Section 1: Additional material

Transcription

1 Section 1: Additional material Section 1 provides you with optional chapters that you can use with your class. Preliminary chapters are intended for use as revision and to reinforce foundation knowledge that is needed in the year. You may wish to assign work from this chapter as remedial activities for students who struggle with the main course work. Enrichment chapters are optional chapters that extend the scope of the work done in the year. If you have time at the end of the year, you can work through this additional chapter with your students. You may wish to assign work from this chapter to stronger students. 2 Section 1: Additional material

2 Preliminary chapter Review of previous coursework Objectives By the end of the chapter, you will be able to recall, from Book 1 of New General Mathematics, the facts and methods that relate to these themes: number and numeration basic operations algebraic processes mensuration and geometry everyday statistics. Teaching and learning materials Teacher: addition and multiplication wall charts number square metre rule measuring tape chalk board instruments (ruler, set-square, compasses) solid shapes. Students: Mathematics set. To make the best use of Book 2 of New General Mathematics, you should be familiar with the contents of Book 1. This chapter contains themes and topics from Book 1 that are necessary to understand Book 2. P 1 Number and numeration Figure P1 hundreds tens units decimal point tenths hundredths thousandths We usually write numbers in the decimal place value system (Figure P1). The symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are called digits. 2 We use the words thousand, million, billion, trillion for large numbers: 1 thousand = million = 1 thousand billion = 1 million trillion = 1 billion Similarly, we use millionth, billionth,, for small decimal fractions: 1 millionth = billionth = When writing large and small numbers, group the digits in threes from the decimal point, for example, and = 4. Number 7 is a whole number that divides exactly into another whole number, 28. This means that 7 is a factor of 28, and 28 is a multiple of 7. 4 A prime number has only two factors, itself and 1. The number 1 is not a prime number. The numbers 2, 3, 5, 7, 11, 13, 17, are prime numbers. They continue without end. The prime factors of a number are those factors that are prime. For example, 2 and 5 are the prime factors of 40. We can write 40 as a product of prime factors: either = 40, or, in index form, = The numbers 18, 24 and 30 all have 3 as a factor. Number 3 is a common factor of the numbers. The highest common factor (HCF) is the largest of the common factors of a given set of numbers. For example, 2, 3 and 6 are the common factors of 18, 24 and 30; 6 is the HCF. The number 48 is a multiple of 4 and a multiple of 6. Number 48 is a common multiple of 4 and 6. The lowest common multiple (LCM) is the smallest of the common multiples of a given set of numbers. For example, 12 is the LCM of 4 and 6. Section 1: Additional material 3

3 6 A fraction is the number obtained when one number (the numerator) is divided by another number (the denominator). The fraction _ 5 8 means 5 8 (Figure P2). Figure P2 5 8 numerator dividing line denominator We use fractions to describe parts of quantities (Figure P3). Figure P3 _ 5 of the circle is shaded 8 The fractions _ 5 8, 10, 15 all represent the same amount; they are equivalent fractions. 5_ is the simplest form of 15 8, that is, 15 in its lowest terms is _ 5 8. P 2 Basic operations To add or subtract fractions, change them to equivalent fractions with a common denominator. For example: 5_ 8 + 2_ 3 = = ( ) = 31 = = (13 10) = To multiply fractions, multiply numerator by numerator and denominator by denominator. For example: 5_ 8 2_ 3 = (5 2) = 10 = 5 in simplest form (8 3) _ 5 8 = _ 8 = (12 5) = 60 (1 8) 8 = 15 2 = 7 1_ 2 To divide by a fraction, multiply by the reciprocal of the fraction. For example: 35 _ 5 8 = _ = (35 8) = (7 8) = 56 5 (1 5) 1 5_ 8 3 3_ = 5_ = 5_ = (5 4) = (8 15) 120 = 1_ 6 1 x% is short for x % means To change a fraction to an equivalent percentage, multiply the fraction by 100. For example: 5_ 8 as a percentage = _ % = 8 % = 62 1_ 2 %. 2 To change a fraction to a decimal fraction, divide the numerator by the denominator. For example: 5_ 8 = When adding or subtracting decimals, write the numbers in a column with the decimal points exactly under each other. For example: Add 2.29, and 4.3, then subtract the result from To multiply decimals: a Ignore the decimal points and multiply the b numbers as if they are whole numbers. Then place the decimal point so that the answer has as many digits after the point as there are in the question. For example, to multiply : 8 3 = 24 There are three digits after the decimal points in the question, so: = To divide by a decimal, change the division so that the divisor becomes a whole number. For example, to divide : = = ( ) = 560 ( ) 7 = 80 3 Numbers may be positive or negative. Positive and negative numbers are called directed numbers. Directed numbers can be shown on a number line (Figure P4) Figure P4 These examples show how to add and subtract directed numbers. 4 Section 1: Additional material

