Download Word - The Further Mathematics Support Programme

Further Mathematics Support Programme
OCR S2 – Scheme of Work Template - 2016-2017
This template is part of a series designed to assist with planning and delivery of further mathematics courses.
It shows how Integral Resources and Live Interactive Lectures can be used to support students and teachers.
Integral
Resources
Integral
Resources
Live Interactive
Lectures
Teacher-level access to the Integral Resources (integralmaths.org/) for
Further Pure and Applied units is available free of charge to all
schools/colleges that register with the Further Mathematics Support
Programme: www.furthermaths.org.uk/
Student-level access to the Integral Resources and the Live Interactive
Lectures for Further Mathematics is available at a moderate cost via:
www.furthermaths.org.uk/lilfm
Integral Resources include a wide range of resources for both teacher and student use in learning and assessment. A selection of these are suggested in the
template below. Sample resources are available via: http://integralmaths.org/help/info.php.
Live Interactive Lectures are available for individual Further Pure and Applied units and take place in the spring and autumn terms. LIL FM is ideal for
schools/colleges teaching Further Mathematics with small groups and/or limited time allocation. It is also useful to support less experienced teachers of
Further Mathematics. See www.furthermaths.org.uk/lilfm
Scheduling will depend on circumstances, but the template below breaks the module down into 7 sections which may be allocated approximately equal
time. Each section corresponds to one Live Interactive Lecture (LIL) and these take place fortnightly to supplement the teaching and tutorial support in
schools/colleges and students' own independent study. FMSP Area Coordinators will be able to offer additional guidance if needed. See
www.furthermaths.org.uk/regions
OCR S2 – Scheme of Work Template - 2016-2017
Topic
Specification statements
Suggested Integral Resources
Continuous
random variables
1: Probability
density functions
 understand the concept of a
continuous random variable,
and recall and use properties
of a probability density
function (restricted to
functions defined over a
single interval)
 use a probability density
function to solve problems
involving probabilities
(explicit knowledge of the
cumulative distribution
function is not included, but
location of the median, for
example, in simple cases by
direct consideration of an
area may be required)
 use a probability density
function to calculate the
mean and variance of a
distribution
► OCR_S2
/ ► Continuous random
variables
/ ► Continuous random
variables 1: Probability density
functions
Continuous
random variables
2: Expectation and
variance
Assessment
(Integral
Resources)
Live Interactive
Lecture
Other resources
Continuous
random variables
1: Probability
density functions
nrich: pdf matcher
 Notes and examples
 Section Test C1
OCR_S2
/ ► Continuous random
variables
/ ► Continuous random
variables 2: Expectation and
variance
 Notes and examples
Continuous
random variables
2: Expectation
and variance
 Section Test C2
► OCR_S2
/ ► Continuous random variables
 Continuous random variables topic assessment
The normal
distribution 1:
Introduction
The normal
distribution 2:
Approximating
distributions
The Poisson
distribution
 understand the use of a
normal distribution to model
a continuous random
variable, and use normal
distribution tables, or
equivalent calculator
functions (knowledge of the
density function is not
expected)
 solve problems concerning a
variable X, where X ~ N(µ,σ2),
including (i) finding the value
of P(X > x1), or a related
probability, given the values
of x1, µ, σ, (ii) finding a
relationship between x1, µ
and σ given the value of P(X
> x1) or a related probability
 recall conditions under which
the normal distribution can
be used as an approximation
to the binomial distribution
(n large enough to ensure
that np > 5 and nq > 5), and
use this approximation, with
a continuity correction, in
solving problems
 calculate probabilities for the
distribution Po(µ) , both
directly from the formula
► OCR_S2
/ ► The normal distribution
/ ► The normal distribution 1:
Introduction
The normal
distribution 1:
Introduction
Making statistics
vital: The coffee
problem
nrich: Into the
normal distribution
 The normal distribution
(Geogebra)
 Normal curves matching
activity (easier)
 Normal curves matching
activity (more challenging)
 Additional exercise
 Section Test N1
► OCR_S1
/ ► Probability
/ ► Probability 2:
Permutations and
combinations
 Normal approximation to the
binomial distribution
(Geogebra)
 Continuity match
 Additional exercise
► OCR_S2
/ ► The Poisson distribution
The normal
distribution 2:
Approximating
distributions
nrich: Over-booking
 Section Test N2
► OCR_S2
/ ► The normal distribution
 The normal distribution topic assessment
The Poisson
Making statistics
distribution
vital: Poisson or not?




