Logarithms

In last week’s post I talked about the work that we had completed on indices and how we were using this to launch logarithms and exponentials this week. The benefits of this approach were shown up during one of the indices lessons when one of the students was tackling the following question (taken from Stuart Price’s Problem Book, @sxpmaths):

He asked if there was an easier way to tackle problems like this, suggesting that if the numbers were much bigger it would make the problem more difficult. This was a perfect opening for introducing logarithms, which we did with a series of similar questions.

When planning this lesson in discussion with Will, we felt that this approach really emphasised the link between indices and logarithms, something that we both felt had been missed by some students in previous years. We felt that the best way to do this was by talking about functions and their inverses. Yes, inverse functions is strictly a second year topic and they wouldn’t need it for their exams in the first year, but it is the connection that has been lost in the past and the reason students struggle with the topic.

In order to build these links we created a Geogebra file that allowed us to turn on an off a series of functions and their inverses. We had it set up to display a couple of linear graphs, a quadratic and then an exponential – the goal being to draw out that the inverses were a reflection in the line y=x. This had students already trying to tell us what shape the inverse of an exponential should be, before we even introduced what the function truly was. I’ve used a similar approach before, but not incorporated the graphs – so the connections were already stronger than they had been for students in the past.

The next part of the lesson formalised the notation of logarithms, after which we went back to these questions and rewrote them as logarithms, solving the later ones using our new calculators as we went.

The next lesson began with a recap starter, but this time we took the students a bit further…

They were all able to calculate the answers on their calculators, but actually explaining why was missing, and there was no real notation or workings out (yet). So when we went back through these questions we modeled taking logs of both sides of the equation, or using both sides as powers with an appropriate base. Then we could discuss that as the two functions that we’d composed were inverses that their effects cancel. All designed to reinforce the links between logarithms and exponentials as well as to lay the groundwork for exponential / logarithmic equations.

In the past we’d found that students can be quite unreliable at remembering the laws of logs, despite the connection to the rules of indices – perhaps down to the split in topics between C1 and C2. On our scheme of work this split is non-existent as we’ve run the topics together. We also decided that to further emphasise the link that we would start with the indices rules and from them actually derive the laws of logs – this might go over the heads of some of the students, but they’d have it to look back and reflect on, plus we knew that there would be a good proportion of our group that would embrace knowing why this rule exists. After doing the product law we let the students try to work out the quotient law, and even to have a go at creating the derivation on their own. This also built on our previous work on proof and on how to construct a solid mathematical argument.

After we’d derived all the rules, the next step would be usually be to work through a series of examples, with students copying them down. As they’d already done a lot of writing I gave them the complete examples and we went through them with the students annotating why things were happening, and what rule was being used to do these things.

When we introduced e^x we got the students to plot graphs in a template that we had created in Geogebra – the idea being that after they had plotted 2^x and 3^x and examined their gradients at each of the points we had given them that they would see that at each point,  e^x has the same gradient as y-value. The group were fairly pleased with their discovery, and it allowed us to give them the reason why e is so special that it has been given its own letter. We did have one query though: “how did they calculate the value of e, so that it is the value that will them give it its own gradient?” – this came from a further mathematician, and that question became his homework.

Much of the rest of the topic was fairly standard – log equations, hidden quadratics involving exponential equations, but with the addition of e^x and ln( ) ≡ log_e( ) for the new spec, however we did make one tweak to what will has been taught in the past. We felt that students could not pick up how to add logs to a term that does not have them in so that they can then combine them using the laws. Even just these 4 questions that we started together were enough to give them something to work from in the future.

Overall we were very pleased with how the teaching of logarithms progressed over the week. In order to assess what we had covered we asked students to complete one of the Integral online assessments. Hopefully as this comes back we will see that students have a greater understanding than we have seen in previous years.

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Will’s Thoughts on the Large Data Set

Will Davies has been working with us on the scheme of work for the new A-level. Over the last few years  he has predominantly taught the statistics content for the A-level courses. Here are his thoughts on the large data set:

“When the new specifications were announced the introduction of these “large data sets” (LDS) left me sceptical, and unsure of exactly how we were going to work with them. With time came a lot more clarity; actually being able to pick over the data sets that were released with the sample assessment material meant we could start to see how they were going to be assessed, and how they might fit into our teaching.