4 Addition Subtraction (+8) + (+3) = +11 (+9) (+4) = +5 (+8) + ( 3) = +5 (+9) ( 4) = +13 ( 8) + (+3) = 5 ( 9) (+4) = 13 ( 8) + ( 3) = 11 ( 9) ( 4) = 5 An integer is any positive or negative whole number as shown in Figure P4. 4 When rounding numbers, round down the digits 1, 2, 3, 4 and round up the digits 5, 6, 7, 8, 9. For example: = to 3 significant figures = to the nearest hundred = 7.28 to 2 decimal places = 7.3 to 1 decimal place = 7 to the nearest whole number 5 The everyday system of numeration uses ten digits and is called a base ten, or denary, system. The base two, or binary, system of numeration uses only two digits, 0 and 1. To convert between bases ten and two, express the given numbers in powers of two: 43 ten = = = = two two = = = 22 ten Use these identities when adding, subtracting and multiplying binary numbers: Addition: = = = = 10 Multiplication: 0 0 = = = 1 Detailed coverage of number, numeration and basic operations is given in Book 1 Chapters 1, 2, 3, 4, 9, 12, 23 and 24. Review test 1 (Number and numeration & basic operations Allow 25 minutes for this test. If you do not understand why some of your answers were incorrect, ask a friend or your teacher. 1 What is the value of: a The 7 in 3.75? b The 5 in 519.2? 2 Find the HCF of: a 6, 15, 36 b 84, 56, 70 3 Find the LCM of: a 4, 5 and 10 b 9, 10 and 12 4 Reduce these fractions to their lowest terms: 45 a 256 b Simplify: 5 a 12 1_ 8 b 5 3_ 3_ 4 5 c 2 2_ d 1 3_ 3_ Express the first quantity as a percentage of the second: a N20, N50 b 4 mm, 5 cm c 20 cm, 5 m d 360 g, 2 kg 7 Each row of this table contains equivalent expressions. Complete the table. (The first row has been done for you.) Common fraction Decimal fraction Percentage % Simplify: a ( 2) + (+6) b (+3) (+4) c 5 + ( 3) d 7 ( 1) 9 Round off these numbers to the nearest: i Thousand ii Hundred iii Ten a b Round off these numbers to the nearest: i Whole number ii Tenth iii Hundredth a b Section 1: Additional material 5

5 10 Round off these numbers to the nearest: i Whole number ii Tenth iii Hundredth a b Write this number in digits, grouping the digits in threes. Sixty one billion, five hundred and seven million, nine hundred and seventy-two thousand, and forty-three 12 Convert: a 58 ten to base two b two to base ten Now try Review test 2. Review test 2 (Number and numeration & basic operations) Allow 25 minutes for this test. If you do not understand why some of your answers were incorrect, ask a friend or your teacher. 1 What is the value of: a The 3 in 0.438? b The 9 in 29.7? 2 Find the HCF of: a 44, 88 and 55 b 144, 90 and 54 3 Find the LCM of: a 3, 4 and 5 b 5, 8 and 9 4 Reduce these fractions to their lowest terms: 24 a 30 b Simplify: a 8_ 9 + 1_ 3 b 4 2_ 5 5_ 5 8 c 2_ of 1 2_ 5 3 d 7 1_ 8 4 3_ 4 6 Express the first quantity as a percentage of the second: a N80, N150 b 140 m, 2 km c 28 cm, 4 m d 26 g, 4 kg 7 Each row of this table contains equivalent expressions. Complete the table. (The first row has been done for you.) Common fraction Decimal fraction Percentage % % 8 Simplify: a ( 8) + (+3) b ( 1) (+7) c 4 + ( 10) d 2 ( 5) 9 Round off these numbers to the nearest: i Thousand ii Hundred iii Ten a b Round off these numbers to the nearest: i Whole number ii Tenth iii Hundredth a b Write these numbers in digits only: a N6 1_ billion b 650 millionths 4 12 Calculate these base two numbers. Give your answers as binary numbers. Check your results in base ten. a b c P 3 Algebraic processes 1 7x 2x + 3y is an example of an algebraic expression. The letters x and y stand for numbers, and 7x, 2x, 3y are the terms of the expression. 3y is short for 3 y. 3 is the coefficient of y. 2 Multiply and divide algebraic terms as follows: 6a 5b = 6 a 5 b = 6 5 a b = 30ab 3p 8p = 3 p 8 p = 3 8 p p = 24p 2 12mn 3m = (12 m n) (3 m) 12n = 3 = 4n 3 Simplify algebraic expressions in these ways. a By grouping like terms. For example: 7x 2x + 3y = 5x + 3y Notice that 7x and 2x are like terms; 5x and 3y are unlike terms. 6 Section 1: Additional material

6 b By removing brackets. For example: a + (2a 7b) = a + 2a 7b = 3a 7b If there is a positive sign before the bracket, the signs of the terms inside the bracket stay the same when removing brackets. For example: 5x (x 9y) = 5x x + 9y = 4x + 9y If there is a negative sign before the bracket, the signs of the terms inside the bracket change when the bracket is removed. 4 x + 8 = 3 is an algebraic sentence containing an equals sign; it is an equation in x. x is the unknown of the equation. If we substitute a value for an unknown, the equation may be true or false. For example, 2y = 6 is true when y = 3 and is false when y = 5. To solve an equation means to find the value of the unknown that makes the equation true. Use the balance method to solve simple equations. For example: 3x 8 = 25 Add 8 to both sides: 3x = That is, 3x = 33. Divide both sides by 3: 3x 3 = 33 3 That is, x = 11. In the balance method, always do the same to both sides of the equation. Detailed overage of algebraic processes is given in Book 1 Chapters 5, 7, 10, 15 and 19. Now try Review test 3. Review test 3 (Algebraic processes) Allow 20 minutes for this test. If you do not understand why some of your answers were incorrect, ask a friend or your teacher. 1 Each sentence is true. Find the number that each letter stands for. a = p b q 14 = 14 c 27 = r + 14 d 12 = s + s + s 2 Find the value of these expressions when x = 6: a 9 + x b 44 x c x 5 + x d 7 x + x 3 Simplify these expressions: a 8p 5p b 8q + 3q c 12r 4 7r + 15 d 6w y + 5z + 9z + 5y 4w 4 a i How many weeks are there in 28 days? ii How many weeks are there in d days? b i A student has five bananas. He eats one of them. How many does he have left? ii Another student has b bananas. She eats two of them. How many does she have left? 5 Remove the brackets and then simplify: a 4 (6a 7) b 3a (5 a) c (8 3a) (3 4a) d (15x + 6y) (8x 7y) 6 Solve these equations: a 4a + 13 = 21 b 5b + 9 = 44 c 9c 11 = 61 d 7 = 10 3k Now try Review test 4. Review test 4 (Algebraic processes) Allow 20 minutes for this test. If you do not understand why some of your answers were incorrect, ask a friend or your teacher. 1 Each sentence is true. Find the number that each letter stands for: a 13 f = 8 b 24 = 15 + g c 30 = m 6 d 18 n = n Section 1: Additional material 7

