Tin Ka Ping Secondary School 05-06 F. Mathematics Syllabus Chapter Rate and Time Guide. Rates. s A. Basic Concept of s B. s of Three Quantities Learn the concept of a rate. Learn the concepts of a ratio and a continued ratio. Learn the properties of ratio and continued ratio. concept of rate through Activity. and daily-life examples. State that expressing the same rate in different units can serve different purposes. meanings of ratio and continued ratio. Learn how to simplify a ratio and a continued ratio. relation ad = bc for two equal ratios a : b and c : d through worked examples. properties of ratio and continued ratio through worked examples. Rates s. Applications of s A. Similar Figures B. Scale Drawings Learn the applications of equal ratios on similar figures and scale drawings. application of ratios on similar figures. application of ratios on scale drawings such as maps and floor plans. Applications of s
Chapter Identities and Factorization Time Guide 4 7. Meaning of Identities. Some Important Algebraic Identities A. Difference of Two Squares B. Perfect Square. Factorization of Simple Algebraic Expressions A. By Taking Out the Common Factors B. By Grouping Terms Method C. By Using Identities Distinguish between equations and identities. Learn how to find the unknown constant(s) in an identity. Learn the identity of the difference of two squares. Learn the identities of the perfect square. Use the above identities to expand expressions or find the values of expressions. Factorize an algebraic expression by taking out the common factors. Factorize an algebraic expression by the grouping terms method. Factorize an algebraic expression by using identities. Guide students to explore the difference between equations and identities through Activity.. Demonstrate how to determine whether an equation is an identity. methods of finding the unknown constants in an identity. Guide students to investigate the identity of the difference of two squares through Activity. and Activity.. use of the identity of the difference of two squares to expand expressions or to find the values of expressions. Guide students to investigate the identities of the perfect squares through Activity.. use of the identities of the perfect squares to expand expressions or to find the values of expressions. Revise the technique of factorization by taking out the common factors. Consolidate the technique of factorization by taking out the common factors with the help of the basic topical worksheet 7. technique of factorization by the grouping terms method. Consolidate the technique of factorization by grouping terms method with the help of the basic topical worksheet 8. technique of factorization using the identities of the difference of two squares and the perfect square. Consolidate the technique of factorization by using identities with the help of the basic topical worksheet 9. Meaning of Identities Some Important Algebraic Identities Animation: Identity of the Difference of Two Squares Activity.: Identity of the difference of two squares. Factorization of Simple Algebraic Expressions. Factorization by Grouping Terms Method and by Using Identities
Chapter Time Algebraic Fractions and Formulas Guide 5 6. Manipulation of Simple Algebraic Fractions A. Multiplication and Division of Algebraic Fractions B. Addition and Subtraction of Algebraic Fractions. Formulas and their Manipulations A. Formula and Substitution B. Change of Subject Learn to manipulate simple algebraic fractions, including addition, subtraction, multiplication and division. Understand the meaning of formulas. Learn to use the method of substitution to find unknowns in a formula. Learn how to change the subject of a formula. meaning of algebraic fraction. Compare simplification of algebraic fractions to simplification of fractions. Compare multiplication and division of algebraic fractions to those of fractions. Consolidate the techniques of multiplication and division of algebraic fractions with the help of the basic topical worksheet. Compare addition and subtraction of algebraic fractions to those of fractions. Consolidate the techniques of addition and subtraction of algebraic fractions with the help of the basic topical worksheet. meaning of formula. use of the method of substitution in finding unknowns in formulas. Illustrate the importance of changing of subject of a formula through Activity.. technique of changing of subject of a formula not involving radical signs. Consolidate the technique of changing subject of a formula with the help of the basic topical worksheet 5. Manipulation of Simple Algebraic Fractions Formulas and their Manipulations
Chapter 4 More about Factorization of Polynomials Time Guide 7 4. Factorization by Cross-method A. Quadratic Polynomials in One Variable x + qx + r B. Quadratic Polynomials in One Variable px + qx + r C. Quadratic Polynomials in Two Variables 4. Sum and Difference of Two Cubes (Non-foundation) Learn to factorize quadratic polynomials in one variable x + qx + r by the cross-method. Learn to factorize quadratic polynomials in one variable px + qx + r by the cross-method. Learn to factorize quadratic polynomials in two variables by the cross-method. Learn to factorize polynomials using the identities of the sum and difference of two cubes. technique of factorization of quadratic polynomials in one variable x + qx + r by the cross-method. technique of factorization of quadratic polynomials in one variable px + qx + r by the cross-method. technique of factorization of quadratic polynomials in two variables by the cross-method. Consolidate the techniques of factorization by cross-method with the help of the basic topical worksheet 0. identities of the sum and the difference of two cubes and the techniques of using them to factorize polynomials. Factorization by Cross-method Drilling Program: Factorization of Quadratic Polynomials Factorization by Using Identities of Sum and Difference of Two Cubes
