ACARA v9 CONTENT DESCRIPTION “apply knowledge of equivalence to compare, order and represent common fractions including halves, thirds and quarters on the same number line and justify their order”
Builds on: Factors, Multiples and Primes (AC9M6N02). Knowing factors and multiples helps with equivalent fractions — the same value written with a larger top and bottom.
Equivalent fractions
Two fractions are equivalent when they describe exactly the same amount, even though they are written differently. Half of a bar is the same length as two quarters or three sixths of that bar; one half, two quarters and three sixths are all equivalent. An equivalent fraction is made by multiplying the top and bottom by the same number, which cuts the same amount into more, smaller pieces without changing its size. Recognising that one value can wear many fraction names is the key that makes comparing and ordering fractions possible.
Equivalent fractions
Cutting the same length into more pieces gives a fraction equal to the first.
1/2 = 2/4 = 3/6 — the same length split into more pieces is an equivalent fraction.
Fractions on one line
Every fraction has a place on the number line between 0 and 1. One half sits exactly halfway; one quarter sits a quarter of the way along; three quarters sits three quarters along. Placing two fractions on the same line settles which is larger at a glance: the one further to the right is greater. The number line turns fractions, which can be hard to compare as written symbols, into positions that can simply be looked at, and it is the central tool this unit uses to compare and order them.
Fractions on one line
Putting two fractions on the same number line shows at a glance which is bigger.
Placed on one line, 1/2 sits to the right of 1/3, so 1/2 is the larger fraction.
Comparing two fractions
To compare two fractions, the surest method is to picture each as a shaded part of the same whole. Shade one half of a bar and one third of an identical bar, and the half is plainly longer, so one half is greater than one third. When the pieces are the same size, like thirds against thirds, the one with more pieces wins; when the number of pieces is the same, like one half against one third, the one with larger pieces wins. Comparing fractions of the same whole removes the guesswork.
Comparing two fractions
Shade each fraction of the same bar; the longer shading is the larger fraction.
Which fraction is larger? Compare the shaded lengths, then pick.
Ordering fractions
Ordering means arranging three or more fractions from smallest to largest, and the number line makes it straightforward. Place each fraction at its position, then read them off from left to right. One quarter, one half and three quarters fall neatly in that order along the line. When fractions have different denominators, rewriting them as equivalent fractions with a common denominator lines them up for easy ordering. Ordering is simply comparing carried out across a whole group at once, and the same number-line thinking does the work.
Ordering fractions
Position each fraction on one line, then read them off from left to right.
Place each fraction on the line, then order them from smallest to largest.
Finding equivalents
Equivalent fractions are the tool that makes comparing different denominators possible. Multiplying the top and bottom of one half by three gives three sixths; the value is unchanged, but now it can be compared directly with other sixths. The reverse also works: dividing top and bottom by a common factor simplifies a fraction to its lowest terms, so two quarters becomes one half. Being able to move freely between equivalent forms, making pieces smaller or larger at will, is what lets any two fractions be brought to a common footing and compared.
Finding an equivalent
Multiply the top and bottom by the same number to get an equivalent fraction.
Which fraction equals 1/2? Pick A, B or C.
Justifying the order
Comparing and ordering fractions is only half the skill; the other half is being able to justify it, to say clearly why one fraction is larger or why two are equal. Halves are larger pieces than thirds, so one half exceeds one third. Two quarters equals one half because it simplifies to it. On a number line, three quarters lies to the right of two thirds, so it is greater. A good justification points to equivalence, to piece size, or to position on the line, turning a correct answer into a reasoned one that can be explained and trusted.
Justifying the order
Comparing fractions is not enough; you should be able to say why one is larger.
Choose the best justification. Pick A, B or C.
Where fractions lead
Comparing, ordering and justifying fractions is the groundwork for all later fraction work. From here come adding and subtracting fractions, which depend on equivalent fractions with common denominators, and the link between fractions, decimals and percentages, which are three ways of naming the same value. The habit of seeing a fraction as a position on a line, and of moving between equivalent forms, carries straight into ratio, proportion and the algebra of later years, where fractions appear constantly.
Quick self-check
1. Which fraction is equivalent to 1/2?
2. On a number line from 0 to 1, which fraction sits furthest to the right?
3. Which fraction is larger, 1/3 or 1/2?
4. Put 1/2, 1/4 and 3/4 in order from smallest to largest. The smallest is...