Aiming for the top

AIMING FOR THE TOP

Activities and exercises to help students aim for the A* grade

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 IntegrationAn 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

VectorsVisualising vectors in two and three dimensions, position vectors, vector equations of lines and the scalar product.

3.30pm

Topic Suggested by TeachersIdeas for teaching a topic suggested by attendees.

4.00pm

Plenary and Feedback

4.30pm

Day ends

Starter

ACHIEVING A* IN A LEVEL MATHS

What do they want?

What is the A* GradeAS 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

A grade: 480 UMS or more

C3 and C4 total 180 UMS or more

Mathematics

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

A grade: 480 UMS or more

Best three A2 units 270 UMS or more

Further Mathematics

Examples

FP1 = 93, FP2 = 91, S1 = 95, S2 = 91, D1 = 30, D2 = 89 Total 489 UMSgrade A*

FP1 = 80, FP2 = 86, M2 = 94, M3 = 85, M4 = 88, M5 = 89 Total 512 UMSgrade 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

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

480 UMS gives an A*grade Exactly 90 UMS in each of C3 and C4 Averaging 75 UMS in the other 4 modules

ResitsRe-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 systemA grade with 480 UMS or more, C3 and C4 total 180 UMS or more

Student AC1 = 91, C2 = 72, C3 = 88, S1 = 71

Student BC1 = 100, C2 = 97, S1 = 58, C3 = 81

Student CC1 = 84, C2 = 86, S1 = 94, C3 = 89

They will be taking M1 and C4 in the final session.

What advice do you give?

Students A, B and C are all approaching their final examination session.They all need A* grades to get into their choice of university

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

mathsDeveloping higher level problem solving skills

A can do approachSpotting the underlying structure of a problemActually doing problemsFinding 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………

....6666

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 non-ambiguous 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.


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.Start with , and Ask students to e.g.find followed by and write this using the correct notationfind followed by and write this using the correct notationfind find etc Repeat using gf until they understand what is meant by gf and fg. Then introduce .

Activity 3 (end of lesson)

Using , and from activity 2Write up , , , , Pick one of them and ask students to work out the value if e.g. Compare the answers to the results found in activity 2. Can they work out if they have or ? Can they identify which composite is which? and should be discussed.Finally, write up 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.


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 Pick one of the pairsAsk the students to calculate and then 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 and 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.


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


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?

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

ϴ

1

Discussion

Look through the trigonometric functions sections of the specifications





1. Students must make sure solutions to an equation are in the right rangeWhen 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 equationIn 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 proveWhen 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 carefullyStudents 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)

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




Differentiation


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

IntegrationCrucial points for students1. 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)

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





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 1The 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 3Points 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

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