7 2 Find the value of these expressions when x = 9: a x + 13 b x 10 c x x d x 5 x 3 Simplify these expressions: a 7f + 7f b 18g 13g c 8m n 5m 3n d 5x + 2y 8 5x a i How many cm in 2 metres? ii How many cm in m metres? b i In Compound 1 there are four goats and three sheep. How many animals are there altogether? ii In Compound 2, there are six goats and n sheep. How many animals are there altogether? 5 Remove the brackets and then simplify: a 2a (a + 8) b 4b (7 b) c (7x + 2y) (3x + y) d (5y 3) (4 3y) 6 Solve these equations: a 4a + 3 = 31 b 3b 14 = 10 c 15 6c = 0 d 18 = 19d 20 Figure P6 face edge vertex The formulae for the surface area and volume of common solids are given in the table on page xxx. An angle is a measure of rotation or turning. 1 revolution = 360 degrees (1 rev = 360 ) 1 degree = 60 minutes (1 = 60 ) The names of angles change with their size. See Figure P7. acute angle (between 0 and 90 ) right angle (90 ) P 4 Mensuration and geometry Figure P5 gives sketches and names of some common solids. All solids have faces; most solids have edges and vertices (Figure P6). straight angle (180 ) reflex angles (between 180 and 360 ) obtuse angle (between 90 and 180 ) Figure P7 cuboid cube cylinder Angles are measured and constructed using a protractor. Figure P8 shows some properties of angles formed when straight lines meet. triangular prism cone squarebased pyramid a a b c sphere the sum of the angles on a straight line is 180 a + b + c = 180 Figure P5 8 Section 1: Additional material

8 b p s r q Figure P10 shows the names and properties of some common quadrilaterals. c vertically opposite angles are equal p = q and r = s square rectangle a b e d c parallelogram rhombus Figure P10 d m Figure P8 a + b + c + d + e = 360 x m = n (alternate angles) x = y (alternate angles) y n Figure P9 shows the names and properties of some common triangles. e a p a = b (corresponding angles) p = q (corresponding angles) b q Figure P11 gives the names of lines and regions in a circle. arc circumference radius diameter centre Figure P11 chord sector semi-circle segment A polygon is a plane shape with three or more straight sides. A regular polygon has all its sides of equal length and all its angles of equal size. Figure P12 gives the names of some common regular polygons. scalene right-angled obtuse-angled 60 equilateral triangle square regular pentagon Figure P9 isosceles equilateral Figure P12 regular hexagon regular octagon Section 1: Additional material 9

9 The formulae for the perimeter and area of plane shapes are given in the table on page 130 of the Student s Book. The SI system of units is given in the tables on pages 229 and 230 of the Student s Book. Use a ruler and set-square to construct parallel lines (Figure P13) and a perpendicular from a point to a line (Figure P14). Figure P13 Parallel lines (ruler and set-square) P l Review test 5 (Mensuration and geometry) Allow 25 minutes for this test. If you do not understand why some of your answers were incorrect, ask a friend or your teacher. 1 a Add these capacities l, 325 ml and 2.04 l. Give the answer in: i litres ii millilitres. b How many minutes in: i 210 seconds? ii 1 day? 2 How many faces, edges and vertices does a hexagonal prism have? 3 In Figure P9, one of the triangles is marked as scalene. Are there any other scalene triangles in Figure P9? 4 Copy this table. Complete the unshaded boxes. P l Shape Length Breadth Perimeter Area a Rectangle 5 cm 9 cm b Rectangle 8 cm 30 cm c Square 28 cm d Square 64 km 2 5 Copy and complete this table of circles. Use the value 3 for π. a Radius Diameter Circumference Area 7 cm b 16 cm c 15 cm d 363 cm 2 6 Copy and complete this table of triangles. Figure P14 Perpendicular from point P to line l (ruler and set-square) Detailed coverage of geometry and mensuration is given in Book 1 Chapters 6, 8, 11, 13, 14, 16, 20 and 21. Now try Review test 5. Base Height Area a 10 cm 7 cm b 9 cm 3 cm c 6 cm 12 cm 2 d 9 m 54 m 2 10 Section 1: Additional material

