7th EDITION NEW SYLLABUS MATHEMATICS TEACHER S RESOURCE BOOK
CONTENTS Syllabus Matching Grid.... Scheme of Work....7 Chapter : Direct and Inverse Proportions Teaching Notes....9 Worked Solutions.... Chapter : Linear Graphs and Simultaneous Linear Equations Teaching Notes.... Worked Solutions.... Chapter : Epansion and Factorisation of Quadratic Epressions Teaching Notes....9 Worked Solutions....9 Chapter : Further Epansion and Factorisation of Algebraic Epressions Teaching Notes.... Worked Solutions.... Revision Eercise A.... Revision Eercise A.... Chapter : Quadratic Equations and Graphs Teaching Notes.... 8 Worked Solutions.... 9 Chapter : Algebraic Fractions and Formulae Teaching Notes.... Worked Solutions.... Chapter 7: Relations and Functions Teaching Notes.... Worked Solutions.... Chapter 8: Congruence and Similarity Teaching Notes.... Worked Solutions.... Chapter 9: Geometrical Transformation Teaching Notes.... 8 Worked Solutions.... 8 Revision Eercise B.... 9 Revision Eercise B.... 9 Chapter : Pythagoras Theorem Teaching Notes.... 97 Worked Solutions.... 98 iii
Chapter : Trigonometric Ratios Teaching Notes.... Worked Solutions.... Chapter : Volume and Surface Area of Pyramids, Cones and Spheres Teaching Notes.... Worked Solutions.... Chapter : Symmetry Teaching Notes.... 9 Worked Solutions.... Revision Eercise C.... 7 Revision Eercise C.... 8 Chapter : Sets Teaching Notes.... Worked Solutions.... Chapter : Probability of Single Events Teaching Notes.... 7 Worked Solutions.... 7 Chapter : Statistical Diagrams Teaching Notes.... 8 Worked Solutions.... 8 Chapter 7: Averages of Statistical Data Teaching Notes.... Worked Solutions.... Revision Eercise D.... 9 Revision Eercise D.... Problems in Real-World Contets.... iv
Syllabus Matching Grid Cambridge O Level Mathematics (Syllabus D) /9. Syllabus for eamination in 8, 9 and. Theme or Topic Subject Content Reference. Number Identify and use: Natural numbers Integers (positive, negative and zero) Prime numbers Square numbers Cube numbers Common factors and common multiples Rational and irrational numbers (e.g. p, ) Real numbers. Set language and notation Use set language, set notation and Venn diagrams to describe sets and represent relationships between sets Definition of sets: e.g. A { : is a natural number}, B {(, y): y m + c}, C { : a < < b}, D {a, b, c, } Book : Chapter Chapter Book : Chapter Book : Chapter. Squares, square roots, cubes and cube roots Calculate Squares Square roots Cubes and cube roots of numbers Book : Chapter Chapter. Directed numbers Use directed numbers in practical situations Book : Chapter. Vulgar and decimal fractions and percentages Use the language and notation of simple vulgar and decimal fractions and percentages in appropriate contets Recognise equivalence and convert between these forms Book : Chapter. Ordering Order quantities by magnitude and demonstrate familiarity with the symbols,, <, >, <, >. 7. Standard form Use the standard form A n, where n is a positive or negative integer, and < A <. 8. The four operations Use the four operations for calculations with: Whole numbers Decimals Vulgar (and mied) fractions including correct ordering of operations and use of brackets. 9. Estimation Make estimates of numbers, quantities and lengths Give approimations to specified numbers of significant figures and decimal places Round off answers to reasonable accuracy in the contet of a given problem Limits of accuracy Give appropriate upper and lower bounds for data given to a specified accuracy Obtain appropriate upper and lower bounds to solutions of simple problems given to a specified accuracy Book : Chapter Chapter Book : Chapter Book : Chapter Book : Chapter Book : Chapter
. Ratio, proportion, rate Demonstrate an understanding of ratio and proportion Increase and decrease a quantity by a given ratio Use common measures of rate Solve problems involving average speed. Percentages Calculate a given percentage of a quantity Epress one quantity as a percentage of another Calculate percentage increase or decrease Carry out calculations involving reverse percentages. Use of an electronic calculator Use an electronic calculator efficiently Apply appropriate checks of accuracy Enter a range of measures including time Interpret the calculator display appropriately Book : Chapter 9 Book : Chapter Book : Chapter 8 Book : Chapter Book : Chapter Chapter Book : Chapter Book : Chapter Book : Chapter. Time Calculate times in terms of the -hour and -hour clock Read clocks, dials and timetables Book : Chapter 9. Money Solve problems involving money and convert from one currency to another Book : Chapter. Personal and small business finance 7. Algebraic representation and formulae Use given data to solve problems on personal and small business finance involving earnings, simple interest and compound interest Etract data from tables and charts Use letters to epress generalised numbers and epress arithmetic processes algebraically Substitute numbers for words and letters in formulae Construct and transform formulae and equations Book : Chapter Book : Chapter Chapter Book : Chapter Book : Chapter 8. Algebraic manipulation Manipulate directed numbers Use brackets and etract common factors Epand product of algebraic epressions Factorise where possible epressions of the form: a + b + kay + kby a b y a + ab + b a + b + c Manipulate algebraic fractions Factorise and simplify rational epressions 9. Indices Understand and use the rules of indices Use and interpret positive, negative, fractional and zero indices Book : Chapter Book : Chapter Chapter Chapter Book : Chapter
