Liu Po Shan Memorial College S4 Mathematics Teaching Scheme (2013-14) Subject Teachers: Chan YT, Chau YY, *Lam YC, Siu KY, Wong CY Lessons per week: 8 / 7 Textbook: New Century Mathematics 4A / 4B (Oxford) Week / Date Chapter week 1 - 2 (3/9 – 13/9) Chapter 1 Number Systems week 3 – 5 (16/9 – 4/10) Chapter 3 Quadratic Equations in One Unknown week 6 – 7 (7/10 – 18/10) Chapter 4 Basic Knowledge of Functions week 8 – 9 (21/10 – 1/11) Chapter 5 Quadratic Functions week 10 – 12 (4/11 – 22/11) Chapter 2 Equations of Straight Lines week 13 – 15 (25/11 –13/12) Chapter 6 More About Polynomials week 16 (16/12 – 19/12) Revision week 17 – 18 (23/12 – 1/1) Christmas and New Year Holiday week 19 – 20 (2/1 – 16/1) S4 First Term Exam week 21 - 22 (20/1 – 27/1) Chapter 8 Exponential Functions week 22 – 23 (28/1 – 9/2) Lunar New Year Holiday week 24 (10/2 – 14/2) Chapter 8 Exponential Functions week 25 – 27 (17/2 – 7/3) Chapter 9 Logarithmic Functions week 28 - 29 (10/3 – 21/3) Chapter 10 Rational Functions week 30 - 32 (25/3 – 11/4) Chapter 11 Basic Properties of Circles week 33 – 34 ( 14/4 – 21/4) Easter Holiday week 35 – 38 (22/4 – 22/5) Chapter 12 week 39 - 40 (26/5 – 5/6) Revision More About Basic Properties of Circles week 41 – 42 (6/6 – 20/6) S4 Final Exam Page 1 Week / Date Week 1 Week 1 - 2 Section Learning Objective Exercises Remark Revision Chapter 1 Number Systems 1.1 The Real Number Systems • Recognize the development of integers. • Recognize the development of rational numbers. • Review the method of converting recurring decimals into fractions. • Recognize the development of irrational numbers. • Recognize the relationships between real numbers and points on the number line. 1.2 Complex Numbers • Understand the needs to extend the real numbers. • Recognize imaginary numbers and imaginary units. • Recognize complex numbers. • Understand the classification in the complex number system. 1A (1-5, 13, __, __) 1.3 Manipulation of Complex Numbers • Recognize the properties of the arithmetic operations of 1B (4-7, 17, 18, __, __) imaginary numbers. • Recognize the arithmetic operations of complex numbers. • Using the property of equality of complex numbers to solve related problems. * Section 1.1D Parts I, II and III: not to be covered in 4B and 4C Page 2 LD Chapter 3 Quadratic Equations in One Unknown 3.1 Solving Quadratic Equations by the •Learn to use different methods to factorize the Factor Method polynomials in quadratic equations so as to solve the equations. Week 3 – 5 3A (3, 7, 9, 13, 15, 23, __, __) 3.2 Solving Quadratic Equations by the Graphical Method • Recognize the use of the x-intercepts of the graph of a quadratic equation in two unknowns to solve the corresponding quadratic equation in one unknown. 3B (1, 3, 5, 7, __, __) 3.3 Solving Quadratic Equations by the Quadratic Formula • Recognize the quadratic formula and understand its usage in solving quadratic equations. 3C (3, 7, 9, 13, 17, 21, __, __) 3.4 Relations between the Discriminant and the Nature of Roots • Recognize the discriminant from the quadratic formula and its relation between the nature of roots of quadratic equations. 3D (7, 9, 11, 13, 15, 19, __, __) 3.5 Practical Problems Leading to Quadratic Equations •Learn to solve practical problems using quadratic equations systematically. 3E (3, 5, 7, 15, 21, __, __) 3.6 Forming Quadratic Equations • Learn to form a quadratic equation from two given roots. 3F (3, 5, 7, 11, 15, __, __) 3.7 Relations between Roots and Coefficients •Understand the relations between the roots and 3G (3, 5, 9, 17, 19, __, __) coefficients of a quadratic equation. • Master the applications of the relations between the roots and coefficients. Page 3 LD LD Chapter 4Basic Knowledge of Functions 4.1 Basic Concept of Functions • Recognize the independent and dependent variables of functions. • Recognize the definition of a function. • Learn to use the definition of a function to determine whether there is a function relation in a given equation. 4A (3, 5, 7, 11, __, __) 4.2 Representations of Functions • Recognize the tabular method to represent a function relation. • Recognize the algebraic method to represent a function relation. • Recognize the graphical method to represent a function relation. Week 6 - 7 4B (5, 7, 9, 11, 15, 23, __, __) 4.3 Notation of a Function: f(x) • Learn the different notations for a function. • Perform calculations to find the value of a function. • Understand the concept of domain and co-domain of a function. • Learn to find the methods of finding the domain of a function and unknowns in a function. • Perform four arithmetic of functions. • Solve practical problems involving functions. * Section 4.3C: not to be covered in 4B and 4C Chapter 5 Basic Knowledge of Functions 5.1 Features of the Graphs of Quadratic • Recognize the graph of y=ax2and its features. Functions •Further recognize the graph of y = ax2 + bx + c and its features. Week 8 – 9 5.2 Finding the Maximum or Minimum •Learn to make a quadratic polynomial a