TEACHER’S RESOURCE BOOK
Published 2025
The Educational Company of Ireland
Ballymount Road
Walkinstown
Dublin 12
www.edco.ie
A member of the Smurfit Westrock Group plc
© Stephanie Mulligan, Brendan Guildea, Megan Kelly, Teresa Carroll 2025
ISBN: 978-1-80230-205-9
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior permission of the publishers or a licence permitting restricted copying in Ireland issued by the Irish Copyright Licensing Agency, 63 Patrick Street, Dún Laoghaire, Co. Dublin.
Copy editor: Sarah Ryan
Proofreader: Christine Bruce
Design and layout: Compuscript
Cover: Design Image
Web references in this book are intended as a guide for teachers. At the time of going to press, all web addresses were active and contained information relevant to the topics in this book. However, The Educational Company of Ireland and the authors do not accept responsibility for the views or information contained in these websites. Content and addresses may change beyond our control and pupils should be supervised when investigating websites. While every care has been taken to trace and acknowledge copyright, the publishers tender their apologies for any accidental infringement where copyright has proved untraceable. They would be pleased to come to a suitable arrangement with the rightful owner in each case.
Introduction to Junior Cycle Maths
The Junior Cycle
Under the new specifications, all subjects take an integrated approach, helping students to develop a wide range of skills and thinking abilities.
All subjects are based on:
Eight principles
Twenty-four statements of learning
Eight key skills.
Eight Principles
The eight principles shown in the diagram and outlined below are applicable to all subjects across the Junior Cycle.
Learning to learn: support independent learning
Choice and flexibility: provide a wide choice of learning experiences
Quality: offer high-quality education
Creativity and innovation: provide opportunities for students to be creative and innovative
Engagement and participation: encourage participation, generate engagement and enthusiasm, and connect with life outside school
Continuity and development: enable students to build on their learning, recognise progress in learning and support future learning
Inclusive education: include all students and provide equality of opportunity, participation and outcomes
Wellbeing: contribute to students’ physical, mental, emotional and social wellbeing and resilience
Twenty-four Statements of Learning (SOLs)
‘The twenty-four statements, underpinned by the eight principles, are central to planning for, the students’ experience of, and the evaluation of the school’s Junior Cycle programme.’ They can be found in the ‘Framework for Junior Cycle 2015’ document available at ncca.ie/en/ resources/framework-for-junior-cycle-2015-2.
SOLs Relevant to Maths
Below are some statements relevant to maths.
SOL 1 The student communicates effectively using a variety of means in a range of contexts in L1.
SOL 14 The student makes informed financial decisions and develops good consumer skills.
SOL 15 The student recognises the potential uses of mathematical knowledge, skills and understanding in all areas of learning.
SOL 16 The student describes, illustrates, interprets, predicts and explains patterns and relationships.
SOL 17 The student devises and evaluates strategies for investigating and solving problems using mathematical knowledge, reasoning and skills.
SOL 18 The student observes and evaluates empirical events and processes and draws valid deductions and conclusions.
SOL 24 The student uses technology and digital media tools to learn, communicate, work and think collaboratively and creatively in a responsible and ethical manner.
Eight Key Skills
The eight key skills are required for successful learning by students inside and outside school.
K E Y S K I L L S
Being literate Developing my understanding and enjoyment of words and language; reading for enjoyment and with critical understanding; writing for different purposes; expressing ideas clearly and accurately; developing my spoken language; exploring and creating a variety of texts, including multi-modal texts.
Managing myself Knowing myself; making considered decisions; setting and achieving personal goals; being able to reflect on my own learning; using digital technology to manage myself and my learning.
Staying well Being healthy and physically active; being social; being safe; being spiritual; being confident; being positive about learning; being responsible, safe and ethical in using digital technology.
Managing information and thinking
Being curious; gathering, recording, organising and evaluating information and data; thinking creatively and critically; reflecting on and evaluating my learning; using digital technology to access, manage and share content.
Being numerate Expressing ideas mathematically; estimating, predicting and calculating; developing a positive disposition towards investigating, reasoning and problem-solving; seeing patterns, trends and relationships; gathering, interpreting and representing data; using digital technology to develop numeracy skills and understanding.
Being creative Imagining; exploring options and alternatives; implementing ideas and taking action; learning creatively; stimulating creativity using digital technology.
