? 93

高中數(shù)學(xué)課程圖

Timeline

Unit/ Theme/

Topic

Contents Objectives

Core Competency

Academic

Proficiency

Level

Teaching

Hours

ATL Skills Homework, Link to TOK,

IM, LP and CAS

10.31-

11.20

Equations,

inequalities

and graphs

4.1 Solving

equations of the

type |axb|=|cx-d|

4.2 Inequalities

4.3Solving modulus inequalities

4.4 Sketching

graphs of cubic

polynomials and

their moduli

4.5 Solving cubic

inequalities

graphically

4.6 Solving more

complex

quadratic

equations, algebraically and

graphically

? solve graphically or algebraically

equations of the type |ax+b|=c and

|ax+b|=|cx+d|

? solve graphically or algebraically

inequalities of the type |ax+b|>c,

|ax+b|<c and ax+b<cx+d

? use substitution to form and solve a

quadratic equation in order to solve a

related equation

? sketch the graphs of cubic

polynomials and their moduli, when

given in form y=k(x-a)(x-b)(x-c)

? solve cubic inequalities in the form

k(x-a)(x-b)(x-c)<d graphically

Inquiring

Reflective

Problem-solving

2 27 Critical-thinking: Draw

reasonable

conclusions and

generalizations;

Test generalizations and

conclusions;

Collaboration

skills- Encourage others

to contribute.

Reflection

skills-Consider

content what do

I learn about

today?

Homework

Assignment on textbook

and Past papers

TOK

Is mathematics a formal

language?

IM

The Sulba Sutras in

ancient India and the

Bakhshali manuscript

contained an algebraic

formula for solving quadratic equations.

11.21-

12.4

Logarithmic and

exponential

functions

6.1 Introduction

to logarithms;

6.2 Rules of

logarithms;

6.3 Common and

natural

logarithms;

6.4 Logarithmic

equations;

6.5 Graphs of

logarithmic and

exponential

functions.

? know simple properties and graphs of

the logarithmic and exponential

functions including ln x and ex

(series expansions are not required).

? know and use the laws of logarithms

(including change of base of

logarithms).

? solve equations of the form ax= b.

? the graphs of functions

2 3 1

,, , , y xy y x y xy x x = = = = =

know the characters of exponential functions

? Monotonicity of exponential and logarithmic functions

? Know the relationship of exponential and logarithmic

functions

? Comparing the increasing rate

of exponential function, linear

function, logarithmic function

Reason

Critical

thinking

Inquiring

Reflective

Problem-solvi

ng

2 18 Critical thinking skills-Analyze complex

concepts and

projects into

their constituent parts and

synthesize them

to create new

understanding;

Creative-thinking

skills-Apply existing

knowledge to

generate new

ideas, products

or process;

Homework

Assignment on textbook

and Past papers

TOK

We are trying to find a

method to evaluate the are

under a curve.

“The main reason

knowledge is produced is

to solve problems.” To

what extent do you agree

with this statement?

IM

The term power was used

by the Greek mathematician Euclid for the square

of a line.

博實(shí)樂(lè)“中外融通課程”

94?

Timeline

Unit/ Theme/

Topic

Contents Objectives

Core Competency

Academic

Proficiency

Level

Teaching

Hours

ATL Skills Homework, Link to TOK,

IM, LP and CAS

12.5-

12.25

Straight-line

graphs

7.1 Gradient of a

line;

7.2 Equation of a

straight line;

7.3 Equation of

parallel and

perpendicular

lines;

7.4 Perpendicular

bisector;

7.5 Linear law.

? interpret the equation of a straight

line graph in the form y = mx + c.

? transform given relationships,

including y = axnand y = Abx, to

straight line form and hence

determine unknown constants by

calculating the gradient or intercept of

the transformed graph.

? solve questions involving mid-point

and length of a line.

? know and use the condition for two

lines to be parallel or perpendicular.

Reason

Inquiring

Reflective

Problem-solving

2 27 Communication skills-Use

and interpret a

range of discipline-specific

terms and symbols

Organization

skills-keep an

organized and

logical system

of information

files/notebook.

Homework

Assignment on textbook

and Past papers

Aliens might not be able

to speak an Earth language but would they still

describe the equation of

a straight line in similar

terms? Is mathematics a

formal language?

12.26-

1.8

Circular

measure

8.1 Radian measure;

8.2 Arc length

and area of a sector and a segment.

? convert degree to radian and radian

to degree

? solve problems involving the arc

length and sector area of a circle,

including knowledge and use of

radian measure.

Inquiring

Reflective

2 18 Critical thinking skills-Propose and evaluate a variety of

solutions, draw

reasonable

conclusions and

generalizations

Creative

-thinking

skills-Apply

existing

knowledge to

generate new

ideas, products

or process;

Homework

Assignment on textbook

and Past papers

TOK

Which is a better measure

of angle: radian or degree?

What are the “best”

criteria by which to decide?

? 95

高中數(shù)學(xué)課程圖

Timeline

Unit/ Theme/

Topic

Contents Objectives

Core Competency

Academic

Proficiency

Level

Teaching

Hours

ATL Skills Homework, Link to TOK,

IM, LP and CAS

2.6-2.12 Trigonometry 9.1 General angles;

9.2 Trigonometric

ratios of any

angles;

9.3 Solving

trigonometric

equations;

9.4 Trigonometric

graphs;

9.5 Trigonometric

identities.

? know the six trigonometric functions

of angles of any magnitude (sine,

cosine, tangent, secant, cosecant,

cotangent).

? understand amplitude and periodicity

and the relationship between graphs

of e.g. sinxand sin2x.

? draw and use the graphs of

y = a sin (bx) + c,

y = a cos (bx) + c,

y = a tan (bx) + c.

where a and b are positive integers

and c is an integer.

? use the relationships

sin A

cos A

= tan A , cos A

sin A

= cot A ,

2 2 sin cos 1 A A + = , sec2 A = 1+ tan2 A , cosec2 A = 1+ cot2 A .

And solve simple trigonometric equations involving the

six trigonometric functions and the above relationships

(not including general solution of trigonometric equations).

? prove simple trigonometric identities.

Reason

Critical

thinking

Inquiring

Reflective

Problem-solving

2 9 Organization

skills-set

goals that are

challenging and

realistic

Transfer skillsChange the

context of an

inquiry to gain

different perspectives

Homework

Assignment on textbook

and Past papers

TOK

What are the platonic

solids and why are they

an important part of the

language of mathematics?

IM

The word “trigonometry”

is derived from the Greek

words “trigon”, meaning triangle and “metria”,

meaning measure.

博實(shí)樂(lè)“中外融通課程”

96?Timeline Unit/ Theme/

Topic

Contents Objectives

Core Competency

Academic

Proficiency

Level

Teaching

Hours

ATL Skills Homework, Link to TOK,

IM, LP and CAS

2.13-3.5 Permutation

and combination

10.1 The basic

counting

principle;

10.2 Permutations;

10.3 Combinations.

? normalize and distinguish between a permutation case

and a combination case.

? know and use the notation n! (with 0! = 1), and the expressions for permutations and combinations of n items

taken r at a time.

? answer simple problems on arrangement and selection

(cases with repetition of objects, or with objects arranged

in a circle or involving both permutations and combinations, are excluded).

Critical

thinking

Reflective

Problemsolving

2 27 Communication

skills—Understand and use

mathematical

notation;

Affective

skills-Practise

strategies to

reduce stress

and anxiety.

