Activities and exercises to help students aim for the A* grade Aiming for the top Phil Chaffé
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?
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
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 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.
Crucial points for students 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 notation find followed by and write this using the correct notation find 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 2 Write 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.
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 Pick one of the pairs Ask 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.
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.