00b-cont.qxd 11/21/2012 10:30 AM Page iv C ONTENTS Expressions and Formulae Chapters 1 - 8 1 Approximation and Estimation . . . . . . . . . . . . . . . . . . . . . . 1 2 Working with Surds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Using Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4 Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5 Algebraic Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6 Gradient of a Straight Line Graph . . . . . . . . . . . . . . . . . . . 40 7 Working with Arcs and Sectors . . . . . . . . . . . . . . . . . . . . . 47 8 Volume of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 SECTION REVIEW - EXPRESSIONS AND FORMULAE SR Non-calculator Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 SR Calculator Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Relationships Chapters 9 - 18 9 Straight Line Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 10 Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 69 11 Simultaneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 81 12 Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 13 Graphs of Quadratic Functions . . . . . . . . . . . . . . . . . . . . . 98 14 Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 15 Pythagoras’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 16 Properties of Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 17 Similar Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 18 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . 144 SECTION REVIEW - RELATIONSHIPS SR Non-calculator Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 SR Calculator Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 iv 00b-cont.qxd 11/21/2012 10:30 AM Page v Applications Chapters 19 - 24 19 Using Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 20 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 21 Percentages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 22 Working with Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 189 23 Comparing Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 197 24 Scatter Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 SECTION REVIEW - APPLICATIONS SR Non-calculator Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 SR Calculator Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Exam Practice EP Non-calculator Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 EP Calculator Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 v 01-appr.qxd 11/15/2012 1:22 AM Page 1 Approximation and Estimation 1 Whole numbers and decimals The numbers 0, 1, 2, 3, 4, 5, … can be used to count objects. Such numbers are called whole numbers. Our number system is made up of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. The position a digit has in a number is called its place value. In the number 5384 the digit 8 is worth 80, but in the number 4853 the digit 8 is worth 800. Numbers and quantities are not always whole numbers. The number system can be extended to include decimal numbers. A decimal point is used to separate the whole number part from the decimal part of the number. 73.26 whole number, 73 This number is read as seventy-three point two six. decimal part, 2 tenths 6 hundredths (which is the same as 26 hundredths) Many measurements are recorded using decimals, including money, time, distance, weight, volume, etc. Approximation In real-life it is not always necessary to use exact numbers. A number can be rounded to an approximate number. Numbers are rounded according to how accurately we wish to give details. For example, the distance to the Sun can be given as 93 million miles. Can you think of other situations where approximations might be used? Rounding using decimal places What is the cost of 1.75 metres of material costing £3.99 a metre? 1.75 3.99 6.9825 The cost of the material is £6.9825 or 698.25p. As you can only pay in pence, a sensible answer is £6.98, correct to two decimal places (nearest penny). This means that there are only two decimal places after the decimal point. Often it is not necessary to use an exact answer. Sometimes it is impossible, or impractical, to use the exact answer. To round a number to a given number of decimal places When rounding a number to one, two or more decimal places: 1. Write the number using one more decimal place than asked for. 2. Look at the last decimal place and l if the figure is 5 or more round up, l if the figure is less than 5 round down. 3. When answering a problem remember to include any units and state the degree of approximation used. Example 1 Example 2 Write 2.76435 to 2 decimal places. Look at the third decimal place. 