Step 4: The result 4 is an

improper fraction and must be reduced to a mixed number, which would be 1 1/4.

6, an equation 1 25 = 13/5 can be recognized as an error step of converting a mixed number to an

improper fraction. Similarly, an equation 1 3/5 = 4/5 also can be recognized as an error step with the aid of Eq.

Students, who have only learned the shortcut algorithm for renaming mixed numbers as

improper fractions, may not know that a mixed number is a sum.

The data from 323 interviews with students at the end of Year 6 (Clarke, Roche, Mitchell & Sukenik, 2006) indicated that students need classroom experiences which assist them to understand more clearly the roles of the numerator and denominator in a fraction, the meaning of

improper fractions, and the relative sizes of fractions.

For example, an environment for solving linear equations can accept both decimal and common fractions or mixed numbers and

improper fractions as answers.

Grade 4 Introduction of fractions;

improper fractions and mixed numbers; comparison of fractions (with like denominators only) Grade 5 Comparison of fractions (unlike denominators); equivalent fractions; addition and subtraction of fractions with like denominators; fractions as quotient; relationships among fractions, decimals & whole numbers Grade 6 Addition and subtraction of fractions with unlike denominators; creating equivalent fractions; multiplication and division of fractions Table 2.

These operations suggest that Joe was developing a scheme for generating

improper fractions.

For example, it could be argued that the number line is better suited to representing

improper fractions than either the area or set models.

For example, in one classroom, the teacher used the Interactive Whiteboard to illustrate how to use simple regional models to explore

improper fractions. The teacher gave the students a fraction (such as 5/3) and the students illustrated 5/3 using circle pieces (3 1/3rds to make one whole plus 2 additional thirds).

Varying the numbers in these problems produces a variety of fractions as solutions, including unit fractions, such as 1/3 for three children sharing one candy bar; nonunit fractions, such as 3/4 in the candy-bar problem;

improper fractions, such as 10/4 for four children sharing ten candy bars; and mixed numbers, such as 2 1/6 for six children sharing thirteen candy bars.

Over the course of the intervention, we gradually introduced more complicated fractions, equivalent fractions,

improper fractions and a wider breadth of representations of fractions.

When one student named a fraction as "fifteen ninths", his partner demonstrated that the fifteen pieces on the twenty-four mat equaled fifteen twenty-fourths, saying, "But we have fifteen pieces and it takes twenty-four to make a whole." This came about because the author previously reviewed that the denominator meant "the number of pieces it takes to make a whole", rather than "the number of pieces in all" to clarify confusion and allow the definition to include

improper fractions.