NSW Syllabuses

# Mathematics K–10 - Stage 2 - Number and Algebra Fractions and Decimals

## Fractions and Decimals 1

### Outcomes

#### A student:

• MA2-1WM

uses appropriate terminology to describe, and symbols to represent, mathematical ideas

• MA2-3WM

checks the accuracy of a statement and explains the reasoning used

• MA2-7NA

represents, models and compares commonly used fractions and decimals

### Content

• Students:
• Model and represent unit fractions, including $$\frac{1}{2},\,~\frac{1}{4},\,~\frac{1}{3}$$ and $$\frac{1}{5}$$ and their multiples, to a complete whole (ACMNA058)
• model fractions with denominators of 2, 3, 4, 5 and 8 of whole objects, shapes and collections using concrete materials and diagrams, eg
• recognise that as the number of parts that a whole is divided into becomes larger, the size of each part becomes smaller (Reasoning)
• recognise that fractions are used to describe one or more parts of a whole where the parts are equal, eg  (Communicating, Reasoning)
• name fractions up to one whole, eg $$\frac{1}{5},\,~\frac{2}{5},\,~\frac{3}{5},\,~\frac{4}{5},\,~\frac{5}{5}$$
• interpret the denominator as the number of equal parts a whole has been divided into
• interpret the numerator as the number of equal fractional parts, eg $$\frac{3}{8}$$ means 3 equal parts of 8
• use the terms 'fraction', 'denominator' and 'numerator' appropriately when referring to fractions
• Count by quarters, halves and thirds, including with mixed numerals; locate and represent these fractions on a number line (ACMNA078)
• identify and describe 'mixed numerals' as having a whole-number part and a fractional part
• rename $$\frac{2}{2},\,~\frac{3}{3},\,~\frac{4}{4},\,~\frac{5}{5}$$ and $$\frac{8}{8}$$ as 1
• count by halves, thirds and quarters, eg $$0, \,~\frac{1}{3}, \,~ \frac{2}{3}, \,~ 1, \,~ 1\!\tfrac{1}{3}, \,~ 1\!\tfrac{2}{3}, \,~ 2, \,~ 2\!\tfrac{1}{3}, \,~ \ldots$$
• place halves, quarters, eighths and thirds on number lines between 0 and 1, eg
• place halves, thirds and quarters on number lines that extend beyond 1, eg
• compare unit fractions using diagrams and number lines and by referring to the denominator, eg $$\frac{1}{8}$$ is less than $$\frac{1}{2}$$
• recognise and explain the relationship between the value of a unit fraction and its denominator (Communicating, Reasoning)

### Background Information

In Stage 2 Fractions and Decimals 1, fractions with denominators of 2, 3, 4, 5 and 8 are studied. Denominators of 6, 10 and 100 are introduced in Stage 2 Fractions and Decimals 2.

Fractions are used in different ways: to describe equal parts of a whole; to describe equal parts of a collection of objects; to denote numbers (eg $$\frac{1}{2}$$ is midway between 0 and 1 on the number line); and as operators related to division (eg dividing a number in half).

A unit fraction is any proper fraction in which the numerator is 1, eg $$\frac{1}{2},\,~\frac{1}{3},\,~\frac{1}{4},\,~\frac{1}{5},~ \ldots$$

#### Three Models of Fractions

Continuous model, linear – uses one-directional cuts or folds that compare fractional parts based on length. Cuts or folds may be either vertical or horizontal. This model was introduced in Stage 1.

Continuous model, area – uses multi-directional cuts or folds to compare fractional parts to the whole. This model should be introduced once students have an understanding of the concept of area in Stage 2.

Discrete model – uses separate items in collections to represent parts of the whole group. This model was introduced in Stage 1.

### Language

Students should be able to communicate using the following language: whole, part, equal parts, half, quarter, eighth, third, fifth, one-third, one-fifth, fraction, denominator, numerator, mixed numeral, whole number, fractional part, number line.

When expressing fractions in English, the numerator is said first, followed by the denominator. However, in many Asian languages (eg Chinese, Japanese), the opposite is the case: the denominator is said before the numerator.

### National Numeracy Learning Progression links to this Mathematics outcome

When working towards the outcome MA2‑7NA the sub-elements (and levels) of Multiplicative strategies (MuS1, MuS4-MuS5), Number patterns and algebraic thinking (NPA4), Comparing units (CoU1) and Interpreting fractions (InF1-InF7) describe observable behaviours that can aid teachers in making evidence-based decisions about student development and future learning.

