343

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Unit/ Theme/

Topic Contents Objectives Core Competency Home- work Teaching Hours

3. Intuitive imagination

4. Mathematics operation

14.Mathemati

cal modeling

activities and

mathematical

inquiry activities

14.1 Understand the basic

process of mathematical

modeling activities.

14.2 Focus on a specific mathematical problem

14.3 Be able to do independent

exploration, cooperative research and finally solve the

problem

1. In the real life situation, from the perspective

of mathematics, explore, bring out questions,

analyze problems, build models, determine

parameters, calculate, test results, improve

models, and finally solve practical problems.

2. Carried out mathematical inquiry activities

and mathematical modeling activities in the

form of subject research.

1. Mathematical abstraction

2. Logical reasoning

3. Mathematical

modeling

4. Intuitive imagination

5. Mathematics operation

6. Data analysis

Related

exercises

5

15.Application of function 15.1 Dichotomy and ap- proximate solution of equa- tion

15.2 Function and mathematical

model

1. Dichotomy and finding approximate solutions

of equations

(1) Understand the relationship between the zero

point of the function and the solution of the

equation.

(2) Understand the existence theorem of zero

point of function and be able to determine the

existence of zero point of monotonic function.

2. Function and mathematical model

(1) Be able to choose the appropriate function to

describe the rule of change of real life problems.

(2) Be able to compare the difference in the increase rate of logarithmic function, Elementary function and exponential function, and

then understand the meaning of the terms \"logarithmic growth\", \"linear rise\", \"exponential

1. Mathematical abstraction

2. Logical reasoning

3. Mathematical

modeling

4. Intuitive imagination

5. Mathematics operation

Related

exercises

4

博实乐“中外融通课程”

344 Unit/ Theme/

Topic Contents Objectives Core Competency Home- work Teaching Hours

explosion\" ,etc.

16.Mathemati

cal induction

16.1 Mathematical induction 1. Understand the principle of mathematical

induction, and be able to prove some simple

propositions in the sequence with mathematical induction.

1. Mathematical abstraction

2. Logical reasoning

3. Mathematics operation

Related

exercises

6

AS P5 (Probability & Statistics 1)

Unit/ Theme/

Topic Contents Objectives Core competence Home- work Teaching Hours

1.Representation of data 1.1 Types of data 1.2 Representation of discrete data: stem-andleaf diagrams

1.3 Representation of

continuous data: histograms

1.4 Representation of

continuous data: cumulative frequency

graphs

1.5 Comparing different

data representations

1. Select a suitable way of presenting raw statistical data,

and discuss advantages and/or disadvantages that particular representations may have.

2. Construct and interpret stem-and-leaf diagrams, boxand-whisker plotshistograms andcumulative frequency

graphs.

3. Understand and use different measures of central tendency (mean, median, mode) and variation (range, interquartile range, standard deviation), e.g. in comparing

and contrasting sets of data.

4. Use a cumulative frequency graph to estimate the median value, the quartiles and the interquartile range of a

set of data.

5. Calculate the mean and standard deviation of a set of

data (including grouped data) either from the data itself

or from given totals such as



x and

 2 x ,or



 ) ( a

x and .

1. Logical reasoning

2. Mathematical

modeling

3. Mathematics

operation

4. Data analysis

Related

exercises

15



 2

) ( a x

Unit/ Theme/

Topic Contents Objectives Core Competency Home- work Teaching Hours

explosion\" ,etc.

16.Mathemati

cal induction

16.1 Mathematical induction 1. Understand the principle of mathematical

induction, and be able to prove some simple

propositions in the sequence with mathematical induction.

1. Mathematical abstraction

2. Logical reasoning

3. Mathematics operation

Related

exercises

6

AS P5 (Probability & Statistics 1)

Unit/ Theme/

Topic Contents Objectives Core competence Home- work Teaching Hours

1.Representation of data 1.1 Types of data 1.2 Representation of discrete data: stem-andleaf diagrams

1.3 Representation of

continuous data: histograms

1.4 Representation of

continuous data: cumulative frequency

graphs

1.5 Comparing different

data representations

1. Select a suitable way of presenting raw statistical data,

and discuss advantages and/or disadvantages that particular representations may have.

2. Construct and interpret stem-and-leaf diagrams, boxand-whisker plotshistograms andcumulative frequency

graphs.

3. Understand and use different measures of central tendency (mean, median, mode) and variation (range, interquartile range, standard deviation), e.g. in comparing

and contrasting sets of data.

4. Use a cumulative frequency graph to estimate the median value, the quartiles and the interquartile range of a

set of data.

5. Calculate the mean and standard deviation of a set of

data (including grouped data) either from the data itself

or from given totals such as



x and

 2 x ,or



 ) ( a

x and .

1. Logical reasoning

2. Mathematical

modeling

3. Mathematics

operation

4. Data analysis

Related

exercises

15



 2

) ( a x

AS P5 (Probability & Statistics 1)

  345

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2.Statistics 2.1 Basic ways to obtain

data and related concepts

2.2 Sampling

2.3 Statistical correlation

of paired data

2.4 One variable linear regression model

2.5 2×2 table

1. Basic methods to obtain data.

(1) Know that statistical reports, yearbooks, social surveys,

experimental design, census and sampling, Internet, etc.

are the basic ways to obtain data.

(2) Understand the concepts of population, sample, sample

size, and understand the randomness of data.

2. Sampling.

(1) Understand the meaning of simple random sampling

and the process of solving problems, and be able to use

two simple random sampling methods: lottery method

and random number method.

(2) Understand the characteristics and scope of application

of stratified random sampling, understand the necessity

of stratified random sampling, and be able to use the

method of proportional distribution.

3. Statistical correlation of paired data.

(1) Understand the statistical meaning of sample correlation coefficient and the relationship between sample

correlation coefficient and standardized data.

(2) Be able to compare the correlations of multiple paired

data from the correlation coefficient.

4. One variable Linear Regression model.

(1) Understand the meaning of one variable linear regression model and the principle of least squares.

(2) Understand the least square estimation method of parameters of one variable linear regression model, and

be able to use relevant statistical software.

5. 2×2 table.

(1) Understand the Statistical significance of 2×2 table

(2) Understand the independence test of 2×2 table and its

application.

1. Logical reasoning

2. Mathematical

modeling

3. Mathematics

operation

4. Data analysis

Related

exercises

7

3. Permutations and

combinations

3.1 The factorial function

3.2 Permutations

3.3 Combinations

1. Understand the terms permutation and combination,

and solve simple problems involving selections.

2. Solve problems about arrangements of objects in a line,

1. Logical reasoning

2. Mathematical

Related

exercises

18

博实乐“中外融通课程”

346 Unit/ Theme/ Topic Contents Objectives Core competence Home- work Teaching Hours

3.4 Problem solving with

permutations and

combinations

including those involving:repetition (e.g. the number of

ways of arranging the letters of the word ‘NEEDLESS’), restriction (e.g. the number of ways several

people can stand in a line if 2 particular people must —

or must not — stand next to each other.

modeling

3. Mathematics

operation

4. Probability 4.1 Experiments, events

and outcomes

4.2 Mutually exclusive

events and the addition

law

4.3 Independent events

and the multiplication

law

4.4 Conditional probability

4.5 Dependent events and

conditional probability

1. Evaluate probabilities in simple cases by means of enumeration of equiprobable elementary events (e.g. for the

total score when two fair dice are thrown), or by calculation using permutations orcombinations.

2. Use addition and multiplication of probabilities, as appropriate, in simple cases.

3. Understand the meaning of exclusive and independent

events, and calculate and use conditional probabilities in

simple cases, e.g. situations that can be represented by

means of a tree diagram.

1. Logical reasoning

2. Mathematical

modeling

3. Mathematics

operation

Related

exercises

20

5. Discrete

random

variables

5.1 The mode and the

modal class

5.2 The mean

5.3 The median

5.4 The range

5.5 The interquartile range

and percentiles

5.6 Variance and standard

deviation

5.7 Discrete random variables

5.8 Probability distributions

5.9 Expectation and variance of a discrete

random variable

5.10 The binomial distribu1. Construct a probability distribution table relating to a given situation involving a discrete random variable X, and calculate E(X) and Var(X). 2. Use formulae for probabilities for the binomial and geometric distributions, and recognize practical situa- tions where these distributions are suitable models (In- cluding the notations B(n, p) and Geo(p). Geo(p) de- notes the distribution in which for r = 1, 2, 3, … .. 3. Use formulae for the expectation and variance of the bi- nomial distribution and for the expectation of the geo- metric distribution. 1. Logical reason- ing 2. Mathematical modeling 3. Mathematics operation 4. Data analysis Related exercises 20

  347

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Topic Contents Objectives Core competence Home- work Teaching Hours

tion

5.11 The geometric distribution

6. The normal

distribution

6.1 Continuous random

variables

6.2 The normal distribution

6.3 Modelling with the

normal distribution

6.4 The normal approximation to the binomial

distribution

1. Understand the use of a normal distribution to model a

continuous random variable, and use normal distribution

tables.

