Area of a Triangle PDF

4 Area of a triangle Derive and know the formula for the area of a triangle. Use the formula to calculate the area of triangles and compound shapes made from rectangles and triangles...

4 Area of a triangle Derive and know the formula for the area of a triangle. Use the formula to calculate the area of triangles and compound shapes made from rectangles and triangles. Triangles One of the angles of this triangle is a right angle; this is a right- angled triangle. Drawing a rectangle around the triangle will help us to work out the formula for the area of a triangle. Height Height Height Height BaseBase length length BaseBase length length Area of rectangle = base length × height The height is The area of the triangle is half the area of always measured the rectangle. at right angles to the base. This Area of triangle = 1 × base length × height is sometimes 2 This also works for other triangles. known as the perpendicular height. Height Base length 19 9781398301948.indb 19 09/02/21 8:45 PM  SECTION 1 When we draw a rectangle around the triangle, we can see that the area of the triangle is still half that of the rectangle. Height Base length Area of a triangle = 1 × base length × perpendicular height 2 There will be times when the base of the triangle is not conveniently given as a horizontal side. However, we can use any side of the triangle as the base. We must make sure, though, that we measure the height at right angles to the side chosen as the base. These diagrams demonstrate this. Height Base Height Base Worked example Calculate the area of the triangle below. 4 cm 3 cm 8 cm The base length = 8 cm The perpendicular height = 3 cm The side length of 4 cm is not needed to calculate the area of the triangle. Area = 1 × 8 × 3 2 = 12 cm2 20 9781398301948.indb 20 09/02/21 8:45 PM  4 Area of a triangle Exercise 4.1 1 Calculate the area of each of the triangles below: a b c 3 cm 8 cm 10 cm 5 cm 11 cm 4 cm 8 cm 2 Use the formula for the area of a triangle to work out the missing values in the table below: Base length Perpendicular height Area a 7.2 cm 4.8 cm b 20 cm 100 cm2 c 15 cm 15 cm2 d 11 cm 55 cm2 3 Three triangles P, Q and R share a common base and lie between two parallel lines. Which of the three triangles (if any) has the biggest P Q R area? Give a reason for your answer. 4 a Draw a triangle similar to the one drawn opposite. i) Measure the length of side AB and let this be C the base of the triangle. B ii) Draw on your diagram a line which represents the height of the triangle. iii) Measure the height of your triangle. iv) Calculate the area of your triangle. A b Draw another triangle identical to the one you drew in part (a). i) Measure the length of side BC and let this be the base of the triangle. ii) Draw on your diagram, a line which represents the height of the triangle. iii) Measure the height of your triangle. iv) Calculate the area of your triangle. c Draw another triangle identical to the ones you drew in parts (a) and (b). i) Measure the length of side AC and let this be the base of the triangle. ii) Draw on your diagram, a line which represents the height of the triangle. iii) Measure the height of your triangle. iv) Calculate the area of your triangle. d What do you notice about the area of each of the three triangles you drew? Comment carefully on the reasons for any differences or similarities in your answers. 21 9781398301948.indb 21 09/02/21 8:46 PM  SECTION 1 5 The line PQ in the grid below is the base of a triangle. Each square in the grid is 1 cm2. a Copy the diagram and mark on your grid a point R, so that triangle PQR has an area of 8 cm2. P Q b Mark on your grid two other possible positions for point R. c Comment on the positions of the three possible positions for R that you have chosen. d Can you form a generalisation from your answers to (b) and (c)? 6 The diagram below shows a sequence of Sierpinski triangles, which are made of black and white equilateral triangles. If the first large triangle has an area of 64 cm2, calculate the total area of black triangles in each of the other patterns. Show your working clearly. Compound shapes A compound shape is one which is made from two or more simpler shapes. To work out the area of a compound shape it is often easier to work out the area of the simpler shapes first. 22 9781398301948.indb 22 09/02/21 8:46 PM  4 Area of a triangle Worked example Calculate the area of the compound shape below: 12 cm 3 cm 8 cm The shape is an example of a compound shape as, although it is a pentagon, it can be split into a rectangle A and a triangle B as shown. 12 cm 3 cm A B 8 cm Area of rectangle A: 3 × 8 = 24 cm2 Area of triangle B: The ‘height’ of the triangle can be deduced as it is the difference between the 12 cm and 8 cm measurements, i.e. 4 cm The area is therefore 1 × 3 × 4 = 6 cm2 2 Total area of compound shape is 24 + 6 = 30 cm2 Exercise 4.2 Calculate the area of each of the shapes below: 1 30 cm 2 4 cm 8 cm 4 cm 22 cm 4 cm 15 cm 3 20 cm 30 cm 23 9781398301948.indb 23 09/02/21 8:46 PM  SECTION 1 4 Two congruent (identical) right-angled triangles are placed inside a square as shown. The square has a side length of 8 cm. If half the area of the square is not covered by the triangles, calculate the length x of each triangle. Show all your reasoning clearly. x cm 8 cm x cm 5 Four congruent right-angled triangles are arranged to form two squares as shown: 4 cm 12 cm a What is the area of the smaller square as a fraction of the larger square? b Assuming the size of the larger square stays the same, what could the dimensions of each of the four congruent triangles be so that the area of the smaller square is half that of the larger square? c Is it possible for the area of the smaller square to be less than half that of the larger square? Give a convincing reason for your answer. Now you have completed Unit 4, you may like to try the Unit 4 online knowledge test if you are using the Boost eBook. 