Chapter 5 Approximation and Errors Time Guide 5. Significant Figures A. Meaning of Significant Figures B. Significant Figures and Rounding Off 5. Errors A. Absolute Error B. Maximum Absolute Error C. Relative Error D. Percentage Error Understand the meaning of significant figures and the skills of rounding off numbers. Understand and calculate different types of error such as absolute error, maximum absolute error, relative error and percentage error. Guide students to explore the most important digit and the meaning of significant figures through Activity 5.. most significant figure and the number of significant figures of a number. Demonstrate how to round off a number to the required number of significant figures through worked examples. concept of absolute error. Guide students to explore the meaning of maximum absolute error through Activity 5. and to find the lower and upper limits of the actual value. Illustrate the shortcoming of absolute error through a daily-life example. Hence, introduce the concept of relative error. Introduce percentage error as an alternative way of expressing relative error for easy comparison.. Meaning of Significant Figures. Significant Figures and Rounding off Drilling Program: Find the Approximate Values. Absolute Error and Maximum Absolute Error. Relative Error and Percentage Error
Chapter 6 Time Angles related to Rectilinear Figures Guide 5 6. Angles and Sides of a Triangle A. Angle Sum of a Triangle and Exterior Angle of a Triangle B. Properties of Isosceles Triangles and Equilateral Triangles C. Identifying Isosceles Triangles and Equilateral Triangles Review the property of angle sum of a triangle and use it to find unknowns in figures. Learn the property of an exterior angle of a triangle and use it to find unknowns in figures. Learn the properties of isosceles triangles and equilateral triangles, and use them to find unknowns in figures. Recognize the conditions for a triangle to be isosceles or equilateral. Revise the property of angle sum of a triangle. property of exterior angle of a triangle. Students may need some time in mastering the concept of exterior angle. Consolidate the concepts of angle sum of a triangle and exterior angle of a triangle with the help of the basic topical worksheet 57. names of different parts of an isosceles triangle. Guide students to explore the properties of isosceles triangles through Activity 6.. properties of equilateral triangles. Teachers may encourage abler students to prove the properties of an equilateral triangle by using the properties of an isosceles triangle. Demonstrate the use of the above properties in finding unknowns in simple figures through worked examples. Consolidate the concepts of isosceles triangles and equilateral triangles with the help of the basic topical worksheet 58. Demonstrate how to determine whether a triangle is isosceles / equilateral.. Angle Sum of a Triangle and Exterior Angle of a Triangle. Properties of Isosceles Triangles and Equilateral Triangles. Identifying Isosceles Triangles and Equilateral Triangles
Time Guide 4 6. Angles of a Polygon A. Sum of Interior Angles of a Polygon B. Sum of Exterior Angles of a Polygon 6. Tessellation Explore the Guide students to property of the sum explore the property of of interior angles of the sum of interior a polygon and use angles of polygons it to find unknowns through Activity 6.. in simple figures. Guide students to Explore the explore the property of property of the sum the sum of exterior of exterior angles angles of polygons of a polygon and through Activity 6.. use it to find Demonstrate the use of unknowns in the above properties in simple figures. finding unknowns in simple figures through Explore which regular polygons can tessellate. worked examples. Guide students to explore which regular polygons can tessellate through Activity 6.4 and Activity 6.. Show students different tessellation patterns formed by more than one type of regular polygons. Encourage students to create their own tessellation patterns through Project Work (p. E.).. Sum of Interior Angles of a Polygon. Sum of Exterior Angles of a Polygon Extra Activity: Sum of exterior angles of polygons Tessellation Activity 6.: Tessellation of different figures
