National Numeracy Learning Progression

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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.

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.

 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)