perfect square Values of Quadratic Functions by the by the method of completing the square. Algebraic Method • Study the properties of quadratic functions using the method of completing the squares. • Learn to use the method of completing the square to find the extreme values of quadratic functions and solve problems related to the features of the graph. • Use the method of completing the square to solve application problems related to quadratic functions. Page 4 5A (3, 5, 9, 12, 15, __, __) 5B (3, 5, 9, 13, 17, __, __) I﹕Use drawing software and its magnification function to investigate the limitations of graphical method and find the way to obtain a more accurate solution. LD Chapter 2 Equations of Straight Lines 2.1 Equations of Straight Lines Week 10 – 12 • Learn to find the equation of a straight line from two given points. 2A (5, 7, 12, 13, __, __) • Learn to find the equation of a straight line from the coordinates of a given point and the slope of the line. 2B (3, 5, 8, 9, 13, __, __) •Learn to find the equation of a given straight line from its slope and y-intercept. •Learn to find the slope and y-intercept of a given straight line from its equation. •Learn to find the equation of a straight line from its x-intercept and y-intercept. •Recognize the equations of vertical lines, horizontal lines and straight lines passing through the origin. 2.2 Number of Points of Intersection of •Understand and determine the number of points of Two Straight Lines intersection of two straight lines. Chapter 6 More About Polynomials 6.1 Division of Polynomials 6.2 Remainder Theorem Week 13 – 15 • Learn to perform divisions of polynomials by long division. 6A (5, 9, 11, 13, 19, __, __) • Recognize polynomials in one variable and their function notations. • Understand the Remainder Theorem. • Know how to use the Remainder Theorem to find the remainders or unknowns. 6B (3, 7, 9, 13, 21, __, __) 6.3 Factor Theorem • Understand the Factor Theorem. • Know how to use Factor Theorem and the known factor 6C (5, 7, 9, 13, 21, __, __) to factorize polynomials. • Know to factorize polynomials with leading coefficient equal to 1. • Know how to factorize polynomials with integral coefficients. Week 16 Revision week 17 – 18 Christmas and New Year Holiday Week 19 – 20 S4 First Term Examination Page 5 Chapter 8 Exponential Functions I 8.1 Rational Indices Week 21 – 22, 24 8.2 Exponential Functions • Understand the meaning of the nth root of a number and know how to express it with radical sign. 8A (3, 7, 9, 23, 29, __, __) • Understand the meaning of rational indices and know 8A (15, 19, 25, 31, 33, 35, __, __) how to use a calculator to find the values of expressions involving rational indices or surds. • Know how to simplify algebraic expressions involving rational indices or surds by applying laws of rational indices. • Recognize the meaning of exponential functions. • Know to plot the graphs of exponential and recognize the properties of this kind of graph. • Be able to apply the exponential functions to solve various real-life problems. * Sections 8.1C and 8.2B: not to be covered in 4B and 4C Week 22 – 23 Lunar New Year Holiday Page 6 8B (1,3, 5, 7, 11, __, __) Chapter 9 Logarithmic Functions 9.1 Meaning of Logarithms Week 25 – 27 • Recognize the meanings of logarithms and logarithmic functions, and understand the relationship between indices and logarithms and know how to find the value of a logarithm or the unknown in a logarithm from its definition. • Understand and know how to apply the properties of logarithms and the formula for change of base. • Learn to find the values of logarithms (including common logarithms) using a calculator. 9A (3, 5, 7, 13, 17, 19, 21, __) 9.2 Logarithmic Functions and Their • Explore and study the graphs of logarithmic functions, Graphs recognize their properties and the relationship between them and the graphs of exponential functions. 9B (3, 5, 7, __, __) 9.3 Exponential Equations and Logarithmic Equations • Learn to use algebraic method, the definition and properties of logarithms to solve exponential equations. • Learn the techniques for solving logarithmic equations. 9C (3, 5, 11, 13, 19, 27, __, __) 9.4 Applications of Logarithms • Appreciate the concept of logarithms being able to be applied to represent the magnitudes of earthquakes, and learn to make use of logarithmic functions to solve application problems involving the magnitudes of earthquakes. • Appreciate the concept of logarithms being able to be applied to represent the intensity level of sound, and learn to make use of logarithmic functions to solve application problems involving the intensity level of sound. • Understand the history of logarithmic tables and slide rules and learn the applications of logarithmic table and antilogarithmic table in calculations. 