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Cambridge IGCSE
Mathematics
Core and Extended
Ric Pimentel
Frankie Pimentel
Terry Wall
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ISBN: 9781398373624
 Ric Pimentel, Frankie Pimentel and Terry Wall 2023
First published in 2023 by
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Contents
Contents
Introduction
4
ESL support and guidance
8
ESL support material
18
How to use the problem-solving videos
29
Suggested Scheme of Work: Core
40
Suggested Scheme of Work: Extended
56
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Introduction
Introduction
The Cambridge IGCSETM Mathematics courses are designed to develop students’ use of
mathematical techniques and their mathematical understanding through reasoning, problem-solving
and analytical thinking. This Teacher’s Guide will help you to plan the course and to enable
students to achieve their potential.
This resource should be used alongside the Cambridge IGCSETM Mathematics Core and
Extended Fifth Edition Student’s Book and Workbook, and Cambridge IGCSETM Core
Mathematics Fifth Edition Student’s Book. The Cambridge IGCSE Mathematics Core and
Extended Student’s Book includes all the Core and Extended content of the syllabus, at a pace
which is appropriate for Extended-level students. The Cambridge IGCSE Core Mathematics
Student’s Book covers the Core content only, at a pace which is more appropriate for Core-level
students. There are many questions in the textbooks and it should be noted that students are not
expected to answer every question in every exercise; you should be selective in using the material
in the best way for your students’ individual needs.
Syllabus and assessment
The following information is taken from the Cambridge IGCSE and IGCSE (9–1) Mathematics
syllabuses (0580 and 0980) for examination from 2025. You should always check the syllabus for
the relevant year, available from the Cambridge Assessment International Education website, for
full information and updates.
The assessment objectives (AOs) are:
AO1 Knowledge and understanding of mathematical techniques
Candidates should be able to:
» recall and apply mathematical knowledge and techniques
» carry out routine procedures in mathematical and everyday situations
» understand and use mathematical notation and terminology
» perform calculations with and without a calculator
» organise, process and present and understand information in written form, tables, graphs and
diagrams
» estimate, approximate and work to degrees of accuracy appropriate to the context and convert
between equivalent numerical forms
» understand and use measurement systems in everyday use
» measure and draw using geometrical instruments to an appropriate degree of accuracy
» recognise and use spatial relationships in two and three dimensions.
AO2 Analyse, interpret and communicate mathematically
Candidates should be able to:
» analyse a problem and identify a suitable strategy to solve it, including using a combination of
processes where appropriate
» make connections between different areas of mathematics
» recognise patterns in a variety of situations and make and justify generalisations
» make logical inferences and draw conclusions from mathematical data or results
» communicate methods and results in a clear and logical form
» interpret information in different forms and change from one form of representation to
another.
Assessment overview for Cambridge IGCSE TM Mathematics
All candidates take two papers. Core candidates are eligible for grades C−G, and the examination
comprises Paper 1 (1 hour 30 minutes, calculators are not allowed) and Paper 3 (1 hour 30
minutes, calculators are allowed). Extended candidates are eligible for grades A*−E and the
examination comprises Paper 2 (2 hours, calculators are not allowed) and Paper 4 (2 hours,
calculators are allowed).
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Introduction
Learner attributes
Cambridge International have developed ‘Cambridge learner attributes’ which summarise the
attitudes and life skills that students need to develop alongside their academic skills. These
attributes will help students to be successful while they are studying and beyond.
The approach in Cambridge IGCSE Mathematics encourages learners to be:
Confident, in using mathematical language and techniques to ask questions, explore ideas and
communicate.
Responsible, by taking ownership of their learning, and applying their mathematical knowledge
and skills so that they can reason, problem solve and work collaboratively.
Reflective, by making connections within mathematics and across other subjects, and in
evaluating methods and checking solutions.
Innovative, by applying their knowledge and understanding to solve unfamiliar problems
creatively, flexibly and efficiently.
Engaged, by the beauty, patterns and structure of mathematics, becoming curious to learn about
its many applications in society and the economy.
Teaching the course
There are two separate sets of resources for teachers within this Teacher’s Guide: one to
accompany each of the Student’s Books. These include:
» suggested schemes of work
» numerical answers to all questions that appear in the Student’s Books
» numerical answers to all questions that appear in the Core and Extended Workbook and
the online Core worksheets (which have not been through the Cambridge International
endorsement process)
» worked solutions to student assessment questions that appear in the Student’s Books
» a bilingual glossary
» practice questions for each chapter with mark schemes
» ESL teaching guidance
» ESL activities
» ESL videos: these videos give more information about the ESL activities. They describe the
pedagogy and relevance of the activities, and provide some ideas for a sample classroom
procedure.
» CPD videos: these videos provide advice and tips on best practice for teaching the Cambridge
International syllabuses in the classroom, covering facilitating group work, learning command
words, techniques for presenting information and how to structure an answer. These videos
will show you how you can guide and support your students through their mathematics course.
» Problem-solving videos: these videos provide a step-by-step guide for students on common
mathematical techniques.
Suggested schemes of work
There are two suggested schemes of work which have been devised to follow a logical route
through the textbooks. One is for students following the Extended syllabus and using the Core
and Extended textbook; the other is for students following the Core syllabus who are using the
Core textbook. The aim is for students to complete the course by the end of the second term in
the second year of study; this will then allow time for revision and exam practice.
The chapters have been divided into a number of blocks, each with an approximate allocation of
teaching time that roughly equates to four weeks’ work, depending upon individual timetables.
The timings are generous to allow for some flexibility in this area. If necessary, the blocks can be
interchanged to allow for local conditions, preferences, etc. Where prior knowledge is required
before starting a block, this is listed in the ‘Notes’ column in the scheme of work. Please read this
carefully to ensure necessary learning has taken place before attempting the work.
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Introduction
Similarly, the order in which each chapter is completed can be rearranged within each block
if resources or timetabling dictates but, once again, some care needs to be taken to ensure the
necessary prior learning has taken place.
Numerical answers
Numerical answers are provided for all questions (and student assessments) that appear in each of
the Student’s Books and Workbook.
Numerical answers can also be downloaded from www.hoddereducation.com/Cambridgeextras,
where they are available as a single file.
Worked solutions
This Teacher’s Guide provides worked solutions for every question in the student assessments
in the Student’s Book. The worked solutions supplement the numerical answers that are also
provided in this guide.
The worked solutions can be used in different ways:
» The teacher can mark in more depth, as students’ mistakes are easier to isolate.
» Students can mark their own work, which encourages them to engage with the solutions and to
see good practice in laying out solutions. Mathematics is about communication and it is essential
that students gain experience in reading and following mathematical lines of enquiry themselves.
» Students can mark each other’s work, which is excellent practice as it helps to expose
misconceptions and allows for students to apply critical thinking when they try to understand
another student’s errors and their reasons for making them. Students often respond better
to their peers’ comments and may be more likely to read and act on criticism from a fellow
student. The ‘marker’ gains experience in communicating mathematically.
» The worked solutions provide a bank of examples which you can display on an interactive
whiteboard to aid their teaching and exposition of a new topic.
Practice questions and mark schemes
There are separate practice questions for every chapter in each Student’s Book. These consist
mainly of Cambridge International past paper questions, identified with a reference to the
original papers in which they appear. There are two instances in Core and Extended where a new
question has been written by the authors, identified with ‘Author-written question’, one in Chapter
7 and the other in Chapter 22. These provide extra practice questions and help students to prepare
for their examinations. Mark schemes, written by the authors, are provided.
ESL support
Many students on this course will have English as a second language. In recognition of this,
this Teacher’s Guide has included a set of printable ESL resources to help support you in your
teaching and your students in their learning.
A bilingual glossary template, with all the key terms populated, is included in the ESL support.
Some teachers may prefer to add the translations before giving students a copy of the glossary. It
is useful for students to learn new mathematical words in both English and their first language, to
ensure that they have a complete understanding of the meanings.
There are also five ESL videos, which provide more information about the ESL activities. The
videos describe the pedagogy and relevance of the activities, and provide some ideas for a sample
classroom procedure. The ESL videos are only available to teachers in the Boost eBook: Teacher
edition.
Knowledge tests and reporting
This Teacher’s Guide includes formative knowledge tests for the Student’s Books. The knowledge
tests are auto marked, with results provided via a numerical score. Results are available to
students straightaway, and teachers will be able to view results via the Boost dashboard, helping
them to see where learners are secure or need more support.
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Introduction
The knowledge tests can be accessed by students at any time. Alternatively, you can assign them
to students in Boost at the time that you wish them to be completed.
Teachers can generate different report types, as well as access a high-level overview of the
assessment data, by selecting ‘View test results’ from the dashboard or selecting ‘Reports’ from
the left-hand side menu.
To find out more about knowledge tests and the reports available, click the Help icon in the topright of Boost.
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ESL support and guidance
Teaching Cambridge IGCSE™ Mathematics to learners with
English as a second language (ESL)
As Cambridge IGCSE Mathematics is an international qualification taught in many different
countries around the world, it is often taught to learners who do not speak English as a first
language. This introduction will highlight some aspects of language challenges that learners may
have and suggest strategies to address them.
We have provided a set of language development activities and supporting resources to help
teachers and learners, as follows:
» Card matching: a game for learners to practise matching key mathematical terms with their
definitions in English.
» Categorisation activity: a game for learners to practise categorising mathematical terms.
» Crosswords: crossword puzzles for practising key mathematical terms.
» Diagram labelling: a diagram for learners to practise labelling to review key concepts. The
idea can be extended to other concepts that lend themselves to labelling.
» Graphic organisers: a set of graphic organisers to help learners practise organising new
terms for review. The idea can be extended to other concepts that lend themselves to graphic
organisers.
» Jigsaw reading: a simple language development activity to help learners practise defining key
mathematical terms.
