# Mathematics

**Introduction**

The Mathematics Department is one of the central departments to a student’s education at Auckland Grammar School. Students in each year are divided by form level depending on their abilities. Gifted and talented students can choose to be placed in the accelerated programme, which consist of the top two or three classes in Forms 3 to 6.

*Form 4*

**Pre-requisites:** Form 3 Mathematics.

**Course description/aims: **The course aims to revise and develop the basics of Form 3 work in the expectation of getting close to Pre-Q and NCEA Level 1 standard, especially in Algebra, so that the students are well prepared for Mathematics after Pre-Q and NCEA Level 1 (Form 5). This course is an introduction course to Pre-Q as many of the skills and tools are essential for progressing to Pre-Q the following year.

**Course outline: **The Form 4 Mathematics course reinforces and extends Form 3 work in Arithmetic, Algebra and Geometry. New topics include:

- Linear and quadratic relations and their graphs
- Simultaneous Equations
- Right-angled triangle Trigonometry
- Transformation Geometry with enlargement/similarity
- Graphs of trig functions
- Statistics
- Probability
- Symmetry Constructions

**Methods of assessment: **Common tests and three School examinations.

**Special equipment and costs:** A calculator, a ruler, protractor and a compass. Costs will only occur if students enter competitions. Optional cost – the continuation of Maths Buddy. All Form 4 students are encouraged to continue with their subscription.

**Continuation of subject: **The course leads on to Form 5 Pre-Q Mathematics and ultimately to Form 7 Cambridge A2 or NCEA Level 3 courses.

*Form 5 (Pre-Q Advanced)*

**Pre-requisites: **Form 3 and 4 Mathematics.

**Course description/aims:** The aim of the course is to introduce and develop basic mathematical skills, concepts, and understandings in the Mathematical Processes, Number, Measurement, Geometry, Algebra and Statistics curriculum strands. This course will prepare students for the Pre-Q Mathematics Advanced examination.

**Course outline: **Topics covered will include:

- Number
- Algebra and Graphs
- Geometry
- Mensuration
- Co-ordinate Geometry
- Trigonometry
- Vectors
- An introduction to Calculus
- Probability and Statistics

**Methods of assessment: **Students presenting for Pre-Q Mathematics Advanced will be assessed on the basis of a single final examination, assessed in Term 4.

**Special equipment and costs:** A protractor, compass, ruler and a scientific calculator of about Casio FX standard. Casio fx-991ES or fx-115ES are recommended (no graphical calculators are permitted).

**Continuation of subject: **The course will prepare students well for the courses on offer at Forms 6 and 7, Cambridge and NCEA courses.

*Form 5 (Pre-Q Core)*

**Pre-requisites: **Form 3 and Form 4 Mathematics. Students performing between 35% and 50% in Form 4.

**Course description/aims:** The aim of the course is to introduce and develop basic mathematical skills, concepts, and understanding in the Mathematical Strands of Processes, Number, Measurement, Geometry, Algebra and Statistics. This course will prepare students for the Pre-Q Mathematics Core examination.

**Course outline: **Topics covered will include:

- Number
- Algebra and Graphs
- Geometry
- Mensuration
- Co-ordinate Geometry
- Trigonometry
- Vectors
- An introduction to Calculus
- Probability and Statistics

**Methods of assessments: **Students presenting for Pre-Q Mathematics Core will be assessed on the basis of a single three-hour final examination assessed in Term 4. Students will also complete 3 internally-assessed NCEA Level 1 standards during the year, worth a total of 10 credits.

**Special equipment and costs: **A protractor, compasses, ruler and a scientific calculator of about Casio fx-standard. Permissible calculators are: Casio fx-991ES and fx-115ES (no graphical calculators are permitted).

**Continuation of subject:** Successful completion of the course will prepare students for all Mathematics courses on offer at NCEA Levels at Forms 6 and 7.

** Form 5 (NCEA Level 1**)

**Pre-requisites:** Form 3 and Form 4 Mathematics. A cohort of students, primarily from 4P and 4R will be selected to study this course.