2. Solve problems concerning a variable X, where

) , ( ~ X 2   N , including finding the value of P(X > x1),

or a relatedprobability, given the values of x1, μ, σ .

3. Finding a relationship between x1, μ and σ given the

value of P(X > x1) or a related probability.

4. Recall conditions under which the normal distribution

can be used as anapproximation to the binomial distribution (n large enough to ensure that np > 5 and nq > 5),

and use this approximation, with a continuity correction,

in solving problems.

1. Mathematical

modeling

2. Mathematics

operation

3. Data analysis

Related

exercises

18

A2 P3 (Pure Mathematics 2 & 3)

博实乐“中外融通课程”

348 

Unit/ Theme/

Topic Contents Objectives Core competence Home- work Teaching Hours

1. Algebra 1.1 The modulus function

1.2 Graphs of y=｜f(x)｜

1.3 where f(x) is linear

1.4 Solving modulus inequalities

1.5 Division of polynomials

1.6 The factor theorem

1.7 The remainder theorem

1.8 Improper algebraic

fractions

1.9 Partial fractions

1.10 Binomial expansion of

(1 + x)n for values of n

that are not positive integers

1.11 Binomial expansion of

(a+x ) n for values of n

that are not positive integers

1.12 Partial fractions and

binomial expansions

1. Understand the meaning of |x|, sketch the graph of y =

|ax + b| and use relationsand use relations such as 2 2 b a | b| | a| 



 and

b

a

x

a

b

b

a

x









 



 | | in the course of

solving equations and inequalities.

2. Divide a polynomial, of degree not exceeding 4, by a

linear or quadratic polynomial, and identify the quotient

and remainder (which may be zero).

3. Use the factor theorem and the remainder theorem, e.g.

to findfactors, solve polynomial equations or evaluate

unknown coefficients.

4. Recall an appropriate form for expressing rational functions in partial fractions, and carry out the decomposition, in cases where the denominator is no more complicated than(ax + b)(cx + d)(ex + f), ,

) c b)(x (ax 2 2 

 , and where the degree of the numerator does not exceed that of the denominator.

5. Use the expansion of n x) (1

 , where n is a rational

number and |x|<1 (finding a general term is not included, but adapting the standard series to expand e.g.

1 ) 1

2 2(  

x is included).

1. Mathematical

abstraction

2. Logical reasoning

3. Mathematics

operation

Related

exercises

18

2. Logarithmic

and exponential functions

2.1 Logarithms to base 10

2.2 Logarithms to base a

2.3 The laws of logarithms

2.4 Solving logarithmic

equations

2.5 Solving exponential

equations

2.6 Solving exponential

1. Understand the relationship between logarithms and

indices, and use the laws of logarithms (excluding

change of base).

2. Understand the definition and properties of x e and In x,

including their relationship as inverse functions and

their graphs.

3. Use logarithms to solve equations and inequalities in

1. Mathematical

abstraction

2. Logical reasoning

3. Mathematical

modeling

4. Intuitive imagination

Related

exercises

12

2 d) b)(cx (ax





A2 P3 (Pure Mathematics 2 & 3)

  349

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Topic Contents Objectives Core competence Home- work Teaching Hours

inequalities

2.7 Natural logarithms

2.8 Transforming a relationship to linear form

which the unknown appears in indices.

4. Use logarithms to transform a given relationship to linear form, and hence determine unknown constants by

considering the gradient and/or intercept.

5. Mathematics

operation

3. Trigonometry 3.1 The cosecant, secant and cotangent ratios

3.2 Compound angle formulae

3.3 Double angle formulae

3.4 Further trigonometric

identities

3.5 Expressing asin θ +

bcos θin the form

Rsin( θ± α ) or

Rcos( θ± α )

1. Understand the relationship of the secant, cosecant and

cotangent functions to cosine, sine and tangent, and use

properties and graphs of all six trigonometric functions

for angles of any magnitude.

2. Use trigonometrical identities for the simplification and

exact evaluation of expressions and in the course of

solving equations, and select an identity or identities appropriate to the context, showing familiarity in particular

with the use of   2 2 tan 1 sec   and

,the expansions of sin(A ± B),

cos(A ± B) and tan(A ±B),the formulae for sin 2A,

cos2A and tan2A, the expressions of asinθ + bcosθ in

the forms Rsin(θ ± α) and Rcos(θ ± α).

1. Mathematical

abstraction

2. Logical reasoning

3. Mathematical

modeling

4. Intuitive imagination

5. Mathematics

operation

Related

exercises

20

4.Differentiati

on

4.1 The product rule

4.2 The quotient rule

4.3 Derivatives of exponential functions

4.4 Derivatives of natural

logarithmic functions

4.5 Derivatives of trigonometric functions

4.6 Implicit differentiation

4.7 Parametric differentiation

1. Use the derivatives of x e , In x, sin x, cos x, tan x, cot

xtogether with constant multiples, sums, differences and

composites.

2. Differentiate products and quotients;

3. Find and use the first derivative of a function which is

defined parametrically or implicitly.

1. Mathematical

abstraction

2. Logical reasoning

3. Mathematical

modeling

4. Mathematics

operation

Related

exercises

25

5. Integration 5.1 Integration of exponential functions 1. Extend the idea of ‘reverse differentiation’ to include the 1. Mathematical abstraction

2. Logical reasonRelated exercises 25

  2 2 cot 1 cos   ec

博实乐“中外融通课程”

350 Unit/ Theme/

Topic Contents Objectives Core competence Home- work Teaching Hours

5.2 Integration of

b ax

1



5.3 Integration of

) sin(

b ax



,

) cos(

b ax

 and

) ( sec2 b ax



5.4 Further integration of

trigonometric functions

5.5 The trapezium rule

5.6 Derivative of tan-1 x

5.7 Integration of

2 2

1

a

x



5.8 Integration of

) (

) (

'

x f

x kf

5.9 Integration by substitution

5.10 The use of partial fractions in integration

5.11 Integration by parts

5.12 Further integration

integration of b ax

e  ,

b ax

1



, ) sin(

b ax

 , ) cos(

b ax



,and ) ( sec2 b ax

 .

2. Use trigonometrical relationships (such as double-angle

formulae) to facilitate the integration of functions such

as

x 2 cos .

3. Integrate rational functions by means of decomposition

into partial fractions (restricted to the types of partial

fractions specified in paragraph 1 above).

4. Recognise an integrand of the form

) (

) (

'

x f

x kf and integrate,forexample,

1 2 

x

x or

x tan .

5. Recognise when an integrand can usefully be regarded

as a product, and use integration by parts to integrate,

for example, x sin 2x, or ln x .

6. Use a given substitution to simplify and evaluate either a

definite or an indefinite integral.

ing

3. Mathematical

modeling

4. Mathematics

operation

6.Numerical

solution of

equations

6.1 Finding a starting point

6.2 Improving your solution

6.3 Using iterative

processes to solve

problems involving

other areas of mathematics

1. Locate approximately a root of an equation, by means of

graphical considerations and/or searching for a sign

change.

2. Understand the idea of, and use the notation for, a sequence of approximations which converges to a root of

an equation.

3. Understand how a given simple iterative formula of the

form relates to the equation being solved,

and use a given iteration, or an iteration based on a giv1. Mathematical abstraction 2. Logical reason- ing 3. Mathematics operation Related exercises 12

2 x

  351

高中数学课程图

Unit/ Theme/

Topic Contents Objectives Core competence Home- work Teaching Hours

en rearrangement of an equation, to determine a root to a

prescribed degree of accuracy.

7. Vectors 7.1 Displacement or translation vectors

7.2 Position vectors

7.3 The scalar product

7.4 The vector equation of

a line

7.5 Intersection of two

lines

1. Use standard notations for vectors, i.e.   xy , ,

  xyz .

2. Carry out addition and subtraction of vectors and multiplication of a vector by a scalar, and interpret these operations in geometrical terms.

