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Teacher’s Book
Anita Straker, Tony Fisher, Rosalyn Hyde,
Sue Jennings and Jonathan Longstaﬀe
5
ii | Exploring maths Tier 5
Introduction
S5.4 Probability 2
Identifying outcomes of an experiment
Comparing experimental with theoretical
probability
4/5 lessons
S5.3 Enquiry 2
Collecting, representing and interpreting data
on paper and using ICT frequency tables and
diagrams for grouped continuous data
Comparing two distributions
Communicating ﬁndings, using ICT
6/7 lessons
Summer
33 lessons
S5.2 Probability 1
Mutually exclusive events
Sum of probabilities
Estimating probabilities from
experimental data; comparing with
theoretical probability
3/4 lessons
Spring
34 lessons
S5.1 Enquiry 1
Designing survey to collect data, including
data collection sheet
Representing and interpreting data, on
paper and using ICT line graphs for time
series, scatter graphs
Calculating statistics, including with a
calculator; comparing distributions
Communicating ﬁndings, using ICT
6 lessons
Autumn
33 lessons
N5.4 Solving problems
Investigating problems using number and
algebra
Proof; ﬁnding a counter-example
History of mathematics
4/5 lessons
R5.2 Revision/support
Number, algebra, geometry
and measures, statistics
5 lessons
R5.1 Revision/support
Number, algebra, geometry
and measures, statistics
5 lessons
N5.3 Calculations and calculators
Multiplying and dividing by powers of 10:
rounding and estimating calculations
Mental and written calculations with decimals,
all four operations
Use of calculator
Solving problems, including with measures
5/6 lessons
N5.2 Proportional reasoning
Fractions four operations and brackets
Percentage change
Ratio and proportion
Mental and calculator calculations
Solving problems
6 lessons
N5.1 Properties and roots
Prime factor decomposition
Using ICT to estimate squares and roots
Simple cases of index laws
3 lessons
100 lessons
A5.5 Equations, formulae and graphs
Adding/subtracting algebraic fractions;
removing/cancelling common factors;
changing subject of simple formulae
Constructing and solving equations by exact
and approximate methods
Graphs of linear functions; direct proportion
and distance-time graphs
7/8 lessons
A5.4 Using algebra
Constructing, plotting and interpreting reallife functions
Solving direct proportion problems
3 lessons
A5.3 Functions and graphs
Linear graphs, gradient and intercepts
Simple properties of quadratic functions
Deriving formulae
5/6 lessons
A5.2 Equations and formulae
Single-term common factors: adding simple
algebraic fractions; substitution in and
changing subject of simple formulae
Constructing and solving linear equations
with integer coeﬃcients
Approximate solutions of equations using ICT
6 lessons
A5.1 Sequences and graphs
Generating sequences on paper and with ICT;
ﬁnding the nth term of an arithmetic sequence
Graphs of linear functions; gradient and
intercepts; interpreting graphs of real-life
functions, including distance–time
6 lessons
G5.4 2-D and 3-D shapes
Visualising 3-D shapes; planes and elevations
Geometrical reasoning angles and shapes
Surface area and volume of prisms
7/8 lessons
G5.3 Transformations
Plane symmetry of 3-D shapes
Simple combinations of rotations, reﬂections and
translations, on paper and using ICT
Enlargement; scale drawings
Exploring transformations with ICT
5/6 lessons
G5.2 Angles and constructions
Angle sum and exterior angle of polygons
Finding unknown angles; geometrical reasoning
Straight edge and compass constructions
Simple loci
9 lessons
G5.1 Measures and mensuration
Using measures to estimate, measure and
calculate; converting measures
Circumference and area of circle
Volume and surface area of simple prisms
5/6 lessons
Mathematical processes and applications are integrated into each unit
Introduction
The materials
The Exploring maths scheme has seven tiers, indicated by the seven colours in the table below.
Each tier has:
a class book for pupils;
a home book for pupils;
a teacher’s book, organised in units, with lesson notes, mental tests (for number units),
facsimiles of resource sheets, and answers to the exercises in the class book and home book;
a CD with interactive books for display, either when lessons are being prepared or in class,
and ICT resources for use in lessons.
Content, structure and diﬀerentiation
The tiers are linked to National Curriculum levels so that they have the maximum ﬂexibility.
They take full account of the 2007 Programme of Study for Key Stage 3, the Secondary
Strategy’s renewed Framework for teaching mathematics in Years 7 to 11, published in 2008, and
the possibility of taking the statutory Key Stage 3 test before the end of Year 9. Standards for
functional skills for Entry Level 3 and Level 1 are embedded in Tiers 1 and 2. Tier 3 begins to lay
the groundwork for level 2.
Labels such as ‘Year 7’ do not appear on the covers of books but are used in the table below to
explain how the materials might be used.
Extra support
For pupils who achieved level 2 or a weak level 3
at KS2, who will enter the level 3–5 test at KS3 and
who are likely to achieve Grade F–G at GCSE.
Support
For pupils who achieved a good level 3 or weak level
4 at KS2, who will enter the level 4–6 test at KS3 and
who are likely to achieve Grade D–E at GCSE.
Core
For pupils who achieved a secure level 4 at KS2,
who will enter the level 5–7 test at KS3 and who are
likely to achieve B–C at GCSE.
Extension
For pupils who achieved level 5 at KS2, who will
enter the level 6–8 test at KS3 and who are likely to
achieve A or A* at GCSE.
Gifted and talented
For gifted pupils who achieved a strong level 5 at
KS2, who may be entered early for the level 6–8 test
for KS3 and who are likely to achieve A* at GCSE.
Year 7
Year 8
Year 9
Tier 1
NC levels 2–3
(mainly level 3)
Tier 2
NC levels 3–4
(mainly level 4)
Tier 3
NC levels 4–5
(both levels 4
and 5)
Tier 2
NC levels 3–4
(mainly level 4)
Tier 3
NC levels 4–5
(both levels 4
and 5)
Tier 4
NC level 5–6
(mainly level 5)
Tier 3
NC levels 4–5
(both levels 4
and 5)
Tier 4
NC level 5–6
(mainly level 5)
Tier 5
NC levels 5–6
(mainly level 6)
Tier 4
NC level 5–6
(mainly level 5)
Tier 5
NC levels 5–6
(mainly level 6)
Tier 6
NC levels 6–7
(mainly level 7)
Tier 5
NC levels 5–6
(mainly level 6)
Tier 6
NC levels 6–7
(mainly level 7)
Tier 7
NC levels 7–8
(mainly level 8)
The Exploring maths scheme as a whole oﬀers an exceptional degree of diﬀerentiation, so that
the mathematics curriculum can be tailored to the needs of individual schools, classes and
pupils.