Information literacy

skills-Process

data and report

results

Homework

Assignment on textbook

and Past papers

TOK

How many different tickets

are possible in a lottery?

What does this tell us

about the ethics of selling

lottery tickets to those

who do not understand the

implications of these large

numbers?

IM

We can not simply that it

for granted that e can multiply out two infinite series

easily. Gustav Dirichlet,

a German mathematician,

proved how this can be

done.

3.6-3.19 Series 11.1 Pascal’s

triangle;

11.2 Binomial

theorem and

general term

formulae.

11.3 Arithmetic

progressions

11.4 Geometric

progressions

11.5 Infinite

geometric

series

11.6 Further

normalize and

geometric

series

? use the Binomial Theorem for expansion of (a + b)n for

positive integral n.

? use the general term , 0<r<n.(knowledge of

the greatest term and properties of the coefficients is not

required).

? normalize arithmetic and geometric progressions

? use the formulae for the nth term and for the sum of the

first n terms to solve problems involving arithmetic and

geometric progression

? use the condition for the convergence of a geometric

progression, and the fomula for the sum to infinity of a

convergent geometric progression

Reason

Inquiring

Reflective

Problemsolving

2 18 Communication skillsMake effective

summary notes

for studying;

Collaboration

skills-Give and

receive meaningful feedback

Media literacy skillsSeek a range of

perspectives for

multiple and

varied sources

Homework

Assignment on textbook

and Past papers

TOK

Is all knowledge concerned with identification

and use od patterns?

IM

The series S=1-1+1-

1+1-1+… is known as

Grand’s series, after the

Italian Grand.

-

? 97

高中數(shù)學(xué)課程圖

Timeline

Unit/ Theme/

Topic

Contents Objectives

Core Competency

Academic

Proficiency

Level

Teaching

Hours

ATL Skills Homework, Link to TOK,

IM, LP and CAS

3.20-

5.14

Calculus 12.1 Differentiation from first

principles;

12.2 Basic

techniques and

rules of

differentiation.

12.3 Tangents

and normal;

12.4 Rates of

change and

small changes;

12.5 Graphical

interpretation

of a function;

12.6 Problems on

maximum and

minimum

values;

12.7 Integration

techniques;

12.8 Definite

integral;

12.9 Kinematics.

? understand the idea of a derived

function.

? use the notations ( ) f x ′ , ( ) f x ′′ , dydx , ( ) d dy dx dx ? ?

=? ? ? ? .

? use the derivatives of the standard

functions xn(for any rational n), sin x,

cosx, tan x, ex, ln x, together with

constant multiples, sums and

composite functions of these.

? differentiate products and quotients of functions.

? apply differentiation to gradients, tangents and normal,

stationary points, connected rates of change, small increments and approximations and practical maxima and

minima problems.

? discriminate between maxima and minima by any

method.

? understand integration as the reverse process of differentiation.

? integrate sums of terms in powers of x.

? integrate functions of the form (ax + b)n(excluding n =

–1), , sin (ax + b), cos (ax + b).

? evaluate definite integrals and apply integration to the

evaluation of plane areas.

? apply differentiation and integration

to kinematics problems that involve displacement, velocity and acceleration of a particle moving in a straight

line with variable or constant acceleration, and the use

of x –t and v –t graphs.

Reason

Critical

thinking

Inquiring

Reflective

Problem-solving

2 54 Critical-thinking

skills-Identify

trends and

forecast possibilities; Draw

reasonable

conclusions and

generalizations;

Communication

skills-Understand and use

mathematical

notation

Media literacy skillsSeek a range of

perspectives for

multiple and

varied sources

Homework

Assignment on textbook

and Past papers

TOK

What value does the

knowledge of limits have?

IM

Maria Agnessi, an

18th-century Italian

mathematician, published

a text on calculus and also

studied curves of the form.

博實(shí)樂(lè)“中外融通課程”

98?

Timeline

Unit/ Theme/

Topic

Contents Objectives

Core Competency

Academic

Proficiency

Level

Teaching

Hours

ATL Skills Homework, Link to TOK,

IM, LP and CAS

5.15-

5.28

Vectors 13.1 Basic concepts of vectors;

13.2 Laws of

vectors;

13.3 Unit vectors

and position

vectors;

13.4 Applications

of vectors.

? use vectors in any form, e.g. a

b

? ? ? ? ? ? , , p

? know and use position vectors and unit vectors.

? find the magnitude of a vector; add and subtract vectors and multiply vectors by scalars. Determine whether

two vectors are perpendicular

? compose and resolve velocities.

? use relative velocity, including solving problems on

interception (but not closest approach).

Critical

thinking

Inquiring

Reflective

Problem-solving

2 18 Organization

skills-Use

appropriate

strategies for

organizing

complex information

Information literacy

skills-Process

data and report

results

Critical

thinking

skills-Revise

understanding

based on new

evidence;

Homework

Assignment on textbook

and Past papers

TOK

Do you think that one form

of symbolic representation

is preferable to another?

IM

Greek philosopher and

mathematician, Aristotle,

calculated the combined

effect of two or more forces

called the Parallelogram

law.

? 99

高中數(shù)學(xué)課程圖

Timeline

Unit/ Theme/

Topic

Contents Objectives

Core Competency

Academic

Proficiency

Level

Teaching

Hours

ATL Skills Homework, Link to TOK,

IM, LP and CAS

5.29-7.1 Probability

and Statistics

14.1Representation data

14.2Measures of

central tendency

14.3Measures of

variation

14.4Probability

using

2-dimensional

grid, Venn

diagram, tree

diagram，two

way tables

14.5Probability

distributions

14.6 The binomial

distribution

14.7 The geometric distribution

14.8 The normal

distribution

14.9 Statistics

from technology

14.10 Standard

deviation

? Types of data, representation of discrete data and

continuous data, compare different data representations.

? Sampling, lottery，random sampling, Stratified sampling, population, sample

? Statistics terminology

? The mode and the modal class, the mean, the median.

? The range, the interquartile range and percentiles.

? Variance and standard deviation.

? Box-and-whisker plots, range, leaf-stem plot

? Statistics from technology

? Experimental probability,

? probability from tabled data

? .Sample space, theoretical probability

? Experiments, sample space, sample, events and

outcomes.? Mutually exclusive events and the addition

law, independent events and the multiplication law,

Condition probability, dependent events and conditional

probability

? The factorial function

? Discrete random variables.

? Probability distributions.

? Expectation and variance of a discrete random variable.

? The binomial distribution

? The geometric distribution

? Continuous random variables.

? The normal distribution

? Modelling with the normal distribution

?The normal approximation to the binomial distribution.

Critical

thinking

Inquiring

Reflective

Problem-solving

2 63 Collaboration

skills-Listen

actively to other

perspectives

and ideas.

Critical thinking skills-Consider ideas

from multiple

perspectives;

draw reasonable conclusion

and generalizations;

Transfer skillsChange the

context of an

inquiry to gain

different perspectives

Homework

Assignment on textbook

and Past papers

TOK

The nature of knowing is

there a difference between

information and data?

IM

Ronald Fisher lived in

the UK and Australia and

has been described as

“a genius who almost

single-handedly created

the foundations for modern

statistical science”. He

used statistics to analyse

problems in medicine,

agriculture and social

sciences.

博實(shí)樂(lè)“中外融通課程”

100?