**Recommended achievement levels:**

- Students with a mark of 35% or more in the Form 4 examination have an appropriate achievement level to study Form 5 NCEA and gain Numeracy
- Students with a mark between 25-35% in the Form 4 Examination may find aspects of Form 5 NCEA challenging
- Students with a mark below 25% in the Form 4 Examination may find the Form 5 NCEA course difficult

**Course description/aims: **The aim of the course is to introduce and develop basic mathematical skills, concepts, and understanding in the Mathematical Strands of Processes, Number, Measurement, Geometry, Algebra and Statistics. The ultimate aim is to gain NCEA Level 1 and University Entrance Numeracy.

Upon the successful completion of the course of study, students may study Form 6 NCEA Level 2 Mathematics. However for some students, this is a terminating course and will provide credits towards Numeracy only.

**Course outline:**

*Arithmetic – Number* [4 credits, Internal]:

- Ratio, proportion and rates
- Primes, factors, multiples, powers and roots
- Integer and fractional powers applied to numbers
- Fractions, decimals and percentages
- Rounding with decimal places and significant figures, degrees of accuracy including standard form

*Algebra I and II*:

- Factorising and expanding
- Simplifying algebraic expressions involving exponents
- Substituting values into formulae
- Manipulating and simplifying expressions
- Rearranging formulae.
- Solving linear equations and inequations
- Solving quadratic equations
- Solving simple equations involving exponents
- Solving pairs of simultaneous linear equations with the same two unknowns

*Measurement (Time, Units, Perimeter, Area and Volume)* [3 credits, Internal]:

- Perimeter, area and surface area
- Volume
- Time and metric units

*Statistics* [4 credits, Internal]:

- Investigating data that has been collected from a survey situation
- Posing an appropriate comparison question using a given multivariate data set
- Selecting and using appropriate data display(s)
- Giving summary stats such as the summary values (minimum, maximum, median, quartiles and percentiles)
- Discussing features of distributions comparatively, such as shape, middle 50%, shift, overlap, spread, unusual or interesting features and outliers
- Communicating findings, such as informal inference and supporting evidence, in a conclusion

*Linear Graphs* [3 credits, Internal]:

- Applying linear algebra in solving problems
- Developing patterns to problems in context

*Geometry* [3 credits, Internal]:

- 2D Co-ordinate systems
- Scale drawings
- Bearings
- Locus
- Construction
- Nets
- 2D representation of 3D objects

*Trigonometry* [3 credits, Internal]:

- Pythagoras’ Theorem
- Trigonometric ratios
- Similar shapes
- Measuring appropriately

**Special equipment and costs:** A protractor, compasses, ruler and a scientific calculator of about Casio fx-standard. A graphical calculator (Casio fx-9750G PLUS) is permissible and highly recommended. Achievement Standard Workbooks may be required for purchase ($45 approximately).

**Continuation of subject:** Primarily, this subject is a terminating course. However, a successful completion of the course may give students an opportunity to proceed with Mathematics courses on offer at Form 6 NCEA Level 2 at the discretion of the Head of Department. To be considered for Level 2 Mathematics, students must complete successfully the Term 4 Algebra examination plus any of the other standards totaling at least 12 credits.

*Form 6 (AS):*

**Pre-requisites:** Students should have gained more than 65% in Form 5 Pre-Q Mathematics Advanced examination; or 80% or more in Form 5 Pre-Q Mathematics Core examination and at the discretion of the Head of Mathematics.

**Recommended achievement levels:**

- Students with a mark of 75% or more in the Form 5 Pre-Q Mathematics Advanced examination have an appropriate achievement level to study Form 6 Cambridge AS Level
- Students with a mark between 65-75% in Form 5 Pre-Q Mathematics Advanced may find aspects of Form 6 Cambridge AS Mathematics difficult
- Students with a mark below 65% in Form 5 Pre-Q Mathematics Advanced will find Form 6 Cambridge AS Mathematics extremely difficult and are recommended to do the NCEA Level 2 Course

**Course description/aims: **The aim of the course is to introduce and develop basic mathematical skills, concepts, and understandings in the Mathematical Processes, Number, Measurement, Geometry, Algebra and Statistics curriculum strands. This course will cover all aspects of the Cambridge AS course and will serve as an introduction to A2 Pure Mathematics, Statistics or Mechanics courses.

