34 minute read
Aug
2 Year Course Outline
See detailed plan on next page for order of content and approximate dates
Year 1 Topic 1: Number and Algebra (29 hours) Topic 2: Functions (42 hours) 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)
Detailed Plan for year 1 (grade 11) Week commencing: 2021
Aug 29 (Aug 30 first day for all students)
Sep 5
Sep 12
Sep 19
Topic Syllabus reference Textbook Reference, TOK links.
Measurements and estimates
Recording measurements, significant digits and rounding 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?
Measurements: exact or approximation? Speaking scientifically 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?
Trigonometry of right-angled triangles and indirect measurements
Angles of elevation and depression 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.
Trigonometry of non-right triangles
Area of a triangle formula: applications of rightand non-right angled trigonometry 3.2 The sine rule: " #$%&
) #$%* The cosine rule: c+ =a+ +b+ − 2abcosC cosC= a+ +b+ −c+ 2ab Area of a triangle as , + absinC
3.4 - The circle: length of an arc; the area of a sector. 3.7 – radian measure. 3.8 – trigonometric identities. 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?
3D geometry: solids, 3.1 – The distance between two points 1.3
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?
Sep 26 surface area and volume
Oct 3
Oct 10 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
Oct 17 Midterm Break
Oct 24 Voronoi diagrams and the toxic waste dump problem
Oct 31 Functions
Nov 7 Linear models 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 �, and �+ 3.1 - The distance between two points in three-dimensional space, and their midpoint
2.1 - Parallel lines �, =�+. Perpendicular lines �, ×�+ =−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 �(�)=��+� 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. 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
Nov 14
Nov 21
Nov 28
Dec 5 December 12 –January 1
January 2, 2022
Jan 9
Jan 16
Jan 23
Jan 30
Feb 6 Arithmetic sequences
Modelling
Commemoration day, National day December 1 - 2
disadvantages of having different forms and symbolic language in mathematics? 1.2 - Arithmetic sequences and series. 4.4 TOK: Is all knowledge concerned with identification and use of patterns? Consider Fibonacci numbers and connections with the golden ratio.
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
Winter Break
Quadratic Models 2.5 – quadratic models �(�)=��+ + ��+�;� ≠0
Quadratic Models 2.3 - The graph of a function; its equation y=f(x); creating a sketch from information given or a context 6.1 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? 6.2
Quadratic Models 2.4 – determine key features of graphs 6.2
Problems involving quadratics
Cubic models, power functions and direct and inverse variation Cubic models, power functions and direct and inverse variation 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 �(�)=��. + ��+ +��+�; direct/inverse variation �(�)=��/ ,�∈ℤ
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. 6.2
6.3
6.4 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 Compound interest, annuities, amortization 1.4 - Financial applications of geometric sequences and series: compound interest, annual depreciation, 1.7 - Amortization and annuities using technology.
Mar 20
March 27 –April 9
April 10
April 17 Systems of equations 1.8 - Use technology to solve: systems of linear equations in up to 3 variables and polynomial equations 9.3
Midterm Break
Optimization
Optimization
Geometric sequences and series 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.
Exponential models 2.5 – exponential models �(�)= ��0 +�; �(�)=��10 +� 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.
Spring Break
Exponential equations and logarithms 1.5 – Introduction to logarithms with base 10 and e. Numerical evaluation of logarithms using technology.
1.9 – Laws of logarithms. 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. 7.3 – 7.5
TOK: Is mathematics invented or discovered? For instance, consider the number e or logarithms–did they already exist before man defined them?
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
May 1
May 8
May 15 Vectors 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.
May 22
May 29 An introduction to periodic functions
2.5 – sinusoidal models �(�)= asin(��)+�, �(�)=acos(��)+� 2.3 - The graph of a function; its equation y=f(x); creating a sketch from information given or a context Sinusoidal functions 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. Complex numbers 1.12 – complex numbers 1.13 – modulus-argument form, exponential form.
Matrices
Graph Theory 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. 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.
International-
mindedness: The “Bridges of Konigsberg” problem; the Chinese postman problem was first posed by the
June 5 Revision
June 12 Mock Exams
June 19
June 26 July 3
July 4 Summer Break Year 2 (Grade 12)
Month commencing: 2022 Topic
September
Limits and derivatives 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 10.1
Limits and derivatives 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.
Equations of tangents and normal 5.4 – tangents and normals at a given point, and their equations 10.2
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”?
Syllabus reference Textbook Reference
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
October
November
Maximum and minimum points and optimization 5.6 – values of x where the gradient of the curve is zero. 5.7 – optimization problems in context. 10.3
Draft Project Due
Finding areas
5.8 – approximating areas using the trapezoidal rule. Integration: the reverse process 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. Integration: the reverse process 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. 11.3
Kinematics and differential equations 5.13 – Kinematic problems. 5.14 – Differential equations, separation of variables.
Slope fields, Euler’s method 5.15 – Slope fields and their diagrams. 5.16 – Euler’s method.
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?
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?
11.1
11.2
TOK: Is it possible for an area of knowledge to describe the world without transforming it?
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? 11.5 TOK: How have notable individuals such as Euler shaped the development
Vector quantities
Motion with variable velocity 3.12 - Vector applications to kinematics.