4 Assessment

4.1External Assessment

Introduction

to assessment

All candidates will take two written papers. The Additional Math、 Probability & Statistics syllabus contents will be assessed by Paper 1 and Paper 2.

Paper 1

Additional Math: 10–12 questions of various lengths. No choice of question. Duration 2 hours. 80 marks. Externally marked. Weighting 50%.

Probability & Statistics:7-8 questions of various lengths. No choice of

question. Duration 1 hours 15minutes. 50 marks. Externally marked.

Weighting 50%.

Paper 2

Additional Math:10–12 questions of various lengths. No choice of question.

Duration 2 hours. 80 marks.Externally marked. Weighting 50%.

Probability & Statistics:7-8 questions of various lengths. No choice of

question. Duration 1 hours 15minutes. 50 marks. Externally marked.

Weighting 50%.

4.2 In-school Assessment

(1) Formative Assessment

1 . Monthly Test (5%+5%)

2. Assignment (20%)

3. Performance (10%) (Attendance, Homework handing in, Note taking)

(2) Summative Assessment

1. Mid-term exam and Final Exam

2. Chapter review test

5 Resources

[1] Additional Mathematics, by Sue Pemberton, ISBN: 9781108411660, published by

Cambridge Press, etc.

[2] Cambridge International AS&A Level Mathematics: Probability & Statistics 1

Coursebook. ISBN:978-1-108-40730-4

[3]http://www.sdlib.com.cn/

[4]www.cambridgeinternational.org

Chinese and International Integrated

Curriculums for Bright Scholar

High School Section

CDP Math AI HL

Curriculum Map

（2022 version）

Complied by Guangdong Country Garden Senior High Section

博實(shí)樂(lè)“中外融通課程”

102?

1 Course Introduction

1.1 Introduction

Mathematics has been described as the study of structure, order and relation that has

evolved from the practices of counting, measuring and describing objects. Mathematics

provides a unique language to describe, explore and communicate the nature of the world

we live in as well as being a constantly building body of knowledge and truth in itself that is

distinctive in its certainty. These two aspects of mathematics, a discipline that is studied for

its intrinsic pleasure and a means to explore and understand the world we live in, are both

separate yet closely linked.

Mathematics is driven by abstract concepts and generalization. This mathematics is

drawn out of ideas, and develops through linking these ideas and developing new ones.

These mathematical ideas may have no immediate practical application. Doing such

mathematics is about digging deeper to increase mathematical knowledge and truth. The

new knowledge is presented in the form of theorems that have been built from axioms and

logical mathematical arguments and a theorem is only accepted as true when it has been

proven. The body of knowledge that makes up mathematics is not fixed; it has grown during

human history and is growing at an increasing rate.

The side of mathematics that is based on describing our world and solving practical

problems is often carried out in the context of another area of study. Mathematics is used

in a diverse range of disciplines as both a language and a tool to explore the universe;

alongside this its applications include analyzing trends, making predictions, quantifying

risk, exploring relationships and interdependence.

DP Curriculum Mapping

Subject DP Math AI HL Level G3&G4 Syllabus Code

Course Code Credit 14 Duration 2 years

Teaching Periods 480 Designer Wang Li Completed Date 2022.9

? 103

高中數(shù)學(xué)課程圖

1.2 Aims

1) enjoy mathematics, and develop an appreciation of the elegance and power of

mathematics.

2) develop an understanding of the principles and nature of mathematics.

3) communicate clearly and confidently in a variety of contexts.

4) develop logical, critical and creative thinking, and patience and persistence in

problem-solving.

5) employ and refine their powers of abstraction and generalization.

6) apply and transfer skills to alternative situations, to other areas of knowledge and

to future developments.

7) appreciate how developments in technology and mathematics have influenced each

other.

8) appreciate the moral, social and ethical implications arising from the work of

mathematicians and the applications of mathematics.

9) appreciate the international dimension in mathematics through an awareness of the

universality of mathematics and its multicultural and

historical perspectives.

10) appreciate the contribution of mathematics to other disciplines, and as a particular

“area of knowledge” in the TOK course.

2 Course Structure

Math AI HL

Topic 1

Number and

Algebra

Functions Geometry and

Trigonometry

Probability and

statistics

Probability and

statistics

Topic 2 Topic 3 Topic 4 Topic 5 IA

博實(shí)樂(lè)“中外融通課程”

104? 3 Course outline

G3

Timel

ine

Unit/

Theme/

Topic

Contents Objectives

Core

Competency

Academic

Proficiency

Level

Teaching

Periods

ATL Skills

Homework, Link to

TOK, IM, LP and CAS

G3

Week

1~We

ek 9

Number

and

Algebra

HL/SL1.1

? Operations with numbers in

the form a ×10k where 1 ≤ a

< 10 and k is an integer.

HL/SL1.2

? Arithmetic sequences and

series.

? Use of the formulae for the

nth term and the sum of the

first n terms of the sequence.

? Use of sigma notation for

sums of arithmetic sequences.

? Applications.

? Analysis, interpretation and

prediction where a model is

not perfectly arithmetic in

real life.

HL/SL1.3

? Modelling

real-life

situations

with the

structure of

arithmetic

and geometric sequences and

series

allows for

prediction,

analysis and

interpretation.

? Different

representations of

numbers

enable

Logical reasoning,

Mathematical

operations,

Mathematical

abstraction

2 58 Creative

thinking,

Reflection,

Information

literacy

TOK:

1. Do the names that we

give things impact how

we understand them?

For instance, what is the

impact of the fact that

some large numbers are

named, such as the

googol and the googolplex, while others are

represented in this

form?

2. Is all knowledge

concerned with identification and use of patterns? Consider Fibonacci numbers and connections with the golden

ratio.

3. How do mathematiTimel ine

Unit/

Theme/

Topic

Contents Objectives

Core

Competency

Academic

Proficiency

Level

Teaching

Periods

ATL Skills

Homework, Link to

TOK, IM, LP and CAS

G3

Week

1~We

ek 9

Number

and

Algebra

HL/SL1.1

? Operations with numbers in

the form a ×10k where 1 ≤ a

< 10 and k is an integer.

HL/SL1.2

? Arithmetic sequences and

series.

? Use of the formulae for the

nth term and the sum of the

first n terms of the sequence.

? Use of sigma notation for

sums of arithmetic sequences.

? Applications.

? Analysis, interpretation and

prediction where a model is

not perfectly arithmetic in

real life.

HL/SL1.3

? Modelling

real-life

situations

with the

structure of

arithmetic

and geometric sequences and

series

allows for

prediction,

analysis and

interpretation.

? Different

representations of

numbers

enable

Logical reasoning,

Mathematical

operations,

Mathematical

abstraction

2 58 Creative

thinking,

Reflection,

Information

literacy

TOK:

1. Do the names that we

give things impact how

we understand them?

For instance, what is the

impact of the fact that

some large numbers are

named, such as the

googol and the googolplex, while others are

represented in this

form?

2. Is all knowledge

concerned with identification and use of patterns? Consider Fibonacci numbers and connections with the golden

ratio.

3. How do mathemati-

? 105

高中數(shù)學(xué)課程圖

Timel

ine

Unit/

Theme/

Topic

Contents Objectives

Core

Competency

Academic

Proficiency

Level

Teaching

Periods

ATL Skills

Homework, Link to

TOK, IM, LP and CAS

? Geometric sequences and

series.

? Use of the formulae for the

nth term and the sum of the

first n terms of the sequence.