3. Calculate the magnitude of a vector, and use unit vectors, displacement vectors and position vectors.

4. Understand the significance of all the symbols used

when the equation of a straight line is expressed in the

form r = a + tb and find the equation of a line, given sufficient information.

5. Determine whether two lines are parallel, intersect or are

skew, and find the point of intersection of two lines

when it exists.

6. Use formulae to calculate the scalar product of two vectors, and use scalar products in problems involving lines

and points.

1. Mathematical

abstraction

2. Logical reasoning

3. Intuitive imagination

4. Mathematics

operation

Related

exercises

25

8.Differential

equations

8.1 The technique of separating the variables

8.2 Forming a differential

equation from a problem

1. Formulate a simple statement involving a rate of change

as a differential equation, including the introduction if

necessary of a constant of proportionality.

2. Find by integration a general form of solution for a first

order differential equation in which the variables are separable.

3. Use an initial condition to find a particular solution;

4. Interpret the solution of a differential equation in the

context of a problem being modelled by the equation.

1. Mathematical

abstraction

2. Logical reasoning

3. Mathematical

modeling

4. Mathematics

operation

Related

exercises

20

博实乐“中外融通课程”

352  Unit/ Theme/ Topic Contents Objectives Core competence Home- work Teaching Hours

9.Complex

numbers

9.1 Imaginary numbers

9.2 Complex numbers

9.3 The complex plane

9.4 Solving equations

9.5 Loci

1. Understand the idea of a complex number, recall the

meaning of the terms real part, imaginary part, modulus,argument,conjugate, and use the fact that two complex numbers are equal if and only if both real and imaginary parts are equal.

2. Carry out operations of addition, subtraction, multiplication and division of two complex numbers expressed in

cartesian form x + iy .

3. Use the result that, for a polynomial equation with real

coefficients, any non-real roots occur in conjugate pairs;

4. Represent complex numbers geometrically by means of

an Argand diagram.

5. Carry out operations of multiplication and division of two

complex numbers expressed in polarform

.

6. Find the two square roots of a complex number;

7. Understand in simple terms the geometrical effects of

conjugating a complex number and of adding, subtracting, multiplying and dividing two complex numbers.

8. Illustrate simple equations and inequalities involving

complex numbers by means of loci in an Argand diagram, e.g. |z

– a| < k, |z

– a| = |z

– b| , arg(z

– a) = α.

1. Mathematical

abstraction

2. Logical reasoning

3. Intuitive imagination

4. Mathematics

operation

Related

exercises

30

A2 P4 (Mechanics)

Unit/ Theme/

Topic Contents Objectives Core competence Home- work Teaching Hours

1.Forces and

equilibrium

1.1 Resolving forces in

horizontal and vertical

directions in equilibrium problems

1.2 Resolving forces at

1. Identify the forces acting in a given situation;

2. Understand the vector nature of force, and find and use

components and resultants.

3. Use the principle that, when a particle is in equilibrium,

the vector sum of the forces acting is zero, or equiva1. Mathematical abstraction 2. Logical reason- ing 3. Mathematical Related exercises 18

 

 i er ) sin i (cos r 



Unit/ Theme/

Topic Contents Objectives Core competence Home- work Teaching Hours

9.Complex

numbers

9.1 Imaginary numbers

9.2 Complex numbers

9.3 The complex plane

9.4 Solving equations

9.5 Loci

1. Understand the idea of a complex number, recall the

meaning of the terms real part, imaginary part, modulus,argument,conjugate, and use the fact that two complex numbers are equal if and only if both real and imaginary parts are equal.

2. Carry out operations of addition, subtraction, multiplication and division of two complex numbers expressed in

cartesian form x + iy .

3. Use the result that, for a polynomial equation with real

coefficients, any non-real roots occur in conjugate pairs;

4. Represent complex numbers geometrically by means of

an Argand diagram.

5. Carry out operations of multiplication and division of two

complex numbers expressed in polarform

.

6. Find the two square roots of a complex number;

7. Understand in simple terms the geometrical effects of

conjugating a complex number and of adding, subtracting, multiplying and dividing two complex numbers.

8. Illustrate simple equations and inequalities involving

complex numbers by means of loci in an Argand diagram, e.g. |z

– a| < k, |z

– a| = |z

– b| , arg(z

– a) = α.

1. Mathematical

abstraction

2. Logical reasoning

3. Intuitive imagination

4. Mathematics

operation

Related

exercises

30

A2 P4 (Mechanics)

Unit/ Theme/

Topic Contents Objectives Core competence Home- work Teaching Hours

1.Forces and

equilibrium

1.1 Resolving forces in

horizontal and vertical

directions in equilibrium problems

1.2 Resolving forces at

1. Identify the forces acting in a given situation;

2. Understand the vector nature of force, and find and use

components and resultants.

3. Use the principle that, when a particle is in equilibrium,

the vector sum of the forces acting is zero, or equiva1. Mathematical abstraction 2. Logical reason- ing 3. Mathematical Related exercises 18

 

 i er ) sin i (cos r 



A2 P4 (Mechanics)

  353

高中数学课程图

Unit/ Theme/

Topic Contents Objectives Core competence Home- work Teaching Hours

other angles in equilibrium problems

1.3 The triangle of forces

and Lami’s theorem

for threeforceequilibrium

problems

1.4 Non-equilibrium

problems for objects

on slopes and known

directions of acceleration

1.5 Non-equilibrium

problems and finding

resultant forces and

directions of acceleration

1.6 Friction as part of the

contact force

1.7 Limit of friction

1.8 Change of direction of

friction in different

stages of motion

1.9 Angle of friction

lently, that the sum of the components in any direction

is zero.

4. Understand that a contact force between two surfaces

can be represented by two components, the normal

component and the frictional component.

5. Use the model of a ‘smooth’ contact, and understand

the limitations of this model.

6. Understand the concepts of limiting friction and limiting equilibrium; recall the definition of coefficient of

friction, and use the relationship F = μR or F ≤ μR, as

appropriate..

7. Use Newton’s third law.

modeling

4. Intuitive imagination

5. Mathematics

operation

2.Kinematics

of motion in a

straight line

2.1 Displacement and

velocity

2.2 Acceleration

2.3 Equations of constant

acceleration

2.4 Displacement–time

graphs and multistage problems

2.5 Velocity–time graphs

and multi-stage prob1. Understand the concepts of distance and speed as scalar quantities, and of displacement, velocity and accelera- tion as vector quantities (in one dimension only). 2. Sketch and interpret displacement-time graphs and velocity-time graphs, and in particular appreciate thatthe area under a velocity-time graph represents dis- placement,the gradient of a displacement-time graph represents velocity,the gradient of a velocity-time graph represents acceleration. 3. Use differentiation and integration with respect to time 1. Mathematical abstraction 2. Logical reason- ing 3. Mathematical modeling 4. Intuitive imagi- nation 5. Mathematics operation Related exercises 15

博实乐“中外融通课程”

354 Unit/ Theme/ Topic Contents Objectives Core competence Home- work Teaching Hours

lems

2.6 Graphs with discontinuities

2.7 Velocity as the derivative of displacement

with respect to time

2.8 Acceleration as the

derivative of velocity

with respect to time

2.9 Displacement as the

integral of velocity

with respect to time

2.10 Velocity as the

integral of acceleration with respect to

time

to solve simple problems concerning displacement, velocity and acceleration (restricted to calculus within the

scope of unit P1).

4. Use appropriate formulae for motion with constant

acceleration in a straight line.

3.Momentum 3.1 Momentum

3.2 Collisions and conservation of momentum

1. Use the defnition of linear momentum and show understanding of its vector nature.

2. Use conservation of linear momentum to solve problems that may be modelled as the direct impact of two

bodies.

1. Mathematical

abstraction

2. Logical reasoning

3. Mathematical

modeling

4. Intuitive imagination

5. Mathematics

operation

Related

exercises

5

4. Newton’s

laws of motion

4.1 Newton’s first law

and relation between

force and acceleration

4.2 Combinations of

forces

4.3 Weight and motion

due to gravity

1. Apply Newton’s laws of motion to the linear motion of

a particle of constant mass moving under the action of

constant forces, which may include friction, tension in

an inextensible stringand thrust in a connecting rod.

