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Runs for 2 years and 4 semesters
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Firat Vaux Lobut
Use of Technology
Number systems and symbols/notations
Finding the general term (nth term)
Simultaneous Equations (including non-linear)
Simplifying Expressions
Parallel and Perpendicular Lines
Solving Equations (including quadratic)
(SL Guide, 2014)
Much of the topic should be covered in pre-DP courses. Fluency is essential in algebraic manipulation for Maths SL. Extra work might be helpful for some students to improve their skills. If possible, it is a good idea to give students a booklet to complete during the summer break before Maths SL begins.
Practices on linear, quadratic and simultaneous equations are useful to gain technology skills. Focus is on graphs of these equations, NOT algebra.
None
None
None
Quadratic Functions and Equations
Functions
Families of Functions and Transformations of Graphs
(SL Guide, 2014)
Students are to factorise, complete the square, use formula and graphical solutions with quadratic equations. They use the discriminant and present graphically. They sketch graphs and retrieve the equation from the graph. Students are to model and optimise quadratic functions.
Students are to learn the concept of function, domain, range, composite function, inverse function, one-to-one and one-to-many functions, and domain restriction.
Students are to understand the general shape and features of Linear, Quadratic and Cubic functions. They do transformations (reflection, translation and stretch).
The discriminant and modeling should be the main aim. Graphing functions and transforming will be visited later in the course. Geogebra (Graphing/mathematics software) can be used to show trasformation and other properties.
Rene Descartes (France), Gottfried Wilhelm Leibniz (Germany) and Leonhard Euler (Switzerland) (SL Guide, 2014)
(Mastin, 2010)
Inquirers: Students work on a mini project which is related to real life. The aim must be displaying results as a graphic of a function with Geogebra (or another graphic software).
Communicators: Students are to sketch or draw mathematical functions and equations. They graph both on paper and using technology and record methods, solutions and conclusions using standardized mathematical notation. Students are to interpret and describe transformations of graphs.
Thinker: Students are to apply their function and equation skills to solve unfamiliar and complex problems. (SL Guide, 2014)
Is zero the same as nothing?
Is mathematics formal language?
How accurate is a visual representation of a mathematical concept? (SL Guide, 2014)
Chemistry (sketching and interpreting graphs, equilibrium law)
Geographic skills
Economics (shifting of supply and demand curves)
Physics (kinematics, simple harmonic motion )
The circle: radian measure of angles; length of an arc; area of a sector
Solution of triangles
Cos, sin and tan and the unit circle
Trig identities tan x = sin x / cos x and sin^2x + cos^2x = 1
Definition of cosθ and sinθ in terms of the
unit circle
(SL Guide, 2014)
Students are to work on the sine rule, cosine rule and area of a triangle.
They are to solve equations with trig identities. The connection between Pythagoras' theorem and the cosine rule needs to be explained.
Students are to solve questions with triangles and to work on real life applications.
They are to understand the relationship between trig ratios. For example, given sinx, find possible values of tanx without finding "x".
Students are to work with radian, find length of an arc and area of a sector.
(SL Guide, 2014)
(Mastin, 2010)
Knowledgeable: Students have a chance to understand and explain how things work around us by using trigonometry. (GoogleMap, Engineering, Construct buildings)
Thinker: Students are to apply their trigonometry skills to solve unfamiliar and complex problems.
(SL Guide, 2014)
Which is a better measure of angle: radian or degree? What are the “best” criteria by which to decide?
Euclid’s axioms as the building blocks of Euclidean geometry. Link to non-Euclidean geometry.
Trigonometry was developed by successive civilizations and cultures. How is mathematical knowledge considered from a sociocultural perspective? (SL Guide, 2014)
Chemistry (sketching and interpreting graphs, equilibrium law)
Geographic skills
Physics (simple harmonic motion )
Sequences and Series
Binomial Theorem and Counting Principles
Elementary treatment of exponents and
logarithms
(SL Guide, 2014)
Students are to review the laws of exponents. They are to learn definitions of logs, change of base and laws of logs. They solve a^x=b using graphical and log methods.
Students are to learn arithmetic sequences and series, finding the nth term, convergent and divergent sequences as well as periodic and oscillating sequences, sum of finite arithmetic series, sigma notation, geometric sequences and series, sum of finite and infinite geometric series. They work on applications.
