Title: Sequences and Series
Start date: 30th August 2021
Duration: 3 weeks
Chapter 5
(Haese Textbook – Core Topics SL)
Prior Knowledge:
• Solve linear equations.
• Perform operations with fractions.
• Substitute into a formula.
Essential understanding: Number and algebra allow us to represent patterns,
show equivalencies and make generalizations which enable us to model realworld situations. Algebra is an abstraction of numerical concepts and employs
variables which allow us to solve mathematical problems.
Aim: The aim of the SL content of the number and algebra topic is to
introduce students to numerical concepts and techniques which, combined
with an introduction to arithmetic and geometric sequences and series, can be
used for financial and other applications. Students will also be introduced to
the formal concept of proof.
Objectives:
•
Find the general term of a given sequence.
•
Define an arithmetic and geometric sequence.
•
Calculate a term or the common difference of an arithmetic sequence.
•
Calculate a term or the common ratio of a geometric sequence.
•
Identify when a sequence is diverging or converging.
•
Use sigma notation for a sum of arithmetic & geometric sequences.
•
Use formulas to calculate the sum of a finite and infinite series.
•
Calculate compound interest, annual depreciation and other applications
of a sequence (e.g. growth & decay).
•
Work with problems inflation and real value of an investment.
•
Use technology for financial models.
•
Create models of real-life applications of sequences and series and
analyse how close it is to real life.
Other contexts: Loans.
Links to other subjects:
Radioactive decay, nuclear physics,
charging and discharging capacitors
(physics). Loans and repayments
(economics and business
management).
Concepts: Patterns and Generalizations
ATL: Communication/Thinking
Links to the Syllabus/Kognity:
SL1.2
• Arithmetic sequences and series.
• Use of the formulae for the 𝑛th term and the sum of the first 𝑛 terms of
the sequence.
• Use of sigma notation for sums of arithmetic sequences.
• Applications.
• Analysis, interpretation and prediction where a model is not perfectly
arithmetic in real life.
SL1.3
• Geometric sequences and series and applications.
• 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 geometric sequences.
• Applications e.g. spread of disease, salary increase and decrease,
population growth.
SL1.4
• Financial applications of geometric sequences and series: compound
interest and annual depreciation.
SL1.8
• Sum of infinite convergent geometric sequences.
TOK:
Is all knowledge concerned with identification and use of patterns? Consider
Fibonacci numbers and connections with the golden ratio. 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. How have technological advances affected the nature and
practice of mathematics? Consider the use of financial packages for instance.
International-mindedness
The chess legend (Sissa ibn Dahir); 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).
Title: Geometry and Trigonometry
Chapter 6, 7, 8 & 9
(Haese Textbook – Core Topics SL)
Start date: 26th September 2021
Duration: 3 weeks
Concepts: Space & relationships
ATL:
Prior Knowledge:
Links to the Syllabus/Kognity:
Use the Pythagoras’ Theorem
Convert units of length, area and volume
Essential understanding: Geometry and trigonometry allows us to quantify the
physical world, enhancing our spatial awareness in two and three dimensions. This
topic provides us with the tools for analysis, measurement and transformation of
quantities, movements and relationships.
Aim: The aim of the SL content of the geometry and trigonometry topic is to
introduce students to geometry in three dimensions and to non-right-angled
trigonometry.
SL3.1*
• The distance between two points in three- dimensional
space, and their midpoint.
• Volume and surface area of three-dimensional solids
including right-pyramid, right cone, sphere, hemisphere
and combinations of these solids.
• The size of an angle between two intersecting lines or
between a line and a plane.
•
•
Objectives:
•
•
•
•
•
•
•
•
•
•
•
•
Apply the distance and mid-point formulas
Find the mid-point given coordinates (2D and 3D)
Calculate the volume and surface area of pyramids, cones, spheres, and 3D
objects made from grouping these.
Calculate missing sides and angles of a right angled triangle using trigonometry.
