NSW Syllabuses

# Mathematics K–10 - Stage 4 - Number and Algebra Fractions, Decimals and Percentages

## Outcomes

#### A student:

• MA4-1WM

communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols

• MA4-2WM

applies appropriate mathematical techniques to solve problems

• MA4-3WM

recognises and explains mathematical relationships using reasoning

• MA4-5NA

operates with fractions, decimals and percentages

Related Life Skills outcomes: MALS-8NA, MALS-9NA

## Content

• Students:
• Compare fractions using equivalence; locate and represent positive and negative fractions and mixed numerals on a number line (ACMNA152)
• determine the highest common factor (HCF) of numbers and the lowest common multiple (LCM) of numbers
• generate equivalent fractions
• write a fraction in its simplest form
• express improper fractions as mixed numerals and vice versa
• place positive and negative fractions, mixed numerals and decimals on a number line to compare their relative values
• interpret a given scale to determine fractional values represented on a number line (Problem Solving)
• choose an appropriate scale to display given fractional values on a number line, eg when plotting thirds or sixths, a scale of 3 cm for every whole is easier to use than a scale of 1 cm for every whole (Communicating, Reasoning)
• Solve problems involving addition and subtraction of fractions, including those with unrelated denominators (ACMNA153)
• add and subtract fractions, including mixed numerals and fractions with unrelated denominators, using written and calculator methods
• recognise and explain incorrect operations with fractions, eg explain why $$\frac{2}{3} + \frac{1}{4} \ne \frac{3}{7}$$ (Communicating, Reasoning)
• interpret fractions and mixed numerals on a calculator display (Communicating)
• subtract a fraction from a whole number using mental, written and calculator methods,
eg $$3 - \frac{2}{3} = 2 + 1 - \frac{2}{3} = 2\!\tfrac{1}{3}$$
• Multiply and divide fractions and decimals using efficient written strategies and digital technologies (ACMNA154)
• determine the effect of multiplying or dividing by a number with magnitude less than one
• multiply and divide decimals by powers of 10
• multiply and divide decimals using written methods, limiting operators to two digits
• compare initial estimates with answers obtained by written methods and check by using a calculator (Problem Solving)
• multiply and divide fractions and mixed numerals using written methods
• demonstrate multiplication of a fraction by another fraction using a diagram to illustrate the process (Communicating, Reasoning)
• explain, using a numerical example, why division by a fraction is equivalent to multiplication by its reciprocal (Communicating, Reasoning)
• multiply and divide fractions and decimals using a calculator
• calculate fractions and decimals of quantities using mental, written and calculator methods
• choose the appropriate equivalent form for mental computation, eg 0.25 of $60 is equivalent to $$\frac{1}{4}$$ of$60, which is equivalent to $60 ÷ 4 (Communicating) • Express one quantity as a fraction of another, with and without the use of digital technologies (ACMNA155) • express one quantity as a fraction of another • choose appropriate units to compare two quantities as a fraction, eg 15 minutes is $$\frac{15}{60} = \frac{1}{4}$$ of an hour (Communicating) • Round decimals to a specified number of decimal places (ACMNA156) • round decimals to a given number of decimal places • use symbols for approximation, eg $$\doteqdot$$ or $$\approx$$ • use the notation for recurring (repeating) decimals, eg $$0.33333\!\ldots = 0.\dot3$$, $$0.345345345\!\ldots = 0.\dot3 4 \dot5$$, $$0.266666\!\ldots = 0.2\dot6$$ • convert fractions to terminating or recurring decimals as appropriate • recognise that calculators may show approximations to recurring decimals, and explain why, eg $$\frac{2}{3}$$ displayed as $$0.666666667$$ (Communicating, Reasoning) • Connect fractions, decimals and percentages and carry out simple conversions (ACMNA157) • classify fractions, terminating decimals, recurring decimals and percentages as 'rational' numbers, as they can be written in the form $$\frac{a}{b}$$ where $$a$$ and $$b$$ are integers and $$b \ne 0$$ • convert fractions to decimals (terminating and recurring) and percentages • convert terminating decimals to fractions and percentages • convert percentages to fractions and decimals (terminating and recurring) • evaluate the reasonableness of statements in the media that quote fractions, decimals or percentages, eg 'The number of children in the average family is 2.3' (Communicating, Problem Solving) • order fractions, decimals and percentages • investigate 'irrational' numbers, such as $$\pi$$ and $$\sqrt 2$$ • describe, informally, the properties of irrational numbers (Communicating) • Find percentages of quantities and express one quantity as a percentage of another, with and without the use of digital technologies (ACMNA158) • calculate percentages of quantities using mental, written and calculator methods • choose an appropriate equivalent form for mental computation of percentages of quantities, eg 20% of$40 is equivalent to $$\frac{1}{5}$$ × $40, which is equivalent to$40 ÷ 5 (Communicating)
• express one quantity as a percentage of another, using mental, written and calculator methods, eg 45 minutes is 75% of an hour
• Solve problems involving the use of percentages, including percentage increases and decreases, with and without the use of digital technologies (ACMNA187)
• increase and decrease a quantity by a given percentage, using mental, written and calculator methods
• recognise equivalences when calculating percentage increases and decreases, eg multiplication by 1.05 will increase a number or quantity by 5%, multiplication by 0.87 will decrease a number or quantity by 13% (Reasoning)
• interpret and calculate percentages greater than 100, eg an increase from $2 to$5 is an increase of 150%
• solve a variety of real-life problems involving percentages, including percentage composition problems and problems involving money
• interpret calculator displays in formulating solutions to problems involving percentages by appropriately rounding decimals (Communicating)
• use the unitary method to solve problems involving percentages, eg find the original value, given the value after an increase of 20% (Problem Solving)
• interpret and use nutritional information panels on product packaging where percentages are involved (Problem Solving)
• interpret and use media and sport reports involving percentages (Problem Solving)
• interpret and use statements about the environment involving percentages, eg energy use for different purposes, such as lighting (Problem Solving)