And I have come to this conclusion: the LDS is my joint-favourite thing about the new A-Level – the other aspect being that we’ve been able to tear up the old order of topics and build a curriculum that we feel teaches maths in the most logical order and in the best manner. Being able to combine the applied topics in with the pure topics they depend on is key: e.g. binomial theorem and binomial expansion, as well as teaching variable acceleration immediately after calculus is taught.

I have read on Twitter a lot of negativity about the LDS, and I am unsure why. My instinct says that the reason is because the LDS is being perceived as a separate topic that needs to be taught in addition to other content (that we’re already unsure whether we can fit it all in satisfactorily). As a department, from very early in the process we realised that this shouldn’t be the case – the LDS is not a separate topic, it is instead the tool that you use to teach all the data-handing parts of the course.

Every time you do an example – it comes from the LDS.

Every time you set an exercise in class – it comes from the LDS.

Every time you set a homework – it comes from the LDS.

The more the students immerse themselves in the LDS the more familiar they become with it. Homeworks can be to do some calculations or create some charts (and email them to use in advance where appropriate) then as a group we can discuss next lesson. My other big idea for embedding the LDS into our lessons is to have at least once per week a Show-me / Tell-me starter (regardless of whether the lesson is going to be on stats or not). Students will be encouraged to do a little investigation themselves, then getting the class to discuss together discuss the potential causes (e.g. our outliers). This will be way in which we can as a class build up a bank of interesting observations of our LDS, just like the observation we made when we were examining the MEI sample assessment material.

This question from the MEI sample A Level assessment – we were drawn to the very long tail at the bottom of the Sub-Saharan Africa box-plot, and wondered which countries were causing this. Looking at the LDS we quickly came up with 3 countries with very low birth rates: Saint Helena, Mauritius, Seychelles – all island nations. Which feels like a nice fact – that the island nations of Sub Saharan Africa have significantly lower birth rates than other countries in that region.

This brings me onto our choice of exam board – the data sets are not provided in the exam, yet students are expected to be able to use some very specific knowledge of them in order to gain some marks in their exams. With the large LDSs (like Edexcel’s weather data) you could study that for a couple of years and maybe still have examined at the key pieces of data.

So, MEI has the smallest large data set (covering information about the 237 countries of the world), and that brings its own advantages – it is printable. The bulk of it fits on 3 A3 pages, and I have created a single A4 page that expands on the Dependency status of relevant countries. So now all our students have a hard copy of their data set to use – meaning that we don’t always have to be in an IT room when we’re working on it. The other major advantage is that on presenting students with the data set they immediately felt that because it actually wasn’t “too big” that knowing it well was going to be achievable.

When it comes to using technology there are various ways in which we plan to incorporate this with the LDS. The ClassWiz calculator is clearly going to be key as, as is learning a bit about Excel. Filters, sorting and a deep look into the inbuilt statistical formulae will all need to take place – not just for the sake of the LDS, but Excel skills are incredibly useful. We’re also going to look to support/enhance teaching & learning by graphing some of the data in Geogebra and Gnumeric. (Gnumeric is apparently a very good tool for creating box-plots although I am yet to explore that any further). I have also built in Excel a sampler tool that will create random samples from the LDs, although it still needs perfecting. When it is complete I will share it here.

When it comes to assessments, starting work on the LDS from lesson 2 means we will be able to include it in assessments from half term 1 – to start with we will make sure we write the assessments so we know that students have seen (in one form or other) what we will be asking about, then we can progressively choose more and more obscure statistics to include. Finally we plan to set students extended projects to do. These like likely asked them to choose some aspect of the data set, be it a group of regions or a groups of fields, calculate some statistics, create some charts, draw some conclusions, and to write up a little report on their findings.