10 7 Copy this table. Complete the unshaded boxes. Shape Length Breadth Height Base area a Cuboid 8 cm 3 cm 11 cm Volume b Cuboid 2 cm 6 cm 60 cm 3 c Cube 4 m d Prism 5.5 cm 6 cm 2 e Prism 28 cm 84 cm 3 f Prism 4 m 2 30 m 3 8 Write down the sizes of the lettered angles in Figure P15. a b c 140 Figure P15 a b c 115 f g d e n m 82 Now try Review test 6. d i h 63 j k 61 l Review test 6 (Mensuration and geometry) Allow 25 minutes for this test. If you do not understand why some of your answers were incorrect, ask a friend or your teacher. 1 a Add these quantities 2.5 tonnes, 850 kg, tonnes. Give the answer in: i kg ii tonnes. b How many seconds in: i 5 minutes? ii 2 hours? 2 How many faces, edges and vertices does a square-based pyramid have? q p 3 One of the angles of an isosceles triangle is 22. What are the sizes of its other two angles? (Note: There are two possible answers; give both.) 4 Copy this table. Complete the unshaded boxes. Shape Length Breadth Perimeter Area a Rectangle 2 km 11 km b Rectangle 12 cm 36 cm 2 c Square 25 mm d Square 34 m 5 Copy and complete this table of circles. Use the value 3 for π. a Radius Diameter Circumference Area 4.5 cm b 10 cm c 84 m d 3 m 2 6 Copy and complete this table of triangles. Base Height Area a 11 cm 8 cm b 5 cm 15 cm c 2 m 20 m 2 d 4 cm 22 cm 2 7 Copy this table. Complete the unshaded boxes. Shape Length Breadth Height Base area a Cuboid 7 cm 4 cm 10 cm Volume b Cuboid 3 cm 6 cm 54 cm 3 c Cube 216 cm 3 d Prism 8 cm 3.5 cm 2 e Prism 12.5 cm cm 3 f Prism 2 m 26 m 3 8 Write down the sizes of the lettered angles in Figure P16. a b a 44 b f 43 c d 146 e Section 1: Additional material 11

11 c l k i 18 h g 70 Figure P16 j P 5 Everyday statistics d n 254 m p 1 Information in numerical form is called statistics. a Statistical data may be given in rank order (that is, in order of size) such as these marks scored in a test out of 5: 0, 1, 1, 2, 2, 2, 3, 3, 5 b Data may also be given in a frequency table (Table P1). Mark Frequency Table P1 c The frequency is the number of times each piece of data occurs. Statistics can also be presented in graphical form. Figure P17 shows the data above in a pictogram, a pie chart and a bar chart. 2 The average of a set of statistics is a number that represents the whole set. The three most common averages are the mean, the median and the mode. For the numbers given in rank order in point 1 above: a Mean = = 2 1_ 9. b The median is the middle number when the data are arranged in size order (= 2). c The mode is number with the greatest frequency (also 2). If a set of data has two modes, we say it is bimodal. Detailed coverage of statistics is given in Book 1 Chapters 17, 18 and 22. Now try Review test 7. Review test 7 (Everyday statistics) Allow 20 minutes for this test. If you do not understand why some of your answers were incorrect, ask a friend or your teacher. 1 The bar chart in Figure P18 shows the rainfall for each month in a year. 50 Pictogram Pie chart 0 marks 1 mark 2 marks 3 marks 4 marks 5 marks 4 frequency marks Rainfall (cm) Figure P18 J F M A M J J A S O N D a Which month had most rainfall? b How many cm of rain fell in the wettest month? Figure P17 12 Section 1: Additional material

12 c Which months had no rainfall? d List the four wettest months in rank order. e Find in cm the total rainfall for the year. 2 Calculate the mean for each of these sets of numbers: a 11, 13, 15 b 8, 9, 13 c 2, 9, 7, 9, 8 d 5, 5, 1, 0, 9 3 Find the mode, median and mean for each of these sets of numbers: a 6, 6, 8, 11, 14 b 1, 2, 5, 5, 6, 6, 6, 7, 7, 8 c 8, 6, 3, 10, 6, 9 d 6, 4, 10, 6, 11, 7, 5, 8, 6 Now try Review test 8. Review test 8 (Everyday statistics) Allow 25 minutes for this test. If you do not understand why some of your answers were incorrect, ask a friend or your teacher. 1 The marks obtained in a test out of 10 are shown in this frequency table. Mark Frequency a What was the highest mark? b How many students scored this mark? c Are the data bimodal? d How many students scored more than 5 marks? e How many students took the test? 2 Calculate the mean for each of these sets of numbers: a 17, 5, 7, 11 b 2, 10, 5, 7 c 4, 11, 2, 8, 9, 2 d 7, 8, 10, 11, 14, 16 3 Find the mode, median and mean for each of these sets of numbers: a 3, 4, 4, 6, 7, 9 b 3, 7, 10, 10, 11, 11, 11 c 2, 0, 16, 0, 7, 11 d 5, 4, 2, 5, 2, 1, 3, 5, 3, 4, 5, 3 Answers to exercises Review test 1 1 a 7 tenths b 5 hundreds 2 a 3 b 14 3 a 20 b a 5_ 7 b a 7 b c 6 d 3_ 4 6 a 40% b 8% c 4% d 18% 7 Common fraction Decimal fraction Percentage 1_ % 2_ % % 5_ % 8 8 a 4 b 1 c 8 d 8 9 a i ii iii b i ii iii a i 1 ii 0.9 iii 0.87 b i 7 ii 7.0 iii a two b 54 ten Review test 2 1 a 3 hundredths b 9 units 2 a 11 b 18 3 a 60 b a 4_ 5 b 5_ 7 5 a 11 9 = 1 2_ 9 b c 2_ 3 d 1 1_ 2 6 a 53.3% b 7% c 7% d 0.65% Section 1: Additional material 13

