mep Mathematics Enhancement Programme Demonstration Project GCSE Mathematics Revision Pack for Schools Teachers and Students CENTRE FOR INNOVATION IN MATHEMATICS TEACHING
The resource pack of copymasters which follows has been developed for secondary schools participating in the Mathematics Enhancement Programme (MEP) and consists of the following components: Recommendations and Advice for Schools, Teachers and Students Foundation, Intermediate, Higher Tier syllabuses referenced to MEP sections Quick Checks with key questions Example Papers and solutions for each tier of entry, Foundation, Intermediate and Higher, and referenced to MEP Units 1-19 Section(s) can be freely copied for pupils in your school. Any comments or queries concerning this material should be addressed to Liz Holland at: Centre for Innovation in Mathematics Teaching St Luke's Campus University of Exeter Heavitree Road EXETER EX1 2LU Phone 01392 217113 Fax 01392 499387 E-mail e.h.holland@ex.ac.uk First produced: April 1998 Revised: April 1999 CIMT, University of Exeter i
Recommendations and Advice for Schools, Teachers and Students The section is divided into three parts, namely advice for Schools Teachers Students We suggest that the Student part is photocopied for all your pupils to read carefully; it might be helpful for students also to discuss this as a whole class. Schools The last few months up to the actual GCSE exams are a vital part of your students' education. What happens at this stage can have a dramatic effect on their final grades and hence future training and career paths. Many pupils (particularly boys) are maturing fast in this period; they can assimilate knowledge and skills very quickly, if encouraged and helped in the right direction. Others however can be distracted by work on other subjects, and disillusioned if revision appears not to be going well. We hope that your school will support pupils, even to the extent that mathematics is given priority over some other subjects. The difference between a grade C and a grade D in this subject can be crucial for the pathways to be followed by your students, so every opportunity should now be given to help them to succeed (whatever they might have done or not done in the past!). In particular, we recommend that: teaching and revision should continue for as long as possible in the summer term; students are encouraged to attend last minute 'interactive' revision sessions, even over half-term and should not be left to just revise at home on their own; sample exam papers should be used extensively, either as practice under exam conditions (and do keep to the conditions, i.e. exact time for papers, no help, etc., so that students can both experience what it is like to take the paper under exam conditions, and appreciate what they still need to do), or 'interactively' with the whole class in MEP style, question by question, and using individual mistakes as teaching points for the whole class. CIMT, University of Exeter ii
Teachers You are responsible for your students' revision, right up to the date of the exam. You can liken your role to that of the trainer of a group of competitive athletes and for them the exam is the equivalent of the Olympic games! In particular, we recommend that you: specifically check the revision plans of every student in your class, and clarify with them exactly what topics are on the syllabus for the tier (the Appendix will be of help here); ensure that their preparation is timed so that they will be at their best on the day of the exam; help your students to revise in a systematic and thorough way just reading notes or even looking at worked examples is of little use they must test to see if they can do the questions at first without help; we hope that the 'Quick Check' questions will be helpful here; do not believe students when they say they have no problems investigate further, until you find the problems! continue to take an interest in their progress, even between their first and second papers; see them after their first exam, listen and discuss the first paper with them, clarify preparation for the next paper; tell your students to try not to panic in an exam, even when they are finding it more difficult than expected this will often mean that everyone is finding the paper difficult which will lead to grade boundaries being lowered accordingly, so it is advisable not to panic but to gain as many marks as possible; reassure your pupils that after a first 'bad' paper, it is still worth continuing to try hard for the second paper; remind them to eat enough before the exam (breakfast/lunch); visit the cloakroom before they begin, and come equipped with a full set of 'tools' for the event (pen, pencil, protractor, compass, rubber, calculator, etc.). CIMT, University of Exeter iii
