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Curriculum

update

A level Mathematics and

Further Mathematics

from September 2017

All A levels will be linear from September 2017

AS can be co-taught with A level, but will have a separate

exam which will not count towards the A level

Content is 100% specified for Mathematics and 50%

specified for Further Mathematics

Mathematics will now contain compulsory Mechanics and

Statistics

General points Specifications must encourage students to:

• reason logically and recognise incorrect reasoning

• generalise mathematically

• construct mathematical proofs…

• decide on the solution strategy

• …solve a problem in context

• …understand the relationship between problems in context

and mathematical models…to solve them

• read and comprehend mathematical arguments…

• read and comprehend mathematical articles…

• use technology such as calculators and computers

effectively…

Content – General (Maths and FM)

Calculators used must include the following features:

an iterative function

the ability to compute summary statistics and access

probabilities from standard statistical distributions

the ability to perform calculations with matrices up to at

least order 3 x 3 (FM only)

“The use of technology, in particular mathematical and

statistical graphing tools and spreadsheets, must

permeate the study of AS and A level mathematics”

Use of technology 50% content specified, all pure

Awarding organisations can decide on

the other 50% so there are significant

differences

All are producing a specification that can

run alongside Mathematics

This is designed to be a separate,

valuable qualification

It can be co-taught with the full A level in

Further Mathematics

20% of AS FM is specified

A level Further Mathematics

AS level Further Mathematics

Awarding bodies

choose,

may be

pure or

applied

Prescribed content, all

pure.

Awarding body

optional

prescribed

content

A level FM AS FM

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FMSP are offering:

A free one-day course ‘Planning for the new A level’

A programme of low-cost one-day courses looking at the

content of the new AS and A Level Maths and at the new

AS and A Level Further Maths.

LOPD courses in all aspects of the new courses

(including using technology)

Talk to your Area Coordinator about what the FMSP can

provide for you.

Professional development An example from the new specification

Binomial Interlude

Distinct arrangements

In your groups take your two colours of cubes and make

all the possible linear arrangements of 2, 3, 4 or 5 cubes

Choose one colour: sort your arrangements in order of

decreasing numbers of that colour

If you finish quickly, you can try to find all the

arrangements for another number of cubes

Linking to the binomial expansion This combination:

represents one of the possible

combinations where 2 of the cubes

are red and 2 are white.

Compare this with the expansion of

(a+b)4

(a+b) (a+b) (a+b) (a+b)

Selecting a from 2 brackets and b

from 2 brackets gives us the term in

a2b2.

How many ways are there of doing

this…?

Consider the 4-cube

combinations.

Before we make our selections,

the cube line could be considered

to look like this:

Each cube is ‘red-or-white’.

Compare this with the expansion

of (a+b)4

(a+b) (a+b) (a+b) (a+b)

Each term in the expansion

contains a different combination

of a and b.

Linking to the binomial expansion

(𝑎 + 𝑏)0 = 1

(𝑎 + 𝑏)1 = 1𝑎 + 1𝑏

(𝑎 + 𝑏)2 = 1𝑎2 + 2𝑎𝑏 + 1𝑎2

(𝑎 + 𝑏)3 = 1𝑎3 + 3𝑎2𝑏 + 3𝑎𝑏2 + 1𝑏3

(𝑎 + 𝑏)4 = 1𝑎4 + 4𝑎3𝑏 + 𝟔𝒂𝟐𝒃𝟐 + 4𝑎𝑏3 + 1𝑏4

(𝑎 + 𝑏)5 = 1𝑎5 + 5𝑎4𝑏 + 10𝑎3𝑏2 + 10𝑎2𝑏3 + 5𝑎𝑏4 + 1𝑏5

#6Concepts

Introducing

the project

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Background

• Huge increase in numbers of students

taking Maths post-16

• Class size increase

• Widening range of ability

• Hub project looking at raising Post-16

participation has gained limited traction

Contributing factors - students

Students are not always open about struggling, being stuck etc.

