Download Chapter

Liu Po Shan Memorial College
S4 Mathematics Teaching Scheme (2011-12)
Subject Teachers: Mr. Chiu K.Y., Mr. Lau C.K., Mr. Lo C.S., Ms Lo Y.T.
Lessons per week: 8 / 7
Textbook: New Century Mathematics 4A / 4B (Oxford)
Week / Date
Chapter
week 1 (1/9 – 2/9)
Revision
week 2 (5/9 – 9/9)
Chapter 1
Number Systems
week 3 – 4 (12/9 – 23/9)
Chapter 3
Quadratic Functions
week 5 – 7 (26/9 – 14/10)
Chapter 4
Quadratic Equations in One Unknown
week 8 – 10 (17/10 – 4/11)
Chapter 5
Basic Knowledge of Functions
week 11 – 13 (7/11 – 25/11)
Chapter 2
Equations of Straight Lines
week 14 – 17 (28/11 –21/12)
Chapter 6
More About Polynomials
week 17 – 19 (23/12 – 2/1)
Christmas and New Year Holiday
week 19 (3/1)
Revision
week 19 – 21 (4/1 – 19/1)
S4 First Term Exam
week 21 – 23 (20/1 – 31/1)
Lunar New Year Holiday
week 23 (1/2 – 3 /2)
Chapter 7
Further Applications of Mathematics
week 24 - 25 (6/2 – 17 /2)
Chapter 8
Exponential Functions
week 26 – 27 (20/2 – 2/3)
Chapter 9
Logarithmic Functions
week 28 - 29 (6/3 – 16/3)
Chapter 10
Rational Functions
week 30 – 32 (19/3 – 3/4)
Chapter 11
Basic Properties of Circles
week 32 – 33 ( 4/4 – 13/4)
Easter Holiday
week 34 – 37 (16/4 – 10/5)
Chapter 12
More About Basic Properties of Circles
week 38 – 40 (14/5 – 1/6)
Chapter 13
Basic Trigonometry
week 41 (4/6 – 7/6)
Chapter 14
Further Applications of Mathematics
week 41 – 43 (8/6 – 22/6)
S4 Final Exam
Page 1
Week / Date
Week 1
(1/9 – 2/9)
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.
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.
Week 2
(5/9 – 9/9)
Page 2
1A (1-5, 13, __, __)
LD
Chapter 3 Quadratic Functions
3.1 Solving Quadratic Equations by the
Factor Method
•Learn to use different methods to factorize the 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 3B (1, 3, 5, 7, __, __)
3.3 Solving Quadratic Equations by the
Quadratic Formula
• Recognize the quadratic formula and understand its 3C (3, 7, 9, 13, 17, 21, __, __)
polynomials in quadratic equations so as to solve the
equations.
quadratic equation in two unknowns to solve the
corresponding quadratic equation in one unknown.
usage in solving quadratic equations.
LD
• Recognize the discriminant from the quadratic formula
Week 3 – 4
(12/9 – 23/9) 3.4 Relations between the Discriminant and and its relation between the nature of roots of quadratic
the Nature of Roots
equations.
•Learn to solve practical problems using quadratic
3.5 Practical Problems Leading to Quadratic equations systematically.
Equations
3D (7, 9, 11, 13, 15, 19, __, __)
3E (3, 5, 7, 15, 21, __, __)
LD
3.6 Forming Quadratic Equations
• Learn to form a quadratic equation from two given roots.
3.7 Relations between Roots and
Coefficients
•Understand the relations between the roots and
coefficients of a quadratic equation.
3G (3, 5, 9, 17, 19, __, __)
• Master the applications of the relations between the roots
and coefficients.
Page 3
3F (3, 5, 7, 11, 15, __, __)
Chapter 4 Quadratic Equations in One
Unknown
• Recognize the independent and dependent variables of
4.1 Basic Concept of Functions
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.
4.2 Representations of Functions
Week 5 - 7
(26/9–14/10)
4.3 Notation of a Function: f(x)
• 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.
5.2 Finding the Maximum or Minimum
Values of Quadratic Functions by the
Algebraic Method
4B (5, 7, 9, 11, 15, 23, __, __)
• 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.
Chapter 5 Basic Knowledge of Functions
5.1 Features of the Graphs of Quadratic • Recognize the graph of y=ax2and its features.
•Further recognize the graph of y = ax2 + bx + c and its
Functions
features.
Week 8 – 10
(17/10 – 4/11)
4A (3, 5, 7, 11, __, __)
•Learn to make a quadratic polynomial a perfect square
by the method of completing the square.
• 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 11 – 13
(7/11 – 25/11)
• Learn to find the equation of a straight line from two
2A (5, 7, 12, 13, __, __)
given points.
• Learn to find the equation of a straight line from the
2B (3, 5, 8, 9, 13, __, __)
coordinates of a given point and the slope of the line.
•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
• Learn to perform divisions of polynomials by long
division.
6A (5, 9, 11, 13, 19, __, __)
6.2 Remainder Theorem
• 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.
Week 14 – 16
(28/11 – 21/12)
• Know how to use Factor Theorem and the known factor
to factorize polynomials.
• Know to factorize polynomials with leading coefficient
equal to 1.