The progression sub-elements and indicators can be viewed by accessing the National Numeracy Learning Progression.

## Fractions and Decimals 2

### Outcomes

#### A student:

• MA2-1WM

uses appropriate terminology to describe, and symbols to represent, mathematical ideas

• MA2-3WM

checks the accuracy of a statement and explains the reasoning used

• MA2-7NA

represents, models and compares commonly used fractions and decimals

### Content

• model, compare and represent fractions with denominators of 2, 4 and 8; 3 and 6; and 5, 10 and 100
• model, compare and represent the equivalence of fractions with related denominators by redividing the whole, using concrete materials, diagrams and number lines, eg
• record equivalent fractions using diagrams and numerals, eg $$\frac{3}{5}=\frac{6}{10}$$
• Recognise that the place value system can be extended to tenths and hundredths, and make connections between fractions and decimal notation (ACMNA079)
• recognise and apply decimal notation to express whole numbers, tenths and hundredths as decimals, eg 0.1 is the same as $$\frac{1}{10}$$
• investigate equivalences using various methods, eg use a number line or a calculator to show that $$\frac{1}{2}$$ is the same as 0.5 and $$\frac{5}{10}$$ (Communicating, Reasoning)
• identify and interpret the everyday use of fractions and decimals, such as those in advertisements (Communicating, Problem Solving)
• state the place value of digits in decimal numbers of up to two decimal places
• use place value to partition decimals of up to two decimal places, eg $$5.37 = 5 + \frac{3}{10} + \frac{7}{100}$$
• partition decimals of up to two decimal places in non-standard forms, eg $$5.37 = 5 + \frac{37}{100}$$
• apply knowledge of hundredths to represent amounts of money in decimal form, eg five dollars and 35 cents is $$5\tfrac{35}{100}$$, which is the same as $5.35 (Communicating) • model, compare and represent decimals of up to two decimal places • apply knowledge of decimals to record measurements, eg 123 cm = 1.23 m (Communicating) • interpret zero digit(s) at the end of a decimal, eg 0.70 has the same value as 0.7, 3.00 and 3.0 have the same value as 3 (Communicating) • recognise that amounts of money are written with two decimal places, eg$4.30 is not written as $4.3 (Communicating) • use one of the symbols for dollars ($) and cents (c) correctly when expressing amounts of money, ie $5.67 and 567c are correct, but$5.67c is not (Communicating)
• use a calculator to create patterns involving decimal numbers, eg 1 ÷ 10, 2 ÷ 10, 3 ÷ 10 (Communicating)
• place decimals of up to two decimal places on a number line, eg place 0.5, 0.25 and 0.75 on a number line
• round a number with one or two decimal places to the nearest whole number

### Background Information

In Stage 2 Fractions and Decimals 2, fractions with denominators of 2, 3, 4, 5, 6, 8, 10 and 100 are studied. Denominators of 2, 3, 4, 5 and 8 were introduced in Stage 2 Fractions and Decimals 1.

Fractions are used in different ways: to describe equal parts of a whole; to describe equal parts of a collection of objects; to denote numbers (eg $$\frac{1}{2}$$ is midway between 0 and 1 on the number line); and as operators related to division (eg dividing a number in half).

Money is an application of decimals to two decimal places.

Refer also to background information in Fractions and Decimals 1.

### Language

Students should be able to communicate using the following language: whole, part, equal parts, half, quarter, eighth, third, sixth, fifth, tenth, hundredth, one-sixth, one-tenth, one-hundredth, fraction, numerator, denominator, whole number, number line, is equal to, equivalent fractions, decimal, decimal point, digit, place value, round to, decimal places, dollars, cents.

The decimal 1.12 is read as 'one point one two' and not 'one point twelve'.

Refer also to language in Fractions and Decimals 1.

### National Numeracy Learning Progression links to this Mathematics outcome

When working towards the outcome MA2‑7NA the sub-elements (and levels) of Quantifying numbers (QuN9-QuN11), Multiplicative strategies (MuS1, MuS4-MuS6), Operating with decimals (OwD1), Interpreting fractions (InF1-InF7) and Understanding units of measurement (Uum8) describe observable behaviours that can aid teachers in making evidence-based decisions about student development and future learning.

The progression sub-elements and indicators can be viewed by accessing the National Numeracy Learning Progression.