? Use of sigma notation for

the sums of geometric sequences.

? Applications.

HL/SL 1.4

? Financial applications of

geometric sequences and

series:

? compound interest

? annual depreciation.

HL/SL 1.5

? Laws of exponents with

integer exponents.

? Introduction to logarithms

with base 10 and e.

? Numerical evaluation of

quantities to

be compared and

used for

computational

purposes

with ease

and accuracy.

? Numbers

and formulae can appear in different, but

equivalent

forms, or

representations, which

can help us

to establish

identities.

cians reconcile the fact

that some conclusions

seem to conflict with

our intuitions? Consider

for instance that a finite

area can be bounded by

an infinite perimeter.

4. How have technological advances affected the nature and

practice of mathematics? Consider the use of

financial packages for

instance.

5. Is mathematics invented or discovered?

For instance, consider

the number e or logarithms–did they already

exist before man defined

them? (This topic is an

opportunity for teachers

博實(shí)樂(lè)“中外融通課程”

106? Timel

ine

Unit/

Theme/

Topic

Contents Objectives

Core

Competency

Academic

Proficiency

Level

Teaching

Periods

ATL Skills

Homework, Link to

TOK, IM, LP and CAS

logarithms using technology.

HL/SL 1.6

? Approximation: decimal

places, significant figures.

? Upper and lower bounds of

rounded numbers.

? Percentage errors.

? Estimation.

HL/SL 1.7

? Amortization and annuities

using technology.

HL/SL 1.8.0

? Use technology to solve:

? Systems of linear equations in

up to 3 variables

? Polynomial equations

HL/SL 1.8.1

? The concept and representation of set. Through exam-

? Formulae

are a generalization

made on the

basis of

specific

examples,

which can

then be

extended to

new examples

? Mathematical

financial

models such

as compounded

growth

allow computation,

evaluation

to generate reflection on

“the nature of mathematics”).

6. Is mathematical reasoning different from

scientific reasoning, or

reasoning in other areas

of knowledge?

7. What role does language play in the accumulation and sharing of

knowledge in mathematics? Consider for

example that when

mathematicians talk

about “imaginary” or

“real” solutions they are

using precise technical

terms that do not have

the same meaning as the

everyday terms.

8. What is meant by the

? 107

高中數(shù)學(xué)課程圖

Timel

ine

Unit/

Theme/

Topic

Contents Objectives

Core

Competency

Academic

Proficiency

Level

Teaching

Periods

ATL Skills

Homework, Link to

TOK, IM, LP and CAS

ples, understand the meaning of set, understand the

element and set \"belong to\".

For specific problems, can

depict sets with symbolic

language on the basis of

natural language and

graphic language. In specific situations, understand the

meaning of the complete set

and the empty set.

? The basic relationships of

the set. Understand the

meaning of inclusion and

equality between sets and

be able to identify subsets

of a given set.

? Basic operation of set. Understand the meaning of

union and intersection of

two sets, and can solve the

union and of two sets Interand interpreta- tion of debt and invest- ment both approx- imately and accurately. ? Approxi- mation of numbers adds uncer- tainty or inaccuracy to calcula- tions, lead- ing to po- tential er- rors but can be useful when handling terms “l(fā)aw” and “the- ory” in mathematics. How does this compare to how these terms are used in different areas of knowledge? 9. Is it possible to know about things of which we can have no experi- ence, such as infinity? 10. How does language shape knowledge? For example do the words “imaginary” and “com- plex” make the concepts more difficult than if they had different names? 11. Why might it be said that eiπ +1 = 0 is beauti- ful? What is the place of beauty and elegance in

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section. Understand the

meaning of the complement

of a subset in a given set,

can solve the complement

of a given subset. Venn diagrams can be used to express the basic relations and

operations of sets. And experience the role of graphics

in understanding abstract

concepts.

HL/SL 1.8.2

? Necessary conditions, sufficient conditions, sufficient and necessary conditions. ① Through sorting

out typical mathematical

propositions, understand

the meaning of necessary

conditions, comprehensibility the relation between

extremely

large or

small quantities.

? Quantities

and values

can be used

to describe

key features

and behaviours of

functions

and models,

including

quadratic

functions.

? Utilizing

complex

numbers

provides a

system to

efficiently

mathematics? What

about the place of creativity?

12. Given the many

applications of matrices

in this course, consider

the fact that mathematicians marvel at some of

the deep connections

between disparate parts

of their subject. Is this

evidence for a simple

underlying mathematical reality? Mathematics, sense, perception

and reason–if we can

find solutions of higher

dimensions, can we

reason that these spaces

exist beyond our sense

perception?

13. Mathematics can be

? 109

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quality theorem and necessary conditions. ②

Understand the meaning of

sufficient conditions and

judgment by combing

typical mathematical

propositions the relation

between definite theorem

and sufficient condition.

③ Through sorting out the

typical mathematical

propositions, understand

the meaning of sufficient

and necessary conditions,

understand the number the

relationship between

learning definition and

necessary and sufficient

conditions.

? Universal and existential

quantifiers. Understand the

meaning of universal

simplify and

solve problems.

? Matrices

allow us to

organize

data so that

they can be

manipulated

and relationships

can be

determined.

?

Representin

g abstract

quantities

using complex numbers in different forms

enables the

used successfully to

model real-world processes. Is this because

mathematics was created to mirror the world

or because the world is

intrinsically mathematical?

International-mindedness:

1. The history of

number from Sumerians and its development to the present

Arabic system.

2. Aryabhatta is

sometimes considered

the “father of algebra”–compare with

alKhawarizmi; the use

of several alphabets in

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quantifiers and existential

quantifiers through known

mathematical examples.

? Negation of universal

quantifier proposition and

existential quantifier

proposition. ① Can correctly use existential quantifiers to negate universal

quantifier propositions.

②Can correctly use universal quantifiers to negate

existential quantifier

propositions.

HL/SL 1.8.3

? The properties of equality

and inequality. Sort out the

properties of equality, understand the concept of inequality, master the properties of

inequality.

? Basic inequality. Undersolution of real-life problems. mathematical notation (for example the use of capital sigma for the sum). 3. The chess legend (Sissa ibn Dahir) 4. Do all societies view investment and interest in the same way? Links to other sub- jects: 1. Chemistry (Avogadro’s number); physics (order of mag- nitude); biology (mi- croscopic measure- ments); sciences (un- certainty and precision of measurement) 2. Radioactive decay,

? 111

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stand the basic inequality

????≤????? ? ? ??????≥

??.Combined with specific

examples, can solve simple

maxima or minima problems.

HL 1.9

? Laws of logarithms:

log axy = log a x + log a y

log a(x/y)=log a x ? log a y

log a x m = mlog a x

for a, x, y > 0

HL 1.10

? Simplifying expressions,

both numerically and algebraically, involving rational

exponents.

HL1.11

? The sum of infinite geometric sequences.

HL1.12

nuclear physics, charging and discharging

capacitors (physics).

3. Loans and repayments (economics and

business management).

4. Calculation of pH

and buffer solutions

(chemistry).

5. Order of magnitudes (physics); uncertainty and precision of

measurement (sciences).

6. Exchange rates

(economics), loans

(business management).

7. Kirchhoff’s laws

(physics).

8. pH, buffer calculations and finding activation energy from experimental data (chem-

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? Complex numbers: the

number i such that i 2 = ? 1.