Modelling linear motion with constant velocity in two and three dimensions.
of mathematics as an area of knowledge? 12.1 12.2
December
January 2023
Exact solutions of coupled differential equations
Approximate solutions to coupled linear equations
Winter Holiday
Presentation of data
Bivariate data 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
Collecting and organizing univariate data
Sampling techniques 4.1 – Concepts of population, sample, random sample, discrete and continuous data. Reliability and bias. Outliers. Sampling techniques. 2.3, 2.4
Measuring correlation
The line of best fit
Interpreting the regression line 4.4 – correlation coefficient 4.4 – scatter diagrams, line of best fit, equation of regression line. 4.5
5.17 – Phase portrait for solutions of differential equations. 5.18 – Solutions of second derivative functions using Euler’s method. 12.3
12.4
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 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?
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?
February
Theoretical and experimental probability 4.5 – probability of events, complementary events, expected number of outcomes.
Representing combined probabilities with diagrams and formulae 4.6 – Venn diagrams, tree diagrams, sample space diagrams, combined events, conditional probability, independent events. 5.2 – 5.4
Complete, concise and consistent representations
Random variables and probability distributions 4.7 – concept of discrete random variables, expected value, E(X), applications. 4.14 – linear transformation of random variables, unbiased estimators. 13.1 – 13.6
Binomial distribution, Poisson distribution. 4.8 – calculations, mean and variance. 4.17 – Poisson distribution, its mean and variance. 4.18 – Critical values and critical regions. 14.2, 14.3
Normal distribution 4.9 – normal and inverse normal calculations. 4.15 – linear combinations of normal random variables, central limit theorem. 4.16 – Confidence intervals.
Spearmans’ rank correlation coefficient 4.10 – appropriateness, limitations and effect of outliers. 5.1 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?
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
April
May �+ test for independence 4.11 – contingency tables, degrees of freedom, critical value. 14.1 – 14.6
�+ goodness of fit test
The �-test
Spring Holiday
Revision Revision Revision Revision
Exams 4.11 4.12 – Reliability and validity. 4.11 – use of p-value, one-tailed and two-tailed tests.
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
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
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)
2 Year Course Outline
See detailed plan on next page for order of content and approximate dates
Year 1 Topic 1: Number and Algebra (16 hours) Topic 2: Functions (31 hours) 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)
Detailed Plan for year 1 (grade 11)
Week commencing: 2021
Aug 30
Topic
GDC Basics involving the use of equations solver, graphing, simulatanous equation function with graphing
Measurements and estimates
Recording measurements, significant digits and rounding
Syllabus reference
1.6 - Approximation: decimal places, significant figures, upper and lower bound of rounded numbers, percentage errors, estimation.
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 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: " #$%& =
) #$%* The cosine rule: c+ = a+ +b+ −2abcosC cosC= a+ +b+ −c+ 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
Textbook Reference
1.1 1.2 1.3
1.4
1.6 2.1 2.2
2.3 TOK: What is an axiomatic system? Are axioms self
TOK links, IM links
TOK: Is mathematical reasoning different from scientific reasoning, or reasoning in other areas of knowledge?
TOK: Do the names that we give things impact how we understand them? Is Mathematics invented or discovered?
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?
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 �, and �+ 3.1 - The distance between two points in three-dimensional space, and their midpoint 2.1 - Parallel lines �, =�+. Perpendicular lines �, ×�+ =−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 �(�)=��+� 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
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: Descartes showed that geometric problems could be solved algebraically and vice-versa. What does this tell us about mathematical representation and mathematical knowledge?
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.
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 �(�)=��+� 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
function algebraically? What are the advantages and disadvantages of having different forms and symbolic language in mathematics? 5.1
5.2
6.1
TOK: Correlation and causation–can we have knowledge of cause and effect relationships given that we can only observe correlation? What TOK: Do you think mathematics or logic should be classified as a language?
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?
The line of best fit Interpreting the regression line
. Collecting and organizing univariate data
Sampling techniques
Presentation of data
Bivariate data 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. 3.3, 3.4
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
factors affect the reliability and validity of mathematical models in describing real-life phenomena? 6.2, 6.3
3.1, 3.2
Quadratic Models 2.5 – quadratic models �(�)=��+ +��+ �;� ≠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 9.1
Problems involving quadratics 2.6 – develop and fit a model; find parameters of the model, test and reflect on the model; use the model.
Cubic models, power functions and direct and inverse variation 2.5 – cubic models �(�)=��. +��+ + ��+�; direct/inverse variation �(�)= ��/ ,�∈ℤ 9.2
9.3
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? 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?
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 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).
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?
Summer Work (Probability)
7.5
Probability and Distributions
7.6 4.8 Binomial Distribution Mean and variance of the binomial distribution
Syllabus reference
• 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.
TOK links, IM links
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
Test and Review
9.3
11.1 11.2 11.3
8.1
Statistical Testing
8.2 8.3
8.4 Assessment: Cumulative Test on all topics covered
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
MOCK EXAMS
Winter Break
was known to the Chinese mathematician Yang Hui much earlier than Pascal.
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 mathematics that makes them effective? Is simplicity a desirable characteristic in models? 3. 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?
Series and Sequences
Financial Mathematics
5.3
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. 10.1 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 10.2 1.4 Financial applications of geometric sequences and series: •compound interest •annual depreciation.
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
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. 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
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.
Differential Calculus Assessment: Cumulative Test
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)
Test and Review
13.1 13.2 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.
TOK
Is it possible for an area of knowledge to describe the world without transforming it?
Integral Calculus
Optimisation problems in context. Assessment: Cumulative Test
Spring Break
Revision for May Exams Mock Exam Practice