Working with others
Developing good relationships and dealing with conflict; co-operating; respecting difference; contributing to making the world a better place; learning with others; working with others through digital technology.
Communicating Listening and expressing myself; performing and presenting; discussing and debating; using language; using number; using digital technology to communicate.
Introduction to Discover Maths 2 & 3
Welcome to Discover Maths, your new three-book Junior Cycle Maths series designed to fully meet your students’ needs and the requirements of the Junior Cycle Maths specification. Discover Maths 1 is a common-level First Year book. Discover Maths 2 covers the entire Ordinary Level syllabus during Second and Third Year. Discover Maths 3 covers the entire Higher Level syllabus during Second and Third Year.
Written in a highly accessible manner and addressing the full requirements of the new JC Maths syllabus, Discover Maths delivers a complete tuition of the four contextual strands and the unifying strand of the specification. The book contains an abundance of excellent and engaging questions to provide a mathematical learning experience that is easy to manage, enjoyable and inclusive for students of all abilities, while providing excellent preparation for the Junior Cycle exam. Discover Maths brings maths to life!
The Discover Maths series includes the following features:
● Well-structured and focused chapters, with clear learning intentions highlighted at the beginning of each chapter.
● Chapters are colour-coded by strand for ease of navigation.
● Definitions, notes, examples for every question type and exercises included, ensuring students’ seamless progression through the content.
● A strong emphasis on unification, delivered in two accessible ways:
✭ A unifying star icon appearing throughout, encouraging students to identify the skills/topics they are connecting in a question/activity.
✭ Dedicated unification chapters, allowing students to revise and connect prior knowledge and skills acquired in preceding chapters.
● Think-Pair-Share, Class Discussion, Class Activity tasks and questions included to encourage individual inquiry, pair work and whole-class learning.
● Inclusive learning in the book’s design, with a wide range of differentiated questions to cater for all students.
● Exam-style questions incorporated into end-of-chapter revision exercises, and graded into three levels, to help assess students’ mathematical knowledge, understanding and skills.
UNIFICATION ?
● Self-assessment opportunities for students to assess and reflect on their learning throughout, including reflection tasks at the end of each chapter.
● Spotlight on mathematicians throughout history included in each chapter, bringing maths to life and highlighting its development, application and relevance to the world and our daily lives.
● Accompanying Teacher’s Resource Book, including editable documents, to support teacher planning.
● Accompanying digital resources, including full worked solutions, to support student learning.
Separate answer booklet with short answers for all questions, to facilitate all levels of student and styles of learning. Teachers can decide if they would like students to have short answers to hand for classwork or homework.
Digital Resources
The Discover Maths 2 & 3 digital resources provide supplementary content to enhance your students’ learning experience. Links to the resources, which include PowerPoints, videos and interactive quizzes, are referenced throughout the textbook using the following icons.
Chapter summary
Editable PowerPoint presentations, highlighting key points in the textbook to aid revision and exam preparation
Video
Author-presented demonstration videos of specific worked examples covering a selection of questions from each chapter, including construction exercises
Worked solutions
Detailed PowerPoint presentations with step-by-step workings for every textbook question
Interactive Quizlet activities for every chapter available at edco.ie/discovermaths2 and edco.ie/discovermaths3
Teachers can access the resources via the Discover Maths 2 & 3 interactive e-books, which are available online at www.edcolearning.ie.
Additional resources for teachers include editable planning documents, chapter tests and much more.
Quiz
Digital Resources Discover Maths 1
Planning in Maths
Planning is a key consideration for teachers of Junior Cycle Maths. Because planning is specific to the school context, there is no ‘one-size-fits-all’ approach. However, on the following pages teachers will find a suggested approach to help them teach the specification using Discover Maths 2 (Ordinary Level) or Discover Maths 3 (Higher Level). The suggested approach to planning covers the following:
● Schemes of Work
● Units of Learning.
Schemes of Work
When preparing yearly plans, teachers could use Schemes of Work to provide a framework on how students are engaging in their learning. The following can be included:
● Unit of Learning
● Duration.
