DP Mathematics Curriculum

Group Five: Mathematics Analysis and Approaches Applications and Interpretations

Mathematics: Analysis and Approaches HL 2021-2023

Course description (Taken from the IB Mathematics: analysis and approaches guide)

This course recognizes the need for analytical expertise in a world where innovation is increasingly dependent on a deep understanding of mathematics. This course includes topics that are both traditionally part of a pre-university mathematics course (for example, functions, trigonometry, calculus) as well as topics that are amenable to investigation, conjecture and proof, for instance the study of sequences and series at both SL and HL, and proof by induction at HL. The course allows the use of technology, as fluency in relevant mathematical software and handheld technology is important regardless of choice of course. However, Mathematics: analysis and approaches has a strong emphasis on the ability to construct, communicate and justify correct mathematical arguments.

Mathematics: analysis & approaches: Distinction between SL and HL Students who choose Mathematics: analysis and approaches at SL or HL should be comfortable in the manipulation of algebraic expressions and enjoy the recognition of patterns and understand the mathematical generalization of these patterns. Students who wish to take Mathematics: analysis and approaches at higher level will have strong algebraic skills and the ability to understand simple proof. They will be students who enjoy spending time with problems and get pleasure and satisfaction from solving challenging problems.

Mathematics and International Mindedness International-mindedness is a complex and multi-faceted concept that refers to a way of thinking, being and acting characterized by an openness to the world and a recognition of our deep interconnectedness to others.

Despite recent advances in the development of information and communication technologies, the global exchange of mathematical information and ideas is not a new phenomenon and has been essential to the progress of mathematics. Indeed, many of the foundations of modern mathematics were laid many centuries ago by diverse civilisations – Arabic, Greek, Indian and Chinese among others.

Mathematics can in some ways be seen as an international language and, apart from slightly differing notation, mathematicians from around the world can communicate effectively within their field.

Mathematics can transcend politics, religion and nationality, and throughout history great civilizations have owed their success in part to their mathematicians being able to create and maintain complex social and architectural structures. Politics has dominated the development of mathematics, to develop ballistics, navigation and trade, and land ownership, often influenced by governments and leaders. Many early mathematicians were political and military advisers and today mathematicians are integral members of teams who advise governments on where money and resources should be allocated.

Science and technology are of significant importance in today’s world. As the language of science, mathematics is an essential component of most technological innovation and underpins developments in science and technology, although the contribution of mathematics may not always be visible. Examples of this include the role of the binary number system, matrix algebra, network theory and probability theory in the digital revolution, or the use of mathematical simulations to predict future climate change or spread of disease. These examples highlight the key role mathematics can play in transforming the world around us.

One way of fostering international-mindedness is to provide opportunities for inquiry into a range of local and global issues and ideas. Many international organisations and bodies now exist to promote mathematics, and students are encouraged to access the resources and often-extensive websites of such mathematical organisations. This can enhance their appreciation of the international dimension of

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mathematics, as well as providing opportunities to engage with global issues surrounding the subject.

Examples of links relating to international-mindedness are given in the “Connections” sections of the syllabus.

Mathematics and Theory of Knowledge The relationship between each subject and theory of knowledge (TOK) is important and fundamental to the DP. The theory of knowledge course provides an opportunity for students to reflect on questions about how knowledge is produced and shared, both in mathematics and also across different areas of knowledge. It encourages students to reflect on their assumptions and biases, helping them to become more aware of their own perspective and the perspectives of others and to become “inquiring, knowledgeable and caring young people” (IB mission statement).

As part of their theory of knowledge course, students are encouraged to explore tensions relating to knowledge in mathematics. As an area of knowledge, mathematics seems to supply a certainty perhaps impossible in other disciplines and in many instances provides us with tools to debate these certainties. This may be related to the “purity” of the subject, something that can sometimes make it seem divorced from reality. Yet mathematics has also provided important knowledge about the world and the use of mathematics in science and technology has been one of the driving forces for scientific advances.

Despite all its undoubted power for understanding and change, mathematics is in the end a puzzling phenomenon. A fundamental question for all knowers is whether mathematical knowledge really exists independently of our thinking about it. Is it there, “waiting to be discovered”, or is it a human creation?

Indeed, the philosophy of mathematics is an area of study in its own right.

Students’ attention should be drawn to questions relating theory of knowledge (TOK) and mathematics, and they should also be encouraged to raise such questions themselves in both their mathematics and TOK classes. Examples of issues relating to TOK are given in the “Connections” sections of the syllabus. Further suggestions for making links to TOK can also be found in the mathematics section of the Theory of knowledge guide.

Mathematics and Creativity, Action, Service CAS experiences can be associated with each of the subject groups of the DP. CAS and mathematics can complement each other in a number of ways. Mathematical knowledge provides an important key to understanding the world in which we live, and the mathematical skills and techniques students learn in the mathematics courses will allow them to evaluate the world around them which will help them to develop, plan and deliver CAS experiences or projects.

An important aspect of the mathematics courses is that students develop the ability to systematically analyse situations and can recognize the impact that mathematics can have on the world around them. An awareness of how mathematics can be used to represent the truth enables students to reflect critically on the information that societies are given or generate, and how this influences the allocation of resources or the choices that people make. This systematic analysis and critical reflection when problem solving may be inspiring springboards for CAS projects.

Students may also draw on their CAS experiences to enrich their involvement in mathematics both within and outside the classroom, and mathematics teachers can assist students in making links between their subjects and students’ CAS experiences where appropriate. Purposeful discussion about real CAS experiences and projects will help students to make these links.

The challenge and enjoyment of CAS can often have a profound effect on mathematics students, who might choose, for example, to engage with CAS in the following ways: • plan, write and implement a “mathematics scavenger hunt” where younger students tour the school answering interesting mathematics questions as part of their introduction to a new school • as a CAS project students could plan and carry out a survey, create a database and analyse the results, and make suggestions to resolve a problem in the students’ local area. This might be, for example, surveying the availability of fresh fruit and vegetables within a community, preparing an

2

•

action plan with suggestions of how to increase availability or access, and presenting this to a local charity or community group taking an element of world culture that interests students and designing a miniature Earth (if the world were 100 people) to express the trend(s) numerically.

It is important to note that a CAS experience can be a single event or may be an extended series of events.

However, CAS experiences must be distinct from, and may not be included or used in, the student’s Diploma course requirements.

Additional suggestions on the links between DP subjects and CAS can be found in the Creativity, activity, service teacher support material.

Textbook Oxford IB Diploma Programme: IB Mathematics: analysis and approaches, Higher Level ISBN: 978-0-19-842716-2 Calculator A Graphical Display Calculator is required for the course.

Texas Intstrument TI-84 (recommended and used in class)

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2 Year Course Outline See detailed plan on next page for order of content and approximate dates Topic 1: Number and Algebra (19 hours) Topic 2: Functions (32 hours) Topic 3: Geometry and Trigonometry (20 hours) Year 1

Topic 5: Calculus (55 hours) Toolkit (22 hours) Revision Mock Exams

Year 2

Topic 1: Number and Algebra (20 hours) Topic 3: Geometry and Trigonometry (31 hours) Topic 4: Statistics and Probability (33 hours) Toolkit (8 hours) Revision Mock Exams and Final Exams

Assessment Outline External assessment (5 hours)

Paper 1 (2 hours) 30% No technology allowed.

Section A: Compulsory short-response questions based on the whole syllabus.

Section B: Compulsory extended-response questions based on the whole syllabus.

(110 marks) Paper 2 (2 hours) 30% Technology required.

Section A: Compulsory short-response questions based on the whole syllabus.

Section B: Compulsory extended-response questions based on the whole syllabus.

(110 marks) Paper 3 (1 hour) 20% Technology required.

Two compulsory extended response problem-solving questions.

(55 marks)

Internal Assessment (15 hours – included in outline as part of toolkit)

Maths Project 20% Internal assessment in mathematics is an individual exploration. This is a piece of written work that involves investigating an area of mathematics. (20 marks)

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Detailed Plan for year 1 (grade 11) Week commencing: 2021

Topic

Proof

1.3

Binomial Theorem

1.4

1.6 Simple deductive proof 1.15 Proof by induction, proof by contradiction, counterexample.

1.9 The binomial theorem: expansion of (a+b),n∈ℕ.

Arithmetic and Geometric sequences

Applications

Sep 5

Sep 12

Use of Pascal’s triangle and nCr. 1.10 Extension to fractional and negative indices.

1.11 Partial fractions.

Sep 19

Functional relationships Functions

2.1 2.2, 2.3

TOK links

TOK: 1. Is all knowledge concerned Arithmetic sequences and series. with and use Use of the formulae for the nth term and the sum of the firstidentification n of patterns? Consider terms of the sequence.

Fibonacci numbers and Use of sigma notation for sums of arithmetic sequences. connections with the golden Applications.

Analysis, interpretation and prediction where a model isratio. not perfectly arithmetic in real life.

2. How do mathematicians 1.3 reconcile the fact that some Geometric sequences and series. conclusions seem to conflict Use of the formulae for the n th term and the sum of the first n intuitions? with our terms of the sequence.

Consider for instance that a Use of sigma notation for the sums of geometric sequences. finite area can be bounded Applications by an infinite perimeter.

1.4 3. How have technological Financial applications of geometric sequences and series: advances affected the - compound interest nature and practice of - annual depreciation. mathematics? Consider the use of financial packages for 1.8 instance.

Sum of infinite convergent geometric sequences.

IM: Aryabhatta is sometimes considered the “father of algebra”–compare with alKhawarizmi; the use of several alphabets in mathematical notation (for example the use of capital sigma for the sum).

1.2

Arithmetic and Geometric series

Sep 26

Syllabus reference

1.1 1.2

Number patterns and sigma notation Aug 29

Textbook Reference

2.1 Equations of straight lines, gradient, intercepts, parallel and perpendicular lines.

2.2 Concept of a function, inverse function.

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TOK: Is mathematical reasoning different from scientific reasoning, or reasoning in other Areas of Knowledge?

TOK: 1. How have notable individuals shaped the development of mathematics as an area of knowledge? Consider Pascal and “his” triangle.