. Solutions of equations and inequalities Solve simple linear equations in one unknown Solve fractional equations with numerical and linear algebraic denominators Solve simultaneous linear equations in two unknowns Solve quadratic equations by factorisation, completing the square or by use of the formula Solve simple linear inequalities Book : Chapter Book : Chapter Chapter Book : Chapter Chapter. Graphical representation of inequalities Represent linear inequalities graphically Book : Chapter. Sequences Continue a given number sequence Recognise patterns in sequences and relationships between different sequences Generalise sequences as simple algebraic statements. Variation Epress direct and inverse variation in algebraic terms and use this form of epression to find unknown quantities. Graphs in practical situations Interpret and use graphs in practical situations including travel graphs and conversion graphs Draw graphs from given data Apply the idea of rate of change to easy kinematics involving distance-time and speed-time graphs, acceleration and deceleration Calculate distance travelled as area under a linear speed-time graph. Graphs in practical situations Construct tables of values and draw graphs for functions of the form a n where a is a rational constant, n,,,,,, and simple sums of not more than three of these and for functions of the form ka where a is a positive integer Interpret graphs of linear, quadratic, cubic, reciprocal and eponential functions Solve associated equations approimately by graphical methods Estimate gradients of curve by drawing tangents. Function notation Use function notation, e.g. f() ; f :, to describe simple functions Find inverse functions f () 7. Coordinate geometry Demonstrate familiarity with Cartesian coordinates in two dimensions Find the gradient of a straight line Calculate the gradient of a straight line from the coordinates of two points on it Calculate the length and the coordinates of the midpoint of a line segment from the coordinates of its end points Interpret and obtain the equation of a straight line graph in the form y m + c Determine the equation of a straight line parallel to a given line Find the gradient of parallel and perpendicular lines Book : Chapter 7 Book : Chapter Book : Chapter Book : Chapter Book : Chapter 7 Book : Chapter Book : Chapter Chapter Chapter Book : Chapter Chapter 7 Book : Chapter 7 Book : Chapter Book : Chapter Book : Chapter Book : Chapter
8. Geometrical terms Use and interpret the geometrical terms: point; line; plane; parallel; perpendicular; bearing; right angle, acute, obtuse and refle angles; interior and eterior angles; similarity and congruence Use and interpret vocabulary of triangles, special quadrilaterals, circles, polygons and simple solid figures Understand and use the terms: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment 9. Geometrical constructions Measure lines and angles Construct a triangle, given the three sides, using a ruler and a pair of compasses only Construct other simple geometrical figures from given data, using a ruler and protractor as necessary Construct angle bisectors and perpendicular bisectors using a pair of compasses as necessary Read and make scale drawings Use and interpret nets. Similarity and congruence Solve problems and give simple eplanations involving similarity and congruence Calculate lengths of similar figures Use the relationships between areas of similar triangles, with corresponding results for similar figures, and etension to volumes and surface areas of similar solids. Symmetry Recognise rotational and line symmetry (including order of rotational symmetry) in two dimensions Recognise symmetry properties of the prism (including cylinder) and the pyramid (including cone) Use the following symmetry properties of circles: (a) equal chords are equidistant from the centre (b) the perpendicular bisector of a chord passes through the centre (c) tangents from an eternal point are equal in length. Angles Calculate unknown angles and give simple eplanations using the following geometrical properties: (a) angles at a point (b) angles at a point on a straight line and intersecting straight lines (c) angles formed within parallel lines (d) angle properties of triangles and quadrilaterals (e) angle properties of regular and irregular polygons (f) angle in a semi-circle (g) angle between tangent and radius of a circle (h) angle at the centre of a circle is twice the angle at the circumference (i) angles in the same segment are equal (j) angles in opposite segments are supplementary. Loci Use the following loci and the method of intersecting loci for sets of points in two dimensions which are: (a) at a given distance from a given point (b) at a given distance from a given straight line (c) equidistant from two given points (d) equidistant from two given intersecting straight line. Measures Use current units of mass, length, area, volume and capacity in practical situations and epress quantities in terms of larger or smaller units Book : Chapter Chapter Book : Chapter 8 Book : Chapter 9 to Chapter Book : Chapter Chapter Book : Chapter 8 Book : Chapter 8 Book : Chapter 8 Book : Chapter Chapter Book : Chapter Book : Chapter Book : Chapter Chapter Book : Chapter Book : Chapter 8 Book : Chapter Chapter
. Mensuration Solve problems involving: (a) the perimeter and area of a rectangle and triangle (b) the perimeter and area of a parallelogram and a trapezium (c) the circumference and area of a circle (d) arc length and sector area as fractions of the circumference and area of a circle (e) the surface area and volume of a cuboid, cylinder, prism, sphere, pyramid and cone (f) the areas and volumes of compound shapes. Trigonometry Interpret and use three-figure bearings Apply Pythagoras theorem and the sine, cosine and tangent ratios for acute angles to the calculation of a side or an angle of a right-angled triangles Solve trigonometrical problems in two dimensions involving angles of elevation and depression Etend sine and cosine functions to angles between 9 and 8 Solve problems using the sine and cosine rules for any triangle and the formula area of triangle ab sin C Solve simple trigonometrical problems in three dimensions Book : Chapter Chapter Book : Chapter Book : Chapter Book : Chapter Chapter Book : Chapter 8 Chapter 9 7. Vectors in two dimensions Describe a translation by using a vector represented by y, AB or a Add and subtract vectors Multiple a vector by a scalar Calculate the magnitude of a vector y as + y Represent vectors by directed line segments Use the sum and difference of two vectors to epress given vectors in terms of two coplanar vectors Use position vectors Book : Chapter 7 8. Matrices Display information in the form of a matri of any order Solve problems involving the calculation of the sum and product (where appropriate) of two matrices, and interpret the results Calculate the product of a matri and a scalar quantity Use the algebra of matrices including the zero and identity matrices Calculate the determinant A and inverse A of a non-singular matri A 9. Transformations Use the following transformations of the plane: reflection (M), rotation (R), translation (T), enlargement (E) and their combinations Identify and give precise descriptions of transformations connecting given figures Describe transformations using coordinates and matrices. Probability Calculate the probability of a single event as either a fraction or a decimal Understand that the probability of an event occurring the probability of the event not occurring Understand relative frequency as an estimate of probability Calculate the probability of simple combined events using possibility diagrams and tree diagrams where appropriate Book : Chapter Book : Chapter 9 Book : Chapter Book : Chapter Book : Chapter
