NSW Syllabuses

# Mathematics K–10 - Stage 3 - Number and Algebra Fractions and Decimals

## Fractions and Decimals 1

### Outcomes

#### A student:

• MA3-1WM

describes and represents mathematical situations in a variety of ways using mathematical terminology and some conventions

• MA3-2WM

selects and applies appropriate problem-solving strategies, including the use of digital technologies, in undertaking investigations

• MA3-3WM

gives a valid reason for supporting one possible solution over another

• MA3-7NA

compares, orders and calculates with fractions, decimals and percentages

• place fractions with denominators of 2, 3, 4, 5, 6, 8, 10 and 12 on a number line between 0 and 1, eg
• compare and order unit fractions with denominators of 2, 3, 4, 5, 6, 8, 10, 12 and 100
• compare the relative value of unit fractions by placing them on a number line between 0 and 1 (Communicating, Reasoning)
• investigate and explain the relationship between the value of a unit fraction and its denominator (Communicating, Reasoning)
• Investigate strategies to solve problems involving addition and subtraction of fractions with the same denominator (ACMNA103)
• identify and describe 'proper fractions' as fractions in which the numerator is less than the denominator
• identify and describe 'improper fractions' as fractions in which the numerator is greater than the denominator
• express mixed numerals as improper fractions and vice versa, through the use of diagrams and number lines, leading to a mental strategy, eg
• model and represent strategies, including using diagrams, to add proper fractions with the same denominator, where the result may be a mixed numeral, eg
• model and represent a whole number added to a proper fraction, eg $$2 + \frac{3}{4} = 2\tfrac{3}{4}$$
• subtract a proper fraction from another proper fraction with the same denominator, eg $$\frac{7}{8} - \frac{2}{8} = \frac{5}{8}$$
• model and represent strategies, including using diagrams, to add mixed numerals with the same denominator, eg
• use diagrams, and mental and written strategies, to subtract a unit fraction from any whole number including 1, eg
• solve word problems that involve addition and subtraction of fractions with the same denominator, eg 'I eat $$\frac{1}{5}$$ of a block of chocolate and you eat $$\frac{3}{5}$$ of the same block. How much of the block of chocolate has been eaten?'
• use estimation to verify that an answer is reasonable (Problem Solving, Reasoning)
• Recognise that the place value system can be extended beyond hundredths (ACMNA104)
• express thousandths as decimals
• interpret decimal notation for thousandths, eg $$0.123 = \frac{123}{1000}$$
• state the place value of digits in decimal numbers of up to three decimal places
• Compare, order and represent decimals (ACMNA105)
• compare and order decimal numbers of up to three decimal places, eg 0.5, 0.125, 0.25
• interpret zero digit(s) at the end of a decimal, eg 0.170 has the same value as 0.17
• place decimal numbers of up to three decimal places on a number line between 0 and 1

### Background Information

In Stage 3 Fractions and Decimals, students study fractions with denominators of 2, 3, 4, 5, 6, 8, 10, 12 and 100. A unit fraction is any proper fraction in which the numerator is 1, eg $$\frac{1}{2},\,~\frac{1}{3},\,~\frac{1}{4},\,~\frac{1}{5}, ~\ldots$$

Fractions may be interpreted in different ways depending on the context, eg two-quarters $$\left(\frac{2}{4}\right)$$ may be thought of as two equal parts of one whole that has been divided into four equal parts.

Alternatively, two-quarters $$\left(\frac{2}{4}\right)$$ may be thought of as two equal parts of two wholes that have each been divided into quarters.

Students need to interpret a variety of word problems and translate them into mathematical diagrams and/or fraction notation. Fractions have different meanings depending on the context, eg show on a diagram three-quarters $$\left(\frac{3}{4}\right)$$ of a pizza, draw a diagram to show how much each child receives when four children share three pizzas.

### Language

Students should be able to communicate using the following language: whole, equal parts, half, quarter, eighth, third, sixth, twelfth, fifth, tenth, hundredth, thousandth, one-thousandth, fraction, numerator, denominator, mixed numeral, whole number, number line, proper fraction, improper fraction, decimal, decimal point, digit, place value, decimal places.