24 9781398301948.indb 24 09/02/21 8:46 PM  5 Order of operations Understand that brackets, positive indices and operations follow a particular order. Order of operations The order in which mathematical calculations are done depends on the operations being used. Look at this calculation: 6+3×2−1 LET’S TALK Carrying out the calculation from left to right would give an answer Why is it of 17. necessary to have However, if you do the calculation on a calculator, the answer it an agreed order gives is 11. for calculations? This is because mathematical operations are carried out in a particular order: Brackets Any operation in brackets is done first. Indices A number raised to a power (index) is done next. Division and/or Multiplication Multiplications and divisions are done next. Their order does not matter. Addition and/or Subtraction Additions and subtractions are carried out last. Again, their order is not important. A way of remembering this order is with the shorthand BIDMAS. The correct answer to the calculation 6 + 3 × 2 − 1 is 11, because the 3 × 2 must be done first, followed by addition of the 6 and subtraction of the 1. The 1 can be subtracted before the 6 is added; the answer is still 11. Worked examples 1 Calculate 7 + 4 × 9 − 8. 7+4×9−8 = 7 + 36 − 8 The multiplication is done first. = 35 25 9781398301948.indb 25 09/02/21 8:46 PM  SECTION 1 2 Calculate 25 − (2 + 3) × 4. 25 − (2 + 3) × 4 The brackets are done first, (2 + 3) = 5. = 25 − 5 × 4 This is multiplied by 4 next, giving 20. = 25 − 20 Lastly this is subtracted from 25. =5 3 Calculate 22 + (3 + 5)2 × 2. 22 + (3 + 5)2 × 2 Calculate what is inside the brackets first. = 22 + 82 × 2 Calculate any powers next. = 22 + 64 × 2 Work out the multiplication next. = 22 + 128 Lastly add the two numbers together. = 150 Exercise 5.1 1 Work out the following. a 4+3×2−1 f 7 × (4 + 2) − 3 b 6×2+4×3 g 8 + (6 + 3) ÷ 3 c 3×5−2−7 h 16 ÷ (2 + 2)2 + 8 d 8 + 4 × 8 − 40 i 4+4×4−4 e 6−2×3×4 j (4 + 4)2 ÷ (8 − 4) 2 Two pupils, Carla and Ibrahim, are discussing the following calculation. 3 × 22 Carla says the answer is 12, whilst Ibrahim says the answer is 36. a Which pupil is right? Give a convincing reason for your answer. b What mistake is the pupil who got it wrong making? 3 In the following calculations, insert brackets in order to make them correct. a 3 + 2 × 5 + 4 = 29 b 3 + 2 × 5 + 4 = 21 4 Insert any of the following symbols, +, −, ×, ÷ or ( ), in between the numbers to make the calculations correct. a 8 6 4 = 10 b 8 6 4 = 6 c 7 6 3 = 9 d 7 6 3 = 39 e 7 6 3 = 21 26 9781398301948.indb 26 09/02/21 8:46 PM  5 Order of operations f 8 4 2 = 4 g 8 4 2 = 2 h 2 3 5 2 = 20 i 2 3 5 2 = 18 j 3 9 8 5 = 36 5 Explain whether the use of brackets in the following calculations are necessary. Give a convincing reason for your answers. a 8 + (2 × 6) = 20 b (3 + 2)2 − 6 = 19 c (42 − 6) + 8 = 18 6 The following questions all contain four 4’s as shown below. 4 4 4 4 a i) Insert any of the following signs, +, −, ×, ÷ or ( ), in between the numbers to make the following calculation correct. 4 4 4 4 =1 ii) Can you find a different way of getting an answer of 1 using four 4’s? b Insert any of the following signs, +, −, ×, ÷ or ( ), in between the numbers to make the following calculation correct. 4 4 4 4 =2 c i) Using just four 4’s each time and any mathematical operation that you know of, can you make each of the answers from 3 to 20? Check your answers using a calculator. ii) Working with a partner, try to find more than one way to calculate each of the numbers 1–20. Check your answers using a calculator. LET’S TALK For part (c), with a partner decide on the different mathematical operations that you will allow. Decide also whether you will allow two of the 4’s to be written as 44. Now you have completed Unit 5, you may like to try the Unit 5 online knowledge test if you are using the Boost eBook. 27 9781398301948.indb 27 09/02/21 8:46 PM  Algebra beginnings – using 6 letters for unknown numbers Understand that letters can be used to represent unknown numbers, variables or constants. Understand that the laws of arithmetic and order of operations apply to algebraic terms and expressions. In algebra letters are used to represent unknown numbers. Often the task is to find the value of the unknown number, but not always. This unit introduces the main forms that algebra can take. Expressions An expression is used to represent a value in algebraic form. For example, x 3 The length of the line is given by the expression x + 3. Here we are not being asked to find the value of the unknown value x because the total length of the line is not given. 5 x The perimeter of the rectangle is given by the expression x + 5 + x + 5 which can be simplified to 2x + 10. KEY The area of the rectangle is given by the expression 5x. INFORMATION In the examples above, x, 2x and 5x are called terms in the Note that 5x expressions. means 5 times x. The numbers in front of the x in each case are called coefficients, When multiplying in algebra i.e. in the term 5x the ‘5’ is the coefficient. the × sign is not An expression is different from an equation. An equation contains used. an equals sign (=), which shows that the expressions either side of it are equal to each other. For example, the equation x+1=y−2 tells us that the expressions x + 1 and y − 2 are equal to each other. 