Chapter 7 Simple Statistical Diagrams and Graphs (II) Time Guide 6 7. Frequency Distribution and its Graphical Representation A. Review on Histograms B. Frequency Polygons and Curves C. Cumulative Frequency Polygons and Curves 7. Choosing an Appropriate Diagram to Present Data 7. Abuses of Statistical Diagrams Review how to construct histograms. Learn how to construct and interpret frequency polygons and curves. Learn how to construct and interpret cumulative frequency polygons and curves. Learn how to obtain percentiles, quartiles and medians from cumulative frequency polygons and curves. Learn to choose an appropriate diagram according to the nature of the data and the characteristics of the diagram. Learn to recognize inappropriate / misleading presentations of statistical diagrams. Revise the construction of histogram. Consolidate the concept of histograms with the help of the basic topical worksheet 68. Introduce what are frequency polygons and curves. Learn the property of frequency polygons through Activity 7.. Learn the procedures of drawing frequency polygons and curves. concept of cumulative frequency through Activity 7.. Introduce what are cumulative frequency polygons and curves. Learn the procedures of drawing cumulative frequency polygons and curves. Learn how to interpret cumulative frequency polygons and curves. Learn how to find percentiles, quartiles and medians from cumulative frequency polygons or curves. Consolidate the concepts of frequency polygons and cumulative frequency polygons with the help of the basic topical worksheet 69. Recognize the importance of choosing different statistical diagrams for different purposes through Activity 7.. By drawing different statistical diagrams to present the same set of data, learn to choose an appropriate diagram according to the nature of the data and the characteristics of the diagram. Recognize that some statistical diagrams will mislead readers through Activity 7.4. Learn how to use statistical diagrams to present data appropriately.. Frequency Polygons and Frequency Curves. Cumulative Frequency Polygons and Cumulative Frequency Curves Tool: Simple Statistical Graphs Software Demonstration: Construction of Statistical Diagrams Choosing an Appropriate Diagram to Present Data Abuses of Statistical Diagrams
Chapter 8 Linear Equations in Two Unknowns Time Guide 8 8. Linear Equations in Two Unknowns and their Graphs A. Basic Concept of Linear Equations in Two Unknowns B. Graphs of Linear Equations in Two Unknowns 8. Simultaneous Linear Equations in Two Unknowns and their Solutions A. Graphical Method B. Algebraic Methods C. Some Special Simultaneous Linear Equations in Two Unknowns (Enrichment) 8. Applications of Simultaneous Linear Equations in Two Unknowns Recognize that a linear equation in two unknowns has an infinite number of solutions. Plot and explore the graphs of linear equations in two unknowns. Understand the concept of simultaneous linear equations in two unknowns. Learn how to solve simultaneous equations in two unknowns by graphical method. Learn how to solve simultaneous equations in two unknowns by algebraic methods the method of substitution and the method of elimination. Learn some special simultaneous equations in two unknowns Learn how to apply simultaneous linear equations to solve daily-life problems. definition of linear equation in two unknowns. Recognize that the graph of linear equation in two unknowns is a straight line through Activity 8.. Consolidate the technique of drawing graphs of linear equations in two unknowns with the help of the basic topical worksheet 6. Explore the effects of changing a and b on the graphs of the linear equation y = ax + b through Activity 8.. definition of simultaneous linear equations in two unknowns. Learn how to find the solution of simultaneous linear equations in two unknowns by graphical method. Remind students that solutions obtained by graphical method is only an approximation. Consolidate the technique of finding the solution of simultaneous linear equations in two unknowns by graphical method with the help of the basic topical worksheet 7. two algebraic methods of solving simultaneous linear equations in two unknowns, method of substitution and method of elimination. Discuss with students how to determine which algebraic method is more effective in solving simultaneous equations. Consolidate the techniques of finding the solution of simultaneous linear equations in two unknowns by algebraic methods with the help of the basic topical worksheet 8. Introduce that some simultaneous linear equations may have an infinite number of solutions, while some of them have no solutions. Learn the steps in solving real-life problems using simultaneous linear equations. Linear Equations in two Unknowns and their Graphs Activity 8.: Investigating the graphs of linear equations in two unknowns. Solve Simultaneous Linear Equations by Graphical Method. Solve Simultaneous Equations by Algebraic Methods. Some Special Simultaneous Linear Equations in Two Unknowns Applications of Simultaneous Linear Equations in Two Unknowns
