## Interpreting fractions description

This sub-element emphasises the development of the fraction concept and the size of fractions rather than the development of procedures or algorithmic skills. Understanding the size of a fraction is an indicator of the depth of a student’s understanding of the fraction concept.^{[1]}

This sub-element describes how a student becomes increasingly able to use fractions as numbers that describe a relationship between two abstract measures of quantity. Rather than representing two numbers, the fraction \(\frac ab\) represents the result of dividing one by the other. That is, \(\frac23\) is the* number* that results from dividing 2 by 3. Although the notation used with fractions is very powerful, its meaning can often remain opaque. A common misconception is thinking of a fraction as two whole numbers and not as a single number.

*Some students will communicate using augmentative and alternative communication strategies to demonstrate their numeracy skills. This may include digital technologies, sign language, braille, real objects, photographs and pictographs.*

*Each sub-element level has been identified by upper-case initials and in some cases lower-case letters of the sub-element name followed by ascending numbers. The abbreviation for this sub-element is InF. The listing of indicators within each level is non-hierarchical. Subheadings have been included to group related indicators. Where appropriate, examples have been provided in brackets following an indicator.*

[1] Post, TP, Behr, MJ, Lesh, R & Wachsmuth, I 1986, Selected results from the Rational Number Project, http://education.umn.edu/rationalnumberproject

### InF1

**Creating halves**

- identifies the part and the whole
- recognises dividing a whole into 2 parts can create equal or unequal parts
- creates equal halves by attending to the linear aspect of a model (folds a paper strip in half to make equal pieces by aligning the edges or makes 2 groups of 3 when halving a collection of 6 counters in a
__linear arrangement__) - distinguishes between halfway and half

### InF2

**Repeated halving**

- recognises quarters and eighths formed by repeated halving of a length (finds halfway then halves each half, or repeatedly halves using a
__linear arrangement__of__discrete items__– 8 counters halved and then halved again into 4 groups of 2)

### InF3

**Repeating fractional parts**

- accumulates fractional parts of a length (knows that two-quarters is inclusive of one-quarter and twice one-quarter, not just the second quarter)
- checks the equality of parts by
__iterating__one part to form the whole (when given a representation of one-quarter of a length and asked, ‘what fraction is this of the whole length?’, compares the size of the unit to the whole)

### InF4

**Re-imagining the whole**

- calculates thirds by visualising or approximating and adjusting (imagines a paper strip in 3 parts, then adjusts and folds)
- identifies examples and non-examples of partitioned representations of thirds and fifths
- recognises the whole can be redivided into different fractional parts for different purposes (a strip of paper divided into quarters can be redivided to show fifths)
- demonstrates that the more parts into which a whole is divided, the smaller the parts become

### InF5

**Equivalence of fractions**

- identifies the need to have equal wholes to compare fractional parts (explains why one-third as a number is larger than one-quarter)
- creates fractions larger than 1 by recreating the whole (when creating four-thirds, recognises that three-thirds corresponds to the whole and the fourth third is part of an additional whole)
- creates
__equivalent fractions__by dividing the same-sized whole into different parts (shows two-sixths is the same as one-third of the same whole) - links partitioning to establish relationships between fractions (creates one-sixth as one-third of one-half)

### InF6

**Fractions as numbers**

- connects the concepts of fractions and division: a fraction is a
__quotient__, or a division statement (two-sixths is the same as 2 ÷ 6 or 2 partitioned into 6 equal parts) - justifies where to place fractions on a number line (to place two-thirds on a number line, divides the space between 0 and 1 into 3 equal parts)
- understands the relationship between a fraction, decimal and percentage as different representations of the same quantity (½ = 0.5 = 50%)
- shows an understanding that a fraction represents a single number, not two separate whole numbers (explains why \(\frac24\) is not halfway between \(\frac13\) and \(\frac35\), although 2 is midway between 1 and 3 and 4 is midway between 3 and 5)

### InF7

**Using fractions**

- uses knowledge of equivalence to compare fractions (when comparing two-thirds and three-quarters, subdivides the whole into twelfths)
- justifies the need for the same denominators to add or subtract fractions
- uses strategies to find a fraction of a quantity (to find two-thirds of 27, finds one-third then doubles)
- demonstrates why dividing by a fraction can result in a larger number
- understands the difference between multiplying and dividing fractions (recognises \(\frac12\times\frac14\) as one-half of a quarter and \(\frac12\div\frac14\) as how many quarters are in one half)