Students are to investigate Pascal's triangle and patterns. They learn about factorial notations and understand the relationship between nCr notation and Pascal's triangle. They do the expansion of (a+b)^n up to the 7th degree and see the pattern for coefficients.
“father of algebra”. Compare with al-Khawarizmi.
(SL Guide, 2014)
(Mastin, 2010)
Knowledgeable: Students are to understand the impact of algebra in computing and how it changes our lives.
Thinker: Students are to apply their algebraic skills to solve unfamiliar and complex problems.
(SL Guide, 2014)
How did Gauss add up integers from 1 to 100? Discuss the idea of mathematical intuition as the basis for formal proof.
Debate over the validity of the notion of “infinity”: finitists such as L. Kronecker consider that “a mathematical object does not exist unless it can be constructed from natural
numbers in a finite number of steps”.
What is Zeno’s dichotomy paradox? How far can mathematical facts be from intuition?
Are logarithms an invention or discovery?
Chemistry (Calculation of pH)
Physics (Motion, Gravity)
(SL Guide, 2014)
Introduction to Statistics
Central tendency - Applications
Cumulative frequency
(SL Guide, 2014)
Students are to cover the following contents: population, sample, random sample, discrete and continuous data. They use frequency distributions (tables), frequency histograms with equal class intervals, box-and-whisker plots to present the data.
Students are to find mean-mode-median-range-quartiles. They use technology to interperet and display data.
Students are to do cumulative frequency.
(Mastin, 2010)
(SL Guide, 2014)
Inquirers: The topic offers a good chance for students to work on a mini project which is related to real life. They collect their own data and conduct inquiry. Students work independently.
Communicators: Students are to write conclusions about their outcomes. They mention assumptions, limitations and strengths of their findings. They are required to give evidence for their outcomes with mathematical facts.
Thinker: Students are to understand that a problem can have more than one answer.
(SL Guide, 2014)
Do different measures of central tendency express different properties of the data? Are these measures invented or discovered? Could mathematics make alternative, equally true, formulae? What does this tell us about mathematical truths?
How easy is it to lie with statistics?
Psychology (descriptive statistics, random sample)
Biology (calculating mean and standard deviation; comparing means and spreads between two or more samples; correlation does not imply causation)
Statistical calculations to show patterns and changes; geographic skills; statistical graphs
Chemistry (Curves of best fit)
Geography (Geographic skills).
(SL Guide, 2014)
Introduction to the Exploration. Explain assessment criteria. Get students to start thinking about a possible exploration topic.
Students are to decide on their exploration titles at the end of the first semester.
Formative: Quiz and Mock Exam
Summative: Mid Semester Exam and End of Semester Exam
(Extended essay guide, 2007)
("EE Timeline", 2012)
(According to West Sound Academy Extended Essay Timeline, 2018)
Concepts of trial, outcome, equally likely
outcomes, sample space (U) and event
Combined events, P(A U B), intersection
Students are to review trial, outcome, equally likely outcomes, sample space (U) and event. They are to work on complementary events, use Venn diagrams, tree diagrams and two-ways tables.
Students are to find answers for combined events, mutually exclusive events, conditional probability, independent events and probabilities with and without replacement.
Students are to make correlations between mathematical notation and English expressions, such as "and", "or", "none", "neither".
Pierre de Fermat (France), Blaise Pascal (France), Jacob Bernoulli (Switzerland), Abraham de Moivre(France)
(Mastin, 2010)
Inquirers: Students might do an experiment to find differences between theoretical and experimental probability.
Communicators: Students are to gain an understanding of probability notations and their corresponding meanings in English.
Thinker: Students are to apply their probability skills to solve unfamiliar and complex problems.
To what extent does mathematics offer models of real life? Is there always a function to model data behaviour?
Is mathematics useful to measure risks?
Can gambling be considered as an application of mathematics?
Biology (Genetics)
Economy
("Probability - GeneEd - Genetics, Education, Discovery", 2017)
(Myerson, 2005)
Exponential functions and their graphs
Applications of graphing skills and solving
equations that relate to real-life situations
The reciprocal function
Rational functions and their graphs
(SL Guide, 2014)
Students are to work on logarithmic functions and their graphs.