. Categorical, numerical and grouped data Collect, classify and tabulate statistical data Read, interpret and draw simple inferences from tables and statistical diagrams Calculate the mean, median, mode and range for individual and discrete data and distinguish between the purposes for which they are used Calculate an estimate of the mean for grouped and continuous data Identify the modal class from a grouped frequency distribution Book : Chapter Book : Chapter 7 Book : Chapter. Statistical diagrams Construct and interpret bar charts, pie charts, pictograms, simple frequency distributions, frequency polygons, histograms with equal and unequal intervals and scatter diagrams Construct and use cumulative frequency diagrams Estimate and interpret the median, percentiles, quartiles and interquartile range for cumulative frequency diagrams Calculate with frequency density Understand what is meant by positive, negative and zero correlation with reference to a scatter diagram Draw a straight line of best fit by eye Book : Chapter Book : Chapter Book : Chapter
Week ( classes min) Direct and Inverse Proportions Chapter Section. Direct Proportion (pp. ). Algebraic and Graphical Representation of Direct Proportion (pp. ) Secondary Mathematics Scheme of Work Specific Instructional Objectives (SIOs) Syllabus Subject Content Activity ICT Eplain the concept of direct proportion Solve problems involving direct proportion Demonstrate an understanding of ratio and proportion Investigation Direct Proportion (p. ) Class Discussion Real-Life Eamples of Quantities in Direct Proportion (p. ) Eplain the concept of direct proportion using tables, equations and graphs Solve problems involving direct proportion Epress direct variation in algebraic terms and use this form of epression to find unknown quantities Construct tables of values and draw graphs for functions of the form a n where a is a rational constant, n,,,,,, and simple sums of not more than three of these and for functions of the form ka where a is a positive integer Thinking Time (p. ) Investigation Graphical Representation of Direct Proportion (pp. 7) Thinking Time (p. 7) Additional Resources Reasoning, Communication and Connection Investigation Direct Proportion (p. ) Class Discussion Real-Life Eamples of Quantities in Direct Proportion (p. ) Thinking Time (p. ) Thinking Time (p. 7) E A Q (iv) (p. ) 7
Week ( classes min) Chapter Section. Other Forms of Direct Proportion (pp. 8). Inverse Proportion (pp. 9 ). Algebraic and Graphical Representations of Inverse Proportion (pp. 9) Specific Instructional Objectives (SIOs) Eplain the concept of direct proportion using tables, equations and graphs Solve problems involving direct proportion Eplain the concept of inverse proportion Solve problems involving inverse proportion Eplain the concept of inverse proportion using tables, equations and graphs Solve problems involving inverse proportion Syllabus Subject Content Demonstrate an understanding of ratio and proportion Epress inverse variation in algebraic terms and use this form of epression to find unknown quantities Construct tables of values and draw graphs for functions of the form a n where a is a rational constant, n,,,,,, and simple sums of not more than three of these and for functions of the form ka where a is a positive integer Activity ICT Investigation Other Forms of Direct Proportion (pp. ) Thinking Time (p. ) Investigation Inverse Proportion (p. 9) Class Discussion Real-Life Eamples of Quantities in Inverse Proportion (p. ) Thinking Time (p. ) Investigation Graphical Representation of Inverse Proportion (pp. ) Thinking Time (p. ) Additional Resources Reasoning, Communication and Connection Investigation Other Forms of Direct Proportion (pp. ) Thinking Time (p. ) Investigation Inverse Proportion (p. 9) Class Discussion Real-Life Eamples o f Quantities in Inverse Proportion (p. ) Thinking Time (p. ) Thinking Time (p. ) Just For Fun (p. ) 8
Week ( classes min) Chapter Section. Other Forms of Inverse Proportion (pp. 9 ) Specific Instructional Objectives (SIOs) Eplain the concept of inverse proportion using tables, equations and graphs Solve problems involving inverse proportion Syllabus Subject Content Activity ICT Worked Eample (p.) Additional Resources Miscellaneous Solutions for Challenge Yourself Linear Graphs and Simultaneous Linear Equations. Gradient of a Straight Line (pp. 9 ) Find the gradient of a straight line State the y-intercept of a straight line Find the gradient of a straight line Calculate the gradient of a straight line from the coordinates of two points on it Investigation Equation of a Straight Line (p. 9) Class Discussion Gradients of Straight Lines (p. ) Investigation Equation of a Straight Line (p. 9) Class Discussion Gradients in the Real World (p. ) Investigation Gradient of a Horizontal Line (p. ) Investigation Gradient of a Vertical Line (p. ) Reasoning, Communication and Connection 9
Week ( classes min) Chapter Section. Further Applications of Linear Graphs in Real-World Contets (pp. ). Horizontal and Vertical Lines (pp. 8). Graphs of Linear Equations in the form a + by k (pp. 8 ) Specific Instructional Objectives (SIOs) State the equation of a horizontal line and of a vertical line Draw graphs of linear equations in the form a + by k Syllabus Subject Content Apply the idea of rate of change to easy kinematics involving distance-time and speed-time graphs Draw graphs from given data Activity ICT Worked Eample (p. ) Investigation Equation of a Horizontal Line (pp. ) Investigation Equation of a Vertical Line (p. 7) Investigation Graphs of a + by k (p. 9) Investigation Graphs of a + by k (p. 9) Additional Resources Reasoning, Communication and Connection Investigation Equation of a Horizontal Line (pp. ) Investigation Equation of a Vertical Line (p. 7) Investigation Graphs of a + by k (p. 9)
