Sep

Detailed Plan for year 1 (grade 11)

Week commencing: 2021

Aug 29 Number patterns and sigma notation

Arithmetic and Geometric sequences

Arithmetic and Geometric series

Applications 1.1 1.2

Topic Textbook Reference

Sep 5

Sep 12

Sep 19 Proof

Binomial Theorem 1.3

1.4

Syllabus reference TOK links

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 n th 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. 1.8

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. 3. How have technological advances affected the nature and practice of mathematics? Consider the use of financial packages for 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).

1.6 Simple deductive proof 1.15 Proof by induction, proof by contradiction, counterexample. 1.9 The binomial theorem: expansion of (a+b),n∈ℕ. Use of Pascal’s triangle and nCr. 1.10 Extension to fractional and negative indices. 1.11 Partial fractions.

TOK:

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

TOK:

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

Sep 26 Functional relationships Functions 2.1 2.2, 2.3 2.1 Equations of straight lines, gradient, intercepts, parallel and perpendicular lines. 2.2 Concept of a function, inverse function.

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

Oct 3 2.3 The graph of a function. 2.4 Key features of graphs. Intersection of 2 functions. 2.16 Graphs of �=|�(�)|.

Oct 10 Reciprocal functions 2.2, 2.3 2.8 Rational functions and their graphs, equations of asymptotes. 2.13 Rational functions with quadratic factors.

Oct 17 Mid term break

Oct 24 Exponential functions 2.3, 2.4 2.9 Exponential functions and their graphs, logarithmic functions. 2.10 Solving equations, graphically and analytically. 2.15 Solutions of �(�)≥�(�).

Oct 31 Transformation s 2.5 2.5 Composite functions, the identity function. 2.11 Transformation of functions – reflections, translations, dilations. 2.14 Odd and even functions.

Nov 7 Quadratic functions 3.1 2.6 The quadratic function. 2.7 Quadratic equations and inequalities.

Nov 14

Nov 21 Polynomial equations and inequalities 3.3

3.4 2.12 Polynomial functions, factor and remainder theorems. Sum and product of the roots of polynomial equations.

Fundamental theorem of algebra

Nov 28

Dec 5 Complex Numbers

National Day, Commemorati on Day Dec 1-3

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?

3.2 1.12 The complex plane

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?

3.5, 3.6 1.16 Solutions of systems of linear equations up to three unknowns

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

Linear equations and inequalities Dec 12 Winter Break Limits and derivatives 4.1, 4.2 5.1 Concept of a limit. Derivative as gradient function and rate of change. 5.12 Continuity, convergence and divergence.

Jan 2 2022

Jan 9 Increasing and decreasing functions

Differentiation 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.

Jan 16 More differentiation 4.3, 4.6 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.

Jan 23

Jan 30 The second derivative 4.4 5.7 Relationship between f, f’ and f’’. Function and Liebniz notation. Graphical behavior. 5.8 Local maxima and minima. Optimization. Points of inflexion.

Feb 6 Tangents and normals 4.5 5.4 Tangents and normal at a given point, and their equations.

Feb 13 Mid term break

Feb 20 Coordinate geometry 6.1 3.1 The distance between 2 points. The midpoint. Volume and surface area. Angles between lines and planes.

Feb 27 Trigonometry and applications 6.2 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.

these spaces exist beyond our sense perception?

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?

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?

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

March 6 Radians 6.3 3.4 Circles. Radian measure. Arc lengths and sector areas.

March 13 Trigonometric functions and identities 6.4 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.

March 20 Circular functions and equations 6.5 3.8 Solve trigonometric equations graphically and analytically.

March 27 Spring Break

April 10 Integration 7.1 5.5 Integration as anti-differentiation. Boundary condition to determine constant. Definite integrals.

April 17 Exponential and log functions 7.2, 7.3 5.15 Derivatives of log functions, power functions and inverse trig functions.

April 24

May 1 Integration 7.4

Definite integration 8.1 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.

May 8

Eid break May 12/13

Kinematics 8.2 5.9 Problems involving displacement, velocity and acceleration. Total distance travelled.

May 15 Differential Equations 8.3 5.18 First order differential equations. Euler’s method. Separation of variables.

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

May 22 Maclaurin Series 8.4 5.13 L’Hopital’s rule. 5.19 Obtain expansions for trigonometric, exponential and logarithmic functions.

May 29 Mocks June 5 Mocks June 12 Toolkit June 19 Toolkit June 26 Toolkit July 3 Summer break 12 12 12 Homogeneous differential equations.

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 commencing: 2022

September

Topic Textbook reference Syllabus reference

Geometrical representation of vectors Intro to vector algebra 9.1

9.2 3.12 Position vectors, displacement vectors, magnitude of a vector.

October Scalar product and its properties 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.

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?

November

December

Vector equation of a plane Lines, planes and angles

Applications of vectors Forms of a complex number 9.6

9.7

9.8 3.17 Vector equation of a plane.

3.18 Intersection of line and plane, 2 planes. Angle between a line and plane, 2 planes.

10.1 1.12 The complex plane

Operations with complex numbers in polar form Powers and roots of complex numbers Review

Mock Exams 10.2 1.13 Polar form and Euler form

10.3 1.14 Complex conjugate roots

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?

January 2021

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.

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?