**Course outline:**

- Algebra (Surds and Indices)
- Graphs
- Quadratic Equations
- Functions and Notations
- Binomial Expansion
- Statistics I (Data display and graphs)
- Arrangements (Perms and Combs)
- Co-ordinate Geometry
- Calculus I (Differentiation) and II (Integration)
- Trigonometry I (Radians) and II (Identities and equations)
- Probability
- Random Variables
- Normal Distribution
- Sequences and Series

**Methods of assessment:** A common testing programme will exist throughout the year. Students will also sit three examinations:

- Term 1 Progress Examination [1 hour]
- Term 2 Examination [2 hours]
- Term 3 Final Examination [3 hours]

Students presenting for AS Mathematics will be assessed on the basis of two external examinations. The external examinations are as follows:

- Cambridge AS Mathematics Paper P1 9709/1 [1 hour, 45 minutes] – Pure Mathematics paper sat in late October/early November.
- Cambridge AS Mathematics Paper S1 9709/6 [1 hour, 15 minutes] – Statistics paper sat in late October/early November.

**Special equipment and costs:** A protractor, compass, ruler and a scientific calculator of about Casio FX- standard (Casio fx-991ES is recommended).

**Continuation of subject: **The course will prepare students well for all the Cambridge courses on offer at Form 7.

*Form 6 (NCEA Level 2):*

**Pre-requisites:** Students should have gained an average of 50% or more in any Form 5 Pre-Q Mathematics examinations and/or a minimum of 10 credits in NCEA Level 1.

**Recommended achievement levels:**

- Students averaging 65% or more in Form 5 Pre-Q Mathematics Core have an appropriate achievement level to study Form 6.
- B. Students averaging between 35% & 65% in Form 5 Pre-Q Mathematics Core may find aspects of Form 6 Maths difficult
- Students averaging below 35% in Pre-Q Mathematics Core will find Form 6 Mathematics extremely difficult and may struggle to get any credits at NCEA Level 2
- Students who achieve at least 12 Credits in Form 5 NCEA Level 1 Mathematics (with a Term 4 examination on Algebra) have an appropriate achievement level to study Form 6 Mathematics. Entry to the course will be at the discretion of the Head of Department

**Course description/aims: **The aim of the course is to introduce and develop basic mathematical skills, concepts, and understanding in the Mathematical Processes, Number, Measurement, Geometry, Algebra and Statistics curriculum strands. This course will cover all the aspects of the New Zealand Mathematics Curriculum for Form 6 and serve as an introduction to all NCEA Level 3 Mathematics Advancing, General and Statistics courses offered in Form 7.

**Course outline:** The outline of the syllabus is as follows:

- Algebra
- Graphs
- Quadratic Equations
- Probability
- Statistics
- Co-ordinate Geometry
- Calculus
- Trigonometry

**Methods of assessment:** This course will be assessed by internally-assessed and externally-assessed NCEA Level 2 Achievement Standards.

- A common testing programme will exist throughout the year.
- Students will sit three School examinations. Grades will be determined by:

– Term 1 Examination [1 hour]

– Term 2 Examination [2 hours]

– Term 3 Examination [3 hours]

Achievement Standards assessed:

- 2.1 – Apply co-ordinate geometry methods in solving problems [2 credits, Internal]
- 2.2 – Apply graphical models in solving problems [4 credits, Internal]
- 2.4 – Apply trigonometric relationships in solving problems [3 credits, Internal]
- 2.6 – Apply algebraic methods in solving problems [4 credits, External]
- 2.7 – Apply calculus methods in solving problems [5 credits, External]
- 2.12 – Apply probability methods in solving problems [4 credits, External]

**Special equipment and costs:** A protractor, compass, ruler and a scientific calculator of about Casio FX- standard. A graphical calculator (Casio fx-9750G PLUS) is permissible and highly recommended. Achievement Standard Workbooks maybe required for purchase ($45 approximately).