A Scheme of Work is an active, working collaborative departmental document, with evidence of this being the case. Your maths subject plan may need to change from year to year, depending on the needs of your students. On the following pages, teachers will find a suggested Scheme of Work for Second and Third Year Maths (Ordinary Level pages 9–14, Higher Level pages 15–20), as well as a blank template for their own planning on page 21. Editable versions of these are available on www.edcolearning.ie, so teachers can adapt them when creating their own Schemes of Work.
Units of Learning
In planning for Junior Cycle Maths, teachers are encouraged to engage with the curriculum through Units of Learning. A Unit of Learning is a set of lessons over a period of time, covering one or more Learning Outcomes taken from the curriculum. The Learning Outcome(s) within a Unit of Learning can be covered in part or in full, and the duration and number of Units of Learning are at the discretion of each maths department.
To support maths departments in creating Units of Learning, teachers can consider the following points in their teaching practice:
● Student Context
● Learning Outcomes
● Links to Other Skills/Topics
● Key Learning
● Ongoing Assessment
● Learning Experiences
● Notes/Reflection.
On page 22 teachers will find a blank Unit of Learning template, as well as unit overviews on subsequent pages. Editable versions of these are available on www.edcolearning.ie.
Unifying Strand
In each Unit of Learning, the diagram below is used to illustrate how the Unifying Strand (see page XX) is connected to each unit.
Representation
Building blocks
Connections
U5, U6 U7, U8, U9, U10 U1, U2, U3 U13
U11, U12
Communication
Generalisation and proof
Problem solving
The six elements of the Unifying Strand incorporating the 13 Learning Outcomes
U4
Suggested Scheme of Work for Ordinary Level Maths
Unit of Learning*
1. Algebra 1 – Recap of First Year
Simplifying Expressions with Addition and Subtraction
Multiplying Terms and Expressions
Multiplying Expressions by Expressions
Evaluating Expressions (Substitution)
Revision Exercises
Reflection Task
2. Algebra 2 – Equations and Inequalities
Solving Linear Equations
Number Sets and their Number Lines
Solving Linear Inequalities
Revision Exercises
Reflection Task
3. Statistics 1
Statistics Overview
Notation and Terminology Dictionary
Collecting Data
Types of Data
Tallying and Frequency Tables
Sampling
Designing a Valid Survey
Revision Exercises
Reflection Task
4. Sets
Revision of First Year Sets
Set Complement
Problem Solving with Sets
Recap of Regions of Venn Diagrams
Recap of Set Notation and Terminology
Revision Exercises
Reflection Task
*There are no Units of Learning for Unification 1–3 (Chapters 7, 15 and 22), or for the CBA and Exam Skills chapters (24 and 25), as these contain no new learning.
**The duration for each Unit of Learning will vary, depending on your students’ needs.
5. Indices
Writing Numbers in the Form ����
Rules of Indices
Scientific Notation
Revision Exercises
Reflection Task
6. Perimeter, Area and Volume
Terminology and Formulae
Revision of Perimeter and Area
Nets of Cuboids and Cubes
Surface Area of Cuboids and Cubes
Volume of a Cube and Cuboid
Volume of a Cylinder
Scaled Diagrams
Revision Exercises
Reflection Task
8. Algebra 3 – Factorising and Quadratic Equations
Factorising Algebraic Expressions
Highest Common Factors (HCF)
Factorising with Difference of Two Squares
Factorising by Grouping
Factorising Quadratic Trinomials
Solving Quadratic Equations with Factors
Revision Exercises
Reflection Task
9. Geometry
Recap of Geometry Concepts Covered in First Year
Examinable Terms
Angles and Parallel Lines
Triangle Theorems and Special Triangles
Quadrilaterals and Parallelograms
Pythagoras’ Theorem
Congruent Triangles
Similar Triangles
Circle Terminology
Angles in a Semicircle
Geometry Overview
Revision Exercises
Reflection Task
10. Algebra 4 – Simultaneous Equations
Simultaneous Linear Equations
Point of Intersection of Two Lines
Problem Solving
Revision Exercises
Reflection Task
11. Patterns
Terminology
Revision of Linear Sequences
Finding the General Term, ����, for a Linear Sequence
Finding the Position of a Term (��)
Linear and Non-linear Sequences
