out, and you may if you followed the previous proof be able to replicate the entire proof for this differentiation from first principles of cos theta, and you should hopefully get to negative sine theta. So if I give you 30 or 40 seconds to do that.

Slide 77 Differentiation: first The formula book has a few useful things, so here are a few useful things 22

principles that are in the formula book. And so we’re going to use that second one, the formula that’s just here, with cos a add b, and so that gives us cos cos subtract sin sin, so we’re going to use that right now, and we get cos cos subtract sine sine, and then we've still got the subtract cos x at the top. Simplify the denominator down to h, and now in this next line I’m going to factorise out cos x. And if I factorise out cos x I get cos h take away 1, inside my bracket, so if you can see that there, sorry not there, from this and this here, and then I’ve got this middle term which I’ve shifted to the end. And then I’m going to break this up into two separate parts, I’m going to take this part as one bit and the second part over here. And now we're going to need to use the part which they gave us as assumed in the SAM earlier, so I’ll leave that for a moment in case you want to copy down that last line there, so this line here. And then in five seconds time I shall just give you that assumption that they have us in the same back again, so that you can then hopefully complete the proof. And hopefully while you’re completing the proof my phone won't ring.

So here’s the assumption, so there's what they told us earlier. Now we're not going to need, we are going to need exactly those, so sin h over h tends to one, as h gets smaller and smaller cos h take away 1 over h tends to zero, as h gets smaller and smaller.

So if you combine these with that last line that I was pointing to a moment ago, then hopefully you'll be able to get right down to the end of the differentiation from first principles of cos theta. If I pop that slide back there for you now, hopefully along with the assumptions that we had you’ll be able to…

OK, so we know that as h gets smaller and smaller, this term here is going to tend to zero. We know that as h gets smaller and smaller this tends to 1. Those are the two things that we just saw on the question, and so as we’ve let h get smaller and smaller we get this thing tending to zero, so there's our zero, this bit becomes 1, which means we're left with just a sine x, so there's our sine x, with the negative, and so we get negative sine x. And we've shown from first principles that the derivative of cos theta is sine theta. Actually I’ve used x in the entire question rather than theta. I’m sure you’ll all survive.

Slide 78 Arising from the SAMs: AS level paper 1 Q7b

OK, let's move on. So I’m going to look at a few issues now arising from the sample assessment martials now, I know Kim’s shown you quite a bit from the sample assessment materials earlier. I’m going to be looking at this from the point of view of people who have actually done the questions. Now if you haven’t done the questions, at some point I imagine you probably will when you have time to, and you'll probably look at the mark schemes and you may then have some questions that come about. Now hopefully we will have pre-empted them all by you doing this course.

So first of all I’m going to look at the as level sample assessment question 7b from paper 1, and in this question they ask a standard binomial question, so they ask you to find the binomial expansion of this, I think just the first three, yeah, the first three terms, and that part is a nice standard easy part that you’d have had in the old spec. And similarly part b looks like the sort of thing that you had in the old spec, except it isn’t because they don’t ask you for an estimate of this value, they ask how would you use the expansion, just an estimate of how you would use it, for one mark,

23

to get an estimate for the value of this. Now you could go off and just do that, and if you've got time to do that and only spend around a minute on it, because you get around a minute per mark, then that's fine.

Here’s one solution I’d like you to, hopefully you can read it, have a look at it and see whether it's worthy of the mark. So I’ll give you 20 seconds.

OK. So here's the mark scheme.

And there are the notes on the mark scheme.

And I'm just going to highlight one part of this.

So we're looking at that and that.

And there's that answer again. So 20 seconds, if you’re with colleagues then you might want to have a little chat with them and decide do you think it would be worth the mark. Just in case you want the question again, explain how you would use your expansion to give an estimate for the value of 1.995 to the power of 7. I’d use my expansion to approximate that by setting two subtract x over two equal to 1.995 and obtain the approximate value, sorry, the appropriate value for x. I would then substitute this value into my expansion.

So, I personally felt that this was worth the mark and I felt that the mark scheme suggested that it didn’t get the mark. We raised this with the examining team, and the examiner team responded to us and said, yes this should be worth the mark, but the mark scheme implies that it isn’t, so we will raise this with the assessment team and amend for future. Their amendment could be multiple types of amendment, their amendment could change the question to demand that you ask, that you do calculate that value of x, so they could amend it by asking explicitly to find the value of x required, or they could amend the mark scheme and take out this word and, and just imply that so long as the explanation states what the student would do then that would be sufficient.

OK, so that would get the mark, just to be clear, that has been confirmed.

