Level 100 Mathematics for Secondary School Students - Mathematics and Statistics - University of Canterbury - New Zealand

Welcome to the Secondary School Programme


100-level mathematics course for secondary school students

This course is designed for students who have been involved in accelerated courses at school and may have passed Level 3 calculus credits in year 12, and for those whose ability in mathematics is such that they would like an enrichment/extension course while concurrently doing their year 13 work.

Entry Requirements

The course coordinator consults closely with schools about the suitability of students on an individual basis. Take as a (flexible) guideline, about 20 credits of mathematics with a good proportion of excellences at Level 2 or Level 3 NCEA, or the equivalent for students not doing a full NCEA programme or those doing alternative programmes (for example, Cambridge A levels). In view of the high-level nature of this course, the School of Mathematics and Statistics reserves the right to exclude students who it decides would not benefit from the course.


Students who successfully complete this course are given a Certificate of Completion. They are graded on their year’s work and this grade appears on their certificate. Students may also apply for an official transcript which has been created specifically for STAR students (MATH 199 is a STAR course) and can be used for credit transfer/credit recognition purposes.

On enrolling at the University of Canterbury students are given 30 points credit towards a Canterbury degree and may enrol in our 200-level mathematics courses. This credit is restricted against our core courses, MATH 101, MATH 102, MATH 103, EMTH 118 and EMTH 119.


Students must enrol in MATH 199 through their school. Enrolment forms are enclosed with the letter sent to the Head of Mathematics or can be downloaded here:

Please fax (03 364 2587) or post (see address below) these forms to Liz Ackerley in the School of Mathematics and Statistics.

Liz Ackerley
School of Mathematics and Statistics
University of Canterbury
Private Bag 4800
Christchurch, 8140

Confirmation of acceptance will normally take a week from receipt of enrolment forms. Enrolments will be considered up to the second week in February (or later, in special circumstances).


The cost per student is $915, GST inclusive. The cut-off date for withdrawal is Friday 5th April 2013. Schools are not charged for students who withdraw before this time.

STAR students will be exempt the Student Services Levy. As part of their enrolment, STAR students will continue to be issued with Canterbury Cards which give them access to essential University services.

Course Details

The course will start on Tuesday 12th February and finish the week beginning Monday 28th October. Actual term times coincide with those of the schools.

WHEN: 4.30pm - 6.30pm Tuesday (a two hour lecture with a break)
4.30pm - 6.30pm Thursday (a one hour lecture followed by a tutorial)
WHERE: University of Canterbury
Room 031 in the basement of the Erskine building (to be confirmed).


On Thursday afternoons, the class is broken up into small groups under the supervision of a tutor. In tutorials students are given an opportunity to ask questions about the course work and to sort out any problems they may be having. To benefit fully from tutorials they must attempt the set exercises (these are given out a week earlier) before the actual tutorial.


The first test is one hour long and is held in the first school term. Students need to be scoring at least 60% in this test to be confident of coping with the demands of the course. The second and third tests are each 1 1/2 hours long. The final exam is two hours long and is held in the last week of the course.

Three Tests:50%
Final Examination:35%

Key Dates

Tests/ExamTest 1Test 2Test 3Exam
Date Held21st Mar21st May6th Aug29th Oct

Texts for Recommended Reading

These texts (any edition) will do for background reading. They are used in our other 100-level courses, but there are many similar texts available.

  • Anton, Calculus, Wiley, any edition.
  • Anton, Elementary Linear Algebra, any edition.

Teaching Topics

Test One Topics

  • Topic One: Linear Equations and Matrices

    Solving systems of linear equations. Elimination and back-substitution. Applications. Matrices and their algebra. Invertible and elementary matrices.

Test Two Topics

  • Topic Two: Preparation for Calculus

    Review: What is a function? Operations on functions. Some basic functions and their inverses. The inverse trigonometric functions. The hyperbolic functions and their inverses. Intuitive idea of a limit, one sided and two sided limits, infinite limits and limits at infinity. Properties of limits. Continuity. Mathematical Induction.

  • Topic Three: Differentiation

    The definition of the derivative, differentiability and continuity. The rules of differentiation. Derivatives of some special functions. Implicit differentiation. Optimisation. Curve sketching. The Mean Value Theorem. L’Hôpital’s Rule.

Test Three Topics

  • Topic Four: Vector Geometry and Vector Functions

    Vector algebra. Dot product, orthogonality and projections. Cross product, areas and volumes. Representing lines and planes in terms of vectors. Intersection and distance problems.

    Vector-valued functions.

  • Topic Five: Integration

    The definite and indefinite integral. The Fundamental Theorem of Calculus. Riemann Sums.

    Techniques of integration (substitution, parts, partial fractions). Trigonometric integrals. Improper integrals. An application - arc length.

Exam Topics

  • Topic Six: Sequences and Series, Taylor Polynomials and Approximations

    Sequences. Geometric, harmonic and alternating harmonic series, power series. Approximation by Taylor polynomials. Taylor series.

  • Topic Seven: An Introduction to Functions of Several Variables

    Graphs and level curves. Partial derivatives. Maxima and minima.

  • Topic Eight: Differential Equations

    First-order differential equations (linear, separable) and applications.

    Second-order linear differential equations with constant coefficients and applications.

  • Topic Nine: Determinants

    Matrices as transformations. Determinants as area and volume. Properties of determinants. Calculating the determinant by elementary row operations. Cofactor expansions. An introduction to eigen problems.

  • Topic Ten: Probability and its Applications to Calculus

    Sets and experiments. Axioms and basic theorems of probability. Conditional probability. Discrete and continuous random variables, and their distributions. Expectations. Transformations of random variables. The delta method for approximating mean and variance.