1. Fraction notation and textbooks in Australia Peter Gould

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

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

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

5. Representing fractions- Australia Students are expected to demonstrate their understanding of fractions by shading in parts of a shape. For example: New Signpost mathematics 7 p. 318 7 10 (b) Shade seven-tenths () of the following shape. (c) Shade four-fifths ( ) of the 4 5 following shape. Maths Plus Unit 4 Stage 2 p. 15 3 8 (a)Colour the correct number of equal parts. Colour Go Maths Stage 2B p. 42

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

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

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

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

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

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

12. Fractions beyond one

14. 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.