This document provides an outline for teaching a stage 5 mathematics unit on surds and indices. It includes key ideas such as defining rational and irrational numbers, performing operations with surds, and using integers and fractions for index notation. Students will learn to distinguish between rational and irrational numbers, perform the four basic operations with surds, expand expressions involving surds, and convert between surd and index form. The unit aims to build students' understanding of the real number system.

1. TOPIC: 4
SURDS & INDICES
OUTCOMES:
Stage 5: NS 5.3.1 Performs operations with surds and indices ( p68 )
SUGGESTED TIME:
CONTENT KNOWLEDGE AND SKILLS RESOURCES TERMINOLOGY
Key Ideas for Stage 5 Students learn about
• defining a rational number:
A rational number is the ratio
b
a
of two integers
where b ≠ 0.
• distinguishing between rational and irrational numbers
• using a pair of compasses and a straight edge to
construct simple rationals and surds on the number line
• defining real numbers:
Real numbers are represented by points on the number
line.
Irrational numbers are real numbers that are not
rational.
• demonstrating that x is undefined for x < 0,
0=x for x = 0, and x is the positive square root
of x when 0>x
• using the following results for x, y > 0:
( )
y
x
y
x
yxxy
xxx
=
=
==
.
22
• using the four operations of addition, subtraction,
multiplication and division to simplify expressions
involving surds
• expanding expressions involving surds such as
( )2
53 + or ( )( )3232 +−
• rationalising the denominators of surds of the form
• New Coarse Maths
Yr9 Int Ch 4
• New Century Maths
Int 9
Ch 1 p42 – 45
• New Century Maths
Yr 9 Adv Ch 5
• New Century Maths
Yr 10 Adv Ch 1
• Maths Works 10 Int
Ch 13
• Mathscape Yr 9
Stage 5·3
Ch 8
surd
rational
irrational
rationalise
binomial
conjugate
radical
1. Define the system of real numbers distinguishing
between rational and irrational numbers
2. Perform operations with surds
3. Use integers and fractions for index notation
4. Convert between surd and index form.
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2. dc
ba
• using the index laws to demonstrate the reasonableness
of the definitions for fractional indices
n mn
m
nn
xx
xx
=
=
1
• translating expressions in surd form to expressions in
index form and vice versa
evaluating numerical expressions involving fractional
indices eg 3
2
27
• using the y
x
1
key on a calculator
• evaluating a fraction raised to the power of –1, leading
to
a
b
b
a
=





−1
WORKING
MATHEMATICALLY
• explain why all integers and
recurring decimals are
rational numbers
(Communicating, Reasoning)
• demonstrate that not all real
numbers are rational
(Communicating, Applying
Strategies, Reasoning)
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