4 This is less than 5, so, round down. Answer 2.76 Write 7.104 to 2 decimal places. 7.104 7.10 to 2 d.p. The zero is written down because it shows the accuracy used, 2 decimal places. Notation: Often decimal place is shortened to d.p. 01-appr.qxd 11/15/2012 1:22 AM Page 1 Approximation and Estimation 20 1 Practice Exercise 1.1 1. 2. 3. 4. Write the number 3.9617 correct to (a) 3 decimal places, (b) 2 decimal places, The display on a calculator shows the result of 34 7. What is the result correct to two decimal places? 847 68.kg d.p. Answer 6. The scales show Gary’s weight. Write Gary’s weight correct to one decimal place. Copy and complete this table. Number 5. (c) 1 decimal place. 2.367 0.964 0.965 15.2806 0.056 4.991 4.996 1 2 2 3 2 2 2 2.4 Carry out these calculations giving the answers correct to (a) 1 d.p. (b) 2 d.p. (c) 3 d.p. (i) 6.12 7.54 (ii) 89.1 0.67 (iv) 98.6 5.78 (v) 67.2 101.45 (iii) 90.53 6.29 In each of these short problems decide upon the most suitable accuracy for the answer. Then calculate the answer. Give a reason for your degree of accuracy. (a) One gallon is 4.54596… litres. How many litres is 9 gallons? (b) What is the cost of 0.454 kg of cheese at £5.21 per kilogram? (c) The total length of 7 equal sticks, lying end to end, is 250 cm. How long is each stick? (d) A packet of 6 bandages costs £7.99. How much does one bandage cost? (e) Petrol costs 133.9 pence a litre. I buy 15.6 litres. How much will I have to pay? Rounding using significant figures Consider the calculation 600.02 7500.97 4500732.0194 To 1 d.p. it is 4500732.0, to 2 d.p. it is 4500732.02. The answers to either 1 or 2 d.p. are very close to the actual answer and are almost as long. There is little advantage in using either of these two roundings. The point of a rounding is that it is a more convenient number to use. Another kind of rounding uses significant figures. The most significant figure in a number is the figure which has the greatest place value. Consider the number 237. Noughts which are used to The figure 2 has the greatest place value. It is worth 200. locate the decimal point and So, 2 is the most significant figure. preserve the place value of In the number 0.00328, the figure 3 has the greatest place value. other figures are not significant. So, 3 is the most significant figure. To round a number to a given number of significant figures When rounding a number to one, two or more significant figures: 1. Start from the most significant figure and count the required number of figures. 2. Look at the next figure to the right of this and l if the figure is 5 or more round up, l if the figure is less than 5 round down. 3. Add noughts, as necessary, to locate the decimal point and preserve the place value. 4. When answering a problem remember to include any units and state the degree of approximation used. Expressions and Formulae 1 01-appr.qxd 11/15/2012 1:22 AM Page 2 Example 3 Write 4 500 732.0194 to 2 significant figures. The figure after the first 2 significant figures 45 is 0. This is less than 5, so, round down, leaving 45 unchanged. Add noughts to 45 to locate the decimal point and preserve place value. So, 4 500 732.0194 4 500 000 to 2 sig. fig. Notation: Often significant figure is shortened to sig. fig. Example 4 Write 0.000364907 to 1 significant figure. The figure after the first significant figure 3 is 6. This is 5 or more, so, round up, 3 becomes 4. So, 0.000364907 0.0004 to 1 sig. fig. Notice that the noughts before the 4 locate the decimal point and preserve place value. Choosing a suitable degree of accuracy In some calculations it would be wrong to use the complete answer from the calculator. The result of a calculation involving measurement should not be given to a greater degree of accuracy than the measurements used in the calculation. Example 5 What is the area of a rectangle measuring 4.6 cm by 7.2 cm? 4.6 7.2 33.12 Since the measurements used in the calculation (4.6 cm and 7.2 cm) are given to 2 significant figures the answer should be as well. 33 cm2 is a more suitable answer. Note: To find the area of a rectangle: multiply length by breadth. Practice Exercise 1.2 1. 