Students are to develop a good understanding of real life applications of functions (NOT only logarithmic).
Geometric series.
Students are to graph rational and reciprocal functions.
Students are to use technology for real life applications' graphs and interpret them.
(SL Guide, 2014)
(SL Guide, 2014)
Inquirers: Students work on a mini project which is related to real life. The aim must be displaying results as a graphic of a function with Geogebra.
Communicators: Students are to sketch or draw mathematical functions and equations. They graph both on paper and using technology and record methods, solutions and conclusions using standardised mathematical notation.
Thinker: Students are to apply their function skills to solve unfamiliar and complex problems.
(SL Guide, 2014)
Is zero the same as nothing?
Is mathematics formal language?
How accurate is a visual representation of a mathematical concept?
Economy (Compound interest, growth and decay)
Physics (Projectile motion; braking distance; electrical
circuits, radioactive decay)
Biology (Half-life)
Radian measure
Cosx, sinx, tanx and unit circle
Trigonometric equations
Double angle formula
Sinx, cosx, tanx and their graphs, composite functions of sinx and cosx
Modelling with trig functions
(SL Guide, 2014)
Students are to understand the relationship between radian and degree.
They can show cosx, sinx and tanx on the unit circle.
Equations leading to quadratic equations in
sin x, cos x or tan x .
Students become confident using double angle formulas.
They are to use technology to show the circular functions sinx, cosx and tanx, their domains and ranges, amplitude, their periodic nature, and their graphs.
They work with composite functions of the form
f (x) = asin (b(x + c) ) + d, its transformation and applications.
(SL Guide, 2014)
Knowledgeable: Students have a chance to understand and explain how things work around us by using trigonometry. (Soundwaves, wave, change of temperature)
Thinker: Students are to apply their trigonometry skills to solve unfamiliar and complex problems.
Communicators: Students are to graph their findings by using technology and interpret the important points.
(SL Guide, 2014)
Which is a better measure of angle: radian or degree? What are the “best” criteria by which to decide?
Euclid’s axioms as the building blocks of Euclidean geometry. Link to non-Euclidean geometry.
Trigonometry was developed by successive civilizations and cultures. How is mathematical knowledge considered from a sociocultural perspective? (SL Guide, 2014)
Chemistry (sketching and interpreting graphs, equilibrium law)
Geographic skills
Physics (simple harmonic motion, motion of sound )
Linear correlation of brivate data
Correlation coefficient r
Scatter diagrams
Line of best fit
Equation of the regression line of y on x
Use of the equation for prediction purposes
(SL Guide, 2014)
Students are to work with bivariate analysis and scatter diagrams. They reflect their findings with the line of best fit.
They predict equations' outcomes by using different methods, for exmaple, least squares regression and measuring correlation.
Students are to understand and work with correlation coefficient r. They draw scatter diagrams and the line of best fit.
(Mastin, 2010)
(SL Guide, 2014)
Inquirers: The topic offers a chance for students to work on a mini project which is related to real life. They collect their own data and conduct inquiry. Students work independently.
Communicators: Students are to write conclusions about their outcomes. They mention assumptions, limitations and strengths of their findings. They are required to give evidence for their outcomes with mathematical facts.
Thinker: Students are to understand that a problem can have more than one answer.
(SL Guide, 2014)
Can we predict the value of x from y, using this equation?
Can all data be modelled by a (known) mathematical function? Consider the reliability
and validity of mathematical models in describing real-life phenomena.
Psychology (descriptive statistics, random sample)
Biology (calculating mean and standard deviation; comparing means and spreads between two or more samples; correlation does not imply causation)
Statistical calculations to show patterns and changes; geographic skills; statistical graphs
Chemistry (Curves of best fit)
Geography (Geographic skills).
(SL Guide, 2014)
Students are to work on exploration in class. Their progress is monitored.
They are requiered to complete their draft.
Formative: Quiz
Summative: Written assignment
Formative: Quiz and Mock Exam
Summative: End of Semester Exam
(Extended essay guide, 2007)
("EE Timeline", 2012)
(According to West Sound Academy Extended Essay Timeline, 2018)
Limits
Differentiation from first principles
Derivative interpreted as gradient function and as rate of change
Tangents and normals, and their equations
Derivatives of polynomial functions
Differentiation of a sum
The chain rule for composite functions
The product and quotient rules
The second derivative
Local maximum and minimum points
Points of inflexion with zero and non-zero
gradients
Optimization
(SL Guide, 2014)
Students are to work with limit and understand limit notation. They learn the meaning and graphics of first principles. Students find the derivative by using first principles.