Exploring maths Tier 5 Introduction | iii
There are at least ﬁve tiers available for each of the year groups 7, 8 and 9. The range of tiers to
be used in Year 7 can be chosen by the school to match the attainment of their incoming pupils
and their class organisation. Teachers of mixed-ability classes can align units from diﬀerent tiers
covering related topics (see Related units, p. xi).
The Results Plus Progress entry test, published separately, guides teachers on placing pupils
in an appropriate tier at the start of Year 7. The test analysis indicates which topics in that tier
may need special emphasis. Similar computer assessments are available for other years (see
Computer-mediated assessments, p. viii).
Pupils can progress to the next tier as soon as they are ready, since the books are not labelled
Year 7, Year 8 or Year 9. Similarly, work on any tier could take more than a year where pupils
need longer to consolidate their learning.
Pupils in any year group who have completed Tier 4 or above successfully could be entered
early for the Key Stage 3 test if the school wishes. Single-level tests for pupils working at
particular national curriculum levels, which pupils can take in the winter or summer of any
calendar year, are currently being piloted in ten local authorities as part of the Making good
progress project. The tiered structure of Exploring maths is ideally suited to any extension of this
pilot.
Each exercise in the class book oﬀers diﬀerentiated questions, so that teachers can direct
individual pupils to particular sections of the exercises. Each exercise starts with easier
questions and moves on to harder questions, identiﬁed by underscored question numbers.
More able pupils can tackle the extension problems.
If teachers feel that pupils need extra support, one or more lessons in a unit can be replaced
with or supplemented by lessons from revision units.
Organisation of the units
Each tier is based on 100 lessons of 50 to 60 minutes, plus 10 extra lessons to use for revision or
further support, either instead of or in addition to the main lessons.
Lessons are grouped into units, varying in length from three to ten lessons. The number of
lessons in a unit increases slightly through the tiers so that there are fewer but slightly longer
units for the higher tiers.
Each unit is identiﬁed by a code: N for number, A for algebra, G for geometry and measures,
S for statistics and R for revision. For example, Unit N4.2 is the second number unit for Tier 4,
while Unit G6.3 is the third geometry and measures unit for Tier 6. Mathematical processes and
applications are integrated throughout.
The units are shown in a ﬂowchart giving an overview for the year (see p. ii). Some units need to
be taught before others but schools can determine the precise order.
Schools with mixed-ability classes can align units from diﬀerent tiers covering related topics.
For example, Unit G4.2 Measures and mensuration in Tier 4 can be aligned with the Tier 3 Unit
G3.1 Mensuration and the Tier 5 Unit G5.2 Measures and mensuration. For more information on
where to ﬁnd related units, see p. xi.
Revision units
Each optional revision unit consists of ﬁve stand-alone lessons on diﬀerent topics. These
lessons include national test questions to help pupils prepare for tests.
Revision lessons can be taught in any order whenever they would be useful. They could be
used with a whole class or part of a class. Schools that are entering pupils for national tests may
wish to use, say, ﬁve of the revision lessons at diﬀerent points of the spring term and ﬁve in the
early summer term.
iv | Exploring maths Tier 5
Introduction
The revision lessons can either replace or be taught in addition to lessons in the main units.
Units where the indicative number of lessons is given as, say, 5/6 lessons, are units where a
lesson could be replaced by a revision lesson if teachers wish.
Balance between aspects of mathematics
In the early tiers there is a strong emphasis on number and measures. The time dedicated
to number then decreases steadily, with a corresponding increase in the time for algebra,
geometry and statistics. Mathematical processes and applications, or using and applying
mathematics, are integrated into the content strands in each tier.
The lessons for each tier are distributed as follows.
Number
Algebra
Geometry and
measures
Statistics
Tier 1
54
1
30
15
Tier 2
39
19
23
19
Tier 3
34
23
24
19
Tier 4
26
28
27
19
Tier 5
20
29
29
22
Tier 6
19
28
30
23
Tier 7
17
29
29
25
TOTAL
209
157
192
142
30%
23%
27%
20%
The teacher’s book, class book and home book
Teacher’s book
Each unit starts with a two-page overview of the unit. This includes:
the necessary previous learning and the objectives for the unit, with the process skills and
applications listed ﬁrst for greater emphasis;
the titles of the lessons in the unit;
a brief statement on the key ideas in the unit and why they are important;
brief details of the assessments integrated into the unit;
common errors and misconceptions for teachers to look out for;
the key mathematical terms and notation used in the unit;
the practical resources required (equipment, materials, paper, and so on);
the linked resources: relevant pages in the class book and home book, resource sheets,
assessment resources, ICT resources, and so on;
references to useful websites (these were checked at the time of writing but the changing
nature of the Internet means that some may alter at a later date).
The overview is followed by lesson notes. Each lesson is described on a two-page spread. There
is enough detail so that non-specialist teachers could follow the notes as they stand whereas
specialist mathematics teachers will probably adapt them or use them as a source of ideas for
teaching.
Exploring maths Tier 5 Introduction | v
Each lesson identiﬁes the main learning points for the lesson. A warm-up starter is followed by
the main teaching activity and a plenary review.
The lesson notes refer to work with the whole class, unless stated otherwise. For example,
where pupils are to work in pairs, the notes make this clear.
All the number units include an optional mental test for teachers to read out to the class, with
answers on the same sheet.
All units in the teacher’s book include answers to questions in the class book, home book,
check ups and resource sheets. The answers are repeated in the answer section at the back of
the teacher’s book.
Class book
The class book parallels the teacher’s book and is organised in units. The overall objectives for
the unit, in pupil-friendly language, are shown at the start of the unit, and the main objective
for each individual lesson is identiﬁed.
Interesting information to stimulate discussion on the cultural and historical roots of
mathematics is shown throughout the units in panels headed ‘Did you know that…?’
The exercises include activities, games or investigations for groups or individuals, practice
questions and problems to solve. Questions are diﬀerentiated, with easier questions at the
beginning of each exercise. Harder questions are shown by underlining of the question
number. Challenging problems are identiﬁed as extension problems. The exercises for each
lesson conclude with a summary of the learning points for pupils to remember.
Answers to exercises and functional skills activities in the class book are given in the teacher’s book.
Each unit ends with a self-assessment section for pupils called ‘How well are you doing?’ to
help them to judge for themselves their grasp of the work. Answers to these self-assessment
questions are at the back of the class book for pupils to refer to.
Home book
Each lesson has an optional corresponding homework task. Homework tasks are designed to
take most pupils about 15 to 20 minutes for Tiers 1 and 2, 25 minutes for Tiers 3, 4 and 5, and 30
minutes for Tiers 6 and 7.
Homework is normally consolidation of class work. It is assumed that teachers will select
from the homework tasks and will set, mark and follow up homework in accordance with the
school’s timetable. Because each school’s arrangements for homework are diﬀerent, feedback
and follow-up to homework is not included in the lesson notes. It is assumed that teachers will
add this as appropriate.