» Listening activities: detailed listening practice to help learners develop their listening skills in
English.
» Note-taking template: provides learners with guidance on how to take notes while listening.
Learners complete the template, record new terms, key information and lesson objectives.
» Terminology record sheet: used as an alternative to a glossary, this record sheet allows for
more information to be recorded.
» Bilingual glossary: a list of key English terms, organised by chapter. Learners fill in
translations in their own language and make notes if needed. This can be used as a reference
by learners.
» Flashcards: a ready-made set of flashcards for learners or teachers to use to practise key
mathematical terms.
You will find the bilingual glossaries and flashcards on Boost (boost-learning.com). There are
some general principles which are likely to support all your learners but also will provide extra
support for any learners with language needs.
Learning objectives
When defining learning objectives for your lessons, give thought to the kind of language that
learners will need to successfully participate in the lesson and to use the knowledge that you plan
to teach. Make sure that you define a language objective; this will help you to plan appropriate
activities and assessment for learners. If you are teaching a monolingual class, it will also help you
to plan when to allow learners to use their first language in lessons.
Contextualise
Take the opportunity at the start of every lesson to link what is to be presented to what has
already been taught and what learners already know. Learners who have spent time in a different
country to other learners may have personal perspectives on some topics. Giving second-language
speakers of English this chance to think about the topic before starting the lesson will make it
easier to understand the language used.
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ESL support and guidance
Managing input
How will information be presented to the class? Are there ways that this can be made more
accessible for learners whose first language is not English? Consider the use of transcripts,
background reading and how long each piece of input lasts. If learners are listening or reading in
their second language, they are likely to benefit from shorter sections of input interspersed with
questions, discussion and collaborative tasks.
Collaborative learning
If learners are obliged to work together and communicate with each other, they will have to put
new language learnt to use, consolidating it in their memory. This means that they will have to
use information they have listened to or read in the lesson, and this provides opportunities for the
teacher to see any gaps in their knowledge or understanding.
Assessment for learning
Regular formative assessment, learning checks and developmental feedback are particularly
important when teaching learners who do not have English as a first language. It allows teachers
to check how well learners have understood and can express lesson content. It also helps learners
to see which aspects of their language skills require development.
Teaching vocabulary to second-language speakers of English
Mathematics is a subject with a considerable specialist vocabulary and a need for precision. These
two factors can make it challenging for learners with English as a second or additional language.
As they are less likely than native speakers to have as large a mathematical vocabulary, they are
likely to need to develop their knowledge of mathematical language. However, mathematics is not
solely expressed in terms of specialist terminology – many parts of a Cambridge IGCSE course
rely on everyday vocabulary. While we can reasonably expect a native speaker of English to
know almost all the general vocabulary used on a course, there may be surprising gaps in secondlanguage English speakers’ general vocabulary knowledge.
The vocabulary of mathematics
Mathematics is a subject that has its own vocabulary. This can be broken down into different
categories, which present different challenges.
Specific concepts
Much of the vocabulary that learners of mathematics have to learn is specific concepts which
are denoted by specific terms. Some examples of these are: factorise, integer and hyperbola.
Some concepts are expressed through combinations of mathematical terms such as: terminating
decimals, inverse proportion and exponential equations, while others may be made up of roots and
suffixes such as: grad- (gradient, gradual) and equi-/equa- (equidistant, equilateral, equation).
General terms with mathematical meaning
Mathematical concepts and operations can also be expressed through general terms which
may have a range of meanings, but acquire a specific meaning when used in the domain of
mathematics. Learners may know inequality as a social concept rather than as an expression
using Boolean operators. Power in “8 to the power of 4” has a different meaning in mathematics
to the meaning that it has generally in English. There are also general terms which indicate that
a particular operator is required, e.g. and, together, all for addition; difference, fewer, remain for
subtraction; by, of, product for multiplication and per, split, cut up, parts for division.
General terms
Mathematical problems are often expressed using general vocabulary, and learners have to be
able to understand this general vocabulary in order to select appropriate mathematical operations
to perform.
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ESL support and guidance
Command words
Learners also need to be aware of the meaning of, and the difference between, the different
command words that are used in practice questions. This knowledge is essential for learners to
be able to address tasks correctly. The command words that can be used in Cambridge IGCSE
Mathematics are:
» Calculate
» Construct
» Describe
» Determine
» Explain
» Give
» Plot
» Show (that)
» Sketch
» State
» Work out
» Write
» Write down
The precise meanings of these command words are set out in the Cambridge IGCSE Mathematics
syllabuses. Learners should be aware that the choice of command word in a question will
determine how they should answer the question.
All of these areas of language could potentially cause difficulties for learners who have English
as a second language. They are unlikely to have learnt specialist mathematical terms in general
language lessons. The terminology used with higher-level mathematical concepts may also be
unknown to native English speakers, and so this language will be taught explicitly to learners as a
matter of course. Teachers should, however, check that learners with English as second language
know the language used to express more basic concepts. The third area described above, where
general language is used to describe the context, is also likely to cause more problems for secondlanguage speakers of English than for learners who have English as a first language.
Vocabulary difficulties
Some learners are not able to understand or use terminology correctly. Some learners may use
general vocabulary where more specific vocabulary would be preferable. For example, they may
describe translation and (negative) enlargement using terms like move and shrink, which are not
appropriate. Learners should use appropriate mathematical terminology.
Learners may sometimes confuse similar terms. These may be terms that relate to similar
concepts, such as confusion between tangents, chords and diameter – all straight lines associated
with circles. Or terms that are made up of similar words, such as highest common factor and
lowest common multiple.
Some concepts associate only with a specific context and learners may sometimes attempt to link
concepts inappropriately. For example, they may use congruent and similar with lines rather than
triangles. It is clear that learners need to be able to not only recall a wide range of mathematical
terms, but that they also need to be able to differentiate between terms that are related in
meaning or similar in form and to understand where they can be used.
Strategies for developing learners’ mathematical vocabulary
Principles
There are a number of principles to consider when looking for effective ways of developing
vocabulary. Firstly, the frequency with which learners encounter new vocabulary is crucial in
ensuring that they are able to easily recognise, recall and use vocabulary items. Teachers should
ensure that learners encounter each new vocabulary item on numerous occasions. Learners
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ESL support and guidance
should also have new vocabulary presented in context – this helps to make the meaning clear, and
also gives a more memorable frame to support recall. Thirdly, it has been suggested that the depth
to which learners engage with new vocabulary increases their ability to memorise it. Teachers
should make sure that learners get opportunities to actively use new vocabulary.
Teacher strategies
Presenting new vocabulary within context is unlikely to be difficult in a Cambridge IGCSE
class. It will be beneficial for learners to expose themselves to as much mathematical language
as possible, both inside and outside the class. Learners should be encouraged to actively identify
useful new vocabulary items whenever they are listening to lessons or reading mathematical texts.
It is also vital for teachers to build in as much terminology review and recap as possible. Time
should be spent at the start of each lesson reviewing key concepts that have been presented in
recent lessons and encouraging learners to recall the terms for themselves. There are various
engaging activities that can be used for this, including using the vocabulary flashcards provided
in Boost. Regular formative assessment will increase the frequency with which learners review
these concepts. Online quiz applications, such as Kahoot or Quizlet, also provide opportunities
for further practice. Additionally, as learners often confuse similar terminology, it is likely to be
beneficial to spend some time getting them to work specifically on identifying the differences
between related terms, as well as distinguishing between them.
It is important to ensure that learners engage with the terms presented and have the opportunity
to use them actively as well as passively. Teachers should build in pair and group work to lessons,
to ensure that learners have the chance to use new terms when communicating with each other.
This is likely to lead to deeper engagement with new vocabulary.
Learner strategies
Learners should be encouraged to keep notes of new vocabulary. There are a number of ways that
this could be done, through language journals, word cards or shared online documents. These
records should also contain information about related terms, how these terms differ and how
they can be used. As well as keeping records, learners could also form study groups where they
talk through the topics studied, ensuring that they use new vocabulary. Learners should also be
encouraged to actively practise recalling language from their records. This could be done through
word games or quizzes similar to those used in class.
Developing second-language learners’ mathematical
reading skills
Reading in Cambridge IGCSE Mathematics
Language learning classes rarely focus on mathematical language development, so learners with
English as their second language may find the reading required by their mathematical studies
challenging.
Although mathematics does not tend to require learners to read large amounts of text, reading
skills are still crucial to success in the subject. Cambridge IGCSE Mathematics learners need
to be able to read the explanations of different concepts and techniques in order to study
independently. They also need be able to understand tasks presented as problems. As well
as being able to understand concepts expressed in prose, learners also have to be skilled in
understanding information presented in tables, charts and diagrams.
Reading skills
Reading is often considered to be made up of two different types of skills – bottom-up skills,
which come from understanding the meaning of the words and structures on the page, and topdown skills, which come from the readers’ own understanding of the topic and situation described
in a particular text. Successful reading involves the interaction of these two different sets of skills.
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ESL support and guidance
Consider this example problem:
In a sale the price of a jacket originally costing $1700 is reduced by $400. Any item not sold on the
last day of the sale is reduced by a further 50%. If the jacket is sold on the last day of the sale:
a calculate the price it is finally sold for.
b calculate the overall percentage reduction in price.
Bottom-up skills
The previous section has detailed the challenges that mathematical vocabulary can present for
second-language speakers of English. Unsurprisingly, vocabulary knowledge is a very important
part of being able to read successfully. In the example above, sale, reduction, further and
percentage are crucial to understanding the meaning of the question. For unknown vocabulary,
learners may be able to look at the prefixes or suffixes of the words and use their knowledge
of these morphological features, e.g. that the -ion ending of reduction suggests that it is a noun
related to the verb reduce used previously. In addition to vocabulary, learners need to understand
the structures and language features used to talk about mathematics. In this case, learners need
to understand that the question is part of a conditional sentence, and that the if indicates that they
need work out the price on the final day of the sale.