Slide 79 Arising from the SAMs: AS level paper 1 Q9

Right, here's a question that looks like it could have appeared on the previous specification, and I’d like to just pause for a moment while you consider whether there’s anything noteworthy in the question at all. I’ll give you a little hint, there might be something noteworthy given that I’m asking you the question.

OK, so nothing major in here apart from this unusual domain for the x value. Something that students may or may not be very experienced with, if you're using old spec papers I think that domain is pretty unusual compared to what they used to ask in the past, so again just important to stress when teaching this sort of content that looking at the domain and just being absolutely clear that they’ve answered exactly what the question asked for is going to be important.

Slide 80 Arising from the SAMs

OK, Kim showed you this question earlier, this was a modelling question and they ask about volume of oil.

So we’re going to be looking at part a and I’m going to try and zoom in a little bit for you, and those are the two particular aspects that I’d like you to have a look at, so the daily volume of oil, v, measured in barrels that the company will extract from this oil field depends upon the time t years after

24

the start of drilling. So use model a to estimate the daily volume of oil that will be extracted exactly three years after the start of drilling.

So here’s a solution. Linear model so estimate for v when t = 2.

I stick t = 2 in and I get 12,500. I put t=3 in and I get 12,500 add 9000 over 2, so I get 10,750.

So would it get the mark? So the, hopefully you saw this at the start of the session, there's an icon at the top of your screen which has a raised hand. If you think this person who wrote, sorry, if you think that the value, 10750 here, sorry, I’ve lost my green pointer, so if you think that this solution or this attempt will get the 1 mark according to the mark scheme then can you put up a tick for agree, and if you think it won’t then put in a disagree. I’m not saying by the way that I think it does or it doesn’t, you’re not agreeing with me, it’s just a tick for yes, one mark, or a cross for no, zero marks.

A few answers coming in, I’ll leave it just a bit longer.

OK, so it looks like a bit of a mixture here. OK, so the question really that I’m asking about is about the units, which Kim did mention earlier. Now this is a bit of a tricky one. Would it get the mark? Well the examiner team, we asked the question and the examiner team got back to us and said well the question asked for an estimate of the daily volume of oil extracted, so in essence the answer should have units. However the principal running the examination board would make the final decision in conjunction with the chief for that unit, we did say that there's a question mark and a bit of a lack of clarity which they would also look at, which is that v was measured in barrels, so does that mean when you get a value for v that barrels is already implied, given that they mention that it’s measured in barrels here. Possibly, so this is unusual, and you might argue that it's not a particularly helpful response from the examiner team. However the message for me as a teacher is, just encourage my students to make sure they're thinking about units all the time so that there's no risk.

Slide 81 Arising from the SAMs: A level paper 1 Q11a

OK, here's another question. So an archer shoots an arrow, oh, I’ve lost a bit of text, there we go. So there’s the question and the answer, and it's a quadratic and they need to solve the quadratic and in order to solve the quadratic there’s no problem with them using a graphical calculator and that's explicitly stated in the mark scheme. That's on paper 1 of the A level SAMs question 11a.

Slide 82 Arising from the SAMs: A level Paper 2 Q12

If we now look at this question you get a similar issue, you get a quadratic here, this is a quadratic, once you'd made some substitutions, but the mark scheme doesn’t mention the graphical calculator at all, so again we checked on delegates' behalves, and the examiner team said there's absolutely no problem using a graphical calculator on this either, it just happens to be explicitly stated on one mark scheme and not the other. So here we have no mention of the scientific or graphical calculator, no mention to calculator use at all, graphic or scientific, but absolutely fine to do so.

Slide 83 Arising from the SAMs: from the specification

OK. Right, there's an important part here, which is about set notation. So it says here being able to express solutions through correct use of and or through set notation.

Slide 84 Arising from the And there's a question here which is an inequality question, if I remember 25

SAMs: A level Paper 2 Q11

rightly. So when we're solving this final part we end up needing to get two values of k, sorry, an interval for k. And in order to find the right interval we need to solve the inequalities and we should get these 2 areas. So there's our solutions that we initially find, and then we've got an answer written explicitly in set notation form.

Now we asked the question does it have to be in set notation form, especially considering the wording, and the examiner team said OK, actually in order for this mark scheme to be valid the question would have had to have said you must state your answer in set notation form. However given that it didn’t they would accept these answers without the set notation part.

Slide 85 Arising from the SAMs: A level Paper 2 Q12c

OK. This is the A level SAMs paper 1 question 12c, and it requires a comment about the limits of extrapolation, i.e. something which is only really been discussed in the past in statistics. So they're asking about the validity of the model, again touching on what Kim said earlier. Explain why the information provided could not be reliably used to estimate the day when the number of microbes in the culture first exceeds one million.