If the homework is other than consolidation (e.g. Internet research, collecting data for use in
class), the lesson notes state that it is essential for pupils to do the homework. The next lesson
refers to the homework and explains how it is to be used.
Answers to the homework tasks are given in the teacher’s book.
The ActiveTeach CD-ROM
The ActiveTeach contains interactive versions of the Teacher’s Book, Class Book, Home Book
and a variety of ICT resources. Full notes on how to use the ActiveTeach are included on the
CD-ROM in the Help tab.
Teachers can use the interactive version of the Teacher’s Book when they are planning or
teaching lessons.
From the contents page of the Teacher’s Book, teachers can navigate to the lesson notes for the
relevant unit, which are then displayed in a series of double page spreads.
vi | Exploring maths Tier 5
Introduction
Clicking on the thumbnail of the PowerPoint slide or the triangular icon shown on the edges
of the pages allows teachers to view ICT resources, resource sheets, and other Microsoft Oﬃce
program ﬁles. All these resources, as well as exercises in the Class Book and tasks in the Home
Book, can be accessed by clicking on the reference to the resource in the main text.
There is also an option for teachers to use a resource palette to put together their own set of
resources ready for a particular lesson, choosing from any of the Exploring maths resources in
any tier, and adding their own if they wish. This option will be especially useful for teachers of
mixed ability classes.
Interactive versions of the Class Book and Home Book can be displayed in class. From the
contents page, teachers can go to the relevant unit, which is then shown in a series of double
page spreads. It is possible to zoom in and enlarge particular worked examples, diagrams or
photographs, points to remember, homework tasks, and so on. Just as in the Teacher’s Book,
clicking on the triangular icon launches the relevant resource.
ICT resources
Each tier has a full range of ICT resources, including: a custom-built toolkit with over 60 tools,
Flash animations, games and quizzes, spreadsheets and slides.
The diﬀerent resources are coded as follows.
Check ups (CU)
Each unit is supplemented by an optional check-up for pupils in the form of a PDF ﬁle to
print and copy (see also the section on Assessment for learning).
Resource sheets (RS)
Some units have PDF ﬁles of resource sheets to print and copy for pupils to write on in class.
Tools (TO)
These general purpose teaching tools can be used in many diﬀerent lessons. Examples are:
– an interactive calculator, similar to an OHP calculator (in most cases, the scientiﬁc
calculator will be needed);
– number lines and grids;
– a graph plotter;
– simulated dice and spinners;
– squared paper and dotty paper;
– drawing tools such as a protractor, ruler and compasses.
Simulations (SIM)
Some of these are animations to play and pause like a video ﬁlm. Others are interactive and
are designed to generate discussion; for example, the teacher may ask pupils to predict an
outcome on the screen.
Quizzes (QZ)
These are quizzes of short questions for pupils to answer, e.g. on their individual
whiteboards, usually at the start or end of a lesson.
Interactive teaching programs (ITP)
These were produced by the Primary Strategy and are included on the CD-ROM with
permission from the DCSF.
PowerPoint presentations (thumbnails)
These are slides to show in lessons. Projected slides can be annotated, either with a
whiteboard pen or with the pen tool on an interactive whiteboard. Teachers without
access to computer and data projector in their classrooms can print the slides as overhead
projector transparencies and annotate them with an OHP pen.
Exploring maths Tier 5 Introduction | vii
Excel ﬁles (XL)
These are spreadsheets for optional use in particular lessons.
Geometer’s Sketchpad ﬁles (GSP)
These are dynamic geometry ﬁles for optional use in particular lessons.
Other ICT resources, such as calculators, are referred to throughout the units.
The table on p. x identiﬁes those lessons where pupils have an opportunity to use ICT for
themselves.
Assessment for learning
There is a strong emphasis on assessment for learning throughout Exploring maths.
Learning objectives for units as a whole and for individual lessons are shown on slides and
in the class book for discussion with pupils.
Potential misconceptions are listed for teachers in the overview pages of each unit.
Key questions for teachers to ask informally are identiﬁed in the lesson notes.
The review that concludes every lesson allows the teacher to judge the eﬀectiveness of the
learning and to stress the learning points that pupils should remember.
The points to remember are repeated in the class book and home book.
A self-assessment section for pupils, ‘How well are you doing?’, is included in each unit in the
class book to help pupils to judge for themselves their grasp of the work.
Optional revision lessons provide extra support in those areas where pupils commonly have
diﬃculty.
Each unit on the CD-ROM includes an optional check-up of written questions.
Each number unit of the teacher’s book includes an optional mental test of 12 questions for
teachers to read to the class.
The mental test could be used as an alternative to part of the last lesson of the unit. About 20
minutes of lesson time is needed to give the test and for pupils to mark it. Answers are on the
same sheet.
The written check-ups include occasional questions from national tests. Teachers could use some
or all of the questions, not necessarily on the same occasion, and pupils could complete them
in class, at home, or as part of an informal test. For example, some written questions could be
substituted for the ﬁnal homework of a unit and the mental test could be used as an alternative to
part of the last lesson. Answers to the written check-ups are given in the teacher’s book.
Computer-mediated assessments
Exploring maths is complemented by Results Plus Progress, a series of stimulating on-line
computer-mediated assessments supporting Key Stage 3 mathematics, available separately.
There is an entry test for Year 7 to guide teachers on placing pupils in an appropriate tier at the
start of the course. For each of Years 7, 8 and 9, there are two end-of-term assessments for the
autumn and spring terms, and an end-of-year assessment.
Each product oﬀers sets of interactive test questions that pupils answer on computers, either in
school or on home computers with internet access. Because the tests are taken electronically,
the products oﬀer instant marking and analysis tools to identify strengths and weaknesses of
individuals or groups of pupils. Future units from Exploring maths that are dependent on the
same skills are identiﬁed so that teachers are aware of the units that they may need to adapt,
perhaps by adding in extra revision or support lessons.
Results Plus Progress has been developed by the Test Development Team at Edexcel, who have
considerable experience in producing the statutory national end-of-key-stage tests and the
optional tests for Years 7 and 8.
viii | Exploring maths Tier 5
Introduction
Where can I ﬁnd…?