**Continuation of subject: **The course will prepare students well for all the NCEA courses on offer at Form 7.

*Form 6 Mathematics with Statistics (NCEA Level 2):*

**Pre-requisites:** Students should have gained a minimum of 10 credits in NCEA Level 1 or Pre-Q Core Mathematics.

**Recommended achievement levels:**

- Students averaging below 35% in Pre-Q Mathematics Core will find Form 6 Mathematics with Statistics their best option at Level 2
- Students who achieve at least 10 Credits in Form 5 NCEA Level 1 Mathematics have an appropriate achievement level to study this course in Form 6. Entry to the course will be at the discretion of the Head of Department

**Course description/aims: **The aim of the course is to introduce and develop basic mathematical skills, concepts, and understanding in the Mathematical Processes, Number, Measurement, Geometry, Algebra and Statistics curriculum strands without the usual heavy Algebra component found in the 6MAT course. This course will cover most aspects of the New Zealand Mathematics Curriculum for Form 6 and serve as an introduction to the NCEA Level 3 Mathematics with Statistics (7MAS) course offered in Form 7.

**Course outline:** The outline of the syllabus is as follows:

- Systems of Equations
- Simulations
- Networks
- Questionnaires
- Co-ordinate Geometry
- Inference
- Graphical Methods

**Methods of assessment:** This course will be comprised of entirely internally-assessed NCEA Level 2 Achievement Standards.

- A common testing programme will exist throughout the year.
- Students will sit three School examinations. Grades will be determined by:

– Term 1 Examination [1 hour]

– Term 2 Examination [2 hours]

– Term 3 Examination [3 hours]

Achievement Standards assessed:

- 2.1 – Apply co-ordinate geometry methods in solving problems [2 credits]
- 2.2 – Apply graphical models in solving problems [4 credits]
- 2.5 – Apply network methods in solving problems [2 credits]
- 2.8 – Design a questionnaire [3 credits]
- 2.9 – Use statistical methods to make an inference [4 credits]
- 2.13 – Investigate a situation involving elements of chance using a simulation [2 credits]
- 2.14 – Apply systems of equations in solving problems [2 credits]

**Special equipment and costs:** A graphical calculator (Casio fx-9750G PLUS) is permissible and highly recommended. Achievement Standard Workbooks maybe required for purchase ($45 approximately).

**Continuation of subject: **Successful completion of this course will prepare students well for the NCEA course Maths with Statistics on offer at Form 7.

*Form 7 (A2):*

**Pre-requisites: **Students must have gained passes in Cambridge AS (Pure Mathematics 1) and Cambridge AS (Statistics 1).

**Recommended achievement levels:**

- Students with ‘A’ or ‘B’ grades from Cambridge AS Mathematics have an appropriate achievement level to continue for a full A level qualification doing the Pure Mathematics and Statistics course
- Students with ‘C’ or ‘D’ grades from Cambridge AS Mathematics may find aspects of Pure Mathematics and Statistics course difficult
- Students with a ‘D’ grade from Cambridge AS Mathematics will find aspects of the Pure Mathematics and Statistics course difficult and are recommended to do the Pure Mathematics and may be directed to do the Mechanics course
- Students with a grade lower than a ‘D’ will find A2 too difficult and should consider repeating Cambridge AS or opting for another subject

**Course description/aims: **The aim of the course is to introduce and develop a range of mathematical skills, particularly mathematics in context, and to develop an understanding of mathematical principles in the areas of Number, Measurement, Geometry, Algebra and Statistics curriculum strands.