2. Use the relationship between mass and weight.

3. Solve simple problems which may be modelled as the

motion of a particle moving vertically or on an inclined

1. Mathematical

abstraction

2. Logical reasoning

3. Mathematical

modeling

4. Intuitive imagiRelated exercises 15

  355

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Topic Contents Objectives Core competence Home- work Teaching Hours

4.4 Normal contact force

and motion in a vertical line

4.5 Newton’s third law

4.6 Objects connected by

rods

4.7 Objects connected by

strings

4.8 Objects in moving

lifts

plane with constant acceleration;

4. Solve simple problems which may be modelled as the

motion of connected particles.

nation

5. Mathematics

operation

5.Energy,

work and

power

5.1 Work done by a force

5.2 Kinetic energy

5.3 Gravitational potential

energy

5.4 The work–energy

principle

5.5 Conservation of energy in a system of conservative forces

5.6 Conservation of energy in a system with

non-conservative

forces

5.7 Power

1. Understand the concept of the work done by a force,

and calculate the work done by a constant force when

its point of application undergoes a displacement not

necessarily parallel to the force (use of the scalar product is not required).

2. Understand the concepts of gravitational potential energy and kinetic energy, and use appropriate formulae;

3. Understand and use the relationship between the

change in energy of a system and the work done by the

external forces, and use in appropriate cases the principle of conservation of energy.

4. Use the definition of power as the rate at which a force

does work, and use the relationship between power,

force and velocity for a force acting in the direction of

motion.

5. Solve problems involving, for example, the instantaneous acceleration of a car moving on a hill with resistance.

1. Mathematical

abstraction

2. Logical reasoning

3. Mathematical

modeling

4. Intuitive imagination

5. Mathematics

operation

Related

exercises

18

博实乐“中外融通课程”

356 

4 Assessment

AS assessments：

Paper 1

Paper 1: Pure Mathematics 1 (P1)

1 hour 50 minutes

About 10 shorter and longer questions

75 marks weighted at 30% of total

Paper 5

Paper 5: Probability and Statistics 1 (S1)

1 hour 15 minutes

About 7 shorter and longer questions

50 marks weighted at 20% of total

A2 assessments：

Paper 3

Paper 3: Pure Mathematics 3(P3)

1 hour 50minutes

About 10 shorter and longer questions

75 marks weighted at 30% of total

Paper 4

Paper 4:Mechanics (M1)

1 hour 15 minutes

About 7 shorter and longer questions

50 marks weighted at 20% of total

5 Resources

Online

resources

www.cie.org.uk/alevel

http://teachers.cie.org.uk

http://www.emaths.co.uk

http://www.waldomaths.com

http://www.ex.ac.uk/cimt/welcome.html

http://www.ex.ac.uk/cimt/mep/index.htm

http://matti.usu.edu/nlvm/nav/vlibrary.html

http://www.cambridgestudents.org.uk

http://www.mathsnet.net/index.html

http://www.supermathsworld.com

http://www.mymaths.co.uk

  357

高中数学课程图

http://www.pbs.org/teachers/mathline/concepts/moremathconcepts.shtm

http://homepage.ntlworld.com/jontreby/Personal/mathsLinks.htm#Investigations

Endorsed

Textbooks

Author Title Publisher ISBN

Julian Gilbey Pure Mathematics 1 Cambridge University

Press 9781108407144

Julian Gilbey Pure Mathematics 2

& 3

Cambridge University

Press 9781108407199

Julian Gilbey Mechanics 1 Cambridge University

Press 9781108407267

Julian Gilbey Probability & Statistics 1

Cambridge University

Press 9781108407304

Suggested

Books

Author Title Publisher ISBN

Backhouse,

Houldsworth

& Horrill

Pure Mathematics 1 Longman, 1985 0582353866

Backhouse,

Houldsworth

& Horrill

Pure Mathematics 2 Longman, 1985 0582353874

Backhouse,

Houldsworth,

Horrill &

Wood

Essential Pure

Mathematics Longman, 1991 0582066581

Bostock &

Chandler

Core Maths for Advanced Level Nelson Thornes, 2000 0748755098

Emanuel, Wood &

Crawshaw

Pure Mathematics 1 Longman, 2001 0582405505

Emanuel, Wood &

Crawshaw

Pure Mathematics 2 Longman, 2001 0582405491

Sadler &

Thorning

Understanding Pure

Mathematics

Oxford University

Press, 1987 0199142432

Smedley &

Wiseman

Introducing Pure

Mathematics

Oxford University

Press, 2001 0199142432

Solomon Advanced Level

Mathematics John Murray, 1995 071955344X

博实乐“中外融通课程”

358 

Chinese and International Integrated

Curriculums for Bright Scholar

High School Section

CALFurther Mathematics

Curriculum Map

（2022 version）

Complied by Guangdong Country Garden Senior High Section

  359

高中数学课程图

ALEVEL Further Mathematics Curriculum Mapping

Subject

ALEVEL

Further Mathematics

Level G3&G4 Syllabus Code

Course Code Credit 8 Duration 2 years

Teaching Hours 560 Designer Lin Bai Completed Date 2022.9

1 Course Introduction

1.1 Introduction

Cambridge International AS and A Level Mathematics is recognized and accepted by

Universities as a proof of understanding of mathematics. Good math learning skills enables

students to acquire lifelong skills, including：

•Deepen the understanding of mathematical principles;

•Further developing mathematics skills, including the application of mathematics in

everyday situations and possibly in studying other subjects;

•The ability to analyze problems logically and to identify when and how to represent a

situation mathematically;

•Using mathematics as a means of communication;

•Lay a solid foundation for further research.

The syllabus gives centre the flexibility to choose from three different approaches to AS

mathematics - pure mathematics or pure mathematics plus mechanics or pure mathematics

plus probability statistics. The centre can choose from three different routes to Cambridge

International A2 Mathematics, depending on the choice of mechanics, probability statistics,

or both in a wide range of ‘applied’ fields.

1.2 Aims

•Develop students’ mathematical knowledge and skills, encourage confidence, and

provide satisfaction and fun;

•Promote the understanding of mathematical principles and the appreciation of

mathematics as a logical and coherent discipline；

博实乐“中外融通课程”

360 

•Mastering a range of math skills, especially those that enable them to use math in

everyday situations and in other subjects they may study；

•Develop the ability to analyze problems logically, to recognize when and how to

represent them mathematically, to identify and interpret relevant factors, and to select

appropriate mathematical methods to solve problems when necessary；

•Use mathematics as a means of communication and emphasize the use of clear

expression；

•Obtain the mathematics background required for further study in this discipline or

related subjects.

2 Course Structure

3

•Mastering a range of math skills, especially those that enable them to use math in

everyday situations and in other subjects they may study；

•Develop the ability to analyze problems logically, to recognize when and how to

represent them mathematically, to identify and interpret relevant factors, and to select

appropriate mathematical methods to solve problems when necessary；

•Use mathematics as a means of communication and emphasize the use of clear

expression；

•Obtain the mathematics background required for further study in this discipline or

related subjects .

2Course Structure

Structure of AL Further

Mathematics

International Advanced

Subsidiaryin Further

Mathematics

International Advanced

level in Further

Mathematics

FP1

FP2, FP3, M1,

M2, M3, S1, S2,

S3, D1

FP1 and either

FP2 or FP3

FP2, FP3, M1,

M2, M3, S1, S2,

S3, D1

Choose 3units Choose 6units

  361

高中数学课程图

3 Course outline

FP1

Unit/

Theme/

Topic

Contents Objectives

Core

competence

Home-wo

rk

Teaching

Hours

Complex

Number

1.1 Imaginary and complex

numbers

1.2 Multiplying complex

numbers

1.3 Complex conjugation

1.4 Argand diagram

1.5 Modulus and Argument of

complex numbers

1.6 Modulus-argument form

of complex numbers

1.7 Roots of quadratic equations

1.8 Solving cubic and quartic

equation

1.Definition of complex numbers in the form ?? ൅????and

?? …‘•?? ൅???? •‹??Ǥ

2.Sum, product and quotient of complex numbers.

3.Geometrical representation of complex numbers in the

Argand diagram.

Geometrical representation of sums, products and quotients of complex numbers.

4.Complex solutions of quadraticequations with real

coefficients

5.Finding conjugate complex roots and a real root of a

cubic equation with integer coefficients.

7.Finding conjugate complex roots and/or real roots of a

quartic equation with real coefficients

Critical

thinking

Inquisitive

Reflective

Related

exercises

5

Roots of

quadratic

2.1 Roots of quadratic

equations

1.Sum of roots and product of roots of a quadratic

equation.

Critical

thinking

Related

exercise

8

博实乐“中外融通课程”

362 Unit/

Theme/

Topic

Contents Objectives

Core

competence

Home-wo

rk

Teaching

Hours

equations 2.2 Forming quadratic equations with new roots 2.Manipulation of expressions involving the sum of roots and product of roots.