? Cartesian form: z = a + bi;

the terms real part, imaginary part, conjugate, modulus and argument.

? Calculate sums, differences,

products, quotients, by hand

and with technology. Calculating powers of complex

numbers, in Cartesian form,

with technology.

? The complex plane.

? Complex numbers as solutions to quadratic equations

of the form ax 2 + bx + c =

0, a ≠ 0, with real coefficients where b 2 ? 4ac < 0.

AH1.13

? Modulus–argument (polar)

form:

z = r cosθ + isinθ = rcisθ.

istry).

9. Stochastic processes, stock market values and trends (business

management).

Homework:

Exercises from the text

books or questions from

the IB exams. Sometimes, may be a summary of what have

learnt.

? 113

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? Exponential form: z = re iθ

? Conversion between Cartesian, polar and exponential

forms, by hand and with

technology.

? Calculate products, quotients and integer powers in

polar or exponential forms.

? Adding sinusoidal functions

with the same frequencies

but different phase shift angles.

Geometric interpretation of complex numbers.

HL1.14

? Definition of a matrix: the

terms element, row, column

and order for m× n matrices.

? Algebra of matrices: equality; addition; subtraction;

multiplication by a scalar

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for m × n matrices.

? Multiplication of matrices.

? Properties of matrix multiplication: associativity, distributivity and

non-commutativity.

? Identity and zero matrices.

? Determinants and inverses

of n × n matrices with

technology, and by hand for

2 × 2 matrices.

? Awareness that a system of

linear equations can be

written in the form Ax = b.

? Solution of the systems of

equations using inverse matrix.

HL1.15

? Eigenvalues and eigenvectors.

? Characteristic polynomial

of 2 × 2 matrices.

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? Diagonalization of 2 × 2

matrices (restricted to the

case where there are distinct

real eigenvalues).

? Applications to powers of 2

× 2 matrices.

Week

10~w

eek

24

Function SL 2.1 ? Different forms of the equation of a straight line.

? Gradient; intercepts.

? Lines with gradients ???and

???

Parallel lines ??? = ??? .

Perpendicular lines m 1

×???= ? 1.

SL 2.2

? Concept of a function, domain, range and graph.

? Function notation, for example ????????????????????.

? Different

representations of

functions,

symbolically and visually as

graphs,

equations

and tables

provide

different

ways to

communiMathematical abstraction, Mathematical modeling, Logical rea- soning, Mathematical operations 2 84 Critical thinking, Transfer, Communi- cation, Re- flection, Information literacy TOK: 1. Descartes showed that geometric problems could be solved alge- braically and vice versa. What does this tell us about mathematical representation and mathematical knowl- edge? 2. Do you think mathematics or logic should be classified as a language?

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? The concept of a function as

a mathematical model.

? Informal concept that an

inverse function reverses or

undoes the effect of a function.

? Inverse function as a reflection in the line ?? ???, and

the notation ???? ????.

HL/SL2.3

? The graph of a function; its

equation ?? ???????.

? Creating a sketch from information given or a context, including transferring a

graph from screen to paper.

? Using technology to graph

functions including their

sums and differences.

cate mathematical

relationships.

? The parameters in a

function or

equation

may correspond to

notable

geometrical

features of a

graph

and can

represent

physical

quantities in

spatial dimensions.

? Moving

between

3. Does studying the

graph of a function contain the same level of

mathematical rigour as

studying the function

algebraically? What are

the advantages and disadvantages of having

different forms and

symbolic language in

mathematics?

4. What role do

models play in mathematics? Do they play a

different role in mathematics compared to their

role in other areas of

knowledge?

5. What is it about

models in mathematics

that makes them effective? Is simplicity a

? 117

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HL/SL 2.4

? Determine key features of

graphs.

? Finding the point of intersection of two curves or

lines using technology.

HL/SL 2.5

? Modelling with the following functions:

? Linear models. ?? ?? ?

???? ????

? Quadratic models.

?? ?? ?????? ????? ???? a ≠

0. Axis of symmetry, vertex, zeros and roots, intercepts on the x-axis and y

-axis.

? Exponential growth and

decay models.

?????? ??????? ???

?????? ???????? ???, (for

?? ? ?)

different

forms to

represent

functions

allows for

deeper understanding

and provides different approaches to

problem

solving.

? Our spatial

frame of

reference

affects the

visible part

of a function and by

changing

this “window” can

show more

desirable characteristic

in models?

6. Is mathematics

independent of culture?

To what extent are we

aware of the impact of

culture on what we believe or know?

7. Is there a hierarchy of areas of knowledge in terms of their

usefulness in solving

problems?

8. Does the applicability of knowledge vary

across the different areas of knowledge? What

would the implications

be if the value of all

knowledge was measured solely in terms of

its applicability?

International-mindedness:

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Week

25~

Week

40

Geometry and

Trigonometry

HL/SL3.1

? The distance between two

points in three dimensional

space, and their midpoint.

? Volume and surface area of

three-dimensional solids including right-pyramid, right

cone, sphere, hemisphere

and combinations of these

solids.

? The size of an angle between two intersecting lines

or between a line and a

plane.

HL/SL3.2

? Use of sine, cosine and

tangent ratios to find the

sides and angles of

right-angled triangles.

? The sine rule:

a/sinA=b/sinB=c/sinC

? The cosine rule:

? The properties of

shapes are

highly dependent on

the dimension they

occupy in

space.

? Volume

and surface

area of

shapes are

determined

by formulae, or general mathematical

relationships

or rules

expressed

Intuitive imagination,

Mathematical

abstraction,

Logical reasoning,

Mathematical

modeling.

2 92 Critical

thinking,

Creative

thinking,

Transfer,

Collaboration,

Information

literacy

TOK:

1. What is an axiomatic system? Are axioms self evident to everybody?

2. Is it ethical that

Pythagoras gave his

name to a theorem that

may not have been his

own creation? What

criteria might we use to

make such a judgment?

3. If the angles of a

triangle can add up to

less than 180°, 180° or

more than 180°, what

does this tell us about

the nature of mathematical knowledge?

4. Does personal

experience play a role in

the formation of

? 119

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c2 = a2 + b2 ? 2abcosC

cosC = a2 + b2? c 2/2ab

? Area of a triangle as: (1/2)

absinC.

HL/SL3.3

? Applications of right and

non-right angled trigonometry, including Pythagoras’ theorem.

? Angles of elevation and

depression.

? Construction of labelled

diagrams from written

statements.

HL/SL3.4

? The circle: length of an arc;

area of a sector.

HL/SL3.5

? Equations of perpendicular

bisectors.

HL/SL3.6

? Voronoi diagrams: sites,

using symbols or variables.

? The relationships

between the

length of the

sides and

the size of

the angles in

a triangle

can be used

to solve

many problems involving

position,

distance,

angles and

area.

? Different

representaknowledge claims in mathematics? Does it play a different role in mathematics compared to other areas of knowl- edge? 5. Is the division of knowledge into disci- plines or areas of knowledge artificial? 6. Which is the better measure of an angle, degrees or radians? What criteria can/do/should mathema- ticians use to make such judgments? 7. To what extent is mathematical knowl- edge embedded in par- ticular traditions or bound to particular cul-

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vertices, edges, cells.

? Addition of a site to an

existing Voronoi diagram.

? Nearest neighbour interpolation.

? Applications of the “toxic

waste dump” problem.

