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Supplementary Materials for Mathematics Audits (Primary Teacher Training) Euros Davies Jan Morgan The aim of the materials is to act as revision guides for trainees who are undertaking a primary teacher training degree programme. It is not intended that the trainees will necessarily work through every sheet, but choose those most appropriate to their needs. Worked examples and exercises are given, and answers can be found at the end of the pack of materials. The Index is cross- referenced to enable students to find the most appropriate material for them. Individual tutors can use the materials as revision guides with a group of students, or the materials can be worked through by individual students. It should be stressed that the sheets are for revision purposes – individual students may need further teaching or assistance if the topic is a new one for them. It may be necessary for the students to be referred to further examples or exercises for extension purposes. Further reading may also be needed. Other useful resources may be: www.bbc.co.uk GCSE Bitesize revision guide Letts Revision Guide CGP Revision Guide This research and supplementary materials pack have been made possible by financial support from ESCalate. Contents 1. Averages 2. Brackets 3. Circles 4. Co-ordinates 5. Disproving a Hypothesis 6. Estimating results of calculations 7. Factorising algebraic expressions 8. Factors 9. Forming algebraic expressions 10. Fractions 11. Gradient 12. Grouped Data 13. Inequalities 14. Interpretation of a Graph 15. Nth terms 16. Order of arithmetic operations 17. Order of size / Rational – irrational numbers 18. Probability 19. Pythagoras’ Theorem 20. Ratio 21. Rounding 22. Straight lines 23. Surface area of 3D shapes 24. Tree diagrams Topic: Averages There are three types of average – mean, median, mode. Consider the set of numbers: 7, 3, 4, 5, 4, 9, 2, 4, 6, 3 Mean – add up all the data and divide by the number of items = 47/10 = 4.7 Median – put the data in order and pick the middle one: 2, 3, 3, 4, 4, 4, 5, 6, 7, 9 (if there are two middle ones, add them and halve) So median = 4 Mode – the most common data item = 4 Each has its advantages and disadvantages. Median and mode are usually one of the data items; mean usually isn’t. Median and mode are unaffected by extreme values. Mean is best for further calculation, but adversely affected by extreme values. Sometimes the data is given in table form – a frequency distribution: Here, the number 3 occurs 4 times in the set, for example. So totalling the 3s is easier by calculating 3x4 = 12. These partial products are then added to give the grand total. So Mean = ∑fx = 28 = 2.333 ∑f 12 Median = 2, Mode = 2, x 1 2 3 4 f 2 5 4 1 12 fx 2 10 12 4 28 (there are 12 items, so middle one is between 6 th and 7th) (1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4) the most common, as there are five of them. Exercises 1. The hourly wages of employees in a firm are £15, £12, £ 20, £8, £15, £25. Work out the average hourly rate (all three of them!) 2. If the employee earning £25 per hour has an increase to £100 per hour, what effect does this now have on the three averages? 3. The mean age of ten boys is 9.6 years and of 8 girls is 10.2 years. What is their combined average age? 4. Work out all averages for the following data: x f 3 2 5 4 6 5 7 2 NB. x = item of data; f = frequency; ∑ = sum of Topic: Brackets See also: Factorising Algebraic Expressions sheet Young children use these ideas when multiplying: 6 x 23 = 6 x (20 + 3) = 6 x 20 + 6 x 3 = 120 + 18 = 138 2a = a + a, so 2(y + z) = y + z + y + z = 2y + 2z 3(w – 2) = w – 2 + w – 2 + w – 2 = 3w – 6 This process is equivalent to multiplying each of the terms inside the bracket by the term outside: 6(c – 3) = 6c – 18 3(p + q - 2r) = 3p + 3q – 6r a(b + c) = ab + ac x(x + y) = x2 + xy Using a grid: y 2 2y z 2z x x x2 y xy If we have two sets of brackets, the approach is the same: (x + 1)(x + 2) = (x + 1)x + (x + 1)2 x = x2 + x + 2x + 2 x x2 = x2 + 3x + 2 2 2x Exercises Multiply out: 2(3 + 2x) = 4(a + 2b + 3c) = x(x – 1) = a(b + c – a) = 3y(x + z) = x(x2 – ab) = 2w(w – 3) = 4(2x –y + 3z) = (x + 3)(x + 4) = (x – 1)(x + 2) = 1 x 2 (p + 2)(p –2) = (m +2)(2m + 1) = Topic: The Circle People often mix up the circumference (or perimeter) and the area of a circle. The circumference is a distance – all the way round the circle - 2πr or πd where r = radius, d = diameter. It is measured in ‘length units’ (centimetres or metres, for example) The area is the space enclosed by the circumference; area = πr 2 It is measured in ‘area units’ (cm2 or m2, for example) Example: r r A semicircle has diameter 20 cm. What is its perimeter? What is its area? 20 cm. It has half the circumference of a full circle For the curved part = ½ x πd = ½ x 3.14159 x 20 = 31.4159 cm. Remember to add the diameter here! So total perimeter = 31.4159 + 20 = 51.4159 cm. Area of semicircle = half area of whole circle = ½πr2 = ½ x 3.14159 x 102 = 157.08 cm2 (NB. d = 20, so r = 10) Exercise. 