MEI Conference
Change a grade C to a grade B?
Change a grade E to a grade C?
Errors in S1 and S2 and advising
students how to avoid them.
• In January 2007 there was just a 6 raw
mark difference between most grades for
S1
• Grade E 26
• Grade D 32
• Grade C 38
• Grade B 44
• Phil Stockton Head of Mathematics
Prince William School
Oundle Northamptonshire
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Change a grade C to a grade B?
Change a grade E to a grade C?
MEI Conference
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3rd July 2008
MEI Conference
Measures of spread
Measures of spread 2
• Can year 12 students remember 4
Errors on root mean square deviation included:
• Confused ∑x2 and (∑
(∑x)2
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∑fx by calculating (∑
(∑fx)2 or even ∑xf2
• Forgetting to multiply by f
• Forgot to square root
• Divided by nn-1 rather than n
formulae?
Probably not, so get them to remember
only 2 but test them on recall and which
uses √
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MEI Conference
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Do not lose the marks they can get by:
• Improving accuracy
• Use sound methods, rather than intuitive
• Students are careful to avoid common
errors highlighted by teachers or in
examiner reports
higher there was again just a 6 or 7 raw mark
difference between most grades for S1
Grade E 29
Grade D 35
Grade C 41
Grade B 48
Grade A 55
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MEI Conference
How can students reach their
target grade?
• In June 2007 where the grade boundaries were
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Mean and standard deviation
Outliers
June 07 Q3
Transforming data:
y = 1.2x - 10
Errors included:
• S. Dev (y) = 1.2 x S.Dev (x) -10
• Attempting to scale summary data such as ∑x
• Errors when using IQR
Highlight correct use by use of boxplot
and ∑x2
• Some students did over 2 pages of calculations
for a 2 mark part
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Errors included
• Median ± 2 x IQR
• Median ± 1.5 x IQR
• UQ + 2 x IQR
• UQ + IQR
• They will have met boxplots at GCSE
…………………………………..
…………………………………..
3rd July 2008
MEI Conference
When do you ignore an outlier?
Scales
• Has the data handling coursework helped
• Heavy penalty if misread scales
• Some students choose a nonnon-linear scale
students get an awareness of outliers?
• Are they familiar with the chances of
getting an unusual value from a large data
set?
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when plotting a histogram
• Some students label the xx-axis
t<5
5 ≤ t <10
……………
•
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Median from discrete data
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• How many ways can you get a total score
Do students get confused about small and large
data sets?
ie from a Cumulative Frequency graph makes
very little difference reading from 100 or 100.5
when 200 pieces of data, but does make a
difference when smaller data sets.
MEI Conference
MEI Conference
Simple combinations
Many students stated that for a data set of 22
items, the median was the 11th, rather than the
11.5th (June 07 Q5)
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of 16 throwing 3 ordinary dice?
• Answers included 2 ways or 12
• Is this a good A01 activity at GCSE?
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2
Why does it take so long for an A
level Maths student to order a
meal?
Why does it take so long for an A
level Maths student to order a
meal? (2)
• Peter and Esther now decide to choose
• 4 starters 5 main courses 3 sweets
• Calculate the number of ways in which Peter
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different dishes from each other
(still 4 starters 5 main courses 3 sweets).
Show that the number of starters is 6.
• Instead of 4C2 the following were
attempted:
3! or 3C1 x 2C1 Well they do get 6
may choose his 3 course meal.
Worrying that some students put 12
Worrying that some students calculated
4! X 5! X 3! = 17280 If each selection took 10
seconds to consider it would take
48 hours, or 2 days to consider all options.
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MEI Conference
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3rd July 2008
MEI Conference
More confidence using conditional
probability
Probability from GCSE to AS level
June 07 Q4 (iii)
Common errors included:
• Jan 2007 Question 5
• To calculate P(T∩
P(T∩J) use
• Forgetting to multiply by 2, ie just considering first
P(T|J)
P(T|J) = P(T∩
P(T∩J)
P(J)
• But students calculated 0.4 x 0.2
or 0.3 x 0.2
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person recycles and second does not
Forgetting to decrease from 50 to 49
A question from this year’
year’s GCSE includes similar work,
but students had been given a partly completed tree
diagram and the probabilities did not change.
Do we need to emphasise at AS cases where we are
sampling without replacement?
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MEI Conference
More confidence on Venn Diagrams
Yet more confidence on Venn
Diagrams
• Students must label the correct section.
• Can they distinguish between
• Jan 05 Q3
• Do students recognise the use of a
J and J ∩ T′ ?
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formula for P(A u B) ?
• When does P(A u B) = P(A) + P(B)?
P(B)?
• Can they label each part of the diagram?
Even the outer section?
• But more students get success on S1
questions if Venn diagram already drawn
for them.
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3
Do students understand the
concept of independence?
Confuse them by an unusual
situation
• June 06 Q2
• Are they confused about students working
• Most students recognise the ways of
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independently?
• Do they confuse with mutually exclusive
events?
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winning tennis matches for 3 or 5 set
games.
• But in Jan 08 Q5 a new situation was
suggested, why?
• Would this new situation increase the
chances of Andy Murray winning
Wimbledon?
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Hypothesis Testing Notation
Hypothesis Testing Using tables
Good or bad?
• H0 = 0.1
• H0 :p= 0.1
• H0 :P(X= 0.1)
• H0 :P(X) = 0.1
• Do not use tables when p=0.12 by
rounding to 0.1
• Use tables when p=0.1 and need to find
P(X<4) rather than do 4 binomial
calculations
• But even if correct did they define p?
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3rd July 2008
MEI Conference
Hypothesis Testing
Calculating Binomials
Hypothesis Testing
Writing a conclusion
Check the use of inequalities from a question in words
June 07 Q8
So at least 4 got the following responses:
• P(X=4)
• P(X>4)
But even if they put P(X≥
P(X≥4) they made mistakes with
• 1- P(X≤
P(X≤ 4)
• 1- P(X<3)
• Accept H0 is not sufficient
• Improve by
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There is insufficient evidence to claim that
there has been a reduction.
Why not draw a number line?
Useful when identifying critical regions.
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4
In S2
In S2
• Using a Poisson approximation to the
• Inappropriate use of continuity correction.
• Not defining μ as the population mean or
Binomial when the exact answer can be
calculated.
(even when the next part of the question
says using a Poisson approximation)
• Missing out a continuity correction.
3rd July 2008
MEI Conference
ρ as the population product moment
correlation coefficient.
In June 07 the mark for correctly defining
μ proves to be rarely given.
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3rd July 2008
MEI Conference
In S2
In S2
• Not working to a suitable degree of accuracy
• Poor work with inequalities, sometimes
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especially in questions where there is a degree
of approximation in the modelling
Over reliance on a calculator and not
interpreting answers correctly, eg on lines of
regression
When given a probability corresponding to a
value k for a Normal distribution confusion over
whether to use a positive or negative value with
the tabulated value.
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heavily penalised at this level.
P(X>2) is not the same as 11- P(X=0) –
P(X=1)
3rd July 2008
MEI Conference
In S2
In S2
• Relate the conclusion to the context of the
• Marks for explanation and interpretation
question
There is no evidence of association between the
2 variables does not get credit
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continue to be elusive to even the most
able candidates.
But there is no evidence to conclude that there
is association between home location and
ambition gets the mark.
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