3.Forming quadratic equations with new roots.

Inquisitive

Reflective

Numerical

Solution of

equations

3.1 Locating roots

3.2 Interval bisection

3.3 Linear interpolation

3.4 The Newton-Raphson

method

1.Equations of the form f(x) = 0

solved numerically by:

(i) interval bisection,

(ii) linear interpolation,

(iii) the Newton-Raphson process.

Critical

thinking

Inquisitive

Reflective

Related

exercise

10

Coordinate

System

4.1 Parametric equations

4.2 The general equation of a

parabola

4.3 The equation for a rectangular hyperbola and the

equations of tangents and

normals

1.Cartesian equations for the parabola and rectangular

hyperbola.

2.Idea of parametric equation for parabola and rectangular hyperbola.

3.The focus-directrix property of the parabola.

4.Tangents and normals to these curves.

Critical

thinking

Inquisitive

Reflective

Related

exercise

10

Matrices 5.1 Introduction to matrices

5.2 Matrix multiplication

5.3 Determinants

1.Addition and subtraction of matrices.

2.Multiplication of a matrix by a scalar.

3.Products of matrices.

Critical

thinking

Related

exercise

15

  363

高中数学课程图

Unit/

Theme/

Topic

Contents Objectives

Core

competence

Home-wo

rk

Teaching

Hours

5.4 Inverting a 2 x 2 matrix 4.Evaluation of 2 × 2 determinants

5.Inverse of 2 × 2 matrices

Inquisitive

Reflective

Transformations using

matrices

6.1 Linear transformation in

two dimensions

6.2 Reflections and rotations

6.3 Enlargements and

stretches

6.4 Successive transformations

6.5 The inverse of a linear

transformation

1.Linear transformations of column vectors in two

dimensions and their matrix representation.

2.Applications of 2 × 2 matrices to represent geometrical

transformations.

3.Combinations of transformations.

4.The inverse (when it exists) of a given transformation

or combination of transformations.

Critical

thinking

Inquisitive

Reflective

Related

exercise

15

Series 7.1 Sums of natural numbers

7.2 Sums of squares and cubes

1.Summation of simple finite series. Critical

thinking

Inquisitive

Reflective

Related

exercise

8

博实乐“中外融通课程”

364 Unit/

Theme/

Topic

Contents Objectives

Core

competence

Home-wo

rk

Teaching

Hours

Proof 8.1 Proof by mathematical

induction

8.2 Roving divisibility results

8.3 Using mathematical induction to produce a proof

8.4 Proving statements involving matrices

1.Proof by mathematical induction Critical

thinking

Inquisitive

Reflective

Related

exercise

10

FP2

Unit/

Theme/

Topic

Contents Objectives

Core competence Home-w ork Teaching Hours

Inequalities 1.1 Algebraic methods

1.2 Using graphs to solve

inequalities

1.3 Modulus inequalities

1.The manipulation and solution of algebraic inequalities

and inequations, including those involving the modulus

sign.

Critical

thinking

Inquisitive

Reflective

Related

exercises

5

Series 2.1 The method of differences 1.Summation of simple finite series using the method of

differences.

Critical

thinking

Related

exercise

8

FP2

  365

高中数学课程图

Unit/

Theme/

Topic

Contents Objectives

Core competence Home-w ork Teaching Hours

Inquisitive

Reflective

Complex

numbers

3.1 Exponential form of complex numbers

3.2 Multiplying and dividing

complex numbers

3.3 De Moivre’s theorem

3.4 Trigonometric identities

3.5 nth root of a complex

number

1.Euler’s relation ??‹?? ൌ …‘•?? ൅ ‹ •‹??.

2.De Moivre’s theorem and its application to trigonometric identities and to roots of a complex number.

3.Loci and regions in the Argand diagram.

4.Elementary transformations from the z-plane to the

w-plane.

Critical

thinking

Inquisitive

Reflective

Related

exercise

10

Further Argand diagram 4.1 Loci in an Argand diagram 4.2 Further loci in an Argand diagram

4.3 Regions in an Argand

diagram

4.4 Further regions in an

Argand diagram

4.5 Transformations of the

complex plane

Critical

thinking

Inquisitive

Reflective

Related

exercise

15

博实乐“中外融通课程”

366 Unit/

Theme/

Topic

Contents Objectives

Core competence Home-w ork Teaching Hours

First-order

differential

equations

5.1 Solving first-order differential equations with separable

variables

5.2 First-order linear differential equations of the form

????

???? ൅??ሺ??ሻ?? ൌ??ሺ??ሻ

5.3 Reducible first-order differential equations

1.Further solution of first order differential equations

with separable variables.

2.First-order linear differential equations of the form

????

???? ൅??ሺ??ሻ?? ൌ??ሺ??ሻ

3.Differential equations reducible to the above types by

means of a given substitution.

Critical

thinking

Inquisitive

Reflective

Related

exercise

20

Second-orde

r differential

equations

6.1 Second-order homogeneous differential equations

6.2 Second-order

non-homogeneous differential

equations

6.3 Using boundary conditions

6.4 Reducible second-order

differential equations

1.The linear second order differential equation:

??

??ʹ??

????ʹ ൅??

????

???? ൅???? ൌ??ሺ??ሻ

where a, b and c are real constants and the particular integral can be found by inspection or trial.

2.Differential equations reducible to the above types by

means of a given substitution.

Critical

thinking

Inquisitive

Reflective

Related

exercise

20

Maclaurin

and Taylor

series

7.1 Higher derivatives

7.2 Maclaurin series

7.3 Series expansion of com1.Third and higher order derivatives 2.Derivation and use of Maclaurin series. 3.Derivation and use of Taylor series. Critical thinking Related exercise 15

  367

高中数学课程图

Unit/

Theme/

Topic

Contents Objectives

Core competence Home-w ork Teaching Hours

pound functions

7.4 Taylor series

7.5 Series solution of differential equations

4.Use of Taylor series method for series solutions of differential equations. Inquisitive

Reflective

Polar coordinates 8.1 Polar coordinates and equations

8.2 Sketching curves

8.3 Area enclosed by a polar

curve

8.4 Tangents to polar curves

1.Polar coordinates ሺ??ǡ?? ሻ, ??≥ Ͳ

2.Use of the formula

ͳ

ʹ ??ʹ????????

for area.

Critical

thinking

Inquisitive

Reflective

Related

exercise

15

FP3

Unit/ Theme/

Topic

Contents Objectives

Core competence Home- work Teaching Hours

Hyperbolic

functions

1.1 Introduction to hyperbolic

functions

1.Definition of the six hyperbolic functions in terms

of exponentials.

Critical

thinking

Related

exer10

FP3

博实乐“中外融通课程”

368 Unit/ Theme/

Topic

Contents Objectives

Core competence Home- work Teaching Hours

1.2 Sketching graphs of hyperbolic functions

1.3 Inverse hyperbolic functions

1.4 Identities and equations

Graphs and properties of the hyperbolic functions.

2.Inverse hyperbolic functions, their graphs, properties and logarithmic equivalents. Inquisitive

Reflective

cises

Further coordinate systems 2.1 Ellipse 2.2 Hyperbolas

2.3 Eccentricity

2.4 Tangents and normals to an

ellipse

2.5 Tangents and normal to a hyperbola

2.6 Loci

1.Cartesian and parametric equations for the ellipse

and hyperbola.

2.The focus-directrix properties of the ellipse and

hyperbola, including the eccentricity.

3.Tangents and normals to thesecurves.

4.Simple loci problems.

Critical

thinking

Inquisitive

Reflective

Related

exercise

10

Differentiation 3.1 Differentiating Hyperbolic functions

3.2 Differentiating inverse Hyperbolic functions

3.3 Differentiating inverse trigonometric functions

1.Differentiation of hyperbolic functions and expressions involving them.

2.Differentiation of inverse functions, including trigonometric and hyperbolic functions.

Critical

thinking

Inquisitive

Reflective

Related

exercise

25

Integration 4.1 Standard integrals

4.2 Integration

1.Integration of hyperbolic functions and expressions

involving them.

Critical

thinking

Related

exercise

25

  369

高中数学课程图

Unit/ Theme/

Topic

Contents Objectives

Core competence Home- work Teaching Hours

4.3 Trigonometric and hyperbolic

substitutions

4.4 Integrating expressions

4.5 Integrating inverse trigonometric and hyperbolic functions

4.6 Deriving and using reduction

formulae

4.7 Finding the length of an arc of

a curve

4.8 Finding the area of a surface

of revolution

2.Integration of inverse trigonometric and hyperbolic

functions.