Graphs and Problem Solving
Revision Exercises
Reflection Task
12. Algebra 5 – Fractions
Adding and Subtracting Algebraic Fractions
Solving Linear Equations that Contain Algebraic Fractions
Simplifying Algebraic Fractions Using Factors
Algebraic Long Division
Revision Exercises
Reflection Task
13. Statistics 2
Introduction to Averages
Spread of Data
Comparison of Data Sets
Frequency Distribution Tables
Mean, Mode and Median of a Frequency Distribution
Given the Mean
Revision Exercises
Reflection Task
14. Applied Arithmetic
Notation and Terminology
Value for Money
Value Added Tax (VAT)
Ratios / Proportions
Direct Proportion
Percentage Profit and Loss
Currency Exchange
Income Tax
Compound Interest
Revision Exercises
Reflection Task
16. Trigonometry
Trigonometry Introduction
The Trig Ratios: Sin ��, Cos �� and Tan ��
Using a Calculator
Inverse Sine, Inverse Cosine and Inverse Tangent
Using the Trig Ratios to Find Unknown Angles
Using the Trig Ratios to Find Unknown Sides
Pythagoras’ Theorem
Complete Guide JC Trigonometry (4 Key Skills)
Real-World Applications
Revision Exercises
Reflection Task
17. Coordinate Geometry
Coordinate Geometry Concepts Covered in First Year
Midpoint Formula
Distance Formula
Slope of a Line (��)
The Equation of a Line (1)
The Equation of a Line (2)
Parallel Lines
Points on a Line
The �� and ��-intercepts and Sketching Lines
The Equations of Horizontal and Vertical Lines
Unit of Learning*
Point of Intersection of Two Lines (Simultaneous Equations)
Revision Exercises
Reflection Task
18. Transformation Geometry
Axes and Symmetry
Translations, Reflections, Rotations and Central Symmetry
Transformation in The Coordinate Plane
Revision Exercises
Reflection Task
19. Statistics 3
Displaying Data
Revision of Bar Charts, Line Plots and Stem and Leaf Plots
Histograms
Pie Charts
Misleading Graphs and Statements
Revision Exercises
Reflection Task
20. Functions
Introduction to Functions
Functions Notation and Terminology
Determining if a Mapping Diagram Represents a Function
Finding Specific Inputs and Outputs
Graphs of Functions
Graphing Functions
Reading Information from Graphs
Practical Questions
Revision Exercises
Reflection Task
21. Probability
Glossary of Probability Terms
Estimating Probability from Experiments
Expected Frequency
Making Lists and The Fundamental Principle of Counting
Duration** Completed
Unit of Learning* Duration** Completed
Probability and Equally Likely Outcomes
Combined Events and Tree Diagrams
Revision Exercises
Reflection Task
23. Geometry Constructions***
Instruments in your Geometry Set
1. Bisector of an Angle
2. Perpendicular Bisector of a Line Segment
4. Perpendicular through a Point on a Line
5. Parallel Line through a Given Point
6. Division of a Line into Two or Three Segments
8. Line Segment of a Given Length
9. Angle of a Given Size
10. Triangle, Given Three Sides
11. Triangle, Given SAS
12. Triangle, Given ASA
13. Right-Angled Triangle, Given Hypotenuse and One Side
14. Right-Angled Triangle, Given One Side and an Acute Angle
15. Rectangle, Given the Side Lengths
Reflection Task
***Note: Constructions 3 and 7 are not on the Ordinary Junior Cycle syllabus; students will learn them at Senior Cycle.
Suggested Scheme of Work for Higher Level Maths
Unit of Learning* Duration**
1. Algebra 1
Glossary of Algebraic Notation and Terminology
Simplifying Expressions with Addition and Subtraction
Multiplying Terms and Expressions
Evaluating Expressions (Substitution)
Revision Exercises
Reflection Task
2. Algebra 2
Solving Linear Equations
Manipulating Equations
Number Sets and their Number Lines
Solving Linear Inequalities
Revision Exercises
Reflection Task
3. Statistics 1
Statistics Overview
Notation and Terminology Dictionary
Collecting Data
Types of Data
Tallying and Frequency Tables
Sampling
Designing a Valid Survey
Revision Exercises
Reflection Task
4. Sets
Revision of First Year Sets
Set Complement
Problem Solving with Sets
Triple Sets
Operations on Sets
Problem Solving with Three Sets
Recap of Regions of Venn Diagrams
Recap of Set Notation and Terminology
Revision Exercises
Reflection Task
*There are no Units of Learning for Unification 1–3 (Chapters 7, 15 and 24), or for the CBA and Exam Skills chapters (26 and 27), as these contain no new learning.