Historical and cultural references
N5.1
Euclid’s Fundamental Theorem of Arithmetic
Class book p.6
A5.1
Oresme and Descartes and the invention of a coordinate system
Class book p.20
A5.1
Graphing calculators
Class book p.24
G5.1
The history of π
Class book p.37
N5.2
The Ancient Egyptians’ use of fractions
Class book p.56
N5.2
Use of the ratio system in the seventeenth century
Class book p.65
N5.2
Exchange rates
Class book p.71
A5.2
Fahrenheit and Celsius
Class book p.100
A5.2
Euler’s formula
Class book p.101
A5.2
The history of solving equations
Class book p.107
G5.2
Regular polygons in man-made and natural environments
Class book p.119
G5.2
The building of the Hagia Sophia
Class book p.124
G5.2
How emperor penguins huddle to keep warm
Class book p.129
N5.3
A googol and googolplex
Class book p.137
S5.2
Abraham de Moivre and the development of the theory of probability
Class book p.155
G5.3
Pyramids and plane symmetry
Class book p.187
G5.3
Islamic patterns
Class book p.194
Home book p.61
A5.4
The Rhind Mathematical Papyrus
Class book p.214
S5.3
William Farr – the ﬁrst chief statistician
Class book p.224
S5.3
The coordinate system developed by Rene Descartes
Class book p.227
G5.4
Carl Friedrich Gauss and the construction of a heptadecagon
Class book p.242
G5.4
Logo and turtle graphics
Class book p.246
A5.5
The origins of algebra
Class book p.272
S5.4
Girolamo Cardano’s work on probability
Class book p.300
S5.4
The Monty Hall Game
Class book p.312
N5.4
Mathematicians who helped to develop our number system
Class book p.316
Home book p.98
N5.4
Ramanujan and the sum of two cubes
Class book p.322
R5.1
The Whetstone of Witte – the ﬁrst algebra book written in English
Class book p.338
R5.2
The Moscow Papyrus
Class book p.355
Exploring maths Tier 5 Introduction | ix
ICT lessons: hands-on for pupils
Pupils have many opportunities for hands on use of ICT.
Lesson 1: Using the xy key on a calculator
Teacher’s book p.4
Lesson 2: Using Excel to construct spreadsheets
Teacher’s book p.6
A5.1
Lesson 3: Generating sequences using spreadsheets
Teacher’s book p.20
G5.1
Lesson 2: Using dynamic geometry software to ﬁnd Teacher’s book p.38
N5.2
Lesson 1: Using a calculator to simplify fractions
Teacher’s book p.54
Lesson 2: Using a calculator to multiply fractions
Teacher’s book p.56
Lesson 3: Using a calculator to divide fractions
Teacher’s book p.58
A5.2
Lesson 6: Using graph-plotting software for trial and improvement
Teacher’s book p.108
G5.2
Lesson 1: Exploring angles and lines with dynamic geometry
software
Teacher’s book p.116
Lesson 7: Using Excel to create spreadsheets and graphs
Teacher’s book p.128
N5.3
Lesson 5: Consolidating and extending calculator skills
Teacher’s book p.152
S5.2
Lesson 4: Using spreadsheets in probability experiments
Teacher’s book p.170
A5.3
Lesson 1: Using graph-plotting software to generate graphs
Teacher’s book p.182
Lessons 2 and 5: Using graph-plotting software to sketch and
transform linear graphs
Teacher’s book p.184
Lesson 6: Using graph-plotting software to generate quadratic
graphs
Teacher’s book p.192
Lesson 2: Creating transformations using dynamic geometry
software
Teacher’s book p.202
Lesson 6: Using dynamic geometry software to explore area and
perimeter
Teacher’s book p.210
Lesson 2: Using graph-plotting software to solve geometry
problems
Teacher’s book p.224
Lesson 3: Using the interactive algebra program Frogs (optional)
Teacher’s book p.226
S5.3
Lesson 5: Producing graphs using Excel
Teacher’s book p.242
G5.4
Lesson 2: Drawing regular polygons with Logo
Teacher’s book p.260
A5.5
Lesson 8: Using graph-plotting software to solve problems
Teacher’s book p.302
N5.4
Lesson 1: Internet research on the history of our number system
Teacher’s book p.328
N5.1
G5.3
A5.4
Functional skills
Standards for functional skills for Entry Level 3 and Level 1 are embedded in Tiers 1 and 2.
Tiers 3 and 4 begin to lay the groundwork for the content and process skills for functional
skills at level 2. This continues in Tier 5.
Activities to encourage the development of functional skills are integrated throughout the Tier
5 class book.
In addition, there are four speciﬁc activities. These can be tackled at any point in the year,
including the beginnings and ends of terms. They are all group activities which lend themselves
to further development and follow-up. Many of the questions are open ended.
x | Exploring maths Tier 5
Introduction
The activities focus on these process skills:
identifying the mathematics in a situation and mathematical questions to ask;
recognising that a situation can be represented using mathematics;
selecting the information, methods, operations and tools to use, including ICT;
making an initial model of a situation using suitable forms of representation;
changing values in the model to see the eﬀects on answers;
examining patterns and relationships;
interpreting results and drawing conclusions;
considering how appropriate and accurate results and conclusions are;
choosing appropriate language and forms of presentation to communicate results and
solutions.
FS1
Should you buy or rent a TV set? Class book p.54
FS2
Where is the mathematics?
Class book p.112
FS3
Cutting it up
Class book p.220
FS4
Shoe sizes
Class book p.298
Related units
Tier 4
Tier 5
Tier 6
N4.1 Properties of numbers
N5.1 Powers and roots
N6.1 Powers and roots
N4.3 Fractions, decimals and
percentages
N5.3 Calculations and calculators
N6.3 Decimals and accuracy
N4.4 Proportional reasoning
N5.2 Proportional reasoning
N6.2 Proportional reasoning
N4.5 Solving problems
N5.4 Solving problems
N6.4 Using and applying maths
A4.1 Linear sequences
A5.1 Sequences and graphs
A6.2 Linear functions and graphs
A4.3 Functions and graphs
A5.3 Functions and graphs
A6.3 Quadratic functions and graphs
A4.5 Using algebra
A5.4 Using algebra
A6.4 Using algebra
A4.2 Expressions and formulae
A4.4 Equations and formulae
A5.2 Equations and formulae
A5.5 Equations, formulae and graphs
A6.1 Expressions and formulae
G4.1 Angles and shapes
G5.2 2D and 3D shapes
G4.4 Constructions
G5.4 Angles and constructions
G6.1 Geometrical reasoning
G4.3 Transformations
G5.3 Transformations
G6.3 Transformations and loci
G4.2 Measures and mensuration
G5.1 Measures and mensuration
G6.4 Measures and mensuration
N4.2 Whole numbers, decimals and
fractions
G6.2 Trigonometry 1
G6.5 Trigonometry 2
S4.2 Enquiry 1
S5.1 Enquiry 1
S6.1 Enquiry 1
S4.3 Enquiry 2
S5.3 Enquiry 2
S6.3 Enquiry 2
S4.1 Probability
S5.2 Probability 1
S6.2 Probability 1
S5.4 Probability 2
S6.4 Probability 2
R4.1 Revision unit 1
R5.1 Revision unit 1
R6.1 Revision unit 1
R4.2 Revision unit 2
R5.2 Revision unit 2
R6.2 Revision unit 2
Exploring maths Tier 5 Introduction | xi
Tier
5
Contents