They use the derivative rules and find the derivative of x^n, sinx, cosx, tanx, e^x and ln x.
They use derivative to find equations of tangents and normals.
Students are to interpret derivative as gradient and rate of change.
They derivate a sum of functions.
They use the chain rule, product and quotient rules.
Second derivative.
They find local maximum and minimum points and test their results.
Students are to work on applications of derivative. Optimization.
They are to show the relationship between behaviour of functions and thier graphs. f, f' and f''.
Leibnitz discovered certain calculus concepts
(SL Guide, 2014)
(Mastin, 2010)
Inquirers: Students are to understand the meaning of differentiation by working on a real situation. For example, they can find the optimum volume of a soft drink can, cost of building a house, area of a farm, possible lifetime of a cell etc.
Knowledgeable: Students are to explore the use of derivative in our lives. For example, the stock exchange, the population growth, how fast someone must run to break the 100m World record etc.
Thinker: Students are to apply their differentiation skills to solve unfamiliar and complex problems.
(SL Guide, 2014)
What value does the knowledge of
limits have? Is infinitesimal behaviour
applicable to real life?
Opportunities for discussing hypothesis
formation and testing, and then the formal proof can be tackled by comparing certain cases, through an investigative approach.
Economics (marginal cost, marginal revenue, marginal profit)
Chemistry (interpreting the gradient of a curve, particle motion)
Phsyics (velocity, acceleration, planetary motion)
Profit, area and volume
(SL Guide, 2014)
Vectors as displacements
Components of vectors, base vectors I, j, k
Algebraic and geometric approaches to vector properties
The scalar product of two vectors
Vector equation of a line
Determining whether two lines intersect
Distinguishing between coincident and parallel lines
Applications
(SL Guide, 2014)
Students are introduced to the components of a vector and work with 2 and 3 dimension vectors. They understand the vectors as displacements in space.
They solve the sum and difference of two vectors, zero vector and the vector -v. They multiply a vector by its scalar.
Students find the magnitude of a vector. They discuss the meaning of magnitude. They develop an understanding of base vectors, i, j and k and position vector.
They are to work with questions on scalar product of two vectors, perpendicular and parallel vectors and find an angle between two vectors.
They are to develop a strong understanding of vector equation of a line.
Students can show the difference between coincident and parallel lines. They can find the intersection of two vectors or determine if two vectors intersect.
(SL Guide, 2014)
JW Gibbs or O Heaviside?
(SL Guide, 2014)
("Vectors", 2017)
Communicators: Students are to sketch vectors. They understand that they can display mathematical results with vectors as well. For example, distance between two points, parallel and perpendicular lines, intersection of two lines.
Thinker: Students are to apply their vector skills to solve unfamiliar and complex problems.
How do we relate a theory to the author?
Are algebra and geometry two separate domains of knowledge?
Physics (vector sums and differences, vector resultants)
(SL Guide, 2014)
Indefinite integration – anti-differentiation
Indefinite integral of of x^n, e^x, sinx, cosx and 1/x
Composites
Definite integrals and boundary conditions
Areas under curves
Volumes of revolution
Students are to understand the relationship between differentiation and integration.
They are to solve questions with indefinite integrations, find c. They work with indefinite integral of of x^n, e^x, sinx, cosx and 1/x.
Students are to integrate by inspection.
They do integration with boundary conditions. They find areas under curves or between curves.
They find the volume of an object or a shape after the revolution around x-axis.
Students are to do definite integrals using technology.
(SL Guide, 2014)
the pyramidal frustum by ancient Egyptians
(Egyptian Moscow papyrus)
cylinder by Chinese mathematician Liu Hui
calculate the integral of a function, in order to
find the volume of a paraboloid.
(SL Guide, 2014)
Inquirers: Students are to understand the meaning of integration by working on a real situation. For example, they can find the exact volume of a soft drink can or the speed of a spaceship.
