Transcript MEI PowerPoint Template

Activities and exercises to help students aim for the A* grade
AIMING FOR THE TOP
Phil Chaffé
9.45am
Registration and coffee
10.00am
Welcome and introduction
10.30am
Functions
Introducing functions and their importance to A-level Maths.
Topics to be covered include composite functions, domains and ranges, inverse functions, the modulus function, exponential
functions and logarithmic functions.
12.00pm
Trigonometric Functions
Introducing secant, cosecant and cotangent, inverse trig functions.
Exploration of teaching and learning of graphs of trig functions, domains and ranges and visualising trig identities.
12.30pm
Lunch
1.30pm
Differentiation and Integration
An introduction to calculus.
Also covered will be differentiation using the product, quotient and chain rules, differentiating trigonometric functions, integration
of trig functions, integration of exponential functions, integration by substitution and integration by parts.
2.30pm
Vectors
Visualising vectors in two and three dimensions, position vectors, vector equations of lines and the scalar product.
3.30pm
Topic Suggested by Teachers
Ideas for teaching a topic suggested by attendees.
4.00pm
Plenary and Feedback
4.30pm
Day ends
Starter
What do they want?
ACHIEVING A* IN A LEVEL
MATHS
What is the A* Grade
AS qualification is graded on a five-point scale: A, B, C, D and E.
Full A level qualification is graded on a six-point scale: A*, A, B, C, D and E.
To achieve an A* in Mathematics, students need
 a grade A on the full A level qualification
 90% of the maximum uniform mark on the aggregate of C3 and C4.
To achieve an A* in Pure Mathematics, candidates will need
 a grade A on the full A level qualification
 90% of the maximum uniform mark on the aggregate of all three A2
units.
To achieve A* in Further Mathematics, candidates need
 a grade A on the full A level qualification
 90% of the maximum uniform mark on the aggregate of the best three of
the A2 units which contributed towards Further Mathematics.
UMS
Mathematics
A grade: 480 UMS or more
C3 and C4 total 180 UMS or more
Examples
C1 = 90, C2 = 79, C3 = 95, C4 = 94, M1 = 87, M2 = 89
Total = 534 UMS
grade A*
C1 = 79, C2 = 78, C3 = 94, C4 = 86, M1 = 77, S1 = 74
Total = 488 UMS
grade A*
C1 = 95, C2 = 98, C3 = 92, C4 = 87, D1 = 87, D2 = 89
Total = 548 UMS
grade A
C1 = 90, C2 = 92, C3 = 91, C4 = 92, M1 = 57, D1 = 58
Total = 479 UMS
grade B
Further Mathematics
A grade: 480 UMS or more
Best three A2 units 270 UMS or more
Examples
FP1 = 93, FP2 = 91, S1 = 95, S2 = 91, D1 = 30, D2 = 89
Total 489 UMS
grade A*
FP1 = 80, FP2 = 86, M2 = 94, M3 = 85, M4 = 88, M5 = 89
Total 512 UMS
grade A*
UMS – some thoughts
It is possible that a student with a high UMS score will
only achieve an A grade if they do not do well on C3
and C4.
Some students with comparatively low UMS scores may
achieve an A* if they do well in C3 and C4.
A* is mainly about C3 and C4 performance .
Bizarre but possible
480 UMS gives an A*grade
 Exactly 90 UMS in each of C3 and C4
 Averaging 75 UMS in the other 4 modules
579 UMS gives an A grade
 89 UMS in C3 and 90 in C4 or vice versa
 100 UMS in all of the other 4 modules
Resits
Re-sits have been accepted for any student cashing-in
for a Mathematics grade from summer 2010.
The highest UMS marks for C3 and C4 in the bank are
used to make the calculation.
For Further Mathematics the best results for three A2
units are used.
Individual universities set their own requirements.
A maths break….
“Every prime number >3 is either one more or one less than a
multiple of 6.
Is this true?
How would you prove it?
Stretch and challenge questions
AQA C4 January 2010
Preparing students for A* questions

Differentiate practice questions
 Make sure the more able students try the harder questions
 Set questions based on what they need to know rather than just
for repetition

Give clear indicators of what must be learnt and why
 Trigonometric identities
 Standard graph shapes

Look for questions that require insight

Keep an ear/eye out for subtle misconceptions
 Vectors

Plan in some good, solid revision for C3
Playing the system
A grade with 480 UMS or more, C3 and C4 total 180 UMS or more
Students A, B and C are all approaching their final examination session.