28 9781398301948.indb 28 09/02/21 8:46 PM  6 Algebra beginnings – using letters for unknown numbers In the earlier rectangle, if we are told that the area of the rectangle is 20 cm2, then the equation 5x = 20 can be formed. If we are asked to solve the equation, then we have to find the value of x that makes the left-hand side of the equation equal to the right- hand side. Worked example A boy is y years old. His sister is 4 years older than him, his younger brother 2 years younger than him and his grandmother 8 times older than him. Write an expression for the ages of his sister, younger brother and grandmother. As his sister is 4 years older than him, the expression for her age is y + 4. As his younger brother is 2 years younger than him, his age is y − 2. As the grandmother is 8 times older than him, her age can be expressed as 8y. Exercise 6.1 1 The lines below have the lengths shown. In each case write an expression for the length of the line. a 7 x c 2x y b 2x 1 d 4x 2y 3 2 i) Write an expression for the distance around the edge of each of these shapes. ii) Simplify your expression where possible. a 4 d x y b y e q x p c 2 m f y+3 y+2 29 9781398301948.indb 29 09/02/21 8:46 PM  SECTION 1 g m+8 h y−1 x m+2 x+5 3 3 A box contains wooden building blocks of two different lengths p cm and q cm as shown. Describe how the following expressions for length can be represented using these blocks. p cm q cm a p+q c p−q b p + 2q d 2p + 2q 4 A girl has an age of x years. Her grandfather is aged y years. Write an expression for: a their combined age b the difference between their ages c the grandfather’s age 10 years ago d the girl’s age when she is twice as old as she is now e their combined age in 5 years’ time f the difference between their ages in 5 years’ time g the grandfather’s age when the girl is twice the age that she is now. 5 A rectangle has dimensions as shown. 6x 4 a Write an expression for the area and perimeter of the rectangle. b The rectangle is split in half as shown. A pupil makes the following statement: ‘Because the large rectangle is halved, the area and perimeter of each of the smaller rectangles will be half that of the larger rectangle’. Is the pupil’s statement completely right, partly right or completely wrong? Give a convincing reason for your answer. 6 A square piece of card is cut in half and the two pieces placed side by side as shown. 4a 30 9781398301948.indb 30 09/02/21 8:46 PM  6 Algebra beginnings – using letters for unknown numbers a If the large square has a side length of 4a, write an expression for the total perimeter of the two rectangles. b One of the two rectangles is halved lengthways and the pieces are also placed side by side. Write an expression for the total perimeter of the three rectangles. c The three pieces are then arranged in different ways with no gaps between them as shown. Explain convincingly, using algebra, why the perimeter of the two arrangements is not the same. d Write an expression for the difference in the perimeters of the two arrangements above. Order of operations when simplifying expressions In Unit 5 you saw that calculations need to be carried out in a particular order. This order is not necessarily from left to right. For example, the calculation 2 + 3 × 4 has the answer 14 (rather than 20) because the multiplication is done before the addition. The order in which operations are carried out is as follows: Brackets Indices Division/Multiplication Addition/Subtraction A useful way of remembering the order is with the shorthand BIDMAS. The same order of operations applies when working with algebraic expressions. 31 9781398301948.indb 31 09/02/21 8:46 PM  SECTION 1 The multiplication Worked example 3 × 4a is done first. Simplify the expression 2a + 3 × 4a − a. 2a + 3 × 4a − a = 2a + 12a − a = 13a Exercise 6.2 1 Simplify the following expressions. a 2a + 3a − 4a c 3c + 2b + 2c − 4b b 2b − 5b + 7b − b d 5d − 4f + 3e − 2d + f − 2e 2 Write an expression for the area of each of the following shapes. a j b 10 h 8 1 2 h 3 2 2 j 3 3 A compound shape is given opposite. a Show that the total perimeter of the shape is given by the 2 4 3x − 5 expression 6x + 12. b Show that the total area of the shape is given by the expression 10x − 6. 2x c A pupil says that he has worked out the total area to be 4 12 + 6x − 10 + 4x − 8. Explain whether this answer is correct. Substitution into expressions and formulae KEY INFORMATION 3p is an expression. The plural of 3p = s is a formula. formula is formulae. 3p = 9 is an equation. LET’S TALK It is important that you understand the differences between these. Can you think of any formulae you An expression is just an algebraic statement. have used in your maths or science lessons? 32 9781398301948.indb 32 09/02/21 8:46 PM  6 Algebra beginnings – using letters for unknown numbers KEY A formula describes a relationship between different variables and INFORMATION can be used to calculate values. If p = 5, you can use the formula 3p = s A variable can take to work out that the value of s is 15. If p = 8, then s = 24, and so on. several values. An equation, however, is only true for specific values of the variable. The equation 3p = 9 is only true when p = 3. Numbers can be substituted for the letters in both expressions and formulae. Substitution into expressions Worked example Evaluate (work out) the expressions below when a = 3, b = 4 and c = 5. a 2a + 3b − c = (2 × 3) + (3 × 4) − (5) The multiplications are done first, = 6 + 12 − 5 then the addition and subtraction. = 13 b 3 ac + b = ( 3 × 3 × 5 ) + 4 The multiplications and division are done first 2 2 = 45 + 2 then the addition. = 47 Exercise 6.3 Evaluate the expressions in questions 1 and 2 when p = 2, q = 3 and r = 5. 