**Course outline:**

Pure Mathematics/Statistics course:

- Algebra
- Trigonometry
- Calculus
- Vectors
- Complex Numbers
- Differential Equations
- Random Variables
- Distributions
- Sampling
- Estimation
- Hypothesis Testing

Pure Mathematics/Mechanics course:

- Algebra
- Trigonometry
- Calculus
- Vectors
- Complex Numbers
- Numerical Methods
- Differential Equations
- Forces and Equilibrium
- Kinematics
- Newton’s Law of Motion and Energy
- Work and Power

**Methods of assessment:** Common tests and three examinations in Terms 1, 2 and 3. Students presenting for Cambridge A2 Mathematics will be assessed on the basis of two external examinations:

- P3/S2 course: Pure Mathematics Paper P3 9709/03 [1 hour 45 minutes] / Statistics 2 S2 9709/07 [1 hour 15 min]
- P3/M1 course: Pure Mathematics Paper P3 9709/03 [1 hour, 45 minutes] / Mechanics 1 M1 9709/04 [1 hour 15 min]

**Special equipment and costs:** A scientific calculator of about Casio FX standard (Casio fx-991ES is recommended).

*Form 7 (NCEA Level 3 Advancing):*

**Pre-requisites: **Form 6 NCEA Mathematics with Achievement Standards 2.4 (Trigonometry), 2.6 (Algebra) and 2.7 (Calculus).

**Recommended achievement levels:**

- Students averaging over 70% in Form 6 Mathematics School examinations or Merits in the above Achievement Standards have an appropriate achievement level to study Form 7 Mathematics – Advancing
- Students averaging 40-70% in Form 6 Mathematics School examinations or Achieved in Achievement Standards above should cope but will find aspects difficult
- Students averaging less than 40% in Form 6 Mathematics School examinations may struggle and will be encouraged to do Mathematics – General

**Course description/aims: **This course is intended for students moving on to university Engineering and Medicine Faculties next year. The course will provide students with a solid base of mathematical knowledge for their tertiary mathematics education.

The aim of this course is to introduce and develop essential mathematical skills, concepts, and understandings in the Mathematical Processes, Number, Measurement, Geometry, Algebra and Statistics curriculum strands to advance with mathematics the following year at a tertiary institution. Students would be expected to do both Form 7 Mathematics Advancing (7MAA) and Form 7 Statistics (7STA) courses.

**Course outline: **The main areas of study are:

- 3.1 – Conics [3 credits, Internal]
- 3.3 – Trigonometry [4 credits, Internal]
- 3.5 – Algebra and Complex Numbers [5 credits, External]
- 3.6 – Calculus: Differentiation [6 credits, External]
- 3.7 – Calculus: Integration [6 credits, External]

**Methods of assessment:**

- Class tests
- Two common tests each term
- Three School examinations in Terms 1, 2 and 3

**Special equipment and costs:** A scientific calculator or graphical calculator (a graphical calculator is highly recommended).

*Form 7 (NCEA Level 3 Statistics):*

**Pre-requisites: **Form 6 NCEA Mathematics with Achievement Standards 2.6 (Algebra) and 2.12 (Probability). A student should be taking Form 7 Mathematics – Advancing (7MAA) to take this course.

**Recommended achievement levels:**

- Students averaging over 70% in Form 6 Mathematics School examinations or Merits in the above Achievement Standards have an appropriate achievement level to study Form 7 Statistics
- Students averaging 40-70% in Form 6 Mathematics School examinations or Achieved in the above Achievement Standards should cope but may find aspects difficult
- Students averaging less than 40% in Form 6 Mathematics School examinations may struggle and will be encouraged to do Mathematics – General

**Course description/aims: **The course is not heavy in Algebra but will still give students the ability to achieve an NCEA Level 3 Mathematics Subject Endorsement. This course is intended for students moving on to university Engineering and Medicine Faculties next year. The course will provide students with a solid base of mathematics with statistical skills for their tertiary mathematics education.

The aim of this course is to introduce and develop essential mathematical skills, concepts, and understandings in the Mathematical Processes, Number, Measurement, Geometry, Algebra and Statistics curriculum strands to advance with mathematics the following year at a tertiary institution. Students would be expected to do both – Form 7 Mathematics Advancing (7MAA) and Form 7 Statistics (7STA) courses.

**Course outline: **The main areas of study are:

- 3.2 – Linear Programming [3 credits, Internal]
- 3.8 – Time Series [4 credits, Internal]
- 3.9 – Bivariate Data [4 credits, Internal]
- 3.10 – Inference [4 credits, Internal]
- 3.13 – Probability [4 credits, External]
- 3.14 – Probability Distributions [4 credits, External]
- 3.15 – Simultaneous Equations [3 credits, Internal]

**Methods of assessment:** Class/common tests plus three examinations.

**Special equipment and costs:** A scientific calculator or graphical calculator is highly recommended.

*Form 7 (NCEA Level 3 General):*

**Pre-requisites: **Form 6 NCEA Mathematics with Achievement Standards 2.6 (Algebra), 2.7 (Calculus) and 2.12 (Probability).

**Recommended achievement levels:**

- Students averaging over 60% in Form 6 Mathematics School examinations or Achieved in all of the above Achievement Standards have an appropriate achievement level to study Form 7 Mathematics – General
- Students averaging 30-60% in Form 6 Mathematics School examinations should cope, but may find aspects difficult
- Students averaging less than 30% in Form 6 Mathematics School examinations may struggle taking this subject

**Course description/aims: **This course is intended for students moving on to university courses next year that need a background in general mathematics. The course will provide students with a broad base of mathematical knowledge for their tertiary mathematics education in Commerce, Humanities and Science courses. The aim of this course is to introduce and develop broad but basic mathematical skills, concepts, and understandings in the Mathematical Processes, Number, Measurement, Geometry, Algebra and Statistics curriculum strands to supplement a tertiary course.

**Course outline: **The main areas of study are:

- 3.2 – Linear Programming [3 credits, Internal]
- 3.6 – Differentiation (Merit Level only) [6 credits, External]
- 3.7 – Integration (Merit Level only) [6 credits, External]
- 3.14 – Probability Distributions [4 credits, External]
- 3.15 – Simultaneous Equations [3 credits, Internal]

**Methods of assessment:** Class/common tests and three examinations.

**Special equipment and costs:** A scientific calculator or graphical calculator is highly recommended.

*Form 7 Mathematics with Statistics (NCEA Level 3):*

**Pre-requisites: **Students should have gained a minimum of 10 credits in NCEA Level 2 with the preference of achieving the Achievement Standards 2.9 (Inference) and 2.14 (Systems of Equations).

**Recommended achievement levels:**

- A. Students averaging below 35% in NCEA Level 2 Mathematics examinations will find Form 7 Mathematics with Statistics their best option at Level 3
- Students who achieve at least 10 Credits in Form 6 NCEA Level 2 Mathematics with Statistics have an appropriate achievement level to study this subject in Form 7. However, entry to the course will be at the discretion of the Head of Department
- Students with less than 10 Credits in Form 6 NCEA Level 2 Mathematics with Statistics should consider repeating their study in this subject in Form 6

**Course description/aims: **The aim of this course is to introduce and develop basic mathematical skills, concepts and understanding in the Mathematical Processes, Number, Measurement, Geometry, Algebra and Statistics curriculum strands without the usual heavy Algebra component found in NCEA Level 3 Mathematics courses of 7MAA and 7MAG. This course will cover some of the aspects of the New Zealand Mathematics Curriculum for Form 7.

**Course outline: **The main areas of study are:

- Systems of Equations
- Linear Programming
- Time Series
- Bivariate Data
- Inference
- Probability methods

**Methods of assessment:**

- A common testing programme will exist throughout the year.
- Students will sit three School examinations. Grades will be determined by:

– Term 1 Examination [1 hour]

– Term 2 Examination [2 hours]

– Term 3 Examination [3 hours]

Internally-assessed Achievement Standards:

- 3.2 – Apply linear programming methods in solving problems [3 credits]
- 3.8 – Investigate time series data [4 credits]
- 3.9 – Investigate bivariate measurement data [4 credits]
- 3.10 – Use statistical methods to make an inference [4 credits]
- 3.13 – Apply probability concepts in solving problems [4 credits]
- 3.15 – Apply systems of simultaneous equations in solving problems [3 credits]

**Special equipment and costs: **A graphical calculator (Casio FX-9750G PLUS) is permissible and highly recommended. Achievement Standard Workbooks maybe required for purchase ($45 approximately).

If you have any questions, please feel free to contact our staff today.