**The duration for each Unit of Learning will vary, depending on your students’ needs.
5. Indices and Surds
Writing Numbers in the Form ����
Rules of Indices
Equations that Include Indices
Surds
Scientific Notation
Revision Exercises
Reflection Task
6. Perimeter, Area and Volume
Terminology
Revision of Perimeter and Area
Scaled Diagrams
Nets
Surface Area
Volume of a Cube and a Cuboid
Volume of a Cylinder
Volume of a Sphere
Volume of a Prism
Problem Solving and Combined Shapes
Glossary of Perimeter, Area and Volume Formulae
Revision Exercises
Reflection Task
8. Algebra 3
Factorising Algebraic Expressions
Highest Common Factors (HCF)
Factorising with Difference of Two Squares
Factorising by Grouping
Factorising Quadratic Expressions
Factorising Expressions with Three Factors
Solving Quadratic Equations with Factors
Solving Quadratic Equations with the Quadratic
Formula
Constructing a Quadratic, Given the Roots
Revision Exercises
Reflection Task
9. Geometry 1
Recap of Geometry Concepts Covered in First Year
Examinable Terms
Angles and Parallel Lines
Triangle Theorems and Special Triangles
Unit of Learning*
Quadrilaterals and Parallelograms
Pythagoras’ Theorem
Congruent Triangles
Similar Triangles
A Line Parallel to a Side in a Triangle
Creating Geometry Proofs
Geometry Overview
Revision Exercises
Reflection Task
10. Algebra 4
Simultaneous Linear Equations
More Intricate Simultaneous Linear Equations
Point of Intersection of Two Lines
Problem Solving
Revision Exercises
Reflection Task
11. Patterns
Terminology
Revision of Linear Sequences
Finding the General Term, ����, of a Linear Sequence
Finding the Position of a Term (��)
Linear and Non-linear Sequences
Finding the General Term, ���� for a Quadratic Sequence
Finding the Position of a Term in a Quadratic Sequence
Graphs and Problem Solving
Revision Exercises
Reflection Task
12. Algebra 5
Adding and Subtracting Algebraic Fractions
Solving Linear Equations that Contain Algebraic Fractions
Solving Quadratic Equations that Contain Algebraic Fractions
Simplifying Algebraic Fractions Using Factors
Algebraic Long Division
Revision Exercises
Reflection Task
Duration** Completed
13. Statistics 2
Introduction to Averages
Spread of Data
Comparison of Data Sets
Frequency Distribution Tables
Grouped Frequency Distributions
Given the Mean
Revision Exercises
Reflection Task
14. Applied Arithmetic
Notation and Terminology
Value for Money
Value Added Tax (VAT)
Ratios / Proportions
Direct and Indirect Proportion
Percentage Profit and Loss
Currency Exchange
Income Tax
Compound Interest
Revision Exercises
Reflection Task
16. Trigonometry
Trigonometry Introduction
The Trig Ratios: Sin ��, Cos �� and Tan ��
Using a Calculator
Inverse Sine, Inverse Cosine and Inverse Tan
Using the Trig Ratios to Find Unknown Angles
Using the Trig Ratios to Find Unknown Sides
Pythagoras’ Theorem
Complete Guide JC Trigonometry (4 Key Skills)
Real-World Applications
Revision Exercises
Reflection Task
17. Coordinate Geometry
Recap of Coordinate Geometry Concepts Covered in First Year
Midpoint Formula
Distance Formula
Slope of a Line (��)
The Equation of a Line (1)
The Equation of a Line (2)
Find the Slope, Given the Equation of a Line
Unit of Learning*
Parallel and Perpendicular Lines
Points on a Line
The �� and ��-intercepts and Sketching Lines
The Equations of Horizontal and Vertical Lines
Point of Intersection of Two Lines (Simultaneous Equations)
Revision Exercises
Reflection Task
18. Transformation Geometry
Axes of Symmetry
Translations, Reflections, Rotations and Central Symmetry
Transformations in the Coordinate Plane
Revision Exercises
Reflection Task
19. Statistics 3
Displaying Data
Revision of Bar Charts, Line Plots and Stem and Leaf Plots
Histograms
Pie Charts
Back-to-Back Stem and Leaf Plots
Misleading Graphs and Statements
Revision Exercises
Reflection Task
20. Functions 1
Introduction to Functions
Functions Notation and Terminology
Determining if a Mapping Diagram Represents a Function
Finding Specific Inputs and Outputs