1. Calculate the perimeter and area of the given shape (Semicircle and rectangle) 6 14 2. A bicycle wheel has diameter 80 cm. How far would I cycle if it made 40 revolutions? d 3. A washer looks like two concentric circles of diameters 10 and 20 mm. respectively. What is the area of the washer ring? Topic: Coordinates Vocab: Axis, axes, coordinate, origin See also: Interpretation of a Graph sheet When position is to be identified on a piece of paper, it has to be referenced to a starting point (origin). On a straight line, we just need to know the distance along the line from the origin, O. 4eg. A is 3 units from O B is –2 units from O B -2 -3 -1 O 1 2 A 3 y In two dimensions, we need to use two reference lines – called axes. These are usually labelled the x-axis and y-axis. • P(4,3) Q The point P has coordinates (4,3). This means It is 4 units in the x-direction and 3 units in the y-direction. Points are marked with a cross. • 0 • R x The ‘horizontal’ coordinate is given first. Q is (-2,1); R is (0, -1) Axes do not have to be labelled x and y. A car is travelling along a straight road and is 1km away after 5 mins and 4km away after 10 mins. We plot Time (t) horizontally and Distance (d) vertically: d 4 1 + (10, 4) + (5, 1) 5 10 t Geographers use this system for map references. (613206: 613 , 206 ; walk in the house, then go up the stairs!) Exercises 1. The following coordinates are three corners of a rectangle. Plot them and find the coordinates of the remaining corner: (3, 4), (5, 4) , (5, -1) . 2. The following points are reflected in the y-axis. Plot them and give the coordinates of their images: (2, 3) , (1, 1) , (3, 0) , (0, 2) , (-2, -1) 3. Which point is furthest from the origin: (3, 5) (6, 0) (-2, 5.5) (-1, -6) ? Topic: Disproving a hypothesis If a given statement has to be disproved, all you have to do is to find one example to contradict the statement. Eg. Hypothesis: all even numbers are divisible by 4 Try 14. This is even. 14 = 3 rem 2; 4 So 14 is not divisible by 4. Thus statement is incorrect. Eg. Hypothesis: 1 + 1 = 2 m n m+n Try m =2 and n = 4; statement give s 1 + 1 = 2 = 1 2 4 6 3 But we know that ½+ ¼ = ¾ So hypothesis is incorrect. You cannot use this approach to prove a hypothesis is correct. Eg. Hypothesis: if m is a whole number, 2m is always even. Try m = 3: m = 4: 2m = 2x3 = 6; even 2m = 2x4 = 8; even This only ‘proves’ the hypothesis is true when m = 3 and m = 4. It confirms nothing else! Exercise. 1. Show that the following statements are incorrect: - the sum of any three consecutive whole numbers is always odd - the square of any number is larger than the original number - (a + b)2 = a2 + b2 for any values of a and b 2. The cube of any odd number is odd, because 3 3 = 27, 53 = 125, 73 = 343. What conclusion can we draw from this? Topic: Estimating results of calculations The ability to obtain a rough estimate to a calculation is quite useful. It gives you an approximate value to the answer which is a useful check before doing the full calculation. Eg. 31.6 x 6.9 this is roughly 30 x 7 = 210 (Actual answer = 218.04) 19.8 7.6 roughly 20 = 2.5 8 (Actual answer = 2.6052631) If you had used a calculator to answer 31.6 x 6.9 and got the answer 2180.4, your estimate (210) would show you something was wrong – possibly an omitted decimal point. This can be used in another way too. Suppose we know 32 x 74 = 2368. what is 3.2 x 74? Using this information, we can work related calculations: 3.2 is roughly 3; 74 is roughly 75, giving an approximated result 3 x 75 = 225. So 3.2 x 74 = 236.8 What is 0.32 x 740? This is roughly 1/3 x 750 = 250. So 0.32 x 740 = 236.8 What is 32 x 0.74? This is roughly 32 x ¾ = 24 So 32 x 0.74 = 23.68 What is 32 x 75? Roughly 30 x 80 = 2400 32 x 75 = 32 x (74 + 1) = 32 x 74 + 32 x 1 = 2368 + 32 = 2400 Exercise. 1. Give approximate answers: 5.1 x 19.4 = 38 x 11.3 = 2. Given 83 x 15 = 1245, underline the correct answer: 8.3 x 15 = 12.45 124.5 8.3 x 150 = 12.45 = 1.5 85 x 15 = 12.45 0.83 1247 1245 124.5 1245 8.3 83 1275 1285 Topic: Factorising algebraic expressions See also: Brackets sheet You can think of this as just the ‘opposite process’ to multiplying out brackets. 