 GCSE MATHEMATICS GCSE 8300 Teaching guidance For teaching from September 2015 onwards For GCSE exams in June 2017 onwards Version 2.0, October 2015 Our specification is published on our website (www.aqa.org.uk). We will let centres know in writing about any changes to the specification. We will also publish changes on our website. The definitive version of our specification will always be the one on our website, this may differ from printed versions. You can get further copies of this Teaching guidance from: The GCSE Mathematics Department AQA Devas Street Manchester M15 6EX Or, you can download a copy from our All About Maths website (http://allaboutmaths.aqa.org.uk/) Contents General information - disclaimer 5 Subject content 5 1 Number – Structure and calculation 6 1 2 Number – Fractions, decimals and percentages 26 1 3 Number – Measures and accuracy 31 1 4 Algebra – Notation, vocabulary and manipulation 39 1 5 Algebra – Graphs 53 1 6 Algebra – Solving equations and inequalities 72 1 7 Algebra – Sequences 85 1 8 Ratio, proportion and rates of change 90 1 9 Geometry and measures – Properties and constructions 115 1 1 0 Geometry and measures – Mensuration and calculation 140 1 1 1 Geometry and measures – Vectors 161 1 1 2 Probability 165 1 1 3 Statistics 182 1 1 4 Appendices: Mathematical formulae 192 1 B Assessment objectives 196 1 Version 2.0 3 4 Version 2.0 GCSE MATHEMATICS 8300 TEACHING GUIDANCE General Information - Disclaimer This teaching guidance will help you plan by providing examples of the content of the specification. It is not, in any way, intended to restrict what can be assessed in the question papers based on the specification. Questions will be set in a variety of formats including both familiar and unfamiliar contexts. Examples given in this teaching guidance illustrate the type of questions which would be asked on a question paper. However, the wording and format used in this guidance do not always represent how questions would appear in a question paper. Questions in this guidance have not been through the same rigorous checking process used in our question papers. All knowledge from the Key Stage 3 and Key Stage 4 programmes of study is subsumed into the content of the GCSE specification. Subject content Students can be said to have confidence and competence with mathematical content when they can apply it flexibly to solve problems. The expectation is that:  All students will develop confidence and competence with the content identified by standard type  All students will be assessed on the content identified by the standard and the underlined type; more highly attaining students will develop confidence and competence with all of this content  Only the more highly attaining students will be assessed on the content identified by bold type. The highest attaining students will develop confidence and competence with the bold content. The distinction between standard, underlined and bold type applies to the content statements only, not to the assessment objectives or to the mathematical formulae in the appendix. Version 2.0 5 1 Number – Structure and calculation N1 Order positive and negative integers, decimals and fractions; use the symbols =, ≠, <, >, ⩽, ⩾ Teaching Guidance Students should be able to:  know and use the word integer and the equality and inequality symbols  recognise integers as positive or negative whole numbers, including zero  order positive and/or negative numbers given as integers, decimals and fractions, including improper fractions. Notes Including use of a number line. Students should know the conventions of an open circle on a number line for a strict inequality and a closed circle for an included boundary. See A22 Examples 1 Put these numbers in order starting with the smallest. 1 1 2 1 1 1.12 1 1.12 1 1 2 2 3 8 2 Write 4.2, 4.02, 4.203 and 4.23 in ascending order. 3 Write these fractions in order of size, starting with the smallest. 5 2 7 6 3 9 18 17 31 4 Which of the improper fractions , or is the greater? 5 6 10 1 5 Which of these is closest to ? 3 3 1 0.35 0.29 10 2 6 Version 2.0 GCSE MATHEMATICS 8300 TEACHING GUIDANCE 6 Write down the integer values of x where 3 < x ⩽ 8 7 Put these numbers in ascending order. 3 4 13 0.83 4 5 20 Version 2.0 7 N2 Apply the four operations, including formal written methods, to integers, decimals and simple fractions (proper and improper), and mixed numbers - all both positive and negative; understand and use place value (eg when working with very large or very small numbers, and when calculating with decimals) Teaching Guidance Students should be able to:  add, subtract, multiply and divide integers using both mental and written methods  add, subtract, multiply and divide decimals using both mental and written methods  add, subtract, multiply and divide positive and negative numbers  interpret a remainder from a division problem  recall all positive number complements to 100  recall all multiplication facts to 12  12 and use them to derive the corresponding division facts  perform money and other calculations, writing answers using the correct notation  apply the four rules to fractions with and without a calculator  multiply and divide a fraction by an integer, by a unit fraction and by a general fraction  divide an integer by a fraction. Notes Students may use any algorithm for addition, subtraction, multiplication and division. Students are expected to know multiplication facts up to 12  12 and squares up to 15  15 Questions will be set in a variety of contexts, both familiar and unfamiliar. For example, in household finance questions, students will be expected to know and understand the meaning of profit, loss, cost price, selling price, debit, credit, balance, income tax, VAT and interest rate. See N8, R9 Examples 1 Write down the place value of 8 in the answer to 2850  10 2 The population of Cambridge is 108 863 The population of Oxford is 153 904 How many more people live in Oxford than in Cambridge? 8 Version 2.0 GCSE MATHEMATICS 8300 TEACHING GUIDANCE 3 There are 75 students travelling in 16-seater mini-coaches. If as many of the mini-coaches as possible are full, how many students travel in the mini-coach that is only partly full? 4 Four cards are numbered 3, 5, 7 and 8 Use each card once to make this calculation work. ……. ……. + ……. ……. = 158 5 The temperature falls 4 °C from 3.5 °C. Work out the new temperature. 6 Here is a bank statement. (a) Write down what you understand by the word ‘balance’. (b) Complete the statement. Date Description Credit Debit Balance Starting balance £63.50 12/12/2013 Cash £120.00 ………….. 16/12/2013 Gas bill £102.50 ………….. 17/12/2013 Electricity bill £220.00 ………….. 7 A builder employs seven bricklayers. Each bricklayer earns £12.60 per hour worked. 1 They each work 37 hours per week. 2 The builder says he needs £33 075 each week to pay his bricklayers. Use a calculator to check if he is correct. Version 2.0 9 8 A builder employs some bricklayers. 1 Each bricklayer works 37 hours per week. 2 He needs the bricklayers to work a total of 500 hours per week. How many bricklayers should he employ? 3 9 Work out 1  4 4 2 3 10 Work out 1 + 5 4 5 1 11 Work out 3  2 6 2 1 12 Work out 2  3 2 7 13 Work out 4  8 6 14 Work out ÷ 3 11 3 8 15 Work out  4 9 4 4 16 Write down the answer to ÷ 15 15 2 17 Work out 8 ÷ 3 18 Work out 0.3  0.2 19 3.5 Work out 0.5 20 Helen earns £8000 per year She pays 20% income tax on this amount. How much income tax does she pay each month? 21 Electricity is 21p per unit. A household uses 450 units. VAT is added at 5% Work out the total cost of the electricity 10 Version 2.0