Identifying the smallest data set, and revisiting it weekly for 2 years will give students the best chance of becoming as familiar as they can be with the LDS (aside from dedicating too much curriculum time to it). I suppose the bottom line is that we feel that using the LDS to teach all data topics is going to be such an improvement on using (essentially random) examples that are using a similar approach with our GCSE statistics. In lessons our year 9s and 10s are currently populating their own data-set (containing information about themselves). They have really enjoyed the data collection (although I did receive a complaint from the English classroom underneath the standing long-jump) – now to analyse it!”

Scheme of work and developing a teaching plan

This post is contributed by Simon Clay who is part of the Teacher Support team at MEI.

Given that the changes to A level mathematics are significant, an overhaul of teaching schemes for the new two-year long qualification is not a trivial task.  During 2016-17 a number of members of staff at MEI developed a Scheme of Work for the new A levels with the aim of trying to produce something useful for as wide a range of audience as possible.  This result is this freely available SoW, accessible via the MEI website.

Some of the thinking behind the design of the SoW units was as follows:

– It aimed to break down the new A level content into manageable units.

– It needed to function as a starting point for discussions in departments and therefore needed to be editable.

– It needed to take seriously the changes in emphasis of the new A levels, including the three overarching themes – Mathematical argument, language and proof; Mathematical problem solving; Mathematical modelling.

– It needed to incorporate useful features such as ideas as to how the use of technology can permeate the teaching of A level mathematics, questions which promote mathematical thinking, etc.

– It needed to be both adaptable and useable in the classroom.

– It needed to exemplify, and give free access to, some high quality teaching resources which can be ‘picked up and used’ in any classroom.

Since its launch in March, we have been pleased with the way the SoW has been received.  A common request, however, was for the provision of a plan for how the units could be linked together in a cohesive way to ensure the content is covered in the time available.  We have therefore worked on producing a series of schedules which show how the units of the first year (or AS content) can be arranged depending on considerations or constraints a department may have e.g. two teachers sharing a group, one of whom teaches pure and mechanics while the other teaches pure and statistics.  (We have so far only tackled Year 1 content but Year 2 will follow in due course!)

The reason for a post in this blog is because Schedule E is as a result of the thinking and work done by Bruce and the team at TGA Redditch.  It has been my privilege to take part in the discussions in which this SoW Schedule has been developed.

Below is an image of Schedule E taken from mei.org.uk/2017-sow and beneath this I describe the key features:

– The team wanted to begin the course with an emphasis on problem-solving and proof in order to set the culture of working in this way from lesson 1.  This means lesson 1 will contain no mathematics beyond GCSE and will instead focus on reasoning, language and proof.  Lesson 2 will look at indices but with an emphasis on reasoning and proof rather than subject content coverage.

– There was a strong desire to get the students working with and becoming familiar with the large data set (LDS) right from the start of the course.  Thus by the end of the first teaching week students will know about the LDS and have done some initial exploratory work using it.

– The team identified some units, namely ‘Problem-solving’ and ‘Graphs and transformations’ as being recurring themes which can be addressed in a number of different units throughout the course rather than taught as discrete topics.

– The team wanted to use a teaching model where the class is shared between two members of staff but essentially runs as a single series of lessons.  This will clearly involve a high level of collaboration between them but they are keen to dovetail their teaching so that the student experience is as coherent and seamless as possible.

– They wanted the applied units to be taught alongside the relevant pure unit so it is clear what mathematics is being applied.  It is hoped that this will also help with fitting in the content in the time available.

– They wanted technology to be used by teachers and students whenever possible and so in the first few weeks there are planned opportunities for this, in particular when analysing the LDS and exploring graphs of exponential functions .

– The school has made a central decision that all students need to be prepared and entered for AS level examinations at the end of Year 12. This means that although at points it would be nice to extend and cover Year 2 topics straightaway these will need to wait.

And now there are only a few weeks until the schedule can be implemented!