Week ( classes min) Chapter Section. Solving Simultaneous Linear Equations Using Graphical Method (pp. 7). Solving Simultaneous Linear Equations Using Algebraic Methods (pp. 7 7) Specific Instructional Objectives (SIOs) Solve simultaneous linear equations in two variables using the graphical method Solve simultaneous linear equations in two variables using the elimination method Solve simultaneous linear equations in two variables using the substitution method Syllabus Subject Content Solve simultaneous linear equations in two unknowns Solve associated equations approimately by graphical methods Construct and transform formulae and equations Solve simultaneous linear equations in two unknowns Activity ICT Investigation Solving Simultaneous Linear Equations Graphically (p. ) Investigation Solving Simultaneous Linear Equations Graphically (p. ) Class Discussion Choices of Appropriate Scales for Graphs and Accuracy of Graphs (p. ) Class Discussion Coincident Lines and Parallel Lines (p. ) Class Discussion Coincident Lines and Parallel Lines (p. ) Thinking Time (p. ) Thinking Time (p. 8) Thinking Time (p. 7) Thinking Time (p. 7) Thinking Time (p. 7) Additional Resources Reasoning, Communication and Connection Investigation Solving Simultaneous Linear Equations Graphically (p. ) Class Discussion Choices of Appropriate Scales for Graphs and Accuracy of Graphs (p. ) Class Discussion Coincident Lines and Parallel Lines (p. ) Thinking Time (p. ) Thinking Time (p. 7) Thinking Time (p. 7)
Week ( classes min) Chapter Section.7 Applications of Simultaneous Equations in Real-World Contets (pp. 77 8) Specific Instructional Objectives (SIOs) Formulate a pair of linear equations in two variables to solve mathematical and real-life problems Syllabus Subject Content Activity ICT Thinking Time (p. 78) Just For Fun (p. 8) Additional Resources Miscellaneous Solutions for Challenge Yourself Epansion and Factorisation of Quadratic Epressions. Quadratic Epressions (pp. 9 9) Recognise quadratic epressions Practise Now (p. 9) Practise Now (p. 9). Epansion and Simplification of Quadratic Epressions (pp. 97 ). Factorisation of Quadratic Epressions (pp. ) Epand and simplify quadratic epressions Use a multiplication frame to factorise quadratic epressions Factorise where possible epressions of the form: a + ab + b a + b + c Class Discussion Epansion of Quadratic Epressions of the Form (a + b)(c + d) (p. ) Main Tet (pp. 7) Practise Now (p. 9) Practise Now (p. 98) Class Discussion Epansion of Quadratic Epressions of the Form (a + b)(c + d) (p. ) Practise Now (p. 7) Miscellaneous Solutions for Challenge Yourself Reasoning, Communication and Connection Just For Fun (p. 8)
Week ( classes min) Further Epansion and Factorisation of Algebraic Epressions Chapter Section. Epansion and Factorisation of Algebraic Epressions (pp. ) 7. Epansion Using Special Algebraic Identities (pp. ) 7. Factorisation Using Special Algebraic Identities (pp. 8) Specific Instructional Objectives (SIOs) Epand and simplify algebraic epressions Use a multiplication frame to factorise algebraic epressions Recognise and apply the three special algebraic identities to epand algebraic epressions Recognise and apply the three special algebraic identities to factorise algebraic epressions Syllabus Subject Content Epand product of algebraic epressions Factorise where possible epressions of the form: a + ab + b a + b + c a + b + kay + kby a b y Activity ICT Thinking Time (p. 8) Class Discussion Special Algebraic Identities (p. ) Additional Resources 8. Factorisation by Grouping (pp. 8 ) Factorise algebraic epressions by grouping Thinking Time (p. ) Class Discussion Equivalent Epressions (p. ) 8 Miscellaneous Solutions for Challenge Yourself Reasoning, Communication and Connection Class Discussion Equivalent Epressions (p. )
Week ( classes min) 8 Quadratic Equations and Graphs Chapter Section. Solving Quadratic Equations by Factorisation (pp. 9 ) 9. Applications of Quadratic Equations in Real-World Contets (pp. ) Specific Instructional Objectives (SIOs) Solve quadratic equations by factorisation Solve mathematical and real-life problems involving quadratic equations algebraically Syllabus Subject Content Solve quadratic equations by factorisation Activity ICT Thinking Time (p. ) Additional Resources 9. Graphs of Quadratic Functions (pp. 9 9) Draw graphs of quadratic functions State the properties of the graphs of quadratic functions Solve mathematical and real-life problems involving quadratic equations graphically Interpret graphs of quadratic functions Investigation Relationship Between the Area of a Square and the Length of its Side (p. 7) Investigation Graphs of y and y (p. 8) Investigation Graphs of y a + b + c, where a (pp. 9 ) Investigation Relationship Between the Area of a Square and the Length of its Side (p. 7) Internet Resources (p. 8) Investigation Graphs of y and y (p. 8) Investigation Graphs of y a + b + c, where a (pp. 9 ) 9 Miscellaneous Solutions for Challenge Yourself Reasoning, Communication and Connection Thinking Time (p. ) Investigation Relationship Between the Area of a Square and the Length of its Side (p. 7) Investigation Graphs of y and y (p. 8) Investigation Graphs of y a, where a (pp. 9 )
Week ( classes min) Algebraic Fractions and Formulae Chapter Section. Algebraic Fractions (pp. ). Multiplication and Division of Algebraic Fractions (pp. 7). Addition and Subtraction of Algebraic Fractions (pp. 8 7). Manipulation of Algebraic Formulae (pp. 7 77) Specific Instructional Objectives (SIOs) Simplify simple algebraic fractions Multiply and divide simple algebraic fractions Add and subtract algebraic fractions with linear or quadratic denominators Change the subject of a formula Find the value of an unknown in a formula Syllabus Subject Content Manipulate algebraic fractions Activity ICT Class Discussion Finding the Value of an Unknown in a Formula (p. 7) Additional Resources Miscellaneous Solutions for Challenge Yourself 7 Relations and Functions 7. Relations (p. 8) 7. Functions (pp. 8 89) Define a function Verify if a given relation is a function Solve problems on functions involving linear epressions Use function notation, e.g. f() ; f :, to describe simple functions Thinking Time (p.8) Miscellaneous Solutions for Challenge Yourself Reasoning, Communication and Connection Class Discussion Finding the Value of an Unknown in a Formula (p. 7) Thinking Time (p.8)
