Notation, order of operations and rational vs irrational numbers in context

1. What this topic actually is, and why it sits under everything else

Area of Study 1 opens with the conventions of formal mathematical terminology and notation — the agreed rules for how a calculation is written, read and evaluated. It looks like the least glamorous part of Foundation Mathematics, and it is the part that quietly decides your Section B mark. Every one of the twelve extended-response questions is worth five marks, working must be shown for any part worth more than one mark, and an assessor can only give marks for working they can follow. Notation is how your working becomes readable.

There are three separate skills bundled together here. The first is writing mathematics correctly: equals signs that mean equal, units attached to quantities, money written as $77.55 rather than 77.55$, and a clear final statement. The second is evaluating a written expression in the correct order, by hand and on a scientific calculator, so that 12 + 8 ÷ 4 gives 14 and not 5. The third is knowing what kind of number you are dealing with — a rational value you can write exactly, or an irrational value such as π that a calculator can only give you as a rounded decimal.

These three skills reappear in every other area of study. In AOS3 you will write a tax calculation as a chain of operations; in AOS4 you will substitute into an area formula containing π; in AOS2 you will divide a sum by a count. If your order of operations is shaky, the error surfaces in a financial question and looks like a finance mistake. That is why the study design puts conventions first: it is the shared language, not a separate chapter.

One practical note about the assessment conditions. You are permitted one scientific calculator — no CAS, no graphics calculator — and one annotated bound reference. A scientific calculator will apply the order of operations correctly to whatever you type, which means the risk is never the calculator being wrong. The risk is you typing an expression that is not the one on the page.

2. Order of operations, by hand and on the calculator

The convention is usually remembered as BODMAS or BIDMAS: Brackets, then Orders (indices, powers and roots), then Division and Multiplication, then Addition and Subtraction. Two details matter more than the acronym.

Division and multiplication share a rank, and so do addition and subtraction. Where two operations of equal rank sit side by side, you work left to right. So 60 ÷ 5 × 2 is 24, not 6 — the division is done first because it comes first, not because D comes before M in the mnemonic. Similarly 20 − 8 + 3 is 15, not 9. This single rule accounts for a large share of avoidable arithmetic slips.

A fraction bar and a square-root sign act as invisible brackets. An expression written as a stacked fraction means the whole numerator is divided by the whole denominator. When you type it into a calculator on one line you must supply the brackets yourself. To evaluate the mean of 14, 19 and 21 you type (14 + 19 + 21) ÷ 3, not 14 + 19 + 21 ÷ 3. To evaluate a root of a sum you must close the bracket around the whole sum before the calculator will do it correctly.

Watch the difference between the subtraction key and the negative (sign) key on a scientific calculator; they are physically different buttons and using the wrong one produces a syntax error or a wrong sign. Watch also that −3² and (−3)² are different: the first squares 3 and then negates, giving −9; the second squares −3, giving 9.

In a multi-step context problem, the safest method is to break the calculation into named lines rather than typing one long expression. Write the subtotal, then the tax, then the total, each on its own line with a label. You get the same answer, but every line is a place where an assessor can award a method mark, and every line is a place where you can spot an error rather than hunt for it inside a forty-character expression. The ANS key lets you carry the exact unrounded value from one line into the next, which is exactly what you want.

3. Rational numbers: fractions, decimals and percentages as one idea

A rational number is any number that can be written as a ratio of two integers, a/b, where b is not zero. That definition is broader than students expect. Every whole number is rational (7 is 7/1). Every terminating decimal is rational (0.375 is 3/8). Every recurring decimal is rational (0.333... is 1/3). Every percentage is rational (17.5% is 17.5/100, or 7/40). Every ratio, rate and probability you meet in this course is rational.

Because fractions, decimals and percentages are three notations for the same kind of number, fluent conversion between them is a core computation skill. To go from a fraction to a decimal, divide the numerator by the denominator. To go from a decimal to a percentage, multiply by 100 and attach the per cent sign. To go from a percentage to a decimal, divide by 100. In a financial or measurement context you will often want the decimal form for calculating and the percentage form for reporting: you compute with 0.0825, and you write the answer as 8.25%.

Know a small set of equivalences cold, because they let you check a calculator answer instantly: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/5 = 0.2 = 20%, 1/8 = 0.125 = 12.5%, 1/3 = 0.333... ≈ 33.3%. If a question asks for one third of $2400 and your calculator shows 720, the equivalence tells you immediately that something is wrong.