Problem Solving and Technology

In our work on revamping the curriculum for the new specification we have been careful to make sure that we are considering the overarching themes of problem solving and use of technology. We are very keen to ensure that use of technology does not become ever more complicated and ‘interactive’ PowerPoint files that are demonstrated from the front of the class room with little chance for students to use and develop their own skills. We also want to introduce this aspect as early as possible to encourage students to think about technology as a vehicle for working on and solving problems when they get stuck. On Friday I met with Simon, Fiona Kitchen (from FMSP) and two colleagues from my department to discuss methods for this.

Our starting point was to use the worksheet “Problem Solving with Geogebra” from MEI’s scheme of work. We looked at solving the problems ourselves, trying to limit techniques to those that year 12 students were able to use. This proved rather difficult! After much wrestling (and a plea to twitter) we managed to create working models in Geogebra for the first three of the problems.

We had a lot of fun working on these problems but concluded that the level was a too high for students who are starting out in year 12. As such we will need to adapt to something closer to GCSE if we are going to introduce Geogebra in this way at the beginning of the course. One thing that struck me during the afternoon was that we persevered with the problems for a long period, around 2.5 hours. This was something that our students would have really struggled to achieve. The same morning one of my year 11 students, when confronted with a difficult question, said “It’s alright for you sir, you are good at maths and can do it easily.” I was unable to make her understand that I don’t find all maths easy and that I enjoy the struggle with harder problems. This perhaps sums up the major problem we have been fighting against with our A-level students over the last few years – the lack of resilience as soon as a problem gets complicated.

Hopefully this process of really concentrating on both problem solving and technology will help to improve this, which is certainly the focus of what we are looking at during this process. For now though here are the problems that we worked on.

Problem 1

With this problem I found it very easy to create a polynomial fixed by the points A, B and C. This initially created a point D that moved as A, B and C moved. To do this I used the measuring tool in Geogebra to calculate how far away the points were from the origin.

My second attempt used a division at the start of the polynomial to allow me to control point D as well. It was at this point that I realised the shortcomings of my method of measuring distances – when I moved the points to negative values Geogebra continued to measure them as positive. To complete the problem Simon showed me how to use just the x-ordinate of A etc. in the calculation. My final solution is at: https://www.geogebra.org/m/bZD6fARj

Problem 2

For this problem I started by drawing a circle centred on the origin with a point on the circumference fixed into the side AC. I then created another circle centred on C which connected to the first. I repeated this methodology to create a third circle centred on point B. The three circles could then be manipulated together until I had a solution that worked. This did not satisfy me – I wanted to be able to change the triangle and the circles to remain a solution to the puzzle.

My instincts for this puzzle were probably from spending time playing the mobile phone game Euclidea – maybe those hours were not completely wasted! I guessed that the points where the circles met on the edges of the triangle were those where the circle inscribed in the triangle also touched the sides of the triangle. My resulting solution can be found here: https://www.geogebra.org/m/jMSThZg8

Problem 3

Problem 3 caused me the most problems. For a long time I was able to either create a line that was perpendicular to the tangent from A or a line that passed through the point B but not both. After a long time trying (and a plea to Twitter), Simon came up with a solution that can be found at: https://www.geogebra.org/m/MFgcJaAd

Problem 4

We ran out of time before tackling problem 4 and I have not yet returned to it. I have some ideas about using a quadratic function with roots that are the x-ordinates of A and B and integrating it but have not progressed any further yet. I leave that one with you…

Thoughts on Large Data Sets

One of the thoughts that came out of my most recent meeting with Simon was that the choice of exam board will be influenced by the large data set. I had previously been of the opinion that I could leave the choice until January 2018, seeing if any more specimen/mock papers became available and analysing question types. However this would mean not spending as long familiarising students with the specific large data set for whichever exam board we choose. As a result of this I have downloaded the data sets for AQA, Edexcel and both specifications of OCR. I should point out that I am not a statistician, I have taught S1 once and try to avoid it if I can!

I have started to look at the data sets to see which is most useable, and which students will be able to best gain insight into for reproduction in their exams. We want to be revisiting the data constantly, so that students are really familiar with it. This means that portability is important as we will not always be able to access computer facilities.