Calculate and angle between a line and a plane.
Calculate an angle between an edge/face and the base.
Solve real life problems involving right –angled triangles.
Calculate the angles of elevation and depression.
Apply the area of a triangle formula.
Calculate missing sides and angles using the Sine and Cosine rule.
Apply the ambiguous case when calculating angles using the sine rule.
Apply trigonometry to real life situations.
Other contexts:
Architecture and design.
Triangulation, map-making.
SL3.2*
•
•
•
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.
•
SL3.3*
•
•
Applications of right and non-right angled trigonometry,
including Pythagoras’ theorem.
Angles of elevation and depression.
Construction of labelled diagrams from written
statements.
Links to other subjects:
TOK:
Design technology; volumes of stars and inverse What is an axiomatic system? Are axioms self-evident to
square law (physics).
everybody? 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? If the angles
of a triangle can add up to less than 180°, 180° or more than
180°, what does this tell us about the nature of mathematical
knowledge?
International-mindedness
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.
Title: Introducing functions
Start date:
Duration:
Chapter 3
(Haese Textbook –AA SL)
Prior Knowledge: Plot and read coordinates.
Essential understanding: Models are depictions of real-life
events using expressions, equations or graphs while a function is
defined as a relation or expression involving one or more variables.
Creating different representations of functions to model the
relationships between variables, visually and symbolically as
graphs, equations and tables represents different ways to
communicate mathematical ideas.
Aim: The aim of the SL content in the functions topic is to
introduce students to the important unifying theme of a function in
mathematics and to apply functional methods to a variety of
mathematical situations.
Objectives:
Define a function
Create and interpret a mapping diagram
•
Apply the vertical line rule
•
Express functions using different functional notations.
•
Draw and sketch a graph of a function
•
Define the domain and range
•
Specifies the domain and range
•
Identify a piecewise function and sketch it
•
Apply the correct notation for composite functions
•
Evaluate composite functions
•
Define and identify a self-inverse function
Other contexts: Temperature Links to other subjects:
and currency conversions.
Currency conversions and cost
functions (economics and
business management); projectile
motion (physics).
•
Concepts: Representation & relationships
ATL:
Links to the Syllabus/Kognity:
SL2.2*
• Concept of a function, domain, range and graph. Function notation, for
example (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−1(𝑥).
SL2.3*
• The graph of a function; its equation 𝑦 = (𝑥).
• 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.
SL2.5
• Composite functions.
• Identity function. Finding the inverse function 𝑓−1(𝑥).
• Define and identify a self-inverse function
•
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?
Do you think mathematics or logic should be classified as a language?
International-mindedness: The development of functions by Rene Descartes (France), Gottfried Wilhelm Leibnitz (Germany) and Leonhard Euler
(Switzerland); the notation for functions was developed by a number of different mathematicians in the 17th and 18th centuries–how did the notation
we use today become internationally accepted?
Title: Linear & Quadratic
Functions
Chapter 1
(Haese Textbook – Core Topics
SL)
Start date:
Duration:
Chapter 2
(Haese Textbook –AA SL)
Concepts: Modelling & relationships
ATL:
Prior Knowledge:
Links to the Syllabus/Kognity:
Solve simple equations and factorise expressions
Essential understanding: Models are depictions of reallife events using expressions, equations or graphs while a
function is defined as a relation or expression involving one
or more variables. Creating different representations of
functions to model the relationships between variables,
visually and symbolically as graphs, equations and tables
represents different ways to communicate mathematical
ideas.
Aim: The aim of the SL content in the functions topic is to
introduce students to the important unifying theme of a
function in mathematics and to apply functional methods to a
variety of mathematical situations.
SL2.1*
• 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.
Objectives:
•
•
•
•
•
•
•
•
•
Define the domain and range of a function
Calculate the inverse and composite functions
Understand the features of a parabola: symmetry, vertex,
intercepts, equations of axis of symmetry.