2a + 2b has a common factor of 2, so 2a + 2b = 2(a + b) Perimeter of a rectangle of dimensions a cm. by b cm. is a + b + a + b = 2a + 2b cm. Both terms contain a ’2’, so it can be regarded as a common factor and written 2(a + b). a We can interpret this as: b the perimeter is (2a + 2b), twice the length plus twice the width or the perimeter is 2(a + b), length plus width all doubled. Similarly, 4xy + 4xw = 4(xy + xw), a 4 is a common factor. But x is also a common factor, so the full factorisation is 4x(y + w) xy + xz = x(y + z), as x is a common factor x3 + x2z = xxx + xxz = x2(x + z), as x2 is a common factor 3x + 6 = 3(x + 2), as 3 is a common factor 4x + 6xy = 2x(2 + 3y), as both 2 and x are common factors 2x + 4y – 6z = 2(x + 2y - 3z) (You can check that each factorisation is correct by multiplying out the brackets: eg.2(x + 2y – 3z) = 2x + 4y - 6z) Exercises Factorise: 6x + 3y = 8 – 2x = xy + 10x2 = 6abc – 8bcd = x3 + x 2 = 6s3 – s = 3xy + 6y2 –9y = Topic: Factors Vocab: Factor, multiple, prime, composite number, proper factor A factor is a whole number that divides exactly into another. So we can say that 6 is a factor of 18; or 18 is a multiple of 6. Factors usually come in pairs eg. as 6 is a factor of 18, so is 3, as 18/6 = 3. So factors of 18 are: 1 and 18; 2 and 9; 3 and 6 ie. {1,2,3,6,9,18} Factors of 30 are 1 & 30, 2 & 15, 3 & 10, 5 & 6 ie. {1,2,3,5,6,10,15,30} Note1: 1, 2, 3, 6 are common factors of both 18 and 30. Note2: a number is a factor of itself: 6 is a factor of 6 Note3: the proper factors of a number do not include the number itself: proper factors of 6 are 1, 2, 3 A prime number has exactly two factors eg. 7 is prime as its only factors are 1 & 7. Composite numbers are non primes: 18 is composite; 19 is prime. All numbers can be written as an unique product of prime numbers 18 = 2x3x3 ; 30 = 2x3x5 ; 72 = 2x2x2x3x3 = 23 x 32 Exercises 1. Put the following numbers into one of two sets – prime or composite numbers: 14, 27, 7, 2, 11, 49, 37, 12 2. Find the factors of the following numbers: 21, 31, 15. Which are prime? 3. Write the following as products of primes: 20 = 32 = 42 = 3. The number 6 is called a perfect number, as it is equal to the sum of its proper factors, ( 1 + 2 + 3 = 6). Show 28 is also perfect. Topic: Forming algebraic expressions m + m = 2m; 2m means ‘twice m’ (or ‘m multiplied by 2’) n + 2n = 3n; 3n means ‘n multiplied by 3’ y2 is read as ‘y squared’, meaning ‘y multiplied by itself’. Algebraic expressions can be created one step at a time: - a number n is doubled and then four is added: n is doubled then add 4 2n 2n + 4 - a number m is cubed, then the result is doubled and five subtracted; then this result is halved: m cubed m3 then double 2m3 subtract 5 2m3 – 5 halved ½(2m3 – 5) - to convert degrees Fahrenheit to Centigrade, subtract 32; then multiply by 5 and divide by 9: degrees Fahrenheit F subtract 32 F – 32 multiply by 5 5(F – 32) divide by 9 5(F – 32)/9 = C Exercise. 1. A boy is x years old. His father is five times older. So father is _____ years old. 2. A boy is m years old. In two years’ time, his father will be six times as old as the boy will be. So the father is _________ years old now. 3. N is a whole, positive, even number. What is the next even number? 4. Write down an expression for the mean (average) of numbers p and q. Topic: Fractions Equivalent fractions If you multiply the top (numerator) and bottom (denominator) of a fraction by the same number, the resulting fraction is equal to the original – they are equivalent to one another; eg. 1= 1x3 = 3; so 1 = 3 2 2x3 6 2 6 2 = 2X4 =8; 5 5 x 4 20 so 2 = 8 5 20 Adding fractions Fractions can only be added if their denominators are the same Eg. 1/5 + 2/5 = 3/5 (one fifth and two fifths give three fifths) So the first stage is to make sure that both fractions have the same denominator – we need to create equivalent fractions. Eg. 1/5 + 2/3 ; we’ll make the denominators 15 So 1 = 1x3 = 3 ; 2 = 2 x 5 = 10 5 1x3 15 3 3x5 15 Hence, 1 + 2 = 3 + 10 = 13 5 3 15 15 15 Subtraction – use the same approach as for addition Multiplication To multiply two fractions together: Either – multiply the numerators, then the denominators, then cancel common factors Or – cancel common factors top and bottom, then multiply numerators, then denominators Example 1: 8 x 3 = 24 = 2 15 4 60 5 or: (Divide by common factor 12) 2 8 x 31 = 2 41 5 5 15 Example 2: mixed fractions have to be converted to top heavy ones first 21/10 x 12/3 = 721 x 51 = 7 = 31/2 31 2 210 Exercise. 1. Equivalent fractions: write 2/9 in three different ways. 4 2. Cancel down to lowest terms: 3. Evaluate: 3 /10 + 5/8 ; /10 , 28 /42 , 111 /141 22/5 x 11/9 Addition of fractions We need the same denominators (bottom part) before fractions can be added. 1 Eg1. /3 + 1/5 look at the lowest common multiple of 3 and 5 (ie. smallest number both 3 and 5 divide into) LCM (3,5) is 15. Convert both fractions into fifteenths: So 1/3 + 1/5 = 5/15 + 3/15 = Eg2. 