N5.1 Powers and roots
1 Integer powers of numbers
2 Estimating square roots
3 Prime factor decomposition
Mental test
Check up
Answers
2
4
6
8
10
11
12
A5.1 Sequences and graphs
1 Generating sequences
2 Making generalisations
3 Using computers
4 Sketching linear graphs
5 Rearranging linear equations
6 Graphs representing real-life contexts
Check up
Answers
14
16
18
20
22
24
26
28
29
G5.1 Measures and mensuration
1 Perimeter and area
2 Finding 3 Area of a circle
4 Solving circle problems and using 5 Volume of prisms
6 Surface area of prisms
Check up
Answers
34
36
38
40
42
44
46
48
49
N5.2 Proportional reasoning
1 Adding and subtracting fractions
2 Multiplying fractions
3 Dividing fractions
4 Percentage change
5 Ratio
6 Direct proportion
Mental test
Check up and resource sheets
Answers
52
54
56
58
60
62
64
66
67
68
S5.1 Enquiry 1
1 Stem-and-leaf diagrams
2 Starting a statistical investigation 1
3 Completing a statistical investigation 1
4 Data collection sheets
5 Starting a statistical investigation 2
6 Completing a statistical investigation 2
Check up and resource sheets
Answers
72
74
76
78
80
82
84
86
88
xii | Exploring maths Tier 5
Introduction
A5.2 Equations and formulae
1 Multiplying out brackets
2 Factorising expressions
3 Substituting into formulae
4 Changing the subject of a formula
5 Solving linear equations
6 Trial and improvement
Check up
Answers
96
98
100
102
104
106
108
110
111
G5.2 2D and 3D shapes
1 Exploring angles and lines
2 Solving problems
3 Solving longer problems
4 Drawing 3D objects
5 Drawing plans and elevations
6 More plans and elevations
7 Solving problems using surface area and volume
8 Surface area and volume of prisms
Check up and resource sheets
Answers
114
116
118
120
122
124
126
128
130
132
134
N5.3 Calculations and calculators
1 Powers of 10
2 Rounding and approximation
3 Mental calculations with decimals
4 Written calculations with decimals
5 Using a calculator
6 Problems involving measures
Mental test
Check up and resource sheets
Answers
142
144
146
148
150
152
154
156
157
158
S5.2 Probability 1
1 Simple probability
2 Equally likely outcomes with two events
3 Mutually exclusive events
4 Practical probability experiments
5 Simulating probability experiments
Check up and resource sheets
Answers
162
164
166
168
170
172
174
176
A5.3 Functions and graphs
1 Generating linear graphs using ICT
2 Sketching graphs
3 Drawing accurate graphs
4 Direct proportion
5 Reﬂecting graphs in y x
180
182
184
186
188
190
6 Simple quadratic graphs using ICT
Check up
Answers
192
194
195
G5.3 Transformations
1 Planes of symmetry
2 Combined transformations
3 Islamic patterns
4 Enlargements
5 Enlargements in real-life applications
6 Length, area and volume
Check up and resource sheets
Answers
198
200
202
204
206
208
210
212
214
A5.4 Using algebra
1 Using graphs to solve problems
2 Using algebra in geometry problems
3 Using algebra in investigations
Check up
Answers
220
222
224
226
228
229
S5.3 Enquiry 2
1 Calculating statistics
2 Line graphs for time series
3 Scatter graphs
4 Collecting and organising data
5 Analysing and representing data
6 Interpreting data
7 Reporting and evaluating
Check up and resource sheets
Answers
232
234
236
238
240
242
244
246
248
251
G5.4 Angles and constructions
1 Angles in polygons
2 Regular polygons
3 Regular polygons and the circle
4 Angle problems and polygons
5 Polygons and parallel lines
6 Constructions
7 Constructing triangles
8 Loci
9 More loci
Check up and resource sheets
Answers
256
258
260
262
264
266
268
270
272
274
276
278
A5.5 Equations, formulae and graphs
1 Factorising
2 Working with algebraic fractions
3 Working with formulae
4 Forming equations
286
288
290
292
294
5 Visualising graphs
6 Interpreting graphs
7 Matching graphs to real-life situations
8 Using graphs to solve problems
Check up
Answers
296
298
300
302
304
305
S5.4 Probability 2
1 Theoretical and experimental probability
2 Mutually exclusive events
3 Using experimental probability
4 Choice or chance?
Check up and resource sheets
Answers
312
314
316
318
320
322
323
N5.4 Solving problems
1 History of our number system and zero
2 Number puzzles based on 3 by 3 grids
3 Exploring fractions
4 Problems involving properties of numbers
5 Using algebra and counter-examples
Slide commentary
Mental test
Check up and resource sheets
Answers
326
328
330
332
334
336
338
341
342
343
R5.1 Revision unit 1
1 Using a calculator
2 Using percentages to compare proportions
3 Sequences, equations and graphs
4 Angles and polygons
5 Charts and diagrams
Mental test
Resource sheet
Answers
346
348
350
352
354
356
358
359
360
R5.2 Revision unit 2
1 Ratio and proportion
2 Solving number problems
3 Expressions, equations and formulae
4 Circles and enlargements
5 Probability
Mental test
Answers
364
366
368
370
372
374
376
377
Schools planning a shortened two-year programme for
Key Stage 3 may not have time to teach all the lessons.
The lessons in black cover the essential material for pupils
taking this route. The lessons in blue provide useful
consolidation and enrichment opportunities. These
should be included wherever possible.
Exploring maths Tier 5 Introduction | xiii
N
5.1
Powers and roots
Previous learning
Objectives based on NC levels 5 and 6 (mainly level 6)
Before they start, pupils should
be able to:
recognise and use
multiples, factors (divisors),
common factor, highest
common factor, lowest
common multiple and
primes
use squares, positive and
negative square roots,
cubes and cube roots, and
index notation for small
positive integer powers.
In this unit, pupils learn to:
identify problems and the methods needed to tackle them
select and apply mathematics to ﬁnd solutions
represent problems and synthesise information in diﬀerent forms
use accurate notation
calculate accurately, selecting mental methods or a calculator as appropriate
use appropriate checking procedures
record methods, solutions and conclusions
make convincing arguments to justify generalisations or solutions
interpret and communicate solutions to problems
and to:
use index notation for integer powers
know and use the index laws for multiplication and division of positive integer
powers
extend mental methods of calculation with factors, powers and roots
use the power and root keys of a calculator
use ICT to estimate square roots and cube roots
use the prime factor decomposition of a number.
Objectives in colour lay the groundwork for Functional Skills.
Lessons
1 Integer powers of numbers
2 Estimating square roots
3 Prime factor decomposition
About this unit
Sound understanding of powers and roots of numbers helps pupils to
generalise the principles in their work in algebra and provides a foundation
for later work on numbers in standard form and surds.
