# YEAR 8 MATHS FOCUS

## NUMBER AND ALGEBRA

## FRACTIONS & DECIMALS

OUTCOME

A student:

MA4-5NA:

operates with fractions, decimals and percentages

TEACHING POINTS | 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 One-cent and two-cent coins were withdrawn by the Australian Government in 1990.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, e.g. 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 |

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. 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. |

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, e.g. 4÷5=45 (or 0.8 or 80%). |

## Expectations of Attainment

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, e.g. 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, e.g. explain why 2/3+1/4≠3/7 {Communicating, Reasoning, Literacy, Critical and creative thinking} | |

– interpret fractions and mixed numerals on a calculator display {Communicating, Critical and creative thinking} | |

subtract a fraction from a whole number using mental, written and calculator methods, e.g. 3−2/3=2+1−2/3=2 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, Critical and creative thinking} | |

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, Literacy} | |

– explain, using a numerical example, why division by a fraction is equivalent to multiplication by its reciprocal {Communicating, Reasoning, Literacy, Critical and creative thinking} | |

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, e.g. 0.25 of $60 is equivalent to 1/4 of $60, which is equivalent to $60 ÷ 4 {Communicating, Critical and creative thinking} |

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, e.g. 15 minutes is 15/60=1/4 of an hour {Communicating, Critical and creative thinking} |

Round decimals to a specified number of decimal places (ACMNA156) | round decimals to a given number of decimal places |

use symbols for approximation, e.g. ≑ or ≈ {Literacy} |

Investigate terminating and recurring decimals(ACMNA184) | use the notation for recurring (repeating) decimals, e.g. 0.33333…=0.3˙, 0.345345345…=0.3˙45˙, 0.266666…=0.26˙ {Literacy} |

convert fractions to terminating or recurring decimals as appropriate | |

– recognise that calculators may show approximations to recurring decimals, and explain why, e.g. 23 displayed as 0.666666667 {Communicating, Reasoning, Critical and creative thinking} |

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 ab where a and b are integers and b≠0 {Literacy} |

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, e.g. ‘The number of children in the average family is 2.3’ {Communicating, Problem Solving, Critical and creative thinking} | |

order fractions, decimals and percentages |

Investigate the concept of irrational numbers, including π (pie) (ACMNA186) | investigate ‘irrational’ numbers, such as π and √2 {Literacy Critical and creative thinking} |

– describe, informally, the properties of irrational numbers {Communicating, Literacy} |

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