3 /4 + 1/6 Converting 8 1 x 5 = 5 and 1 x 3 = 3 3 5 15 5 3 15 /15 LCM (4,6) = 12; (could use 24, but 12 is lowest ) 3x3 = 9 4 3 12 1x2 = 2 6 2 12 So 3/4 + 1/6 = 9/12 + 2/12 = 11/12 [If we’d used 24 as denominator, result would be the same, but a common factor would have to be cancelled out at the end: Converting: 3 x 6 = 18 4 6 24 So 3/4 + 1/6 = 18/24 + 4/24 = Eg3. 5 /12 + 1/6 1x4 = 4 6 4 24 22 /24 = Here LCM is 12, etc. Multiplication of fractions 11 /12 ] Here common factors between numerator and denominator (top and bottom) can be cancelled out – the direction of the cancelling lines indicating which pairs of numbers have been affected: 2 6 x 51 = 2 5 25 93 15 So factor of 3 cancelled from 6 & 9 factor of 5 cancelled from 5 & 25 Topic: Gradient Vocab: Gradient, slope, coordinates, axis See also: Straight Lines sheet The gradient or slope of a straight line is used to determine how steep the line is. It is possible to have a positive, a negative or a zero gradient. Definition: The gradient of a line indicates how fast a quantity is changing. A working definition is: Gradient = increase vertically (parallel to y-axis) increase horizontally (parallel to x-axis) Process: Pick two points on the line. Complete a right angled triangle with short sides parallel to the two axes and the hypotenuse as part of the line. Work out the coordinates of both points. Find vertical and horizontal differences. Calculate gradient. Example1: Let A(2,3) and B(6,5) As you move A B, both coordinates increase. y B A 2 4 Gradient = 2/4 = ½ x Example2: Let A(2,4) and B(6,3). As A B, one coordinate increases and one decreases, so a negative slope. y A -1 B 4 Gradient = -1/4 x Example3: A horizontal line has no vertical increase, so gradient = 0. Example4: Plotting Distance vertically and Time horizontally would give Gradient = increase in distance = speed. increase in time So the greater the slope, the faster the speed. A Exercises 1. y B Work out the gradient of these lines: (2,2) 1 x A y 2. Which of these lines has: a negative slope a zero gradient ? B x C 3. On the distance-time graph shown, how can you interpret the gradients indicated by line OA, line AB? d A B t O Topic: Grouped data See also: Averages sheet Data is often grouped into classes to give a frequency distribution, as it is easier for the reader to understand when there is a large amount of data. If we had scores: 3,4,9,5,4,3,1,7,8,11,19,4,12,14,15,6,9,9,11,4,3,14,18,16,18 , then, with little effort, we could work out the actual mean, median, and mode (9.1, 9, 4 resp). If we had over 100 items though, it might be quite a long task. So we group the data into classes and indicate how many items fall into each class (the frequency). Class 0 - <5 5 - <10 10 - <15 15 - <20 f 8 7 5 5 ∑f = 25 mid value(x) fx 2.5 20 7.5 52.5 12.5 62.5 17.5 87.5 ∑fx = 222.5 So, although we can see from the table that there are eight scores between 0 and under 5, we cannot actually say what these scores are unless the original data is still available. If it isn’t, then we take the mid-point of the class as a representative value for the whole class – so we have 8 scores of 2.5, giving 20 in total, when the actual total is 26. Obviously, this can lead to a level of inaccuracy, but it’s the best we can do. Mean = Total (approx) sum ∑fx = 222.5 = 8.9 Number of scores ∑f 25 Mode = we can only say that the ‘modal class’ is 0 - <5. Median – we need to create an ogive (cumulative frequency graph), so we need two more columns in our table. Class f upper bound cum f 0 - <5 5 - <10 10 - <15 15 - <20 8 7 5 5 <5 <10 <15 <20 8 15 20 25 We can now draw the graph: Using the graph, we can work out the median – at the 50% mark reading off horizontal axis. We can also work out the 25% and 75% values; the difference between these gives the interquartile range. = 32% = 60% = 80% = 100% %cf 75 25 LQ M UQ Upper Bd Topic: Inequalities We use the symbols ‘=’ (equals, is equal to) and ‘ = ’ (not equal to) quite often, but there are four others that you need to understand too: > greater than, < less than, ≥ greater than or equals, ≤ less than or equals x > 3 means the value of x is greater than 3 (so x could be 3.1, 4, 11, 15.6, . . . ) º 3 x ≥ 3 means that x could also equal 3 itself • 3 Sometimes we want to limit x to a small range of values – say between 2 and 5 – so x > 2 and x < 5; this can be written as 2 < x < 5 2 5 Examples 1. 2 < x ≤ 5 means 2<x and x ≤ 5 , x is between 2 (not included) and 5 (included) 2. If x is an integer (whole positive or negative number) and -3 ≤ x < 2, Then x could be -3, -2, -1, 0, 1. 