TOK: Descartes showed that geometric problems could be solved algebraically and vice versa. What does this tell us about mathematical

2.3 The graph of a function.

2.4 Key features of graphs. Intersection of 2 functions.

2.16 Graphs of 𝑦 = |𝑓(𝑥)|.

Oct 3

Oct 10

Oct 17

Reciprocal functions

Mid term break Exponential functions

2.2, 2.3

2.8 Rational functions and their graphs, equations of asymptotes.

2.13 Rational functions with quadratic factors.

2.3, 2.4

2.9 Exponential functions and their graphs, logarithmic functions.

2.10 Solving equations, graphically and analytically.

2.15 Solutions of 𝑔(𝑥) ≥ 𝑓(𝑥).

2.5 Composite functions, the identity function.

2.11 Transformation of functions – reflections, translations, dilations.

2.14 Odd and even functions.

2.6 The quadratic function.

2.7 Quadratic equations and inequalities.

1.12 The complex plane

Oct 24

2.5 Oct 31

Nov 7

Nov 14

Nov 21

Nov 28 Dec 5

Transformation s

Quadratic functions

3.1

Complex Numbers

3.2

Polynomial equations and inequalities

3.3

3.4 National Day, Commemorati on Day Dec 1-3

3.5, 3.6

2.12 Polynomial functions, factor and remainder theorems.

Sum and product of the roots of polynomial equations.

Fundamental theorem of algebra 1.16 Solutions of systems of linear equations up to three unknowns

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representation and mathematical knowledge?

TOK: 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?

TOK: What are the implications of accepting that mathematical knowledge changes over time?

TOK: What role do “models” play in mathematics? Do they play a different role in mathematics compared to their role in other areas of knowledge?

TOK: Do you think mathematics or logic should be classified as a language?

TOK: What are the key concepts that provide the building blocks for mathematical knowledge?

TOK: How does language shape knowledge? For example, do the words “imaginary” and “complex” make the concepts more difficult than if they had different names?

TOK: Is it an oversimplification to say that some areas of knowledge give us facts whereas other areas of knowledge give us interpretations?

TOK: Mathematics, Sense, Perception and Reason: If we can find solutions in higher dimensions can we reason that

Dec 12

Linear equations and inequalities Winter Break Limits and derivatives

these spaces exist beyond our sense perception?

4.1, 4.2

5.1 Concept of a limit. Derivative as gradient function and rate of change.

5.12 Continuity, convergence and divergence.

4.3

5.2 Increasing and decreasing functions, graphical interpretation of f’(x)>0, f’(x)=0, f’(x)<0.

5.3 Derivatives of polynomial functions.

More differentiation

4.3, 4.6

The second derivative

4.4

Tangents and normals

4.5

5.6 Derivatives of trigonometric functions, exponential and natural logarithmic functions. Chain rule, product rule, quotient rule.

5.14 Implicit differentiation, related rates of change.

5.7 Relationship between f, f’ and f’’. Function and Liebniz notation. Graphical behavior.

5.8 Local maxima and minima. Optimization.

Points of inflexion.

5.4 Tangents and normal at a given point, and their equations.

Jan 2 2022

Increasing and decreasing functions Jan 9

Jan 16

Differentiation

Jan 23

Jan 30

Feb 6

Feb 13

Feb 27

TOK: When mathematicians and historians say that they have explained something, are they using the word “explain” in the same way?

TOK: 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?

Mid term break 6.1

Feb 20

TOK: What value does the knowledge of limits have? Is infinitesimal behaviour applicable to real life? Is intuition a valid way of knowing in mathematics?

TOK: The seemingly 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?

TOK: What is the role of convention in mathematics? Is this similar or different to the role of convention in other areas of knowledge?

Coordinate geometry Trigonometry and applications

6.2

3.1 The distance between 2 points. The midpoint.

Volume and surface area. Angles between lines and planes.

3.2 Sides and angles in right-angled triangles.

The sine rule and cosine rule. Area of a triangle.

3.3 Angles of elevation and depression.

Bearings.

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TOK: What is an axiomatic system? Are axioms self evident to everybody?

TOK: 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?

TOK: If the angles of a triangle can add up to less than 180°, 180° or more than 180°, what does this tell

Radians

6.3

3.4 Circles. Radian measure. Arc lengths and sector areas.

Trigonometric functions and identities

6.4

Circular functions and equations

6.5

3.5 Definition of sin, cos and tan. Exact values.

Ambiguous case of sine rule.

3.9 Definition of reciprocal trigonometric ratios, sec, cosec and cot. Inverse functions arccos, arcsin, arctan.

3.6 The Pythagorean identity. Double angle identities. Relationship between trigonometric ratios.

3.10 Compound angle identities. Double angle identity for tan.

3.8 Solve trigonometric equations graphically and analytically.

March 6

March 13

March 20 March 27

Spring Break Integration

7.1

April 10

April 17

Exponential and log functions Integration

7.2, 7.3

Definite integration

8.1

Eid break May 12/13 Kinematics

8.2

Differential Equations

8.3

7.4

April 24

May 1

May 8

May 15

5.5 Integration as anti-differentiation.

Boundary condition to determine constant.

Definite integrals.

5.15 Derivatives of log functions, power functions and inverse trig functions.

5.10 Indefinite integrals of trigonometric, exponential and reciprocal functions.

Integration by inspection.

5.16 Integration by substitution, integration by parts.

5.11 Areas under curves, areas bounded by 2 curves.

5.17 Volumes of revolution.

5.9 Problems involving displacement, velocity and acceleration. Total distance travelled.

5.18 First order differential equations. Euler’s method. Separation of variables.

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us about the nature of mathematical knowledge?

TOK: Which is a better measure of angle: radian or degree? What criteria can/do/should mathematicians use to make such decisions?

TOK: Trigonometry was developed by successive civilizations and cultures. To what extent is mathematical knowledge embedded in particular traditions or bound to particular cultures? How have key events in the history of mathematics shaped its current form and methods?

TOK: What is the relationship between concepts and facts? To what extent do the concepts that we use shape the conclusions that we reach?

TOK: Music can be expressed using mathematics. What does this tell us about the relationship between music and mathematics?

TOK: Is it possible for an area of knowledge to describe the world without transforming it?

TOK: Can a mathematical statement be true before it has been proven?

TOK: An infinite area sweeps out

a finite volume. Can this be reconciled with our intuition? Do emotion and intuition have a role in mathematics?

TOK: Is mathematics independent of culture? To what extent are we people aware of the impact of culture on what we they believe or know?

TOK: Does personal experience play a role in the formation of knowledge claims in mathematics? Does it play a

Homogeneous differential equations.

May 22 May 29 June 5 June 12 June 19 June 26 July 3

Maclaurin Series

Mocks Mocks Toolkit Toolkit Toolkit Summer break

8.4

5.13 L’Hopital’s rule.

5.19 Obtain expansions for trigonometric, exponential and logarithmic functions.

12 12 12

IA introduction

different role in mathematics compared to other areas of knowledge?

TOK: Is there always a trade-off between accuracy and simplicity?

Year 2 (Grade 12) Week Topic Textbook Syllabus reference commencing: reference 2022 Geometrical 9.1 September representation of vectors Intro to vector 3.12 algebra Position vectors, displacement vectors, 9.2 magnitude of a vector.

9.3

3.13 Scalar product. Angle between 2 vectors.

Perpendicular and parallel vectors.

Vector equation of a line

9.4

Vector product and properties

9.5

3.14 Vector equation of a line.

3.15 Coincident, parallel, intersecting and skew lines.

3.16 Vector product.

Scalar product and its properties

October

9

TOK Links

TOK: The nature of mathematics: why this definition of scalar product?

TOK: How can there be an infinite number of discrete solutions to an equation? What does this suggest about the nature of mathematical knowledge and how it compares to knowledge in other disciplines?

TOK: Mathematics and the knower: are symbolic representations of threedimensional objects easier to deal with than visual representations?

What does this tell us about our knowledge of mathematics in other dimensions?

Vector equation of a plane Lines, planes and angles

November

3.17 Vector equation of a plane.

9.7

3.18 Intersection of line and plane, 2 planes.

Angle between a line and plane, 2 planes.

Applications of 9.8 vectors Forms of a 10.1 complex number Operations with complex numbers in polar form Powers and roots of complex numbers Review

December

9.6

10.2

10.3

1.12 The complex plane

TOK: How does language shape knowledge? For example, do the words “imaginary” and “complex” make the concepts more difficult than if they had different names?

1.13 Polar form and Euler form 1.14 Complex conjugate roots

Mock Exams Winter Break Sampling

5.1

4.1 Populations, samples, reliability, bias, outliers.

Descriptive statistics

5.2

4.2 Presentation of data, histograms, box and whisker diagrams.

Justification of statistical techniques

5.3

4.3 Measures of centre, measures of dispersion.

Quartiles.

Correlation, causation and linear regression

5.4

4.4 Correlation of bivariate data. Scatter diagrams, lines of best fit. Regression.

5.4

4.10 Equation of regression line of x on y.

Predictions.

January 2021

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TOK: 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?

TOK: What is the difference between information and data?

Does “data” mean the same thing in different areas of knowledge?

TOK: Could mathematics make alternative, equally true, formulae?

What does this tell us about mathematical truths? Does the use of statistics lead to an overemphasis on attributes that can be easily measured over those that cannot?

TOK: Correlation and causation– can we have knowledge of cause and effect relationships given that we can only observe correlation?

What factors affect the reliability and validity of mathematical models in describing real-life phenomena?

TOK: Is it possible to have knowledge of the future?

February

Axiomativ probability systems

11.1

Combined events

March

Probability distributions, continuous random variables The binomial distribution The normal distribution

11.2 11.3

11.4 11.5

4.5 Probability of an event. Complementary events. Expected number of outcomes.

4.6 Venn diagrams, tree diagrams, sample space diagrams. Combined events, independent events.

4.11 Conditional probability.

4.13 Bayes theorem.

4.7 Discrete random variables and applications.

4.14 Continuous random variables. Linear transformations.

4.8 Binomial distribution.

4.9 Normal distribution and inverse normal.

4.12 Standardized normal variables.

Revision April

Spring Break Revision Revision Revision Revision

May Exams

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TOK: To what extent are theoretical and experimental probabilities linked? What is the role of emotion in our perception of risk, for example in business, medicine and travel safety?

TOK: Can calculation of gambling probabilities be considered an ethical application of mathematics?