They all need A* grades to get into their choice of university
Student A
C1 = 91, C2 = 72, C3 = 88, S1 = 71
Student B
C1 = 100, C2 = 97, S1 = 58, C3 = 81
Student C
C1 = 84, C2 = 86, S1 = 94, C3 = 89
They will be taking M1 and C4 in the final session.
What advice do you give?
More than that…..
Developing a wider interest in mathematics
 Mathematics is exciting, interesting, beautiful, elegant etc.
 Mathematics is about more than just computation
 Mathematics hasn’t all been discovered
 There are a lot of very interesting books about maths
Developing higher level problem solving skills
A can do approach
Spotting the underlying structure of a problem
Actually doing problems
Finding problems interesting for their own sake
Developing analytical skills
 Methods of proof
 Questioning methods and techniques
 Mathematical fluency and accuracy of “language”
 Efficiency in application
Maths break………
6  6  6  6  ....
Activities and exercises to help students aim for the A* grade
FUNCTIONS
Introducing functions
“The function concept is one of the most fundamental concepts of modern
mathematics. It did not arise suddenly. It arose more than two hundred years
ago out of the famous debate on the vibrating string and underwent profound
changes in the very course of that heated polemic. From that time on this
concept has deepened and evolved continuously, and this twin process
continues to this very day. That is why no single formal definition can include
the full content of the function concept. This content can be understood only by
a study of the main lines of the development that is extremely closely linked
with the development of science in general and of mathematical physics in
particular.”
Nikolai Luzin
Three important ideas for A* students
 Precision – functions allow mathematical relations/correspondence to be
defined in a much more precise way. The notation can be made nonambiguous and is incredibly useful in a variety of applications.
 Universality – functions are present throughout A level mathematics even
when there is no overt mention of them. The terms sequence, measure,
length, volume, vector and so on are all functions in disguise.
Functions are present throughout mathematics far beyond A level.
 Definition – the act of carefully defining a mathematical object is something
very new to A level mathematicians. The concepts of a domain, region
(range) and rule all acting together is unfamiliar to most students and not
something they appreciate at A level.
Discussion
Look through the functions sections of the specifications
What do students need to be confident in?
What are students going to find difficult?
How are the ideas linked to mathematics that the students
will encounter in future
Defining a function
 Students need to be able to see why a function is defined in a certain way.
 Examples are needed to show the problem with one to many mappings.
 Strong analogies help – number machines from Key Stage 3
 Students need to be familiar with multiple representations of mappings and
multiple notations.
 Students should try to reflect on areas where they have encountered functions in
the past without really referring them to as functions.
 Students need to realise that when the square root is used in a function it has to be
the principal square root.
Domain and Range
 Graphical representation is the key to this for C3.
 Students should be very familiar with graphs of the standard
functions in C3/C4
 The use of standard transformations is vitally important for the type
of questions asked.
Crucial points for students
1. Make sure that students know what all of the terminology means
Check that students know the meaning of all the terminology relating to mappings
and functions, and in particular, when a mapping is a function.
2. Students should know what effect a transformation has on the equation and graph
Make sure that students know the effect on the equation of a graph of translations,
stretches and reflections.
3. Students need to take care when doing multiple transformations
Make sure students are careful when using more than one transformation.
Students need to realise that changing the order can sometimes give a different result.
Activity 1
Use the mappings sheet (A4 – enlarge to A3 in practice)
and the mappings cards.
The mappings cards show mappings in a variety of forms.
Students sort the cards into groups
Activity 2
Match the graphs to the domains and ranges
The function cards can be used as an extension. Match
these to the graphs justifying the choice. There are some
‘red herrings’
Activity 3
What can you say about this function?
Students try to say as much about each function as
possible using the terms given in the corner.
Activity 4
This function ……
Students try to find a function that can match the
description on the card.
Activity 5
Explain why?
Either using the cards or a sheet, the students try to
explain the statement.
Composite Functions
 Students need to be confident in their ability to use algebraic
substitution.
 Demonstrate a composite function by having a two stage “machine”.
 Demonstrate problems with this by having the range of the first function
being partially incompatible with the domain of the second.
 Use Geogebra or a similar graphing package to experiment with
functions and composite functions.