So in order to realise why it's not a good way to do this they have to realise that log base 10 of n is greater than or equal to 6, which is way off the graph, especially because the graph doesn’t even get to five, which is the end of the axis here, and so using it for six would be certainly risky at the very least.

Slide 86 Arising from the SAMs: A level Paper 2 Q3

OK, here’s A level SAMs paper 2 question 3, and this time they give us a product that they want us to differentiate, and this could have appeared in the previous spec, so I’ll give you another 20 seconds to have a think about what if anything is noteworthy.

OK, so hopefully when you look at this you'll see that the final answer here has been differentiated, so they’ve differentiated this using a bit of product rule and chain rule together. And then they would get an answer which wouldn’t necessarily initially look like this, and they would then need to factorise. And it's very rare in the old spec that that was the case, whereas now because of this being a show that question, they would have to go that step further and actually put it into factorised form.

Slide 87 Your turn… Right, we've got about ten minutes left, just a touch over, so now I’m going to give you guys a chance to put into practice some of the things you might have learned during this hour and three quarters that we've had so far.

So these are going to be some A level SAMs questions, you'll see that there's two hours for the paper, so two hours with 100 marks, a bit of proportional reasoning, so that gives you 1.2 minutes per mark, which is 1 minute and 12 seconds.

Slide 88 Your turn: AS level SAMs, Paper 1, Q6

So 4 mark question, gives you 4 minutes 48 seconds, me being the very kind and generous soul that I am, I’m happy to give you 5 minutes. So here's your 5 minute countdown, off you go.

OK, you’ve had about one third of your time so far. If you are having any issues then you could send in a question to the host and they could send it to me or Kim to give you some support with this. I will go through the

26

mark scheme at the end of your five minutes, and you can mark your own attempts and see how you've got on.

Just a short amount of time now.

OK, so let's have a look at this question. So there's the mark scheme for it, so you can have a look at your work, don’t worry whether you’ve used hs or deltas or a combination of the two, you’ll get marks for that either way. You’re looking in the first scenario to be considering this fraction, so if there’s any evidence of that, that's where your first mark’s going to appear. We're then looking to expand the brackets at the top, into 3x squared add 6xh add 3h squared, so that’s that there.

Then we're looking to simplify.

And when we simplify we'll get these extra terms that have hs or delta x in it, and as that h or delta gets closer and closer to zero, we end up with just 6x as our final answer, because we'll have that or that and these terms over here will end up tending to zero. So that's your four marks.

Slide 89 Your turn: AS level SAMs, Paper 1, Q10

OK, let's look at another one. So this time it’s another four mark question, the equation kx squared add 4kx add 3 equals zero where k is a constant has no real roots. Prove that zero, sorry, k is between zero and three quarters. So here's your five minutes, starting now.

If you’re happy for me to go through the mark scheme for this question when the time gets to three quarters of the way through, could you raise your hand please on your icons using the facility you used earlier? If you'd rather I waited, if you could either put in a disagree or don’t do anything, I shall take that to mean you’d rather I didn’t.

OK, I’m going to go through the mark scheme in just a few seconds' time.

So here we go, so this time, this first part I think is quite important. So k equals zero will give no real roots as the equation becomes 3 equals zero, so if you go back to the original question and then substitute in k equals zero, you get zero term here, zero term here, 3 equals zero. 3 equals zero doesn’t work, so clearly k can’t equal zero.

Then for the remainder of the question it is pretty much exactly the same as the old spec, solving in equalities or, sorry, quadratic inequalities by looking at a discriminant for a quadratic, so the rest of it I think you can do pretty much as normal.

OK. And just before I move on, there's a question that's come across about the script service, and the script service is only available where you can get all the scripts back for free until the end of September, so that runs out very soon, so if you are looking to do that you’ve got a very short amount of time that you can speak to Edexcel and get scripts back which you could then use for your spot the mistake kind of questions that Kim was mentioning earlier on today.

Slide 90 Free support Right, I’m not going to go through these other questions with you which I put in if there was time allowing, but what I am going to go through is the other areas of free support that Edexcel provide for you.

So first of all there's a whole heap of stuff online, this is all through the emporium site and the Edexcel qualifications website. There’s also the exams wizard and the results plus sites. And all of these you can access

27

through an Edexcel online log in. If you don’t have one of those then it’s usually your exams officer that I think can organise that for you, and it’s probably worth knowing about the maths emporium, and also the emporium mailing list, which Graham regularly sends emails out from, or someone from the same team, although I gather Graham is much more active on Twitter these days.

OK, and then if you have any questions the session will be ending

28

qualifications.pearson.com Level... · Web viewSlide. Purpose of Slide. Additional Information....

Documents

Transcript of qualifications.pearson.com Level... · Web viewSlide. Purpose of Slide. Additional Information....