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

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 MA3‑7NA the sub-elements (and levels) of Quantifying numbers (QuN10-QuN11), Operating with decimals (OwD3) 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:

• MA3-1WM

describes and represents mathematical situations in a variety of ways using mathematical terminology and some conventions

• MA3-2WM

selects and applies appropriate problem-solving strategies, including the use of digital technologies, in undertaking investigations

• MA3-3WM

gives a valid reason for supporting one possible solution over another

• MA3-7NA

compares, orders and calculates with fractions, decimals and percentages

• model, compare and represent fractions with denominator of 2, 3, 4, 5, 6, 8, 10, 12 and 100 of a whole object, a whole shape and a collection of objects
• compare the relative size of fractions drawn on the same diagram, eg
(Reasoning)
• compare and order simple fractions with related denominators using strategies such as diagrams, the number line, or equivalent fractions, eg write $$\frac{3}{5},\,~\frac{3}{{10}},\,~1\tfrac{1}{{10}},\,~\frac{4}{5}$$ and $$\frac{7}{{10}}$$ in ascending order
• find equivalent fractions by re-dividing the whole, using diagrams and number lines, eg
• record equivalent fractions using diagrams and numerals
• develop mental strategies for generating equivalent fractions, such as multiplying or dividing the numerator and the denominator by the same number,

eg $$\frac{1}{4} = \frac{{1 \times 2}}{{4 \times 2}} = \frac{{1 \times 3}}{{4 \times 3}} = \frac{{1 \times 4}}{{4 \times 4}} = ~\ldots$$, ie $$\frac{1}{4} = \frac{2}{8} = \frac{3}{{12}} = \frac{4}{{16}} = ~\ldots$$
• explain or demonstrate why two fractions are or are not equivalent (Communicating, Reasoning)
• write fractions in their 'simplest form' by dividing the numerator and the denominator by a common factor, eg $$\frac{4}{16} = \frac{4 \div 4}{16 \div 4} = \frac{1}{4}$$
• recognise that a fraction in its simplest form represents the same value as the original fraction (Reasoning)
• apply knowledge of equivalent fractions to convert between units of time, eg 15 minutes is the same as $$\frac{15}{60}$$ of an hour, which is the same as $$\frac{1}{4}$$ of an hour (Problem Solving)
• Solve problems involving addition and subtraction of fractions with the same or related denominators (ACMNA126)
• add and subtract fractions, including mixed numerals, where one denominator is the same as, or a multiple of, the other, eg $$\frac{2}{3} + \frac{1}{6}$$,   $$2\tfrac{3}{8} - 1\tfrac{1}{2}$$,   $$2\tfrac{3}{8} - \frac{3}{4}$$
• convert an answer that is an improper fraction to a mixed numeral (Communicating)
• use knowledge of equivalence to simplify answers when adding and subtracting fractions (Communicating, Reasoning)
• recognise that improper fractions may sometimes make calculations involving mixed numerals easier (Communicating)
• solve word problems involving the addition and subtraction of fractions where one denominator is the same as, or a multiple of, the other, eg 'I ate $$\frac{1}{8}$$ of a cake and my friend ate $$\frac{1}{4}$$ of the cake. What fraction of the cake remains?'
• multiply simple fractions by whole numbers using repeated addition, leading to a rule,
eg $$\frac{2}{5} \times 3 = \frac{2}{5}+\frac{2}{5}+\frac{2}{5} = \frac{6}{5} = 1\tfrac{1}{5}$$ leading to $$\frac{2}{5} \times 3 = \frac{2 \times 3}{5} = \frac{6}{5} = 1\tfrac{1}{5}$$
• Find a simple fraction of a quantity where the result is a whole number, with and without the use of digital technologies (ACMNA127)
• calculate unit fractions of collections, with and without the use of digital technologies, eg calculate $$\frac{1}{5}$$ of 30
• describe the connection between finding a unit fraction of a collection and the operation of division (Communicating, Problem Solving)
• calculate a simple fraction of a collection/quantity, with and without the use of digital technologies, eg calculate $$\frac{2}{5}$$ of 30
• explain how unit fractions can be used in the calculation of simple fractions of collections/quantities, eg 'To calculate $$\frac{3}{8}$$ of a quantity, I found $$\frac{1}{8}$$ of the collection first and then multiplied by 3' (Communicating, Reasoning)
• solve word problems involving a fraction of a collection/quantity
• Add and subtract decimals, with and without the use of digital technologies, and use estimation and rounding to check the reasonableness of answers (ACMNA128)
• add and subtract decimals with the same number of decimal places, with and without the use of digital technologies
• add and subtract decimals with a different number of decimal places, with and without the use of digital technologies
• relate decimals to fractions to aid mental strategies (Communicating)
• round a number of up to three decimal places to the nearest whole number
• use estimation and rounding to check the reasonableness of answers when adding and subtracting decimals
• describe situations where the estimation of calculations with decimals may be useful, eg to check the total cost of multiple items when shopping (Communicating, Problem Solving)
• solve word problems involving the addition and subtraction of decimals, with and without the use of digital technologies, including those involving money
• use selected words to describe each step of the solution process (Communicating, Problem Solving)
• interpret a calculator display in the context of the problem, eg 2.6 means $2.60 (Communicating) • Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating decimals, with and without the use of digital technologies (ACMNA129) • use mental strategies to multiply simple decimals by single-digit numbers, eg 3.5 × 2 • multiply decimals of up to three decimal places by whole numbers of up to two digits, with and without the use of digital technologies, eg 'I measured three desks. Each desk was 1.25 m in length, so the total length is 3 × 1.25 = 3.75 m' • divide decimals by a one-digit whole number where the result is a terminating decimal, eg 5.25 ÷ 5 = 1.05 • solve word problems involving the multiplication and division of decimals, including those involving money, eg determine the 'best buy' for different-sized cartons of cans of soft drink • Multiply and divide decimals by powers of 10 (ACMNA130) • recognise the number patterns formed when decimals are multiplied and divided by 10, 100 and 1000 • multiply and divide decimals by 10, 100 and 1000 • use a calculator to explore the effect of multiplying and dividing decimals by multiples of 10 (Reasoning) • Make connections between equivalent fractions, decimals and percentages (ACMNA131) • recognise that the symbol % means 'percent' • represent common percentages as fractions and decimals, eg '25% means 25 out of 100 or $$\frac{1}{4}$$ or 0.25' • recognise fractions, decimals and percentages as different representations of the same value (Communicating) • recall commonly used equivalent percentages, decimals and fractions, eg 75%, 0.75, $$\frac{3}{4}$$ (Communicating) • represent simple fractions as decimals and as percentages • interpret and explain the use of fractions, decimals and percentages in everyday contexts, eg $$\frac{3}{4}$$ hour = 45 minutes, percentage of trees in the local area that are native to Australia (Communicating, Reasoning) • represent decimals as fractions and percentages, eg $$1.37 = 137\% = \frac {137}{100} = 1 \tfrac {37}{100}$$ • Investigate and calculate percentage discounts of 10%, 25% and 50% on sale items, with and without the use of digital technologies (ACMNA132) • equate 10% to $$\frac{1}{10}$$, 25% to $$\frac{1}{4}$$ and 50% to $$\frac{1}{2}$$ • calculate common percentages (10%, 25%, 50%) of quantities, with and without the use of digital technologies • choose the most appropriate equivalent form of a percentage to aid calculation, eg $$25\%~\text{of}~\200 = \frac{1}{4}~\text{of}~\200 = \200 \div 4 = \50$$ (Problem Solving) • use mental strategies to estimate discounts of 10%, 25% and 50%, eg '50% off the price of$122.70: 50% is the same as $$\frac{1}{2}$$, so the discount is about \$60'
• calculate the sale price of an item after a discount of 10%, 25% and 50%, with and without the use of digital technologies, recording the strategy and result