Should mathematicians be held responsible for unethical applications of their work?

TOK: What do we mean by a “fair” game? Is it fair that casinos should make a profit?

TOK: What criteria can we use to decide between different models?

TOK: 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?

Mathematics: Analysis and Approaches SL 2021-2023

Course description (Taken from the IB Mathematics: applications and interpretation guide)

This course recognizes the need for analytical expertise in a world where innovation is increasingly dependent on a deep understanding of mathematics. This course includes topics that are both traditionally part of a pre-university mathematics course (for example, functions, trigonometry, calculus) as well as topics that are amenable to investigation, conjecture and proof, for instance the study of sequences and series at both SL and HL, and proof by induction at HL. The course allows the use of technology, as fluency in relevant mathematical software and handheld technology is important regardless of choice of course. However, Mathematics: analysis and approaches has a strong emphasis on the ability to construct, communicate and justify correct mathematical arguments.

Mathematics: applications and interpretation: Distinction between SL and HL Students who choose Mathematics: analysis and approaches at SL or HL should be comfortable in the manipulation of algebraic expressions and enjoy the recognition of patterns and understand the mathematical generalization of these patterns. Students who wish to take Mathematics: analysis and approaches at higher level will have strong algebraic skills and the ability to understand simple proof. They will be students who enjoy spending time with problems and get pleasure and satisfaction from solving challenging problems.

Mathematics and International Mindedness International-mindedness is a complex and multi-faceted concept that refers to a way of thinking, being and acting characterized by an openness to the world and a recognition of our deep interconnectedness to others.

Despite recent advances in the development of information and communication technologies, the global exchange of mathematical information and ideas is not a new phenomenon and has been essential to the progress of mathematics. Indeed, many of the foundations of modern mathematics were laid many centuries ago by diverse civilisations – Arabic, Greek, Indian and Chinese among others.

Mathematics can in some ways be seen as an international language and, apart from slightly differing notation, mathematicians from around the world can communicate effectively within their field.

Mathematics can transcend politics, religion and nationality, and throughout history great civilizations have owed their success in part to their mathematicians being able to create and maintain complex social and architectural structures. Politics has dominated the development of mathematics, to develop ballistics, navigation and trade, and land ownership, often influenced by governments and leaders. Many early mathematicians were political and military advisers and today mathematicians are integral members of teams who advise governments on where money and resources should be allocated.

Science and technology are of significant importance in today’s world. As the language of science, mathematics is an essential component of most technological innovation and underpins developments in science and technology, although the contribution of mathematics may not always be visible. Examples of this include the role of the binary number system, matrix algebra, network theory and probability theory in the digital revolution, or the use of mathematical simulations to predict future climate change or spread of disease. These examples highlight the key role mathematics can play in transforming the world around us.

One way of fostering international-mindedness is to provide opportunities for inquiry into a range of local and global issues and ideas. Many international organisations and bodies now exist to promote

12

mathematics, and students are encouraged to access the resources and often-extensive websites of such mathematical organisations. This can enhance their appreciation of the international dimension of mathematics, as well as providing opportunities to engage with global issues surrounding the subject.

Examples of links relating to international-mindedness are given in the “Connections” sections of the syllabus.

Mathematics and Theory of Knowledge The relationship between each subject and theory of knowledge (TOK) is important and fundamental to the DP. The theory of knowledge course provides an opportunity for students to reflect on questions about how knowledge is produced and shared, both in mathematics and also across different areas of knowledge. It encourages students to reflect on their assumptions and biases, helping them to become more aware of their own perspective and the perspectives of others and to become “inquiring, knowledgeable and caring young people” (IB mission statement).

As part of their theory of knowledge course, students are encouraged to explore tensions relating to knowledge in mathematics. As an area of knowledge, mathematics seems to supply a certainty perhaps impossible in other disciplines and in many instances provides us with tools to debate these certainties. This may be related to the “purity” of the subject, something that can sometimes make it seem divorced from reality. Yet mathematics has also provided important knowledge about the world and the use of mathematics in science and technology has been one of the driving forces for scientific advances.

Despite all its undoubted power for understanding and change, mathematics is in the end a puzzling phenomenon. A fundamental question for all knowers is whether mathematical knowledge really exists independently of our thinking about it. Is it there, “waiting to be discovered”, or is it a human creation?

Indeed, the philosophy of mathematics is an area of study in its own right.

Students’ attention should be drawn to questions relating theory of knowledge (TOK) and mathematics, and they should also be encouraged to raise such questions themselves in both their mathematics and TOK classes. Examples of issues relating to TOK are given in the “Connections” sections of the syllabus. Further suggestions for making links to TOK can also be found in the mathematics section of the Theory of knowledge guide.

Mathematics and Creativity, Action, Service CAS experiences can be associated with each of the subject groups of the DP. CAS and mathematics can complement each other in a number of ways. Mathematical knowledge provides an important key to understanding the world in which we live, and the mathematical skills and techniques students learn in the mathematics courses will allow them to evaluate the world around them which will help them to develop, plan and deliver CAS experiences or projects.

An important aspect of the mathematics courses is that students develop the ability to systematically analyse situations and can recognize the impact that mathematics can have on the world around them. An awareness of how mathematics can be used to represent the truth enables students to reflect critically on the information that societies are given or generate, and how this influences the allocation of resources or the choices that people make. This systematic analysis and critical reflection when problem solving may be inspiring springboards for CAS projects.

Students may also draw on their CAS experiences to enrich their involvement in mathematics both within and outside the classroom, and mathematics teachers can assist students in making links between their subjects and students’ CAS experiences where appropriate. Purposeful discussion about real CAS experiences and projects will help students to make these links.

The challenge and enjoyment of CAS can often have a profound effect on mathematics students, who might choose, for example, to engage with CAS in the following ways: • plan, write and implement a “mathematics scavenger hunt” where younger students tour the school answering interesting mathematics questions as part of their introduction to a new school

13

•

•

as a CAS project students could plan and carry out a survey, create a database and analyse the results, and make suggestions to resolve a problem in the students’ local area. This might be, for example, surveying the availability of fresh fruit and vegetables within a community, preparing an action plan with suggestions of how to increase availability or access, and presenting this to a local charity or community group taking an element of world culture that interests students and designing a miniature Earth (if the world were 100 people) to express the trend(s) numerically.

It is important to note that a CAS experience can be a single event or may be an extended series of events.

However, CAS experiences must be distinct from, and may not be included or used in, the student’s Diploma course requirements.

Additional suggestions on the links between DP subjects and CAS can be found in the Creativity, activity, service teacher support material.

Textbook Oxford IB Diploma Programme: IB Mathematics: analysis and interpretation, Standard Level.

ISBN: 978-0-19-842711-7

Calculator A Graphical Display Calculator is required for the course.

Texas Intstrument TI-84 (recommended and used in class)

14

2 Year Course Outline See detailed plan on next page for order of content and approximate dates Topic 1: Number and Algebra (19 hours) Topic 2: Functions (21 hours) Year 1

Topic 5: Calculus (28 hours) Toolkit (22 hours) Revision Mock Exams

Year 2

Topic 3: Geometry and Trigonometry (25 hours) Topic 5: Statistics and Probability (27 hours) Toolkit (8 hours) Revision Mock Exams and Final Exams

Assessment Outline External assessment (3 hours)

Paper 1 (1 hour 30 minutes) 40% Compulsory short-response questions based on the whole syllabus. (80 marks) Paper 2 (1 hour 30 minutes) 40% Compulsory extended-response questions based on the whole syllabus. (80 marks)

Internal Assessment (15 hours – included in outline as part of toolkit)

Maths Project 20% Internal assessment in mathematics is an individual exploration. This is a piece of written work that involves investigating an area of mathematics. (20 marks)

15

Detailed Plan for year 1 (grade 11) Week commencing: 2021

Topic

Textbook Reference

Prior Learning

Setting Expectations Distribute pior learning practice pack 9.1

Aug 30th (3 weeks)

Exponent s and Logarithm s

9.2

1.5 Laws of exponents with integer exponents.

Introduction to logarithms with base 10 and e.

Numerical evaluation of logarithms using technology.

1.7 Laws of exponents with rational exponents.

Laws of logarithms.

Change of base of a logarithm.

Solving exponential equations, including using logarithms.

TOK links, IM links

Tell students, prior learning assessment 2 weeks from today! ( September 13th )

TOK: How have seminal advances, such as the development of logarithms, changed the way in which mathematicians understand the world and the nature of mathematics?

Assessment: Prior Learning Test (10%) First lesson

Sep 13th Binomial Theorem

1.5

1.9 The binomial theorem: expansion of (a+b)n,n∈ℕ.

Sep 19th (3 weeks)

Oct 10th

Syllabus reference

Use of Pascal’s triangle and Cnr.

Toolkit Test and Review

TOK: How have notable individuals shaped the development of mathematics as an area of knowledge? Consider Pascal and “his” triangle.

Assessment: Exponents, Logarithm and Binomial Test First lesson

Series and Sequence s

1.1

Mid Term Break October 17th – 21st

1.3

1.2

Oct 10th (3 weeks)

1.4

TOK: 4. Is all knowledge Arithmetic sequences and series. concerned with Use of the formulae for the nth term and the sum identification and use of patterns? Consider of the first n terms of the sequence.

Use of sigma notation for sums of arithmetic Fibonacci numbers and connections with sequences. the golden ratio.

Applications.

5. How do Analysis, interpretation and prediction where amathematicians model is not perfectly arithmetic in real life. reconcile the fact that 1.3 some conclusions

1.2

16

seem to conflict with Geometric sequences and series. our intuitions?

Use of the formulae for the n th term and the sum Consider for instance of the first n terms of the sequence. that a finite area can Use of sigma notation for the sums of geometric be bounded by an sequences. infinite perimeter.

Applications 6. How have 1.4 technological affected the Financial applications of geometric sequences advances and nature and practice of series: mathematics?

- compound interest Consider the use of - annual depreciation. financial packages for instance.

1.6

1.8 IM: Aryabhatta is Sum of infinite convergent geometric sequences.

Assessment: Mini IA report, Pension Plan (10%) Time in class given for 1 week

Proofs

1.6 Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof.

The symbols and notation for equality and identity.

sometimes considered the “father of algebra”– compare with alKhawarizmi; the use of several alphabets in mathematical notation (for example the use of capital sigma for the sum).