HL/SL 3.6.1

? Basic three-dimensional

graph. ① Observe spatial

graphics by physical objects

and computer software, and

understand the structural

characteristics of column,

cone, table, ball and simple

combination, and can use

these characteristics to describe the structure of simple objects in real life. ②

Know the formulas for calculating the surface area

tions of

trigonometric expressions help to

simplify

calculations.

? Systems of

equations

often, but

not always,

lead to intersection

points.

? In two

dimensions,

the Voronoi

diagram

allows us to

navigate,

path-find or

establish an

optimum

tures? How have key

events in the history of

mathematics shaped its

current form and methods?

8. When mathematicians and historians say

that they have explained

something, are they

using the word “explain” in the same way?

9. Vectors are used

to solve many problems

in position location.

This can be used to save

a lost sailor or destroy a

building with a laser-guided bomb. To

what extent does possession of knowledge

carry with it an ethical

obligation?

? 121

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and volume of spheres,

prisms, pyramids and

prisms, and be able to use

the formulas to solve simple

practical problems.③ Can

draw simple space graphics

(cuboid, ball, cylinder,

cone, prism and its simple

combination) with oblique

two measurement method.

? Position relation of basic

graph. ① With the help of

cuboids, on the basis of an

intuitive understanding of

the position relations of

space points, lines and

planes, the definition of

the position relations of

space points, lines and

planes is abstracted, and

several facts and theorems

are understood. ② Starting

position.

? Different

measurement systems can be

used for

angles to

facilitate

ease of calculation.

? Vectors

allow us to

determine

position,

change of

position

(movement)

and force in

two and

threedimensional

space.

10. Mathematics and

the knower: Why are

symbolic representations of

three-dimensional objects easier to deal with

than visual representations? What does this

tell us about our knowledge of mathematics in

other dimensions?

11. What counts as

understanding in

mathematics? Is it more

than just getting the

right answer?

12. Mathematics and

knowledge claims.

Proof of the four-colour

theorem. If a theorem is

proved by computer,

how can we claim to

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from the definition and

basic facts, with the aid of

cuboids and through intuitive perception, we can

understand the relations

between straight lines,

straight lines and planes,

and the parallel and vertical relations between

planes in space, and conclude the following property theorems and prove

them.③ Starting from the

definition and basic facts,

with the aid of cuboids and

through intuitive perception, we can understand

the relationship between

straight lines, straight lines

and planes, and the parallel and vertical relations

between planes in space,

? Graph

theory algorithms allow

us to

represent

networks

and to model complex

real-world

problems.

? Matrices

are a form

of notation

which allow

us to show

the parameters or

quantities of

several

linear equations simultaneously.

know that it is true?

13. What practical

problems can or does

mathematics try to

solve? Why are problems such as the travelling salesman problem

so enduring? What does

it mean to say the travelling salesman problem

is “NP hard”?

International-mindedness:

1. Diagrams of Pythagoras’ theorem occur

in early Chinese and

Indian manuscripts. The

earliest references to

trigonometry are in

Indian mathematics; the

use of triangulation to

find the curvature of the

? 123

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and conclude the judgment

theorem.④Using the obtained results to prove the

simple propositions of the

position relation of basic

spatial figures

HL3.7

? The definition of a radian

and conversion between

degrees and radians.

? Using radians to calculate

area of sector, length of arc.

HL3.8

? The definitions of cosθ and

sinθ in terms of the unit circle.

? The Pythagorean identity:

cos 2 θ + sin 2 θ = 1

? Definition of tanθ as

sinθ/cosθ

? Extension of the sine rule to

Earth in order to settle a

dispute between England and France over

Newton’s gravity.

2. The use of triangulation to find the curvature of the Earth in

order to settle a dispute

between England and

France over Newton’s

gravity.

3. Seki Takakazu

calculating π to ten

decimal places; Hipparchus, Menelaus and

Ptolemy; why are there

360 degrees in a complete turn? Why do we

use minutes and seconds

for time?; Links to

Babylonian mathematics.

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the ambiguous case.

? Graphical methods of solving trigonometric equations

in a finite interval.

HL3.9

? Geometric transformations

of points in two dimensions

using matrices: reflections,

horizontal and vertical

stretches, enlargements,

translations and rotations.

? Compositions of the above

transformations.

? Geometric interpretation of

the determinant of a transformation matrix.

HL3.10

? Concept of a vector and a

scalar.

? Representation of vectors

4. The origin of the

word “sine”; trigonometry was developed by

successive civilizations

and cultures; how is

mathematical knowledge considered from a

sociocultural perspective?

5. The “Bridges of

Konigsberg” problem.

6. The “Bridges of

Konigsberg” problem;

the Chinese postman

problem was first posed

by the Chinese mathematician Kwan Mei-Ko

in 1962.

Link to other subjects:

1. Design technology; volumes of stars

and inverse square law

? 125

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using directed line segments.

? Unit vectors; base vectors

i, j, k.

? Components of a vector;

column representation;

v = ????????? = v1i + v2 j + v3 k

? The zero vector 0, the vector ?v.

? Position vectors OA= a

? Rescaling and normalizing

vectors.

HL3.11

? Vector equation of a line in

two and three dimensions:

? r = a + λb, where b is a

direction vector of the line.

HL3.12

? Vector applications to kinematics.

? Modelling linear motion

(physics).

2. Vectors (physics).

3. Vectors, scalars,

forces and dynamics

(physics); field studies

(sciences).

4. Diffraction patterns and circular motion (physics).

5. Vector sums, differences and resultants

(physics).

6. Magnetic forces

and fields, and dynamics

(physics).

Homework:

Exercises from the text

books or questions from

the IB exams. Sometimes, may be a summary of what have

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with constant velocity in

two and three dimensions.

? Motion with variable velocity in two dimensions.

HL3.13

? Definition and calculation

of the scalar product of two

vectors.

? The angle between two

vectors; the acute angle

between two lines.

? Definition and calculation

of the vector product of two

vectors.

? Geometric interpretation of

|v × w|.

? Components of vectors.

HL3.14

? Graph theory: Graphs, vertices, edges, adjacent vertices, adjacent edges. Degree of a vertex.

learnt.

? 127

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? Simple graphs; complete

graphs; weighted graphs.

? Directed graphs; in degree

and out degree of a directed

graph.

? Subgraphs; trees.

HL3.15

? Adjacency matrices.

? Walks.

? Number of k -length walks

(or less than k -length

walks) between two vertices.

? Weighted adjacency tables.

? Construction of the transition matrix for a strongly-connected, undirected or

directed graph.

HL3.16

? Tree and cycle algorithms

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with undirected graphs.

? Walks, trails, paths, circuits,

cycles.

? Eulerian trails and circuits.

? Hamiltonian paths and

cycles.

? Minimum spanning tree

(MST) graph algorithms:

Kruskal’s and Prim’s algorithms for finding minimum

spanning trees.

? Chinese postman problem

and algorithm for solution,

to determine the shortest

route around a weighted

graph with up to four odd

vertices, going along each

edge at least once.

? Travelling salesman problem to determine the Hamiltonian cycle of least

weight in a weighted com-

? 129

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Level

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Periods

ATL Skills

Homework, Link to

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plete graph.

? Nearest neighbour algorithm for determining an

upper bound for the travelling salesman problem.

? Deleted vertex algorithm for

determining a lower bound

for the travelling salesman

problem.