It also helps pupils to be aware of the relationships between numbers and
to know at a glance which properties they possess and which they do not.
Calculators vary in the ways that powers and roots are entered. You may
need to point out these diﬀerences to pupils and explain how to adapt the
information in the class book or home book for their own calculators.
Assessment
2 | N5.1
Powers and roots
This unit includes:
an optional mental test which could replace part of a lesson (p. 10);
a self-assessment section (N5.1 How well are you doing? class book p. 9);
a set of questions to replace or supplement questions in the exercises or
homework tasks, or to use as an informal test (N5.1 Check up, CD-ROM).
Common errors and
misconceptions
Look out for pupils who:
__
n;
think that n2 means n 2, or that √n means __
2
wrongly apply the index laws, e.g. 103 104 107, or 103 104 1012;
think that 1 is a prime number;
include 1 in the prime factor decomposition of a number;
confuse the highest common factor (HCF) and lowest common multiple
(LCM);
assume that the lowest common multiple of a and b is always a b.
Key terms and notation
problem, solution, method, pattern, relationship, expression, solve, explain,
systematic, trial and improvement
calculate, calculation, calculator, operation, multiply, divide, divisible,
product, quotient
positive, negative, integer
factor, factor pair, prime, prime factor decomposition,
power,
root, square,
__
__
cube, square root, cube root, notation n2 and √n , n3 and 3√n
Practical resources
Exploring maths
Useful websites
scientiﬁc calculators for pupils
individual whiteboards
computers with spreadsheet
software,
e.g. Microsoft Excel, or graphics
calculators
Tier 5 teacher’s book
N5.1 Mental test, p. 10
Answers for Unit N5.1, pp. 12–13
Tier 5 CD-ROM
PowerPoint ﬁles
N5.1 Slides for lessons 1 to 3
Excel ﬁle
N5.1 Square root
Tools and prepared toolsheets
Calculator tool
Tier 5 programs
Multiples and factors quiz
Ladder method
HCF and LCM
Tier 5 class book
N5.1, pp. 1–10
Tier 5 home book
N5.1, pp. 1–3
Tier 5 CD-ROM
N5.1 Check up
Topic B: Indices: simplifying
www.mathsnet.net/algebra/index.html
Factor tree
nlvm.usu.edu/en/nav/category_g_3_t_1.html
Grid game
www.bbc.co.uk/education/mathsﬁle/gameswheel.html
N5.1 Powers and roots | 3
1 Integer powers of numbers
Learning points
A number a raised to the power 4 is a4 or a a a a.
The number that expresses the power is its index, so 2, 5 and 7 are the indices of a2, a5 and a7.
To multiply two numbers in index form, add the indices, so am an am n.
To divide two numbers in index form, subtract the indices, so am an am n.
When a negative number is raised to an even power, the result is positive;
when a negative number is raised to an odd power, the result is negative.
Starter
Tell pupils that in this unit they will learn more about powers and roots of
numbers. This ﬁrst lesson is about ﬁnding positive and negative integer powers.
Remind pupils that when a number is multiplied by itself the product is called
a power of that number. So a a, or a squared, is the second power of a and is
written as a2; a a a, or a cubed, is the third power of a and is written as a3;
and so on. For a4 we say a to the power of 4, and similarly with higher powers.
The number that expresses the power is its index. So 5 and 7 are the indices of a5
and a7. When the index is 1, it is usually omitted: we write a, rather than a1.
Ask pupils to calculate mentally some powers of integers, e.g.
8 2,
(8)2,
2 5,
(2)5,
34 ,
(3)4,
5 3,
(5)3
Record answers on the board, and ask pupils what they notice. Draw out that for
negative numbers even powers are positive and odd powers are negative.
What is (ⴚ1)123?
What is (ⴚ1)124?
As an optional extension, include some decimals, e.g.
(0.1)3,
Main activity
TO
(0.7)2,
(0.2)4
If appropriate, use the Calculator tool to show pupils how to use the xy keys of
their calculators.
Explore raising a number to the power 0 to show that this always results in 1.
Discuss negative indices. Ask pupils to consider the
pattern on the right.
How is each number found from the one above it?
What is the pattern of the indices?
10 000 104
1000 103
100 102
10 101
1 100
What are the next few lines of the pattern?
Establish that:
1
–1
0.1 __
10 10
1
–2
0.01 ___
100 10
1
–3
0.001 ___
1000 10
Similarly _12 2–1, _14 2–2 and _18 2–3.
Slide 1.1
4 | N5.1
Powers and roots
Show the table of powers of 2 on slide 1.1. Ask pupils to work in pairs to make up
and record some multiplications using the table. Ask questions to help pupils to
discover for themselves the rules for calculations with indices.
What do you notice about the indices in these calculations?
What is a quick way of multiplying powers of 2? Why does it work?
What is this calculation in index form?
32 256 8192 [25 28 213]
4096 512 [212 29 23]
16 384 64 [214 26 28]
What do you notice about the indices in these calculations?
What is a quick way of dividing one power of 2 by another?
Why does it work?
Repeat with the powers of 4 on slide 1.2.
Now generalise. Write on the board m2 m3.
Slide 1.2
What will this simplify to? Explain why.
[m2 m3 (m m) (m m m) m m m m m m5]
Stress that the indices have been added, so that:
m 2 m3 m 2 3 m 5
Repeat with m5 m2. Stress that for division the indices are subtracted, so that:
m5 m2 m5 2 m3
Discuss negative indices, e.g.
1
m3 m7 m3 7 m–4 ___
m4
m5 m–3 m5 3 m2.
Select individual work from N5.1 Exercise 1 in the class book (p. 1).
Review
Show slide 1.3.
Point to two diﬀerent powers of 10. Ask pupils to multiply or divide them and to
write the answer on their whiteboards.
Stress that the rules for multiplying and dividing numbers in index form apply to
both positive and negative indices.
Slide 1.3
Homework
Ask pupils to remember the points on slide 1.4.
Slide 1.4
Homework
Ask pupils to do N5.1 Task 1 in the home book (p. 1).
N5.1 Powers and roots | 5
2 Estimating square roots
Learning points
3
___
__
is the square root of n, e.g. √81 9.
__
is the cube root of n, e.g. √ 125 5, √27 3.
√n
√n
3
____
3
____
Trial and improvement can be used to estimate square roots when a calculator is not available.
Starter
Tell pupils that in this lesson they will be estimating the value of square roots.
__
__
Remind them that the square root of a is denoted by 2√ a , or more simply as √a ,
and that a square
root of a positive number can be positive or negative, e.g. if
__
2
√
a 9, a 3 .
Show the grid on slide 2.1. Write on the board: x 1, z 4.
Point to an expression on the grid. Ask pupils to work out its value mentally and
to write the answer on their whiteboards. Ask someone to explain how they
calculated it. After a while, change the values for x and z to x 9 and z 25.