Students not ready to work hard, not expecting larger amount of

practice work.

Students used to spiral curriculum.

Not all students have a sound understanding of fundamentals from

GCSE.

Presentation of extended responses is often poor.

Students may be becoming more independently responsible for their

own work, resulting in parents being more ‘hands off’ when we need

them to be particularly engaged.

Many students take Maths at A level because they need it for their

chosen career, not because they enjoy it!

Contributing factors - school

Setting from KS3&4 replaced with mixed ability

Large AS sets make assessment for learning, marking

etc difficult and lengthy.

Students from lower sets may not have been taught

material in a way which supports progression to A level;

more concentrating on GCSE results.

Teachers’ varying approaches to key topics at KS4

results in inconsistent experience of fundament

Trigger case 1

Basic Differentiation:

𝑦 =1

5 𝑥3=1

5𝑥−3

2

𝑑𝑦

𝑑𝑥=−3

2×1

5𝑥−5

2

Trigger case 2

Solving pairs of non-linear simultaneous equations:

Requires use of rearrangement, algebraic fractions,

factorising, substitution, zero products, method selection,

multi-stage working, multiple solutions etc…

4𝑥 + 3𝑦 = 8

𝟑𝒚 = 𝟒𝒙 − 𝟖

…because ‘change the side, change the sign’.

BIG Expectations! We EXPECT:

Y12 students to work in a large mixed ability set

All students who take A level Maths, want to do A level Maths

Students to be independent but that parents will still support and

guide them

Much more homework

Sound and consistent understanding of topics which previously

required repeated revision

Students to know which topics from GCSE will be assumed

knowledge for AS Maths

Students are able to identify their areas of misunderstanding so that

they can be tackled

Students to realise that there is not enough time in Year 12 to keep

reviewing topics over and over as they have in the past.

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Some strategies:

Use GCSE Suitability assessments to help students

make sensible decisions about AS Maths.

Extension courses to bridge the perceived gap between

GCSE and A Level (e.g. AQA L2 FM).

Summer tasks for Y11 students who plan to take A level.

Assessment close to the start of Y12; followed up with

support sessions where students can (must?) come

along and address problem areas.

Keep parents fully informed about issues arising.

… these are all administrative measures.

Departmental development – the plan:

In discussion between all AS Maths teachers, select

our big ‘6 Concepts’.

Plan together for the teaching of these topics

throughout KS3&4, all sets, to ensure consistency.

Produce/gather/collate shared materials for teaching

these topics.

Plan assessment which assures students’ confidence

in these methods.

Departmental development – the approach:

For each concept, we need to strive for consistency in:

Vocabulary – teachers and students

Layout of work – model it and expect it

Methods taught – if there are multiple approaches,

select one for all to use

Assessment and feedback

Understanding of why this concept is important –

clearly explain that is a foundation for other topics

Departmental development – the outcome?

We hope:

The result would be that students starting AS Maths

courses would all have been taught these crucial

foundational topics in the same way.

This will mean that teachers taking a Y12 class will

know exactly what the experience of their students has

been to date and will be able to build on it efficiently in

the first crucial months of the course.

Discussion: What would you choose for your 6Concepts?

Think about teaching AS Maths – where are the weak points?

What do you need to revise before you can move on?

Developing one of the 6Concepts:

Vocabulary – teachers and students

Layout of work – model it and expect it

Methods taught – if there are multiple approaches, select

one for all to use

Good resources to share

Assessment and feedback

Understanding of why this concept is important – clearly

explain that is a foundation for other topics

A level sample

lesson

MaxBox!

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The task:

A piece of paper which measures 18cm28cm

A square is cut from each corner and the edges folded up to form an

open tray as shown in the picture:

What’s the maximum possible volume of the box?

Ideas to start: Questions:

• Why does the cut-out have to be square?

• What are the extremes?

• Can we get some sample values?