• Know how to factorize polynomials with integral
coefficients.
week 17 – 19
(23/12 – 2/1)
Week 19 – 21
(4/1 – 19/1)
Week 21 – 23
(20/1 – 31/1)
Christmas and New Year Holiday
S4 First Term Examination
Lunar New Year Holiday
Page 5
6C (5, 7, 9, 13, 21, __, __)
Week 23
(1/2 – 3/2)
Chapter 7 Further
Applications
of
Mathematics
7.1 Calculation of Profit and Selling Price • Understand more sophisticated real-life problems related
to profit and selling price and their functional relation.
• Appreciate the connections between the following areas:
—Setting up a functional relation between 2 variables
from the conditions given in a question
—A function and its domain
—Finding the maximum/minimum value of a quadratic
function
—Maximum/Minimum value of a function and its
boundary conditions
Chapter 8 Exponential Functions I
8.1 Rational Indices
Week 24 - 25
(6/2 –17/2)
8.2 Exponential Functions
• Understand the meaning of the nth root of a number and
8A (3, 7, 9, 23, 29, __, __)
know how to express it with radical sign.
8A (15, 19, 25, 31, 33, 35, __, __)
• Understand the meaning of rational indices and know
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.
Page 6
8B (1,3, 5, 7, 11, __, __)
Chapter 9 Logarithmic Functions
9.1 Meaning of Logarithms

• 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.
Explore and study the graphs of logarithmic
9.2 Logarithmic Functions and Their •
functions,
recognize their properties and the relationship
Graphs
between them and the graphs of exponential functions.
Week 26 – 27
(20/2 – 2/3)
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.
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.
Page 7
9A (3, 5, 7, 13, 17, 19, 21, __)
9B (3, 5, 7, __, __)
9C (3, 5, 11, 13, 19, 27, __, __)
9D (5, 7, 9, 11, __, __)
LD
Chapter 10 Rational Functions
10.1 Highest Common Factor and Least • Learn to make use of factorization to find the Highest
Common Factor (H.C.F.) of polynomials in one or two
Common Multiple
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
(6/3 – 16/3)
10.2 Addition and Subtraction of Rational • Recognize the rational function and its definition.
Functions
• 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 Rational
• Be able to perform multiplication and division of rational
Functions
10C (7, 11, 15, 19, 27, __, __)
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
(19/3 – 3/4)
• Recognize the names of various parts relating to a circle.
• Recognize the names and the meanings of some special 11A (3, 5, 15, 23, __, __)
11A (7, 9, 13, 17, __, __)
circles.
• 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.
11B (3, 5, 13, 15, 19, __, __)
•Understand the meanings and the properties of equal
chords and distance of chord from centre and know how 11B (7, 9, 11, 17, 21, __, __)
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
• 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 and
• Comprehend the equal relationships among angles, arcs
Chords
and chords.
• Understand the proportion relations between arcs and
angles at the centre and between arcs and angles at the
circumference.
week 32 – 33
( 4/4 – 13/4)
Easter Holiday
Page 9
11C (5, 7, 11, 17, __, __)
11C (9, 13, 15, 21, __, __)
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 34 – 37
(16/4 – 10/5)
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
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
• 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.
Page 10
LD
LD
Chapter 13 Basic Trigonometry
13.1 Trigonometric Ratios of Any Angle
13.2 Trigonometric Identities
• Learn the basic trigonometric identities.
• Know to use the identities to simplify trigonometric
ratios of n·90°±θ to those of θ.
13.3 Graphs of Trigonometric Functions
• Recognize the graph of y = sin x.
•Recognize the graph of y = cos x.
•Recognize the graph of y = tan x.
Week 38 – 40
(14/5 – 1/6)
13.4 Trigonometric Equations
Chapter 14 Further Application of
Mathematics
 14.1 Ptolemy’s Theorem and Its
Applications
Week 41
(4/6 – 7/6)
Week 41 – 43
(8/6 – 22/6)
• Understand angles of rotation and the quadrants in which 13A (5, 7, 9, 11, 15, __, __)
they lie.
• Understand the definitions of the trigonometric ratios of
any angle.
• Explore the change in the signs of the trigonometric
ratios in different quadrants.
• Explore the patterns of the values of trigonometric ratios.
• Know how to use a calculator to find the values of
trigonometric ratios of any angle.
• Learn to find the corresponding angle from the value of a
given trigonometric ratio.
• Learn the method for solving equations involving one
kind of trigonometric ratio.
• Learn the method for solving equations involving two
kinds of trigonometric ratios.
• Recognize and understand the Ptolemy’s Theorem.
• Explore and investigate the applications of Ptolemy’s
Theorem.
• Appreciate the connections between the following areas:
— properties of a cyclic quadrilateral
— angle properties of a circle
— Pythagoras’ Theorem
— properties of a regular pentagon
— solving quadratic equations in one unknown
S4 Final Examination
Page 11
13B (1, 3, 5, 9, 11, 15, __, __)
I﹕Use drawing software to plot
trigonometric functions and
investigate period, maximum
and minimum values.
13C (1, 5, 9, 11, 15, 17, __, __)
I﹕Use Graphmatica to
investigate the translation,
reflection and magnification of
images.