Outcomes
A student:

 MA5.31WM
uses and interprets formal definitions and generalisations when explaining solutions and/or conjectures

 MA5.32WM
generalises mathematical ideas and techniques to analyse and solve problems efficiently

 MA5.33WM
uses deductive reasoning in presenting arguments and formal proofs

 MA5.36NA
performs operations with surds and indices
Content
 Students:
 Define rational and irrational numbers and perform operations with surds and fractional indices (ACMNA264)
 define real numbers: a real number is any number that can be represented by a point on the number line
 define rational and irrational numbers: a rational number is any number that can be written as the ratio \( \frac{a}{b} \) of two integers \(a\) and \(b\) where \( b \ne 0 \); an irrational number is a real number that is not rational
 recognise that all rational and irrational numbers are real (Reasoning)
 explain why all integers, terminating decimals and recurring decimals are rational numbers (Communicating, Reasoning)
 explain why rational numbers can be expressed in decimal form (Communicating, Reasoning)
 use a pair of compasses and a straight edge to construct simple rational numbers and surds on the number line (Problem Solving)
 distinguish between rational and irrational numbers
 demonstrate that not all real numbers are rational (Problem Solving)
 use the term 'surd' to refer to irrational expressions of the form \(\sqrt[n]{x}\) where \(x\) is a rational number and \(n\) is an integer such that \(n \ge 2\)

write recurring decimals in fraction form using calculator and noncalculator methods,
eg \( 0.\dot{2} \), \( 0.\dot{2}\dot{3} \), \( 0.2\dot{3} \)
 justify why \( 0.\dot{9} = 1 \) (Communicating, Reasoning)
 demonstrate that \( \sqrt{x} \) is undefined for \(x < 0\) and that \( \sqrt{x} = 0 \) for \(x= 0\)
 define \( \sqrt{x} \) as the positive square root of \(x\) for \(x > 0\)

use the following results for \(x > 0\) and \(y > 0\):
\( \begin{align}\left(\sqrt{x}\right)^2 &= x = \sqrt{x^2}\\\sqrt{xy} &= \sqrt{x}\times\sqrt{y}\\ \sqrt{\frac{x}{y}} &= \frac{\sqrt{x}}{\sqrt{y}}\end{align} \)  apply the four operations of addition, subtraction, multiplication and division to simplify expressions involving surds
 explain why a particular sentence is incorrect, eg explain why \( \sqrt{3} + \sqrt{5} \ne \sqrt{8} \) (Communicating, Reasoning)
 expand expressions involving surds, eg expand \( \left(\sqrt{3}+\sqrt{5}\right)^2 \), \( \left(2\sqrt{3}\right) \left(2+\sqrt{3}\right) \)
 connect operations with surds to algebraic techniques (Communicating)
 rationalise the denominators of surds of the form \( \frac{a\sqrt{b}}{c\sqrt{d}} \)
 investigate methods of rationalising surdic expressions with binomial denominators, making appropriate connections to algebraic techniques (Problem Solving)
 recognise that a surd is an exact value that can be approximated by a rounded decimal
 use surds to solve problems where a decimal answer is insufficient, eg find the exact perpendicular height of an equilateral triangle (Problem Solving)
 establish that \( \left(\sqrt{a}\right)^2 = \sqrt{a}\times\sqrt{a} = \sqrt{a \times a} = \sqrt{a^2} = a\)

apply index laws to demonstrate the appropriateness of the definition of the fractional index representing the square root, eg
\( \begin{align}\left(\sqrt{a}\right)^2 &= a\\\textrm{and} \; \left(a^{\frac{1}{2}}\right)^2 &= a \\\therefore \; \sqrt{a} &= a^{\frac{1}{2}}\end{align} \)
 explain why finding the square root of an expression is the same as raising the expression to the power of a half (Communicating, Reasoning)
 apply index laws to demonstrate the appropriateness of the following definitions for fractional indices: \( x^{\frac{1}{n}} = \sqrt[n]{x} \), \( x^{\frac{m}{n}} = \sqrt[n]{x^m} \)
 translate expressions in surd form to expressions in index form and vice versa
 use the or equivalent key on a scientific calculator
 evaluate numerical expressions involving fractional indices, eg \( 27^{\frac{2}{3}} \)
Background Information
Operations with surds are applied when simplifying algebraic expressions.
Having expanded binomial products and rationalised denominators of surds of the form \( \frac{a\sqrt{b}}{c\sqrt{d}} \), students could rationalise the denominators of surds with binomial denominators.
Early Greek mathematicians believed that the length of any line could always be given by a rational number. This was proved to be false when the Greek philosopher and mathematician Pythagoras (c580–c500 BC) and his followers found that the length of the hypotenuse of an isosceles rightangled triangle with side length one unit could not be given by a rational number.
Some students may enjoy a demonstration of the proof by contradiction that \( \sqrt{2} \) is irrational.
Language
There is a need to emphasise to students how to read and articulate surds and fractional indices, eg \( \sqrt{x} \) is 'the square root of \(x\)' or 'root \(x\)'.