3.Integration using hyperbolic and trigonometric

substitutions.

4.Use of substitution for integrals involving quadratic

surds.

5.The derivation and use of simple reduction formulae.

6.The calculation of arc length and the area of a surface of revolution.

Inquisitive

Reflective

Vectors 5.1 Vector product

5.2 Finding areas

5.3 Scalar triple product

5.4 Straight line

5.5 Vector planes

5.6 Solving geometrical problems

1.The vector product a × b and the triple scalar

product a . b × c.

2.Use of vectors in problems involving points, lines

and planes.

The equation of a line in the form

(r − a) × b = 0.

3.The equation of a plane in the

forms

r.n = p, r = a + sb + tc.

Critical

thinking

Inquisitive

Reflective

Related

exercise

25

Further matrix 6.1 Transposing a matrix 1.Linear transformations of column Critical Related 35

博实乐“中外融通课程”

370 Unit/ Theme/

Topic

Contents Objectives

Core competence Home- work Teaching Hours

algebra 6.2 The determinant of a 3×3 Matrix

6.3 The inverse of a 3×3 Matrix

where it exists

6.4 Using matrices to represent

linear transformations in 3 dimensions

6.5 Using inverse matrices to

reverse the effect of a linear

transformation

6.6 The eigenvalues and eigenvectors of 2×2 and 3×3 matrices

6.7 Reducing a symmetric matrix

to diagonal form

vectors in two and three dimensions and their matrix

representation.

2.Combination of transformations.

Products of matrices.

3.Transpose of a matrix.

4.Evaluation of 3 × 3 determinants

5.Inverse of 3 × 3 matrices.

6.The inverse (when it exists) of a given transformation or combination of transformations.

7.Eigenvalues and eigenvectors of 2 × 2 and 3 × 3

matrices.

8.Reduction of symmetric matrices to diagonal form.

thinking

Inquisitive

Reflective

exercise

S2

  371

高中数学课程图

S2

Unit/ Theme/

Topic

Contents Objectives Core compe- tence Home- work Teaching Hours

Binomial distribution 1.1 Conditions and definition of binomial distributions

1.2 Mean and variance of

binomial distributions

1.3 Calculations of the

probability of binomial

distributions

1.The binomial and Poisson distributions.

(Students will be expected to use these

distributions to model a real-world situation and

to comment critically on their appropriateness.

Cumulative probabilities by calculation or by

reference to tables.)

2.The mean and variance of the binomial and

Poisson distributions.

3.The use of the Poisson distribution as an

approximation to the binomial distribution.

Critical

thinking

Inquisitive

Reflective

Related

exercises

8

Poisson distribution 2.1 Conditions and definition of Poisson distributions

2.2 Mean and variance of

Poisson distributions

2.3 Calculations of the

probability of Poisson

distributions

Critical

thinking

Inquisitive

Reflective

Related

exercises

10

Approxima- 3.1 The use of the Poisson Critical Related 15

博实乐“中外融通课程”

372 Unit/ Theme/

Topic

Contents Objectives Core compe- tence Home- work Teaching Hours

tions distribution as an approximation

to the binomial distribution

3.2 The use of the Normal

distribution as an approximation

to the binomial distribution

3.3 The use of the Normal

distribution as an approximation

to the Poisson distribution

thinking

Inquisitive

Reflective

exercises

Continuous

distributions

4.1 Properties of probability

density function

4.2 Properties of cumulative

distribution function

4.3 Calculations of

mean/variance/mode/ quartiles

of a continuous random variable

1.The concept of a continuous random variable.

2.The probability density function and the

cumulative distribution function for a continuous

random variable.

3.Relationship between density and distribution

functions.

4.Mean and variance of continuous random

variables.

5.Mode, median and quartiles of continuous

random variables.

Critical

thinking

Inquisitive

Reflective

Related

exercises

20

Continuous

uniform distribution

5.1 Concepts of continuous

uniform distribution and related

calculations

1.The continuous uniform (rectangular)

distribution.

2.Use of the Normal distribution as an

Critical

thinking

Inquisitive

Related

exercises

10

  373

高中数学课程图

Unit/ Theme/

Topic

Contents Objectives Core compe- tence Home- work Teaching Hours

approximation to the binomial distribution and the

Poisson distribution, with the application of the

continuity correction.

Reflective

Sampling 6.1 Population and sampling

6.2 Concepts of statistic

6.3 Sample distribution of a

statistic

1.Population, census and sample Sampling unit,

sampling frame .

2.Concepts of a statistic and its sampling distribution.

3.Concept and interpretation of a hypothesis test.

Null and alternative hypotheses.

4.Critical region.

5.One-tailed and two-tailed tests.

6.Hypothesis tests for the parameter p of a binomial distribution and for the mean of a Poisson

distribution.

Critical

thinking

Inquisitive

Reflective

Related

exercises

10

Hypothesis

tests

7.1 Hypothesis test

7.2 Finding critical values

7.3 One-tailed test

7.4 Two-tailed test

7.5 Hypothesis test of Poisson

distributions

7.6 Using approximations in

hypothesis tests

Critical

thinking

Inquisitive

Reflective

Related

exercises

15

S3

博实乐“中外融通课程”

374  S3

Unit/ Theme/

Topic

Contents Objectives Core compe- tence Home- work Teaching Hours

Sampling 1.1 Sampling

1.2 Using a random number

table

1.3 Random sampling

1.4 Non-random sampling

1.Methods for collecting data. Simple random

sampling. Use of random numbers for sampling.

2.Other methods of sampling:

stratified, systematic, quota.

Critical

thinking

Inquisitive

Reflective

Related

exercises

10

Combinations

of random

variables

2.1 Combinations of random

variables

1.Distribution of linear combinations of independent Normal random variables. Critical thinking

Inquisitive

Reflective

Related

exercises

10

Estimators

and confidence intervals

3.1 Estimators, bias and standard error

3.2 Confidence intervals

1.Concepts of standard error, estimator, bias.

2.The distribution of the sample mean.

3.Concept of a confidence interval and its interpretation.

4.Confidence limits for a Normal mean, with variance known.

5.Hypothesis tests for the mean of a Normal distribution with variance known.

6.Use of Central Limit theorem to extend hypothesis tests and confidence intervals to samples

from non-Normal distributions. Use of large samCritical thinking Inquisitive Reflective Related exer- cises 15 Central limit theorem and testing the mean 4.1 The central limit theorem 4.2 Applying the central limit theorem to other distributions 4.3 Confidence intervals using the central limit theorem 4.4 Hypothesis testing the mean Critical thinking Inquisitive Reflective Related exer- cises 25

  375

高中数学课程图

Unit/ Theme/

Topic

Contents Objectives Core compe- tence Home- work Teaching Hours

4.5 Hypothesis testing for the

difference between means

4.6 Use of large sample results

for an unknown population

ple results to extend to the case in which the variance is unknown.

7.Hypothesis test for the difference between the

means of two Normal distributions with variances

known.

8.Use of large sample results to extend to the case

in which the population variances are unknown.

Correlation 5.1 Spearman’s rank correlation

coefficient

5.2 Hypothesis testing for zero

correlation

1.Spearman’s rank correlation coefficient, its use,

interpretation and limitations.

2.Testing the hypothesis that a correlation is zero.

Critical

thinking

Inquisitive

Reflective

Related

exercises

15

Goodness of

fit and contingency

tables

6.1 Goodness of fit

6.2 Degrees of freedom and the

chi-squared (χ 2) family of distributions

6.3 Testing a hypothesis

6.4 Testing the goodness

Of fit with discrete data

6.5 Testing the goodness of fit

with continuous data

6.6 Using contingency tables

1.The null and alternative hypotheses.

The use of

????−???? ʹ????????ൌͳ

as an approximate χ2 statistic.

2.Degrees of freedom.

Critical thinking

Inquisitive

Reflective

Related

exercises

15

博实乐“中外融通课程”

376  M2 M2

Unit/ Theme/

Topic

Contents Objectives Core compe- tence Home- work Teaching Hours

Projectiles 1.1 Horizontal projection

1.2 Horizontal and vertical

components

1.3 Projection at any angle

1.4 Projectile motion formulae

1.Motion in a vertical plane with constant acceleration, e.g. under gravity.

2.Simple cases of motion of a projectile.

3.Velocity and acceleration when the displacement is a function of time.

4.Differentiation and integration of a vector with

respect to time.

