ACARA v9 CONTENT DESCRIPTION “use the 4 operations with integers and with rational numbers, choosing and using efficient strategies and digital tools where appropriate”
Working across the whole number system
The number system you work with now reaches in both directions from zero. It includes the positive whole numbers, their negative partners, and all of the rational numbers that sit between them, written either as fractions or as decimals. Integers and rational numbers share a single number line: every one of them has a place on it, with negatives to the left of zero and positives to the right. The same four operations, addition, subtraction, multiplication and division, work across all of these numbers. This unit is about carrying them out fluently and choosing an efficient route through each calculation.
One number line for every number
Integers and rationals, positive and negative, share one line.
integers and rational numbers (fractions and decimals, positive and negative) all live on one number line, and the same four operations apply across all of them.
Adding and subtracting integers
Adding and subtracting integers becomes clear when you picture movement along the number line. You start at the first number and let the operation point you in a direction. Adding moves you to the right, so -3 + 5 means start at -3 and step five places to the right, landing on 2. Subtracting moves you to the left, so 2 - 6 means start at 2 and step six places to the left, landing on -4. Crossing zero along the way is perfectly normal, and the result can be negative. One handy fact ties the two together: adding a negative is the same as subtracting, so 5 + (-2) = 5 - 2 = 3.
Adding and subtracting as movement
Adding steps right; subtracting steps left.
adding moves right, subtracting moves left; crossing 0 is normal, and the result can be negative.
Multiplying and dividing with signs
Multiplying and dividing integers follow a single rule about signs. When the two numbers share the same sign, the answer is positive; when they have different signs, the answer is negative. So -3 x 4 = -12, because the signs differ, while -2 x -6 = 12, because the signs match. Division behaves in exactly the same way: -12 / -3 = 4, since the signs are the same, and 20 / -5 = -4, since they differ. The size of the answer comes from the ordinary product or quotient of the digits, and the sign rule simply decides whether a minus sign belongs in front. Because the rule is identical for multiplication and division, it is easy to remember.
The sign rules for x and /
Same signs give plus; different signs give minus.
same signs give a positive answer; different signs give a negative answer, for both multiplication and division.
The four operations with fractions
Rational numbers written as fractions follow steady procedures. To add or subtract, rewrite both fractions over a common denominator and then combine the numerators, so 1/2 + 1/3 becomes 3/6 + 2/6 = 5/6. To multiply, multiply the numerators together and the denominators together, so 2/3 x 3/4 = 6/12, which simplifies to 1/2. To divide, multiply by the reciprocal of the second fraction, that is, turn it upside down, so 3/4 / 1/2 = 3/4 x 2/1 = 3/2. Always simplify the result where you can. The very same calculations can be carried out on the decimal forms of these numbers, and a sound answer agrees whichever form you choose.
One worked example of each kind
Add over a common denominator, multiply across, divide by the reciprocal.
add and subtract fractions over a common denominator; multiply across; divide by multiplying by the reciprocal.
Order of operations and efficient strategies
When an expression mixes these operations, the order in which you work matters. Deal with brackets first, then multiplication and division, and finally addition and subtraction. For 2 + 3 x (8 - 6), the bracket gives 2, then 3 x 2 gives 6, and 2 + 6 gives 8; working in any other order would give the wrong total. The same order applies when negatives and fractions appear in the expression. Alongside the order, choose an efficient strategy: simplify or reduce a fraction before you multiply, look for friendly numbers that combine neatly, and reach for a calculator or other digital tool when the values are awkward. A tidy route keeps the work short and reliable.
Working in the right order
Brackets, then multiply and divide, then add and subtract.
do brackets first, then multiplication and division, then addition and subtraction; choosing a tidy order keeps the work efficient and correct.
Why this matters
Being fluent and accurate with operations across integers and rational numbers underpins almost all of the mathematics that follows. Algebra, measurement, geometry and statistics all lean on confident arithmetic with positive and negative numbers and with fractions and decimals. Choosing an efficient strategy saves effort and reduces mistakes, while getting the signs and the order of operations right is what makes a longer calculation trustworthy from start to finish. Treat each calculation as a short, clear sequence of steps, and confirm the result in a second form when you can. This unit stays with pure numeric computation; the use of percentages in financial and modelling contexts is the focus of the next unit.