Week ( classes min) 8 Congruence and Similarity Chapter Section 8. Congruent Figures (pp. 9 ) 8. Similar Figures (pp. ) Specific Instructional Objectives (SIOs) Eamine whether two figures are congruent Solve simple problems involving congruence Eamine whether two figures are similar State the properties of similar triangles and polygons Solve simple problems involving similarity Syllabus Subject Content Use and interpret the geometrical term: congruence Solve problems and give simple eplanations involving congruence Use and interpret the geometrical term: similarity Solve problems and give simple eplanations involving similarity Calculate lengths of similar figures Activity ICT Investigation Properties of Congruent Figures (p. 9) Internet Resources (p. 9) Thinking Time (p. 9) Main Tet (p. 9) Class Discussion Congruence in the Real World (p. 9) Class Discussion Similarity in the Real World (p. ) Investigation Properties of Similar Polygons (p. ) Thinking Time (p. ) Class Discussion Identifying Similar Triangles (p. ) Additional Resources Reasoning, Communication and Connection Investigation Properties of Congruent Figures (p. 9) Thinking Time (p. 9) Class Discussion Congruence in the Real World (p. 9) Class Discussion Similarity in the Real World (p. ) Investigation Properties of Similar Polygons (p. ) Thinking Time (p. ) Class Discussion Identifying Similar Triangles (p. )
Week ( classes min) Chapter Section 8. Similarity, Enlargement and Scale Drawings (pp. 8) Specific Instructional Objectives (SIOs) Make simple scale drawings with appropriate scales Interpret scales on maps Syllabus Subject Content Read and make scale drawings Activity ICT Just For Fun (p. ) Main Tet (p. ) Additional Resources Main Tet (p. ) Main Tet (p. ) Performance Task (p. ) Miscellaneous Solutions for Challenge Yourself 9 Geometrical Transformation 9. Reflection (pp. ) 9. Rotation (pp. ) 9. Translation (pp. ) Reflect an object and find the line of reflection by construction Rotate an object and find the centre of rotation by construction Use the following transformations of the plane: reflection (M), rotation (R) and translation (T) Identify and give precise descriptions of transformations connecting given figures Thinking Time (p. 7) Thinking Time (p. ) Thinking Time (p. ) Translate an object Thinking Time (p. ) Thinking Time (p. 8) Journal Writing (p. 8) Miscellaneous Solutions for Challenge Yourself Reasoning, Communication and Connection Performance Task (p. ) Attention (p. ) Thinking Time (p. 7) Thinking Time (p.) Thinking Time (p. ) Thinking Time (p. ) Thinking Time (p. 8) 7
Week ( classes min) Pythagoras Theorem Chapter Section. Pythagoras Theorem (pp. 9 8) Specific Instructional Objectives (SIOs) Solve problems using Pythagoras Theorem Syllabus Subject Content Apply Pythagoras theorem to the calculation of a side or an angle of a rightangled triangle Activity ICT Investigation Pythagoras Theorem The Secret of the Rope-Stretchers (pp. ) Performance Task (p. ) Investigation Pythagoras Theorem The Secret of the Rope-Stretchers (pp. ) Performance Task (p. ) Additional Resources 7. Applications of Pythagoras Theorem in Real-World Contets (pp. 9 7) 7. Converse of Pythagoras Theorem (pp. 7 77) Solve problems using Pythagoras Theorem Determine whether a triangle is a rightangled triangle given the lengths of three sides Internet Resources (p. ) 7 Miscellaneous Solutions for Challenge Yourself Reasoning, Communication and Connection Investigation Pythagoras Theorem The Secret of the Rope- Stretchers (pp. ) Performance Task (p. ) 8
Week ( classes min) 8 Trigonometric Ratios Chapter Section. Trigonometric Ratios (pp. 8 88) Specific Instructional Objectives (SIOs) Eplain what trigonometric ratios of acute angles are Syllabus Subject Content Activity ICT Investigation Trigonometric Ratios (p. 8) Additional Resources 8. Applications of Trigonometric Ratios to Find Unknown Sides of Right-Angled Triangles (pp. 89 9) 9. Applications of Trigonometric Ratios to Find Unknown Angles in Right-Angled Triangles (pp. 9 ) 9. Applications of Trigonometric Ratios in Real- World Contets (pp. ) Find the unknown sides in right-angled triangles Find the unknown angles in right-angled triangles Apply trigonometric ratios to solve problems in realworld contets Use of an electronic calculator efficiently Apply the sine, cosine and tangent ratios for acute angles to the calculation of a side of a right-angled triangle Apply the sine, cosine and tangent ratios for acute angles to the calculation of an angle of a right-angled triangle Solve trigonometrical problems in two dimensions involving angles of elevation and depression Thinking Time (p. 8) Worked Eample (p. 87) Investigation Using a Clinometer to Find the Height of an Object (pp. ) 9 Miscellaneous Solutions for Challenge Yourself Reasoning, Communication and Connection Investigation Trigonometric Ratios (p. 8) Thinking Time (p. 8) 9
Week ( classes min) Volume and Surface Area of Pyramids, Cones and Spheres Chapter Section. Volume and Surface Area of Pyramids (pp. 7 8). Volume and Surface Area of Cones (pp. 8 7) Specific Instructional Objectives (SIOs) Identify and sketch pyramids Draw and use nets of pyramids to visualise their surface area Use formulae to calculate the volume and the surface area of pyramids Identify and sketch cones Draw and use nets of cones to visualise their surface area Use formulae to calculate the volume and the surface area of cones Syllabus Subject Content Solve problems involving the surface area and volume of a pyramid Solve problems involving the surface area and volume of a cone Activity ICT Class Discussion What are Pyramids? (p. 7) Internet Resources (p. ) Thinking Time (p. 9) Journal Writing (p. 9) Investigation Volume of Pyramids (pp. ) Class Discussion What are Cones? (p. 8) Journal Writing (p. 9) Investigation Comparison between a Cone and a Pyramid (p. ) Thinking Time (p. ) Investigation Curved Surface Area of Cones (pp. ) Thinking Time (p. ) Additional Resources Reasoning, Communication and Connection Class Discussion What are Pyramids? (p. 7) Thinking Time (p. 9) Journal Writing (p. 9) Class Discussion What are Cones? (p. 8) Journal Writing (p. 9) Investigation Comparison between a Cone and a Pyramid (p. ) Thinking Time (p. ) Investigation Curved Surface Area of Cones (pp. )
Week ( classes min) Chapter Section. Volume and Surface Area of Spheres (pp. 8 ) Specific Instructional Objectives (SIOs) Identify and sketch spheres Use formulae to calculate the volume and the surface area of spheres Syllabus Subject Content Solve problems involving the surface area and volume of a sphere Activity ICT Thinking Time (p. 8) Class Discussion Is the King s Crown Made of Pure Gold? (pp. 8 9) Investigation Volume of Spheres (pp. 9 ) Additional Resources Investigation Surface Area of Spheres (p. ). Volume and Surface Area of Composite Solids (pp. 9) Solve problems involving the volume and the surface area of composite solids made up of pyramids, cones, spheres, prisms and cylinders Solve problems involving the surface area and volume of compound shapes Thinking Time (p. ) Miscellaneous Solutions for Challenge Yourself Reasoning, Communication and Connection Thinking Time (p. 8) Class Discussion Is the King s Crown Made of Pure Gold? (pp. 8 9) Investigation Surface Area of Spheres (p. )