9D (5, 7, 9, 11, __, __) * Sections 9.2 and 9.4: not to be covered in 4B and 4C Page 7 LD Chapter 10 Rational Functions 10.1 Highest Common Factor and Least • Learn to make use of factorization to find the Highest Common Multiple Common Factor (H.C.F.) of polynomials in one or two variables. • Learn to make use of factorization to find the Least Common Multiple (L.C.M.) of polynomials in one or two variables. Week 28 - 29 10.2 Addition and Rational Functions Subtraction of • Recognize the rational function and its definition. • Be able to perform addition and subtraction of rational functions in one or two variables. • Be able to find the unknowns in an identity of rational functions. 10A (7, 9, 11, 17, 22, __, __) 10B (3, 5, 9, 13, 17, __, __) 10.3 Multiplication and Division of • Be able to perform multiplication and division of rational 10C (7, 11, 15, 19, 27, __, __) Rational Functions functions in one or two variables. • Be able to perform mixed arithmetic operations of rational functions in one or two variables. Page 8 LD Chapter 11 Basic Properties of Circles 11.1 Basic Knowledge about Circles 11.2 Properties of Chords of a Circle 11.3 Angle Properties of a Circle Week 30 - 32 • Recognize the names of various parts relating to a circle. 11A (3, 5, 15, 23, __, __) • Recognize the names and the meanings of some special 11A (7, 9, 13, 17, __, __) circles. LD • Understand various properties of perpendiculars from centre of a circle to chords, and know how to make use of those properties to solve problems involving circles. • Understand the meanings and the properties of equal chords and distance of chord from centre and know how to make use of those properties to solve problems involving circles. • Understand the fact that when any three non-collinear points are given, there can be one and only one circle passing through these three points. • Recognize the definitions of angles at the centre and angles at the circumference, and understand the relationship between them. • Recognize the definition of angle in a semi-circle and understand the properties of angle in a semi-circle I﹕Use Geometer Sketchpad to investigate the relationship between tangent and radius. LD 11B (3, 5, 13, 15, 19, __, __) 11B (7, 9, 11, 17, 21, __, __) • Know how to make use of the properties of angles at the centre, angles at the circumference and angle in a semi-circle to solve problems involving circles. • Recognize the meaning of angles in the same segment and understand the property of angles in the same segment. • Know how to make use of the property of angles in the same segment to solve problems involving circles. 11.4 Relationships among Angles, Arcs • Comprehend the equal relationships among angles, arcs 11C (5, 7, 11, 17, __, __) and Chords and chords. 11C (9, 13, 15, 21, __, __) • Understand the proportion relations between arcs and angles at the centre and between arcs and angles at the circumference. week 33 – 34 Easter Holiday Page 9 I﹕Use Geometer Sketchpad to investigate the relationship between tangents and radius. I﹕Use Geometer Sketchpad to investigate the relationship between tangents and radius. LD I﹕Use Geometer Sketchpad to investigate the relationship between tangent and radius. LD Chapter 12 More About Basic Properties of Circles 12.1 Properties of Cyclic Quadrilaterals • Recognize the definition of cyclic quadrilaterals and 12A (3, 7, 8, 9, 15, 19, __, __) understand the properties of opposite angles in cyclic quadrilaterals. • Know how to make use of the property of opposite angles of a cyclic quadrilateral to solve problems involving circles. • Recognize the definition of exterior angles of a cyclic 12B (3, 5, 7, 11, 13, __, __) quadrilateral and their relationship with interior opposite angles. 12.2 Tests for Cyclic Quadrilaterals 12.3 Tangents to Circles Week 35 – 38 12.4 Angles in the Alternate Segments • Know how to test whether a set of given points (four points) are concyclic (or form a cyclic quadrilateral) according to the properties of cyclic quadrilaterals. the alternate segment and know how to perform related calculations using their properties. 12.5 Geometric Proofs on Circles • Understand the working principles of induction and deduction, and recognize their differences. • Using the basic properties of circles learnt in Chapter 11 and in this chapter, prove more geometric properties related to circles by deduction. * Section 12.5: not to be covered in 4B and 4C Week 39 – 40 Week 41 – 43 12C (7, 9, 11, 13, 17, __, __) 12D (3, 5, 11, 17, 21, __, __) • Recognize the tangents to circles and the theorems of testing for tangent lines. • Know to use the basic properties of tangents to find unknowns in circles. 12E (1, 5, 7, 11, 13, __, __) • Understand the properties of tangents drawn from an external point to a circle and know how to do calculation according to the related theorems. • Understand the concepts of common tangent and two 12F (3, 7, 11, 13, __, __) circles touching each other and know how to solve problems involving circles by applying the properties of common tangent and two circles touching each other. • Recognize the definition and properties of an angle in Revision S4 Final Examination Page 10 LD LD