Top-down skills
In order to understand mathematical problems, learners need to use a certain amount of
background knowledge to be able to fully understand the problem. This background knowledge
is sometimes referred to as schematic knowledge or schemata. Being able to use this background
knowledge while reading allows learners to read more effectively.
Learners typically use their understanding of how mathematical problems are usually structured
to help them to understand how to find the key information that they need. In the example above,
the situation is presented, followed by further relevant information and then finally, the question.
This is very different from general prose, which usually starts with a topic sentence giving the
main content, followed by supporting information.
Learners will use their understanding of particular mathematical functions to anticipate the
significance of the information presented in a text. In a question about percentages, readers will
realise that they need to find out which figures they need to find percentages of. Once they see
that they are dealing with percentages, a skilled reader will ask themselves relevant questions as
they read, for example:
Which number is reduced by 50%?
Is the jacket reduced by $400 or to $400?
Do I need to find the percentage of the price, or the percentage off the price?
Where problems refer to a real-world situation, an understanding of situations similar to those
described in the task will support learners in understanding the problem and finding the solution.
If learners come from a culture where organised sales do not happen, they may find it harder to
process the mechanics of the situation. Understanding the context also helps learners to be able to
use common-sense checks to make sure that the problem has been understood correctly.
Strategies to support learners
Bottom-up reading skills are dependent on language knowledge, which can be developed
using the strategies described in the vocabulary section above. Extensive practice of reading
mathematical texts will help learners to consolidate this knowledge, as well as identify new
language items they need to learn. Learners should be encouraged to exploit reading texts to find
useful new vocabulary items. However, as well as developing learners’ language knowledge, it is
also important to support them in using their top-down skills and becoming strategic readers.
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ESL support and guidance
Consider what learners already know
When presenting a new text in class, teachers should ask learners what they already know about
a topic. This helps them to find a purpose when reading the text – they can confirm, add to, or
refute their current understanding. This review of prior knowledge also provides an opportunity
to recap key vocabulary related to the text.
Use text features
Once learners are thinking about the subject of a text, features in the text can be used to get them
to predict what they will find when they read it. Learners should look at the title, subheadings and
diagrams or tables in order to inform these predictions.
Skimming and scanning
Learners can be asked to skim a text to confirm their predictions. Skimming is reading a text
quickly to get the general meaning. In order to get learners to skim effectively, teachers should set
a short time limit and communicate this to the class. Once learners have found the main meaning
of the text, they should be encouraged to scan for specific information. Scanning is similar to
skimming, but learners are required to look quickly through the text to find a specific piece of
information. Again, it is important to set a short time limit for these activities and ensure that
learners stick to it.
Close reading
On subsequent readings of a text, learners could be asked to identify any command words, relevant
and irrelevant information, and language indicating the need for the use of specific processes. In
the example above, it is important for learners to identify which sums the figures refer to.
If learners do not know the meaning of something, ask questions to support them to work it out
rather than giving them the answer straight away – it is important to encourage them to develop
strategies for doing this independently in the future. Learners should be encouraged to check
their understanding of the problem as they go through it with both their real-world knowledge
of the context, to check that it makes logical sense, and with their knowledge of a range of
mathematical concepts and techniques.
Vocabulary focus
To help learners to develop their vocabulary knowledge, they could be asked to find words in a
text that fit particular definitions. They could be encouraged to find synonyms in the text. They
could also be asked to complete gapped sentences with words from the text.
Comprehension focus
Questions that check learners’ comprehension of a text, through direct questions, by identifying
true and false sentences, or by something like listing the steps of a process in a flow chart, will
give learners a focus for reading. This will also provide teachers with an opportunity to assess
learners’ reading skills.
Talking through texts
Another useful technique is to ask learners to talk through problems as they read them. This gives
them the opportunity to see how problems and processes are linked, and gives guidance on how
they should approach these problems in future. This may be a particularly useful technique to use
as part of formative assessment to ensure that teachers are able to make an accurate judgement of
a learner’s mathematical knowledge without it being affected by their English language level.
Note-taking practice
Learners can be trained to take notes more effectively; they should set themselves a purpose
before reading, note any important new vocabulary to look up and any questions that the text
raises. One way these skills can be developed in learners is to provide different templates which
support different note-taking strategies. Learners could also practise taking notes in class from
short extracts, and then compare them with the notes taken by their classmates.
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Developing second-language learners’ speaking skills in
mathematics classes
The importance of speaking in mathematics lessons
While it is not a part of the Cambridge IGCSE Mathematics assessment, being able to speak
about mathematics is important for learners who have English as a second language. We have
already seen that there is a considerable amount of language that is specific to mathematics,
or that is used in a specific way in mathematics. In order to be able to use a new language item
confidently and accurately, learners have to be able to use it in context. There are generally few
opportunities to talk about many of the concepts required for Cambridge IGCSE in everyday
life, and it is also unlikely that learners who speak English as a second language will have been
taught mathematical language in their language classes. For this reason, it is important to provide
opportunities for learners to use them in class.
There are more general benefits to speaking in the mathematics classroom. Speaking aids
reasoning and makes thought processes more transparent. This makes it easier for peers
and teachers to support learners to find the most appropriate reasoning for a mathematical
concept or problem. As mentioned in the Reading section, if learners are able to talk through
their understanding of a problem or concept, it is easier for the teacher to identify any
misunderstandings and support the learner to correct them.
Being able to talk confidently about mathematics in English may also be very important for
learners in their future studies or employment.
Features to develop in speech
Strategies for speaking development usually address one of two areas: accuracy and fluency. Both
of these can be developed in mathematics classes.
Accuracy
We have already seen how accuracy is very important in mathematical language; this is also
important in speech. It is important that learners are able to use appropriate mathematical
terminology to describe concepts. Pronunciation is a key aspect of this as learners need other
people to be able to understand them. However, a focus on accuracy needs to be balanced with a
focus on fluency.
Fluency
Fluency is the second area that mathematics teachers need to consider when supporting speaking
skills. Learners may have less confidence in talking about concepts they are less familiar with, or
when speaking in their second language. This means that teachers should adopt strategies which
will help learners’ confidence. Teachers should be aware of the interaction between accuracy and
fluency. If too much focus is placed on accuracy, learners may become reluctant to speak; as a
result, it is important to decide which errors to correct when feeding back to the class.
Principles for developing speaking skills
Repetition
When teaching new language items, it is vital for learners to be given the opportunity to repeat
what they are taught. This provides the chance to support learners’ accuracy in speech. When
you present new terminology, take the time to drill pronunciation so that any difficulties can be
addressed and to give learners a better chance to remember the new language.
Building confidence
Learners may need to build up their confidence in speaking English. This means that it may be
beneficial for them to speak in pairs or small groups before speaking in front of the whole class.
It also means that teachers should be careful when correcting errors, so as not to discourage
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learners. Care should also be taken to ensure that more confident learners do not monopolise
opportunities to speak during the lesson.
Wait time
When questioning learners, it is important to give them enough time to think through their
answers. This is particularly the case for learners who speak English as a second language. Not
only do they have to consider the mathematical concepts and how to apply them, but they have to
work out how to formulate their reply in English.
Learner talking time
When allocating time for learners to talk, teachers should consider how much time each learner
has to speak. If few learners speak at any one time, this means that each individual will only
speak for a limited time in the lesson.
Strategies to support learners’ speaking skills
Questioning techniques
When questioning learners to check understanding, consider how to ensure that as many of them
as possible are included. Use nominated questions, rather than waiting for the first answer. Ask
open questions that allow space for discussion. Consider the use of strategies such as pose-pausepounce-bounce questioning that allows a number of learners to comment on an answer, before it
is finally given. These strategies ensure that a wider range of learners participates actively.
Modelling language
Before asking learners to speak, provide an example answer for them to use as a model. You may
also want to write parts of the model on the board to serve as a prompt.
Recasting
A strategy for supporting learners’ accuracy, either with terminology or pronunciation, is to repeat
what a learner has said, correcting any error made. To be effective, learners should be encouraged
to repeat the correction.
Prompt cards
Prompt cards with language for common functions, such as hypothesising, can be used to support
discussion of mathematical concepts. Encourage learners to use the functional language cards
as a scaffold for their spoken English. Providing support like this will increase both learners’
confidence and their accuracy.
Pair and group discussions
Giving learners the opportunity to speak in pairs and groups increases the amount of time that
they spend speaking in class. It also allows them to build up their confidence before having to
speak in front of the whole class. It is important to consider how to organise group discussions in
order to ensure that all learners participate.
Snowball discussions
A snowball discussion starts as a pair discussion of a concept or problem and groups are then
joined together to form progressively larger groups. It allows learners to compare a range of
different views, without initially having to speak in front of a large number of peers.
Jigsaw discussions
Learners work in groups on separate aspects of a topic, one aspect per group. They then form new
groups with members of other groups, and share the knowledge gained to complete a task which
requires the use of content from all aspects of the topic.
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Developing listening skills with learners who have
English as a second language
Listening in lessons is clearly an important skill for learners of mathematics, and naturally this
can be challenging for those who have English as a second language.
Listening skills
As with reading skills, we should consider the interaction between different kinds of skills and
knowledge when we are listening to a class.
Bottom-up skills
In order to understand a lesson, learners need to be able to use knowledge of language. When
listening, there are two aspects to this. First, learners need to have sufficient knowledge of the
grammar and vocabulary that will feature in the lesson. As we have seen, there is a considerable
amount of this which is either specific to mathematics or is used in a specific way in mathematics.