Week ( classes min) Geometrical Transformation Chapter Section. Line symmetry (p. 7 7). Rotational Symmetry in Plane Figures (p. 8 7). Symmetry in Triangles, Quadrilaterals and Polygons (p. 7 78) Specific Instructional Objectives (SIOs) Identify line symmetry of plane figures Identify rotational symmetry of plane figures Make use of the symmetrical properties of triangles, quadrilaterals and regular polygons Syllabus Subject Content Recognise rotational and line symmetry (including order of rotational symmetry) in two dimensions Activity ICT Investigation Line Symmetry in Two Dimensions (p. 7) Thinking Time (p. 8) Thinking Time (p. 9) Investigation Rotational Symmetry in Two Dimensions (p. 8) Class Discussion Line and Rotational Symmetry in Circles (p. 9) Investigation Symmetry in Triangles (pp. 7 7) Investigation Symmetry in Special Quadrilaterals (pp. 7 7) Investigation Symmetry in Regular Polygons (pp. 77 78) Additional Resources Reasoning, Communication and Connection Thinking Time (p. 8) Thinking Time (p. 9) Class Discussion Line and Rotational Symmetry in Circles (p. 9) Investigation Symmetry in Triangles (pp. 7 7) Investigation Symmetry in Special Quadrilaterals (pp. 7 7) Investigation Symmetry in Regular Polygons (pp. 77 78)
Week ( classes min) Chapter Section. Symmetry in Three Dimensions (p. 79 8) Specific Instructional Objectives (SIOs) Use symmetrical properties of simple solids Syllabus Subject Content Recognise symmetry properties of the prism (including cylinder) and the pyramid (including cone) Activity ICT Thinking Time (p. 79) Thinking Time (p. 8) Additional Resources Investigation Symmetry in Cylinders and Cones (p. 8) Miscellaneous Solutions for Challenge Yourself Sets. Introduction to Set Notations (p. 9 98). Venn Diagrams, Universal Set and Complement of a Set (p. 99 ) Describe a set in words, list all the elements in a set, and describe the elements in a set State and use the terms set, element, equal sets, empty set State and use the terms universal set, complement of a set, subset, proper subset Use Venn diagrams to represent sets, including universal sets, complement of a set and proper subsets Use set language, set notation and Venn diagrams to describe sets and represent relationships between sets Definition of sets: e.g. A { : is a natural number}, B {(, y): y m + c}, C { : a b}, D {a, b, c, } Class Discussion Well-defined and Distinct Objects in a Set (p. 9) Thinking Time (p. 9) Thinking Time (p. ) Class Discussion Understanding Subsets (p. ) Reasoning, Communication and Connection Thinking Time (p. 79) Thinking Time (p. 8) Investigation Symmetry in Cylinders and Cones (p. 8) Class Discussion Well-defined and Distinct Objects in a Set (p. 9) Attention (p. 9) Thinking Time (p. 9) Thinking Time (p. ) Class Discussion Understanding Subsets (p. )
Week ( classes min) Chapter Section. Intersection of Two Sets (p. 7). Union of Two Sets (p. 8 9). Combining Universal Set, Complement of a Set, Subset, Intersection and Union of Sets (p. 9 ) Specific Instructional Objectives (SIOs) State and use the term intersection of two sets, Use Venn diagrams to represent sets, including intersection of two sets State and use the term union of two sets Use Venn diagrams to represent sets, including union of two sets Solve problems using set notations and Venn diagrams Syllabus Subject Content Activity ICT Worked Eample 9 (pp. ) Thinking Time (p. ) Performance Task (p. ) Additional Resources Miscellaneous Solutions for Challenge Yourself Probability of Single Events. Introduction to Probability (p. ). Sample Space (pp. ) Define probability as a measure of chance List the sample space of a probability eperiment Understand relative frequency as an estimate of probability Thinking Time (p. ) Main Tet (p. ) Reasoning, Communication and Connection Thinking Time (p. ) Performance Task (p. ) Thinking Time (p. )
Week ( classes min) Chapter Section. Probability of Single Events (pp. ) Specific Instructional Objectives (SIOs) Find the probability of a single event Syllabus Subject Content Calculate the probability of a single event as either a fraction or a decimal Understanding that the probability of an event occurring probability of the event not occurring Activity ICT Investigation Tossing a Coin (pp. ) Investigation Rolling a Die (pp. 7) Thinking Time (pp. 9) Performance Task (pp. 9) Internet Resources (p. ) Investigation Rolling a Die (pp. 7) Internet Resources (p. ) Additional Resources 7. Further Eamples on Probability of Single Events (pp. ) Solve problems involving the probability of single events Just for Fun (pp. ) 7 Miscellaneous Solutions for Challenge Yourself 7 Statistical Diagrams. Statistical Diagrams (p. ) 7. Dot Diagrams (pp. 8) 8. Stem-and-Leaf Diagrams (pp. 8 ) Construct and interpret data from dot diagrams Construct and interpret data from stem-and-leaf diagrams Main Tet (pp. 8) Main Tet (pp. 8 ) Reasoning, Communication and Connection Investigation Tossing a Coin (pp. ) Investigation Rolling a Die (pp. 7) Thinking Time (p. 9) Performance Task (p. 9)
Week ( classes min) Chapter Section 8. Scatter Diagrams (pp. ) 9. Histograms for Ungrouped Data (pp. 9) Specific Instructional Objectives (SIOs) Construct and interpret data from scatter diagrams Construct and interpret data from histograms Evaluate the purposes and appropriateness of the use of different statistical diagrams Eplain why some statistical diagrams can lead to a misinterpretation of data Syllabus Subject Content Construct and interpret scatter diagrams Understand what is meant by positive, negative and zero correlation with reference to a scatter diagram Draw a straight line of best fit by eye Construct and interpret simple frequency distributions Activity ICT Main Tet (pp. ) Thinking Time (p. ) Class Discussion Scatter Diagram with No Correlation (p. 9) Thinking Time (p. 9) Journal Writing (p. 9) Main Tet (pp. 8) Journal Writing (p. ) Class Discussion Evaluation of Statistical Diagrams (pp. 8 9) Additional Resources Reasoning, Communication and Connection Thinking Time (p. ) Class Discussion Scatter Diagram with No Correlation (p. 9) Thinking Time (p. 9) Journal Writing (p. 9) Journal Writing (p. ) Class Discussion Evaluation of Statistical Diagrams (pp. 8 9)