13 7 Common fraction 1_ _ Decimal fraction Percentage % % % % _ % 40 8 a 5 b 8 c 6 d 3 9 a i ii iii b i ii iii a i 2 ii 2.4 iii 2.39 b i 4 ii 4.3 iii a N b a ( = 32) b (27 5 = 22) c (27 5 = 135) Review test 3 1 a p = 22 b q = 28 c r = 13 d s = 4 2 a 15 b 38 c 7 d 7 3 a 3p b 11q c 5r + 11 d 2w + 4y + 14z 4 a i 4 ii d_ 7 b i 4 ii b 2 5 a 11 6a b 4a 5 c 5 + a d 7x + 13y 6 a a = 2 b b = 7 c c = 8 d k = 1 Review test 4 1 a f = 5 b g = 9 c m = 36 d n = 9 2 a 22 b 1 c 27 d 5 3 a 14f b 5g c 3m 4n d 2y 5 4 a i 200 ii 100m b i 7 ii 6 + n 5 a a 8 b 5b 7 c 4x + y d 8y 7 6 a a = 7 b b = 8 c c = 2 1_ 2 d d = 2 Review test 5 1 a i 4.25 litres ii ml b i 3.5 ii faces, 18 edges, 12 vertices 3 Yes: the right-angled triangle and obtuseangled triangle are also scalene (their three sides are of different lengths). 4 Shape Length Breadth Perimeter Area a Rectangle 5 cm 9 cm 28 cm 45 cm 2 b Rectangle 8 cm 7 cm 30 cm 56 cm 2 c Square 7 cm 28 cm 49 cm 2 d Square 8 km 32 km 64 km 2 5 Radius Diameter Circumference Area a 7 cm 14 cm 42 cm 147 cm 2 b 8 cm 16 cm 48 cm 192 cm 2 c 2.5 cm 5 cm 15 cm cm 2 d 11 cm 22 cm 66 cm 363 cm 2 6 Base Height Area a 10 cm 7 cm 35 cm 2 b 9 cm 3 cm 13.5 cm 2 c 6 cm 4 cm 12 cm 2 d 12 m 9 m 54 m 2 7 Shape Length Breadth Height Base area Volume a Cuboid 8 cm 3 cm 11 cm 264 cm 3 b Cuboid 2 cm 6 cm 60 cm 3 c Cube 4 m 64 m 3 d Prism 5.5 cm 6 cm 2 33 cm 3 e Prism 28 cm 3 cm 2 84 cm 3 f Prism 7.5 m 4 m 2 30 m 3 8 a = 115, b = 65, c = 65, d = 65, e = 115, f = 65, g = 115, h = 119, i = 61, j = 119, k = 61, l = 119, m = 40, n = 58, p = 63, q = 54 Review test 6 1 a i kg ii tonnes b i 300 ii faces, 8 edges, 5 vertices 3 Either 22 and 136 or 79 and Section 1: Additional material

14 4 Shape Length Breadth Perimeter Area a Rectangle 2 km 11 km 26 km 22 km 2 b Rectangle 3 cm 12 cm 30 cm 36 cm 2 c Square 25 mm 100 mm 625 mm 2 d Square 8.5 m 34 m m 2 5 Radius Diameter Circumference Area a 4.5 cm 9 cm 27 cm cm 2 b 5 cm 10 cm 30 cm 75 cm 2 c 14 m 28 m 84 m 588 m 2 d 1 m 2 m 6 m 3 m 2 6 Base Height Area a 11 cm 8 cm 44 cm 2 b 5 cm 15 cm 37.5 cm 2 c 20 m 2 m 20 m 2 d 4 cm 11 cm 22 cm 2 7 Shape Length Breadth Height Base area Volume a Cuboid 7 cm 4 cm 10 cm 280 cm 3 b Cuboid 3 cm 6 cm 3 cm 54 cm 3 c Cube 6 cm 216 cm 3 d Prism 8 cm 3.5 cm 2 28 cm 3 e Prism 8 cm 12.5 cm cm 3 f Prism 2 m 13 m 2 26 m 3 8 a = 136, b = 44 ; c = 34, d = 137, e = 43, f = 34, g = 110, h = 52, i = 18, j = 110, k = 52, l = 110, m = 106, n = p = 37 Review test 7 1 a May b 50 cm c January, October, November, December d May, April, June, July e 195 cm 2 a 13 b 10 c 7 d 4 3 a 6, 8, 9 b 6, 6, 5.3 c 6, 7, 7 d 6, 6, 7 Review test 8 1 a 8 b 2 c Yes (the marks 4 and 6 have a frequency of 9) d 16 e 35 2 a 10 b 6 c 6 d 11 3 a 4, 5, 5.5 b 11, 10, 9 c 0, 4.5, 6 d 5, 3.5, 3.5 Section 1: Additional material 15

15 Enrichment chapter Calculators and tables Objectives By the end of the chapter, you will be able to: use a simple four-function calculator to perform everyday calculations use the memory and square-root functions on a calculator use tables to find squares and square roots of numbers. Teaching and learning materials Teacher: Calculators (tables are provided in this chapter and at the end of the book). Students: Calculator (tables are provided). This enrichment chapter provides help and guidance in using calculators and tables. Its aim is to: develop efficient and effective use of calculating aids ensure that students become skillful in using calculators so that they know how, when and when not to use them. Calculator skills are not part of the national curriculum and their use is not currently allowed in JSCE examinations. However, they can assist studies, for example, by helping to check whether answers are correct or not. All JSS leavers should be able to use calculators sensibly, whether as an aid to further study or as a desirable life skill. E 1 Know your calculator There are various kinds of calculators: hand-held calculators (such as that in Figure E1) desk calculators used in homes, shops and offices electronic calculators that you will find on all computers and on most mobile phones. Most calculators operate in much the same way. But you will find small differences. So learn about your calculator. Find out what it can do. Figure E1 shows the main parts of a simple fourfunction calculator with percentage and square-root keys as well as a memory. Other calculators, such as scientific calculators and programmable calculators, have many more keys and functions. memory keys MC = memory clear MR = memory recall M = take from memory M+ = add to memory percentage key square root key Figure E1 clear last entry all clear number keys and decimal point equals key solar power cell (if fitted) display arithmetical function keys All calculators, however, have the basic functions shown in Figure E1. And these are the functions you will use most often. Power Most calculators get their power from a solar cell. This powers the calculator as long as light is available (daylight, electric bulb or even candlelight). Display The display shows the answers. The digits in the display are usually made of small line segments as shown in Figure E1. Keyboard The keyboard has four main sets of keys or buttons: 1 Number keys Press,,,,,,,,, and the decimal point key (usually shown as a dot ) to enter numbers into the calculator. 16 Section 1: Additional material