Graphs of Functions
Graphing Functions
Reading Information from Graphs
Practical Questions
Revision Exercises
Reflection Task
21. Probability
Glossary of Probability Terms
Estimating Probability from Experiments
Expected Frequency
Making Lists and The Fundamental Principle of Counting
Duration** Completed
Unit of Learning* Duration** Completed
Probability and Equally Likely Outcomes
Combined Events and Tree Diagrams
Revision Exercises
Reflection Task
22. Geometry 2
Circle Terminology
Circle Theorem and Corollaries
The Circle and Trigonometry
Geometry Overview
Combining Theorems and Axioms Within the Circle
Creating Geometry Proofs with Circle Theorem
Revision Exercises
Reflection Task
23. Functions 2
Solving Linear Functions Algebraically
Solving Quadratic Functions Algebraically
The Intercepts
Points on a Functions and Unknown Coefficients
Transformations on Graphs
More Challenging Algebraic Questions
Revision Exercises
Reflection Task
25. Geometry Constructions
Instruments in your Geometry Set
1. Bisector of an Angle
2. Perpendicular Bisector of a Line Segment
3. Perpendicular through a Point not on a Line
4. Perpendicular through a Point on a Line
5. Parallel Line through a Given Point
6. Division of a Line into Two or Three Segments
7. Division of a Line into Any Number of Segments
8. Line Segment of a Given Length
9. Angle of a Given Size
10. Triangle, Given Three Sides
11. Triangle, Given SAS
12. Triangle, Given ASA
13. Right-Angled Triangle, Given Hypotenuse and One Side
14. Right-Angled Triangle, Given One Side and an Acute Angle
15. Rectangle, Given the Side Lengths
Reflection Task
Blank Planning Templates
Scheme of Work Template
Unit of Learning Template
Student Context
Anticipation Exercise
Please read each statement below and tick Agree or Disagree as you feel appropriate: Before
Learning Outcomes
Links to Other Skills/Topics
Key Learning
Ongoing Assessment
Learning Experiences
Notes/Reflection
Units of Learning
Unit of Learning: 5 Indices and Surds
Any content in the Unit of Learning which is specific to higher level is indicated in bold text.
Student Context
Students with prior knowledge of:
● algebraic notation and terminology
● the order of operations (BIRDMAS) on integers
● how to solve linear equations
Anticipation Exercise
Please read each statement below and tick Agree or Disagree as you feel appropriate:
I understand the difference between a base and a power.
Writing numbers in index notation allows us to work with very large numbers more efficiently.
I understand and can apply the rules of indices such as adding the powers when we multiply like bases.
I understand what is meant by “scientific notation” and can convert numbers between decimal form and scientific notation.
I can use my indices skills in practical questions to solve problems.
I will use skills from previous chapters to help me progress through this chapter because maths is all interconnected.
Learning Outcomes
Students should be able to:
N.1 investigate the representation of numbers and arithmetic operations so that they can:
b. perform the operations of addition, subtraction, multiplication, and division and understand the relationship between these operations and the properties: commutative, associative and distributive in ℕ, ℤ, and ℚ and in ℝ \ ℚ, including operating on surds
c. explore numbers written as ���� (in index form) so that they can:
I. flexibly translate between whole numbers and index representation of numbers
II. use and apply generalisations such as
for
and
and ��,
and for
III. use and apply generalisations such as
=
and
��, �� ∈ ℤ; and �� ∈ ℚ
LEARNING INTENTIONS
After this unit of learning, you will be able to:
● Write numbers and expressions in the form ����.
● Understand and apply the rules of indices.
● Convert negative powers to fractions and vice versa.
● Solve equations that involve indices.
● Understand scientific notation and write numbers in the form �� × 10��.
● Add, subtract, multiply and simplify surds.