TOK: Is mathematical reasoning different from scientific reasoning, or reasoning in other Areas of Knowledge?

Nov 14th (1 week)

Nov 21st (3 weeks)

Toolkit Test and Review Functions Introducti on National day

Assessment: Series and Sequences (Cumulative) 2.1 2.2 2.3 2.4 2.5 2.6

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

Informal concept that an inverse function reverses or undoes the effect of a function.

Inverse function as a reflection in the line y=x, and the notation f−1(x).

2.3 The graph of a function; its equation y=f(x).

17

IM: The development of functions by Rene Descartes (France), Gottfried Wilhelm Leibnitz (Germany) and Leonhard Euler (Switzerland); the notation for functions

Decembe r1-2

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.

2.4 Determine key features of graphs.

Finding the point of intersection of two curves or lines using technology.

2.5 Composite functions.

Identity function. Finding the inverse function f−1(x).

December 12 – January 1

was developed by a number of different mathematicians in the 17th and 18th centuries– how did the notation we use today become internationally accepted?

TOK: 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?

Winter Break Quadratic Equations

Jan 2nd (3 weeks)

Transfor mations

3.1 3.2 3.3 3.4 3.5 3.6 3.7

2.1 Different forms of the equation of a straight line.

Gradient; intercepts.

Lines with gradients m1 and m2 Parallel lines m1=m2.

Perpendicular lines m1×m2=−1.

2.6 The quadratic function f(x)=ax2+bx+c: its graph, y -intercept (0,c). Axis of symmetry.

The form f(x)=a(x−p)(x−q), x- intercepts (p,0) and (q,0).

The form f(x)=a(x−h)2+k, vertex (h,k).

2.7 Solution of quadratic equations and inequalities.

The quadratic formula.

The discriminant Δ=b2−4ac and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots.

2.11 Transformations of graphs.

Translations: y=f(x)+b;y=f(x−a).

Reflections (in both axes): y=−f(x);y=f(−x).

Vertical stretch with scale factor p: y=pf(x).

Horizontal stretch with scale factor 1q: y=f(qx).

Composite transformations.

18

IM: The Babylonian method of multiplication: ab=((a+b)2−a2−b2)/2.

Sulba Sutras in ancient India and the Bakhshali Manuscript contained an algebraic formula for solving quadratic equations.

TOK: What are the key concepts that provide the building blocks for mathematical knowledge?

Assessment: Mini IA PowerPoint, Mathematical Modelling (10%) Rational/ Exponenti al and Logarithm ic functions

4.1 4.2 4.3

Jan 23rd (2 weeks)

Toolkit Test and Review Feb 6th

Mid Term Break Feb 13th – 17th

Feb 20th (4 weeks)

Calculus – Differenti ation and its Applicatio n (Focus only on polynomi als)

5.1 5.2 5.3 5.4 5.5

2.8 The reciprocal function f(x)=1x,x≠0: its graph and self-inverse nature.

Rational functions of the form f(x)=(ax+b)/(cx+d) and their graphs.

Equations of vertical and horizontal asymptotes.

2.9 Exponential functions and their graphs: Logarithmic functions and their graphs: 2.10 Solving equations, both graphically and analytically.

Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.

Applications of graphing skills and solving equations that relate to real-life situations.

Assessment: Functions (Cumulative)

TOK: 1. What role do “models” play in mathematics? Do they play a different role in mathematics compared to their role in other areas of knowledge?

5.1 Introduction to the concept of a limit.

Derivative interpreted as gradient function and as rate of change.

5.2 Increasing and decreasing functions.

Graphical interpretation of fʹ(x)>0,fʹ(x)=0,fʹ(x)<0.

IM: Attempts by Indian mathematicians (5001000 CE) to explain division by zero.

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 abstract concept of calculus allows us to create mathematical models that permit human feats such as

5.3 Derivative of f(x)=axn is fʹ(x)=anxn−1, n∈ℤ The derivative of functions of the form f(x)=axn+bxn−1+... where all exponents are integers.

5.4 Tangents and normals at a given point, and their equations.

5.6 Differentiation of a sum and a multiple of these functions.

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2. What assumptions do mathematicians make when they apply mathematics to real-life situations?

Mar 21st Mar 27th – Apr 7th April 10th (4 weeks)

The chain rule for composite functions.

The product and quotient rules.

5.7 The second derivative.

Graphical behaviour of functions, including the relationship between the graphs of f,fʹ and fʺ.

5.8 Local maximum and minimum points.

Testing for maximum and minimum.

Optimization.

Points of inflexion with zero and non-zero gradients.

Assessment: Differentiation (Cumulative)

Toolkit Test and Review

getting a man on the Moon. What does this tell us about the links between mathematical models and reality?

Spring Break Calculus – Integratio n and its applicatio n (focus only on polynomi als)

10.1 10.2 10.3 10.4 10.5 5.5

5.5 Introduction to integration as antidifferentiation of functions of the form f(x)=axn+bxn−1+...., where n∈ℤ, n≠−1 Anti-differentiation with a boundary condition to determine the constant term.

Definite integrals using technology.

Area of a region enclosed by a curve y=f(x) and the x -axis, where f(x)>0.

5.10 Indefinite integral of xn(n∈ℚ) and ex.

The composites of any of these with the linear function ax+b.

Integration by inspection (reverse chain rule) or by substitution for expressions of the form: ∫kgʹ(x)f(g(x))dx.

5.11 Definite integrals, including analytical approach.

Areas of a region enclosed by a curve y=f(x) and the x-axis, where f(x) can be positive or negative, without the use of technology.

Areas between curves.

5.9 Kinematic problems involving displacement s, velocity v, acceleration a and total distance travelled.

20

IM: 1. Accurate calculation of the volume of a cylinder by Chinese mathematician Liu Hui; Ibn Al Haytham: first mathematician to calculate the integral of a function, in order to find the volume of a paraboloid.

2. Does the inclusion of kinematics as core mathematics reflect a particular cultural heritage? Who decides what is mathematics?

TOK: 1. Consider f(x)=1/x,1≤x. An infinite area sweeps out a finite volume.

Can this be reconciled with our intuition? Do emotion and intuition have a role in mathematics?

2. Is mathematics independent of

culture? To what extent are we people aware of the impact of culture on what we they believe or know?

May 8th (1 week)

May 15th to July 3rd

Toolkit Test and Review

Assessment: Integration (Cumulative)

Mock Exam Revision (2 weeks) G11 Mocks(from end May onwards) Review Mock Paper IA preparation

21

Year 2 (Grade 12) Week commencing: 2022

Topic

Geometry Toolkit

Textbook Reference 11.1 11.2 11.3 11.4 11.5

Sep (4 weeks) Calculus – Differentiation and Integration and its application (focus only on trig functions)

Oct (5 weeks)

Trigonometric functions Toolkit

Syllabus reference

TOK links, IM links

3.1 The distance between two points in threedimensional space, and their midpoint.

Volume and surface area of threedimensional 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.

3.2 Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles.

The sine rule The cosine rule Area of a triangle 3.3 Applications of right and non-right angled trigonometry, including Pythagoras’s theorem.

Angles of elevation and depression.

Construction of labelled diagrams from written statements.

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?

5.6 Derivative of xn(n∈ℚ), sinx, cosx, ex and lnx.

5.10 Indefinite integral of xn(n∈ℚ),sinx,cosx,1/x and ex.

The composites of any of these with the linear function ax+b.

12.1 12.2 12.3 12.4

3.4 The circle: radian measure of angles; length of an arc; area of a sector.

3.5 Definition of cosθ, sinθ in terms of the unit circle.

Definition of tanθ as sinθcosθ.

Exact values of trigonometric ratios of 0,

22

IM: 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 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.

IM: Seki Takakazu calculating π to ten decimal places; Hipparchus, Menelaus and Ptolemy; Why are there 360 degrees in a complete turn? Links to Babylonian mathematics.

π/6, π/4, π/3, π/2and their multiples.

Extension of the sine rule to the ambiguous case.

3.6 The Pythagorean identity cos2θ+sin2θ=1.

The relationship between trigonometric ratios.

3.7 The circular functions sinx, cosx, and tanx; amplitude, their periodic nature, and their graphs Composite functions Transformations.

Real-life contexts.

3.8 Solving trigonometric equations in a finite interval, both graphically and analytically.

Equations leading to quadratic equations in sinx,cosx or tanx.

Nov

TOK: Which is a better measure of angle: radian or degree?

What criteria can/do/should mathematicians use to make such decisions?

Review and test Toolkit IA Final Submission

Winter Break Mock Exam Statistics

January 2021

6.1 6.2 6.3

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.

4.2 Presentation of data (discrete and continuous): frequency distributions (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.

4.3 Measures of central tendency (mean, median and mode).

Estimation of mean from grouped data.

Modal class.

Measures of dispersion (interquartile

23

IM: 1. The Kinsey report– famous sampling techniques.

2. Discussion of the different formulae for the same statistical measure (for example, variance).

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?

2. What is the difference between information and data? Does “data” mean the same thing in

Linear Regression

7.1 7.2 7.3 7.4

Venn Diagrams and Probability

8.1 8.2 8.3 8.4

Probability Distributions

14.1 14.2 14.3

February

March

range, standard deviation and variance).

Effect of constant changes on the original data.

Quartiles of discrete data.

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 linear regression y=ax+b.

4.10 Equation of the regression line of x on y.

Use of the equation for prediction purposes.

4.5 Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space (U) and event.

The probability of an event A The complementary events A and Aʹ Expected number of occurrences.

4.6 Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities.

Combined events Mutually exclusive events Conditional probability Independent events 4.7 Concept of discrete random variables and their probability distributions.

Expected value (mean), for discrete data.

Applications.

4.11 Formal definition and use of the formulae: P(A|B)=P(A∩B)P(B) for conditional probabilities, and P(A|B)=P(A)=P(A|Bʹ) for independent events.

4.8 Binomial distribution.

Mean and variance of the binomial distribution.

4.9 The normal distribution and curve.

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different areas of knowledge?

TOK: 1. Is it possible to have knowledge of the future?

2. Correlation and causation–can we have knowledge of cause and effect relationships given that we can only observe correlation?

What factors affect the reliability and validity of mathematical models in describing real-life phenomena?