1 a 3p + 2q b 4p − 3q c p−q−r d 3p − 2q + r pq + 2 p p 4q 2 a pq + 10 r b r c 24 + p × r q d 4 × 2r − 3 Substitution into formulae The perimeter, P, of a rectangle is the distance around it. l b b l 33 9781398301948.indb 33 09/02/21 8:46 PM  SECTION 1 For this rectangle the perimeter is l+b+l+b or 2l + 2b This can be written as: P = 2l + 2b We obtained this formula from the diagram. This is one way to derive a formula. The area, A, of the rectangle can be written as: A = l × b = lb Worked example Using the formula P = 2l + 2b, calculate the perimeter of a rectangle if l = 3 cm and b = 5 cm. P=2×3+2×5 P = 6 + 10 P = 16 cm Exercise 6.4 1 Calculate the perimeter and area of each of these rectangles of length l and breadth b. Write the units of your answers clearly. a l = 4 cm, b = 7 cm e l = 0.8 cm, b = 40 cm b l = 8 cm, b = 12 cm f l = 1.2 cm, b = 0.5 cm c l = 4.5 cm, b = 2 cm g l = 45 cm, b = 1 m d l = 8 cm, b = 2.25 cm h l = 5.8m, b = 50 cm 2 Marta and Raul are studying the two rectangles A and B below. 3x x A 2 B 5 a They calculate that the perimeter of A is given by the formula P = 6x + 4 and that the perimeter of B is given by the formula P = 2x + 10. Show that they are correct. 34 9781398301948.indb 34 09/02/21 8:46 PM  6 Algebra beginnings – using letters for unknown numbers KEY b Marta says that the perimeter of A will always be greater than the INFORMATION perimeter of B because it has a ‘6x’ in its formula which is bigger There are 1000 than ‘2x’. Raul disagrees. He says that the perimeter of B will milliamps in 1 amp. always be greater than that of A because it has a ‘+10’ in its In other words, formula which is bigger than ‘+4’. Explain whether either of them 1 milliamp is is correct. 1/1000th of an amp. 3 In physics this formula is used in calculations about electricity: V = IR V is the voltage in a circuit in volts LET’S TALK I is the current in amps What other R is the resistance in ohms. examples use Without using a calculator, calculate the voltage V when ‘milli’ to represent a I = 7 amps, R = 60 ohms 1/1000th of b I = 8 amps, R = 400 ohms another measure? c I = 0.3 amps, R = 2000 ohms d I = 80 milliamps, R = 5000 ohms Now you have completed Unit 6, you may like to try the Unit 6 online knowledge test if you are using the Boost eBook. 35 9781398301948.indb 35 09/02/21 8:46 PM  Properties of three-dimensional 8 shapes Identify and describe the combination of properties that determine a specific 3D shape. Derive and use a formula for the volume of a cube or cuboid. Use knowledge of area and properties of cubes and cuboids to calculate their surface area. Three-dimensional shapes A three-dimensional shape is a solid figure or object with three dimensions, often described as length, width and height. Some common three-dimensional shapes are: Cube Cylinder Pyramid The different parts of a three-dimensional shape have specific mathematical names. An edge is where two faces meet A face is a flat or curved surface on the shape A vertex is a corner where edges meet. The plural of vertex is vertices 54 9781398301948.indb 54 09/02/21 8:47 PM  8 Properties of three-dimensional shapes Worked example Count the number of faces, edges and vertices on each of the following shapes. A cube A cylinder Number of faces = 6 Number of faces = 3 (2 flat and Number of edges = 12 1 curved) Number of vertices = 8 Number of edges = 2 Number of vertices = 0 Exercise 8.1 1 For the pyramids below count the number of faces, edges and vertices. a Square-based pyramid b Hexagonal-based pyramid 2 Sketch and name each of the three-dimensional shapes with the KEY following properties: INFORMATION a Only one face and no edges or vertices. This is an example b Two faces, one edge and one vertex. of proof by c Five faces, nine edges and six vertices. counter example, 3 Two friends are discussing the properties of three-dimensional shapes. i.e. you find a case One states that if two different three-dimensional shapes have the which contradicts same number of faces, then they must have the same number of edges. the original Prove convincingly that this is not true, by sketching two three- statement. dimensional shapes with the same number of faces but with a different number of edges. 55 9781398301948.indb 55 09/02/21 8:47 PM  SECTION 1 4 Four different types of pyramid are shown below: a It is stated that for every pyramid, there is always an even number LET’S TALK of edges. With a friend, i) Is this true for the four different pyramids above? discuss your ii) Explain whether you think the statement is always true. Give a answers to these convincing reason for your answer. questions. b It is also stated that every pyramid has the same number of faces as vertices. i) Is this true for the four different pyramids above? ii) Explain whether you think the statement is always true. Give a convincing reason for your answer. KEY Volume of a cuboid INFORMATION The volume of a three-dimensional object or shape refers to the The units of amount of space it occupies. volume include mm3, cm3 and m3 By taking a basic cube with side lengths of 1 cm, it is possible to work for solid shapes. out the volume of other cubes and cuboids. This cube has a volume of 1 cm3. LET’S TALK 1 cm By putting more of these cubes together, it is All the units of possible to form larger cubes and cuboids. volume mentioned so far use the 1 cm 1 cm ‘cubed’ notation 3. Can you think of Height = 2 cm units of volume which don’t use this? Width = 2 cm Length = 3 cm As this cuboid is made up of 12 × 1 cm3 it has a volume of 12 cm3. 