IV. generalise numerical relationships involving operations involving numbers written in index form
V. correctly use the order of arithmetic and index operations including the use of brackets
e. present numerical answers to the degree of accuracy specified, for example, correct to the nearest hundred, to two decimal places, or to three significant figures
f. convert the number p in decimal form to the form �� × 10��, where 1 ≤ �� < 10, �� ∈ ℤ �� ∈ ℚ, and �� ≥ 1 and �� < �� < ��
U.1 recall and demonstrate understanding of the fundamental concepts and procedures that underpin each strand
U.2 apply the procedures associated with each strand accurately, effectively, and appropriately
U.3 recognise that equality is a relationship in which two mathematical expressions have the same value
U.4 represent a mathematical situation in a variety of different ways, including: numerically, algebraically, graphically, physically, in words; and to interpret, analyse, and compare such representations
U.5 make connections within and between strands
U.6 make connections between mathematics and the real world
U.7 make sense of a given problem, and if necessary mathematise a situation
U.8 apply their knowledge and skills to solve a problem, including decomposing it into manageable parts and/or simplifying it using appropriate assumptions
U.9 interpret their solution to a problem in terms of the original question
U.13 communicate mathematics effectively: justify their reasoning, interpret their results, explain their conclusions, and use the language and notation of mathematics to express mathematical ideas precisely
Links to Other Skills/Topics
This topic links to Patterns and Algebra.
Key Learning
After this Unit of Learning, students will:
Knowledge
● know what indices are
● know that surds are irrational numbers
● know how to write numbers and expressions in the form ����
● know the rules of indices
Understanding
● understand and apply the rules of indices
● understand scientific notation and write numbers in the form �� × 10��
Skills
● be able to convert negative powers to fractions and vice versa
● be able to use the rules of indices to simplify expressions
● be able to add, subtract and multiply surds
● be able to convert numbers between decimal form and scientific notation
● be able to perform operations on numbers in scientific notation
● be able to solve equations that involve indices
Values
● appreciate that applications of indices can be seen in computer games, physics, pH and Richter measuring
● appreciate that indices are used in areas such as science, engineering, economics and finance
● appreciate that surds are irrational numbers and can’t be expressed as whole or rational numbers
Ongoing Assessment
● Revision Exercises
● Reflection Task
After this Unit of Learning, can students:
● write numbers and expressions in the form ����?
● understand and apply the rules of indices?
● add, subtract and multiply surds?
● solve equations that involve indices?
● convert negative powers to fractions and vice versa?
● understand scientific notation and write numbers in the form �� × 10��?
Learning Experiences
● Exercises 5.1–5.3 (Ordinary Level) Exercises 5.1–5.5 (Higher Level)