IM: The St Petersburg paradox; Chebyshev and Pavlovsky (Russian).

TOK: 1. To what extent are theoretical and experimental probabilities linked?

What is the role of emotion in our perception of risk, for example in business, medicine and travel safety?

2. Can calculation of gambling probabilities be considered an ethical application of mathematics? Should mathematicians be held responsible for unethical applications of their work?

IM: De Moivre’s derivation of the normal distribution and Quetelet’s use of it to describe l’homme moyen.

TOK:

Properties of the normal distribution.

Diagrammatic representation.

Normal probability calculations.

Inverse normal calculations 4.12 Standardization of normal variables (zvalues).

Inverse normal calculations where mean and standard deviation are unknown.

Revision

25

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?

Mathematics: applications and interpretation HL 2021-2023

Course description (Taken from the IB Mathematics: applications and interpretation guide) This course recognizes the increasing role that mathematics and technology play in a diverse range of fields in a data-rich world. As such, it emphasizes the meaning of mathematics in context by focusing on topics that are often used as applications or in mathematical modelling. To give this understanding a firm base, this course also includes topics that are traditionally part of a pre-university mathematics course such as calculus and statistics.

The course makes extensive use of technology to allow students to explore and construct mathematical models. Mathematics: applications and interpretation will develop mathematical thinking, often in the context of a practical problem and using technology to justify conjectures.

Mathematics: applications and interpretation: Distinction between SL and HL Students who choose Mathematics: applications and interpretation at SL or HL should enjoy seeing mathematics used in real-world contexts and to solve real-world problems. Students who wish to take Mathematics: applications and interpretation at higher level will have good algebraic skills and experience of solving real-world problems. They will be students who get pleasure and satisfaction when

Mathematics and International Mindedness International-mindedness is a complex and multi-faceted concept that refers to a way of thinking, being and acting characterized by an openness to the world and a recognition of our deep interconnectedness to others.

Despite recent advances in the development of information and communication technologies, the global exchange of mathematical information and ideas is not a new phenomenon and has been essential to the progress of mathematics. Indeed, many of the foundations of modern mathematics were laid many centuries ago by diverse civilisations – Arabic, Greek, Indian and Chinese among others.

Mathematics can in some ways be seen as an international language and, apart from slightly differing notation, mathematicians from around the world can communicate effectively within their field.

Mathematics can transcend politics, religion and nationality, and throughout history great civilizations have owed their success in part to their mathematicians being able to create and maintain complex social and architectural structures. Politics has dominated the development of mathematics, to develop ballistics, navigation and trade, and land ownership, often influenced by governments and leaders. Many early mathematicians were political and military advisers and today mathematicians are integral members of teams who advise governments on where money and resources should be allocated.

Science and technology are of significant importance in today’s world. As the language of science, mathematics is an essential component of most technological innovation and underpins developments in science and technology, although the contribution of mathematics may not always be visible. Examples of this include the role of the binary number system, matrix algebra, network theory and probability theory in the digital revolution, or the use of mathematical simulations to predict future climate change or spread of disease. These examples highlight the key role mathematics can play in transforming the world around us.

One way of fostering international-mindedness is to provide opportunities for inquiry into a range of local and global issues and ideas. Many international organisations and bodies now exist to promote mathematics, and students are encouraged to access the resources and often-extensive websites of such mathematical organisations. This can enhance their appreciation of the international dimension of mathematics, as well as providing opportunities to engage with global issues surrounding the subject.

26

Examples of links relating to international-mindedness are given in the “Connections” sections of the syllabus.

Mathematics and Theory of Knowledge The relationship between each subject and theory of knowledge (TOK) is important and fundamental to the DP. The theory of knowledge course provides an opportunity for students to reflect on questions about how knowledge is produced and shared, both in mathematics and also across different areas of knowledge. It encourages students to reflect on their assumptions and biases, helping them to become more aware of their own perspective and the perspectives of others and to become “inquiring, knowledgeable and caring young people” (IB mission statement).

As part of their theory of knowledge course, students are encouraged to explore tensions relating to knowledge in mathematics. As an area of knowledge, mathematics seems to supply a certainty perhaps impossible in other disciplines and in many instances provides us with tools to debate these certainties. This may be related to the “purity” of the subject, something that can sometimes make it seem divorced from reality. Yet mathematics has also provided important knowledge about the world and the use of mathematics in science and technology has been one of the driving forces for scientific advances.

Despite all its undoubted power for understanding and change, mathematics is in the end a puzzling phenomenon. A fundamental question for all knowers is whether mathematical knowledge really exists independently of our thinking about it. Is it there, “waiting to be discovered”, or is it a human creation?

Indeed, the philosophy of mathematics is an area of study in its own right.

Students’ attention should be drawn to questions relating theory of knowledge (TOK) and mathematics, and they should also be encouraged to raise such questions themselves in both their mathematics and TOK classes. Examples of issues relating to TOK are given in the “Connections” sections of the syllabus. Further suggestions for making links to TOK can also be found in the mathematics section of the Theory of knowledge guide.

Mathematics and Creativity, Action, Service CAS experiences can be associated with each of the subject groups of the DP. CAS and mathematics can complement each other in a number of ways. Mathematical knowledge provides an important key to understanding the world in which we live, and the mathematical skills and techniques students learn in the mathematics courses will allow them to evaluate the world around them which will help them to develop, plan and deliver CAS experiences or projects.

An important aspect of the mathematics courses is that students develop the ability to systematically analyse situations and can recognize the impact that mathematics can have on the world around them. An awareness of how mathematics can be used to represent the truth enables students to reflect critically on the information that societies are given or generate, and how this influences the allocation of resources or the choices that people make. This systematic analysis and critical reflection when problem solving may be inspiring springboards for CAS projects.

Students may also draw on their CAS experiences to enrich their involvement in mathematics both within and outside the classroom, and mathematics teachers can assist students in making links between their subjects and students’ CAS experiences where appropriate. Purposeful discussion about real CAS experiences and projects will help students to make these links.

The challenge and enjoyment of CAS can often have a profound effect on mathematics students, who might choose, for example, to engage with CAS in the following ways: • plan, write and implement a “mathematics scavenger hunt” where younger students tour the school answering interesting mathematics questions as part of their introduction to a new school • as a CAS project students could plan and carry out a survey, create a database and analyse the results, and make suggestions to resolve a problem in the students’ local area. This might be, for example, surveying the availability of fresh fruit and vegetables within a community, preparing an

27

•

action plan with suggestions of how to increase availability or access, and presenting this to a local charity or community group taking an element of world culture that interests students and designing a miniature Earth (if the world were 100 people) to express the trend(s) numerically.

It is important to note that a CAS experience can be a single event or may be an extended series of events.

However, CAS experiences must be distinct from, and may not be included or used in, the student’s Diploma course requirements.

Additional suggestions on the links between DP subjects and CAS can be found in the Creativity, activity, service teacher support material.

Textbook Oxford IB Diploma Programme: IB Mathematics: applications and interpretation, Higher Level

ISBN: 978-0-19-842704-9

Calculator A Graphical Display Calculator is required for the course.

Texas Intstrument TI-84 (recommended and used in class)

28

2 Year Course Outline See detailed plan on next page for order of content and approximate dates Topic 1: Number and Algebra (29 hours) Topic 2: Functions (42 hours) Year 1

Topic 3: Geometry and Trigonometry (46 hours) Toolkit (25 hours) Revision Mock Exams

Year 2

Topic 4: Statistics and Probability (52 hours) Topic 5: Calculus (41 hours) Toolkit (5 hours) Revision Mock Exams and Final Exams

Assessment Outline External assessment (5 hours)

Paper 1 (2 hours) 30% Compulsory short-response questions based on the whole syllabus. Technology required.

(110 marks) Paper 2 (2 hours) 30% Compulsory extended-response questions based on the whole syllabus. Technology required.

(110 marks) Paper 3 (60 minutes) 20% Two compulsory extended response problem-solving questions. Technology required.

(55 marks)

Internal Assessment (15 hours – included in outline as part of toolkit)

Maths Project 20% Internal assessment in mathematics is an individual exploration. This is a piece of written work that involves investigating an area of mathematics. (20 marks)

29

Detailed Plan for year 1 (grade 11) Week commencing: 2021

Topic

Measurements and estimates Aug 29 (Aug 30 first day for all students)

Recording measurements, significant digits and rounding Measurements: exact or approximation?

Speaking scientifically

Sep 5

Trigonometry of right-angled triangles and indirect measurements Sep 12

Angles of elevation and depression

Trigonometry of non-right triangles

Sep 19

Syllabus reference

Textbook Reference, TOK links.

1.6 - Approximation: decimal places, significant figures, upper and lower bound of rounded numbers, percentage errors, estimation.

1.1 TOK: Is mathematical reasoning different from scientific reasoning, or reasoning in other areas of knowledge?

1.1 - Operations with numbers in the form a× 10! where 1 ≤ a < 10 and k is an integer, 1.5 - Laws of exponents with integer exponents.

1.10 – Simplifying expressions involving rational exponents

1.1 TOK: Do the names that we give things impact how we understand them?

Is Mathematics invented or discovered?

1.2 TOK: Is it ethical that Pythagoras gave his name to a theorem that may not have been his own creation?

If angles of a triangle can add up to less than, or more than, 180 degrees, what does this tell us about the nature of mathematical knowledge?

1.2 TOK: Does personal experience play a role in the formation of knowledge claims in mathematics? Does it play a different role in mathematics compared to other areas of knowledge?

3.2 – use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles.

3.3 - Applications of right and nonright angled trigonometry, including Pythagoras’ theorem.

Angles of elevation and depression.

Construction of labelled diagrams from written statements.

3.2 -

"

'

)

Area of a triangle formula: applications of rightand non-right angled trigonometry

The sine rule: #$%& = #$%( = #$%* The cosine rule: c + = a+ + b+ − 2abcosC a+ + b+ − c + cosC = 2ab , Area of a triangle as + absinC

3D geometry: solids,

3.4 - The circle: length of an arc; the area of a sector.

3.7 – radian measure.

3.8 – trigonometric identities.

3.1 – The distance between two points

30

1.3

Sep 26

surface area and volume

Coordinates, distance and midpoint in 2D and 3D Oct 3

Oct 10 Oct 17

Oct 24

Oct 31

Gradient of a line and its applications Equations of straight lines in different forms Parallel and Perpendicular lines Midterm Break Voronoi diagrams and the toxic waste dump problem

Functions

Linear models

Nov 7

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.

