Boosting achievement in AS level Core Mathematics. Supporting lower ability students through the C1 and C2 modules. Phil Chaffé 2012. Programme. 10.00 – 10.45am Boosting initial subject knowledge and understanding 10.45 – 11.15am Boosting algebra skills
Supporting lower ability students through the C1 and C2 modules
Phil Chaffé 2012
10.00 – 10.45am Boosting initial subject knowledge and understanding
10.45 – 11.15am Boosting algebra skills
11.15 – 11.30am Discussion: coffee break
11.30 – 12.15pm Boosting coordinate geometry skills
12.15 – 1.00pm Boosting sequences and series skills
1.00 – 2.00pm Lunch and informal discussion
2.00 – 2.45pm Boosting trigonometry skills
2.45 – 3.00pm Discussion: afternoon tea
3.00 – 3.45pm Boosting calculus skills
Preparation for initial discussion:
Find the exact value of
A proper factor of an integer N is a positive integer, not including 1 or N, that divides N.
(i) show that has exactly 10 proper factors
(ii) find how many other integers of the form (where and are integers) have exactly 10 factors.
When is sundered by , the residue is . Use the detritus principium to find the appraisal of .
You have just started an AS level mathematics course and have been presented with these questions…..
“Hey, just wanted to know:
How difficult did you find module C1 (in AS level maths) compared to GCSE Maths?
What grade did you get in C1 module and did you do the exam in January? If so, how much revision do you think you did for the C1 module up until you had to take the exam in January?
What books did you find helpful to revise from if you're one of the people who got a grade A in the C1 module maths AS level exam?
Was the transition from GCSE Maths to C1 module AS level maths hard/difficult? or easy or the same difficulty as with GCSE (when you did GCSE)?
ESPECIALLY MEAN EDEXCEL C1 MODULE AS LEVEL”
I did Edexcel... Umm... At 1st I struggled because I didn't work much for GCSE and so I was used to being lazy. lol. With some effort the transition should bridge up nicely. The transition really depends on the person. It was different for everyone in my class and we were a Further Maths Class so we had to finish the spec in 8 weeks. A lot of people who had nearly perfect scores at GCSE thought it was a little challenging at the time but when we did C3 near the end of the year 12 we realised that C1 is not difficult at all. I found I had to put some effort in at AS and A2, it was not plain sailing to say the least. I think differentiation and integration are the hardest things that come up in C1 and you MUST get rid of ALL your silly mistakes by past papering. I used to make so many really dumb mistakes and think to myself wth am I doing but it's all part of the learning process. Maths is a thoroughly enjoyable A-Level.
I’m finding it hard! One day left to revise and I’m bricking it, I got an A at GCSE, I was in the bottom set for Maths then. All my class now have done additional maths and I’m finding it hard, I’m actually scared for Monday!
I thought C1 was ok at the time. I got 81/100 first time round, but I didn’t revise much. Second time round I got 94 I think. If you’re going to do maths, make sure you know your stuff. Revise as early as possible.
One crucial thing is to perfect your exam technique. This applies to any maths exam, as they can be rather challenging!
Do the Solomon papers; they’re kinda difficult, but will make the normal papers look easy. The more examples you do the better is for you, and the higher your chances of getting an A.
I know I’m against the majority here, but I thought it was a pretty big step up from gcse
Personally, I did find the shift difficult at first, but I don't think it was anything like GCSE Maths. Much more like Add Maths, but I did the CCEA exams for GCSE, so who knows. It seems daunting at first (especially since my teacher liked to mix in C2 stuff with it, ah~) but after about a month or so I just learned to go with it. Personally, I hated C1, but that's because I hate non-calc papers; can get the hard stuff but then can't add 7 and 5 properly!
I did my exams in June and so had to remember it and C2 after doing my mechanics module, but I did survive and I think I did reasonably well. I found myself using the revision books much more than the textbooks for revision, if any of that helps.
C1 is basically gcse maths + differentiation. There's little to no jump in difficulty, it's just slightly different I found. As long as you do plenty of past papers and ask about anything you're unsure about it should be fine. I did it in January- (ocr, so I know nothing about Edexcel I’m afraid), and got 100%
I didn't do any maths for almost 18 months and got 96/100 in C1 with around 4 months of preparation - but it took around 2 months to "click" and for me to start understanding things (I imagine that 2 months figure would be lower had I started it straight from GCSE) the trick is just to do lots and lots of practice questions and past papers
I did OCR. A lot of C1 duplicated GCSE maths (surds, quadratics etc) so it was fairly easy to move from GCSE to A-level. I did the exam in January and got 96%. I found the best way to revise was to spend a few weeks before the exam doing lots of past papers to get used to the style of questions you would be asked.
C1 is most probably going to be the easiest maths module you will take in AS/A2 Maths.
Only a little bit tougher than GCSE A/A* maths questions - and you only have to learn basic differentiation, integration and some stuff on arithmetic series, but the rest is pretty simple stuff.
However, don't be put off if you're not getting 95%+ in the first month of sixth form - I wasn't reaching that until Nov/Dec. Although C1 is an easy module, you'll find out that at times you'll make silly mistakes that could cost you a few marks especially as it’s a non-calculator paper.
“The approaches and methods used for teaching mathematics in schools can have a huge impact on how much students learn in the classroom as well as on the quality of the learning that takes place.”