56 9781398301948.indb 56 09/02/21 8:47 PM  8 Properties of three-dimensional shapes Exercise 8.2 1 In the following table several cuboids are drawn. Each cuboid is made from 1 cm3 cubes. For each cuboid write down the length, width and height and work out its volume. Cuboid Length Width Height Volume a b c d e 2 a Describe in words the relationship between the length, width and height of a cuboid and its volume. b Write the relationship you described in part (a) as a formula. 3 Calculate the volume of each of these cuboids, where L = length, W = width and H = height. Give your answers in cm3. a L = 4 cm W = 2 cm H = 3 cm b L = 5 cm W = 5 cm H = 6 cm Height c L = 10 cm W = 10 mm H = 4 cm d L = 40 cm W = 0.2 m H = 50 cm e L = 50 mm W = 30 cm H = 0.1 m Width Length 57 9781398301948.indb 57 09/02/21 8:47 PM  SECTION 1 4 This cuboid has a volume of 360 cm3. Calculate the length (in cm) of the edge marked x. x cm 5 cm 12 cm 5 The volume of this cuboid is 180 cm3. Calculate the length (in cm) of the edge marked y. 6 cm 6 cm y cm 6 a This cuboid has volume 768 cm3 and the edges marked a are equal in length. Calculate the value of a. a cm b In another cuboid of length 12 cm and volume 768 cm3, the width and height are not equal. Give a pair of possible values for their length. a cm 7 A tank of water in the shape of a cuboid has a length of 50 cm, 12 cm a width of 25 cm and a height of 20 cm. The depth of water in the tank is 12 cm as shown below. 20 cm 20 cm d cm 12 cm 25 cm 25 cm 50 cm 50 cm a A cube of side length 10 cm is placed at the bottom of the tank and the water level rises to a depth d cm as shown. Calculate the depth of the water d cm. b How many cubes can be placed in the tank before the water starts to spill over the top? Show all your working clearly. Composite three-dimensional shapes A composite shape is one which is made from other shapes. Here we will look at composite shapes which can be broken down into cubes and cuboids. 58 9781398301948.indb 58 09/02/21 8:47 PM  8 Properties of three-dimensional shapes KEY Worked example INFORMATION A prism is a 3D shape that, Calculate the volume of this ‘n’-shaped prism. when sliced in a particular direction, is the same all the B way through. It 3 cm A 5 cm has a constant C cross-section. 2 cm 8 cm 2 cm 2 cm The shape can be broken down into three cuboids, labelled A, B and C. The volume of the prism is therefore the sum of the volumes of the three cuboids. Volume of cuboid A = 2 × 8 × 5 = 80 cm3 Volume of cuboid B = 3 × 8 × 3 = 72 cm3 Volume of cuboid C = 2 × 8 × 5 = 80 cm3 Total volume of prism = 80 + 72 + 80 = 232 cm3 Exercise 8.3 Calculate the volume of the composite shapes in Questions 1–3. 1 2 cm 2 cm 2 5 cm 5 cm 6 cm 4 cm 8 cm 5 cm 5 cm 15 cm 6 cm 5 cm 59 9781398301948.indb 59 09/02/21 8:47 PM  SECTION 1 8 cm 3 8 cm 4 cm 4 cm 8 cm 4 Design your own composite shape which can be split into two cuboids, with a total volume of 200 cm3. 5 Design your own composite shape which can be split into three cuboids, with a total volume of 500 cm3. 6 The cuboid below has dimensions as shown: 8 cm 8 cm x cm 4 cm 4 cm x cm 10 cm 10 cm A cuboid with a depth of 4 cm but a length and width of x cm is cut out from one corner of the original cuboid as shown. The remaining shape has a volume of 199 cm3. Calculate the value of x. Show all your working clearly. Surface area of a cuboid The surface area of a cuboid refers to the total area of the six faces of the cuboid. As each face is a rectangle, the total surface area involves finding the area of the six rectangles. Worked example Calculate the surface area of this cuboid. 5 cm 3 cm 6 cm 60 9781398301948.indb 60 09/02/21 8:47 PM  8 Properties of three-dimensional shapes There are two ways of solving this problem. Method 1: By calculating the area of each rectangular face: As the front is the same as the back, the top the same as the bottom and the two sides equal to each other, the area can be worked out in pairs. Area of front and back = 6 × 5 × 2 = 60 cm2 Area of top and bottom = 6 × 3 × 2 = 36 cm2 Area of both sides = 3 × 5 × 2 = 30 cm2 Total surface area = 60 + 36 + 30 = 126 cm2 Method 2: By calculating the area of the net of the cuboid. The net of a cuboid is the two-dimensional shape which, when folded up, forms the cuboid. For example: 6 cm 5 cm 5 cm 5 cm 3 cm C B 3 cm 6 cm A 5 cm 3 cm Area of large rectangle A = 6 × 16 = 96 cm2 Area of rectangle B = 5 × 3 = 15 cm2 Area of rectangle C = 5 × 3 = 15 cm2 Therefore total surface area = 96 + 15 + 15 = 126 cm2 61 9781398301948.indb 61 09/02/21 8:47 PM  SECTION 1 Exercise 8.4 1 Calculate the surface area of the following cuboids. a b 2 cm 2 cm 8 cm 5 cm 10 cm 9 cm c 25 cm 1 cm 1.5 cm 2 The net of a cube is shown below. 5 cm Calculate a the surface area of the cube b the volume of the cube. 3 A cube has an edge length of x cm. a Show that the total surface area (A) can be calculated using the formula A = 6x2. b Use the formula to calculate the total surface area of a cube of edge length 10 cm. 4 A cube has an edge length of 3 cm. a Calculate its total surface area. b If the edge length is doubled, how many times bigger does the surface area become? c If the edge length of the original cube is trebled, how many times bigger does the surface area become? d Predict, without calculating the surface area, how many times bigger the total surface area becomes if the edge length of the original cube is 10 times bigger. Justify your answer. 62 9781398301948.indb 62 09/02/21 8:47 PM  8 Properties of three-dimensional shapes 5 For the cuboid drawn below: a draw two possible nets for the cuboid b calculate the total surface area of both nets, showing clearly the dimensions of each part of the net. 3 cm 15 cm 12 cm.6 A room in the shape of a cuboid is shown below. The room has two identical square windows and a door with the dimensions given. 