The fraction key on a scientific calculator keeps values exact through a chain of operations, which avoids the rounding drift you get if you convert 1/3 to 0.33 early and then multiply by a large number. Use it when a question involves thirds, sixths, sevenths or ninths — anything that produces a recurring decimal. Convert to a decimal only at the point where you write the final answer, and only to the accuracy the question asks for.

Finally, note the difference between an exact value and a displayed value. A calculator screen shows a rounded version of whatever it is holding internally. The internal value is what carries forward when you press ANS; the displayed value is not. This is the mechanism behind most rounding-error problems, and it is why the instruction is always to round at the final step only.

4. Irrational numbers and where they turn up in measurement

An irrational number cannot be written as a ratio of two integers. Written as a decimal it never terminates and never settles into a repeating block. The two families you meet in this course are both measurement-related, which is why the study design words it as rational numbers and measurement related irrational numbers.

The first is π, which is the ratio of a circle's circumference to its diameter. It appears in every circle, cylinder, cone and sphere calculation. Its value begins 3.14159..., and your calculator has a dedicated π key holding far more digits than the screen shows. Use the key rather than typing 3.14: typing 3.14 introduces an error of about 0.05% at the start of the calculation, which can be enough to miss a rounded answer on a large volume.

The second is a surd — the square root of a whole number that is not a perfect square, such as √2, √3, √5 or √10. These arise from Pythagoras-style right-angle work and from any situation where you take the square root of an area to recover a side length. The diagonal of a square of side 1 is exactly √2, roughly 1.414. Note that √9 = 3 and √0.25 = 0.5 are rational; only roots that do not resolve to a ratio are irrational.

The practical consequence in a context question is this: any answer involving π or a surd is an approximation the moment you write it as a decimal. That is fine — measurement answers are always reported to a stated accuracy — but it means two things for your working. Keep the irrational value inside the calculator for the whole chain of operations, and only round once, at the end, to the accuracy the question demands. And do not be surprised when the answer sheet value differs in the last digit from a friend's; the difference will come from where each of you rounded.

A question may also ask you to classify a number. Be precise: 0.75 is rational, 22/7 is rational (it is a well-known approximation to π, but it is a ratio of integers and therefore not irrational), 2π is irrational, and a measured length such as 4.7 cm is a rational value regardless of what the true underlying length is.

5. Writing conventions the assessors expect: units, money, symbols and layout

Foundation Mathematics is assessed on real-world contexts, so a bare number is rarely a complete answer. Get these conventions automatic.

Money. Write the dollar sign first and two decimal places: $77.55, $1240.00, $0.85. Do not write 77.55$, do not write $77.5, and do not write $77.55c. If a question says round to the nearest dollar, write $78 — the instruction overrides the two-decimal habit. If an amount is given in cents, either keep it in cents with the cent sign or convert fully to dollars; do not mix.

Units. Attach the unit to the answer, and make sure it is the unit the question asked for. A volume computed in cubic centimetres that the question wants in litres is not finished until you convert. Metric symbols are case-sensitive: km, m, cm, mm, kg, g, mg, L, mL, m², m³. Write 5 kg, not 5 Kg or 5 kgs. Rates take a slash or the word per: 12 L/100 km, $28.50 per hour.

Symbols. Use = only between two things that are genuinely equal — do not string a running commentary together with equals signs. Use ≈ when you have rounded. Use % for per cent. Use < and > correctly if a question asks you to compare, and read them carefully in a stem: at least 40 means 40 or more, and more than 40 means 41 or more when the quantity is a whole number.

Layout. Answer in the space provided, one idea per line, with the operation you are doing visible. A useful shape for a five-mark question part is: write the formula or relationship, substitute the numbers, evaluate, then write a sentence stating the answer with its unit. That structure earns method marks even when the arithmetic slips, and it makes a Show that part straightforward because the marker can see each step landing on the stated value.

Where a question supplies a variable — say C for cost or n for the number of items — use that letter. Introducing your own symbol without defining it forces the assessor to guess, and guesses do not earn marks.

6. Using the scientific calculator so it does not create errors

Your calculator will follow the order of operations perfectly. Every calculator error in this subject is really an input error, so build habits that make input errors visible.

Bracket everything you would bracket by hand. If the expression on the page has a fraction bar, put brackets around the numerator and around the denominator. If it has a root over a sum, bracket the sum. If it has a negative number being raised to a power, bracket the negative number. Most scientific calculators show the expression you typed on a line above the answer — read that line back against the question before you accept the result.