Crucial points for students
1.
For composite functions, make sure students are applying the
functions in the right order
Students need to be careful to apply functions in the correct order when
finding composite functions. They must remember that the function fg
means “first apply g, then apply f to the result”.
Activity 1
Use the Excel file Composite Functions 1
Use individual whiteboards.
Activity 2 (start of lesson)
Using individual whiteboards to check students have understood the
idea of composite functions and the order in which they are
performed.
1
2
Start with f 𝑥 = 2𝑥 + 1, g 𝑥 = 𝑥 and h 𝑥 =
𝑥
Ask students to e.g.
find g(2) followed by f g(2) and write this using the correct
notation
find g(−1) followed by f g(−1) and write this using the correct
notation
find fg(3)
find fg(−2) etc
Repeat using gf until they understand what is meant by gf and fg.
Then introduce h(𝑥).
Activity 3 (end of lesson)
1
𝑥
Using f 𝑥 = 2𝑥 + 1, g 𝑥 = 𝑥 2 and h 𝑥 = from activity 2
2
1
1
Write up 2𝑥 2 + 1, 2𝑥 + 1 2 , + 1,
,
𝑥
2𝑥+1 𝑥 2
Pick one of them and ask students to work out the value if e.g. 𝑥 =
3
Compare the answers to the results found in activity 2. Can they
work out if they have fg(𝑥) or gf(𝑥)?
Can they identify which composite is which?
gh(𝑥) and hg(𝑥) should be discussed.
1
Finally, write up 2
2𝑥 +1
Can they write down what this s as a composite of the three
functions?
Activity 4
How do I get to?
Give students the how do I get to cards.
They should provide a clear account of how to get the given
composite function
This could be made into a poster
Activity 5
Find two functions?
Give students the Find two functions cards.
They should provide a clear account of how to get the given
composite function
This could be made into a poster
Inverse Functions
 Students need to be clear that the inverse of a function is only a function
itself if the original function is a one to one mapping.
 Clear links to some of the key mathematical skills needed to find an
inverse need to be made. These include rearranging formulae and
factorising as well as the index and logarithm laws.
 Graphical interpretation is important here so links between graphs and
transformations of graphs need to be secure.
Crucial points for students
1. Students need to remember that only a one-to-one function has an
inverse function
Sometimes a function can be defined with a restricted domain so that it
does have an inverse function: for example, f(x) = x² is a many-to-one
function for x ∈ R, and so does not have an inverse, but if the domain is
restricted to x ≥ 0, then the function is one-to-one and the inverse
function f −1(x) = √x
2. When finding the domain or range for f-1, students should look at the
limits of the original function
Students need to notice that the domain of an inverse function f-1 is the
same as the range of f, and the range of f-1 is the same as the domain of f.
Activity 1
Function matching sheet
Students match up the functions so that fg(x) = x by drawing a line
between them.
They should then say something about gf(x) = x
Activity 2
Using Geogebra to investigate inverse functions
Activity 3
Using the Intro to inverses sheet (Activity 1), students pair up
functions such that fg 𝑥 = 𝑥
Pick one of the pairs
Ask the students to calculate g 3 and then f g 3 = fg 3
Do this for a few values to make the point.
Activity 4
Using the pairs from activity 1, students use a graphical calculator,
Geogebra or Autograph to draw graphs of 𝑦 = f 𝑥 and 𝑦 = g 𝑥
They should say what they find out and try to explain why this
happens.
The Modulus Function
 This should be introduced as an opportunity to use the definition of a
function.
 Graphical representation is again vital and students should be encouraged
to experiment with different functions based on the modulus function.
 Students should be encouraged to find ‘critical’ points and use these to
sketch graphs that use the modulus function.
Crucial points for students
1. Students must check that they have the right number of solutions
They need to be careful when solving equations involving a modulus function
that they have the correct number of solutions. Sketching a graph is always
helpful.
They should also check their solution(s) by substituting back into the original
equation.
2. Students need to take care with inequality signs, especially when they involve
negative numbers
When solving inequalities involving a modulus sign, students need to be very
careful with the inequality symbol. They need to remember to reverse it if they
are multiplying or dividing through by a negative number.
Students should check their answer by substituting a number from within the
solution set into the original inequality.
Activity 1
Investigation using Geogebra
Activity 2
Match the function to the graph giving justification
Activity 3
Use the two solutions, one solution, no solutions sheet.