2.1 - Different forms of the equation of a straight line.

Gradient; intercepts.

Lines with gradients 𝑚, and 𝑚+ 3.1 - The distance between two points in three-dimensional space, and their midpoint

2.1 - Parallel lines 𝑚, = 𝑚+ .

Perpendicular lines 𝑚, × 𝑚+ = −1 3.5 – Equations of perpendicular bisectors 3.6 - Voronoi diagrams: sites, vertices, edges, cells. Addition of a site to an existing Voronoi diagram. Nearest neighbour interpolation. Applications of the “toxic waste dump” problem.

2.2 - Concept of a function, domain, range and graph.

Function notation, for example f(x),v(t),C(n).

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 y=x, and the notation f -, (x).

2.7 – composite functions.

2.8 – transformation of functions.

2.5 – linear models 𝑓(𝑥) = 𝑚𝑥 + 𝑐 2.3 - The graph of a function; its equation y=f(x); creating a sketch from information given or a context 2.4 – determine key features of graphs 2.6 – develop and fit a model; find parameters of the model, test and reflect on the model; use the model.

31

TOK: What is an axiomatic system? Are axioms self evident to everybody?

3.1 TOK: Descartes showed that geometric problems could be solved algebraically and vice-versa. What does this tell us about mathematical representation and mathematical knowledge?

3.2

3.3 TOK: Is the division of knowledge into disciplines or areas of knowledge artificial?

4.1 TOK: Do you think mathematics or logic should be classified as a language?

4.2, 4.3

TOK: 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?

Arithmetic sequences

1.2 - Arithmetic sequences and series.

4.4

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, approximations.

4.4

2.5 – quadratic models 𝑓(𝑥) = 𝑎𝑥 + + 𝑏𝑥 + 𝑐; 𝑎 ≠ 0

6.1

Quadratic Models

2.3 - The graph of a function; its equation y=f(x); creating a sketch from information given or a context

6.2

Quadratic Models

2.4 – determine key features of graphs

6.2

Problems involving quadratics

2.6 – develop and fit a model; find parameters of the model, test and reflect on the model; use the model.

2.5 – cubic models 𝑓(𝑥) = 𝑎𝑥 . + 𝑏𝑥 + + 𝑐𝑥 + 𝑑; direct/inverse variation 𝑓(𝑥) = 𝑎𝑥 / , 𝑛 ∈ ℤ

6.2

2.3 - The graph of a function; its equation y=f(x); creating a sketch from information given or a context 2.9 – logarithmic models, logistic models, piecewise models.

6.4

Nov 14

Nov 21

Nov 28 Dec 5 December 12 – January 1

Modelling

Commemoration day, National day December 1 - 2 Winter Break Quadratic Models

January 2, 2022

Jan 9 Jan 16 Jan 23

Jan 30

Feb 6

TOK: Is all knowledge concerned with identification and use of patterns? Consider Fibonacci numbers and connections with the golden ratio.

Cubic models, power functions and direct and inverse variation Cubic models, power functions and direct and inverse variation

32

TOK: What role do models play in mathematics? Do they play a different role in mathematics compared to their role in other areas of knowledge?

6.3

TOK: What is it about models in mathematics that makes them effective? Is simplicity a desirable characteristic in models?

February 13 Feb 20 Feb 27

Mar 6

Mar 13

Midterm Break Optimization Optimization

Geometric sequences and series

Compound interest, annuities, amortization

Exponential models

Mar 20

March 27 – April 9

April 10

2.4 – determine key features of graphs 2.6 – develop and fit a model; find parameters of the model, test and reflect on the model; use the model.

1.3 - 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.

1.11 – The sum of infinite geometric sequences.

1.4 - Financial applications of geometric sequences and series: compound interest, annual depreciation, 1.7 - Amortization and annuities using technology.

7.1

TOK: How do mathematicians 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.

7.2

TOK: How have technological advances affected the nature and practice of mathematics?

Consider the use of financial packages for instance.

2.5 – exponential models 𝑓(𝑥) = 𝑘𝑎 0 + 𝑐; 𝑓(𝑥) = 𝑘𝑒 10 + 𝑐 2.3 - The graph of a function; its equation y=f(x); creating a sketch from information given or a context 2.4 – determine key features of graphs 2.6 – develop and fit a model; find parameters of the model, test and reflect on the model; use the model.

7.3 – 7.5

1.5 – Introduction to logarithms with base 10 and e. Numerical evaluation of logarithms using technology.

TOK: Is mathematics invented or discovered?

For instance, consider the number e or logarithms– did they already exist before man defined them?

Spring Break Exponential equations and logarithms

1.9 – Laws of logarithms.

Systems of equations

1.8 - Use technology to solve: systems of linear equations in up to 3 variables and polynomial equations

April 17

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9.3

TOK: 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.

April 24

An introduction to periodic functions

Sinusoidal functions May 1 Complex numbers May 8

Vectors May 15

Matrices

May 22

Graph Theory

2.5 – sinusoidal models 𝑓(𝑥) = asin(𝑏𝑥) + 𝑑, 𝑓(𝑥) = acos(𝑏𝑥) + 𝑑 2.3 - The graph of a function; its equation y=f(x); creating a sketch from information given or a context 2.4 – determine key features of graphs 2.6 – develop and fit a model; find parameters of the model, test and reflect on the model; use the model.

1.12 – complex numbers 1.13 – modulus-argument form, exponential form.

3.10 – Vectors and scalars.

3.11 – Vector equations in 2 and 3 dimensions.

3.12 – Vector applications to kinematics.

3.13 – Scalar product and vector product.

1.8 - Use technology to solve: systems of linear equations in up to 3 variables and polynomial equations 1.14 – Matrices and operations, identity and zero matrices.

1.15 – Eigenvalues and eigenvectors.

3.9 – Geometric transformations using matrices.

4.19 – Transition matrices, markov chains.

3.14 – Graph theory 3.15 – Adjacency matrices, weighted adjacency tables.

3.16 – Applications.

May 29

8.1, 8.2

8.3, 8.4

8.5 TOK: How does language shape knowledge? For example do the words “imaginary” and “complex” make the concepts more difficult than if they had different names?

3.4, 3.5, 3.6

9.1 – 9.7

15.1 – 15.5 Other contexts: Using GPS to find the shortest route home; describe current and voltage in circuits as cycles; vehicle routing problems.

Internationalmindedness: The “Bridges of Konigsberg” problem; the Chinese postman problem was first posed by the

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Chinese mathematician Kwan Mei-Ko in 1962.

TOK: 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”?

June 5 June 12 June 19 June 26 July 3 July 4

Revision Mock Exams

Summer Break

Year 2 (Grade 12) Month commencing: 2022

September

Topic

Limits and derivatives

Limits and derivatives

Equations of tangents and normal

Syllabus reference

Textbook Reference

5.1 – introduction to the concept of a limit, derivative interpreted as gradient function and as rate of change 5.2 – increasing and decreasing functions 5.3 – derivatives of polynomial functions 5.9 – derivatives of trigonometric functions, exponential function, logarithmic functions.

The chain rule, product rule and quotient rule.

Related rates of change.

5.10 – The second derivative.

10.1

5.4 – tangents and normals at a given point, and their equations

10.2

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TOK: What value does the knowledge of limits have? Is infinitesimal behaviour applicable to real life? Is intuition a valid way of knowing in mathematics?

10.1

TOK: The seemingly 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?

TOK: 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?

Maximum and minimum points and optimization October

5.6 – values of x where the gradient of the curve is zero.

5.7 – optimization problems in context.

10.3

5.8 – approximating areas using the trapezoidal rule.

5.5 – integration as antidifferentiation. Boundary condition to determine constant. Definite integrals using technology.

5.11 – Definite and indefinite integration of functions including trigonometric and exponential. Integration by inspection.

5.5 – integration as antidifferentiation. Boundary condition to determine constant. Definite integrals using technology.

5.12 – Areas enclosed by a curve and an axis, volumes of revolution.

5.13 – Kinematic problems.

5.14 – Differential equations, separation of variables.

11.1

5.15 – Slope fields and their diagrams.

5.16 – Euler’s method.

11.5 TOK: How have notable individuals such as Euler shaped the development

TOK: Is it possible for an area of knowledge to describe the world without transforming it?

How can the rise in tax for plastic containers, for example plastic bags, plastic bottles etc be justified using optimization?

Draft Project Due Finding areas Integration: the reverse process

Integration: the reverse process

Kinematics and differential equations November

Slope fields, Euler’s method

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11.2

TOK: Is it possible for an area of knowledge to describe the world without transforming it?

11.3

11.4 TOK: What is the role of convention in mathematics?

Is this similar or different to the role of convention in other areas of knowledge?

of mathematics as an area of knowledge?

Vector quantities

Motion with variable velocity

December

Exact solutions of coupled differential equations Approximate solutions to coupled linear equations

3.12 - Vector applications to kinematics.

12.1 12.2

Modelling linear motion with constant velocity in two and three dimensions.

5.17 – Phase portrait for solutions of differential equations.

5.18 – Solutions of second derivative functions using Euler’s method.

12.3

4.2 – presentation of data. Construction and use of histograms, cumulative frequence graphs, box and whisker diagrams.

4.3 – Measures of centre and dispersion, modal class.

2.1, 2.2

4.1 – Concepts of population, sample, random sample, discrete and continuous data.

Reliability and bias.

Outliers. Sampling techniques.

4.4 – correlation coefficient 4.4 – scatter diagrams, line of best fit, equation of regression line.

2.3, 2.4

12.4

Winter Holiday Presentation of data Bivariate data

January 2023

Collecting and organizing univariate data Sampling techniques

Measuring correlation The line of best fit Interpreting the regression line

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TOK: What is the difference between information and data? Does “data” mean the same thing in different areas of knowledge?

Could mathematics make alternative, equally true, formulae? What does this tell us about mathematical truths? Does the use of statistics lead to an overemphasis on attributes that can be easily measured over those that cannot?

TOK: 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?

4.5

TOK: Correlation and causation–can we have knowledge of cause and effect relationships given that we can only observe correlation? What factors affect the reliability and validity of mathematical models in describing real-life phenomena?