Critical

thinking

Inquisitive

Reflective

Related

exercises

10

Variable

Acceleration

2.1 Functions of time

2.2 Using differentiation

2.3 Using integration

2.4 Differentiating vectors

2.5 Integrating vectors

2.6 Constant acceleration

2.7 Formulae

Critical

thinking

Inquisitive

Reflective

Related

exercise

10

Centres of

Mass

3.1 Centre of mass of a set of

particles on a straight line

3.2 Centre of mass of a set of

particles arranged in a plane

3.3 Centres of mass of standard

1.Centre of mass of a discrete mass distribution in

one and two dimensions.

2.Centre of mass of uniform plane figures, and

simple cases of composite plane figures.

3.Simple cases of equilibrium of a plane lamina.

Critical thinking

Inquisitive

Reflective

Related

exercise

15

  377

高中数学课程图

Unit/ Theme/

Topic

Contents Objectives Core compe- tence Home- work Teaching Hours

uniform plane laminas

3.4 Centre of mass of a composite lamina

3.5 Centre of mass of a

framework

3.6 Laminas in equilibrium

3.7 Frameworks in equilibrium

3.8 Non-uniform composite

laminas and frameworks

Work and

Energy

4.1 Work done

4.2 Kinetic and potential energy

4.3 Conservation of mechanical energy and work–energy

principle

4.4 Power

1.Kinetic and potential energy, work and power.

The work-energy principle. The principle of conservation of mechanical energy.

Critical

thinking

Inquisitive

Reflective

Related

exercise

10

Impulses and

collisions

5.1 Momentum as a vector

5.2 Direct impact and newton’s

law of restitution

5.3 Direct collision with a

smooth plane

1.Momentum as a vector. The impulse-momentum principle in vector form. Conservation of linear momentum.

2.Direct impact of elastic particles.

Newton’s law of restitution. Loss of mechanical

Critical

thinking

Inquisitive

Reflective

Related

exercise

10

博实乐“中外融通课程”

378 Unit/ Theme/

Topic

Contents Objectives Core compe- tence Home- work Teaching Hours

5.4 Loss of kinetic energy

5.5 Successive direct impacts

energy due to impact

3.Successive impacts of up to three particles or

two particles and a smooth plane surface.

Statics of rigid bodies 6.1 Static rigid bodies 1.Moment of a force. 2.Equilibrium of rigid bodies. Critical thinking

Inquisitive

Reflective

Related

exercise

10

4Assessment

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高中数学课程图

4 Assessment

博实乐“中外融通课程”

380 

5 Resources

[1] www.cie.org.uk/alevel

[2] http://teachers.cie.org.uk

[3] http://www.emaths.co.uk/

[4] http://matti.usu.edu/nlvm/nav/vlibrary.html

[5] http://www.waldomaths.com/

[6] http://www.supermathsworld.com/

[7] http://www.mymaths.co.uk/

[8] http://homepage.ntlworld.com/jontreby/Personal/mathsLinks.htm#Investigations

[9] http://www.pbs.org/teachers/mathline/concepts/moremathconcepts.shtm

Chinese and International Integrated

Curriculums for Bright Scholar

High School Section

CAP Mathematics

Curriculum Map

（2022 version）

Complied by Guangdong Country Garden Senior High Section

博实乐“中外融通课程”

382 

AP Curriculum Map

Subject AP Pre-calculus Level G1&2 Syllabus Code

Course Code Credit 10 Duration 1Year

Teaching Periods 200 Designer XiongXiaomin Completed

Date

1 Course Introduction

1.1 Introduction

In Advanced Precalculus, students will develop a deeper understanding of various

types of functions on the knowledge they have already learned. Topics studied include The

concepts and the property theorems of parallelograms, rectangles, rhombuses, and squares,

Quadratic equations and quadratic functions, Exponential function, logarithmic function

and logistic function. Trigonometric functions,inverse trigonometric functions, laws of sines

and cosines, polar coordinates, parametric functions. Analytical geometry of plane, polar

coordinates, and parametric functions,and an introduction to Calculus.

1.2 Aims

a) Develop logical, critical and creative thinking；

b) Improve the ability of understanding and expression, study the skills of

communication;

c) Apply and transfer skills to alternative situations, and patience and persistence in

problem-solving;

d) Understand mathematics and enjoy the beauty of mathematics;

e) Develop an appreciation of calculus as a coherent body of knowledge and as a

human accomplishment .

2 Course Structure

Numberand

quantity

real

number

complex

number equation inequality special

parallelograms

Quadraticfunctions,

Exponential function,

Logarithmic function and

logistic function.

Trigonometric functions

Polar coordinates,

Parametric functions.

circle

CH Pre-calculusG1

Algebra Geometry Functions

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高中数学课程图

3 Course Outline

G1

Unit/ Theme/

Topic

Contents Objectives

Core

Competency

Academic

Proficiency

Level

Assignment Teaching Periods

1.Quadratic

equation in

one variable

1.1 List the formula

1.2 Solve quadratic

equations in one

variable

1.3 The discriminant of

the roots of a quadratic

equation in one variable

1.4 Relationship

between roots and

coefficients

1.5 Application

1.Able to use quantitative relations to list

quadratic equations of one variable;

2.Understand the matching method, and be able

to use the matching method, formula method, and

factorization method to solve quadratic equations

in one variable;

3. The discriminant of roots can be used to

determine how many real roots the equation has;

4. Understand the relationship between roots and

coefficients;

5. Can check whether the roots of the equation

are valid according to the actual meaning of the

actual problem.

Reason

Critical

thinking

Reflective

Problem-solvi

ng

2 Ninth grade

mathematics

published by

Beijing

Normal

University

Chapter2

exercise

10

2.Quadratic

function

2.1 List quadratic

function expressions

2.2 The graphs and

properties of the

quadratic function

1. Ability to use tables, relational expressions,

and images to represent the quadratic function

relational expressions between variables;

2. According to the specific problem, the

appropriate method can be selected to express the

Reason

Critical

thinking

Reflective

Problem-solvi

ng

2 Ninth grade

mathematics

published by

Beijing

Normal

10

博实乐“中外融通课程”

384  Unit/ Theme/

Topic

Contents Objectives

Core

Competency

Academic

Proficiency

Level

Assignment Teaching Periods

2.3 The relationship

between quadratic

function and straight

line

2.4 Find the quadratic

function expression

2.5 Application of

quadratic function

quadratic function relationship between the

variables;

3. Can make the image of the quadratic function,

and can analyze the nature of the quadratic

function according to the image;

4. According to the expression of the quadratic

function, determine the opening direction,

symmetry axis and vertex coordinates of the

quadratic function;

5. Use the relationship between the quadratic

function and the one-dimensional quadratic

equation to find the approximate solution of the

equation;

6. Use quadratic functions to solve practical

problems.

University

Chapter2

exercise

3.

Figures and

Geometry

3.1 Recognize special

parallelograms;

3.2 The properties and

theorems of special

parallel four sides;

3.3 Proof of special

1. Understand the concepts of parallelogram,

rectangle, rhombus, and square, and the

relationship between them, and understand the

instability of quadrilateral;

2. Explore and prove the property theorems of

rectangles, rhombuses and squares;

Reason

Critical

thinking

Reflective

Problem-solvi

ng

2 Ninth grade

mathematics

published by

Beijing

Normal

University

10

  385

高中数学课程图

Unit/ Theme/

Topic

Contents Objectives

Core

Competency

Academic

Proficiency

Level

Assignment Teaching Periods

parallelism;

3.4 The basic nature of

the ratio;

3.5 The nature of

similar graphics；

3. Understand the basic nature of the ratio, the

ratio of line segments, the golden ratio, etc.;

4. Prove the similarity of graphics through

specific examples, and learn the definition and

properties of similar graphics;

5. Explore and understand the judgment theorem

of similar triangles, understand the property

theorem of similar triangles, understand the

similarity of graphics, and solve some practical

problems.

Chapter1

exercise

4.

Circle

4.1 Definition of circle

4.2 Symmetry of circle,

vertical diameter

theorem;

4.3 The relationship

between the

circumferential angle

and the central angle;

4.5 Determine the

conditions of the circle;

4.6 The positional

1. Students improve their mathematical thinking

ability through the process of exploring circles

and related conclusions;

2. Recognize the axis symmetry and center

symmetry of the circle;

3. Understand the relationship between arcs,

chords, central angles, and circumferential angles,

and combine other methods to explore the

vertical radius theorem, the relationship between

circumferential angles and central angles, and the

characteristics of the circumferential angles

Reason

Critical

thinking

Reflective

Problem-solvi

ng

2 Ninth grade

mathematics

published by

Beijing

Normal

University

Chapter3

exercise

15

博实乐“中外融通课程”

386  Unit/ Theme/

Topic

Contents Objectives

Core

Competency

Academic

Proficiency

Level

Assignment Teaching Periods

relationship between the

straight line and the

circle;

4.7 Tangent Length

Theorem

4.8 Regular polygon

inscribed in circle

4.9 Formulas for arc

length and sector

facing straight lines;

4. Explore and know the positional relationship

between points and circles, straight lines and

circles, and circles and circles;

5. Know the concept of tangents, draw tangents

and use the tangent length theorem.