In listening, it is also important for learners to be able to decode the words spoken from the
stream of sound that is heard. This can be challenging because spoken language does not always
closely reflect the written form, and gaps in the stream of speech do not fall in the same place
as gaps between words. Consider how acute angle could sound like a cute angle or even a queue
tangle. Learners may hear letters that are not in the written word, e.g. the word ward can be
often be heard in co-ordinates, so learners may expect to find the letter w. Many words have
different pronunciations depending on their position and significance in a sentence, e.g. of may
be pronounced more like uv when it is used in something like the difference of (diff-ren-suv) two
squares.
Top-down skills
In addition to being able to decode spoken sound, listeners also use background knowledge
to build understanding when they are listening. In mathematics lessons, learners will use their
knowledge of mathematical concepts to predict what will be described and to check their
understanding of the lesson. Effective listeners will also use their understanding of the format of
a lesson and what they expect the teacher to do to support their understanding of what is being
described.
Lesson stages
Pre-listening stage
Both top-down and bottom-up skills can be supported before learners have to listen to English in
the lesson. It is useful to remind learners of relevant topics that have already been studied. This
will allow learners to be better able to use their top-down schematic knowledge when they are
listening to the lesson. Where possible, learners should do some background reading before the
lesson about the topic to be presented, and should be encouraged to predict what they will learn
about during the lesson. Teachers can also support learners’ knowledge of language by providing
examples and definitions for any new language before it is presented. It would also be useful for
learners to hear these words, said with natural pronunciation.
During listening
Train learners to use different kinds of note-taking strategies in lessons. Giving learners templates
of graphic organisers for this can help them to effectively consider both their top-down schematic
knowledge and their bottom-up knowledge of the language and sounds of English. Learners could
be encouraged to note down linked ideas, or to consider possible applications for a mathematical
technique to improve their prediction skills. At the same time, they should also be encouraged to
note down new language as it is presented.
Teachers can also make it simpler for learners when they are presenting to the class. While it is
important for teachers to use appropriate mathematical language, they should consider how fast
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and clearly they are speaking. They should also try to avoid using language that is too complex or
likely to be unfamiliar to learners. When speaking, teachers should take care to write down key
terminology on the board, which will support learners to understand these words as they hear
them through the lesson.
After listening
It is important to check understanding after any period of extended listening, as learners may
have only partially understood the content presented. Consider how the learning checks that you
use include all learners. You may want to give learners the opportunity to compare notes after
listening so they can check whether they have understood correctly. Nominated questions to check
understanding are also useful for this.
Language development activities
Dictation activities can be particularly useful in helping learners to develop their ability to
recognise words for a stream of English speech. The aim of these activities is to raise awareness of
features of connected speech in English to make it easier for learners to understand. So it is best
to focus on these aspects when correcting, rather than spelling or grammar, etc.
How many words?
Dictate short sentences of mathematical language and ask learners to listen for how many words
are in each sentence. Ask learners to suggest their answers and what they heard before revealing
the answers. Discuss with the class reasons for any misheard sentences.
Gapped dictation
Give learners a short extract of mathematical language with groups of words blanked out. Dictate
the whole text to learners and ask them to fill in the gaps with the groups of words that they hear.
Ask learners to suggest their answers and what they heard before revealing the correct answers.
Discuss with the class reasons for any misheard sentences.
Dictation comparison
There are many variations of dictation activities, where learners listen and attempt to write
down exactly what was said. These can be very valuable tools to get learners to think about their
listening. Dictate a short extract of mathematical language (repeating as necessary), and then
ask learners to compare what they have written to the text itself. By looking at the kind of errors
made, they can work out what common chunks of mathematical language sound like.
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Card matching
In card matching activities, learners can be asked to match cards with definitions. Examples are
shown below. Alternatively, a worksheet could be provided, where learners can simply match
words and definitions by drawing a line between them or by numbering them, etc. Making card
sets also allows you to play games such as ‘snap’ or ‘Pelmanism’, which provide engaging games
for learners.
acute angle
An angle that lies between 0° and 90°.
acute-angled triangle
A triangle, where all three angles are less than 90°.
height
The perpendicular distance of a triangle from its base to
its third vertex.
iscoceles triangle
A triangle with two equal angles and two sides of equal
length.
obtuse angle
An angle that lies between 90° and 180°.
obtuse-angled triangle
A triangle with one angle that is greater than 90°.
polygon
A closed two-dimensional shape made up of straight lines.
quadrilateral
A 4-sided polygon.
reflex angle
An angle that lies between 180° and 360°.
regular polygon
A polygon with all sides of equal length and all angles of
equal size.
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Categorisation activity
Copy and cut out one set of the cards with the words shown below for each group of learners.
Ask learners to work in groups to sort the cards according to whether the terms refer to earnings
or profit and loss.
Net pay
Overtime
Cost price
Gross earnings
Piece work
Selling price
Bonus
Earnings
Discount
Basic pay
Profit & Loss
Average
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Crosswords
Crosswords are an engaging way to give learners practice in recalling new vocabulary during the
course. There are a number of online tools for creating these activities quickly and easily. Here is
an example.
Across
2 The type of number that is the result when an integer is multiplied by itself.
3 The part of the circumference of a circle between two radii.
5 The type of number (positive or negative) that can be written as a fraction.
6 A type of angle that lies between 0° and 90°.
8 A positive or negative whole number (including zero).
9 Any number (positive or negative) that cannot be written as a fraction.
10 A number with exactly two factors: one and itself.
Down
1
3
4
6
7
Any factor of a number that is also a prime. (Two words)
The side of a right-angled triangle that is next to a specific non-right-angle.
A number that divides into another number exactly.
The point of a pyramid where the triangular faces of the pyramid meet.
The whole non-negative numbers (integers) used in counting (0, 1, 2, 3, …).
1
2
3
4
5
6
8
7
9
10
Down: 1 prime factor; 3 adjacent; 4 factor; 6 apex; 7 natural
Across: 2 square; 3 arc; 5 rational; 6 acute; 8 integer; 9 irrational; 10 prime
KEY
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Diagram labelling
Where concepts can be easily illustrated through diagrams, diagram labelling activities can be a
way of giving learners practice at recalling terminology. In the example below, learners could be
given a list of terminology to use, or could be asked to recall it from memory. The activity could
be done individually or in groups.
e
b
c
a
f
g
segment
diameter
chord
sector
d
radius
centre
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Graphic organisers: Venn diagram
Ask learners to draw a Venn diagram, like the one shown below, to compare similar concepts.
You could give learners the characteristics in a list for them to match to the concept, or just ask
them to complete the diagram from scratch.
Rhombus
Paralellogram
All sides are equal.
Opposite
angles are
equal.
Diagonals intersect
at right angles.
Two pairs
of parallel
sides.
Opposite sides
are equal.
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Jigsaw reading
Jigsaw reading is an activity that enables learners to practise using a range of skills, but it can be
particularly useful in developing speaking skills, as learners are required to teach each other the
information they have read.
In the following example, learners would be divided into groups (AAAA, BBBB, CCCC, DDDD,
etc.) and each group would be asked to read about how to carry out a separate mathematical
operation on fractions, using the jigsaw cards below. Learners would then use the notes sheet to
make notes about the operation they have read about.
Learners are then regrouped. This time each group should have one person from each of the
previous groups in their group (ABCD, ABCD, ABCD, etc.). Learners should then explain
the mathematical operation they have read about to their new group, while the other learners
complete their notes sheet.
Finally, learners work together to complete a set of questions, requiring knowledge from all of the
original groups.
Jigsaw cards
Adding fractions
In order to add fractions, they need to have the same denominator. If the fractions that you wish to add have
different denominators, find the lowest common denominator and rewrite the fractions with this denominator, e.g.:
1+1= 4 + 3
6 8 24 24
Once the denominators are the same, the fractions can be added by adding the numerators together, e.g.:
4 + 3 = 7
24 24 24
Subtracting fractions
In order to subtract fractions, they need to have the same denominator. If the fractions that you wish to subtract have
different denominators, find the lowest common denominator and rewrite the fractions with this denominator, e.g.:
1− 1 = 5 − 3
3 5 15 15
Once the denominators are the same, the fractions can be subtracted by subtracting the numerators, e.g.:
5 − 3 = 2
15 15 15
Multiplying fractions
To multiply fractions, multiply the numerators together to find the numerator of the product. Similarly, the
denominators of the fractions are multiplied to find the denominator of the product. If any of the fractions are
expressed as mixed numbers, they should be rewritten as improper fractions before the multiplication.
3 1 × 2 1 = 7 × 9 = 7 × 9 = 63 = 7 7
2
4 2 4 2×4 8
8
Dividing fractions
In order to divide one fraction by another fraction, invert the second fraction and then multiply the fractions
together. To multiply fractions, multiply the numerators together to find the numerator of the product. Similarly,
the denominators of the fractions are multiplied to find the denominator of the product. If any of the fractions are
expressed as mixed numbers, they should be rewritten as improper fractions first.
1 1 ÷ 2 1 = 4 ÷ 11 = 4 × 5 = 4 × 5 = 20
3
5 3 5 3 11 3 × 11 33
Questions
Work with the other members of your group to find the answers to these questions:
a 11 + 2 1 ÷ 35 × 4 1
b 73 − 21 × 21 ÷ 1
2
6
2
5
4
4 2
3
2
1
7
1
2
3
c 6 +4 −3 ×3
d 3 ÷1 + 7 1 − 42
3
5
8
3
4
3
5
7
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Notes sheet
Adding fractions
Subtracting fractions
Numerators
Numerators
Denominators
Denominators
Notes
Notes
Multiplying fractions
Dividing fractions
Numerators
Numerators
Denominators
Denominators
Notes
Notes
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Listening activities
When learners have to listen to information in lessons, it is good practice to ask questions to
check their understanding of the topics taught, but it is also beneficial to use activities that help to
develop learners’ ability to understand spoken English. Activities based on dictation are useful for
this. Below are some example activities, with suggested content to give to the learners.
