Levels 1 to 4

This set of exemplars describes a possible progression in number: the progression in students' knowledge, understanding, and use of fractions.

The progression in the development of students' understanding of non-whole numbers (fractions, decimal numbers, and percentages) tells us a lot about their development in number as a whole. Fractions of numbers play an important role in our lives, because not everything that needs to be expressed in number can be represented by integers.

Identify half or quarter of a region

At this stage in the fractions progression, students are able to state what unit fraction of a region is shaded, and/or shade a stated unit fraction of a region.

Assign unit fraction symbols to regions

At this stage in the fractions progression, students are able to write the correct symbol to represent a unit fraction.

Order unit fractions and fractions with like denominators

At this stage in the fractions progression, students are able to rank unit fractions or fractions with common denominators from smallest to largest.

Order fractions with unlike numerators and denominators

At this stage in the fractions progression, students are able to identify equivalent fractions and to order all fractions from smallest to largest.

Convert fractions, decimals, and percentages

At this stage in the fractions progression, students are able to convert fractions to decimals or percentages and vice versa. They are able to use this ability to identify equivalent fractions and to rank all fractions in order of size.

Background to the task

The task chosen as a context for this set of exemplars gives the teacher sufficient scope to observe students' knowledge, understanding, and use of fractions through levels 1 to 4.

The teacher asks the students to:

• complete a series of tasks requiring them to name fractions
• rank fractions in order from smallest to largest.

The teacher increases the difficulty of the task by altering the types of fraction provided, according to the students' level of development.

The task

The teachers asked the students whose work is shown in these exemplars to complete one (or more) of the following tasks, according to their level of development.

• Level 1: The teacher showed the students a region with a fraction shaded ( or ) and asked them to identify how much of it was shaded. If they could identify this verbally, the teacher asked them to write down their answer.
• Levels 2, 3, and 4: The teachers gave the students some sets of fractions and asked them to rank each set from smallest to largest. The sets included:
  • level 2: a set of unit fractions (,, , , ) and a set of fractions with a common denominator (, , , , )
  • levels 2, 3, and 4: a mixed set of fractions (, , , , , )
  • level 4: a set of percentages and decimals to include in the range (30%, 0.5, 0.6, 75%) if the students could order the mixed set of fractions correctly.

Parallel tasks

This progression could be explored in the context of any kind of activity where students are required to use fractions.

References

Department of Education (1985-1989). Beginning School Mathematics: Cycles 1-8. Wellington: School Publications.

Ministry of Education (1992). Beginning School Mathematics: Cycles 9-11. Wellington: Learning Media, Ministry of Education.

Ministry of Education (1993). Beginning School Mathematics: Cycle 12. Wellington: Learning Media, Ministry of Education.

Ministry of Education (2002). Figure It Out, Level 2. Wellington: Learning Media.

Ministry of Education (1999). Figure It Out, Levels 2-3. Wellington: Learning Media.

Ministry of Education (2000). Figure It Out, Level 3. Wellington: Learning Media.

Ministry of Education (2001). Figure It Out, Levels 3-4. Wellington: Learning Media.

Ministry of Education (2002). Figure It Out, Levels 7-8. Wellington: Learning Media.

Note: Each level of Figure It Out consists of a set of student books, with an accompanying Answers and Teachers Notes for each book.

Ministry of Education (1992). Mathematics in the New Zealand Curriculum. Wellington: Learning Media.

Ministry of Education (1996). Te Whāriki: He Whāriki Mātauranga mō ngā Mokopuna o Aotearoa/Early Childhood Curriculum. Wellington: Learning Media.