G4

Week

41~

week

60

Probability

and

Statistics

HL/SL 4.1

? Concepts of population,

sample, random sample,

discrete and continuous data.

? Reliability of data sources

and bias in sampling.

? Interpretation of outliers.

? Sampling techniques and

their effectiveness.

? Organizing,

representing

, analysing

and interpreting data,

and utilizing

different

statistical

tools

Logical reasoning，

Data analysis，

Mathematical

operations.

2 104 Reflection,

Information

literacy,

Critical

thinking

Transfer

Collaboration,

Reflection

TOK:

1. Why have mathematics and statistics

sometimes been treated

as separate subjects?

How easy is it to be

misled by statistics? Is it

ever justifiable to purposely use statistics to

mislead others?

Timel

ine

Unit/

Theme/

Topic

Contents Objectives

Core

Competency

Academic

Proficiency

Level

Teaching

Periods

ATL Skills

Homework, Link to

TOK, IM, LP and CAS

plete graph.

? Nearest neighbour algorithm for determining an

upper bound for the travelling salesman problem.

? Deleted vertex algorithm for

determining a lower bound

for the travelling salesman

problem.

G4

Week

41~

week

60

Probability

and

Statistics

HL/SL 4.1

? Concepts of population,

sample, random sample,

discrete and continuous data.

? Reliability of data sources

and bias in sampling.

? Interpretation of outliers.

? Sampling techniques and

their effectiveness.

? Organizing,

representing

, analysing

and interpreting data,

and utilizing

different

statistical

tools

Logical reasoning，

Data analysis，

Mathematical

operations.

2 104 Reflection,

Information

literacy,

Critical

thinking

Transfer

Collaboration,

Reflection

TOK:

1. Why have mathematics and statistics

sometimes been treated

as separate subjects?

How easy is it to be

misled by statistics? Is it

ever justifiable to purposely use statistics to

mislead others?

G4

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Competency

Academic

Proficiency

Level

Teaching

Periods

ATL Skills

Homework, Link to

TOK, IM, LP and CAS

HL/SL 4.2

? Presentation of data (discrete and continuous): frequency distribution(tables).

? Histograms.

? Cumulative frequency;

cumulative frequency

graphs; use to find median,

quartiles, percentiles, range

and interquartile range

(IQR).

? Production and understanding of box and whisker diagrams.

HL/SL 4.3

? Measures of central tendency (mean, median and

mode).

? Estimation of mean from

grouped data.

? Modal class.

? Measures of dispersion

facilitates

prediction

and drawing

of conclusions.

? Different

statistical

techniques

require

justification

and the

identification of their

limitations

and

validity.

? Approximation in

data can

approach

the truth but

may not

2. What is the difference between information and data? Does

“data” mean the same

thing in different areas

of knowledge?

3. Could mathematics make alternative,

equally true, formulae?

What does this tell us

about mathematical

truths? Does the use of

statistics lead to an

over-emphasis on attributes that can be easily measured over those

that cannot?

4. Correlation and

causation–can we have

knowledge of cause and

effect relationships

given that we can only

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Periods

ATL Skills

Homework, Link to

TOK, IM, LP and CAS

(interquartile range, standard deviation and variance).

? Effect of constants changes

on the original data.

? Quartiles of discrete data.

HL/SL 4.4

? Linear correlation of bivariate data.

? Pearson’s product-moment

correlation coefficient, r.

? Scatter diagrams; lines of

best fit, by eye, passing

through the mean point.

? Equation of the regression

line of y on x.

? Use of the equation of the

regression line for prediction purposes.

? Interpret the meaning of the

parameters, a and b, in a lialways achieve it. ? Correla- tion and regression are power- ful tools for identifying patterns and equivalence of systems. Syllabus content 50 Mathe- matics: applications and inter- pretation guide ? Modelling and finding structure in observe correlation? What factors affect the reliability and validity of mathematical models in describing real-life phenomena? 5. To what extent are theoretical and experi- mental probabilities linked? What is the role of emotion in our per- ception of risk, for ex- ample in business, medicine and travel safety? 6. Can calculation of gambling probabilities be considered an ethical application of mathe- matics? Should mathe- maticians be held re- sponsible for unethical

博實(shí)樂(lè)“中外融通課程”

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Level

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Periods

ATL Skills

Homework, Link to

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near regression y=ax+b.

HL/SL 4.5

? Concepts of trial, outcome,

equally likely outcomes,

relative frequency, sample

space (U) and event.

? The probability of an event

A is?? ?? ?????????????.

? The complementary events

A and A’ (not A).

? Expectation number of occurrences.

HL/SL 4.6

? Use of Venn diagram, tree

diagrams, sample space diagrams and tables of outcomes to calculate probabilities.

? Combined events:

seemingly

random

events facilitates prediction.

? Different

probability

distributions

provide a

representation of the

relationship

between the

theory and

reality, allowing us to

make predictions

about what

might happen.

? Statistical

applications of their

work?

7. What do we mean

by a “fair” game? Is it

fair that casinos should

make a profit?

8. What criteria can

we use to decide between different models?

9. To what extent

can we trust mathematical models such as

the normal distribution?

How can we know what

to include, and what to

exclude, in a model?

10. Does correlation

imply causation?

Mathematics and the

world. Given that a set

of data may be approximately fitted by a

? 133

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Level

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Periods

ATL Skills

Homework, Link to

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?? ??∪?? ??? ?? ??? ??

??????

∩???

? Mutually exclusive events:

?? ??∩?? ? ?

? Independent events:

?? ??∩?? ?????????????

HL/SL 4.7

? Concept of discrete random

variables and their probability distributions.

? Expected value(mean),

E(X) for discrete data.

? Application.

HL/SL 4.8

? Binomial distribution.

? Mean and variance of the

binomial distribution.

HL/SL 4.9

? The normal distribution and

curve.

literacy

involves

identifying

reliability

and validity

of samples

and whole

populations

in a

closed system.

? A systematic approach to

hypothesis

testing allows statistical inferences to be

tested for

validity.

? Represenrange of curves, where would a mathematician seek for knowledge of which equation is the “true” model? 11. Why have some research journals “banned” p -values from their articles because they deem them too misleading? In practical terms, is saying that a result is significant the same as saying it is true? How is the term “significant” used dif- ferently in different areas of knowledge? 12. What are the strengths and limitations of different methods of data collection, such as

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Competency

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Proficiency

Level

Teaching

Periods

ATL Skills

Homework, Link to

TOK, IM, LP and CAS

? Properties of the normal

distribution.

? Diagrammatic representation.

? Normal probability calculation.

? Inverse normal calculations.

HL/SL 4.10

? Spearman’s rank correlation

coefficient, rs.

? Awareness of the appropriateness and limitations of

Pearson’s product moment

correlation coefficient and

Spearman’s rank correlation

coefficient, and the effect of

outliers on each.

HL/SL 4.11

? Formulation of null and

alternative hypotheses, H0

and H1.

? Significance levels.

tation of

probabilities

using transition matrices enables

us to efficiently predict

long-term

behaviour

and outcomes.

questionnaires?

13. Mathematics and

the world: In the absence of knowing the

value of a parameter,

will an unbiased estimator always be better

than a biased one?

14. The central limit

theorem can be proved

mathematically (formalism), but its truth can be

confirmed by its applications (empiricism).

What does this suggest

about the nature and

methods of mathematics?