Count the words
Listen to your teacher read out five sentences. Count the words in each sentence that you hear
and write the number in the first column. Then try to remember what you heard and write the
sentence in the second column. When your teacher gives you the answer, write this in the third
column.
Circle any differences. Why do you think you heard something different? Discuss with a partner.
No. of words
Sentence heard
Answer
1
2
3
4
5
Possible sentences to use
1 An equation is formed when the value of an unknown quantity is needed.
2 The two most common ways of solving simultaneous equations algebraically are by elimination
and by substitution.
3 A function is a particular type of relationship between two variables.
4 A column vector describes the movement of the object in both the x direction and the y
direction.
5 Combined events look at the possibility of two or more events.
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Gapped dictation
Read the text below. Now listen to your teacher read the complete text. In the gaps, write the
groups of missing words that you hear in the text. Each gap needs to be completed with two to
four words.
In many instances
using a calculator produce answers which are not
. A calculator will give the answer to as many
is not needed. Unless
as will fit on its screen. In most cases this
an answer is exact or a different accuracy is specifically asked for in a question, answers should be given to
significant figures. Angles should
decimal place and money should be given
be given correct to
to
decimal places.
Solution
In many instances calculations carried out using a calculator produce answers which are not whole
numbers. A calculator will give the answer to as many decimal places as will fit on its screen. In
most cases this degree of accuracy is not needed. Unless an answer is exact or a different accuracy
is specifically asked for in a question, answers should be given to 3 significant figures. Angles
should be given correct to 1 decimal place and money should be given to 2 decimal places.
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Note-taking template
Many learners will benefit from guidance on how to take notes when they are listening in lessons.
One way to do this is to provide them with a template that supports them to listen actively.
Lesson objectives:
What do I want to learn in this lesson?
Questions to ask
Key information
New terminology
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Terminology record sheet
As an alternative to a glossary, learners could use terminology record sheets when they come
across new words. This will help learners to study and remember new vocabulary.
Term:
Translation:
Definition:
Related terms:
Example/diagram:
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How to use the problem-solving
videos
The most essential skill for Cambridge IGCSE Mathematics students to acquire is the ability to
read a problem, understand what the question is asking and know which mathematical approaches
they need to solve it.
.
These videos have been created with the aim of helping learners develop this skill. They have
all been developed to encourage the student to reflect on what they have done and understand
how the strategies developed in solving the problem can be applied to other similar types of
question.
Below is a list of the videos and which syllabuses, topics and chapters they support. Each one
works through a problem in a step-by-step way to encourage the learner to think about what skills
and processes to use. Each one comes with an introduction designed to encourage the learner
to think about how to approach the problem and how they can apply what they learn to other
questions.
Video title
Cambridge IGCSE Core
Mathematics
Cambridge IGCSE Core and
Extended Mathematics
Forming equations
Topic 2: Algebra and graphs
Chapter 13: Equations
Topic 2: Algebra and graphs
Chapter 13: Equations and
inequalities
From equations to graphs
Topic 2: Algebra and graphs
Chapter 18: Graphs of functions
Parallel lines and angles
Topic 4: Geometry
Chapter 21: Angle properties
Topic 4: Geometry
Chapter 25: Angle properties
Distance-time graphs (1)
Topic 2: Algebra and graphs
Chapter 15: Graphs in practical
situations
Topic 2: Algebra and graphs
Chapter 17: Graphs in practical
situations
Distance-time graphs (2)
Topic 2: Algebra and graphs
Chapter 15: Graphs in practical
situations
Topic 2: Algebra and graphs
Chapter 17: Graphs in practical
situations
Matching sequences
Topic 2: Algebra and graphs
Chapter 14: Sequences
Topic 2: Algebra and graphs
Chapter 15: Sequences
Sequences from patterns
Topic 2: Algebra and graphs
Chapter 14: Sequences
Topic 2: Algebra and graphs
Chapter 15: Sequences
Using graphs to solve problems
Topic 2: Algebra and graphs
Chapter 15: Graphs in practical
situations
Topic 2: Algebra and graphs
Chapter 17: Graphs in practical
situations
Highest Common Factor and Lowest
Common Multiple
Topic 1: Number
Chapter 1: Number and language
Topic 1: Number
Chapter 1: Number and language
Reverse price calculations with
percentages
Quadrilaterals
Topic 1: Number
Chapter 5: Further percentages
Topic 2: Algebra and graphs
Chapter 11: Algebraic representation
and manipulation.
Topic 4: Geometry
Chapter 21: Angle properties.
Topic 2: Algebra and graphs
Chapter 11: Algebraic representation
and manipulation
Topic 4: Geometry
Chapter 25: Angle properties
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How to use the problem-solving videos
Forming equations
Questions
1 In a cutlery drawer in a restaurant, the total mass of 42 forks is 1974 g.
What will be the total mass of the forks when all 60 forks are in the drawer?
2 A loaf of bread costs x cents.
A cake costs (x−5) cents.
The total costs of 6 loaves of bread and 11 cakes is $13.56.
Find the cost of a loaf of bread and the cost of a cake.
These are two different sorts of problems that require you to make an equation to find the answer.
In the first case, you need to do two separate calculations − finding the mass of one fork, and then
multiplying that answer by 60 to find the mass of 60 forks.
The second problem is different. The cost of the cake is described in terms of the cost of a loaf of
bread (x−5).
The video also shows you how to approach these problems.
It helps you to think about:
» What the problem would actually look like − the knives in the drawer or the bread and cakes
on the shelf.
» What you have been asked to find out − the mass of 60 forks rather than 42 or the cost of one
loaf and the cost of one cake.
» What other things you will need to bring to the problem − how you can describe the 6 loaves
and 11 cakes.
» The need to have just one unknown fact on one side of an equation.
» The rules about rearranging the parts of an equation so that you do the same action to both
sides of the equation.
So you could write this process down as:
» What is this problem asking me?
» What would it look like in reality?
» What do I know?
» What am I being asked to find out?
» What knowledge about this area of maths do I have that I can bring to the problem?
You might need to go through those steps several times at different stages of solving the problem,
but this is an approach that will help you to work out the answer. Remember to write down your
thinking as you go through the problem, as sometimes the method is just as important as the final
answer.
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How to use the problem-solving videos
From equations to graphs
Question
Sketch the graph of 2x2 − 16 − y − 4x = 0.
This type of problem is asking you to draw a graph that would be produced by the equation.
The video shows you how to approach this problem and takes a simpler example to remind you of
the process and thinking involved. It requires:
» you to know that it must be a quadratic expression because it has a term in x2, a term in y and a
term in x
» just one unknown on one side of an equation − in this case you can rearrange the equation to
show y in terms of x
» understanding the rules about rearranging the parts of an equation so that you do the same
action to both sides of the equation
» factorisation and how to find values of x and y in relation to 0.
So you could write this process down as:
» What is this problem asking me?
» Work out the value of x = 0 for the y-intercept.
» Then write y in terms of x and make it as easy to read as possible.
» Complete the square for a quadratic by collecting all the x together for the minimum value.
y
6
4
2
(−2,0)
0
–3 –2 –1
–2
(4, 0)
1
2
3
4
5
6
x
–4
–6
–8
–10
–12
–14
–16
–18
–20
(0, −16)
(−1, −18)
You will need to make sure that you understand each of the steps in this process. Remember that
you have been asked to sketch the graph, so you are not being asked to work out the exact values
of many points, just to show its shape as accurately as you can.
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How to use the problem-solving videos
Parallel lines and angles
C
A
E
B
D
This type of diagram will be used in problems which draw on your knowledge of parallel lines and
lines crossing them, but also the properties of triangles. This video will show you how to approach
diagrams like this and takes facts you know about parallel lines and angles of lines crossing them
to work out the values of the angles.
When you come to solving this problem, it will help you to think about these questions:
» What do you know about parallel lines?
» How can you label each of the points where lines cross?
» What can you say about the angles at each of the points?
» How does knowing some of the facts about triangles help you?
So you could write this process down as:
» What is this problem asking me?
» What information have I been given about the values of any of the angles?
» The intersection of two lines with a parallel line means which angles in the diagram must have
the same value?
» The diagram has two triangles in it. What does that mean about the value of the angles inside
the triangles?
It always helps to write down the facts you know from theorems about geometrical problems.
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How to use the problem-solving videos
Distance-time graphs
Question
The grid shows the travel graph for a car travelling from Ambleside to Brownsford, 14 kilometres
away.
y
Distance from Ambleside (km)
16
14
12
10
8
6
4
2
0
11:00
11:10
11:20
11:30
Time
11:40
11:50
x
a Calculate the average speed, in kilometres per hour, for the journey from Ambleside to
Brownsford.
b The car waits at Brownsford for 8 minutes before returning home at a constant speed of 70
km/h. Complete the travel graph.
Graphs like this are often used to plot journey times or changes in temperature. It is really
important to make sure that you know what is being shown on the different axes, and also the
units that are being used.
When you come to solving this problem, it will help to think about these questions:
» What does each part of the graph show?
» What does each vertical interval represent?
» What does each horizontal unit represent? How does that relate to the units you need to use to
give the answer?
» To solve part a of the question you need to work out the average speed of the journey. What
formula do you need to use?
So, you could write this process down as:
» Say what you see. Describe the diagram to yourself.
» Make sure you know what units are being used, and how they relate to the units you need to
use to give the answer.
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How to use the problem-solving videos
Sequences
Question
Below is a sequence of diagrams constructed using square tiles.
1
2
3
a How many square tiles would there be in diagram 4?
b Which diagram would contain 42 square tiles?
c Find an expression for the number of squares in diagram n, where n is the same integer.
Matching sequences explores the way in which sequences of numbers can be described using
mathematical expressions.
When you come to solving this problem, it will help you to think about these steps:
» What can you see in each of the diagrams?
» What are you being asked to find?
» Break down the diagram into parts.
