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# Fractions: Theoretical Background

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Tweet## Stevie Cassidy

on 7 October 2012#### Transcript of Fractions: Theoretical Background

A fraction is part of a whole and less than one (Jameson-Proctor, R. 2012) Fractions are quite different from the concepts of whole numbers that students have had experience with. Common fractions are their first encounter with a set of numbers not based on counting

(Behr & Post, 1998, cited by Baroody & Coslick, p.9-7).

It is important that students develop a comprehensive introduction and exploration of fractions as many student difficulties in algebra can be traced back to an incomplete understanding of earlier fraction ideas. (Behr et al., 1993, p.3) Children partake in the concept of fractions during early childhood and have an insistence on fairness (Perry & Conroy, 1994), developing their informal ideas of partitioning, sharing, and measuring. Their experience in sharing equal amounts can provide an introduction to the formal understanding of fractions. In some ways, sharing can play the role for rational numbers that counting does for whole numbers. (National Research Council, 2001. p232) Research suggests that fractions is a topic which many teachers find difficult to understand and teach and many students find difficult to learn. Among the factors that make rational numbers in general, and fractions in particular difficult to understand are their many representations and interpretations.” (Clarke, D.M., Mitchell., A & Roche, A. 2005. p1) Fractions: Theoretical Background According to Van De Wall (2004), providing students with the opportunity to experience fractions in a variety of different ways, including such contexts as division, ratio and parts of a whole to really develop a comprehensive understanding. There are three categories of models for working with fractions;

Length

Set or quantity

Area All fractions have the same attributes:

1.Fractions are part of a whole and as such smaller than the whole

2.All sections of a fraction are of equivalent size

3.As the NUMBER of sections or fractions gets LARGER in quantity, the SIZE of the sections or fractions gets SMALLER.

(Van de Walle, J. 2004)

This differs to what student’s are familiar with in Whole Number i.e. in whole number, as the NUMBER gets LARGER so does its value. Teachers need to be mindful of this when teaching student’s fractions, ensuring they use correct language and move through each of the language stages thoroughly to ensure student’s have ample opportunity to really understand and work through the fractions concepts, skills and strategies. The Australian Curriculum introduces fractions beginning in Year 1, moving through to year 7.

Year 1

Recognise and describe one-half as one of two equal parts of a whole. (ACMNA016)

Year 2

Recognise and interpret common uses of halves, quarters and eighths of shapes and collections (ACMNA033)

Year 3

Model and represent unit fractions including 1/2, 1/4, 1/3, 1/5 and their multiples to a complete whole (ACMNA058)

Year 4

Investigate equivalent fractions used in contexts (ACMNA077)

Count by quarters halves and thirds, including with mixed numerals. Locate and represent these fractions on a number line (ACMNA078)

Recognise that the place value system can be extended to tenths and hundredths. Make connections between fractions and decimal notation (ACMNA079)

Year 5

Compare and order common unit fractions and locate and represent them on a number line(ACMNA102)

Investigate strategies to solve problems involving addition and subtraction of fractions with the same denominator(ACMNA103)

Year 6

Compare fractions with related denominators and locate and represent them on a number line(ACMNA125)

Solve problems involving addition and subtraction of fractions with the same or related denominators(ACMNA126)

Find a simple fraction of a quantity where the result is a whole number, with and without digital technologies (ACMNA127)

Add and subtract decimals, with and without digital technologies, and use estimation and rounding to check the reasonableness of answers (ACMNA128)

Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating decimals, with and without digital technologies (ACMNA129)

Multiply and divide decimals by powers of 10 (ACMNA130)

Make connections between equivalent fractions, decimals and percentages (ACMNA131)

Year 7

Compare fractions using equivalence. Locate and represent positive and negative fractions and mixed numbers on a number line(ACMNA152

Solve problems involving addition and subtraction of fractions, including those with unrelated denominators (ACMNA153)

Multiply and divide fractions and decimals using efficient written strategies and digital technologies (ACMNA154)

Express one quantity as a fraction of another, with and without the use of digital technologies (ACMNA155)

Round decimals to a specified number of decimal places (ACMNA156)

Connect fractions, decimals and percentages and carry out simple conversions (ACMNA157)

Find percentages of quantities and express one quantity as a percentage of another, with and without digital technologies. (ACMNA158)

Recognise and solve problems involving simple ratios (ACMNA173)

(ACARA, n.d.) Big Ideas about Fractions!

Fractions are smaller, equal sized parts of a whole.

These parts can have unique names like thirds, or denominator, depending on which context you are referring to them in.

Because Fractions are much more complicated than Whole Number, students need lots of time to work with fractions in a tangible and meaningful way, across a variety of contexts.

Students can make meaning in fractions by working with partitioning and iterating, with particular emphasis placed on work around the numerator and denominator.