Theoretical and experimental probability

Representing combined probabilities with diagrams and formulae

Complete, concise and consistent representations February

Random variables and probability distributions

Binomial distribution, Poisson distribution.

Normal distribution

Spearmans’ rank correlation coefficient

4.5 – probability of events, complementary events, expected number of outcomes.

5.1

4.6 – Venn diagrams, tree diagrams, sample space diagrams, combined events, conditional probability, independent events.

5.2 – 5.4

4.7 – concept of discrete random variables, expected value, E(X), applications.

4.14 – linear transformation of random variables, unbiased estimators.

4.8 – calculations, mean and variance.

4.17 – Poisson distribution, its mean and variance.

4.18 – Critical values and critical regions.

4.9 – normal and inverse normal calculations.

4.15 – linear combinations of normal random variables, central limit theorem.

4.16 – Confidence intervals.

4.10 – appropriateness, limitations and effect of outliers.

13.1 – 13.6

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TOK: To what extent are theoretical and experimental probabilities linked? What is the role of emotion in our perception of risk, for example in business, medicine and travel safety?

TOK: Can calculation of gambling probabilities be considered an ethical application of mathematics?

Should mathematicians be held responsible for unethical applications of their work?

TOK: What do we mean by a “fair” game? Is it fair that casinos should make a profit?

14.2, 14.3

TOK: What criteria can we use to decide between different models?

TOK: 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?

TOK: Does correlation imply causation? Mathematics and the world. Given that a set of data may be approximately fitted by a range of curves, where would a mathematician seek for knowledge of which equation is the “true” model?

March

𝜒 + test for independence 𝜒 + goodness of fit test The 𝑡-test

4.11 – contingency tables, degrees of freedom, critical value.

4.11 4.12 – Reliability and validity.

4.11 – use of p-value, one-tailed and two-tailed tests.

Spring Holiday

April Revision Revision Revision Revision May

Exams

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14.1 – 14.6

TOK: 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 differently in different areas of knowledge?

Mathematics: Application and Interpretation SL 2021-2023

Course description (Taken from the IB Mathematics: applications and interpretation guide) This course recognizes the increasing role that mathematics and technology play in a diverse range of fields in a data-rich world. As such, it emphasizes the meaning of mathematics in context by focusing on topics that are often used as applications or in mathematical modelling. To give this understanding a firm base, this course also includes topics that are traditionally part of a pre-university mathematics course such as calculus and statistics.

The course makes extensive use of technology to allow students to explore and construct mathematical models. Mathematics: applications and interpretation will develop mathematical thinking, often in the context of a practical problem and using technology to justify conjectures.

Mathematics: applications and interpretation: Distinction between SL and HL Students who choose Mathematics: analysis and approaches at SL or HL should be comfortable in the manipulation of algebraic expressions and enjoy the recognition of patterns and understand the mathematical generalization of these patterns. Students who wish to take Mathematics: analysis and approaches at higher level will have strong algebraic skills and the ability to understand simple proof. They will be students who enjoy spending time with problems and get pleasure and satisfaction from solving challenging problems.

Mathematics and International Mindedness International-mindedness is a complex and multi-faceted concept that refers to a way of thinking, being and acting characterized by an openness to the world and a recognition of our deep interconnectedness to others.

Despite recent advances in the development of information and communication technologies, the global exchange of mathematical information and ideas is not a new phenomenon and has been essential to the progress of mathematics. Indeed, many of the foundations of modern mathematics were laid many centuries ago by diverse civilisations – Arabic, Greek, Indian and Chinese among others.

Mathematics can in some ways be seen as an international language and, apart from slightly differing notation, mathematicians from around the world can communicate effectively within their field.

Mathematics can transcend politics, religion and nationality, and throughout history great civilizations have owed their success in part to their mathematicians being able to create and maintain complex social and architectural structures. Politics has dominated the development of mathematics, to develop ballistics, navigation and trade, and land ownership, often influenced by governments and leaders. Many early mathematicians were political and military advisers and today mathematicians are integral members of teams who advise governments on where money and resources should be allocated.

Science and technology are of significant importance in today’s world. As the language of science, mathematics is an essential component of most technological innovation and underpins developments in science and technology, although the contribution of mathematics may not always be visible. Examples of this include the role of the binary number system, matrix algebra, network theory and probability theory in the digital revolution, or the use of mathematical simulations to predict future climate change or spread of disease. These examples highlight the key role mathematics can play in transforming the world around us.

One way of fostering international-mindedness is to provide opportunities for inquiry into a range of local and global issues and ideas. Many international organisations and bodies now exist to promote mathematics, and students are encouraged to access the resources and often-extensive websites of such mathematical organisations. This can enhance their appreciation of the international dimension of

40

mathematics, as well as providing opportunities to engage with global issues surrounding the subject.

Examples of links relating to international-mindedness are given in the “Connections” sections of the syllabus.

Mathematics and Theory of Knowledge The relationship between each subject and theory of knowledge (TOK) is important and fundamental to the DP. The theory of knowledge course provides an opportunity for students to reflect on questions about how knowledge is produced and shared, both in mathematics and also across different areas of knowledge. It encourages students to reflect on their assumptions and biases, helping them to become more aware of their own perspective and the perspectives of others and to become “inquiring, knowledgeable and caring young people” (IB mission statement).

As part of their theory of knowledge course, students are encouraged to explore tensions relating to knowledge in mathematics. As an area of knowledge, mathematics seems to supply a certainty perhaps impossible in other disciplines and in many instances provides us with tools to debate these certainties. This may be related to the “purity” of the subject, something that can sometimes make it seem divorced from reality. Yet mathematics has also provided important knowledge about the world and the use of mathematics in science and technology has been one of the driving forces for scientific advances.

Despite all its undoubted power for understanding and change, mathematics is in the end a puzzling phenomenon. A fundamental question for all knowers is whether mathematical knowledge really exists independently of our thinking about it. Is it there, “waiting to be discovered”, or is it a human creation?

Indeed, the philosophy of mathematics is an area of study in its own right.

Students’ attention should be drawn to questions relating theory of knowledge (TOK) and mathematics, and they should also be encouraged to raise such questions themselves in both their mathematics and TOK classes. Examples of issues relating to TOK are given in the “Connections” sections of the syllabus. Further suggestions for making links to TOK can also be found in the mathematics section of the Theory of knowledge guide.

Mathematics and Creativity, Action, Service CAS experiences can be associated with each of the subject groups of the DP. CAS and mathematics can complement each other in a number of ways. Mathematical knowledge provides an important key to understanding the world in which we live, and the mathematical skills and techniques students learn in the mathematics courses will allow them to evaluate the world around them which will help them to develop, plan and deliver CAS experiences or projects.

An important aspect of the mathematics courses is that students develop the ability to systematically analyse situations and can recognize the impact that mathematics can have on the world around them. An awareness of how mathematics can be used to represent the truth enables students to reflect critically on the information that societies are given or generate, and how this influences the allocation of resources or the choices that people make. This systematic analysis and critical reflection when problem solving may be inspiring springboards for CAS projects.

Students may also draw on their CAS experiences to enrich their involvement in mathematics both within and outside the classroom, and mathematics teachers can assist students in making links between their subjects and students’ CAS experiences where appropriate. Purposeful discussion about real CAS experiences and projects will help students to make these links.

The challenge and enjoyment of CAS can often have a profound effect on mathematics students, who might choose, for example, to engage with CAS in the following ways: • plan, write and implement a “mathematics scavenger hunt” where younger students tour the school answering interesting mathematics questions as part of their introduction to a new school • as a CAS project students could plan and carry out a survey, create a database and analyse the results, and make suggestions to resolve a problem in the students’ local area. This might be, for example, surveying the availability of fresh fruit and vegetables within a community, preparing an

41

•

action plan with suggestions of how to increase availability or access, and presenting this to a local charity or community group taking an element of world culture that interests students and designing a miniature Earth (if the world were 100 people) to express the trend(s) numerically.

It is important to note that a CAS experience can be a single event or may be an extended series of events.

However, CAS experiences must be distinct from, and may not be included or used in, the student’s Diploma course requirements.

Additional suggestions on the links between DP subjects and CAS can be found in the Creativity, activity, service teacher support material.

Textbook Oxford IB Diploma Programme: IB Mathematics: analysis and interpretation, Standard Level.

ISBN: 978-0-19-842711-7

Calculator A Graphical Display Calculator is required for the course.

Texas Intstrument TI-84 (recommended and used in class)

42

2 Year Course Outline See detailed plan on next page for order of content and approximate dates Topic 1: Number and Algebra (16 hours) Topic 2: Functions (31 hours) Year 1

Topic 5: Calculus (18 hours) Toolkit (25 hours) Revision Mock Exams

Year 2

Topic 3: Geometry and Trigonometry (36 hours) Topic 5: Statistics and Probability (19 hours) Toolkit (5 hours) Revision Mock Exams and Final Exams

Assessment Outline External assessment (3 hours)

Paper 1 (1 hour 30 minutes) 40% Compulsory short-response questions based on the whole syllabus. (80 marks) Paper 2 (1 hour 30 minutes) 40% Compulsory extended-response questions based on the whole syllabus. (80 marks)

Internal Assessment (15 hours – included in outline as part of toolkit)

Maths Project 20% Internal assessment in mathematics is an individual exploration. This is a piece of written work that involves investigating an area of mathematics. (20 marks)

43

Detailed Plan for year 1 (grade 11) Week commencing:

Topic

Syllabus reference

Textbook Reference

TOK links, IM links

2021 GDC Basics involving the use of equations solver, graphing, simulatanous equation function with graphing

Aug 30

Measurements and estimates

1.6 - Approximation: decimal places, significant figures, upper and lower bound of rounded numbers, percentage errors, estimation.

1.1 1.2 1.3

TOK: Is mathematical reasoning different from scientific reasoning, or reasoning in other areas of knowledge?

1.1 - Operations with numbers in the form a× 10! where 1 ≤ a < 10 and k is an integer, 1.5 - Laws of exponents with integer exponents.

3.2 Angles of elevation ad " The sine rule: #$%& =

1.4

TOK: Do the names that we give things impact how we understand them?

Is Mathematics invented or discovered?

1.6 2.1 2.2

TOK: Does personal experience play a role in the formation of knowledge claims in mathematics?