Week ( classes min) Chapter Section 9. Histograms for Grouped Data (pp. 9 8) Specific Instructional Objectives (SIOs) Construct and interpret data from histograms Evaluate the purposes and appropriateness of the use of different statis-tical diagrams Eplain why some statistical diagrams can lead to a misinterpretation of data Syllabus Subject Content Construct and interpret histograms with equal and unequal intervals Activity ICT Journal Writing (p. 7) Performance Task (p. 7) Class Discussion Histograms for Grouped Data with Unequal Class Intervals (p. 7 7) Additional Resources Construct and interpret frequency polygons Main Tet (p. 77) 9 Miscellaneous Solutions for Challenge Youself Reasoning, Communication and Connection Journal Writing (p. 7) Performance Task (p. 7) Class Discussion Histograms for Grouped Data with Unequal Class Intervals (p. 7 7) 7
Week ( classes min) 7 Averages of Statistical Data Chapter Section 7. Mean (pp. 9 ) 7. Median (pp. ) 7. Mode (pp. 8) 7. Mean, Median and Mode (p. 9 ) Specific Instructional Objectives (SIOs) Find the mean of a set of data Calculate an estimate for the mean Find the median of a set of data Find the class interval where the median lies Find the mode of a set of data State the modal class of a set of grouped data Evaluate the purposes and appropriateness of the use of mean, median and mode Syllabus Subject Content Calculate the mean for individual and discrete data Calculate an estimate of the mean for grouped and continuous data Calculate the median for individual and discrete data Calculate the mode for individual and discrete data Identify the modal class from a grouped frequency distribution Distinguish between the purposes for which the mean, median and mode are used Activity ICT Thinking Time (p. ) Class Discussion Creating Sets of Data with Given Conditions (p. ) Thinking Time (p. 7) Thinking Time (p. ) Class Discussion Comparison of Mean, Median and Mode (p. ) Additional Resources Miscellaneous Solutions for Challenge Yourself Reasoning, Communication and Connection Thinking Time (p. 7) Thinking Time (p. 7) Class Discussion Comparison of Mean, Median and Mode (p. ) 8
TEACHING NOTES Suggested Approach Chapter Direct and Inverse Proportion In Secondary One, students have learnt rates such as $. per egg, or. km per litre of petrol etc. Teachers may wish to epand this further by asking what the prices of, or eggs are, or the distance that can be covered with, or litres of petrol, and leading to the introduction of direct proportion. After students are familiar with direct proportion, teachers can show the opposite scenario that is inverse proportions. Section.: Direct Proportion When introducing direct proportion, rates need not be stated eplicitly. Rates can be used implicitly (see Investigation: Direct Proportion). By showing how one quantity increases proportionally with the other quantity, the concept should be easily relatable. More eamples of direct proportion should be discussed and eplored to test and enhance thinking and analysis skills (see Class Discussion: Real-Life Eamples of Quantities in Direct Proportion). Teachers should discuss the linkages between direct proportion, algebra, rates and ratios to assess and improve students understanding at this stage (see page of the tetbook). Teachers should also show the unitary method and proportion method in the worked eample and advise students to adopt the method that is most comfortable for them. Section.: Algebraic and Graphical Representations of Direct Proportion By recapping what was covered in the previous section, teachers should easily state the direct proportion formula between two quantities and the constant k. It is important to highlight the condition k as the relation would not hold if k (see Thinking Time on page ). Through studying how direct proportion means graphically (see Investigation: Graphical Representation of Direct Proportion), students will gain an understanding on how direct proportion and linear functions are related, particularly the positive gradient of the straight line and the graph passing through the origin. The graphical representation will act as a test to determine if two variables are directly proportional. Section.: Other Forms of Direct Proportion Direct proportion does not always involve two linear variables. If one variable divided by another gives a constant, then the two variables are directly proportional (see Investigation: Other Forms of Direct Proportion). In this case, although the graph of y against will be a hyperbola, the graph of one variable against the other will be a straight line passing through the origin. Teachers may wish to illustrate the direct proportionality clearly by replacing variables with Y and X and showing Y kx, which is in the form students learnt in the previous section. Section.: Inverse Proportion The other form of proportion, inverse proportion, can be eplored and studied by students (see Investigation: Inverse Proportion). When one variable increases, the other variable decreases proportionally. It is the main difference between direct and inverse proportion and must be emphasised clearly. Students should be tasked with giving real-life eamples of inverse proportion and eplaining how they are inversely proportional (see Class Discussion: Real-Life Eamples of Quantities in Inverse Proportion). Teachers should present another difference between both kinds of proportions by reminding students that y is a constant in direct proportion while y is a constant in inverse proportion (see page of the tetbook). Section.: Algebraic and Graphical Representations of Inverse Proportion Similar to direct proportion teachers can write the inverse proportion formula between two quantities and the constant k. It is important to highlight the condition k as the relation would not hold if k (see Thinking Time on page ). Although plotting y against gives a hyperbola, and does not provide any useful information, teachers can show by plotting y against and showing direct proportionality between the two variables (see Investigation: Graphical Representation of Inverse Proportion). 9
Section.: Other Forms of Inverse Proportion Inverse proportion, just like direct proportion, may not involve two linear variables all the time. Again, teachers can replace the variables with Y and X and show the inverse proportionality relation Y k. Challenge Yourself Question involves stating the relations between the variables algebraically and manipulating the equations. For Question, teachers may wish to state the equation relating z, and y outright initially, and eplain that when z is directly proportional to, y is treated as a constant. The same applies to when z is inversely proportional to y, will be considered as a constant. Question follows similarly from Question.