8m Take care, as some dimensions are 100 cm in centimetres 3.5 m whilst others are given in metres cm 250 cm 60 3m Vladimir wants to decorate the room, including the ceiling, with two coats of paint (he will not paint the floor, windows or door). If each paint pot claims it can cover 60 m2 of surface area, how many pots will Vladimir need to buy? Show your working clearly. 7 Two students, Beatriz and Fatou, are discussing the relationship between a cuboid’s volume and its total surface area. Beatriz states that cuboids with the same volume must have the same surface area. Fatou thinks Beatriz is wrong. Which student is correct? Give a convincing reason for your answer. Now you have completed Unit 8, you may like to try the Unit 8 online knowledge test if you are using the Boost eBook. 63 9781398301948.indb 63 09/02/21 8:47 PM  9 Multiples and factors Understand lowest common multiple and highest common factor (numbers less than 100). Use knowledge of tests of divisibility to find factors of numbers greater than 100. Factors of a number are all the whole numbers which divide exactly into that number. For example, the factors of 12 are all the numbers which divide into 12 exactly. They are 1, 2, 3, 4, 6 and 12. Multiples of a number are all the whole numbers which are in that number’s times table. For example, the multiples of 3 are all the numbers in the 3× table. The first five are 3, 6, 9, 12 and 15, but there are in fact an infinite number of multiples of 3. LET’S TALK How can there be an infinite number of multiples of 3 if not all numbers are multiples of 3? Does this mean that there are more than an infinite number of numbers? Highest common factors and lowest common multiples The factors of 12 are 1, 2, 3, 4, 6 and 12. The factors of 18 are 1, 2, 3, 6, 9 and 18. KEY The highest common factor (HCF) of 12 and 18 is therefore 6, INFORMATION as it is the largest factor to appear in both groups. The highest The multiples of 6 are those numbers in the 6× table, i.e. 6, 12, 18, common factor 24, 30 etc. can also be called the greatest The multiples of 8 are those numbers in the 8× table, i.e. 8, 16, 24, common divisor. 32, 40 etc. The lowest common multiple (LCM) of 6 and 8 is therefore 24, as it is the smallest multiple to appear in both groups. 64 9781398301948.indb 64 09/02/21 8:47 PM  9 Multiples and factors Exercise 9.1 1 Find the highest common factor of the following numbers: a 8, 12 b 10, 25 c 12, 18, 24 d 15, 21, 27 e 36, 63, 108 2 Three cards each have a different factor of 18 written on them. Two of the numbers are shown, the third is hidden. If the number on the third card is not a multiple of 3, what must it be? 6 1 3 Find the lowest common multiple of the following numbers: a 6, 14 b 4, 15 c 2, 7, 10 d 3, 9, 10 e 3, 7, 11 4 The lowest common multiple of two numbers is 60. a What two numbers could they be? b Is another pair of numbers possible? If so, what numbers are they? 5 The factors of 24 are arranged in a 3 × 3 grid, leaving one square blank. The totals of each row and column are shown below. 4 22 34 29 14 17 a Explain why the number 24 must appear in the bottom left square of the grid. b Explain why the blank square must appear in the top row. c Copy and complete the grid by inserting the factors of 24 in the correct squares. Divisibility It is useful to be able to check whether a number is divisible (can be divided) by any of the numbers from 2 to 10 without having to use a calculator. There are quick methods that can be used to check for divisibility. 65 9781398301948.indb 65 09/02/21 8:47 PM  SECTION 1 Divisible by 2, 5 or 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 91 92 93 94 95 96 97 98 99 100 In the first grid all the numbers between 1 and 100 that are divisible by 2 have been shaded. We can generalise that multiples of 2 all end with either a 0, 2, 4, 6 or 8. In the second grid all the numbers that are divisible by 5 have been shaded. All multiples of 5 end with either a 0 or 5. Also, numbers divisible by 10 all end with a 0. Divisible by 3 or 9 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 91 92 93 94 95 96 97 98 99 100 66 9781398301948.indb 66 09/02/21 8:47 PM  9 Multiples and factors Adding the digits of each number together gives a total that is divisible by 3. For example, 54 is divisible by 3 as 5 + 4 = 9 and 9 is divisible by 3. This rule applies to larger numbers too: 372 is divisible by 3 as 3 + 7 + 2 = 12 and 12 is divisible by 3. Adding the digits of each number together gives a total that is divisible by 9. For example, 99 is divisible by 9 as 9 + 9 = 18 and 18 is divisible by 9. This rule applies to larger numbers too. 5877 is divisible by 9 as 5 + 8 + 7 + 7 = 27 and 27 is divisible by 9. Divisible by 4, 6, 7 or 8 These are slightly trickier and you will need to be familiar with your times tables to use these rules efficiently. LET’S TALK A number is divisible by 4 if the last two digits of the number are Why do we only divisible by 4. need to consider For example, the number 128 is divisible by 4 as the last two digits the last two digits are 28 and 28 is divisible by 4. of the number to test whether it is An alternative method would be to halve the number and then halve divisible by 4? it again. If the result is a whole number then the original number is divisible by four. For example, the number 128 is divisible by 4 because: LET’S TALK 128 ÷ 2 = 64 and 64 ÷ 2 = 32. As 32 is a whole number 128 is divisible by 4. Why do we only need to consider The number 246 is not divisible by 4 because: the last three digits of the number to 246 ÷ 2 = 123 and 123 ÷ 2 = 61.5, which is not a whole number. test whether it is Similarly, a number is divisible by 8 if the last three digits of the divisible by 8? number are divisible by 8. Hint: Is 1000 divisible by 8? For example, the number 2432 is divisible by 8 as the last three digits are 432, and 432 is divisible by 8. 