### Where are they now?

Recognises and names halves of familiar objects (e.g. half an apple)

### Where to next?

- Recognises and names halves of familiar collections (e.g. half the class)
- Recognises that the ‘whole’ is bigger than one half
*Recognises that equal parts are needed*

### Purpose

**Building Fraction and decimal skills:**Even before they come to school many young children exhibit an awareness of fraction names such as half and quarter. During the first years of schooling, most will be able to halve a piece of paper, identify 3 quarters of an orange and talk about parts of recognised wholes (e.g. blocks of chocolate, pizza, Smarties etc.). While to an adult ear, this sounds like children understand the relationship inherent in fraction representatives, for many they are simply using these items to describe and/or enumerate well-known objects. Such children may not be aware of or even attending to the key ideas involved in a more general understanding of fractions; that is, that equal parts are involved, the number of parts names the parts, and that as the number of parts of a given whole are increased, the size of the parts (or shares) gets smaller.

The use of fraction words to name and recognize parts of recognized wholes lulls adults (teachers and parents) into thinking that many of these children are able to understand and use the fraction symbol in the early years of schooling despite the fact that most curriculum advice now advocates a delay in the formalisation of fractions. When children are introduced to the fraction symbol without a deep understanding of what each part of the symbol refers to they are inclined, quite naturally, to view both the numerator and the denominator as counting or ‘how many numbers’. This leads ultimately to such misconceptions as ‘3/12 is bigger than 3/8 because 12 is bigger than 8’. It also leads to the well-known ‘Freshman’s error’, that is the tendency to add denominators when adding fractions

*(Siemon, 2003).*Imagine a student encountering the symbols we use to record fractions. She is told that ¾ is the same as three out of four. The student then demonstrates her understanding of fraction notation by stating that three people out of four people is the same as ¾, two people out of five people is the same as 2/5 and five people out of nine people is the same as 5/9. All appears well until this student surprises you by writing ¾ + 2/5 = 5/9. For this student and many others, fractions are “two numbers” rather than a single thing.

The rapid transition from modeling fractions to recording fractions in symbolic form, numerator over denominator, can contribute to many students’ confusion. Recording fractions in symbolic form needs to build on an underpinning conceptual framework of units (or parts) and collections of parts that form new units. Emphasizing numeric rules too soon without underlying meaning discourages students from attempting to see rational numbers as something sensible

*(CMIT LFN, 2007, p,47).**Specifically moving from step 1-3 to step 4:*The first goal in the development of fractions should be to help children construct the idea of

*fractional parts of the whole*– the parts that result when the whole or unit has been partitioned in equal-sized portions or fair shares. Children seem to understand the idea of separating a quantity into two or more parts to be shared fairly among friends. They eventually make connections between the idea of fair shares and fractional parts. Sharing tasks are, therefore good places to begin the development of fractions*(Van De Walle, 2006).*The process of equal sharing or division underpins the fraction concept

*(CMIT LFN, 2007, p.47).*

### References and links to other resources

** Count me in too: Learning framework in number**, (CMIT LFN), NSW Department of Education and Training, Professional Support and Curriculum Directorate, 2007.

** Developing Efficient Numeracy Strategies Stage 1** (DENS1) p.34, NSW Department of Education and Training, Professional Support and Curriculum Directorate 2003.

** Elementary and Middle School Mathematics: Teaching developmentally**, John Van De Walle (7

^{th}edn), 2010.

** Partitioning – The missing link in building fraction knowledge and confidence, **Di Siemon, RMIT University Vic, 2003.

** Fractions: pikelets and lamingtons**, NSW Department of Education and Training, Professional Support and Curriculum Directorate, 2003.

** Teaching developmentally**, John Van De Walle (7

^{th}edn), 2010.

** Teaching student-centered mathematics grades K-3**, John Van De Walle & LouAnn Lovin, 2006.

**Online resources:**

http://www.whatworks.edu.au/upload/1282272495845_file_4Numeracy.pdf

*This article provides a clear overview of what works in improving mathematics outcomes for indigenous students.

**Online activities:**

http://www.illuminations.nctm.org

### Activities to move students from Step 4 to Step 5

**Halving the group**

**Focus: **That the halving idea works with collections as well as with physical objects. Extension of ‘equal parts’ idea to collections. Equal sharing and ‘splitting of twos’ as useful methods of halving collections

**How: **See *Halving the group instruction sheet
*

**Beehive activity – sharing to make two equal groups**

**Focus: **Consolidating the idea of halving collections using physical objects. Consolidation of ‘equal parts’ or ‘equal sharing’ idea.

**Halving story**

A halving activity which uses a story shell to help engage the students.

**Focus:** The language of halving. Consider how this language can be made explicit in teaching team planning time so that student L1 explanations can be planned. Equal sharing and ‘splitting of twos’ as useful methods of halving collections

**How: **See *Halving Story instruction sheet
*

**Halve the count**

**Focus:** Halving groups of items visually, then counting and adjusting to ensure halves. Estimation and adjustment (guessing and checking) as useful methods of halving collections

**How: **See* Halve the Count instruction sheet
*