15. Mathematics and

the world. Claiming

brand A is “better” on

average than brand B

? 135

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Proficiency

Level

Teaching

Periods

ATL Skills

Homework, Link to

TOK, IM, LP and CAS

? p-values.

? Expectation and observed

frequencies.

? The ??? test for independence: contingency tables,

degrees of freedom, critical

value.

? The??? goodness of fit test.

? The t-test.

? Use of the p-value to compare the means of two populations.

? Using one-tailed and

two-tailed tests.

HL 4.12

? Design of valid data collection methods, such as surveys and questionnaires.

? Selecting relevant variables

from many variables.

? Choosing relevant and appropriate data to analyse.

can mean very little if

there is a large overlap

between the confidence

intervals of the two

means.

16. To what extent

can mathematical models such as the Poisson

distribution be trusted?

What role do mathematical models play in

other areas of knowledge?

17. Mathematics and

the world. In practical

terms, is saying that a

result is significant the

same as saying that it is

true? Mathematics and

the world. Does the

ability to test only certain parameters in a

博實(shí)樂(lè)“中外融通課程”

136?Timel

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Theme/

Topic

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Core

Competency

Academic

Proficiency

Level

Teaching

Periods

ATL Skills

Homework, Link to

TOK, IM, LP and CAS

? Categorizing numerical data

in a ??? table and justifying

the choice of categorization.

? Choosing an appropriate

number of degrees of freedom when estimating parameters from data when carrying out the ??? goodness

of fit test.

? Definition of reliability and

validity .

? Reliability tests.

? Validity tests.

HL4.13

? Regression with non-linear

functions.

? Evaluation of least squares

regression curves using

technology.

? Sum of square residuals (SS

res ) as a measure of fit for a

population affect the

way knowledge claims

in the human sciences

are valued? When is it

more important not to

make a Type I error and

when is it more important not to make a Type

II error?

International-Mindedness:

1. The Kinsey report–famous sampling

techniques.

2. Discussion of the

different formulae for

the same statistical

measure (for example,

variance).

3. The benefits of

sharing and analysing

data from different

? 137

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Proficiency

Level

Teaching

Periods

ATL Skills

Homework, Link to

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model.

? The coefficient of determination (R 2 ).

? Evaluation of R 2 using

technology.

HL4.14

? Linear transformation of a

single random variable.

? Expected value of linear

combinations of n random

variables.

? Variance of linear combinations of n independent random variables.

?

?? as an unbiased estimate

of μ.

? Sn ? 12 as an unbiased estimate of σ 2.

HL4.15

? A linear combination of n

countries; discussion of

the different formulae

for variance.

4. The St Petersburg

paradox; Chebyshev and

Pavlovsky (Russian).

5. The so-called

“Pascal’s triangle” was

known to the Chinese

mathematician Yang

Hui much earlier than

Pascal.

6. De Moivre’s derivation of the normal

distribution and

Quetelet’s use of it to

describe

l’hommemoyen.

Link to other subjects:

1. Descriptive statistics and random samples

(biology, psychology,

博實(shí)樂(lè)“中外融通課程”

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Proficiency

Level

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Periods

ATL Skills

Homework, Link to

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independent normal random

variables is normally distributed. In particular,

X~N( μ, σ 2) ? ?? ~N (μ,

σ2/n).

? Central limit theorem.

HL4.16

? Confidence intervals for the

mean of a normal population.

HL4.17

? Poisson distribution, its

mean and variance.

? Sum of two independent

Poisson distributions has a

Poisson distribution.

HL4.18

? Critical values and critical

regions.

sports exercise and

health science, environmental systems and

societies, geography,

economics; business

management); research

methodologies (psychology).

2. Presentation of

data (sciences, individuals and societies).

3. Descriptive statistics (sciences and individuals and societies);

consumer price index

(economics).

4. Curves of best fit,

correlation and causation (sciences); scatter

graphs (geography).

5. Theoretical genetics and Punnett squares

? 139

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Competency

Academic

Proficiency

Level

Teaching

Periods

ATL Skills

Homework, Link to

TOK, IM, LP and CAS

? Test for population mean

for normal distribution.

? Test for proportion using

binomial distribution.

? Test for population mean

using Poisson distribution.

? Use of technology to test

the hypothesis that the population product moment

correlation coefficient (ρ) is

0 for bivariate normal distributions.

? Type I and II errors including calculations of their

probabilities.

HL4.19

? Transition matrices.

? Powers of transition matrices.

? Regular Markov chains.

? Initial state probability ma-

(biology); the position

of a particle (physics).

6. Normally distributed real-life measurements and descriptive

statistics (sciences,

psychology, environmental systems and

societies).

7. Fieldwork (biology, psychology, environmental systems and

societies, sports exercise

and health science).

8. Fieldwork (biology, psychology, environmental systems and

societies, sports exercise

and health science, geography).

9. Data collection in

field work (biology,

博實(shí)樂(lè)“中外融通課程”

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Level

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Periods

ATL Skills

Homework, Link to

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trices.

? Calculation of steady state

and long-term probabilities

by repeated multiplication

of the transition matrix or

by solving a system of linear equations.

psychology, environmental systems and

societies, sports exercise

and health science, geography, business management and design

technology); data from

social media and marketing sources (business

management)

10. Evaluation of R2

in graphical analysis

(sciences).

11. Data from multiple samples in field

studies (sciences, and

individuals and societies).

12. Analysis of data

from field studies (sciences and individuals

and societies).

? 141

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Teaching

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Homework, Link to

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13. Field studies (sciences and individuals

and societies).

Homework:

Exercises from the text

books or questions from

the IB exams. Sometimes, may be a summary of what have

learnt.

Week

61

~

Week

75

Calculus HL/SL 5.1

? Introduction to the concept

of a limit.

? Derivative interpreted as

gradient function and as rate

of change.

HL/SL5.2

? Increasing and decreasing

? Students

will understand the

links between the

derivative

and the rate

of change

and interpMathematical abstraction， Logical rea- soning， Mathematical operations， Mathematical modeling 2 82 Critical thinking, Communi- cation, Organiza- tion, Information liter- acy,Transfer TOK: 1. What value does the knowledge of limits have? Is infinitesimal behaviour applicable to real life? Is intuition a valid way of knowing in mathematics? 2. The seemingly

博實(shí)樂(lè)“中外融通課程”

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Homework, Link to

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functions. Graphical interpretation of f′(x) > 0, f′(x)

= 0, f′(x) < 0.

? Derivative of f(x) = axn is

f′(x) = anxn ? 1 , n ∈?.

? The derivative of functions

of the form f (x) = axn +

bxn ? 1 +... where all exponents are integers.

HL/SL5.4

? Tangents and normals at a

given point, and their equations.

HL/SL5.5

? Introduction to integration

as anti-differentiation of

functions of the form f(x) =

axn + bxn ? 1 + ....,

where n ∈?, n ≠ ? 1.

? Anti-differentiation with a

ret the

meaning of

this in context.

? Students

will understand the

relationship

between the

integral and

area and

interpret the

meaning of

this in context.

? Finding

patterns in

the derivatives of

polynomials

and their

behavior,

,

Media literacy

abstract concept of calculus allows us to create

mathematical models

that permit human feats

such as getting a man on

the Moon. What does

this tell us about the

links between mathematical models and

reality?

3. In what ways has

technology impacted

how knowledge is produced and shared in

mathematics? Does

technology simply allow

us to arrange existing

knowledge in new and

different ways, or

should this arrangement

itself be considered

knowledge?