Journeys School
Upper School Mathematics
Advanced Math (Math HLA)/Calculus (Math HLB)
Curriculum Overview – Series and Differential Equation Option
Introduction: This overview provides a general scope and sequence of the content covered in
the class. Variation may occur due to special projects and unique events.
Text: Mathematics for the International Student (HL Mathematics) by Haese and Harris
Year 1 (Advanced Math/Math HLA)
Quarter 1
1.
Algebra
a.
b.
c.
d.
e.
f.
g.
h.
Arithmetic and geometric sequences and series; sigma notation
Exponents and logarithms; laws of logarithms; change of base
Binomial Theorem; counting principles, including permutations and combinations
Proof by mathematical induction; forming conjectures to be proved by mathematical induction
Complex numbers; conjugate, modules and argument; Cartesian form
Sums, products and quotients of complex numbers
De Moivre’s theorem; powers and roots of a complex number
Conjugate roots of polynomial equations with real coefficients
Quarter 2
2.
Circular Functions and Trigonometry
a. Circles, radian measure, arc length, sector area
b. Define sin & cos terms of unit circle
c. Compound angle identities and double angle identities
d. Periodic nature of trig functions, their domains, ranges and graphs
e. Solutions to trig equations over finite interval; Use of trig identities and factorisation to transform
equations
f. Solution of triangles, cosine rule, sine rule, area of triangle
Quarter 3
3.
4.
Matrices
a. Definition – element, row, column, and order
b. Matrix algebra -- =, +, -, X scalar, X by another matrix, identity & zero matrices
c. 2x2 & 3x3 determinants, inverse of 2x2 matrix, conditions for inverse existence
d. Solution to systems of linear equations to a maximum of 3 equations in 3 unknowns; Conditions
for the existence of a unique solution, no solution and an infinity of solutions
Vectors
a. Vectors as displacements in 2 and 3 dimensions
b. Scalar product of two vectors – Algebraic properties of the scalar product; perpendicular vectors,
parallel vectors, and the angle between vectors
c. Line as r = a +  b, the angle between two lines
d. Distinguishing between coincident, parallel intersecting and skew lines, points of intersection
e. Vector product of two vectors, v x w; determinant representation; geometric interpretation of v x
w
f. Vector equation of a plane r = a + b + c; use of normal vector to obtain the form r  n = a  n;
Cartesian equation of a plane ax + by + cz = d
g. Intersections of: a line with a plane; two planes; three planes. Angle between: a line and a plane;
two planes
Quarter 4
5.
Statistics and Probability
a. Concept of population, sample, random sample and frequency distribution of discrete and
continuous data
b. Presentation of data: frequency tables and diagrams, box and whisker plots
c. Measures of central tendency: mean, median, ode; quartiles, percentiles; range; interquartile
range; variance; standard deviation
d.
e.
f.
g.
h.
i.
j.
k.
Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles.
Concepts of trial, outcome, equally likely outcomes, sample space (U) and event; probability of an
event, complementary events
Combined events
Conditional probability; independent events; use of Bayes’ theorem for two events
Venn diagrams, tree diagrams and tables of outcomes to solve problems
Concept of discrete and continuous random variables and their probability distributions; definition
and use of probability density functions; expected value (mean), mode, median variance and
standard deviation
Binomial distribution; its mean and variance. Poisson distribution; its mean and variance
Normal distribution; its properties; standardization of normal variables
Year 2 (Calculus/Math HLB)
6.
7.
8.
Functions and Equations
a. Concept of function, domain and range, image, composite functions, inverse function
b. Graph of a function, use of GDC to investigate properties of graphs and their solutions
c. Transformations of graphs: translations, stretches; reflections in the axes inverse functions—
reflection across y = x; graph of the reciprocal of a function; absolute value graphs
d. Reciprocal function, its graph; its self-inverse nature
e. Quadratic function, graph, axis of symmetry, vertex form, factored form—x-intercepts
f. Solutions to quadratic functions, quadratic formula, use of its discriminant
g. Exponential and logarithmic functions
h. The exponential function of e raised to the x power and its inverse lnx, x > 0
i. Inequalities in one variable, using their graphical representation
j. Polynomial functions; the factor and remainder theorems, with application to the solution of
polynomial equations and inequalities
Calculus
a. Limits and convergence
b. Differentiation of a sum and a real multiple of the functions in VII. A, chain rule, application of
the chain rule to related rates of change; product rule, quotient rule, second derivative; awareness
of higher derivatives
c. Local maxima & minima points, first & second derivatives in optimization problems
d. Indefinite integration as anti-differentiation; indefinite integrals of exponential, and trig functions,
the composite of any of these with the linear function ax + b
e. Anti-differentiation with a boundary condition to determine the constant term; definite integrals;
areas under curves; areas between curses; volumes of revolution
f. Kinematic problems involving displacement, s, velocity, v, and acceleration, a
g. Graphical behavior of functions; tangents and normals, behavior for large x , asymptotes;
significance of the second derivative; distinction between maximum and minimum points; points
of in flexion with zero and non-zero gradients.
h. Implicit differentiation
i. Further integration: integration by substitution; integration by parts
j. Solution of first order differential equations by separation of variables
Statistics and Probability – Option
a. Expectation algebra, linear transformation of a single random variable, mean and variance of
linear combinations of two independent random variables, extension to linear combinations of n
independent random variables
b. cumulative distribution functions; discrete distributions: uniform, Bernoulli, binomial, negative
binomial, Poisson, geometric, hypergeometric; continuous distributions: uniform, exponential,
normal
c. Distribution of a sample mean; distribution of linear combinations of independent normal random
variables; central limit theorem; approximate normality of the proportion of successes in a large
sample
d. Finding confidence intervals for the mean of a population; finding confidence intervals for the
proportion of successes in a populations
e. significance testing for a mean; significance testing for a proportion; null and alternative
hypotheses H0 and H1; type I and type II errors; significance levels; critical region, critical values;
p values; one-tailed and two tailed tests.
f. chi-squared distribution; degrees of freedom, v; the chi-squared statistic; chi-squared goodness of
fit test; contingency tables - chi-squared test for the independence of two variables