WORKED SOLUTIONS Investigation (Direct Proportion). The fine will increase if the number of days a book is overdue increases. Fine when a book is overdue for days. Fine when a book is overdue for days 9 The fine will be doubled if the number of days a book is overdue is doubled. Fine when a book is overdue for days. Fine when a book is overdue for days 9 The fine will be tripled if the number of days a book is overdue is tripled. Fine when a book is overdue for days. Fine when a book is overdue for days 7. The fine will be halved if the number of days a book is overdue is halved. Fine when a book is overdue for days Fine when a book is overdue for 9 days The fine will be reduced to of days a book is overdue is reduced to of the original number if the number of the original number. Class Discussion (Real-Life Eamples of Quantities in Direct Proportion) The following are some real-life eamples of quantities that are in direct proportion and why they are directly proportional to each other. In an hourly-rated job, one gets paid by the number of hours he worked. The longer one works, the more wages he will get. The wages one gets is directly proportional to the number of hours he worked. A Singapore dollar coin weights approimately g. As the number of coins increases, the total mass of the coins will increase proportionally. The total mass of the coins is directly proportional to the number of coins. The circumference of a circle is equivalent to the product of p and the diameter of the circle. As the diameter increases, the circumference increases proportionally. The circumference of the circle is directly proportional to the diameter of the circle. The speed of a moving object is the distance travelled by the object per unit time. If the object is moving at a constant speed, as the distance travelled increases, then the time spent in travelling increases proportionally. The distance travelled by the object is directly proportional to the time spent in travelling for an object moving at constant speed. The length of a spring can be compressed or etended depending on the force applied on it. The force required to compress or etend a spring is directly proportional to the change in the length of the spring. This is known as Hooke s Law, which has many practical applications in science and engineering. Teachers may wish to note that the list is not ehaustive. Thinking Time (Page ) If we substitute k into y k, then y. This implies that for all values of, y. y cannot be directly proportional to in this case. Investigation (Graphical Representation of Direct Proportion) y in this contet means that for any additional number of a day a book is overdue, the fine will increase by cents.. Fine (y cents) 9 7 7 8 9 Fig... The graph is a straight line.. The graph passes through the origin. Thinking Time (Page 7). Since y is directly proportional to, y k k y Number of days () Since k, then we can rename k k where k is another constant. Hence, k y, where k and is directly proportional to y.. k y is the equation of a straight line. When y,. We will get a straight line of against y that passes through the origin.. If the graph of y does not pass through the origin, then y k + c, when c. Since and y are not related in the form y k, y is not directly proportional to.. As increases, y also increases. This does not necessarily conclude that y is directly proportional to. It is important that when increases, y increases proportionally. Also, when, y. y k + c is an eample of how increases and y increases, but y is not directly proportional to.
Investigation (Other Forms of Direct Proportion). y is not directly proportional to. The graph of y against is not a straight line that passes through the origin.. y.. Time taken when speed of the car is km/h Time taken when speed of the car is km/h The time taken will be doubled when the speed of the car is halved. Time taken when speed of the car is km/h Time taken when speed of the car is km/h The time taken will be tripled when the speed of the car is reduced to of its original speed. 8 Fig.. y is directly proportional to. The graph of y against is a straight line that passes through the origin. Thinking Time (Page ). C.n + C.n C. n n C n Since is a constant, then n is directly proportional C to C. The variable is C.. y y Since y is a constant, then y is directly proportional to. Investigation (Inverse Proportion). The time taken decreases when the speed of the car increases. Time taken when speed of the car is km/h. Time taken when speed of the car is km/h. The time taken will be halved when the speed of the car is doubled. Time taken when speed of the car is km/h Time taken when speed of the car is km/h Class Discussion (Real-Life Eamples of Quantities in Inverse Proportion) The following are some real-life eamples of quantities that are in inverse proportion and why they are inversely proportional to each other. Soldiers often dig trenches while serving in the army. The more soldiers there are digging the same trench, the faster it will take. The time to dig a trench is therefore inversely proportional to the number of soldiers. The area of a rectangle is the product of its length and breadth. Given a rectangle with a fied area, if the length increases, then the breadth decreases proportionally. Therefore, the length of the rectangle is inversely proportional to the breadth of the rectangle. The density of a material is the mass of the material per unit volume. For an object of a material with a fied mass, the density increases when the volume decreases proportionally. The density of the material is inversely proportional to the volume of the material. The speed of a moving object is the distance travelled by the object per unit time. For the same distance, when the speed of the object increases, the time to cover the distance is decreased proportionally. The speed of the object is inversely proportional to the time to cover a fied distance. For a fied amount of force applied on it, the acceleration of the object is dependent on the mass of the object. When the mass of the object increases or decreases, the acceleration of the object decreases or increases proportionally. This is known as Newton s Second Law and has helped to eplain many physical phenomena occurring around us. Teachers may wish to note that the list is not ehaustive. Thinking Time (Page ) If we substitute k into y k, then y. This implies that for all values of, y y cannot be inversely proportional to in this case. The time taken will be reduced to when the speed of the car is tripled. of the original number