Sort the equation cards into the appropriate columns.
Activity 4
Find the errors (SW)
Find the errors in a number of calculations on the sheet.
Exponential and logarithmic functions
Crucial points for students
1. Students need to learn and be confident using the laws of indices and
logarithms
Make sure that students know the rules of logarithms and of indices so they
can manipulate expressions involving exponentials and logarithms
confidently.
2. Make sure that students remember that the exponential and logarithm
functions are the inverses of each other
Students need to remember that the exponential function and the natural
logarithm function are inverse functions; so they can “undo” an exponential
function by using natural logarithms, and “undo” a natural logarithm by using
exponentials.
Activity 1
‘Live’ Geogebra activity – what is an exponential function?
Activities and exercises to help students aim for the A* grade
TRIGONOMETRIC
FUNCTIONS
Introducing secant, cosecant and cotangent
Secants, cosecants and cotangents cause a great deal of unnecessary
problems for students.
Students aiming for the A* grade should be confident when dealing with
these trigonometric functions in the following ways
 As the reciprocal functions of cosine, sine and tangent.
 As a way of writing trigonometric expressions on one line
1
ϴ
Discussion
Look through the trigonometric functions sections of the
specifications
What do students need to be confident in?
What are students going to find difficult?
How are the ideas linked to mathematics that the students
will encounter in future
Crucial points for students
1. Students must make sure solutions to an equation are in the right range
When solving an equation make sure that students check:
 what range the solutions should lie in
 whether the solutions should be in radians or degrees.
2. Students should never cancel a factor in an equation
In an equation such as sinθ − sinθ cosθ = 0 students should never cancel out the term sinθ
because they will lose the roots to the equation sinθ = 0.
They should never cancel – always factorise.
3. Students should work from one side of the identity which they are trying to prove
When trying to prove an identity students should only ever work with one side of the
identity. They should never try to rearrange it and cancel out terms.
4. Students should read the question carefully
Students should always check which form of r sin(θ ± α) or r cos(θ ± α) the question is
looking for
Activity 1
Exact values (SW)
Activity 2
True or false trig equations sheet
Activity 3
Sometimes, always, never true (SW)
Activity 4
Solving trigonometric equations (SW)
Activity 5
Brackets (SW)
Activities and exercises to help students aim for the A* grade
DIFFERENTIATION AND
INTEGRATION
Differentiation and Integration
There are seemingly a large number of methods used to differentiate and integrate the
functions encountered in the C3 and C4 modules. Students struggle to find the correct
methods to apply, particularly when put under the pressure of being in an examination.
Students aiming for the A* grade should have a clear idea of where each of the methods
used for differentiation and integration are used. They should, hopefully, have enough
experience of each method used to realise that there aren’t really that many methods in
reality.
Discussion
Look through the differentiation and integration sections of the
specifications
What do students need to be confident in?
What are students going to find difficult?
How are the ideas linked to mathematics that the students
will encounter in future
Differentiation
Crucial points for students
1. Students must make sure they don’t mix up the derivative of ex with that of xn
2. Students must make sure they don’t mix up the integral and differential of ekx
3. Students should remember that they cannot integrate across an asymptote when
evaluating a logarithmic integral
4. Make sure students remember the du/dx part of the chain rule
5. Make sure students recognise situations when the chain rule should be used
Students should know that the chain rule is used for functions which can be
written in the form y = f(u), where u is a function of x. They should be clear that
it cannot be used to differentiate functions which are a product of two functions
– and that requires the product rule.
6. Make sure students use the product rule correctly
7. Make sure students use the quotient rule correctly
They must make sure they don’t get ‘u’ and ‘v’ mixed up and remember the
negative sign in the numerator
8. Students must be careful when finding stationary points of quotient
functions
9. Students must remember that when differentiating trigonometric functions
the derivative results rely on measuring x in radians
10. Students must be careful not to mix up the derivatives and integrals of sin x
and cos x
11. Students must make sure that they understand the process of differentiating
an equation implicitly
Activity 1
Match up the differentiations (exponential)
Match up. There should be one card of each colour at the end
Activity 2
True or False (SW)
Activity 3
Product and quotient rule domino chain (SW)
Activity 4
Product, quotient and chain rule Venn diagram.