Understanding equivalent fractions is imperative. It allows students to work with operations in fractions, simplifying of fractions, and describing the same amount using a variety of different fractions.

(Van De Walle, J. 2004;

Jamieson-Proctor, R. 2012)

Full transcript(Behr & Post, 1998, cited by Baroody & Coslick, p.9-7).

It is important that students develop a comprehensive introduction and exploration of fractions as many student difficulties in algebra can be traced back to an incomplete understanding of earlier fraction ideas. (Behr et al., 1993, p.3) Children partake in the concept of fractions during early childhood and have an insistence on fairness (Perry & Conroy, 1994), developing their informal ideas of partitioning, sharing, and measuring. Their experience in sharing equal amounts can provide an introduction to the formal understanding of fractions. In some ways, sharing can play the role for rational numbers that counting does for whole numbers. (National Research Council, 2001. p232) Research suggests that fractions is a topic which many teachers find difficult to understand and teach and many students find difficult to learn. Among the factors that make rational numbers in general, and fractions in particular difficult to understand are their many representations and interpretations.” (Clarke, D.M., Mitchell., A & Roche, A. 2005. p1) Fractions: Theoretical Background According to Van De Wall (2004), providing students with the opportunity to experience fractions in a variety of different ways, including such contexts as division, ratio and parts of a whole to really develop a comprehensive understanding. There are three categories of models for working with fractions;

Length

Set or quantity

Area All fractions have the same attributes:

1.Fractions are part of a whole and as such smaller than the whole

2.All sections of a fraction are of equivalent size

3.As the NUMBER of sections or fractions gets LARGER in quantity, the SIZE of the sections or fractions gets SMALLER.

(Van de Walle, J. 2004)

This differs to what student’s are familiar with in Whole Number i.e. in whole number, as the NUMBER gets LARGER so does its value. Teachers need to be mindful of this when teaching student’s fractions, ensuring they use correct language and move through each of the language stages thoroughly to ensure student’s have ample opportunity to really understand and work through the fractions concepts, skills and strategies. The Australian Curriculum introduces fractions beginning in Year 1, moving through to year 7.

Year 1

Recognise and describe one-half as one of two equal parts of a whole. (ACMNA016)

Year 2

Recognise and interpret common uses of halves, quarters and eighths of shapes and collections (ACMNA033)

Year 3

Model and represent unit fractions including 1/2, 1/4, 1/3, 1/5 and their multiples to a complete whole (ACMNA058)

Year 4

Investigate equivalent fractions used in contexts (ACMNA077)

Count by quarters halves and thirds, including with mixed numerals. Locate and represent these fractions on a number line (ACMNA078)

Recognise that the place value system can be extended to tenths and hundredths. Make connections between fractions and decimal notation (ACMNA079)

Year 5

Compare and order common unit fractions and locate and represent them on a number line(ACMNA102)

Investigate strategies to solve problems involving addition and subtraction of fractions with the same denominator(ACMNA103)

Year 6

Compare fractions with related denominators and locate and represent them on a number line(ACMNA125)

Solve problems involving addition and subtraction of fractions with the same or related denominators(ACMNA126)

Find a simple fraction of a quantity where the result is a whole number, with and without digital technologies (ACMNA127)

Add and subtract decimals, with and without digital technologies, and use estimation and rounding to check the reasonableness of answers (ACMNA128)

Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating decimals, with and without digital technologies (ACMNA129)

Multiply and divide decimals by powers of 10 (ACMNA130)

Make connections between equivalent fractions, decimals and percentages (ACMNA131)

Year 7

Compare fractions using equivalence. Locate and represent positive and negative fractions and mixed numbers on a number line(ACMNA152

Solve problems involving addition and subtraction of fractions, including those with unrelated denominators (ACMNA153)

Multiply and divide fractions and decimals using efficient written strategies and digital technologies (ACMNA154)

Express one quantity as a fraction of another, with and without the use of digital technologies (ACMNA155)

Round decimals to a specified number of decimal places (ACMNA156)

Connect fractions, decimals and percentages and carry out simple conversions (ACMNA157)

Find percentages of quantities and express one quantity as a percentage of another, with and without digital technologies. (ACMNA158)

Recognise and solve problems involving simple ratios (ACMNA173)

(ACARA, n.d.) Big Ideas about Fractions!

Fractions are smaller, equal sized parts of a whole.

These parts can have unique names like thirds, or denominator, depending on which context you are referring to them in.

Because Fractions are much more complicated than Whole Number, students need lots of time to work with fractions in a tangible and meaningful way, across a variety of contexts.

Students can make meaning in fractions by working with partitioning and iterating, with particular emphasis placed on work around the numerator and denominator.

Understanding equivalent fractions is imperative. It allows students to work with operations in fractions, simplifying of fractions, and describing the same amount using a variety of different fractions.

(Van De Walle, J. 2004;

Jamieson-Proctor, R. 2012)