2. 2 Write these numbers correct to one significant figure. (a) 17 (b) 523 (c) 350 (f) 0.083 (g) 0.086 (h) 0.00948 (d) 1900 (i) 0.0095 (e) 24.6 Copy and complete this table. Number 456 000 454 000 7 981 234 0.000567 0.093748 0.093748 sig. fig. 2 2 3 2 2 3 Answer 460 000 3. This display shows the result of 3400 7. What is the result correct to two significant figures? 4. Carry out these calculations giving the answers correct to (a) 1 sig. fig. (b) 2 sig. fig. (c) 3 sig. fig. (i) 672 123 (ii) 6.72 12.3 (iv) 7.19 987.5 (v) 124 65300 (iii) 78.2 12.8 5. A rectangular field measures 18.6 m by 25.4 m. Calculate the area of the field, giving your answer to a suitable degree of accuracy. 6. In each of these short problems decide upon the most suitable accuracy for the answer. Then work out the answer, remembering to state the units. Give a reason for your degree of accuracy. (a) The area of a rectangle measuring 13.2 cm by 11.9 cm. (b) The area of a football pitch measuring 99 m by 62 m. (c) The total length of 13 tables placed end to end measures 16 m. How long is each table? (d) The area of carpet needed to cover a rectangular floor measuring 3.65 m by 4.35 m. 01-appr.qxd 11/15/2012 1:22 AM Page 3 Approximation and Estimation 20 1 Estimation It is always a good idea to find an estimate for any calculation. An estimate is used to check that the answer to the actual calculation is of the right magnitude (size). If the answer is very different to the estimate then a mistake has possibly been made. Estimation is done by approximating every number in the calculation to one significant figure. The calculation is then done using the approximated values. Example 6 Estimate 421 48. Round 421 to one significant figure: 400 Round 48 to one significant figure: 50 400 50 20 000 Use a calculator to calculate 421 48. Comment on your answer. Example 7 78.5 0.51 Use estimation to show that is close to 2. 18.7 Approximating: 78.5 80 to 1 sig. fig. 0.51 0.5 to 1 sig. fig. 18.7 20 to 1 sig. fig. 80 0.5 4200 2 (estimate) 20 Remember: When you are asked to estimate, write each number in the calculation to one significant figure. 78.5 0.51 Using a calculator 18.7 To do the calculation enter the following sequence into your calculator. 78.5 0.51 2.140909… 18.7 Is 2.140909 reasonably close to 2? Yes. Practice Exercise 1.3 1. John estimated 43 47 to be about 2000. Explain how he did it. 2. Make estimates to these calculations by using approximations to one significant figure. (a) (i) 39 21 (ii) 115 18 (iii) 797 53 (iv) 913 59 (b) (i) 76 18 (ii) 597 29 (iii) 889 61 (iv) 3897 82 3. Lilly ordered 39 prints of her holiday photographs. Each print cost 52 pence. Use suitable approximations to estimate the total cost of the prints. Show your working. 4. (a) When estimating the answer to 29 48 the approximations 30 and 50 are used. How can you tell that the estimate must be bigger than the actual answer? (b) When estimating the answer to 182 13 the approximations 200 and 10 are used. Will the estimate be bigger or smaller than the actual answer? Explain your answer. 5. Bernard plans to buy a conservatory costing £8328 and furniture costing £984. (a) By using approximations, estimate the total amount Bernard plans to spend. (b) Find the actual cost. 6. Kath uses her calculator to work out the value of 396 0.470. The answer she gets is 18.612. Use approximations to show that her answer is wrong. Expressions and Formulae 3 01-appr.qxd 11/15/2012 1:22 AM Page 4 49.7 10.6 7. (a) Calculate 9.69 3.04 (b) Do not use your calculator in this part of the question. By using approximations show that your answer to (a) is about right. 8. Find estimates to these calculations by using approximations to 1 significant figure. Then carry out the calculations with the original figures. Compare your estimate to the actual answer. 7.9 3.9 (a) 4.8 400 0.29 (b) 6.2 81.7 4.9 (c) 1.9 10.3 4.12 49.7 (d) 0.096 78 .9 9. (a) Calculate 0.037 5.2 (b) Show how you can use approximations to check your answer is about right. 