● Demonstration/worked example videos
Notes/Reflection
Teacher reflection – what worked/did not work well/improvements?
Ordinary Level Solutions
Chapter 5: Indices
Exercise 5.1
Index notation Expansion Answer
31 3 3
32 (3)(3) 9
33 (3)(3)(3) 27
34 (3)(3)(3)(3) 81
35 (3)(3)(3)(3)(3) 243
Index notation Expansion Answer
21 2 2
22 (2)(2) 4
23 (2)(2)(2) 8
24 (2)(2)(2)(2) 16
25 (2)(2)(2)(2)(2) 32
26 (2)(2)(2)(2)(2)(2) 64
Index notation Expansion Answer 21 2 2
22 (2)(2) 4
23 (2)(2)(2) 8
24 (2)(2)(2)(2) 16
25 (2)(2)(2)(2)(2) 32
26 (2)(2)(2)(2)(2)(2) 64
Index notation Expansion Answer
71 7 7
72 (7)(7) 49
73 (7)(7)(7) 343
74 (7)(7)(7)(7) 2401
75 (7)(7)(7)(7)(7) 16 807
5. (a) 4 = 22 (b) 9 = 32 (c) 25 = 52 (d) 49 = 72 (e) 27 = 33 (f) 8 = 23 (g) 81 = 34 (h) 125 = 53 (i) 625 = 54 (j) 169 = 132 (k) 16 = 24 (l) 243 = 35
Exercise 5.2
1. (a) 32 × 33 = 35 (b) 34 × 33 = 37 (c) 3 × 34 × 35 = 310 (d) 32 × 32 × 32 = 36
2. (a) 23 × 25 × 22 = 210 (b) 2(22)(23) = 26 (c) 25 23 = 22 (d) 27 22 = 25
3. (a) 106 103 = 103 (b) 103 105 = 10−2 (c) 10 103 = 10−2 (d) (102)3 = 106
4. (a) (54)2 = 58 (b) (53)4 = 512 (c) 54 57 = 5−3 (d) 5(56)(54) = 511
5. (a) 23 × 22 = 25 (b) 104 × 103 = 107 (c) 32(35) = 37 (d) 5(52) = 53
6. (a) 2(29) = 210 (b) 51 × 52 × 53 = 56 (c) 102(102)(102) = 106 (d) 7(72)(73) = 76
Exercise 5.2 continued
7. (a) 35 33 = 32 (b) 57 54 = 53 (c) 23 24 = 2−1 (d) 105 107 = 10−2 8. (a) 74 72 = 72 (b) 25 24 = 2 (c) 36 33 = 33 (d) 58 55 = 53
9. (a) (23)2 = 26 (b) (102)4 = 108 (c) (32)5 = 310 (d) (72)2 = 74
10. (a) (53)2 = 56 (b) (112)4 = 118 (c) (32)5 = 310 (d) (22)2 = 24
11. (a) √ 5 = 5 1 2 (b) √ 3 = 3 1 2 (c) √ = 2 1 2 (d) √ 7 = 7 1 2
12. (a) 3(35)(36) = 312 (b) 310 34 = 36 (c) (73)5 = 715 (d) 6(6)(6)(6) = 64
13. (a) √ 10 = 10 1 2 (b) (23)3 = 29 (c) 5 53 = 5−2
(d) (102)6 = 1012
14. 44 × 43 42 = 47 42 = 45
15. 35(33) 32 = 38 32 = 36
16. 25 (22)2 = 25 24 = 2
Exercise 5.3
1. (a) 2 × 102 = 200 (b) 3 × 104 = 30 000 (c) 2.7 × 103 = 2700 (d) 1.64 × 105 = 164 000 (e) 1.73 × 103 = 1730 (f) 2.217 × 106 = 2 217 000 (g) 4 × 106 = 4 000 000 (h) 1.898 × 107 = 18 980 000 (i) 7 × 10−1 = 0.7 (j) 5 × 10−2 = 0.05 (k) 3 × 10−6 = 0.000003 (l) 7.28 × 10−3 = 0.00728 (m) 2.64 × 10−2 = 0.0264 (n) 1.6 × 10−4 = 0.00016
(o) 9.6 × 10−1 = 0.96 (p) 1.787 × 10−5 = 0.00001787
2. (a) 4300 = 4.3 × 103 (b) 75 000 = 7.5 × 104
(c) 917 000 = 9.17 × 105 (d) 310 = 3.1 × 102
(e) 15 000 000 = 1.5 × 107 (f) 672 = 6.72 × 102
(g) 130 000 = 1.3 × 105 (h) 54 000 000 = 5.4 × 107
(i) 0.03 = 3 × 10−2 (j) 0.005 = 5 × 10−3
(k) 0.00641 = 6.41 × 10−3 (l) 0.017 = 1.7 × 10−2
(m) 0.026 = 2.6 × 10−2 (n) 0.0009 = 9 × 10−4
(o) 0.000000212 = 2.12 × 10−7 (p) 0.000185 = 1.85 × 10−4
3. (a) 256 740 000 000 ÷ 1000 (b) 2.5674 × 108 = 256 740 000 km
4. 1 100 000 = 1 × 10−5
5. 2 400 000 = 2.4 × 106
6. 17 000 = 1.7 × 104
7. (a) 1.5 × 108 × 1000 = 150 000 000 × 1000 = 150 000 000 000
(b) 1.5 × 1011
8. 6.02214 × 1023
9. 56 000 000 = 5.6 × 107
10. (a) 1417 000 000 = 1.417 × 109
(b) one billion four hundred and seventeen million
11. (a) (5.6 × 102) + (4 × 103)
560 + 4000
4560
4.56 × 10³
(b) (2.3 × 104) − (1.9 × 103)
23 000 − 1900
21 100 2.11 × 104
Higher Level Solutions
Chapter 5: Indices and Surds
Exercise 5.1 continued
4 (7)(7)(7)(7) 2401
5 (7)(7)(7)(7)(7) 16807
Exercise 5.2