Chapter 9 Laws of Integral Indices Time Guide 9. Laws of Positive Integral Indices A. Review on the Laws of Indices for a m a n and a m a n B. Laws of Indices for (a m ) n C. Laws of Indices for (ab) n and n a b 9. Zero and Negative Integral Indices 9. Scientific Notation A. Meaning of Scientific Notation B. Applications of Scientific Notation 9.4 Different Numeral Systems A. Denary System B. Binary System C. Hexadecimal System D. Conversions between Denary System and Other Numeral Systems (Non-foundation) Learn the laws of indices with positive integral indices. Understand the meanings of zero and negative integral indices. meaning of scientific notation. Use scientific notation to express and manipulate extremely large or small numbers. Learn the denary system and the place values in denary numbers. Learn the binary system and the place values in binary numbers. Learn the hexadecimal system and the place values in hexadecimal numbers. Learn the conversions between the denary system and other numeral systems. Revise the following two laws of indices learnt in Book B Chapter : a m a n = a m + n a m a n = a m n Guide students to explore the laws of indices for (a m ) n through Activity 9.. Guide students to explore the laws of indices for (ab) n a and through b Activity 9.. Learn to simplify expressions by using laws of indices through worked examples. Consolidate the concepts of laws of integral indices with the help of the basic topical worksheets 0 &. Study the meanings of zero and negative integral indices through Activity 9.. Consolidate the concepts of zero and negative integral indices with the help of the basic topical worksheet. method to express numbers as powers of 0 through Activity 9.4. Learn to use scientific notation to express and evaluate extremely large or small numbers through worked examples. Consolidate the techniques of the conversion between numbers in scientific notation and integers / decimals with the help of the basic topical worksheet. By grouping objects in different ways, introduce the concepts of denary system, binary system and hexadecimal system. Study the place value of each digit in denary numbers, binary numbers and hexadecimal numbers. Learn to express denary numbers, binary numbers and hexadecimal numbers in expanded form. Learn the conversions between the denary system and other numeral systems. n Laws of Positive Integral Indices Zero and Negative Integral Indices Drilling Program: Laws of Integral Indices (II) Scientific Notation Different Numeral Systems
Chapter 0 Introduction to Deductive Geometry Time Guide 0. Basic Concept of Deductive Reasoning 0. Deductive Geometry A. Euclid and Elements B. Definition, Axiom and Theorem Realize the shortcomings of making judgement by intuition. Hence, lead students to learn the importance of deductive reasoning in studying mathematics. Learn the story about Euclid, his book Elements and his learning attitude. Understand the meanings of definition, axiom and theorem, and their relations. Guide students to realize the shortcomings of making judgement by intuition through Activity 0.. Point out the importance of deductive reasoning in studying mathematics. Share with students the story about Euclid and highlight his learning attitude. Discuss the contribution of Euclid s Elements on deductive reasoning. meanings of definition, axiom and theorem and their relations with mathematical examples. Teachers may provide daily-life examples to help students understanding. The Importance of Deductive Reasoning and Deductive Geometry
Time Guide 4 7 0. Deductive Proofs about Angles related to Lines and Triangles A. Angles related to Lines B. Angles related to Triangles 0.4 Deductive Proofs about Congruent, Isosceles and Similar Triangles A. Congruent Triangles B. Isosceles Triangles C. Similar Triangles Learn how to present simple proofs about angles related to lines and triangles. Understand and use the conditions for congruent triangles to perform simple proofs. Understand and use the properties and conditions of isosceles triangles to perform simple proofs. Understand and use the conditions for similar triangles to perform simple proofs. Clarify which part of a statement is the given condition, and which part is the conclusion that needs to be proved. Discuss the thinking method in performing proofs. Remind students to give reasons for each step in their proofs. Guide students to perform simple proofs by applying some properties about angles related to lines learnt in Book B Chapter. Guide students to perform simple proofs by applying some properties about angles related to triangles learnt in Book A Chapter 6. Revise the conditions for congruent triangles: SSS, SAS, ASA, AAS and RHS. Learn to perform proofs about congruent triangles. Guide students to prove the properties of isosceles triangles using congruent triangles. Revise the conditions for similar triangles: AAA, sides prop. and ratio of sides, inc.. Learn to perform proofs about similar triangles. Deductive Proofs about Angles related to Lines and Triangles. Deductive Proofs about Congruent Triangles and Isosceles Triangles. Deductive Proofs about Similar Triangles
Time Guide 0.5 Construction Using Compasses and Straight Edge A. Copying an Angle B. Construction of Angle Bisector C. Construction of Perpendicular Bisector D. Construction of Special Angles: 0, 45, 60 and 90 E. Construction of Special Regular Polygons (Non-foundation) Learn to use compasses and straight edge to copy an angle, construct the angle bisector of an angle, construct the perpendicular bisector of a line segment, construct the special angles 0, 45, 60 and 90, construct some special regular polygons. Illustrate how to do different construction using a straight edge and compasses. concepts of angle bisectors and perpendicular bisectors. Demonstrate how to construct some regular polygons. Teachers may encourage abler students to construct regular -sided polygons and 4-sided polygons. Animation:. Use Compasses and Straight Edge to Copy an Angle and Construct Bisectors. Construction of Regular Polygons