Convert between different forms of a quadratic function:
general form, vertex and intercept form.
Factorise and solve a quadratic, finding the roots
Complete the square and find the vertex
Calculate and interpret the discriminant
Apply stretches, translations and reflections to a function
and sketch it.
Describe a transformation from one function to another.
Other contexts: Gradients
of mountain roads,
gradients of access ramps.
Temperature and currency
conversions.
SL2.4*
•
•
Determine key features of graphs.
Finding the point of intersection of two curves or lines using technology.
SL2.6
•
•
The quadratic function (𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐: its graph, 𝑦-intercept 0, 𝑐. Axis of
symmetry.
The form (𝑥) = (𝑥 − 𝑝)(𝑥 − 𝑞), 𝑥 intercepts (𝑝, 0) and (𝑞, 0). The form (𝑥) = (𝑥 − ℎ)2 +
𝑘, vertex (ℎ, 𝑘).
SL2.7
•
•
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.
SL2.10
•
•
•
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.
SL2.11
• Transformations of graphs. Translations: 𝑦 = (𝑥) + 𝑏; 𝑦 = (𝑥) − 𝑎.
• Reflections (in both axes): 𝑦 = −(𝑥); 𝑦 = 𝑓(−𝑥).
• Vertical stretch with scale factor 𝑝: 𝑦 = (𝑥).
• Horizontal stretch with scale factor 1/𝑞: 𝑦 = (𝑞𝑥).
• Composite transformations.
TOK: Descartes showed that geometric problems could be solved algebraically and vice
versa. What does this tell us about mathematical representation and mathematical
knowledge? Do you think mathematics or logic should be classified as a language? Are
there fundamental differences between mathematics and other areas of knowledge? If
so, are these differences more than just methodological differences?
Links to other subjects:
Exchange rates and price and
income elasticity, demand and
supply curves (economics);
graphical analysis in
experimental work (sciences
group subjects).
International-mindedness: Bourbaki group analytical approach versus the Mandlebrot visual approach.
Title: Rational Functions
Start date:
Duration:
Chapter: 3D
(Haese Textbook –AA SL)
Concepts: Representations and Equivalence
ATL:
Prior Knowledge:
Links to the Syllabus/Kognity:
Solve simple equations
Sketch horizontal and vertical lines (e.g. 𝑥 = 2, 𝑦 = −3).
Essential understanding: Models are depictions of real-life
events using expressions, equations or graphs while a function is
defined as a relation or expression involving one or more variables.
Creating different representations of functions to model the
relationships between variables, visually and symbolically as
graphs, equations and tables represents different ways to
communicate mathematical ideas.
Aim: The aim of the SL content in the functions topic is to
introduce students to the important unifying theme of a function in
mathematics and to apply functional methods to a variety of
mathematical situations.
SL2.8
• The reciprocal function (𝑥) =1/𝑥, 𝑥 ≠ 0: its graph and self-inverse nature.
• Rational functions and their graphs.
• Equations of vertical and horizontal asymptotes.
•
•
Objectives:
Identify the domain and range of a rational function
Describe the features of a reciprocal and rational function
including the symmetry, intercepts, horizontal and vertical
asymptotes.
•
Create a model with reciprocal and rational functions.
Other contexts:
Links to other subjects:
TOK:
Radioactive decay and population Sketching and interpreting
Does studying the graph of a function contain the same level of mathematical rigour
growth and decay, compound graphs and identification and
as studying the function algebraically? What are the advantages and
interest, projectile motion, braking interpretation of key features
disadvantages of having different forms and symbolic language in mathematics?
distances.
of graphs (sciences group
What are the implications of accepting that mathematical knowledge changes over
subjects, geography,
time? What assumptions do mathematicians make when they apply mathematics
economics).
to real-life situations?
International-mindedness
Bourbaki group analytical approach versus the Mandlebrot visual approach.
