Resources:  Mathematics and Science

Mathematics, Physics, and Chemistry

Many textbooks and internet sites are available to explain technical matters. However, what can also make a difference in tests and exams are things like:

The first point is mainly improved by practising (learning by experience, recognising patterns). Some thoughts on the latter point are compiled in the PDF file linked above. 

As discussed in the "Checking your answers in mathematics and physics" document, taking care to ensure consistency with dimensions (e.g. length, mass, time) and units (e.g. cm, kg, min) is extremely important to avoid mistakes, and indeed to catch accidental missteps.  Therefore knowledge of units and ability to perform unit conversions are vital.  The following document is focussed on metric / SI units, with brief coverage of imperial lengths;  however, the basic methodology can be used for any unit conversion. 

Units and unit conversions:  length, area, volume, time, speed & concentration, with tables for memorisation/practice.

Mathematics: fundamentals

In lower years of school a 'fluency' in mental arithmetic is required.  Even in later years strong competency and speed in mental arithmetic is highly beneficial, as it means that the student's attention and time can be focused on the more challenging concepts, rather than the basic arithmetic. 

Arithmetic:  fundamental principles, strategies for mental calculation, and tables for memorisation/practice.

Tips to sound like you know what you're talking about:

  • In Australia the accepted abbreviation of "mathematics" is "maths".  (Americans say "math".)
  • The "+" symbol.
    • "+" should be read as "plus" when used as a preposition:  "4 + 3 = 7" is read as "Four plus three equals seven".  
      • This process is commonly known as "addition".  Hence we can describe the previous operation as "adding" two numbers;  specifically, in the above example three was "added" to four.  
      • This process is occasionally known as "summation".  Hence we can describe the previous operation as "summing" two numbers, or "finding the sum" of two numbers;  specifically, in the above example the "sum" of four and three was found.  
      • This process should not be called "plussing".  
    • "+" should be read as "positive" when used as an adjective:  "+5" on a number line is read as "positive five".  
  • The "−" symbol.  
    • "−" should be read as "minus" when used as a preposition:  "4 − 3 = 1" is read as "Four minus three equals one".  
      • This process is commonly known as "subtraction".  Hence we can describe the previous operation as "subtracting" two numbers;  specifically, in the above example three was "subtracted" from four.  
      • This process is related to "finding the difference", except that when finding a difference we typically only care about the size (magnitude) of the difference, and not the sign.  
      • This process should not be called "minussing".  
    • "−" should be read as "negative" when used as an adjective:  "−5" on a number line is read as "negative five".  
    • The correct symbol is a minus sign ("−"), not a hyphen ("-").  
  • The "×" symbol.
    • "×" should be read as "times" when used as a preposition:  "4 × 3 = 12" is read as "Four times three equals twelve".  
      • In casual usage it is acceptable to say "by", as in "Four by three equals twelve".
      • This process is commonly known as "multiplication".  Hence we can describe the previous operation as "multiplying" two numbers;  specifically, in the above example four was "multiplied" by three.  
      • This process is occasionally known as "finding the product".  Hence we can describe the previous operation as "finding the product" of two numbers;  specifically, in the above example the "product" of four and three was found.  
      • This process should not be called "timesing".  
    • The correct symbol is a multiplication sign ("×"), not the letter x ("x"), nor an asterisk ("*").  
      • Occasionally a centred dot ("·") is used.  
  • The "÷" symbol.
    • "÷" should be read as "divided by" when used as a prepositional phrase:  "12 ÷ 3 = 4" is read as "Twelve divided by three equals four".  
      • It is not recommended to abbreviate this to just "by", as it is easily confused for the casual usage for multiplication.  (Indians sometimes say just "by".)  
      • This process is commonly known as "division".  Hence we can describe the previous operation as "dividing" two numbers;  specifically, in the above example twelve was "divided" by three.  
      • This process is occasionally called "finding the quotient". Hence we can describe the previous operation as "finding the quotient" of two numbers;  specifically, in the above example the "quotient" of twelve and three was found.   
    • Occasionally the solidus (/) — sometimes called a slash — is used as an alternative symbol:  "12 / 3 = 4".  This reflects the interpretation of the division operation as equivalent to evaluation of a fraction, as in "12/3 = 4", usually read as either "Twelve over three equals four" or "Twelve thirds equals four".  