16 2 Basic calculation keys Press,,,,, and to do mathematical operations on the numbers you have entered, and to display answers. 3 Clearing keys The key clears the last number you entered. Press if you enter a wrong number by mistake. On some calculators, you may see instead of. The key clears the whole calculation that you are working on. Use this key if you want to start from the beginning again. Often the key is linked to the calculator s on key and appears as or just as, as in Figure E1. Press before starting any calculation. This gives 0 on the display. 4 Memory keys a Memory plus key Press to store the displayed number in the memory of the calculator. If there is any previous number in the memory, it adds the displayed number to it. b Memory minus key Press to subtract the displayed number from the number in the memory. The result of adding or subtracting the number will be the new number in the memory. c Memory recall key If there is a number in the memory, the calculator usually shows a small M in one corner of the display. Press to display the number in the memory. d Memory clear key Press to clear the number stored in the memory. Key sequence Guess Result Notes Exercise Ea (Class activity) Copy and complete Table E1. For each key sequence: a First guess the outcome. b Then use your calculator to get a result. c If anything unexpected happens, make a note in the right-hand column. Table E1 Section 1: Additional material 17

17 Display on your calculator: a the highest possible number b the lowest positive number. To find what a snail lives in: a Calculate b Turn your calculator upside down and read the display. To find out what plants grow in: a Calculate b Turn your calculator upside down and read the display. a Use your calculator to complete Table E2. Table E2 Powers of 7 Value b Look at the final digits of the values displayed in Table E2. Is there a pattern? If so, what is it? c Is there a recognisable pattern in the final two digits? d Try the above with a different starting number, For example, 3, 6, 11 or 13. Are there any patterns? 100 up is a game for one person. To start: Enter any 2-digit prime number into your calculator. Aim: To get the calculator to display a number in the form 100. *****, where * may be any digit. Rule: You must multiply the number shown in the calculator display by any number of your choice. Scoring: Record each multiplication as a trial. Try to achieve your aim in as few trials as possible, that is, you should keep your score as low as possible. Here is a sample game: Display Press keys Trial no. (score) Start Finish The score for this game is 5. Starting with 29, can you do better? Try to beat 5, then play some games starting with other prime numbers. E 2 The operations,,, Addition and subtraction Use the and keys to add and subtract numbers and to display the result. Example 1 Calculate Keystrokes: Display: (answer) Rough check: = It is a good idea to start any new calculation by pressing the key. This clears any previous calculation or data that the calculator may contain. When using a calculator, it is possible to make keying-in mistakes. So make a habit of doing a rough check. Do the check mentally. Example 2 Calculate Keystrokes: Display: (answer) Rough check: = 20 In Example 2, notice the value 35 in the display. This is an intermediate result (89 54 = 35). It appears when the second operation is entered. 18 Section 1: Additional material

18 Example 3 Calculate Keystrokes: Display: (answer) Rough check: = 10 Notice that calculators give a negative outcome when subtracting a larger number from a smaller number. Thus 9 16 gives 7 as an intermediate result during the above calculation. Exercise Eb Do these calculations on your calculator. Write down what appears in the display when you press each key and underline the final answer. a b 9 5 c d e f g h i j Look at these calculations. Six of them are incorrect. i = 14 ii = 912 iii = 10 iv = 53 v = vi = 13.4 vii = 82 viii = 12 a Decide which ones appear to be incorrect. b Use your calculator to correct them. Look at these calculations before doing them. What kind of answer do you expect? Do the calculations. a 2 7 b 5 15 c d e 8 75 f g h An athlete buys some clothes. Figure E2 shows the bill. Check that the shop assistant has added up everything correctly. Sports World Tracksuit N Sweatshirt N Shorts N Running shoes N Total N Figure E2 What will the items in Figure E3 cost in total? Bottle orange juice N 429 Jar coffee N 582 Packet of tea N 195 Sugar N 180 Margarine N 299 Jar of peanut butter N 265 Oranges N 389 Meat N 850 Shampoo N 405 Bottle of apple juice N 376 Chicken N Figure E3 Multiplication and division Use the and keys to multiply and divide numbers and to display the result. Given a multiplication (or a division) in the form a b (or a b), press the keys (or ) then either,,, or will display the answer. Example 4 Calculate Keystrokes: Display: (answer) Rough check: = Section 1: Additional material 19