67 9781398301948.indb 67 09/02/21 8:47 PM  SECTION 1 You may need to do a quick division to realise this! An alternative method would be to halve the number, halve it again and then again (i.e. halve the number three times). If the result is a whole number then the original number is divisible by 8. For example, the number 2432 is divisible by 8 because: 2432 ÷ 2 = 1216, 1216 ÷ 2 = 608 and 608 ÷ 2 = 304 As 304 is a whole number 2432 is divisible by 8. For a number to be divisible by 6 it must be divisible by both 2 and 3, for example, 1104 is divisible by 6 as it is divisible by 2 (the number ends in a 4) and divisible by 3 (1 + 1 + 0 + 4 = 6 which is divisible by 3). KEY INFORMATION Divisibility by 7 involves the following steps: To see why this For example, to test whether 1078 is divisible by 7: method works, take the last digit off the number and double it (8 × 2 = 16) you may want to subtract this from the remaining digits (107 − 16 = 91). look it up on the internet. If the answer is divisible by 7, then the original number is too (91 is divisible by 7, therefore 1078 is divisible by 7). Exercise 9.2 1 Copy the following table and tick the squares when the number is divisible by a number written along the top. One example has been started for you. 2 3 4 5 6 7 8 9 10 25 100 a 50 ✓ ✓ b 270 c 1120 d 135 e 302400 2 Four cards are arranged below to form a four-digit number. 5 2 3 8 68 9781398301948.indb 68 09/02/21 8:47 PM  9 Multiples and factors a Arrange the four cards so that the number is: i) divisible by 5 ii) divisible by 9 iii) divisible by 6 iv) divisible by 8. b In one of the questions in part (a) the order of the cards does not matter. Which one is it? Justify your answer. 3 Type this formula into cell A1 in a spreadsheet: =RANDBETWEEN(0,500). This will generate a random integer (whole number) between 0 and 500. Copy the formula down to cell A20 to generate 20 random numbers in the first column of the spreadsheet, for example as shown here. a Use divisibility tests to find out which of your random numbers are divisible by either 2, 3 or 6. b Which of your random numbers are divisible by either 5, 10 or 100? c Which of your random numbers are divisible by 9? d Is it true that any number which is divisible by 9 is also divisible by 3? Explain your answer. e Is it true that any number which is divisible by 3 is also divisible by 9? Explain your answer. Now you have completed Unit 9, you may like to try the Unit 9 online knowledge test if you are using the Boost eBook. 69 9781398301948.indb 69 09/02/21 8:47 PM  Probability and the likelihood 10 of events Use the language associated with probability and proportion to describe, compare, order and interpret the likelihood of outcomes. Understand and explain that probabilities range from 0 to 1, and can be represented as proper fractions, decimals and percentages. Probability is the study of chance, or the likelihood of an event happening. In everyday language we use words that are associated with probability all the time. For example, “I might see that film”, “I’m definitely going to win this race” or “It’s unlikely that I’ll pass this test”. Some other words include: Likely Certain Rarely Possible Even chance Probably When using these words we are usually indicating how likely something is to happen. For example, an event that is ‘likely’ is more probable than an event that ‘rarely’ happens. In this unit we will be looking at theoretical probability, that is, what you would expect to happen in theory. But, because probability is based on chance, what should happen in theory does not necessarily happen in practice. With an ordinary coin, there are two possible outcomes (two results that could happen). These are heads or tails. Each of these possible outcomes is an equally likely outcome if the coin is fair. This means that the coin is equally likely to land on heads or tails. The probability of getting a head when the coin is flipped is 21. The probability is 21 because getting a head is only one outcome out of two possible outcomes. This probability could also be written in decimal form as 0.5, or in percentage form as 50%. 70 9781398301948.indb 70 09/02/21 8:47 PM  10 Probability and the likelihood of events Worked example 1 An unbiased spinner, numbered 1–4, is spun. 4 2 3 a i) Calculate the probability of getting a 2. The probability of getting a 2 when the spinner is used is 41. ii) Write the probability of getting a 2 as a decimal and a percentage. 1 4 = 0.25 = 25% b Calculate the probability of getting a 7. 0 The probability of getting a 7 is 4 as there is no number 7 on the spinner. (Note: 04 = 0) Calculate the probability of getting either a 1, 2, 3 or 4. The probability of getting a 1, 2, 3 or 4 is 44 as there are four numbers out of four possible outcomes. It is certain that we will spin one of those numbers. (Note 44 = 1) If an outcome has a probability of 0, it means the outcome is impossible. If an outcome has a probability of 1, it means the outcome is certain. KEY The probability of an event can be placed on a probability scale from INFORMATION 0 to 1 like this. All probabilities Impossible lie in the range Certain 0–1. Evens 0 1 — 1 2 Exercise 10.1 1 a Write down at least 15 words which are used in everyday language to describe the likelihood of an event happening. b Draw a probability scale similar to the one above. Write each of your words from part (a) where you think they belong on the probability scale. 