Chapter nal and Irrational Numbers Time Guide. Square Roots and Surds A. Square Roots B. Surds C. Estimating the Values of Square Roots. nal and Irrational Numbers A. nal Numbers B. Irrational Numbers. Manipulation of Surds A. Properties of Surds B. nalization of Denominators C. Surd in its Simplest Form D. Addition, Subtraction, Multiplication and Division of Surds (Non-foundation) meanings of square roots and surds. Learn how to estimate the values of square roots. definitions of rational numbers and irrational numbers. properties of surds. Learn how to rationalize the denominators of expressions. Learn how to express surds in their simplest forms. Learn the addition, subtraction, multiplication and division of surds. Realize the relation between squares and square roots using integral examples. symbol of radical sign and the meaning of surds. Guide students to estimate the values of square roots. Understand that all integers, fractions, terminating decimals and recurring decimals can be written in the form Learn the definitions and concepts of rational numbers and irrational numbers. m. n Guess the following properties of surds through Activity.: a b a b a ab b Realize the advantage of rationalization of the denominators. Consolidate the technique of the rationalization of denominators with the help of the basic topical worksheet 4. Learn to express surds in their simplest forms. Consolidate the technique of expressing surds in their simplest forms with the help of the basic topical worksheet 5. Learn the addition, subtraction, multiplication and division of surds. Consolidate the techniques of the addition, subtraction, multiplication and division of surds with the help of the basic topical worksheets 6 & 7. Square Roots and Surds Activity.: The value of nal Numbers and Irrational Numbers Manipulation of Surds
Chapter Pythagoras Theorem Time Guide 4. Pythagoras Theorem and its Proofs A. Pythagoras Theorem B. Proofs of Pythagoras Theorem C. Surds and Irrational Numbers (Non-foundation topics: The First Irrational Numbers: ). Converse of Pythagoras Theorem. Applications of Pythagoras Theorem and its Converse Learn Pythagoras theorem. Admire different proofs of Pythagoras theorem (including the method used in proving Kou-ku theorem ). Learn the story about the first crisis in Mathematics. Learn the representation of irrational numbers on a number line. Learn the converse of Pythagoras theorem. Use Pythagoras theorem and its converse to solve problems in real life. Deduce the Pythagoras theorem through Activity.. Apply Pythagoras theorem to find unknowns in simple right-angled triangles. Guide students to perform proof of Pythagoras theorem through Activity.. Teachers may use the teaching aid Pythagoras Theorem Board to demonstrate a proof of the Pythagoras theorem. Introduce some other proofs of Pythagoras theorem. Tell the story about the first crisis in Mathematics. Learn how to represent an irrational number on a number line with a pair of compasses. Deduce the converse of Pythagoras theorem by performing simple proofs of congruent triangles. Apply the converse of Pythagoras theorem to solve problems. Learn to apply Pythagoras theorem and its converse to solve real-life problems. Consolidate the use of Pythagoras theorem with the help of the basic topical worksheet 59.. Pythagoras Theorem and its Proofs. Surds and Irrational Numbers Activity.: Proofs of Pythagoras theorem Animation: Proofs of Pythagoras Theorem The Converse of Pythagoras Theorem Applications of Pythagoras Theorem
B Chapter Areas and Volumes (II) Time Guide. Circles A. Circumferences of Circles B. Areas of Circles. Arcs and Sectors A. Lengths of Arcs B. Areas of Sectors. Cylinders A. Volumes of Cylinders B. Total Surface Areas of Cylinders Review the formula for calculating the circumferences of circles. Explore the formula for calculating the areas of circles. Apply the formulas to calculate the circumferences and areas of circles. Introduce arcs and angles at the centre. Learn the formula for calculating arc lengths. Introduce sectors and the angles of sectors. Learn the formula for calculating the areas of sector. Understand and use formulas to calculate the volumes and total surface areas of cylinders. Revise the formula for calculating the circumferences of circles. Explore the formula for calculating the areas of circles through Activity.. Teachers may use the teaching aid Area of Circles Model to explore the formula for calculating the areas of circles. meanings of arcs and angle at the centre. Explore the formula for calculating the lengths of arcs through Activity.. meanings of sectors and angle of sectors. Learn the formula for calculating the areas of sectors. Deduce the formula for calculating the volumes of cylinders by considering the formula for calculating the volumes of prisms. Teachers may use the teaching aid Volume of Cylinders Model to explore the formula for calculating the volumes of cylinders. Explore the formula for calculating the curved surface areas of cylinders through Activity.. Consolidate the techniques of finding the lengths of arcs, areas of sectors, volumes and total surface areas of cylinders with the help of the basic topical worksheet 4. Circles Arcs and Sectors Software Demonstration: Construction of Circles, Arcs and Sectors Cylinders