3. Suppose we know that x is in the set {-5, -4, -3, . . . ., 2, 3} and x < -3 or x > 1. Then x could be –5, -4, 2, 3. If we asked for values such that x < -3 and x > 1, that’s impossible, as both inequalities cannot be true at the same time. 4. The ‘opposite’ (or complement) of x > 3 is not x < 3. The numbers, which are not in the set of numbers greater than 3, must include all those less than 3 and 3 itself. So, if x > 3, then the complement is x ≤ 3. Topic: Interpretation of a graph See also: Coordinates Sheet speed Given - a graph of an object’s speed at certain times. At time t = 0, object is stationary (ie. speed = 0) A B OA: speed increases/ object accelerates for 2 mins. AB: speed constant for 2 mins. BC: object slows down (decelerates) until it stops O 2 4 after 7 mins. (NB. the graph does not tell us the object returns to its start point.) C 7 time Distance Given – a graph of distance (from Wrexham) against time. OA: car moves off from Wrexham for 1 min. AB: distance from Wrexham doesn’t change Stationary Or moving in a circle, centre Wrexham! BC: returns to Wrexham in 30 secs. A C O (BC is steeper than OA, so return speed is greater than outward speed.) Exercise 1 3 4 time h Graph represents man climbing ladder, plotting height (m.) versus time (secs). What could be happening between - A and B ? - C and D ? B C A B O D t Topic: Nth terms If a sequence of numbers changes in a regular way, it is often possible to create a general (or nth ) term for the whole sequence. If we know the form of this general term, we can use it to predict new terms of the sequence. Suppose the nth term is (4n + 3). By substituting values of n into this formula, we can generate the sequence itself: When n = 1, 4n + 3 = 4x1 +3 = 7 When n = 2, 4n + 3 = 4x2 + 3 = 11 When n = 3, 4n + 3 = 4x3 + 3 = 15 etc So the sequence begins 7, 11, 15, . . . . Suppose the sequence begins: 2 Write term number (n) underneath: 1 So the term value is double the term number. General (nth) term = 2n Sequence: 3 5 7 9 11 Term number (n): 1 2 3 4 5 The difference between successive terms is 2. So the general term is (2n + k); we need to work out the value of k. When n = 1, term value = 3 , so 2n + k = 2x1 + k = 3, giving k = 1. So general term is (2n + 1). Sequence: 2 5 8 11 14 17 Term number: 1 2 3 4 5 6 Difference is 3 each time. So general term is (3n + k). When n = 1, term value is 2, so 3n +k = 3x1 + k = 2, giving k = -1. So general term is (3n – 1). If we have a diagram to work with, rather than just a sequence, we can use a different approach. 4 2 6 3 8 4 10 5 Term No. 1 • • • 2 • • • • • 3 ..... • • • • • • • n Term value 3 5 7 ? Vertical dots Horizl dots 2 1 3 2 4 3 Exercise 1. Find the nth terms: 3, 6, 9, 12, . . . . . 6, 10, 14, 18, 22, . . . . 2½, 3¼, 4, 4¾, . . . . 2. Find number of matches in the n th picture: n+1 n General term = 2n + 1 Topic: Order of arithmetic operations In mathematics, there are certain operational conventions that need to be followed in order to avoid confusion. For example, does 3 + 4 x 5 give 35 or 23? The order in which operations should be performed is easily remembered with the mnemonic BODMAS Brackets Of Divide, Multiply (With two operations on the same level, Add, Subtract work from left to right through expression) Examples 3 + 4 x 5 = 3 + 20 = 23 (3 + 4) x 5 = 7 x 5 = 35 12 ÷ 3 x 4 = 4 x 4 = 16 1 /5 of 10 + 3 = 2 + 3 = 5 Exercise Evaluate: 14 + 6 ÷3 = 3 x (4 – 2) = 7+4x3= (4.6 + 7.1) x 2 – 1.3 = 5 + 3 –2 + 1 = 8 –2 + 1 = 6 + 1 = 7 10 ÷ 2 + 3 x 2 = 5 + 6 = 11 10 ÷ (2 + 3) x 2 = 10 ÷ 5 x 2 = 2 x 2 = 4 Topic: Order of size To compare numbers, it is best to convert them to decimals: 1 /5 = 0.2 ; 3/4 = 0.75 ; 1/8 = 0.125; so 0.125 < 0.2 < 0.75 or 1 /8 < 1/5 < 3/4 To convert a fraction into a decimal, divide numerator by denominator Eg. 0. 75 3 /4 = 4 )3.00 Exercise. 1. Put in numerical order: 2. Put in numerical order: 3 /8 0.3 1 5 /8 61 2 /100 /3 /3 2 /5 65% 4 /9 0.62 Topic: Rational/Irrational numbers Rational numbers can be written as the ratio of two whole numbers (ie fractions). So any number that can be written in this way is rational: 2 = 2/1 = 4/2 ; -5 = -5/1 Irrational numbers cannot be written as fractions eg. π, √3, 2 + √5 The decimal expansion does not repeat, but does ‘go on for ever’: π = 3.1415926535897932384626433832795. . . . √3= 1.73205080756887729352744634150587. . . (NB1. Whilst the square roots of most numbers are irrational, some are not eg √9 = 3 exactly, so is rational. NB2. Recurring decimals are rational: 1/3 = 0.33333333333. . . . . . 