Do not retype an intermediate value. Retyping is where rounding error and transcription error enter. Use ANS to reuse the previous result, or store a value in memory (usually STO followed by a letter key) and recall it. If a superannuation or area calculation needs the same subtotal three times, store it once.

Know your modes. If the display shows a fraction when you want a decimal, there is a toggle key (often marked S⇔D) that switches between exact and decimal forms. If numbers are appearing in scientific notation with a small exponent on the right, you are working with a very large or very small value — read the exponent rather than the mantissa alone. Clearing properly between questions (AC, not just DEL) avoids inheriting a stray memory value.

Estimate before you press equals. This is the single most valuable habit in the subject and it is explicitly in the key skills: use estimation and other approaches to check the outcomes, including for accuracy and reasonableness of results. Before evaluating 4.85 × 312, know that the answer is near 5 × 300 = 1500. If the screen shows 15.1 or 15 132, you have a decimal-point or a keystroke problem, and you have caught it in two seconds rather than at the end of the paper.

Sanity-check against context. A weekly wage of $1 950 000, a room 43 metres wide, or a tax bill larger than the income are all signals to go back. Section B questions are set in believable situations; an implausible number is nearly always an arithmetic slip, not a surprising truth.

7. Worked pattern: turning a wordy context into a clean calculation

Consider a typical multi-step Section B opening. A community centre hires a hall for a function. The hire fee is $185 for the first three hours and $46.50 for each additional hour. Catering is charged at $23.80 per guest, and a cleaning deposit of $150 is added and later refunded. The organiser books the hall for seven hours for 64 guests. Calculate the total amount payable before the deposit is refunded, correct to the nearest dollar.

Step 1 — separate the quantities. Additional hours = 7 − 3 = 4. This is the step students skip, and it is where marks are lost: the 7 in the stem is not the number you multiply by $46.50.

Step 2 — write each component on its own line. Hall base = $185.00. Extra hours = 4 × $46.50 = $186.00. Catering = 64 × $23.80 = $1523.20. Deposit = $150.00.

Step 3 — add, then round only now. Total = 185.00 + 186.00 + 1523.20 + 150.00 = $2044.20, which is $2044 to the nearest dollar.

Step 4 — check by estimating. Catering is roughly 60 × $24 = $1440; hire is roughly $185 + $190 = $375; plus $150 gives about $1965. The exact answer of $2044 sits comfortably near that, so the magnitude is right.

Notice what the layout achieves. Each line is a named quantity with a unit, so if the extra-hours line were wrong the catering line would still earn its mark. The rounding instruction is obeyed once, at the end. And the estimate is written down, not merely thought — where a question says Estimate or asks you to check reasonableness, the written estimate is itself worth a mark. Had the question instead said Show that the total is $2044.20, the same four lines would constitute a complete response, because a Show that part requires visible working that arrives at the given value rather than a restatement of it.

8. Examining conventions: where these marks are actually distributed

Conventions and computation are almost never the whole question. They are examined inside questions about money, data and measurement, which means the marks are distributed across the whole paper rather than concentrated in one place. In Section A you will meet one-mark items that turn on a single convention: which of four expressions is evaluated correctly, which number is irrational, which amount is written properly as money, what 3/8 is as a percentage. These take seconds if the conventions are automatic and cost you a mark if they are not. Because no marks are deducted for a wrong multiple-choice answer, never leave one of the twenty blank.

In Section B, conventions are examined through the marking scheme rather than through the wording. A part worth two or more marks expects visible working, and the assessment reports repeatedly note that answer-only responses lose the method marks. The rounding instruction in the stem is not decoration: correct to two significant figures, to the nearest dollar, in litres are all instructions that carry a mark, and rounding at an intermediate step frequently pushes the final digit out of range.

What separates a top-band response is discipline rather than difficulty. The strongest scripts show four things. They define and label — each line of working says what it is calculating. They keep full precision through the chain and round once, at the point of reporting. They attach units and currency in the required form, and they use the letters the question supplied. And they state the answer in a sentence that answers the question that was asked, so a part beginning Determine the total cost ends with a total cost, not with a bare number floating in the margin.

The habit worth building in the last weeks before the exam is to re-read your own working as if you were the assessor. If a line does not make it obvious what quantity you are computing, add three words. Those three words are frequently the difference between a method mark awarded and a method mark withheld.