2 a For the spinner shown calculate the probability of getting: i) yellow ii) light blue iii) any blue. 71 9781398301948.indb 71 09/02/21 8:47 PM  SECTION 1 b The spinner is spun 60 times. i) Estimate the number of times you would expect to get the colour red. ii) Would you definitely get red that number of times? Explain your answer. 3 Five cards are numbered with a different number from 1 to 10 as shown below. One card is covered. LET’S TALK ‘At random’ means that each choice is 1 5 8 9 equally likely. A card is chosen at random. What could be the number on the covered card if: a the probability of picking an even number is 0.4 b the probability of picking a number less than five is 20% c the probability of picking 10 is 15 ? 4 The letters T, C and A can be written in several different orders. a Write the letters in as many different orders as possible. b If a computer writes these three letters in a random order, calculate the probability that: i) the letters will be written in alphabetical order ii) the letter T is written before both the letters A and C iii) the letter C is written after the letter A iv) the computer will spell the word CAT. 5 In one school, students have the option of studying either French (F) or Spanish (S) or neither. The incomplete Venn diagram below shows the number of students studying each language. ξ F S 12 8 32 A student is chosen at random. If the probability of choosing a student who studies French is 41 , calculate: a the total number of students in the year group b the probability of choosing a student who studies neither language. Give your answer as a fraction, decimal and percentage. 6 500 balls numbered from 1 to 500 are placed in a large container. A ball is picked at random. a Are the numbers an example of equally likely outcomes? Justify your answer. b Calculate the probability that the ball: i) has the number 1 on it ii) has one of the numbers 1 to 50 on it iii) has one of the numbers 1 to 500 on it iv) has the number 501 on it. Now you have completed Unit 10, you may like to try the Unit 10 online knowledge test if you are using the Boost eBook. 72 9781398301948.indb 72 09/02/21 8:47 PM  Section 1 – Review 1 In the magic square on the right, the numbers 1–9 2 are arranged so that all the rows, columns and main diagonals add up to the same total. 9 5 1 a Copy and complete the magic square by filling 3 in the missing numbers. b Use your answer to (a) to complete a magic square for the numbers 11–19. c i) What was the total of each row, column and main diagonal in (b) above? ii) Explain the difference between this total and the total for each row, column and main diagonal in the original magic square. 2 You will need isometric dot paper for this question. Part of a pattern using four rhombuses is drawn on isometric dot paper below. By drawing two more rhombuses, complete the pattern so that it has a rotational symmetry of order 3. 3 a Give one advantage and one disadvantage of using a questionnaire for collecting data. b Give one advantage and one disadvantage of using an interview for collecting data. 73 9781398301948.indb 73 09/02/21 8:47 PM  SECTION 1 4 A triangle is drawn inside a rectangle as shown. If the area of triangle 1 is half the area of triangle 2, calculate the length x. x cm 1 2 6 cm 12 cm 5 A student wants to work out the answer to the following calculation: 3 + 5 × 4 He writes the following steps: 3 + 5 × 4 =8×4 = 32 a i) Explain what mistake he has made. ii) What should the correct answer be? b By inserting any brackets necessary, rewrite the calculation so that the answer is 32. 6 The formula for the area (A) of a trapezium is given as A = 21 (a + b) h, where a, b and h are the lengths shown in the diagram. Calculate the area of the trapezium if a = 8 cm, b = 13 cm and h = 5 cm. a h b 74 9781398301948.indb 74 09/02/21 8:47 PM  Section 1 – Review 7 Eduardo collects data about the number of people in cars passing his house one afternoon. He presents the results both as a pie chart and a waffle diagram as shown. Explain whether the diagrams are showing the same results. Justify your answer fully. Number of people in cars Key 1 person 2 people 3 people 4 people 5 people 8 Two cubes P and Q are placed on top of each other. The volume of P is eight times the volume of Q. If the combined volume of the two cubes is 243 cm3, calculate the side length of each of the two cubes. Q P 75 9781398301948.indb 75 09/02/21 8:47 PM  SECTION 1 9 A four-digit number is given below. Unfortunately one of the digits is covered with ink. 3 4 8 a i) If the four-digit number is divisible by 4, what digit can the missing one be? ii) Justify your answer. b i) If the four-digit number is also divisible by 9, what number must the missing one be? ii) Justify your answer. 10 A class of students is asked what their favourite sport is. The results are shown in the table. Football Hockey Tennis Basketball Other Boys 5 1 2 4 3 Girls 3 3 5 1 3 A student is chosen at random. What is the probability that: a it is a girl b it is a basketball player c it is a boy who likes hockey d it is a girl whose favourite sport is not tennis? 76 9781398301948.indb 76 09/02/21 8:47 PM  SECTION 2 History of mathematics − The development of algebra The roots of algebra can be traced to the ancient Babylonians, who used formulae for solving problems. However, the word algebra comes from the Arabic language. Muhammad ibn Musa al-Khwarizmi (AD790–850) wrote Kitab al-Jabr (The Compendious Book on Calculation by Completion and Balancing), which established algebra