2 /7 = 0.2857142857142857. . . . . ) Topic: Probability Probability is an indication of the chance that something will happen. It can be worked out accurately in some cases, but, in others, it can only be approximated by experiment. Theoretically Probability of event = number of ways in which event can happen total number of possible ways things can happen Example: What is the probability of getting an even number with an ordinary six sided die? The die can fall in six ways altogether Number of ways to get even number = 3 (2 or 4 or 6) So Prob (even) = 3 = ½ 6 Experimentally, Relative frequency = number of times we get required result total number of trials In many cases, we cannot work out the probability of an event occurring theoretically, so we have to conduct trials to calculate the relative frequency. After a great number of trials, the relative frequency should ‘settle down’ to a particular value – which is then taken to be the probability that the required event will occur. Example: What is the chance a drawing pin lands tip up? not This cannot be determined theoretically, so we will have to drop the pin many times and count the number of times it lands tip up. No. of throws 10 No. tip up 5 Rel. freq. ½ = 0.5 25 100 1000 2500 9 44 421 1051 0.36 0.44 0.421 0.4204 So relative frequency seems to be settling down to about 0.42, which is taken as the probability that the pin ends tip up. With only a small number of trials, it is unlikely that the RF will be anywhere near the final value, but, after a large number of trials, the RF should be fairly steady. Topic: Pythagoras’ Theorem Hypotenuse Side1 This is an important result which only holds true when we have a right angled triangle. Side2 If we know the lengths of the three sides, then: (Hypotenuse)2 = (Side1)2 + (Side2)2 2 2 5 2 eg. 5 = 25 ; 3 + 4 = 9 + 16 So a 3, 4, 5 triangle is right angled as 52 = 32 + 42 3 4 Example 1: Calculate the length of the diagonal of a rectangle with sides 2 by 5 cm. Let diagonal length be y cm. Pythagoras’ theorem: y2 = 22 + 52 = 4 + 25 =29 2 5 So y = √29 = 5.385 to 3 dec pl. Example 2: A 10m. long ladder is leaning against a wall with its top touching the wall 9m. above the ground. How far is the base of the ladder from the wall? 9 Let base be d metres from the wall. Pythagoras’ theorem: 102 = 92 + d2 100 = 81 + d2 10 d2 = 100 – 81 = 19 d = √19 = 4.36 to 3 sig figs d Exercise. y 3 Find m. 7 3 m (Hint: find y first) Qu. Why can’t we just say m2 = 32 + 72 ? Topic: Ratio This is a way of comparing two quantities, provided they are given in the same units: Eg. Two trees are 6m and 10m tall. Ratio of their heights is 6 : 10 (or 3 : 5); first is 3/5 the height of the second In a class, there are 20 boys and 15 girls. So ratio of boys to girls is 20 : 15 (or 4 : 3) Ratios are usually expressed in the form a : b and cancelled down to lowest terms, using common factors. The numbers are usually left as whole numbers; thus we usually write 2 : 3, not 1 : 1.5 Maps use scale factors – say 1cm on map is 1km on the ground. Then the ratio is given as 1: 100000. (NB. same units) Example 1: A model of a classic car is built in the ration 1 : 20. If the model is 25cm long, how long is the actual car? Every 1cm on the model is equivalent to 20cm on actual car So 25cm on the model is equivalent to 20 x 25 cm = 500cm = 5m. on car. Example 2: The height of the actual car above is 1.8m, what is the height of the model? Ratio is 1 : 20, so model is 1/20 the size of the car. So height of model = 1/20 of 1.8m = 9 cm. (NB. ratio only affects linear dimensions – number of wheels does not increase in ratio 1 : 20, nor angles between spokes!) Example 3: The ratio of boys to girls in a group is 3 : 5. If there are 56 children, how many of them are boys? 3:5 means 3 boys for every 5 girls, so 8 children, 3/8 are boys. So out of 56 children, there are 3/8 of 56 = 21 boys. Example 4: 120 gm of a mixture contains substances A, B, and C in the ratio 2:3:1. How much of each is there in the mixture? 2:3:1 means 2 parts A, 3 parts B and 1 part C – a total of 6 parts. So each part is 120/6 gm = 20 gm. Thus, 40gm of A, 60gm of B, 20gm of C. Exercise 1. Express in lowest terms: 4:6 10 : 25 45 : 27 2. A model of a wedge is made in the ratio 1 : 12. If the length and the slope of the actual wedge are 108cm and 18º respectively, what are the corresponding values on the model? Topic: Rounding If asked your age, you are unlikely to say 19 years 3 months 2 weeks and 4 days. You usually ‘round’ (approximate) to the nearest whole year (ie. you’re 19). Of course, if you are 19 years 10 months, you would say ‘nearly 20’. We also do this with other types of data: Whole numbers 246 246 is 250 to the nearest 10 is 200 to the nearest 100 Decimal numbers 6.236 is 6.2 to 1 decimal place 6.236 is 6.24 to 2 dec pl If the digit in the place just beyond what is required is 5 or more, we round up by 1: 3.2783 is 3.278 to 3 dec pl 3.2783 is 3.28 to 2 dec pl 3.2783 is 3.3 to 1 dec pl 3.885 3.885 14.697 14.697 is 3.89 to 2dec pl is 3.9 to 1 dec pl is 14.70 to 2 dec pl (NB. Zero needed here) is 14.7 to 1 dec pl Another system uses significant figures – this will include all digits, not just those after the decimal point: Eg. 3.146 3.155 30.08 = 3.15 to 3 sig figs = 3.2 to 2 sig figs = 30.1 to 3 sig figs NB. If you have leading zeros, these are included, but not counted: 0.01278 = 0.0128 to 3 sig figs 0.00219 = 0.0022 to 2 sig figs Exercise 1. Write the following correct to 3 decimal places: 2. 