Mathematics: OC Test

The N.S.W. Opportunity Class Placement Test (“OC Test” for short) presents challenging questions to primary school students from Year 4 and Year 5. 

Generally it is not recommended for students to be attempting this type of selective test if they are not already performing well academically.  (It would likely be unhelpful to force those young students to try to cram for such a test, taking time away from more healthy learning activities, physical activities and social activities.  Even if the student can scrape in to the Opportunity Class system, they may then face the stress of having to maintain that level of intense study to keep up.)  

Study (and optionally tutoring) for the OC Test is more suitable for students who are already performing well academically.  This extra attention can help the student to become more familiar with the style of questions, the format of the test, strategies to allocate time wisely, and so on.  And, of course, they can advance their learning of mathematics and logical reasoning (Mathematical Reasoning and Thinking Skills) as well as English comprehension (Reading) too.  

A wide variety of questions can appear on the OC Test, so broad study would be appropriate.  Here only a limited number are discussed.  

OC Test:  discussion of a small selection of example questions, with additional practice questions.

Mathematics: Year 11 & Year 12

There are several options that students can choose for studying mathematics in Year 11 and Year 12 in N.S.W..  The official source of detailed information on these options is the N.S.W. Education Standards Authority ("NESA"), supported by resources provided by your local high school. 

A convenient a summary of the syllabus for each mathematics offering at Year 11 and Year 12 level in N.S.W. was prepared in 2018, including information on options introduced in 2018 and 2019.  It has been updated in 2024, with the main changes being that Numeracy has replaced Mathematics General as a Content-Endorsed Course.  Despite being updated in 2024, that document may not have all of the latest information, as NESA is releasing new syllabi for Y11–12 mathematics courses in 2024 for implementation in 2026 as part of a "comprehensive reform of the NSW school curriculum".  An updated summary document may be issued here in future as NESA releases more information.  

Tips to sound like you know what you're talking about:

  • The word "locus" sounds like "focus", with only the initial consonant changing.  LOW-kus (/ˈləʊkəs/), not LOCK-us.  The plural is "loci" (analogous to "foci"), pronounced LOW-kai (/ˈləʊkaɪ/). 
  • To "derive" means to obtain ("We derived the following proof by applying Pythagoras's Theorem");  "derivation" is the process or result of deriving something ("The derivation required seven steps"). 
  • To "differentiate" means to obtain the so-called gradient function (an expression representing the gradient of the original function);  "differentiation" is the process of differentiating a function;  a "derivative" is the result of performing differentiation. 

Mathematics: Year 11

Exposition of various methods to factorise quadratic polynomials: 

Methods for factorising quadratics.

Computation of trigonometric functions of angles in each of the four quadrants:

Trigonometric functions of angles in each quadrant.

Below is a short multiple-choice exam intended to test students' knowledge of material associated with the Year 11 ('Preliminary') "2-unit" mathematics syllabus. 

Year 11 2-Unit Mathematics review test. 

Section I is easiest, Section II is somewhat harder, and Section III is the most challenging. 
Topics and some questions are adapted from Maths in Focus. Mathematics Preliminary Course (Margaret Grove, 2009, 2nd Edition).

Description of the problems that can arise when squaring (both sides of) an equation: 

Perils of squaring equations.

Mathematics: Year 12 (HSC)

Exposition of four methods to find the overall outcome with the highest probability in problems involving binomial probability: 

Binomial probabilities from binomial theorem.

Finance: Bachelor-degree level

Expanded discussion of scenarios in which a worker values their employment status according to how much they are paid (compared to their income without a job), offset by any equivalent 'opportunity cost' (which could arise even from feelings about the job, i.e. "affect"): 

Employment rent & Disutility of work.