» Draw the next item by building up the legs, the seat and the back.
» This will give you the answer for part a.
» Solving part b can be achieved by drawing the diagrams until you get to 42, but it is much
better solved by using mathematic sequences to help you.
» What do you know about the way sequences are described mathematically?
So, you could write this process down as:
» Say what you see. Describe the diagram to yourself.
» Break down the diagram into parts.
» Look for a mathematical relationship between your answers.
» Write down the expression that sums up that relationship.
There are six different sequences.
The initial five terms of five of the sequences are shown in the table on the left.
Expressions representing five of the sequences are shown in the table on the right.
A
3
6
9
12
15
1
3n
B
−3
−6
−9
−12
−15
2
−3n − 10
10 − 3n
C
13
16
19
22
25
3
D
−13
−16
−19
−22
−25
4
3n + 10
E
7
4
1
−2
−5
5
−3n
Match up the lists of terms with the correct expressions.
One set of terms will need its expression to be written.
One expression will need its set of terms to be written.
Notice that in the question you are told that the set of terms on the left does not have a correctly
matching expression on the right. You are also told that one of the expressions on the right does
not match the list of terms on the left.
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How to use the problem-solving videos
Initial term
Expression
A 3, 6, 9, 12, 17
1 3n
Creating a table to put the answers in is very important, so that there is no doubt which initial
term you have matched to each expression.
When you come to solving this particular problem, it helps to think about these steps:
» Looking for relationships between the initial terms − positive and negative, for example.
» Looking for relationships between the sets of expressions can be helpful too.
» Make sure that when you find a sequence of terms that does not have an expression you work
out the expression, because there is one missing!
» And there also is an extra expression for which you’ll need to create the sequence of terms.
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How to use the problem-solving videos
Using graphs to solve problems
Question
Emily is hiring a bike. Which is the better deal?
Chris’s Bikes
$4 per hour
Joe’s Wheels
$7 hire fee
plus $2 per hour
This is an interesting question, because it does not give us enough information to answer straight
away. It will depend on how long Emily wants to hire the bike for. It is best to read the answer
from a graph so that a decision can be made based on what the graph shows.
When you come to solving this problem, it will help to think about these steps:
» What are you being asked to find out?
» The two deals are different, so how can you make them comparable?
» Two sets of data can be plotted on a graph and you can then compare them.
» How can you tell that the graph will be a straight line graph?
So you could write this process down as:
» Say what you see. Describe the problem to yourself.
» How can you describe each pricing system?
» What do you then have to do to compare them?
» Make sure you label the axes of the graph so that you know what it is showing.
Visualising what the answer will look like is important to help you solve the problem. Ask
yourself questions as you go through the process − what do I know, how can I show this, what I am
expecting to see?
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How to use the problem-solving videos
Highest Common Factor and Lowest Common Multiple
Questions
1 A party has 50 guests who all shake hands with each other. How many handshakes are there
altogether?
2 This question is suitable for students following either the Core or Extended syllabus.
Safi has a piece of paper is 24 cm wide and 30 cm long.
Safi cuts the paper into equal sized squares, without any paper left over.
Find the smallest number of squares that Safi can cut the paper into.
3 This question is for students following the Extended syllabus.
Find the Highest Common Factor (HCF) of 64, 48 and 72.
4 Tom and Lindsay set the alarms on their phones to sound at 5:30 am. Both alarms sound
at 5:30 am.
Tom’s alarm then sounds every 8 minutes. Lindsay’s alarm then sounds every 6 minutes.
At what time will both alarms then sound together?
These four questions all use your knowledge of numbers, especially the Highest Common Factor
and the Lowest Common Multiple.
The video also shows you how to approach each problem in turn.
For question 1, start by thinking about 4 people shaking each other’s hands.
For question 2, start by drawing a diagram.
You could try cutting the paper into 30 × 24 = 720 squares of size 1 cm by 1 cm.
However, the question asks for the smallest number of squares.
You need to find the Highest Common Factor of 24 and 30.
Factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24
Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30
So, the highest common factor is 6.
4 × 6 = 24 (width)
and 5 × 6 = 30 (length)
Safi can cut 4 squares along the width of the paper and 5 squares along the length.
So, Safi cuts 4 × 5 = 20 squares altogether.
For question 3, work out the factors for one of the numbers and then compare it with the second.
For question 4, make a chart of each phone alarm time. You can then compare them to find out
the answer.
To solve these problems, it will help you to think about these questions:
» How can you simplify the problem?
» Have you read the question carefully?
» How can you record your thinking? For example, writing down the steps or labelling the parts
of the problem.
When you watch the video, think about how the problem is simplified:
» What is changed by taking a simpler example?
» How did the solution use labels or layout to help prepare for the bigger question?
» What were the key words in the question to make sure you could find the right answer?
» What knowledge about this area of maths do I have that I can bring to the problem?
It is often best to simplify practical questions to make sure you understand what sort of answer
you should expect.
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How to use the problem-solving videos
Reverse price calculations with percentages
Question
In a sale all prices are reduced by 20%. The sale price of a coat is $54.
What was the original price?
We all buy things in the sales and it is good to know how much we are saving when we buy
something that is reduced by a percentage figure.
When you come to solving this problem, it will help you to think about these steps:
» What are you being asked to find out?
» What do you know now?
» How can you use the price and the size of the discount to find out the original price?
» Finding out what 1% of the item is worth always enables you to find the answer.
» This is also true if you know that something has gone up by 6%, as in another example shown
in the video.
So you could write this process down as:
» Whenever you have to find out about a percentage change, find out what 1% is worth.
» If there has been a reduction in price, then you will need to divide by a number less than 100. If
there has been an increase, then you will need to divide by a number greater than 100.
» You can always check your answer by working out the value of the reduction and subtracting
this from the full amount.
» You can do the same to check an increase, by adding the value of the increase to the original
price.
Visualising what the answer will look like is important to help you solve the problem. Ask
yourself questions as you go through the process − what do I know, how can I show this, what I am
expecting to see?
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How to use the problem-solving videos
Quadrilaterals
Question
Solve for p in the quadrilateral below.
2p°
p° + 30°
p° + 70°
110° − p°
This type of problem is asking you to use your knowledge of angle facts in quadrilaterals – and
your knowledge of algebra – to find the value of p.
The video shows you how to approach this problem, using a standard problem-solving strategy
that requires you to ask:
» What do I know from the information given in the diagram?
» What do I want to find out?
» What can I introduce?
It will help to think about these steps:
» You know that any quadrilateral can be split into two triangles.
» You also know that the three angles in a triangle add up to 180 degrees – and therefore the
angles in a quadrilateral will add up to 360 degrees.
» You now need to use algebraic manipulation to find the solution.
» This means collecting common terms and simplifying.
» Once you have done that, finding the value of p is straightforward.
So you could write this process down as:
» What is this problem asking me?
» What information is given in the diagram that I can use?
» Apply knowledge of angles in quadrilaterals.
» Apply algebraic manipulation to find the answer.
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This Scheme of Work has been devised to follow a logical route through the textbook for students following the Core content of the syllabus and using
the Core textbook. Its aim is for students to complete the course by the end of the second term in the second year of study; this will then allow time for
revision and preparation for their exams. The chapters have been divided into fifteen blocks each with 14 hours of teaching time; this roughly equates to
four weeks’ work, depending upon individual timetables. The timings are generous to allow for some flexibility in this area.
If necessary, the blocks can be interchanged to allow for local conditions, preferences, etc. Where prior knowledge is required before starting a block, this
is listed in the ‘Notes’ column in the Scheme of Work; please read this carefully to ensure necessary learning has taken place before attempting the work.
Similarly, the order in which each chapter is completed can be rearranged within each block if resources or timetabling dictates but, once again, some
care needs to be taken to ensure the necessary prior learning has taken place.
Learning objectives included in the schemes of work below are reproduced from the Cambridge IGCSETM and IGCSE (9–1) Mathematics syllabuses
(0580/0980) for examination from 2025. This Cambridge International copyright material is reproduced under licence and remains the intellectual
property of Cambridge Assessment International Education.
Suggested Scheme of Work: Core
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Resources in Cambridge IGCSE Core Mathematics Fifth Edition
Block 1: Total time 14 hours
Subject area
Approx. time
allocation
Learning objectives
Vocabulary
Mathematical investigations
and ICT
Notes
Chapter 1
Number and
language
9 hours
C1.1 Types of number
Identify and use:
• natural numbers
• integers (positive, zero and negative)
• prime numbers
• square numbers
• cube numbers
• common factors
• common multiples
• rational and irrational numbers
• reciprocals.
cube number; cube root;
factor; highest common factor;
integer; irrational number;
lowest common multiple;
multiple; natural number;
negative number; positive
number; power; prime factor;
prime number; rational
number; reciprocal; square
number; square root
Mystic Rose, Pages 95–97
This fully worked example
takes students through the
process of carrying out a
mathematical investigation
and the value of systematic
working. Students should work
through the problem and then
compare their methods with
the worked solution.
Primes and squares, Page 98
This is an investigation into
which prime numbers can
be written as the sum of two
squares.
This chapter covers the
different types of number and
vocabulary that students need
to be familiar with.
In Exercise 1.9 (Page 8),
students need to recall some
work from Lower Secondary
including Pythagoras’ theorem
and the formula for the
circumference and area of a
circle.
The chapter covers noncalculator work as well
as giving the students the
opportunity to practise using
their calculator to find powers
and roots.
C1.3 Powers and roots
Calculate with the following:
• squares
• square roots
• cubes
• cube roots
• other powers and roots of numbers.
Chapter 2
Accuracy
5 hours
C1.10 Limits of accuracy
Give upper and lower bounds for data
rounded to a specified accuracy.
accuracy; decimal place;
estimate; lower bound;
rounding; significant figure;
upper bound
41
This chapter involves rounding
to powers of 10, decimal places
and significant figures. It also
includes using an appropriate
degree of accuracy and
estimation. It is important
for students to use estimation
as a means of checking their
calculations.