19 Example 5 Calculate Keystrokes: Display: (answer) Rough check: = 350 In Example 5, notice that = 29. appears in the display at an intermediate stage of the calculation. Calculators have a limited number of spaces in the display (usually eight spaces). Therefore, there is a limit to the size of answer they can display. Try the calculation on a calculator. Figure E4 shows what happens on some eight-digit calculators. a b c Figure E4 Note that = (requiring 10 digits). The calculator displays in a and b show the digits but are unable to display the full number properly. So they print a small (error) to warn the user. The calculator displays in c and d group the display into two parts: and. This is short for Note the nine zeros in the second number. They correspond to the. This type of calculator is called a scientific calculator. Scientific calculators can cope with very large numbers because they give the outcome in standard form: = d Example 6 How many seconds are there in a 31-day month? Number of seconds = = (calculator) Exercise Ec Do these calculations on your calculator. Write down what appears in the display as you press each key. Underline your final answer. a b c d 462 e f g h Do these calculations. Give each answer: i As displayed on the calculator. ii Rounded off to 2 decimal places. a b c d e f g h Look at these calculations. Six of them are incorrect. i 5 9 = 30 ii = iii = iv = 45 v = 33 vi = vii = 4 viii = 5 a Decide which ones appear to be incorrect. b Use your calculator to correct them. Multiply by on your calculator. What is displayed? 5 a Calculate b Calculate How many seconds are there in a 365-day year? 7 a Write your age to the nearest year. b Calculate how many days you have lived (assume 365 days in a year). c Calculate how many hours you have lived. d Calculate how many minutes you have lived. e Is it possible for your calculator to calculate the number of seconds you have lived? 20 Section 1: Additional material

20 A health inspector gets a salary of N per month. How much does this represent: a per year? b per day? Assume a 30-day month. Give answers to the nearest naira. Twelve members of a club hired a bus to go to a match. If the bus company charged N31 500, how much did each member have to pay? 0 An aeroplane travels 550 km in 1 h. a How many km does it travel in 1 min? b How many metres does it travel in 1 min? c How many metres does it travel in 1 s? E 3 Further calculator techniques Mixed operations Look again at Exercise Ea, Question 1, part g. In some cases, calculators appear to give two answers to the same problem. According to the rules of precedence in arithmetic: = = 17 And = = 17. However, for the first expression, the calculator gives this result: Keystrokes: Display: (answer) For the second expression, the calculator gives this result: Keystrokes: Display: (answer) This is because the calculator follows the operations in the order that it receives them. Example 7 Calculate There are no brackets, but always do multiplication before addition. Rearrange the numbers: = = (8 52) + 34 = 450 (calculator) Brackets Example 8 Calculate 2.3 ( ). The brackets tell us to do subtraction first. Rearrange the numbers, so that the subtraction comes before the multiplication: 2.3 ( ) = ( ) 2.3 = (calculator) In the above examples, it is possible to turn the calculation round because in general, a b = b a. However, with division this is not possible. Read Example 9 carefully. Example 9 Calculate 84 (37 23). Do the subtraction in the brackets first. Store the outcome in memory. Recall it when needed. The sequence of working is: 6 Check the above sequence on your own calculator and note the changes in display. It is important to enter numbers and operations in an order that will allow the calculator to give the correct results. This means doing calculations in brackets first and storing them if necessary. Then, do multiplications and divisions before additions and subtractions. Exercise Ed In six of these cases, calculators will give incorrect results if you enter the numbers and operations in the given order. In those cases rearrange the numbers so that calculators will give the correct results. a b c d e 18 (17 15) f (19 + 9) 7 g 487 (6 + 3) h 100 (31 14) Do these calculations, rearranging the order where necessary. a b Section 1: Additional material 21

21 c d e 49 (19 3) f ( ) 13 g h 8.8 ( ) All of these calculations require part of the calculation to be stored (either on paper or in the calculator s memory). a b 216 (25 7) c 708 ( ) d e 46.7 ( ) f Squares and square roots Squares Many calculators have a squares key, usually shown as. However, if your calculator does not have a squares key, then it is usually possible to get squares by pressing the key twice. For example, try answer of 81. Square-roots Simply enter a number and use the For example gives Exercise Ee ; this should give an key. Use your calculator to find the value of these numbers. Round your numbers to 3 s.f. a 67 2 b 76 2 c d e f Use your calculator to find the value of these numbers. Round your answers to 3 s.f. a 3 b 5 c 760 d 6.82 e 24.9 f Calculator fun a Find Turn the calculator upside down. Is the display on your hand or on your foot? b Find Turn the calculator upside down. Is the display male or female? c Make up more like these. E 4 Using tables Table of squares If calculators are not available, then use tables to do calculations. See page 231 of the Student s Book. Use the table on page 231 of the Student s Book to convert 2-digit numbers to the squares of those numbers. Example 10 Use the table of squares to find is the first digit. Look for 6 in the left-hand column of the squares table. 7 is the second digit. Look for the column headed.7 of the squares table. Find the number across from 6 and under.7. See Figure E5. 6 Figure E5 The number is Thus, = Example 11 Use the table of squares to find: a 19 2 b a 19 = = (1.9 10) 2 = From the table: = 3.61 Thus, 19 2 = = 361 b 190 = = ( ) 2 = = = Section 1: Additional material

