ACARA v9 CONTENT DESCRIPTION “compare and order fractions with the same and related denominators including mixed numerals, applying knowledge of factors and multiples; represent these fractions on a number line”
Builds on a first understanding of fractions as equal parts of a whole, and on the factors and multiples met earlier this year. Year 5 sharpens this into comparing and ordering fractions: deciding which of two fractions is larger, and arranging several in order. When fractions share a denominator the comparison is direct; when their denominators are related, equivalent fractions and a common denominator make them comparable. Placing fractions on a number line ties it all together, and prepares the way for adding, subtracting and working with fractions in the units that follow.
Equivalent fractions name the same amount
Different fractions can describe exactly the same amount. One half, two quarters and three sixths all shade the same part of a whole, so they are equivalent fractions. An equivalent fraction is found by multiplying the top and bottom by the same number: multiplying one half by two over two gives two quarters, and by three over three gives three sixths. This is where factors and multiples come in, because the new denominator is always a multiple of the old one. Recognising equivalent fractions is the key to comparing fractions that look different but may be closer than they seem, or even equal.
Equivalent fractions name the same amount
Different fractions can shade the same part.
Reveal the equivalent fractions: each shades the same amount.
Comparing fractions with the same denominator
When two fractions have the same denominator, comparing them is straightforward. The denominator tells how many equal parts the whole is cut into, and since the parts are the same size, the fraction with more of them is larger. Five eighths is greater than three eighths because five parts are more than three parts of the same size. The numerators alone decide the order. This simple case is the foundation for every harder comparison, because the strategy for fractions with different denominators is to rewrite them so that they share one.
Comparing fractions with the same denominator
Same parts: the larger numerator wins.
With the same denominator, which is bigger: 3/8 or 5/8?
Fractions on a number line
A number line gives fractions a place to live. Marking zero at one end and one at the other, the line can be divided into equal steps to locate any fraction: one quarter sits one step of four along, and three quarters three steps along. Placing fractions on the line shows their size at a glance and makes their order visible, since a fraction further to the right is larger. The number line also makes equivalent fractions obvious, because one half and two quarters land on exactly the same point. It is a picture of comparison that works for every fraction.
Fractions on a number line
A fraction further right is larger.
Place each fraction on the line from 0 to 1.
Comparing fractions with related denominators
Fractions with different denominators can still be compared when the denominators are related, meaning one is a multiple of the other. To compare two thirds and five sixths, notice that six is a multiple of three, so two thirds can be rewritten as four sixths. Now both fractions share the denominator six, and the comparison is easy: five sixths is greater than four sixths. Finding a common denominator using factors and multiples turns an awkward comparison into the simple same-denominator case. The amounts never change; they are simply renamed so they can be measured against each other.
Comparing fractions with related denominators
Rewrite to a shared denominator, then compare.
Which is bigger: 2/3 or 5/6? Their denominators are related.
Ordering fractions and mixed numerals
Ordering means arranging several fractions from smallest to largest, and the same tools apply. Fractions with a common denominator are ordered by their numerators; fractions with related denominators are first rewritten to share one. Mixed numerals, like one and a quarter, are ordered by their whole-number part first and then by the fraction, so one and a quarter is larger than three quarters. Placing each fraction on a number line is a reliable check, since the order along the line is the order of size. Ordering pulls together equivalence, common denominators and the number line into a single skill.
Ordering fractions and mixed numerals
Order by size; whole part first for mixed numerals.
Arrange 3/4, 1/4, 1/2 from smallest to largest.
Comparing fractions with confidence
Comparing and ordering fractions comes down to making them comparable. If the denominators match, compare the numerators; if they are related, rewrite to a common denominator using factors and multiples; and use a number line whenever a picture helps. Equivalent fractions are the thread running through all of it, showing when fractions are equal and how to rename them. With these habits a child can compare any two fractions, order a whole set including mixed numerals, and is ready for the adding and subtracting of fractions that the next units bring.
Quick self-check
1. Two fractions have the same denominator. The bigger one has...
2. Which fraction is equivalent to 1/2?
3. To compare 2/3 and 5/6, a good first step is to...
4. On a number line from 0 to 1, the fraction 3/4 sits...
5. Ordered from smallest to largest, 1/2, 1/4 and 3/4 are...