In Exercise 2.4 on Pages 19–20,
they need to find area and
volume of simple compound
2D and 3D shapes.
Remind students to round any
inexact answers to 3 s.f. Also
when working with angles, give
inexact angles correct to 1 d.p.
– see Block 11 Chapter 25.
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Suggested Scheme of Work: Core
C1.9 Estimation
1 Round values to a specified degree of
accuracy.
2 Make estimates for calculations involving
numbers, quantities and measurements.
3 Round answers to a reasonable degree of
accuracy in the context of a given problem.
Block 2: Total time 14 hours
Subject area
Approx. time
allocation
Learning objectives
Vocabulary
Mathematical investigations
and ICT
Notes
Chapter 3
Calculations
and order
8 hours
C1.5 Ordering
Order quantities by magnitude and
demonstrate familiarity with the symbols =, ≠,
>, < , ⩾, ⩽ .
addition; division; indices;
inequality; multiplication;
order of operations;
subtraction
Football leagues, Page 98
Students use systematic
working to investigate how
many games there are in total
when t teams play each other
twice.
This chapter focuses on
ordering decimals and
fractions, and order of
operations with integers.
C1.6 is split into two parts. In
Block 3, Chapter 4 students
will learn to: use the four
operations for calculations
with fractions and decimals,
including correct ordering of
operations and use of brackets.
average; frequency; mean;
median; mode; range
Reading age, Page 352
Students compare the reading
ages of two newspaper articles.
Students learn about measures
of spread and types of average.
They learn to calculate
averages for raw, frequency
and grouped data and how to
determine which average is
the most suitable for a given
data set.
C9.3 is also covered in Block6
Chapter 2.
area; capacity; centimetre;
gram; kilogram; kilometre;
length; litre; mass; metre;
millilitre; millimetre; volume
Fountain borders, Page 224
This investigation looks at
the number of tiles needs
to border different sized
fountains. Students need to
work systematically to solve
the problem.
This chapter focuses on units
and conversions.
C1.6 The four operations
Use the four operations for calculations with
integers, fractions and decimals, including
correct ordering of operations and use of
brackets.
C1.14 Using a calculator
1 Use a calculator efficiently.
2 Enter values appropriately on a calculator.
3 Interpret the calculator display
appropriately.
C2.6 Inequalities
Represent and interpret inequalities,
including on a number line.
Chapter 28
Mean, median,
mode and range
3 hours
C9.2 Interpreting statistical data
1 Read, interpret and draw inferences from
tables and statistical diagrams.
2 Compare sets of data using tables, graphs
and statistical measures.
3 Appreciate restrictions on drawing
conclusions from given data.
C9.3 Averages and range
Calculate the mean, median, mode, and range
for individual data and distinguish between
the purposes for which these are used.
Chapter 22
Measures
3 hours
C5.1 Units of measure
Use metric units of mass, length, area, volume
and capacity in practical situations and
convert quantities into larger or smaller units.
Suggested Scheme of Work: Core
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Resources in Cambridge IGCSE Core Mathematics Fifth Edition
Block 3: Total time 14 hours
Approx. time
allocation
Learning objectives
Vocabulary
Chapter 4
Integers,
fractions,
decimals and
percentages
9 hours
C1.4 Fractions, decimals and percentages
1 Use the language and notation of the
following in appropriate contexts:
• proper fractions
• improper fractions
• mixed numbers
• decimals
• percentages.
2 Recognise equivalence and convert
between these forms.
decimal; denominator;
equivalent fraction; fraction;
improper fraction; mixed
number; numerator; order of
operations; percentage; proper
fraction; simplest form
In this chapter C1.6 is
revisited.
In Block 2, Chapter 3, Pages
27–30 students learnt to:
C1.6 Use the four operations
for calculations with integers,
including correct ordering of
operations and use of brackets.
This objective is revisited to
include a greater focus on
non-calculator methods when
working with larger integers
and calculations with fractions.
expand; expression; factorise;
formula; subject; substitute
The rest of C2.5 is covered in
Block 7, Chapter 13:
1 Construct simple
expressions, equations and
formulas.
2 Solve linear equations in
one unknown.
3 Solve simultaneous
linear equations in two
unknowns.
C1.6 The four operations
Use the four operations for calculations with
integers, fractions and decimals, including
correct ordering of operations and use of
brackets.
Chapter 11
Algebraic
representation
and
manipulation
5 hours
C2.1 Introduction to algebra
1 Know that letters can be used to represent
generalised numbers.
2 Substitute numbers into expressions and
formulas.
C2.2 Algebraic manipulation
1 Simplify expressions by collecting like
terms.
2 Expand products of algebraic expressions.
3 Factorise by extracting common factors.
C2.5 Equations
4 Change the subject of simple formulas.
Mathematical investigations
and ICT
Notes
43
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Suggested Scheme of Work: Core
Subject area
Block 4: Total time 14 hours
Subject area
Approx. time
allocation
Learning objectives
Vocabulary
Mathematical investigations
and ICT
Notes
Chapter 18
Geometrical
vocabulary
7 hours
C4.1 Geometrical terms
1 Use and interpret the geometrical terms:
• point
• vertex
• line
• parallel
• perpendicular
• bearing
• right angle
• acute, obtuse and reflex angles
• interior and exterior angles
• similar
• congruent
• scale factor.
2 Use and interpret the vocabulary of:
• triangles
• special quadrilaterals
• polygons
• nets
• simple solids.
3 Use and interpret the vocabulary of a
circle.
acute; bearing; centre;
circle; circumference; cone;
congruent; construction; cube;
cuboid; cylinder; decagon;
diameter; edge; equilateral
triangle; exterior angle; face;
frustum; hemisphere; hexagon;
interior angle; irregular
polygon; isosceles triangle;
kite; line; net; obtuse and
reflex angles; octagon; parallel;
parallelogram; pentagon;
perpendicular; perpendicular
bisector; plane; point; polygon;
prism; pyramid; quadrilateral;
radius (plural radii); rectangle;
regular polygon; rhombus;
right angle; right-angled
triangle; scale factor; scalene
triangle; similar; solid shape;
sphere; square; surface;
trapezium; vertex
Tiled walls, Page 225
Students can investigate the
number of spacers (T shaped
or + shaped) used to separate
the tiles in different tiling
patterns.
This chapter is an introduction
to geometrical vocabulary and
properties of shapes.
index; powers; rules of indices;
standard form
Towers of Hanoi, Page 292
Students investigate the classic
problem of the Towers of
Hanoi.
The rule for the number of
moves to move n discs is 2 n − 1
Core candidates are only
expected to calculate
with standard form on the
calculator paper.
C4.4 Similarity
Calculate lengths of similar shapes.
Chapter 7
Indices and
standard form
7 hours
C1.7 Indices I
1 Understand and use indices (positive, zero
and negative integers).
2 Understand and use the rules of indices.
C1.8 Standard form
1 Use the standard form A × 10 n where n is a
positive or negative integer, and 1 ⩽ A < 10.
2 Convert numbers into and out of standard
form.
3 Calculate with values in standard form.
Suggested Scheme of Work: Core
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Resources in Cambridge IGCSE Core Mathematics Fifth Edition
Block 5: Total time 14 hours
Subject area
Approx. time
allocation
Learning objectives
Vocabulary
Mathematical investigations
and ICT
Notes
Chapter 19
Geometrical
constructions
and scale
drawings
11 hours
C4.2 Geometrical constructions
1 Measure and draw lines and angles.
2 Construct a triangle, given the lengths
of all sides, using a ruler and pair of
compasses only.
3 Draw, use and interpret nets.
construct; net; plan; scale
ICT activity 1, Pages 98–99
Students explore growth tiling
patterns.
Block 4 must be completed
first.
Part of C4.3 is covered in
Block 10, Chapter 24:
2 Use and interpret
three-figure bearings.
index; laws of indices; powers
Chequered boards, Page 153
This is an investigation into
the total number of black
and white squares on an m
by n chequered board. It is a
variation of the problem ‘How
many square are there on a
chess board?’
Block 4 must be completed
first.
C4.3 Scale drawings
1 Draw and interpret scale drawings.
Chapter 12
Algebraic
indices
3 hours
C2.4 Indices II
1 Understand and use indices (positive, zero
and negative).
2 Understand and use the rules of indices.
Suggested Scheme of Work: Core
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Block 6: Total time 14 hours
Subject area
Approx. time
allocation
Learning objectives
Vocabulary
Mathematical investigations
and ICT
Notes
Chapter 29
Collecting,
displaying and
interpreting
data
14 hours
C9.1 Classifying statistical data
Classify and tabulate statistical data.
bar chart; composite bar
chart; correlation; discrete
data; dual bar chart; grouped
frequency table; line of best fit;
pictogram; pie chart; scatter
diagram; stem and leaf; tally
table; two-way table
ICT activity, Pages 352–353
In this activity students use a
spreadsheet and graphing tools
to make a timetable of their
day.
This block focuses on the
collection, display and
interpretation of data, and
continues work on averages
from Chapter 28 (covered in
Block 2).
C9.2 Interpreting statistical data
1 Read, interpret and draw inferences from
tables and statistical diagrams.
2 Compare sets of data using tables, graphs
and statistical measures.
3 Appreciate restrictions on drawing
conclusions from given data.
C9.3 Averages and range
Calculate the mean, median, mode and range
for individual data and distinguish between
the purposes for which these are used.
C9.4 Statistical charts and diagrams
Draw and interpret:
a bar charts
b pie charts
c pictograms
d stem-and-leaf diagrams
e simple frequency distributions.
C9.5 Scatter diagrams
1 Draw and interpret scatter diagrams.
2 Understand what is meant by positive,
negative and zero correlation.