Students Revision Plans Revision is an important part of your preparation for the exams (unless you are a mathematical genius!), and you can easily improve things by a full grade. But revision is not easy you will often be working on your own, and for it to be really successful, there are some key points to act on. First of all, the do NOTS. Do NOT think that you know it all. Do NOT read things through for revision and think that you will be able to do them in an exam Do NOT rely on looking it up on the formula sheet you waste time doing this: all the formulae on the sheet and key results should be at your fingertips learn them Do NOT expect to ever finish your revision there will always be more to do Do NOT try to cover anything in a limited time it is better to know some things well and get all the marks for those parts, rather than to have a smattering of wide knowledge Do NOT try to master more awkward topics when you have not understood the basic concepts. Now the DO's. DO work consistently hard throughout the revision period DO base your revision on tackling exam style questions particularly those from past papers DO learn by heart the basic formulae needed for your paper DO ask for help, no matter how silly you may think it seems. This pack includes a set of Quick Check revision questions, based on each Unit of work in the MEP scheme. Candidates for the Foundation Tier should use all the Foundation Tier Quick Check Intermediate Tier should use all the Foundation Tier and Intermediate Tier Quick Checks Higher Tier should use all the Intermediate Tier and Higher Tier Quick Checks. CIMT, University of Exeter iv
Although revision is important, so is your exam technique and although the following points are common sense, we will highlight them many candidates, under exam conditions, do not follow them! Work steadily through the paper you are expected to attempt all questions, but do not spend long on questions that give you difficulties return to these questions at the end; if you are not careful, you can get 'bogged down' with earlier questions and then not have time to even attempt questions at the end of the paper which you could have scored marks on. Read each question carefully so often, candidates skim the questions, particularly those placed in a context, and think that they know what the question wants, rather than actually finding out. Set out work correctly and concisely there are some 'method' marks on each paper, and, particularly for questions which state "show all your working" you will not get any marks for a correct answer without the working; the examiner will find it helpful when giving marks if working is shown clearly, even in situations where you go wrong and do not achieve the correct final answer. Give your answers to an appropriate degree of accuracy if the question asks for '1 decimal place' or '2 significant figures' then do so! there will often be an accuracy mark for giving the answer correctly if no accuracy is asked for in the question, then give your answer correctly to 3 significant figures (or more) and state the accuracy, e.g. '27.1 to 3 s.f.'. Do not round calculated answers too early often answers to part (a) of a question are used again in part (b), or later, and even if you are asked to give the answer to part (a) correct to, say, 1 decimal place, you should use the non-rounded answer in part (b) otherwise your answer to part (b) will not be accurate. Check numerical answers carefully, even repeating the calculations if you have time at the end of the exam never leave the exam early! CIMT, University of Exeter v
MEP Year 10-11 UNITS referenced to current GCSE Tiering UNIT 1 Indices F I H 1.1 Multiplication and Division - - 1.2 Squares, Cubes, Square Roots and Cube Roots - 1.3 Index Notation 1.4 Factors - 1.5 Prime Factors - 1.6 Further Index Notation 1.7 Standard Form 1.8 Calculations with Standard Form UNIT 2 Formulae F I H 2.1 Using Formulae - - 2.2 Construct and Use Simple Formulae 2.3 Revision of Negative Numbers - - 2.4 Substitution into Formulae - 2.5 More Complex Formulae 2.6 Changing the Subject 2.7 Further Change of Subject 2.8 Expansion of Brackets 2.9 Factorisation 2.10 Algebraic Manipulation 2.11 Algebraic Fractions UNIT 3 Angles F I H 3.1 Measuring Angles - - 3.2 Line and Rotational Symmetry - - 3.3 Angle Geometry 3.4 Angles with Parallel and Intersecting Lines - 3.5 Angle Symmetry in Polygons 3.6 Symmetry Properties of 3-D Shapes - 3.7 Compass Bearings - 3.8 Angles and Circles 1 3.9 Angles and Circles 2 3.10 Circles and Tangents CIMT, University of Exeter vi