5.

Inverse

proportional

functions

5.1 List inverse

proportional function

expressions

5.2 The graphs and

properties of the inverse

proportional function

5.3 Find the inverse

proportional function

expression from graph

5.4 The transformation

of the inverse

proportional function

1. Draw the graphs of inverse functions.

2. Master the main properties of inverse

functions

3. Model the inverse proportional functions and

solve problems in real life context

Reason

Critical

thinking

Reflective

Problem-solvi

ng

2 Ninth grade

mathematics

published by

Beijing

Normal

University

Chapter5

exercise

10

  387

高中数学课程图

Unit/ Theme/

Topic

Contents Objectives

Core

Competency

Academic

Proficiency

Level

Assignment Teaching Periods

5.5 Application of the

inverse proportional

function

6. Sets 6.1 Theconcept and

representation of sets

6.2 Basic relationships

of collections 6.3 Set of

basic operations

1. Understand the meaning of sets, complete sets

and empty sets; understand the \"belonging\"

relationship between elements and sets; and can

describe collections in natural language, graphic

language and symbolic language.

2. Understand the equivalence of sets;

candetermine a subset of a given set.

3. Understand the meaning of the union,

intersection and complement of two sets; you can

find the union, intersection and complement of

two sets by combining graphics.

Reason

Critical

thinking

Reflective

Problem-solvi

ng

2 Elective

1-1mathemati

cs published

by Beijing

Normal

University

Chapter1

exercise

10

7.

Functions

and graphs

7.1 Modeling and

solving equations

According to the

situation in real life,

establish numerical,

algebraic or graphical

1. Understand the definition of a function, the

relationship between a set and a function,

understand the elements of a function, be able to

find the domain and range of the function, and

understand the constituent elements of a function;

2. In actual situations, functions can be expressed

Reason

Critical

thinking

Reflective

Problem-solvi

ng

2 Precalculus

Chapter 1

exercise

10

博实乐“中外融通课程”

388  Unit/ Theme/

Topic

Contents Objectives

Core

Competency

Academic

Proficiency

Level

Assignment Teaching Periods

models, list equations,

and solve equations

7.2 Definition and

properties of functions

7.3 Functions and

operations between

functions

7.4 Composite

functions,

Parametric equations

and inverse functions

7.5 Image

transformation

in different ways;

3. Understand the monotonicity and extrema

value of the function;

4. Learn to use the graph of the function to study

and analyze the nature of the function;

5. Evaluation composite functions,parametric

equations and inverse functions

8.

Polynomial

functions,

power

functions

and rational

functions

8.1 Linear function,

quadratic function and

modeling

8.2 Power function

properties and graphs

8.3 Properties and

graphs of polynomial

functions

1.Understand the meaning of rational exponent

power, master the operation of power, and be able

to perform simple analysis using the image of the

power function.

2.Understand the relationship between the zero

point of the function and the solution of the

equation; 3.Understand the function zero

existence theorem, and will be used to judge the

Reason

Critical

thinking

Reflective

Problem-solvi

ng

2 Precalculus

Chapter 2

exercise

15

  389

高中数学课程图

Unit/ Theme/

Topic

Contents Objectives

Core

Competency

Academic

Proficiency

Level

Assignment Teaching Periods

8.4 Real roots of

polynomial functions

8.5 Complex Roots of

Polynomial Functions

and Fundamental

Theorems of Algebra

8.6 Image and

properties of rational

functions

8.7 Solving equations in

one variable

8.8 Solving inequalities

in one variable

8.9 The binary method

seeks the approximate

solution of the equation

zero existence of the monotone function.

9.

Exponential

function,

logarithmic

function and

9.1 Exponential

function and logistic

function images and

properties

9.2 Exponential

1. Understand the actual background of the

exponential function model;

2. Understand the meaning of rational exponent

power and the operation of power;

3. Understand the concept and meaning of

Reason

Critical

thinking

Reflective

Problem-solvi

ng

2 Precalculus

Chapter 3

exercise

15

博实乐“中外融通课程”

390  Unit/ Theme/

Topic

Contents Objectives

Core

Competency

Academic

Proficiency

Level

Assignment Teaching Periods

logistic

function

function and logistic

function model

9.3 Image and

properties of

logarithmic function

9.4 Solving and

modeling equations

9.5 Application of

Mathematics in Finance

exponential function, be able to draw images with

the help of a calculator, explore and understand

the monotonicity and special points of

exponential function;

4. Can solve practical problems.

10.

Trigonometr

ic function

10.1 Angle and

measurement

10.2 Trigonometric

functions of acute

angles

10.3 Trigonometric

function and unit circle

of arbitrary angle

10.4 Graphs of sine and

cosine functions

10.5 Tangent function

graph

1. Understand the expansion of angles,

understand the radian system and angle values,

and be able to convert between them;

2. Use the unit circle to understand the theorems

of trigonometric functions at any angle, and be

able to judge the signs of trigonometric functions

in each quadrant;

3. Using the transformation of graphics, master

the transformation into, and be able to find the

amplitude, period, phase, frequency, etc.;

4. Able to determine the value of a, b, c;

5. Master images and functions;

Reason

Critical

thinking

Reflective

Problem-solvi

ng

2 Precalculus

Chapter 4

exercise

20

  391

高中数学课程图

Unit/ Theme/

Topic

Contents Objectives

Core

Competency

Academic

Proficiency

Level

Assignment Teaching Periods

10.6 Inverse

trigonometric functions

10.7 The application of

trigonometric functions

in real life

6. Can solve practical problems.

11.

Solve the

triangle

11.1 Basic identities

11.2 Proof of Triangular

Identity

11.3 Triangular

formulas for sum and

difference of two angles

11.4 Double angle

formula

11.5 The Law of Sines

11.6 The Law of

Cosines

1. Discover the relationship between the angle of

a triangle and the length of a side;

2. Master the theorem of sine and cosine and be

able to solve related problems;

3. Master the triangular relationship formula of

two angles and difference, be able to use and

calculate, and master the formula of product and

difference;

4. Master the double angle formula and

application.

Reason

Critical

thinking

Reflective

Problem-solvi

ng

2 Precalculus

Chapter 5

exercise

20

12.

Analytical

Geometry of

Plane and

12.1 Graphs and

properties of parabola

12.2 Circles and ellipses

12.3 Parabola and

1. Understand the definition of ellipse, hyperbola,

and parabola;

2. Understand the graphs and properties of the

three, and be able to find the equations of ellipse,

Reason

Critical

thinking

Reflective

Problem-solvi

2 Precalculus

Chapter 8

exercise

15

博实乐“中外融通课程”

392  Unit/ Theme/

Topic

Contents Objectives

Core

Competency

Academic

Proficiency

Level

Assignment Teaching Periods

Space hyperbola

12.4 Polar coordinate

equation of conic

section

12.5 Polar curves

hyperbola and parabola according to known

conditions;

3. Determine whether the given equation is

ellipse, hyperbola or parabola according to

known conditions;

4. Ability to use the discriminant of the root to

find the positional relationship between the conic

section and the circle and the coordinates of the

intersection point, etc.;

ng

13.

Discrete

Mathematics

13.1 Permutation and

Combination

13.2 Binomial Theorem

13.3 Sequence

13.4 Number of levels

13.5 Probability

13.6 Describing the

Distribution of a

Quantitative Variable

13.7 Introduction to

Planning a Study

13.8 Random Sampling

1. Master the difference between permutation and

combination, and choose the corresponding

method to solve the given problem;

2. Master the formula and related characteristics

of the binomial theorem, and be able to find out

the binomial coefficients or coefficients with

special requirements such as constant terms;

3. Master the arithmetic sequence and geometric

sequence, and can perform calculations;

4. Master the sum formula of equal difference

and equal ratio;

5. Learn to master the method of classical

Reason

Critical

thinking

Reflective

Problem-solvi

ng

2 Precalculus

Chapter 10

exercise

10