22 In Example 11, notice that: = = = When multiplying a number by increasing powers of 10, you multiply its square by increasing powers of 100. Example 12 Use the table of squares to find: a b a 0.8 = (0.8) 2 = ( ) 2 = From the table: 8 2 = Thus, 0.82 = = 0.64 b 0.25 = (0.25) 2 = ( ) 2 = From the table: = 6.25 Thus, = = Examples 14, 15 and 16 on pages 24 and 25 explain how to use the square root table on pages 232 and 233 of the Student s Book. Notice this about the square root table. 1 There are no decimal points. Use inspection to place them correctly. 2 There are two sets of digits for each number. Exercise Ef Use the table of squares (on page 231 of the Student s Book) for this exercise. Find the value of these numbers: a b c d e f g h i j k l Find the value of these numbers: a 18 2 b 31 2 c 32 2 d 15 2 e 29 2 f 44 2 g 70 2 h 20 2 i 62 2 j 9 2 k 81 2 l 99 2 Round off these numbers to 2 s.f. Then find the approximate square of each number. a b c d e f g h i Find the value of these numbers: a b c d e f Look at this pattern: = 2.25 = = 6.25 = = = Does the pattern continue in the same way? 6 Find the squares of: a 12 and 21 b 13 and 31 c What do you notice? A square has a side of length 4.3 cm. Calculate its area in: a cm 2 b m 2. A square plot has a side of length 240 m. Calculate its area in: a m 2 b hectares. (1 hectare = m 2 ) A square has a perimeter of 30 cm. Find: a the length of one of its sides b its area. Find out which of these statements are true: a 65 2 = b 65 2 = c 65 2 = d 65 2 = e 65 2 = f 65 2 = Find the values of these numbers: a b c d e f g h i j k l Approximate square roots A perfect square is a whole number with an exact square root. 5 2 = 5 5 = 25 Thus, 25 = 5. We can find the approximate square root of a number by knowing the perfect squares immediately before and after it. Section 1: Additional material 23

23 Example 13 Between which whole numbers does the square root of these numbers lie? a 3.6 b 9.4 c 40 d 78 a lies between 1 and 4. Thus, 3.6 lies between 1 and 4. That is, 3.6 lies between 1 and 2. b lies between 9 and 16 Thus, 9.4 lies between 9 and 16. That is, 9.4 lies between 3 and 4. c lies between 36 and 49 Thus, 40 lies between 36 and 49. That is, 40 lies between 6 and 7. d lies between 64 and 81. Thus, 78 lies between 64 and 81. That is, 78 lies between 8 and 9. Square root tables Use the tables on pages 232 and 233 to find the square roots of 2-digit numbers. Examples 14 and 15 explain how to put in the decimal point and how to use approximations to choose the correct set of digits. Example 14 Use the square root table to find: a 5.7 b R57 a lies between 4 and 9. Thus, 5.7 lies between 4 and 9. That is, 5.7 lies between 2 and 3. Thus, 5.7 = 2 point something. In the square root table, look for 5 in the lefthand column. The next digit is 7. Look for the column headed 7. Find the digits across from 5 and under 7. See Figure E6. 5 Figure E6 This gives As the required result begins with a 2, the required digits are 239. You can ignore the 755. Thus, 5.7 = b lies between 49 and 64. Thus, 57 lies between 49 and 57. That is, 57 lies between 7 and 8. Thus, 57 = 7 point something. From the table: The required digits are 755. You can ignore the 239. Thus, 57 = 7.55 Notice that the square root table gives values rounded to 3 significant figures. Thus: 5.7 = 2.39 to 3 s.f. On a calculator 5.7 = However, 3 significant figures are accurate enough for most purposes. Example 15 Use the square root table to find: a 940 b a 940 = = = is just over 9. Thus, 9.4 is just over 3. Thus, 9.4 = 3 point something. From the table: 9.4 = 3.07 Thus, 940 = = 30.7 to 3 s.f. b We can only use the square root table for 2-digit numbers. So, first round off to 2 s.f = to 2 s.f. Thus, Section 1: Additional material

24 40 is between 36 and 49. Thus, 40 is between 6 and 7. That is, 40 = 6 point something. From the table: 40 = 6.32 Thus, Notice again that the final answers are not exact. For example, = (calculator). Example 16 Use the square root table to find: a 0.28 b 0.47 a = = = 28 R10 2 = lies between 25 and 36. Thus, 28 lies between 5 and 6. From the table: 28 = 5.29 Thus 0.28 = = 5.29 = b 0.47 = = = = lies between 36 and 49. Thus, 47 lies between 6 and 7. From the table: 47 = 6.86 Thus 0.47 = = 6.86 = Exercise Eg Use the tables at the back of the Student s Book for this exercise. Between which whole numbers does the square root of these numbers lie? a 7.6 b 9.6 c 17 d 27 e 34 f 30 g 48 h 60 i 75 j 69 k 54 l 83 Find the square roots of these numbers: a 9 b 90 c 2.8 d 28 e 4.7 f 47 g 36 h 3.6 i 25 j 2.5 k 6.3 l 63 Find the square roots of these numbers: a 7 b 70 c 700 d e 2.9 f 29 g 290 h i j 250 k l 38 m 380 n o p 10 q 100 r s 2 t 430 u 500 v w x 960 Round off these numbers to 2 s.f. Then find their approximate square roots. a 9.28 b 78.3 c 463 d 8.45 e 61.3 f 613 g 59.4 h 5.86 i j k l Is 10 a good approximation for π? 6 A square has an area of about 45 cm 2. a Find the length of one of its sides. b Hence find its perimeter to the nearest cm. a Use the square root table to find m, if m = 41. b Using the value of m, use the squares table to find the value of m 2. c What do you notice? Find the square roots of these numbers: a 0.21 b 0.34 c 0.43 d 0.62 e 0.55 f 0.73 g 0.86 h 0.92 i 0.69 j 0.78 k 0.59 l 0.98 Summary In many everyday situations, people use calculators to do calculations. Where calculators are not available, it is possible to use tables to make calculations easier. Whether using calculators or tables, it is very useful to be able to estimate the size of the likely outcome of a calculation and where the decimal point is likely to be. This helps to detect errors (either from pressing the wrong key or from using a table incorrectly). Section 1: Additional material 25