Mathematics Education in Europe (Eurydice 2012)
“it is not possible to identify a single best method, but found that there are many different types of learning and many different methods that should be applied, appropriate to the learner and the particular learning outcome required”
Mathematics Matters (Swann, NCETM 2008)
“different methods are appropriate in developing these different types of
learning, including, as an example, the use of higher order questions, encouraging reasoning rather than 'answer getting', and developing mathematical language through communicative activities”
“particular methods are not, in general, effective or ineffective. All teaching methods are ‘effective for something’ ”
“different methods are appropriate in developing these different types of learning, including, as an example, the use of higher order questions, encouraging reasoning rather than 'answer getting', and developing mathematical language through communicative activities”
Two important features of teaching when developing conceptual understanding:
“when developing skill efficiency…… clear and fast paced presentation and modelling by the teacher, followed by practice by the students, worked well.”
“this is not a simple dichotomy…… it is not true that one approach works
in one area only”
“a weighted balance between the two teaching approaches might
be appropriate, with a heavier emphasis on the features related to conceptual understanding”
It is important that students develop an ability to cope with learning styles that are not necessarily their favoured ones.
In other words – let the situation dictate the learning style that is encouraged…
BUT make sure that there are situations that address different learning styles…
REMEMBER to give each student the opportunity to achieve some success – limiting learning styles may not let this happen.
Individualized learning is about tailoring the approach to meet a student’s needs rather than wants.
Notes page 17 – example copied from a transition booklet given to year 11 students
The aim of the activity is for students (and the teacher) to asses their ability to find equivalent indices.
This game works particularly well on a smart board but any projection method is fine.
Two teams line up, one each side of the whiteboard.
The teacher clicks on the cloud in the centre of the screen and a ‘question’ comes up.
The first player in each team has to tap the equivalent expression from their ‘answers’ list.
The teacher taps the board again and the correct answer is highlighted.
The front player moves to the back of the queue and the process is repeated.
This runs until all of the questions have been asked.
* Square numbers maze
* Number ladder game
* Surd maze
* Odd one out activity
* Initial activity – classifying functions – polynomials and other functions (Excel)
* Algebraic division
* The factor theorem – Excel file
* Different forms of quadratics – sorting activity
* Investigating quadratic curves
Coordinate geometry forms a large part of the core mathematics course at AS level.
What are the key ideas in coordinate geometry?
What skills should the students already possess?
What problems do you anticipate that students will have with coordinate geometry?
What do you eventually expect students to be able to achieve?
* What is gradient?
* Matching gradients to lines.
* Using gradient to find points.
What is the equation of a straight line?
* Straight lines and their gradients (perpendicular lines) activity.
* Excel student self test
* Group work activity – what does the design look like?
* Fine tuning – find the equations of the “objects” in each design
* Transformations of the unit circle.
* Group work activity – what does the design look like?
* Fine tuning – find the equations of the “objects” in each design.
* Tangents and normals (introducing ideas for differentiation).
* Sorting out the sequences activity
Generating a sequence by description
* Definitions matching game
How many clues do you need?
* Excel file - the sum of an arithmetric series
* Excel file - the sum of a geometric series
* Excel file - convergence and divergence
* Demonstration using geogebra
* Degrees and radians game
* Circular functions using geogebra
* Symmetries of trig functions - activity
* Card sort – true, false, sometimes
* True or false trig solutions cards
Before the lesson, students are given the student sheet and spend a couple of minutes reading it.
They should be told that the idea of the checklist is to make sure they are confident that everything that should be covered in the lesson has been covered by the end.
They can tick off the boxes as and when they think that section of the lesson is finished.
They should also read the points to consider and listen out for section thoroughly.
A teacher led reminder of how to calculate the gradient of straight line
A demonstration of what is meant by the tangent to the curve using Geogebra(or an equivalent piece of software) and showing that the gradient of the tangent changes as the point of contact is moved along the curve
A discussion of the way that a chord can be used to find a gradient close to that of a tangent – measuring the gradient of the tangent directly using Geogebra is not done as that particular shortcut will not help with understanding that we are looking for a limit
An activity in which the students use Geogebra to draw a polynomial curve (different curves given to different groups) and calculate the gradient of a chord between a fixed point and a variable point further up the curve. The variable point is moved closer to the fixed point in equal steps until the students in the group can predict the limit. Other points are then considered.
A collection of each group’s results with a discussion of what they seem to be showing
A plenary showing (or hopefully summarising) the algebraic method for differentiating a polynomial and a question session to mop up any areas the students do not feel have been covered
When is the gradient of the tangent to a curve positive?
When is the gradient of the tangent to a curve negative?
What is a chord?
Why are chords being used to find the gradient of a tangent?
What is meant by a limit?
Some things to consider during the activity
Why are you finding the gradient of a set of chords from a set starting point?
Why are you making the distance between the start and end point of the chord closer together each time?
Are you able to predict the limit of the gradient of the chord?
Why are you changing the starting point?
Why have other groups been given different curves?
What do these curves have in common with your curve?
Some things to listen out for or consider during the follow up to the activity
Why are each group’s results being placed in a table?
Can you see how the limit relates to the equation of the original curve in each case?
Can you think of a way to predict what the gradient of the tangent will be?
Some things to listen out for or consider during the plenary
Do you know what is meant by ‘differentiation’?
Do you know how to find the gradient of the tangent to a simple curve at a given point?
Could you write a rule that gives the algebraic method for differentiating a polynomial function?
If you do not know the answers to any of these questions at the end of the lesson, please ask about them.
* Tangents and chords Geogebra activity
* Using tangents and chords to get the idea of differentiation Geogebra and Activ Inspire
* Tangents of curves to check gradient formulae
Teaching indefinite integration
* Using tangent fields – Activ Inspire (or Autograph)
Teaching definite integration
Why does the “antiderivative” give the area under a curve?