Slide 2.1
Main activity
Discuss how to estimate
that is
___the positive square root
___
___not a perfect
___ of a number
√
√
√
√
square. For example, 70 must lie between
___ 64 and 81 , so 8 70 9. Since
70 is closer to 64 than to 81, we expect √70 to be closer to 8 than to 9, perhaps
about 8.4.
64
70
8
81
9
__
Show the class how they could ﬁnd √7 if they had only a basic calculator with no
square-root key. Tell them that the process is called trial and improvement.
__
Explain that √7 must lie between 2 and 3, because 7 lies between 22 and 32.
Try 2.52 6.25
Try 2.62 6.76
Try 2.72 7.29
Try 2.652 7.0225
Try 2.642 6.9696
Try 2.6452 6.986025
too small
too small
too big
very close but a little bit too big
very close but too small
still too small
The answer lies between 2.645 and 2.65.
All numbers
between 2.645 and 2.65 round up to 2.65.
__
So √7 2.65 correct to two decimal places.
Ask
___pupils to work in pairs and, using only the key on their calculator, to ﬁnd
√12 to two decimal places [answer: 3.46]. Establish ﬁrst that it must lie between
3 and 4.
Show the class how they could use a spreadsheet for this activity, without using
the square-root function, e.g. use the Excel ﬁle N5.1 Square root.
XL
6 | N5.1
Powers and roots
Point out that the strategy here is diﬀerent. We work systematically in tenths from
3 to 4, then in hundredths from 3.4 to 3.5, then in thousandths from 3.46 to 3.47.
You can use this ﬁle to estimate other square roots by overtyping 3, 3.4 and 3.46.
If possible, pupils should develop similar spreadsheets, using either a computer
or a graphics calculator.
Select individual work from N5.1 Exercise 2 in the class book (p. 4).
Review
Introduce root notation. Explain that
____ if 729 is the cube of 9, then 9 is the cube
3
√729 9. The cube root, fourth root, ﬁfth root, …
root of 729, which is written
as
__ __ __
of a are denoted by 3√a , 4√a , 5√a , …
Use the Calculator tool to demonstrate how to ﬁnd roots. You may need to
explain that some calculators have a cube root key as well as a general key like
x _
√ , or other variations, e.g.
3
____
To ﬁnd the value of √216 , key in 3
[Answer: 6]
___
x _
√
TO
2 1 6.
_____
Ask pupils to work out √64 and √125 . Explain that the cube root of a positive
number is positive, and the cube root of a negative number is negative.
3
3
______
Show how to use a calculator with an example such as √32 768 8.
5
Sum up the lesson by reminding pupils of the learning points.
Homework
Ask pupils to do N5.1 Task 2 in the home book (p. 2).
N5.1 Powers and roots | 7
3 Prime factor decomposition
Learning points
Writing a number as the product of its prime factors is called the prime factor decomposition of
the number.
You can use a tree method or a ladder method to ﬁnd a number’s prime factors.
To ﬁnd the highest common factor (HCF) of a pair of numbers, ﬁnd the product of all the prime
factors common to both numbers.
To ﬁnd the lowest common multiple (LCM) of a pair of numbers, ﬁnd the smallest number that is a
multiple of each of the numbers.
Starter
Tell pupils that in this lesson they will be ﬁnding the prime factors of numbers
and using them to ﬁnd common factors and multiples of a pair of numbers.
Remind them of the deﬁnitions of multiple, factor, factor pair and prime number.
Launch Multiples and factors quiz. Ask pupils to answer on their whiteboards.
Use ‘Next’ and ‘Back’ to move through the questions at a suitable pace.
QZ
Main activity
Write on the board three products such as:
11 5 3
2 2 13
2355
What do you notice about the numbers in these products?
Establish that they are all prime numbers. Explain that when a number is
expressed as the product of its prime factors it is called the prime factor
decomposition of a number. Stress that because 1 is not a prime number it is not
included in the decomposition.
How can we ﬁnd the prime factor decomposition of 80?
First explain the tree method, i.e. split 80 into a product such as 20 4, then
continue factorising any non-prime number in the product. Repeat with 300.
80
2
4
20
4
2
300
5
2
10
2
5
30
2
15
2
5
3
Launch Ladder method. Use it to show the alternative method, where the
number is repeatedly divided by any prime that will divide into it exactly.
SIM
Demonstrate with 63, dragging numbers from the grid to the
relevant positions. Drag 63 to the box contained in the sentence.
Drag prime numbers to the circles on the right. Continue to divide by
prime numbers until completed. The bottom square will become a circle.
Express the answer as 63 3 3 7 32 7.
3 63
3 21
7 7
1
Repeat with 80, either on the board or using the simulation.
Show how to ﬁnd the highest common factor (HCF) and lowest common multiple
(LCM) of a pair of numbers. Launch HCF and LCM.
SIM
8 | N5.1
Powers and roots
Select ‘Find the lowest common multiple’. Select 8 and 6 using the arrows by
the numbers. Drag multiples of 8 and multiples of 6 from the 100-square to the
answer boxes. (Numbers snap back to the 100-square if dragged from answer
box.) Numbers common to both boxes change colour to blue.
Which numbers are both multiples of 8 and multiples of 6?
Which is the lowest number that is both a multiple of 8 and 6?
Drag the LCM into the box below the 100-square.
Repeat with diﬀerent numbers, then change to ‘highest common factor’, which
works similarly. Select 24 and 18, then drag factors to the answer boxes.
Which numbers are both factors of 24 and factors of 18?
Which is the highest number that is both a factor of 24 and 18?
Repeat with diﬀerent numbers.
Show how to use a Venn diagram to ﬁnd the HCF and LCM of a pair of numbers
such as 36 and 30. Explain that:
the overlapping prime factors give the HCF
(2 3 2 3 6);
all the prime factors give the LCM
(2 2 3 3 5 22 32 5 180).
36
30
3
2
2
5
3
Repeat with 18 and 24.
Select individual work from N5.1 Exercise 3 in the class book (p. 7).
Review
Sum up the lesson by stressing the points on slide 3.1.
Round oﬀ the unit by referring again to the objectives. Suggest that pupils ﬁnd
time to try the self-assessment problems in N5.1 How well are you doing? in the
class book (p. 9).
Slide 3.1
Homework
Ask pupils to do N5.1 Task 3 in the home book (p. 3).
N5.1 Powers and roots | 9
N5.1 Mental test
Read each question aloud twice.
Allow a suitable pause for pupils to write answers.
1
What number is ﬁve to the power three?
2
Write all the prime factors of forty-two.
3
Write down a factor of thirty-six that is greater than ten and less than twenty. 2005 KS3
4
What is the next number in the sequence of square numbers?
One, four, nine, sixteen, …
5
Look at the numbers.
Write down each number that is a factor of one hundred.