AQA – Purchased quantities of household food & drink by Government Office Region and Country

The data given is split into 10 regions (under separate tabs), with the average amounts of various foods and drinks per person per week. There is also a tab with averages for the whole of England. Having spent some time in Excel playing around with the data it is possible to fit each region onto a single sheet of A3 paper (total of 11 sheets).

Looking at the questions in the specimen paper, students are expected to be able to recall information about the average amounts of certain food groups from different regions. This is something that could only be known by someone who has done extensive work with the data set before, and given the sheer scale of the data is unlikely to be something that you could repeat for all of the different food groups.

Later questions involving the data set give a small excerpt and ask questions about these. These are much more accessible to students who do not have as much familiarity, but will be easier for those who are aware of the context. For example there is question about the total amount of confectionery purchased, which does not state that it is based on averages.

Total Marks based on Large Data Set in AS Spec Paper: 9 (Out of 80 on paper 2, 160 across the AS)

OCR A – Method of Travel / Age Structure

The OCR A specification looks at the methods of travel to work, broken down into regions, taken from the national census in 2001 and 2011 (separated into two sheets). There is also data about the ages of the residents of the regions (2 further separate sheets). Each tab can be set to cover three A3 pages, so a total of 12 will be needed for a portable copy.

In the question pictured here it would be advantageous to be familiar with the data set, particularly for part (ii), as there are different codes for the authorities based on their type. If you knew this then you would know how to separate the authorities further and would merely have to explain this.

For the other question based on the data set (not pictured), a summary table has been created. It is not as obvious what the benefits to knowing that data are here, although general familiarity and having looked at possible summary statistics will help.

Total Marks based on Large Data Set in AS Spec Paper: 8 (Out of 75 on paper 1 and 150 across the AS)

OCR B (MEI) – Population data and Olympic success

The first thing to note here is that the MEI specification (OCR B) has taken a very different position to the other boards. There will be three different data sets that will be used in rotation. The data sets that will be used for ‘live’ specifications are not available yet.

The data set that is available for the specimen papers is far less ‘large’ than the others, reducing to two A3 sheets. The question included here really grabbed me as being interesting – what were the outliers in Sub-Saharan Africa? On inspection, the data that stood out was that from islands, rather than countries on the continent.

This data set seems much more manageable than the others, and over two years I would expect students to be able to become very familiar with it.

Total Marks based on Large Data Set in AS Spec Paper: 7 (Out of 70 on paper 2 and 140 across the AS)

Edexcel – Weather Data

Edexcel’s weather data consists of 5 weather stations in the UK and 3 from abroad, with readings from both 1987 and 2015. I have been able to fit the data for each station, for a single year, on one A3 sheet (total 16 sheets).

The questions based on this data set again seemed to not require much detailed knowledge of the readings. In the question shown here it is only the fact that there is one reading per day that will help with part (b).

Of course, as Edexcel has not been accredited yet, this may change.

Total Marks based on Large Data Set in AS Spec Paper: 11 (Out of 60 on paper 2 and 160 across the AS)

Summary

While the use of the data set will only form part of my decision on which exam board to use, I have found the process of sifting through the data sets, and the questions that relate to them, extremely useful. It has also shown me the benefits of this approach. In starting to look at the data sets it is already noticeable how the data is starting to feel familiar. I think that this will develop much more ownership of the data and make structuring easier. Now students know they are expected to know the data set, they are more likely to see the value in using it as part of exercises.

First Attempt at a Framework

On Friday I met with Simon and my head of department Pete to try and create an initial framework of topics for the scheme of work. The target was to have a loose order of topics to cover the first year of an AS course, changing from our previous structure of 3 teachers each teaching individual modules, to a linear structure that will probably be taught by two teachers.

One of the real benefits of moving to the linear scheme will be how much time it frees up compared to our old structure by removing some of the assessment. Previously we have tested students each half term in all three modules. This has been in the form of a one hour assessment based on past exam questions, starting off quite narrow and expanding as more content has been covered. By the time these assessments had been completed and feedback given we were looking at 6 hours of teaching time lost per half term. In a linear system I would anticipate that the assessment could be reduced to a single one hour paper initially, allowing us at least 4 hours more time each half term.