10. Niamh calculates 5 967 000 0.029. She gets an answer of 2 057 586 207. Use approximations to check whether Niamh’s answer is of the right magnitude. Key Points 䉴 In real-life it is not always necessary to use exact numbers. A number can be rounded to an approximate number. Numbers are rounded according to how accurately we wish to give details. For example, the distance to the Sun can be given as 93 million miles. 䉴 You should be able to approximate using decimal places. Write the number using one more decimal place than asked for. Look at the last decimal place and l if the figure is 5 or more round up, l if the figure is less than 5 round down. 䉴 You should be able to approximate using significant figures. Start from the most significant figure and count the required number of figures. Look at the next figure to the right of this and l if the figure is 5 or more round up, l if the figure is less than 5 round down. Add noughts, as necessary, to preserve the place value. 䉴 You should be able to choose a suitable degree of accuracy. The result of a calculation involving measurement should not be given to a greater degree of accuracy than the measurements used in the calculation. 䉴 You should be able to use approximations to estimate that the actual answer to a calculation is of the right order of magnitude. Estimation is done by approximating every number in the calculation to one significant figure. The calculation is then done using the approximated values. Review Exercise 1 4 1. Write these numbers correct to 2 decimal places. (a) 28.714 (b) 6.91288 (c) 12.397 (d) 0.0418 (e) 0.00912 2. Write these numbers correct to 3 significant figures. (a) 2313 (b) 23.58 (c) 36.97 (d) 503.89 (e) 0.0005646 01-appr.qxd 11/15/2012 1:22 AM Page 5 Approximation and Estimation 20 1 3. The display shows the result of 179 7. What is the result correct to: (a) two decimal places, (b) one decimal place, (c) one significant figure? 4. Calculate 7.25 0.79 (a) to 1 decimal place, (b) to 2 decimal places, (c) to 3 decimal places. 5. Calculate 107.9 72.5 (a) to 1 significant figure, (b) to 2 significant figures. 6. Find estimates to these calculations by using approximations to one significant figure. (a) 86.5 1.9 (b) 2016 49.8 7. Aimee uses her calculator to multiply 18.7 by 0.96. Her answer is 19.752. Without finding the exact value of 18.7 0.96, explain why her answer must be wrong. 8. Daniel has a part-time job in a factory. He is paid £36 for each shift he works. Last year he worked 108 shifts. Estimate Daniel’s total pay for the year. You must show all your working. 9. Flour costs 78p per kilogram from the flour mill. Rachel bought 306 kg of flour from the mill. She shared the flour equally between 18 people and calculated that each person should pay £1.14. (a) Without using a calculator, show that Rachel’s calculation must be wrong. (b) Roughly how much should Rachel charge each person? Show your working. 10. Estate Agents sometimes quote the floor area of a flat in square metres. They quote an estimate so that buyers can easily compare one flat with another. Write down the lengths and widths of each room to 1 significant figure. (a) Obtain an estimate of the total floor area for each of the two flats. Meadow View Flat Park View Flat Reception 1 4.1 m 6.9 m Reception 1 3.9 m 5.1 m Reception 2 3.9 m 5 m Reception 2 4 m 3.8 m Bedroom 1 3.2 m 3.7 m Bedroom 1 4.1 m 3.9 m Bedroom 2 2.9 m 2.1 m Bedroom 2 3.1 m 2.9 m (b) Work out the actual floor area of each flat. Compare the estimates. 11. By using approximate values, find estimates for these questions. (a) 5.98 3.04 (b) 17.742 9.81 (d) 0.0049 0.000 97 (e) 39.5 0.14 (c) (0.0275)2 (f) 237.4 38.65 12. (a) A rectangular lawn measures 27 metres by 38 metres. A firm charges £9.95 per square metre to turf the lawn. Estimate the charge for turfing the lawn. (b) Is your estimate too large or too small? Give a reason for your answer. 13. The floor of a lounge is a rectangle which measures 5.23 m by 3.62 m. The floor is to be carpeted. (a) Calculate the area of carpet needed. Give your answer to an appropriate degree of accuracy. (b) Explain why you chose this degree of accuracy. 14. 兹3 苶5苶.4 苶 is approximately equal to 6. 7.91 兹3苶5苶.4 苶 Use this to estimate the value of 5.2 1.85 Expressions and Formulae 5

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertising