# 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*

**Prerequisites:** 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. This course is an introduction course to Pre-Q and the work covered consolidates the essential skills required to continue with Pre-Q Mathematics. Learning these key skills and tools are essential for progressing to Pre-Q.**

**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 and 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)*

**Prerequisites: Forms 3 and 4 Mathematics. Students need to perform above 50% in Form 4.**

**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.

- Apply mathematical skills, concepts, and understandings to familiar and unfamiliar problems arising in real and simulated situations
- Demonstrate the ability to select and use appropriate mathematical techniques in problem solving
- Demonstrate the ability to reason logically and systematically
- Demonstrate the ability to communicate mathematical ideas

At the conclusion of this course of study, a student will be well prepared to continue his studies in Form 6 Cambridge AS Mathematics.

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

- Number
- Algebra and graphs
- Geometry
- Mensuration
- Co-ordinate Geometry
- Trigonometry
- Vectors
- An introduction to Calculus
- Probability
- 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 successful completion of 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)*

**Prerequisites: Form 3 and 4 Mathematics. Students need to perform 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. The course objectives are that the candidate will be expected to:

- Apply mathematical skills, concepts, and understandings in real and simulated problems
- Demonstrate the ability to select and use appropriate mathematical techniques in problem solving
- Demonstrate the ability to reason logically and systematically
- Demonstrate the ability to communicate mathematical ideas

At the successful conclusion of the course of study, a student will be well prepared to continue his studies in Form 6 NCEA Level 2 Mathematics.

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

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

**Methods of assessments: **Students presenting for Pre-Q Mathematics Core will be assessed on the basis of a single final examination, assessed in Term 4. Students will also complete three 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:** The 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**)

**Prerequisites:** Form 3 and 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 final 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 final Form 4 examination may find aspects of Form 5 NCEA challenging
- Students with a mark below 25% in the final 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 course objectives are that the candidate will be expected to:

- Apply mathematical skills, concepts, and understandings in real and simulated problems
- Demonstrate the ability to select and use appropriate mathematical techniques in problem solving
- Demonstrate the ability to reason logically and systematically
- Demonstrate the ability to communicate mathematical ideas

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]*

Students will be expected to be familiar with methods related to:

- 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*

Students will be expected to be familiar with methods related to:

- Factorising and expanding
- Simplifying algebraic expressions involving exponents
- Substituting values into formulae
- Manipulating and simplifying expressions
- Rearranging formulae

*Algebra II*

Students will be expected to be familiar with methods related to:

- 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] *

Students will be expected to be familiar with methods related to:

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

*Statistics [4 Credits, Internal]*

Students will be expected to be familiar with the statistical enquiry cycle which involves:

- 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]*

Students will be expected to be familiar with methods related to:

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

*Geometry [3 Credits, Internal]*

Students will be expected to be familiar with methods related to:

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

*Trigonometry [3 Credits, Internal]*

Students will be expected to be familiar with methods related to:

- 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 (approximately $45).

**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 successfully complete the Term 4 Algebra examination plus any of the other standards totalling at least 12 credits.

*Form 6 (AS):*

Recent experience shows that Pre-Q Mathematics provides good preparation for Cambridge AS Mathematics.

- Students with a mark of 55% in Pre-Q Advancing Mathematics are capable of scoring at least a C grade in Form 6 Cambridge AS Level
- The majority of students scoring between 45% and 55% in Pre-Q Advancing Mathematics continue on to Form 6 Cambridge AS Level and pass, achieving the University Entrance Numeracy requirement in Form 6

Cambridge AS Mathematics in Form 6 is also available to students who have studied Pre-Q Core 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 cover all aspects of the Cambridge AS course and will serve as an introduction to A2 Pure Mathematics, Statistics or Mechanics courses. **The course objectives are that candidates will be expected to:

- Apply mathematical skills, concepts, and understandings to familiar and unfamiliar problems arising in real and simulated situations
- Demonstrate the ability to select and use appropriate mathematical techniques in problem solving
- Demonstrate the ability to reason logically and systematically
- Demonstrate the ability to communicate mathematical ideas

**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)
- Trigonometry I (radians and circular measure)
- Trigonometry II (graphs, identities and equations)
- Probability
- Calculus II (integration)
- 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:

- Paper P1 9709/ [1 hour, 50 mins] – Pure Mathematics paper sat in late October/early November
- Paper S1 9709/5 [1 hour, 15 mins] – 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 Plus 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):*

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

**Course description/aims: **The aim of the NCEA Level 2 Mathematics 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 serve as an introduction to the NCEA Level 3 Mathematics courses offered in Form 7.

Students will be expected to:

- Apply mathematical skills, concepts, and understandings to familiar and unfamiliar problems arising in real and simulated situations
- Demonstrate the ability to select and use appropriate mathematical techniques in problem solving
- Demonstrate the ability to reason logically and systematically
**Demonstrate the ability to communicate mathematical ideas**

**Course outline: **The course content will include six to eight topics, to be selected from the list below:

- Systems of equations
- Simulations
- Networks
- Questionnaires
- Co-ordinate Geometry
- Inference
- Graphical methods
- Algebra
- Graphs
- Probability and Statistics
- Calculus
- Trigonometry

**Methods of assessment: **The NCEA Level 2 Mathematics course is assessed by internal and external Achievement Standards. A common testing programme will operate throughout the year and there will be three School examinations.

Students will also be assessed by internal Achievement Standards as Common Assessment Tasks (CATs), to be selected from the following list:

- 2.1 – apply co-ordinate geometry methods in solving problems [2 Credits]
- 2.2 – apply graphical models in solving problems [4 Credits]
- 2.4 – apply trigonometric relationships in solving problems [3 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]

Students presenting for NCEA Level 2 Mathematics will be assessed on the basis of a three-hour external examination. This examination will contain 3 booklets, each one examining an externally-examined Achievement Standard as follows:

- 2.6 – apply algebraic methods in solving problems [4 Credits]
- 2.7 – apply calculus methods in solving problems [5 Credits]
- 2.12 – apply probability methods in solving problems [4 Credits]

**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 may be required for purchase (approximately $50).

**Continuation of subject: Successful completion of this course will prepare students for NCEA Level 3 Mathematics courses offered at Form 7.**

*Form 7 (A2):*

**Prerequisites: **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 (P3) and Statistics (S2) course
- Students with C or D grades from Cambridge AS Mathematics may find aspects of Pure Mathematics (P3) and Statistics (S2) course difficult
- Students with a D grade from Cambridge AS Mathematics will find aspects of the Pure Mathematics (P3) and Statistics (S2) course difficult and are recommended to do the Pure Mathematics (P3) and may be directed to do the Mechanics (M1) 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 [P3 (Paper 3)/S2 course (Paper 6)]:

- Algebra
- Trigonometry
- Calculus
- Vectors
- Complex numbers
- Differential equations
- Random variables
- Distributions
- Sampling
- Estimation
- Hypothesis testing

Pure Mathematics/Mechanics course [P3 (Paper 3)/M12 course (Paper 4)]:

- 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:

- Paper P3 9709/03 [1-hour 45 mins] / Paper S2 9709/07 [1 hour 15 min]
- Paper P3 9709/03 [1-hour, 50 mins] / Paper M 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):*

**Prerequisites: **The NCEA Level 3 Maths course is differentiated. Students will be placed in the appropriate class, based on performances in the Form 6 Level 2 Mathematics courses. It is possible to combine Form 7 Level 3 Mathematics and Form 7 Statistics courses.

**Course description/aims: **All classes will provide students with a solid grounding in mathematical skills, concepts and understanding in the Mathematical Processes, Number, Measurement, Geometry, Algebra and Statistics curriculum strands.

Most students will study courses that prepare them for university-level courses requiring a mixture of Algebra, Probability and Calculus. Level 3 Mathematics students will be expected to:

- Apply mathematical skills, concepts, and understandings to familiar and unfamiliar problems arising in real and simulated situations.
- Demonstrate the ability to select and use appropriate mathematical techniques in problem-solving.
- Demonstrate the ability to reason logically and systematically
**Demonstrate the ability to communicate mathematical ideas.**

**Course outline:** The course will include five to six topics to be selected from the below:

- Conics
- Linear programming
- Trigonometry
- Algebra and complex numbers
- Differentiation
- Integration
- Probability distributions
- Simultaneous equations
- Systems of equations
- Time series
- Bivariate data
- Inference
- Critical paths analysis

**Methods of assessment: **The NCEA Level 3 Mathematics course is assessed by internal and external Achievement Standards. Class tests and a common testing programme will operate throughout the year, along with three School examinations. Grades will be determined by:

- AS 91573 – apply the geometry of conic sections in solving problems [Term 3]
- AS 91574 – apply linear programming methods in solving problems [3 Credits, Internal]
- AS 91575 – apply trigonometric methods in solving problems [Term 2]
- AS 91576 – use critical path analysis in solving problems [2 Credits, Internal]
- AS 91580 – investigate time series data [4 Credits, Internal]
- AS 91581 – investigate bivariate measurement data [4 Credits, Internal]
- AS 91582 – use statistical methods to make an inference [4 Credits, Internal]
- AS 91587 – apply systems of simultaneous equations in solving problems [3 Credits, Internal]

Students sitting NCEA Level 3 Mathematics will be assessed on the basis of a three-hour external examination containing external Achievement Standards:

- AS 91577 – apply algebra of complex numbers in solving problems
- AS 91578 – apply differentiation methods in solving problems
- AS 91579 – apply integration methods in solving problems
- AS 91586 – apply probability distributions in solving problems

**Special equipment and costs:** A scientific or graphical calculator (a graphical calculator is highly recommended). Achievement Standard Workbooks may be required for purchase (approximately $45).

*Form 7 (NCEA Level 3 Statistics):*

**Prerequisites: Form 6 NCEA Mathematics with Achievement Standards 2.6 (Algebra) and 2.12 (Probability).**

**Recommended achievement levels:**

- Students averaging over 70% in the final Form 6 Mathematics School examination (or Merits in the above Achievement Standards) have an appropriate achievement level to study Form 7 Statistics
- Students averaging 40-70% in the final Form 6 Mathematics School examinations (or Achieved in the above Achievement Standards) and achieved above 50% or above in English should cope but may find aspects difficult
- Students averaging less than 40% in Form 6 Mathematics School examinations and scored less than 50% in English may struggle and will be encouraged to do Mathematics – General next year

**Course description/aims: **A student taking this course should consider taking the Form 7 Mathematics with Calculus alongside. 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 but will cover key skills for other Faculties. The course will provide students with a solid base of mathematics with statistical skills for their tertiary mathematics education.

With the standards covered in this course, there is an expectation of good report writing skills.

**The general aim of this course is to introduce and develop essential mathematical skills, concepts, and understandings in the Mathematical Processes, Number, Measurement, Algebra and Statistics curriculum strands to advance with mathematics the following year at a tertiary institution. It is permissible for a student to do both Form 7 Mathematics with Calculus and Form 7 Statistics courses.**

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

- 3.2 – Linear Programming [3 Credits, Internal]
- 3.8 – Time Series [4 Credits, Internal]
- 3.10 – Inference [4 Credits, Internal]
- 3.13 – Probability Concepts [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 School examinations in Terms 1, 2 and 3.

**Special equipment and costs:**

Scientific calculator or graphical calculator (a graphical calculator is highly recommended). Achievement Standard Workbooks may be required for purchase (approximately $45).

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