Using the AS topic headings from the freely available MEI SoW we began to organise the topics into a coherent order, focussing on pre-requisite knowledge, and links between topics.

Having a hard copy of the MEI SoW to hand (http://mei.org.uk/2017-sow) was useful as we moved topics around.  It is designed to be editable for any specification and allowed us to focus on the connections between mathematics topics

One of the striking things that came up in the conversation was how we had previously compartmentalised topics. Surds and Indices is a C1 topic, whereas Logarithms and Exponentials is a C2 topic. Yet they are different ways of looking at the same thing and surely if taught together would allow a much better understanding of where logarithms come from, something that I have always struggled to get students to see. As such we have decided that the first thing we will teach is logarithms and exponentials, while at the same time revising the surds and indices materials students should have met at GCSE. This means that students will be meeting something new straight away, hopefully catching interest, but also brings in a link to previous learning.

A provisional model is shown in the diagram below, pure units in green, statistics in blue and mechanics in orange.

The model we have come up with looks very heavily weighted to the first half term. However of the five pure elements, four should be revision from GCSE. Historically we have taught these as the first half term of C1, a third of our teaching time across the whole course. While thinking about the links in topic areas we discussed how some of the topics (see the right-hand columns of the grid above) might be better spread over the course, with pieces put into different topics to improve connections. An example would be that for transformation of graphs in the past we have taught completing the square early in the course and touched back to how this links to transformations much later. We feel that by expecting to make links with transformations at appropriate points throughout the course as it naturally arises the links should be much clearer and stronger for the students.

Coordinate geometry is another topic that we felt was better split across the year. Tangents and normal will fit in as an introduction to differentiation and circles has strong links to trigonometry.

With the statistics elements of the courses we decided that the large data set should be introduced as early as possible. This meant that we inserted data collection, which is largely about sampling, into the first block of topics. This also got me thinking – I had previously decided that I would not make the decision on which exam board to use until much later. However in order to introduce the large data set I need to have made the decision so that students are used to working with the relevant data.

Mechanics fits in very well with elements of the ‘pure’ maths, particularly with calculus and variable acceleration. This has always been something that I have felt is a missed opportunity in the teaching of A-level maths, it should create a connection and allows us to show the roots of these skills in a real life situations.

This of course is only a first attempt and will continue to evolve as we move forward. At our next meeting with Simon we are going to look at the individual content statements for each topic and to order those, whether within the current structure or moved to further emphasise links.

Beginning to Order

One of the first things I am considering when putting together my scheme of work is the order in which we will teach. It feels like I have performed this task repeatedly in the last few years. My first scheme of work included modular examinations in January, then those were removed so I shuffled C1 and C2 into a single ‘core’ unit. When I moved schools to my current position I wrote a new scheme of work, within which a teacher took control of a single module, meaning that each group had three teachers. Last year I had a slightly smaller job of transferring my existing scheme of work into a new format so that it was on the correct templates.

Should we teach the topics broadly in the order MEI have placed them in their scheme of work (albeit with the statistics and mechanics spread throughout) or re-arrange so that similar content is taught together. In the past, when teaching C1 and C2 as a single ‘core’ unit I have rearranged the content so that, for example, all of the differentiation is together. This has already been assumed in the scheme of work, but should integration be taught immediately after. Perhaps it would be even better to teach them at the same time? This would hopefully create a much better understanding of the inverse nature of differentiation and integration – students could differentiate a curve, then integrate to get back to it, using a point on the original to find c.

A different strategy would be to separate similar content, allowing more structured interleaving. As students come back to a topic, they revise the original content and then build on it. This has obvious benefits of seeing things more than once, but is likely to lead to too much time being spent going back over previously learned content and falling behind as a result. As time is already tight we cannot afford too much slippage.

On Friday I will be meeting Simon and my head of department to start to build up the scheme of work, beginning with this process of ordering topics. I have loosely grouped topics – we now need to come up with an answer to the questions posed above. I am ready with the topics on cards to move around, string to make links between the topics and a blank timeline…