2004 KS3
1999 Y7
[Write on the board: 10 15 20 25 30 35 40 45 50]
6
Write two factors of twenty-four which add to make eleven.
2005 KS2
7
What is the square root of eighty-one?
2001 KS3
8
What number is ﬁve cubed?
2003 KS3
9
The volume of a cube is sixty-four centimetres cubed.
What is the length of an edge of the cube?
2002 KS3
10 What is the square of three thousand?
2001 KS3
11 To the nearest whole number, what is the square root of
2004 KS3
eighty-three point nine?
12 I think of a number. I square my number and get the answer
one thousand six hundred. What could my number be?
Key:
KS2
Key Stage 2 test
Y7
Year 7 optional test (1999)
KS3
Key Stage 3 test
Questions 3 to 7 are at level 5; 8 to 11 are at level 6; 12 is at level 7
Answers
10 | N5.1
1
125
2 2, 3, 7
3 12 or 18
4 25
5
10, 20, 25, 50
6 3 and 8
7 9
8 125
9
4 cm
Powers and roots
10 9 000 000
11 9
12 40
2007 KS3
N5.1 Check up
Check up
N5.1
Write your answers in your book.
Powers and roots (no calculator)
1
2001 level 6
Which two of the numbers below are not square numbers?
24
2
25
26
27
28
David says that 211 2048.
What is 210?
3
To the nearest whole number, what is the square root of 93.7?
4
If √81 n √144 , then n could be which of the following numbers?
5
Year 8 Optional Test level 6
___
____
9
11
Terry has 24 centimetre cubes.
He uses them to make a cuboid that is one
cube high.
Tina has 24 centimetre cubes.
She uses them to make a solid cuboid that
is two cubes high.
12
13
1 cm high
1 cm wide
1 cm long
What could the dimensions of her cuboid be?
6
What is the biggest number that is a factor of both 105 and 135?
7
What is the smallest number that is a multiple of both 12 and 27?
Powers and roots (calculator allowed)
8
1996 level 6
Mary thinks of a number.
First I subtract 3.76
Then I find the square
root of what I get
My answer is 6.80
Which number did Mary think of?
© Pearson Education 2008
Tier 5 resource sheets | N5.1 Powers and roots | 1.1
N5.1 Powers and roots | 11
N5.1 Answers
Class book
3 a 2
Exercise 1
4 Each answer is correct to 1 d.p.
1 a 23
a 2.4
b 45
c 38
d (1)4
e 52
f
c 256
d 128
e 1
f
1
g __
16
1
h ___
125
d 19 683
e 1024
f
g 32 157.43
h 11 272.96
7 3.87 metres to 2 d.p.
11.39
Extension problem
8 2982 88 804
888 is not a perfect square. There is no whole
number between the square root of 8880 and
the
square root______
of 8889 but 298 lies between the
______
√88 800 and √88 899 .
b 35
c 104
d a8
e 54
f
g 80
h b3
124
2
1
6
1
Exercise 3
1
2
1
1 a 3 22
c 37
d 23 3
2
7
e 33
f
3
4
5
6 a 19 32 32 12
b 41 62 22 12
c 50 5 4 3
d 65 6 5 2
e 75 72 52 12
f
2
2
2
2
2
b 35
2 33
2 a 1, 2, 5, 10, 25, 50
b 2 52
3 a 1, 3, 5, 9, 15, 45
b 5 32
2
94 72 62 32
or 92 32 22
7 Rachel and Hannah are 14 and 11 years old.
4 e.g. 63 (with factors 1, 3, 7, 9, 21, 63)
5 a 72 and 30: HCF 6, LCM 360
Extension problem
b 50 and 80: HCF 10, LCM 400
8 The smallest whole numbers are 6 and 10:
102 62 100 36 64 43
103 63 1000 216 784 282
c 48 and 84: HCF 12, LCM 336
Exercise 2
1 a x 3
c x 12
2 a 1.41 to 2 d.p.
120
b x 7
2
2
75
5
5
3
d x 1
b 2.15 to 2 d.p.
d 0.2
e 5
f
g 1.73 to 2 d.p.
h 1
Powers and roots
6 HCF 15, LCM 600
2
c 4
12 | N5.1
d 8.4
c a 20.4 to 1 d.p.
c 1331
1
c 10.7
b a 12.3 to 1 d.p.
b 15 625
5
b 6.7
d 8
6 a a 9.7 to 1 d.p.
1
3 a 2401
4 a 28
c 11
5 Between 700 and 750 slabs will be used. 26 26
is 678, which is too few, and 28 28 is 784,
which is too many. So the exact number of slabs
is 27 27 729.
61
b 243
2 a 64
b 7
1.22 to 2 d.p.
7 HCF 10, LCM 360
40
90
2
2
2
5
3
3
8 a 2 and 3
b 6
b (11)2 121
(19)2 361
(21)2 441
(29)2 841
(31)2 961
c 378
9 a 28 and 40: HCF 4, LCM 280
b 200 and 175: HCF 25, LCM 1400
c 36 and 64: HCF 4, LCM 576
c (6)3 216
10 1050
d (13)3 2197
Extension problems
Task 2
11 a 22 4 (with factors 1, 2 and 4)
1 a 19
b 24 16 (with factors 1, 2, 4, 8, 16)
c 7 factors: 26 64
9 factors: 28 256
11 factors: 210 1024
13 factors: 212 4096
c 9
d 5
e 24
f
g 8.67 to 2 d.p.
h 4.24 to 2 d.p.
2
2 18.8 cm to 1 d.p.
12 420 days from now, since 420 is the lowest
common multiple of 1, 2, 3, 4, 5, 6 and 7.
How well are you doing?
3 a 3
c 6
b 4
d 10
Task 3
1 a 32, 24, 52, 33 or 9, 16, 25, 27
b 57 55 52 3125 25 78 125
2 a 34 81 is the largest. 34 92
b 25 and 27 are not square numbers.
3 a 32 9
b 9
1 a 23 3 7
b 35
2 a 2 32 52
b 5 7 17
3 45
b 27 128
4 936
c 32 2 18
5 7, 13 and 17
4 a a3
b b2
5 a HCF is 12
b LCM is 144
CD-ROM
Check up
6 Suzy’s number is 4.9.
1 25 and 27 are not square numbers.
7 5 11 19 1045
2 2048 2 1024
Home book
3 10
Task 1
1 a 3
e 4
11
3
4 11
b 2
f
6
10
c 11
4
2 a 28 33 13
b 72 43 23
c 1125 103 53
3 a (15)2 225
(25)2 625
g 8
1
10
d x
h z
6
5 2 cm high by 1 cm wide by 12 cm long
2 cm high by 2 cm wide by 6 cm long
2 cm high by 3 cm wide by 4 cm long
6 15
7 108
8 50
N5.1 Powers and roots | 13