4567 43.23589 0.00245 1.2945 2. Write the following correct to 3 significant figures: 2.3456 3.00763 0.012875 0.0024961 Topic: Straight lines See also: Gradient sheet In general, equations of the form 2x + 3y = 6 and y = 3x – 1 give graphs which are straight lines. They can always be written in the form y = mx + c. (The equation 2x + 3y = 6 can be re-written 3y = -2x +6, then y = -2/3 x + 2 ) When in this form, m gives the slope or gradient of the line and c gives the intercept on the yaxis. y B A A: y = 2x –1 B: y = x + 2 D C: y = 1 – x (y = -x + 1) D: y = x E: 2x + 3y +6 (y = -2/3 x + 2) E x C A is the steepest line, as it has a slope of 2. B has slope of 1; C has slope of –1. Lines intersect the y-axis (the intercepts on the y-axis): A: at –1 B: at 2 C: at 1 D: at 0 E: at 2 Lines with the same slope (m value) are parallel – look at B and D above. Example: Find the equation of the line passing through (0, 1) and (3, 7). 6 =2 3 Intercept on y-axis, c = 1 So equation is y = 2x + 1 y (3, 7) Slope, m = 6 1 3 x To plot a straight line, we usually choose three values of x and work out the corresponding y coordinates by substituting in the equation. Eg. When y = 2x + 1, if x = 0, y = 2x0 + 1 = 1 x = 1, y = 2x1 + 1 = 3 x = 2, y = 2x2 + 1 = 5 Now plot the points (0,1), (1,3), (2,5) on graph paper. Exercises 1. Write in the form y = mx + c : Which line is the steepest? y – 2x = 1 2y + 4 = 3x 1 + 3x = 3y 2. Where do the lines in Qu.1 cross the y-axis? 3. What is the equation of the line that has slope 2, passing through the point (0, 2)? 3. What is the equation of the line passing through the points (0, -1) and (2, 5)? Topic: Surface area of 3D shapes We need to think of the net of the shape. The ‘faces’ of 3D shapes are either flat [or curved]. Flat surfaces form polygons – triangles, rectangles, circles . . . . [Curved surfaces could be on spheres, cylinders, cones, . . . . . ] Area of triangle = ½x base x height Area of rectangle = length x height Area of circle = πr2 r Curved surface of cylinder: opens out to give a rectangle (and two circles) r h = 2πrh + 2πr2 2πr r h r ] Example1: Cuboid 6 x 4 x 3 cm. Surface area = 2A + 2B + 2C A: 3 x 4 = 12 cm2 B: 3 x 6 = 18 cm2 4 3 6 C B A C: 4 x 6 = 24 cm2 Example2: So SA = 108cm2. Wedge 3 x 4 x 5cm., 4 cm wide 5 3 Net would have 3 rectangles and two right angles triangles So SA = 3x4 + 4x5 + 4x4 + 2x(½x3x4) = 12 + 20 + 16 + 12 = 60 cm2 4 4 NB. Volume of 3D shape: Cuboid = length x width x height cm3 Wedge = ½ x L x B x H cm3 (Note units) Topic: Tree diagrams See also: Fractions Sheet This is a way to represent the probabilities of combined/successive events. Each route represents a particular sequence of events. Probabilities on the route are multiplied to find the overall probability of that sequence. When more than one path satisfy the conditions of the problem, their overall probabilities are added. Example 1: Tossing two coins (or one coin twice) 1st toss 2nd toss ½ H P(HH) = ½x½=¼ ½ ½ T P(HT) = ¼ ½ ½ H P(TH) = ¼ ½ T P(TT) = ¼ 1 H T (NB. These should add to 1) So probability of having one head and one tail = P(HT or TH) = P(HT) + P(TH) = ½ Example 2: A bag contains 2 red and 3 white balls; draw two balls without replacement R 2 /5 ¼ R P(RR) = 2/20 ¾ W P(RW) = 6/20 (NB. Probability on second draw is over 4 as we have one ball in the bag) less 3 /5 W 2 /4 R P(WR) = 6/20 2 /4 W P(WW) = 6/20 20 /20 = 1 (Check!) So probability of no white = P(RR) = 2/20 Probability of at least one white = 1 – P(no white) = 1 – 2/20 = 18/20 [or = P(RW or WR or WW) = 6/20 + 6/20 +6/20 = 18/20 ] Example 3: A Win B Throw die (1 – 6); if 6, A moves one square and wins if 1,2,3,4,5, B moves one square. Continue throwing die. Who is more likely to win? P (B wins) = P (B moves 5 times) = 5 x 5 x 5 x 5 x 5 = 0.4 (approx) 6 6 6 6 6 P ( A wins) = 1 – P (B wins) = 0.6 (approx) 1/6 5/6 1st throw 6 (A wins) not 6 (B moves) 1/6 5/6 6 (A wins) not 6 (B moves) etc….. 2nd throw 3rd throw Answers Averages 1. 2. 3. 4. Mode - £15; Mean -£15.83; Median - £15 Mode - £15; Mean -£28.33; Median - £15 9.86 years 5.385 Brackets 6 + 4x x²- x 3xy + 3yz 2w²- 6w x²+ 7x + 12 x²+ x – 2 p² - 4 2m² + 5m + 2 Circle 1. 48cm 238cm² 2. 100.5m 3. 942.5mm² Co-ordinates 1. (3, -1) 4a + 8b + 12c ab + ac - a² x³ - abx 8x – 4y + 12z 2. (2, 3) – (-2, 3) (1, 1) – (-1, 1) (3, 0) – (-3, 0) (0, 2) – (0, 2) (-2, -1) – (2, -1) Disproving a Hypothesis There are alternative answers but the following are some examples: 1. 