Hypothesis tests 1:
Sampling and
continuous
variables
and also by using tables of
cumulative Poisson
probabilities (or equivalent
calculator functions)
use the result that if X ~
Po(µ) then the mean and
variance of X are each equal
to µ
understand informally the
relevance of the Poisson
distribution to the
distribution of random
events, and use the Poisson
distribution as a model
use the Poisson distribution
as an approximation to the
binomial distribution where
appropriate (n > 50 and np <
5, approximately)
use the normal distribution,
with continuity correction, as
an approximation to the
Poisson distribution where
appropriate (µ > 15,
approximately)
 understand the distinction
between a sample and a
population, and appreciate
the benefits of randomness
in choosing samples
 explain in simple terms why a
given sampling method may
/ ► The Poisson distribution 1:
Introduction
 Poisson dominoes
 Poisson matching activity
 Additional exercise
Making statistics
vital: Parameter
gaps
 Section Test P1
► OCR_S2
/ ► The Poisson distribution
/ ► The Poisson distribution 2:
Approximating distributions
Making statistics
vital: Raindrops onto
paving slabs
Making statistics
vital: A close
approximation
 Poisson approximation to the
binomial distribution
(Geogebra)
 Normal approximation to the
Poisson distribution
(Geogebra)
 Approximation dominoes
 Additional exercise
 Section Test P2
► OCR_S2
/ ► The Poisson distribution
► OCR_S2
/ ► Hypothesis tests
/ ► Hypothesis tests 1:
Sampling
 Notes and examples
 The Poisson distribution topic assessment
Hypothesis tests
Making statistics
1: Sampling and
vital: Sampling
continuous
variables
Making statistics
vital: Generating
random numbers
 Section Test H1
be unsatisfactory and
suggest possible
improvements (knowledge of
particular methods of
sampling, such as quota or
stratified sampling, is not
required, but candidates
should have an elementary
understanding of the use of
random numbers in
producing random samples)
 recognise that a sample
mean can be regarded as a
random variable, and use the
facts that 𝐸(𝑋̅) = 𝜇 and that
𝜎2
𝑉𝑎𝑟(𝑋̅) =
𝑛
 use the fact that 𝑋̅ has a
normal distribution if X has a
normal distribution
 use the Central Limit
Theorem where appropriate
 calculate unbiased estimates
of the population mean and
variance from a sample,
using either raw or
summarised data (only a
simple understanding of the
term ‘unbiased’ is required);
 understand the nature of a
hypothesis test, the
difference between one-tail
and two-tail tests, and the
terms ‘null hypothesis’,
‘alternative hypothesis’,
‘significance level’, ‘rejection
region’ (or ‘critical region’),
► OCR_S2
/ ► Hypothesis tests
/ ► Hypothesis tests 2:
Continuous variables
 Hypothesis test for the mean
(Geogebra)
 Additional exercise
Making statistics
vital: Sample mean
gap-filler
 Section Test H2
Hypothesis tests 2:
Discrete variables
and errors in
hypothesis testing




‘acceptance region’ and ‘test
statistic’
formulate hypotheses and
carry out a hypothesis test of
a population proportion in
the context of a single
observation from a binomial
distribution, using either
direct evaluation of binomial
probabilities or a normal
approximation with
continuity correction;
formulate hypotheses and
carry out a hypothesis test of
a population mean in the
following cases: (i) a sample
drawn from a normal
distribution of known
variance, (ii) a large sample,
using the Central Limit
Theorem and an unbiased
variance estimate derived
from the sample, (iii) a single
observation drawn from a
Poisson distribution, using
direct evaluation of Poisson
probabilities;
understand the terms ‘Type I
error’ and ‘Type II error’ in
relation to hypothesis tests
calculate the probabilities of
making Type I and Type II
errors in specific situations
involving tests based on a
normal distribution or
approximation, or on direct
► OCR_S2
/ ► Hypothesis tests
/ ► Hypothesis tests 3:
Discrete variables
 Hypothesis tester (Excel)
 Hypothesis testing using the
binomial distribution
(Geogebra)
 Hypothesis testing using the
Poisson distribution
(Geogebra)
 Additional exercise
Hypothesis tests
2: Discrete
variables and
errors in
hypothesis
testing
 Section Test H3
/ ► OCR_S2
/ ► Hypothesis tests
/ ► Hypothesis tests 4: Errors
in hypothesis testing
 Notes and examples
 Errors in hypothesis testing
(Geogebra)
 Section Test H4
Making statistics
vital: Significance
levels
evaluation of binomial or
Poisson probabilities
► OCR_S2
/ ► Hypothesis tests
 Hypothesis tests topic assessment
Consolidation and
revision
FMSP - Revision
Videos
The study plans available on Integral Resources refer to Statistics 2 for OCR (Cambridge Advanced Level Mathematics) (ISBN 9780521548946). Other
textbooks covering this course may be available, and Integral Mathematics Resources does not endorse any particular set of textbooks.