UNIT 4 Trigonometry F I H 4.1 Squares and Triangles - - 4.2 Pythagoras' Theorem - 4.3 Further Work with Pythagoras' Theorem 4.4 Sine, Cosine and Tangent 4.5 Finding Lengths in Right Angled Triangles 4.6 Finding Angles in Right Angled Triangles 4.7 Mixed Problems with Trigonometry 4.8 Sine and Cosine Rules 4.9 Angles Larger than 90 UNIT 5 Probability F I H 5.1 Probabilities - - 5.2 Simple Probability - 5.3 Outcome of Two Events 5.4 Finding Probabilities Using Relative Frequency 5.5 Determining Probabilities 5.6 Probability of Two Events 5.7 Use of Tree Diagrams 5.8 Multiplication for Independent Events 5.9 Mutually Exclusive Events 5.10 Tree Diagrams and Conditional Probability 5.11 Using Venn Diagrams to Find Probabilities UNIT 6 Number System F I H 6.1 Decimals - - 6.2 Multiplying and Dividing with Decimals - - 6.3 Fractions and Decimals - 6.4 Long Multiplication and Division - 6.5 Estimating Answers 6.6 Using Brackets and Memory on a Calculator 6.7 Upper and Lower Bounds 6.8 Number System 6.9 Surds CIMT, University of Exeter vii
UNIT 7 Mensuration F I H 7.1 Units and Measuring - - 7.2 Estimating Areas - - 7.3 Making Solids Using Nets - - 7.4 Constructing Nets - 7.5 Conversion of Units - 7.6 Squares, Rectangles, Triangles - 7.7 Area and Circumference of Circles - 7.8 Volumes of Cubes, Cuboids, Cylinders and Prisms - 7.9 Plans and Elevations - 7.10 Using Isometric Paper 7.11 Discrete and Continuous Measures 7.12 Areas of Parallelograms, Trapeziums, Kites and Rhombuses 7.13 Surface Area 7.14 Maximum Volume and Density 7.15 Volumes, Areas and Lengths 7.16 Dimensions 7.17 Areas of Triangles UNIT 8 Data Handling F I H 8.1 Tables and Timetables - 8.2 Pictograms and Bar Charts - 8.3 Pie Charts - 8.4 Line Graphs - 8.5 Questionnaires and Surveys 8.6 Frequency Graphs 8.7 Histograms with Unequal Class Intervals 8.8 Sampling UNIT 9 Data Analysis F I H 9.1 Mean, Median, Mode and Range - 9.2 Finding the Mean from Tables and Tally Charts - 9.3 Calculations with the Mean - 9.4 Mean, Median and Mode for Grouped Data - 9.5 Cumulative Frequency 9.6 Standard Deviation CIMT, University of Exeter viii
UNIT 10 Equations F I H 10.1 Negative Numbers - - 10.2 Arithmetic with Negative Numbers - 10.3 Simplifying Expressions 10.4 Simple Equations - - 10.5 Solving Equations 10.6 Trial and Improvement Method 10.7 Expanding Brackets 10.8 Simultaneous Linear Equations 10.9 Factorisation 1 10.10 Factorisation 2 10.11 Solving Quadratic Equations by Factorisation 10.12 Solving Quadratic Equations Using the Formula 10.13 Algebraic Fractions 10.14 Completing the Square 10.15 Algebraic Fractions and Quadratic Equations UNIT 11 Fractions and Percentages F I H 11.1 Fractions, Decimals, Percentages - - 11.2 Fractions and Percentages of Quantities - - 11.3 Quantities as Percentages - 11.4 More Complex Percentages - 11.5 Percentage Increase and Decrease 11.6 Addition and Subtraction of Fractions 11.7 Multiplication and Division of Fractions 11.8 Compound Interest and Depreciation 11.9 Reverse Percentage Problems UNIT 12 Number Patterns F I H 12.1 Simple Number Problems - - 12.2 Recognising Number Patterns - 12.3 Extending Number Patterns - 12.4 Formulae and Number Patterns 12.5 General Laws 12.6 Quadratic Formulae CIMT, University of Exeter ix
UNIT 13 Graphs F I H 13.1 Positive Coordinates - - 13.2 Coordinates - 13.3 Plotting Straight Lines - 13.4 Plotting Curves - 13.5 Gradient 13.6 Applications of Graphs - 13.7 Scatter Plots and Lines of Best Fit 13.8 The Equation of a Straight Line 13.9 Horizontal and Vertical Lines - - 13.10 Solutions of Simultaneous Equations by Graphs 13.11 Graphs of Common Functions 13.12 Graphical Solutions of Equations UNIT 14 Loci and Transformations F I H 14.1 Drawing and Symmetry - - 14.2 Scale Drawing - - 14.3 Constructing Triangles and Other Shapes 14.4 Enlargements 14.5 Reflections 14.6 Construction of Loci 14.7 Enlargements which Reduce 14.8 Further Reflections 14.9 Rotations 14.10 Translations 14.11 Combined Transformations 14.12 Congruence 14.13 Similarity 14.14 Enlargement with Scale Factors UNIT 15 Variation F I H 15.1 Simple Ratios - - 15.2 Proportion and Ratio - 15.3 Map Scales and Ratios - 15.4 Proportional Division - 15.5 Direct Proportion CIMT, University of Exeter x
15.6 Inverse Proportion 15.7 Functional and Graphical Representations 15.8 Further Functional Representations UNIT 16 Inequalities F I H 16.1 Inequalities on a Number Line - - 16.2 Solution of Linear Inequalities - 16.3 Inequalities Involving Quadratic Terms 16.4 Graphical Approach to Inequalities - 16.5 Dealing With More Than One Inequality UNIT 17 Using Graphs F I H 17.1 Transformations of Graphs 17.2 Areas Under Graphs 17.3 Tangents to Curves 17.4 Finding Coefficients UNIT 18 3-D Geometry F I H 18.1 Using Pythagoras' Theorem and Trigonometry in Three Dimensions 18.2 Angles and Planes UNIT 19 Vectors F I H 19.1 Vectors and Scalars 19.2 Applications of Vectors 19.3 Vectors and Geometry 19.4 Further Work With Vectors 19.5 Commutative and Associative Properties KEY not in syllabus topic in syllabus - not examined directly, but assumed knowledge CIMT, University of Exeter xi