• How can we record that?

• Can we predict anything about the next result?

• How do we make greater improvements?

More depth:

Nudge towards:

• Using a single variable (maybe x?

maybe something else?)

• Spreadsheet to gather data

• Graphing – plot results and extend

• How do we optimise the volume?

• Can you find a formula for the volume?

Generalising:

Generalising:

• What happens near the maximum?

(nudge towards changing gradient)

Steep here

What happens at the top? Shallower here

Two paths:

GCSE:

Think about tangents

to graph, consider

where gradient is

horizontal.

Extending to A level:

Think about tangents to graph,

consider where gradient was

horizontal. We can estimate this

from the graph but how can we do

this accurately?

A level path:

The graph of 𝑦 = 2 − 𝑥2 is on the work sheet.

Draw a series of tangents to the curve.

Measure the gradient g of the tangent at (x, y) and then plot

the point (x, g) on the same axes.

What do you notice? Nudge towards:

• straight line

• 𝑦 = −2𝑥

• crosses x-axis at (0, 0)

Thinkers:

𝑦 = 4 − 𝑥2?

𝑦 = 4 − 2𝑥2?

A level path:

The graph of 𝑦 = 3𝑥 − 𝑥3 is on the work sheet.

Draw a series of tangents to the curve.

Measure the gradient g of the tangent at (x, y) and then plot

the point (x, g) on the same axes.

What do you notice? Nudge towards:

• parabola

• 𝑦 = 3 − 3𝑥2

• crosses x-axis at (1, 0)

and (-1, 0)

Thinkers:

𝑦 = 5 + 3𝑥 − 𝑥3 ?

𝑦 = 2 3𝑥 − 𝑥3 ?

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A level path:

Can you predict what will happen with the OpenBox graph?

𝑦 = 𝑥 28 − 2𝑥 18 − 2𝑥

= 4𝑥(14 − 𝑥)(9 − 𝑥)

= 4𝑥(126 − 23𝑥 + 𝑥2)

= 504𝑥 − 176𝑥2 + 4𝑥3

Extension:

An open cylindrical water tank

has surface area 6m2. What is

the optimal radius of the tank

to give the greatest volume?

Using

technology in

the classroom to

support

transition from

KS4 to KS5

Planning a

Linear Scheme

of Work:

Tracking Bivariate

Data from GCSE to A

level

drawing a scatter diagram

independent and dependent variables,

explanatory and response variables

non-linear relationships in bivariate

data

using a line of best fit to estimate

results

using a regression equation to estimate

results

hypothesis testing

interpolation and extrapolation

Think about these ideas: Pearson/PMCC

Rank correlation

drawing a line of best fit by

eye

significance levels

sample size

calculating the equation of a

line of regression

sampling methods

correlation causation

Requirements from DfE document

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Requirements from DfE document Requirements from DfE document

All bold

AS level

Requirements from DfE document

All bold

AS level

Requirements from DfE document

All bold

AS level

Requirements from DfE document

Non-bold

text A

level only

Scatter diagrams

Mayfield High Age and Height

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Hypothesis testing The ideal hypothesis test (from MEI Statistics 1 textbook):

1. Establish the null and alternative hypothesis.

2. Decide on the significance level [and whether test is

one- or two-tailed].

3. Collect suitable data using a random sampling

procedure that ensures the items are independent.

4. Conduct the test, doing the necessary calculations.

5. Interpret the result in terms of the original claim,

theory or problem.

Hypotheses and significance Null hypothesis, H0: nothing interesting here. This result

could be obtained randomly.

Alternative hypothesis, H1: the result is interesting,

unlikely* to have been attained accidentally.

Level of significance: how unlikely do you want the result

to be in order to consider it interesting?

Hypothesis testing for correlation

“We think there will be correlation”= 2-tailed test

“We think there will be a positive correlation”= 1-tailed test

“We think there will be a negative correlation”= 1-tailed test

The critical value is determined based on sample size

and significance level.