Knowledgeable: Students are to explore the use of integration in life. For example, volume of production in factories, electric charges and universal gravitation.
Thinker: Students are to apply their integration skills to solve unfamiliar and complex problems.
(SL Guide, 2014)
Mathematics and knowledge claims. Euler was able to make important advances in mathematical analysis before calculus had been put on a solid theoretical foundation by Cauchy and others. However, some work was not possible until after Cauchy’s work. What does this tell us about the importance of proof and the nature of mathematics?
Mathematics and the real world. 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 physical reality?
(Eastaugh & Sternal-Johnson, 2013)
Other Disciplines
Physics (velocity, acceleration, universal gravitation)
(SL Guide, 2014)
Students are to complete the exploration by the end of semester 3.
Formative: Quiz and Mock Exam
Summative: Mid Semester and End of Semester Exam
(Extended essay guide, 2007)
("EE Timeline", 2012)
(According to West Sound Academy Extended Essay Timeline, 2018)
Derivatives of trig functions
Integrals of sinx and cosx
Kinematics
(SL Guide, 2014)
Students are to revisit and derivate and integrate trig functions in more detail.
They work with displacement s, velocity v and acceleration a to solve kinematic problems.
(SL Guide, 2014)
(SL Guide, 2014)
(Mastin, 2010)
Inquirers: Students are to understand the meaning of differentiation by working on a real situation. For example, they can find the distance of a particle after a certain time, speed of a car or acceleration of a spaceship.
Knowledgeable: Students are to explore the use of derivative and integral in life. For example, finding the necessary force to leave the earth with a spaceship or the distance required to stop while travelling at a certain speed.
Thinker: Students are to apply their skills to solve unfamiliar and complex problems.
(SL Guide, 2014)
Mathematics and the real world. 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 physical reality?
(Eastaugh & Sternal-Johnson, 2013)
Physics (kinematics)
Discrete random variables
Binomial distribution
Normal distributions and curves
Standardization of normal variables (z-values, z-scores)
(SL Guide, 2014)
Students are to work on discrete random variables and their probability distributions.
They are to do expected value, E(x) for discrete data. They work under real life application, games of chance, fair game etc.
Students are to understand the concept of binomial distribution.
They do mean, variance and standard deviation.
Continuous random variables and their probability distributions.
Students are to work with Normal and Standard Normal distributions and their curves.
(SL Guide, 2014)
(Mastin, 2010)
(SL Guide, 2014)
Inquirers: The topic offers a chance for students to work on a mini project which is related to real life. They collect their own data and conduct inquiry. Students work independently.
Thinker: Students are to apply their statistics skills to solve unfamiliar and complex problems.
(SL Guide, 2014)
How easy is it to lie with statistics?
Biology (links to normal distribution)
Psychology (descriptive statistics)
Mock Exams
External Exam - Exam Block
From 27 October to November 17
(According to IB Diploma Programme November 2017 examination schaedule)
(Extended essay guide, 2007)
("EE Timeline", 2012)
(According to West Sound Academy Extended Essay Timeline, 2018)
Semester 1 IB 1
Semester 2 IB 2
Semester 1 IB 1
Semester 2 IB 2
Algebra 9 hours
Functions and equations 24 hours
Circular functions and trigonometry 16 hours
Vectors 16 hours
Statistics and probability 35 hours
Calculus 40 hours
Exploration 10 hours
Total 150 hours
The items in the timeline are arranged in chronicle order.
Presumed Knowledge - Use of Technology
Functions and Equations 1
Trigonometry 1
Summative Exam 1
Algebra
Statistics 1
The Exlporation
Summative Exam 2
Extended Essay
The Formative and Summative Assessments
The items in the timeline are arranged in chronicle order.
Probability 1
Summative assessment 3 - Written assignment
Functions 2
Trigonometry 2
Statistics 2
The Exlporation
Summative assessment 4
Extended Essay
The Formative and Summative Assessments
The items in the timeline are arranged in chronicle order.
Calculus 1
Vectors
Summative Assessment 5
Calculus 2
The Exlporation
Summative Assessment 6
Extended Essay
The Formative and Summative Assessments
The items in the timeline are arranged in chronicle order.
Calculus 3
Statistics 3
Extended Essay
Revision
Mock Exams
External Exams - Block of Exam Time