### Background Information

In Stage 3, the study of fractions is limited to denominators of 2, 3, 4, 5, 6, 8, 10, 12 and 100 and calculations involve related denominators only.

Students are unlikely to have had any experience with rounding to a given number of decimal places prior to Stage 4. The term 'decimal place' may need to be clarified. Students should be aware that rounding is a process of 'approximating' and that a rounded number is an 'approximation'.

All recurring decimals are non-terminating decimals, but not all non-terminating decimals are recurring.

The earliest evidence of fractions can be traced to the Egyptian papyrus of the scribe Ahmes (about 1650 BC). In the seventh century AD, the method of writing fractions as we write them now was invented in India, but without the fraction bar (vinculum), which was introduced by the Arabs. Fractions were widely in use by the twelfth century.

One-cent and two-cent coins were withdrawn by the Australian Government in 1990. When an amount of money is calculated, it may have 1, 2, 3 or more decimal places, eg when buying petrol or making interest payments. When paying electronically, the final amount is paid correct to the nearest cent. When paying with cash, the final amount is rounded correct to the nearest five cents, eg

$25.36,$25.37 round to $25.35$25.38, $25.39,$25.41, $25.42 round to$25.40
$25.43,$25.44 round to $25.45. #### Purpose/Relevance of Substrand There are many everyday situations where things, amounts or quantities are 'fractions' or parts (or 'portions') of whole things, whole amounts or whole quantities. Fractions are very important when taking measurements, such as when buying goods (eg three-quarters of a metre of cloth) or following a recipe (eg a third of a cup of sugar), when telling the time (eg a quarter past five), when receiving discounts while shopping (eg 'half price', 'half off'), and when sharing a cake or pizza (eg 'There are five of us, so we'll get one-fifth of the pizza each'). 'Decimals' and 'percentages' represent different ways of expressing fractions (and whole numbers), and so are other ways of representing a part of a whole. Fractions (and decimals and percentages) are of fundamental importance in calculation, allowing us to calculate with parts of wholes and to express answers that are not whole numbers, eg $$4 \div 5 = \frac{4}{5}$$ (or 0.8 or 80%). ### Language In questions that require calculating a fraction or percentage of a quantity, some students may benefit from first writing an expression using the word 'of', before replacing it with the multiplication sign (×). Students may need assistance with the subtleties of the English language when solving word problems. The different processes required by the words 'to' and 'by' in questions such as 'find the percentage increase if$2 is increased to $3' and 'find the percentage increase if$2 is increased by \$3' should be made explicit.

When solving word problems, students should be encouraged to write a few key words on the left-hand side of the equals sign to identify what is being found in each step of their working.

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

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 MA4‑5NA the sub-elements (and levels) of Operating with decimals (OwD2-OwD4), Operating with percentages (OwP2-OwP6), Comparing units (CoU2) and Interpreting fractions (InF5-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.