Fraction notation and textbooks in Australia Peter Gould

$What is meant by a fraction? Subconstructs: 1. Part-whole 2. Quotient 3. Ratio number$

What is meant by a fraction? Subconstructs: 1. Part-whole 2. Quotient 3. Ratio number 4. Operator 5. Measure

$What is meant by a fraction? Schemes: 1. Simultaneous partitioning 2. Part-whole 3. Equi-partitioning$

What is meant by a fraction? Schemes: 1. Simultaneous partitioning 2. Part-whole 3. Equi-partitioning 4. Partitive fractional 5. Reversible partitive fractional 6. Iterative fractional

$Two types of fractions 1. Partitioned A partitioned fraction is always a fraction of$

Two types of fractions 1. Partitioned A partitioned fraction is always a fraction of something. 2. Quantity A quantity fraction references a universal measurement unit – a whole that doesn’t change size.

$Representing fractions- Australia Students are expected to demonstrate their understanding of fractions by shading$

Representing fractions- Australia Students are expected to demonstrate their understanding of fractions by shading in parts of a shape. For example: (b) Shade seven-tenths ( of the following shape. New Signpost mathematics 7 7 10 ) (a) Colour the correct number of equal parts. Colour 3 8 Go Maths Stage 2 B p. 42 (c) Shade four-fifths ( 54) of the following shape. p. 318 Maths Plus Unit 4 Stage 2 p. 15

What do students think? A Year 5 student showing one-third as 3 equal parts

What do students think? A Year 6 student’s representation of number of parts

Which is larger, 1 3 1 6 or ? Number of parts rather than area of the parts

$What fraction is the shaded part? Do you think this is one-third or three-eighths?$

What fraction is the shaded part? Do you think this is one-third or three-eighths?

Visual discrimination • aligned lengths, • non-aligned length, • angle, • area, • slope. Easier Harder The Elements of Graphing Data (Cleveland, 1994).

Interpreting regional models Children cannot use subtended angle or area to visually interpret regional models. Many count parts.

Fractions beyond one

Teacher materials • Although regional models such as pieces of circles are often used to introduce fractions, some students focus on the number of regions and not the area of the regions compared to the whole shape. • Teaching materials using bar models will be developed and trialed in upcoming lesson studies.