Activity 5
Solving Problems (SW)
Activity 6
Which is which and why? (SW)
Activity 7
Build an implicit function
Integration
Crucial points for students
1. When using the integration by parts formula, students must remember to integrate
to find ‘v’ rather than differentiating.
2. Students must be careful with signs when using the integration by parts formula
3. Students need to remember to substitute for dx in the integral when integrating by
substitution
4. Students must remember to change the limits of a definite integral when making a
substitution
When students change the variable in an integration (from x to u say) by making a
substitution, they must change the limits of the integration from values of x to the
equivalent values of u.
5. Students need to be careful with signs when substituting values into definite
integrals
6. Students should always check their integration by differentiating
It is easy for students to make mistakes when integrating. Differentiating the result
is a quick and comparatively easy way of checking their work.
7. Students should learn to look out for the standard patterns
Students should look for any integrals which they should be able to integrate by
inspection. They should make sure that they adjust any constants if necessary.
8. Students need to remember when to use logarithms in integration
Some students make the mistake of wrongly using logarithms when integrating
inverse powers of linear functions of x.
9. Students should be careful to use the correct integration technique when dealing
with products Some products require integration by substitution, other need
integration by parts.
10. Students should remember the ‘π’ in the volume of revolution formula
11. Students must make sure that they use the correct limits of integration for
volumes of rotation
Students need to remember that if they are rotating about the x-axis, the limits of
integration must be x-coordinates, and if they are rotating about the y-axis, the
limits of integration must be y-coordinates.
12. Students must remember to integrate with respect to the correct variable for
volumes of revolution
They need to correctly substitute for x² or y² to do this.
Activity 1
Mark and correct the sheet (SW).
Activity 2
Types of integration Venn diagram (SW)
Activity 3
True or false activity (SW)
Activities and exercises to help students aim for the A* grade
VECTORS
 Vectors are used in a vast number of applications of mathematics.
 There are two ways that vectors are used. They can be thought of as points in
a coordinate system corresponding to points in space, or as objects with
magnitude and direction.
 These two definitions of vectors cause students some real problems. The
most perceptive and mathematically able school students often feel they don't
understand the use of vectors and they are absolutely right to question this
because school textbooks often switch between the different sorts of vectors
without justifying what they are doing.
 A student aiming for an A* grade needs to be fluent in the use of all types of
vectors at C4 level. They need to be able to switch between the various uses
and notations for vectors with ease.
Discussion
Look through the vectors sections of the specifications
What do students need to be confident in?
What are students going to find difficult?
How are the ideas linked to mathematics that the students
will encounter in future
Crucial points for students
1. Students must make sure they use vector notation correctly
They should remember that in handwriting they should underline vectors, or in
the case of a vector joining two points, use an arrow above, e.g. AB
2. Students must make sure they know how to find the resultant of two vectors
3. Students must know how to find the vector joining two points
4. Students should know how to find a unit vector
To find a unit vector in the same direction as a given vector, a, they should
divide by the magnitude of a
5. Students need to understand the relationship between vector and cartesian
equations of lines
6. Students should always read the question carefully
They should check whether the question is asking for the angle or the cosine of the
angle.
7. Students should know how to find the angle between two lines
They should know that to find the angle between two lines simply find the angle
between the two direction vectors.
8. Students need to remember that the scalar product of perpendicular vectors is zero
To show that two vectors are perpendicular they should just show that the scalar
(or dot) product of the vectors is 0.
9. Students should draw diagrams to make sure that you are using the right vectors
10. Students should be careful with signs when converting between the vector and
cartesian equations of a line.
11. Students must be careful when writing down the Cartesian equation of a line
which has one or two zeros in the direction vector.
12. Students should make sure they know the form of the equation of a plane
13. Students should be able to check whether a point lies on a plane by substituting
the coordinates into the equation of the plane.
Activity 1
The card set ‘Representation of Vectors’ shows the different ways in
which vectors can be represented. Students have to link one of each
form together. Although some of the connections are very easy it
does reinforce the need to be flexible in the way vector information
is recorded.
As a follow up students could be given one piece of information
and asked to construct the other forms.
Activity 2
Target grid
Activity 3
Points on lines
Given the parameter, which point goes with which line?
Activity 4
Match the equations to the lines
Follow up with the statements and justify
Activity 5
3D Lines Venn diagram (SW)
Activity 6
Complete the statements about vector a
Activity 7
Just 3 points – the vector equation of a plane