The development of functions, Rene Descartes (France), Gottfried Wilhelm Leibniz (Germany) and Leonhard Euler (Switzerland).
•
•
Title: Exponents & Logarithms
Start date:
Duration:
Chapter: 3 (Core Textbook),
5 & 6 (AA SL Textbook)
Prior Knowledge:
•
•
•
Concepts: Representations
ATL:
Links to the Syllabus/Kognity:
Basic laws of exponents
Basic algebraic manipulation
Solving linear and quadratic equations
SL1.5
• Laws of exponents with integer exponents.
Essential understanding: Number and algebra allow us to represent patterns, • Introduction to logarithms with base 10 and e.
show equivalencies and make generalizations which enable us to model real- • Numerical evaluation of logarithms using technology.
world situations. Algebra is an abstraction of numerical concepts and employs
variables which allow us to solve mathematical problems.
SL1.7
• Laws of exponents with rational exponents.
Aim: The aim of the SL content of the number and algebra topic is to introduce
• Laws of logarithms.
students to numerical concepts and techniques which, combined with an
• Change of base of a logarithm.
introduction to arithmetic and geometric sequences and series, can be used for
• Solving exponential equations, including using logarithms.
financial and other applications. Students will also be introduced to the formal
concept of proof.
Objectives:
Apply basic laws of indices to simplify algebraic expressions.
Apply laws of indices involving fractional and negative exponents.
•
Solve exponential equations not requiring logs (common base).
•
Properties of exponential functions (including transformations).
•
Calculations involving ‘e’.
•
Properties of log functions.
•
Understanding the concept of logs.
•
Applying laws of logs.
•
Working with natural logs.
•
Solving log equations
Other contexts: Richter scale
Links to other subjects:
and decibel scale.
Calculation of pH and buffer solutions
(chemistry) and finding activation energy
from experimental data (chemistry).
Radioactive decay, charging and
discharging capacitors (physics); first order
reactions and activation energy (chemistry);
growth curves (biology).
International-mindedness
Do all societies view investment and interest in the same way?
•
•
SL2.9
• Exponential functions and their graphs
f(x)=ax, a>0, f(x)=ex
• Logarithmic functions and their graphs:
f(x)=logax, x>0, f(x)=lnx, x>0.
TOK:
Is mathematics invented or discovered? For instance, consider the
number e or logarithms–did they already exist before man defined
them? How have seminal advances, such as the development of
logarithms, changed the way in which mathematicians understand
the world and the nature of mathematics? What role do “models”
play in mathematics? Do they play a different role in mathematics
compared to their role in other areas of knowledge?
Title: Trig Functions
Start date:
Duration:
Chapter: 7 (AA SL Textbook)
Prior Knowledge:
•
•
Concepts: Equivalence & Space
ATL: Communication
Links to the Syllabus/Kognity:
Use the Pythagoras’ Theorem and right-angled trigonometry ratios.
Transformation of basic functions.
SL3.4
• The circle: radian measure of angles; length of an arc; area of a
sector.
Essential understanding: Geometry and trigonometry allows us to
quantify the physical world, enhancing our spatial awareness in two and
three dimensions. This topic provides us with the tools for analysis,
measurement and transformation of quantities, movements and
relationships.
Aim: Students will explore the circular functions and use properties and
identities to solve problems in abstract and real-life contexts.
Objectives:
•
•
•
Converting between radians and degrees.
Calculate arc length and sector area.
Solve problems involving arc length and sector area.
Other contexts:
Architecture and design.
Triangulation, map-making,
music waves.
Links to other subjects:
Diffraction patterns and circular motion
(physics). Simple harmonic motion
(physics).
TOK:
Which is a better measure of angle: radian or degree? What criteria
can/do/should mathematicians use to make such decisions?
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?
International-mindedness
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. The first work to refer explicitly to the sine as a function of an angle is the Aryabhatiya of Aryabhata (ca 510).