Does it play a different role in mathematics compared to other areas of knowledge?

Recording measurements, significant digits and rounding Measurements: exact or approximation?

Percentage error Speaking scientifically

Trigonometry of non-right triangles Area of a triangle formula: applications of rightand non-right angled trigonometry

3D geometry: solids, surface area and volume

'

)

= #$%* The cosine rule: c + = a+ + b+ − 2abcosC a+ + b+ − c + cosC = 2ab Area of a triangle as , absinC + #$%(

3.4 - The circle: length of an arc; the area of a sector.

3.1 – The distance between two points in three-dimensional space, and their midpoint. Volume and

44

2.3 TOK: What is an axiomatic system? Are axioms self

Toolkit – Al Dar Mini IA Coordinates, distance and midpoint in 2D and 3D Gradient of a line and its applications Equations of straight lines in different forms Parallel and Perpendicular lines

Voronoi diagrams and the toxic waste dump problem

Linear models

surface area of threedimensional solids including rightpyramid, 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.

evident to everybody?

2.1 - Different forms of the equation of a straight line.

Gradient; intercepts.

Lines with gradients 𝑚, and 𝑚+ 3.1 - The distance between two points in three-dimensional space, and their midpoint 2.1 - Parallel lines 𝑚, = 𝑚+ .

Perpendicular lines 𝑚, × 𝑚+ = −1 3.5 – Equations of perpendicular bisectors 3.6 - Voronoi diagrams: sites, vertices, edges, cells.

Addition of a site to an existing Voronoi diagram. Nearest neighbour interpolation.

Applications of the “toxic waste dump” problem.

2.5 – linear models 𝑓(𝑥) = 𝑚𝑥 + 𝑐 2.3 - The graph of a function; its equation y=f(x); creating a sketch from information given or a context

4.1 4.2 4.3

45

TOK: Descartes showed that geometric problems could be solved algebraically and vice-versa.

What does this tell us about mathematical representation and mathematical knowledge?

4.4

4.5

5.2

TOK: Does studying the graph of a function contain the same level of mathematical rigour as studying the

TOK: Is the division of knowledge into disciplines or areas of knowledge artificial?

Functions

Linear models

Measuring correlation

2.4 – determine key features of graphs 2.6 – develop and fit a model; find parameters of the model, test and reflect on the model; use the model.

function algebraically?

What are the advantages and disadvantages of having different forms and symbolic language in mathematics?

2.2 - Concept of a function, domain, range and graph.

Function notation, for example f(x),v(t),C(n).

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 y=x, and the notation f -, (x).

2.5 – linear models 𝑓(𝑥) = 𝑚𝑥 + 𝑐 2.3 - The graph of a function; its equation y=f(x); creating a sketch from information given or a context 2.4 – determine key features of graphs 2.6 – develop and fit a model; find parameters of the model, test and reflect on the model; use the model.

4.4 – correlation coefficient

5.1

TOK: Do you think mathematics or logic should be classified as a language?

5.2

TOK: 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?

46

6.1

TOK: Correlation and causation–can we have knowledge of cause and effect relationships given that we can only observe correlation? What

factors affect the reliability and validity of mathematical models in describing real-life phenomena?

The line of best fit Interpreting the regression line . Collecting and organizing univariate data Sampling techniques

Presentation of data Bivariate data

Quadratic Models

Problems involving quadratics

Cubic models, power functions and direct and inverse variation

4.4 – scatter diagrams, line of best fit, equation of regression line.

4.1 – Concepts of population, sample, random sample, discrete and continuous data.

Reliability and bias.

Outliers. Sampling techniques 4.2 – presentation of data. Construction and use of histograms, cumulative frequence graphs, box and whisker diagrams.

4.3 – Measures of centre and dispersion, modal class.

6.2, 6.3

2.5 – quadratic models 𝑓(𝑥) = 𝑎𝑥 + + 𝑏𝑥 + 𝑐; 𝑎 ≠ 0 2.3 - The graph of a function; its equation y=f(x); creating a sketch from information given or a context 2.4 – determine key features of graphs 2.6 – develop and fit a model; find parameters of the model, test and reflect on the model; use the model.

2.5 – cubic models 𝑓(𝑥) = 𝑎𝑥 . + 𝑏𝑥 + + 𝑐𝑥 + 𝑑; direct/inverse variation 𝑓(𝑥) = 𝑎𝑥 / , 𝑛 ∈ ℤ

9.1

47

3.1, 3.2

TOK: 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?

3.3, 3.4

TOK: What is the difference between information and data? Does “data” mean the same thing in different areas of knowledge?

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?

TOK: What role do models play in mathematics? Do they play a different role in mathematics compared to their role in other areas of knowledge?

9.2

9.3

TOK: What is it about models in mathematics that makes them effective? Is simplicity a desirable characteristic in models?

2.3 - The graph of a function; its equation y=f(x); creating a sketch from information given or a context

Detailed Plan for year 2 (grade 12) Week commencing: 2021

Topic

Textbook Reference 7.1 7.2 7.3 7.4

Summer Work (Probability)

7.5

Probability and Distributions 7.6

Syllabus reference

4.5 Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space (U) and event.

The probability of an event A is P(A)=n(A)n(U).

The complementary events A and Aʹ (not A).

Expected number of occurrences.

4.6 Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities.

Combined events: P(A∪B)=P(A)+P(B)−P(A∩B).

Mutually exclusive events: P(A∩B)=0.

Conditional probability: P(A|B)=P(A∩B)/P(B).

Independent events: P(A∩B)=P(A)P(B).

• IA draft check-up • Consolidate summer work on probability with past year questions 4.7 Concept of discrete random variables and their probability distributions.

Expected value (mean), E(X) for discrete data Applications.

4.8 Binomial Distribution Mean and variance of the binomial distribution

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TOK links, IM links

TOK: 1. To what extent are theoretical and experimental probabilities linked?

What is the role of emotion in our perception of risk, for example in business, medicine and travel safety?

2. Can calculation of gambling probabilities be considered an ethical application of mathematics?

Should mathematicians be held responsible for unethical applications of their work?

3. What do we mean by a “fair” game?

Is it fair that casinos should make a profit?

4. What criteria can we use to decide between different models?

5. To what extent can we trust mathematical models such as the normal distribution? How can we know what to includem and what to exclude, in a model?

IM: 1. The St Petersburg paradox; Chebyshev and Pavlovsky (Russian).

2. De Moivre’s derivation of the normal distribution and Quetelet’s use of it to describe l’homme moyen.

3. The so-called “Pascal’s triangle”

Test and Review Probability and Distributions IA draft due 3rd October

7.7

10.3 10.4

Functions 9.3 11.1 11.2 11.3

Test and Review 8.1

Statistical Testing

was known to the Chinese mathematician Yang Hui much earlier than Pascal.

Assessment: Cumulative Test on all topics covered

8.2 8.3

8.4

4.9 The normal distribution and curve Properties of the normal distribution.

Diagrammatic representation.

Normal probability calculations Inverse normal calculations.

2.5 Exponential growth and decay models Equation of a horizontal asymptote 1.5 Laws of exponents with integer exponents.

Introduction to logarithms with base 10 and e.

Numerical evaluation of logarithms using technology.

2.5 Direct/Inverse variation 2.5 Sinusoidal models

Assessment: Cumulative Test on all topics covered 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.

4.11 Formulation of null and alternative hypotheses, H0and H1.

Significance levels. p -values.

Expected and observed frequencies.

The χ2 test for independence: contingency tables, degrees of freedom, critical value.

The χ2 goodness of fit test.

4.11 The t -test.

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

Using one-tailed and two-tailed tests.

Mock Exam Review Winter Break

MOCK EXAMS

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TOK 1. What role do models play in mathematics? Do they play a different role in mathematics compared to their role in other areas of knowledge?

2. What is it about models in

3.

mathematics that makes them effective? Is simplicity a desirable characteristic in models?

Is mathematics invented or discovered? For instance, consider the number e or logarithms–did they already exist before man defined them?

TOK: 1. Does correlation imply causation?

Mathematics and the world. Given that a set of data may be approximately fitted by a range of curves, where would a mathematician seek for knowledge of which equation is the “true” model?

2. 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 differently in different areas of knowledge?

5.3

Series and Sequences 10.1

10.2

Financial Mathematics

1.2 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.

1.3 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 1.4 Financial applications of geometric sequences and series: •compound interest •annual depreciation.

TOK: 1. Is all knowledge concerned with identification and use of patterns?

Consider Fibonacci numbers and connections with the golden ratio.

2. How do mathematicians 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.

IM 1. Aryabhatta is sometimes considered the “father of algebra”–compare with alKhawarizmi; the use of several alphabets in mathematical notation 2. The chess legend (Sissa ibn Dahir) TOK How have technological advances affected the nature and practice of mathematics? Consider the use of financial packages for instance.

IM: Do all societies view investment and interest in the same way?

Test and Review

Assessment: Cumulative Test 12.1 12.2 12.3

5.1 Introduction to the concept of a limit.

Derivative interpreted as gradient function and as rate of change.

5.2 Increasing and decreasing functions.

Graphical interpretation of fʹ(x)>0,fʹ(x)=0,fʹ(x)<0.

Differential Calculus

5.3 Derivative of f(x)=axn is fʹ(x)=anxn−1, n∈ℤ The derivative of functions of the form f(x)=axn+bxn−1+... where all exponents are integers.

5.4 Tangents and normals at a given point, and their equations.

5.6 Values of x where the gradient of a curve is zero.

Solution of fʹ(x)=0.

Local maximum and minimum points.

5.7

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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 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?

IM Attempts by Indian mathematicians (500-1000 CE) to explain division by zero.

Optimisation problems in context.

Assessment: Cumulative Test

Test and Review 13.1 13.2

Integral Calculus

5.5 Introduction to integration as antidifferentiation of functions of the form f(x)=axn+bxn−1+... where n∈ℤ, n≠−1.

Anti-differentiation with a boundary condition to determine the constant term.

Definite integrals using technology.

Area of a region enclosed by a curve y=f(x) and the x-axis, where f(x)>0.

5.8 Approximating areas using the trapezoidal rule.

Spring Break Revision for May Exams Mock Exam Practice

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TOK Is it possible for an area of knowledge to describe the world without transforming it?