Hypothesis testing for correlation

We’ll look at the boys’ data only.

We suspect that the boys’ heights increase with age i.e.

that there is a positive correlation between the two

variables.

This is a 1-tailed test; we’ll use a significance level of 5%

and a sample size of 60.

H0: there is no correlation between age and height; r=0.

H1: there is positive correlation between age and height; r>0.

Hypothesis testing for correlation

We suspect there is a positive correlation between age

and height.

1-tailed test; significance level of 5%; sample size 60.

H0: there is no correlation between age and height; r=0.

H1: there is positive correlation between age and height; r>0.

Critical value = 0.2144

Only 5% of randomly generate results would exceed this

value.

Hypothesis testing for correlation

H0: there is no correlation between age and height; r=0.

H1: there is positive correlation between age and height; r>0.

Critical value = 0.2144 (would be given in the question)

PMCC = 0.4038 (calculated by spreadsheet)

The calculated statistic is above the critical value. This

suggests that there is positive correlation between height

and age at the 5% level of significance.

Mayfield High Age and Height

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Planning a

Linear Scheme

of Work:

Tracking Kinematics

from GCSE to A level

Kinematics and motion graphs

Gradient/tangent questions

Constant acceleration (suvat) formulae

Deriving the suvat formulae

Applications of calculus to displacement, velocity,

acceleration

Extended to 2D and 3D with use of vectors

Gravity, friction, resistance forces, projectile motion

Think about these ideas:

Graphs of motion

More realistic Graphs

Interpreting kinematics graphs Understand, use and interpret graphs in kinematics for

motion in a straight line:

– displacement against time and interpretation of

gradient;

– velocity against time and interpretation of gradient

and

– area under the graph

Mathematics AS and A level content

Department for Education, December 2014

(m/s)

u

v

t

v - u

The suvat equations:

𝑣 = 𝑢 + 𝑎𝑡

𝑠 = 𝑢𝑡 +1

2𝑎𝑡2

𝑣2 = 𝑢2 + 2𝑎𝑠

𝑠 =1

2(𝑢 + 𝑣)𝑡

𝑠 = 𝑣𝑡 −1

2𝑎𝑡2

What if the acceleration is not

constant?

We can still use gradient

and area …

time (s)

velocity (m/s)

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Kinematics at A level with calculus (where acceleration can be variable)

Displacement (or position vector)

Velocity

Acceleration

Differentiate

w.r.t. time

Integrate

w.r.t. time (use constant or

definite

integration)

r (position vector)

v (velocity vector)

a (acceleration)

Differentiate

w.r.t. time

Integrate

w.r.t. time (use constant or

definite

integration)

xr

y

dxdt

dy

dt

v

2

2

2

2

d x

dt

d y

dt

a

jyixr

The vector can

be written using

i, j notation

z

y

x

r

A third

component is

needed to

work in three

dimensions

Using vectors

Projectile motion Kinematics in GCSE

Gradient/tangent questions in GCSE and M1

Constant acceleration formulae in GCSE Physics

Deriving suvat in AS Mathematics (M1)

Applications of calculus to displacement, velocity,

acceleration in A level Mathematics (M2)

Extended to 2D with use of vectors at A level (M2)

Projectiles (M1 or M2)

Where are these topics taught?

In future, the integrated nature of the course and the linear

structure of the exam will mean that kinematics topics will

be in:

GCSE Mathematics

A Level Mathematics

Schemes of work can adopt a thematic rather than modular

approach and examiners will be able to set synoptic

questions which test both pure and applied mathematics.

Where will these topics be taught? The Further Mathematics

Support Programme Our aim is to increase the uptake of

AS and A level Further Mathematics

to ensure that more students reach

their potential in mathematics.

To find out more please visit

www.furthermaths.org.uk

The FMSP works closely with

school/college maths departments to

provide professional development

opportunities for teachers and maths

promotion events for students.