### Background Information

In Stage 3 Fractions and Decimals, students study fractions with denominators of 2, 3, 4, 5, 6, 8, 10, 12 and 100. A unit fraction is any proper fraction in which the numerator is 1, eg $$\frac{1}{2},\,~\frac{1}{3},\,~\frac{1}{4},\,~\frac{1}{5}, ~\ldots$$

The process of writing a fraction in its 'simplest form' involves reducing the fraction to its lowest equivalent form. In Stage 4, this is referred to as 'simplifying' a fraction.

When subtracting mixed numerals, working with the whole-number parts separately from the fractional parts can lead to difficulties, particularly where the subtraction of the fractional parts results in a negative value, eg in the calculation of $$2\tfrac{1}{3}-1\tfrac{5}{6}$$,   $$\frac{1}{3} - \frac{5}{6}$$ results in a negative value.

### Language

Students should be able to communicate using the following language: whole, equal parts, half, quarter, eighth, third, sixth, twelfth, fifth, tenth, hundredth, thousandth, fraction, numerator, denominator, mixed numeral, whole number, number line, proper fraction, improper fraction, is equal to, equivalent, ascending order, descending order, simplest form, decimal, decimal point, digit, round to, decimal places, dollars, cents, best buy, percent, percentage, discount, sale price.

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

The word 'cent' is derived from the Latin word centum, meaning 'one hundred'. 'Percent' means 'out of one hundred' or 'hundredths'.

A 'terminating' decimal has a finite number of decimal places, eg 3.25 (2 decimal places), 18.421 (3 decimal places).

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

When working towards the outcome MA3‑7NA the sub-elements (and levels) of Operating with decimals (OwD1-OwD4), Operating with percentages (OwP1-OwP3), Number patterns and algebraic thinking (NPA4) 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.