3 + 4 + 5 = 8, not odd; (½)²= ¼, which is less than ½, so it is smaller; (1 + 1)²= 4; 1 + 1 = 2 2. Only that the cubes of the three numbers shown are odd. Estimating results of calculations 1. 5 x 20 = 100 Various answers can be given, including: 40 x 10 = 400 40 x 11 = 440 38 x 10 = 380 2. 124.5 1245 8.3 1275 Factorising algebraic expressions 3 (2x + y) 2 (4 – x) x ( y + 10x) 2bc (3a – 4d) x²(x + 1) s (6s² - 1) 3y (x + 2y - 3) Factors 1. Prime – 2, 7, 11, 37 Composite – 12, 14, 27, 49 2. 21 = 1 x 21 3x7 31 = 1 x 31 PRIME 15 = 1 x 15 3x5 3. 20 = 2²x 5 32 = 25 42 = 2 x 3 x 7 3. 28 = 1 + 2 + 4 + 7 + 14 4. Forming algebraic expressions 1. 2. 3. 4. 5x 6( m + 2) –2 N+2 p+q 2 Fractions 1. 2 9 2. 2 5 4 18 2 3 6 27 8 (Answers may vary) 36 37 47 3. 37 40 4. 2 2 3 Gradient 1. m = 1; m = ½ 2. C B 3. OA –object is moving away from origin over a period of time AB – object is staying at the same distance from the origin, so either stopped or circling Interpretation of a Graph A–B C–D Man is stationary on the ladder Man is coming down ladder very quickly, or falling off. Nth terms 1. 3n 4n + 2 3/4n + 1.75 or 3/4n + 7/4 = 3n + 7 4 Order of arithmetic operations 16 6 19 22.1 Order of size 3 = 0.375 8 0.3 1 = 0.3 (recurring) 3 2 = 0.4 5 4 = 0.4 (recurring) 9 So order of size is: 0.3 1/3 3/8 2/5 4/9 Pythagoras’ Theorem M = 8.7 Because the triangle is not a right-angled triangle. Ratio 1. 2:3 2:5 5:3 2. 9cm length 18º slope Rounding 1. 2.457 43.236 0.002 1.296 2. 2.35 3.01 0.0129 0.00250 Straight lines 1. Y = 2x + 1 Y = 1.5x – 2 or y = 3/2x - 2 Y = x + 1/3 So the first one is the steepest. 2. a ( 0 , 1 ) 3. y = 2x + 2 b ( 0, -2 ) 4. y = 3x - 1 c ( 0, 1/3 ) Index Topic Algebraic expressions Approximate answers Area of circle Area of rectangle Area of triangle Axes / Axis BODMAS Brackets (factorising) Brackets (multiplying) Circumference Common factor Complement Composite number Coordinates Cumulative frequency graph Decimal place Decimals (rounding) Decimals (fraction conversion) Denominator Diameter Sheet Forming algebraic expressions Estimating results of calculations Surface area of 3D shapes / Circles Surface area of 3D shapes Surface area of 3D shapes Coordinates / Gradient / Straight lines Order of arithmetic operations Factorising algebraic expressions Brackets Circles Factorising algebraic expressions / Ratio Inequalities Factors Coordinates / Gradient Grouped data Rounding Rounding Order of size Fractions Circles Digits Distance / speed graph Distance / time graph Equations of straight lines Equivalent fractions Estimating answers Factorisation Factor Fractions Fractions (addition) Fractions (decimal conversion) Fractions (multiplication) Fractions (subtraction) Frequency Distribution Gradient Graph interpretation Greater than / equals to Hypotenuse Hypothesis Inequalities Integer Interception on y – axis Interquartile range Irrational numbers Less than / equals to Lowest term Map references Mean Median Mid value Mode Multiple Net Nth term Numerator Operational conventions Origin Perfect number Perimeter Plotting straight line graph Estimating results of calculations Interpretation of a graph Coordinates/Interpretation of a graph Straight lines Fractions Estimating results of calculations Factorising algebraic expressions Factors Fractions Fractions Order of size Fractions / Tree diagrams Fractions Averages / Grouped data Gradient / Straight lines Interpretation of a graph Inequalities Pythagoras’ Theorem Disproving a hypothesis Inequalities Inequalities Straight lines Grouped data Rational / Irrational numbers Inequalities Fractions / Ratio Coordinates Averages / Grouped data Averages / Grouped data Grouped data Averages / Grouped data Factors Surface area of 3D shapes Nth terms Fractions Order of arithmetic operations Coordinates Factors Circles Straight lines Polygon Prime number Probability Proper factor Proving a hypothesis Pythagoras’ theorem Radius Ratio Rational numbers Recurring decimals (conversion) Reflection of points Relative frequency Right angled triangle Rounding Sequence of numbers Scale factors Sigma Significant figure Straight line graph Surface area Surface area of cuboid Surface area of cylinder Surface area of wedge Term number Tree diagrams Volume of 3D shape Y = mx + c Y – axis Surface area of 3D shapes Factors Probability / Tree diagrams Factors Disproving a hypothesis Pythagoras’ Theorem Circles Ratio Rational / Irrational numbers Rational / Irrational numbers Coordinates Probability Pythagoras’ Theorem Rounding Nth terms Ratio Averages / Grouped data Rounding Straight lines Surface area of 3D shapes Surface area of 3D shapes Surface area of 3D shapes Surface area of 3D shapes Nth terms Tree diagrams Surface area of 3D shapes Straight lines Straight lines