3 Draw by eye, interpret and use a straight
line of best fit.
Suggested Scheme of Work: Core
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Block 7: Total time 14 hours
Subject area
Approx. time
allocation
Learning objectives
Vocabulary
Mathematical investigations
and ICT
Notes
Chapter 13
Equations
8.5 hours
C2.5 Equations
1 Construct simple expressions, equations
and formulas.
2 Solve linear equations in one unknown.
3 Solve simultaneous linear equations in
two unknowns.
elimination; linear equation;
simultaneous equation;
substitution
ICT activity 1, Pages 154–155
Students explore the use a
graphing software to solve
linear simultaneous equations.
Block 6 must be completed
first.
Part of C2.5 is covered in
Block 3, Chapter 11:
4 Change the subject of
simple formulas.
Chapter 6
Ratio and
proportion
5.5 hours
C1.11 Ratio and proportion
Understand and use ratio and proportion to:
• give ratios in their simplest form
• divide a quantity in a given ratio
• use proportional reasoning and ratios in
context.
average speed; compound
measure; density; direct
proportion; inverse proportion;
population density; pressure;
rate; ratio
Modelling: Stretching a spring,
Page 154
In this activity students carry
out a practical experiment to
explore how the extension of
a spring is proportional to the
mass suspended from it.
This chapter involves solving
problems involving direct and
inverse proportion and the use
of compound measures.
C1.12 Rates
1 Use common measures of rate.
2 Apply other measures of rate.
3 Solve problems involving average speed.
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Block 8: Total time 14 hours
Subject area
Approx. time
allocation
Learning objectives
Vocabulary
Mathematical investigations
and ICT
Notes
Chapter 23
Perimeter, area
and volume
11 hours
C5.2 Area and perimeter
Carry out calculations involving the
perimeter and area of a rectangle, triangle,
parallelogram and trapezium.
C5.3 Circles, arcs and sectors
1 Carry out calculations involving the
circumference and area of a circle.
2 Carry out calculations involving arc
length and sector area as fractions of the
circumference and area of a circle, where
the sector angle is a factor of 360°.
C5.4 Surface area and volume
Carry out calculations and solve problems
involving the surface area and volume of a:
• cuboid
• prism
• cylinder
• sphere
• pyramid
• cone.
C5.5 Compound shapes and parts of shapes
1 Carry out calculations and solve problems
involving perimeters and areas of:
• compound shapes
• parts of shapes.
2 Carry out calculations and solve problems
involving surface areas and volumes of:
• compound solids
• parts of solids.
arc; area; circumference;
compound shape; cone;
cuboid; cylinder; diameter;
parallelogram; perimeter;
prism; pyramid; radius,
rectangle; sector; sphere;
surface area; trapezium;
triangle; volume
Metal trays, Page 269
This is an investigation into
a maximum box for the same
surface area.
Answers may need to be given
in terms of π.
Formula for
• area of a triangle
• area of a circle
• circumference of a circle
• curved surface area of a
cylinder
• curved surface area of a
cone
• surface area of a sphere
• volume of a sphere
• volume of a pyramid
• volume of a cone
• volume of a cylinder
• volume of a prism
will be given.
Suggested Scheme of Work: Core
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Resources in Cambridge IGCSE Core Mathematics Fifth Edition
Block 9: Total time 14 hours
Subject area
Approx. time
allocation
Learning objectives
Vocabulary
Mathematical investigations
and ICT
Notes
Chapter 21
Angle
properties
14 hours
C4.6 Angles
1 Calculate unknown angles and give
simple explanations using the following
geometrical properties:
• sum of angles at a point = 360°
• sum of angles at a point on a straight
line = 180°
• vertically opposite angles are equal
• angle sum of a triangle = 180° and
angle sum of a quadrilateral = 360°.
2 Calculate unknown angles and give
geometric explanations for angles formed
within parallel lines:
• corresponding angles are equal
• alternate angles are equal
• co-interior (supplementary) angles
sum to 180°.
3 Know and use angle properties of regular
polygons.
alternate angles; centre;
circumference; corresponding
angles; cyclic quadrilateral;
exterior angle; interior
angle; parallel; polygon;
radius; segment; semicircle;
supplementary; tangent;
vertically opposite angles
ICT activity, Page 225
Students use a spreadsheet to
help them explore interior and
exterior angles in polygons.
It is important that students
learn to give formal reasons
for each step in their working,
for example, use terms such
as alternate angles instead of
Z-angles.
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C4.7 Circle theorems
Calculate unknown angles and give
explanations using the following geometrical
properties of circles:
• angle in a semicircle = 90°
• angle between tangent and radius = 90°.
Block 10: Total time 14 hours
Subject area
Approx. time
allocation
Learning objectives
Vocabulary
Mathematical investigations
and ICT
Notes
4 hours
Chapter 10
Set notation and
Venn diagrams
C1.2 Sets
Understand and use set language, notation
and Venn diagrams to describe sets.
complement; element;
intersection; set; union;
universal set; Venn diagram
Probability drop, Page 325
An investigation into Pascal’s
triangle.
Venn diagrams will be limited
to two sets only.
Chapter 27
Probability
C8.1 Introduction to probability
1 Understand and use the probability scale
from 0 to 1.
2 Calculate the probability of a single event.
3 Understand that the probability of an
event not occurring = 1 – the probability of
the event occurring.
event; expected frequency;
outcome; probability scale;
relative frequency; sample
space diagram; tree diagram;
Venn diagram
ICT activity, Page 327
This is a practical activity
exploring relative frequencies.
Dice sum, Page 326
Students explore the most
likely outcome from rolling
different sized dice.
Probability notation is not
required. Combined events
will be with replacement only.
three-figure bearings
Pythagoras and circles,
Page 291
An activity that investigates
Pythagoras’ theorem using
area of squares, semicircles
and equilateral triangle.
The rest of C4.3 is covered in
Block 5, Chapter 19:
1 Draw and interpret scale
drawings.
7 hours
C8.2 Relative and expected frequencies
1 Understand relative frequency as an
estimate of probability.
2 Calculate expected frequencies.
C8.3 Probability of combined events
Calculate the probability of combined events
using, where appropriate:
• sample space diagrams
• Venn diagrams
• tree diagrams.
Chapter 24
Bearings
3 hours
C4.3 Scale drawings
2 Use and interpret three-figure bearings.
Suggested Scheme of Work: Core
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Resources in Cambridge IGCSE Core Mathematics Fifth Edition
Block 11: Total time 14 hours
Subject area
Approx. time
allocation
Learning objectives
Vocabulary
Chapter 25
Right-angled
triangles
14 hours
C6.1 Pythagoras’ theorem
Know and use Pythagoras’ theorem.
adjacent; cosine hypotenuse;
Pythagorean triples, Page 293
opposite; Pythagoras’ theorem; An internet activity exploring
sine; tangent
Pythagorean triples.
C6.2 Right-angled triangles
1 Know and use the sine, cosine and tangent
ratios for acute angles in calculations
involving sides and angles of a rightangled triangle.
2 Solve problems in two dimensions using
Pythagoras’ theorem and trigonometry.
Mathematical investigations
and ICT
Notes
Block 10 must be completed
first.
Angles will be given in
degrees. Answers should
be written in degrees and
decimals to one decimal place.
Suggested Scheme of Work: Core
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Block 12: Total time 14 hours
Subject area
Approx. time
allocation
Learning objectives
Vocabulary
Chapter 5
Further
percentages
5.5 hours
percentage; percentage
C1.13 Percentages
1 Calculate a given percentage of a quantity. increase / decrease
2 Express one quantity as a percentage of
another.
3 Calculate percentage increase or decrease.
Mathematical investigations
and ICT
Blocks 1, 4 and 10 must be
completed first.
Part of C1.13 is covered in
Chapter 8:
4 Calculate with simple and
compound interest.
C1.4 Fractions, decimals and percentages
1 Use the language and notation of the
following in appropriate contexts:
• proper fractions
• improper fractions
• mixed numbers
• decimals
• percentages.
2 Recognise equivalence and convert
between these forms.
Chapter 8
Money and
finance
8.5 hours
C1.13 Percentages
4 Calculate with simple and compound
interest.
C1.14 Using a calculator
1 Use a calculator efficiently.
2 Enter values appropriately on a calculator.
3 Interpret the calculator display
appropriately.
C1.16 Money
1 Calculate with money.
2 Convert from one currency to another.
compound interest; cost price;
currency conversion; deposit;
discount; earnings; profit
and loss; selling price; simple
interest
Notes
ICT activity 2, Page 99
This activity involves using
a spreadsheet to make a
currency converter.
It is important that students
are confident with the work
from Chapter 5 before moving
onto this chapter.
Suggested Scheme of Work: Core
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Resources in Cambridge IGCSE Core Mathematics Fifth Edition
Block 13: Total time 14 hours
Subject area
Approx. time
allocation
11 hours
Chapter 17
Coordinates
and straight-line
graphs
Learning objectives
Vocabulary
Mathematical investigations
and ICT
Notes
C3.1 Coordinates
Use and interpret Cartesian coordinates in
two dimensions.
axes; coordinates; gradient;
intercept; origin; parallel
Plane trails, Page 175
Students investigate the ways
in which vapour trails from
planes intersect each other.
Block 2 must be completed
first.
12-hour clock; 24-hour clock;
distance; speed; time
Painted cube, Page 307
Students investigate how many
faces of small cubes making up
a larger cube are painted when
the outside of the larger cube
is painted.
Ensure students understand
that say 1.25 hours is not 1
hour 25 minutes.
Students may need to solve
problems involving different
time zones.
C3.2 Drawing linear graphs
Draw straight-line graphs for linear equations.
C3.3 Gradient of linear graphs
Find the gradient of a straight line.
C3.5 Equations of linear graphs
Interpret and obtain the equation of a
straight-line graph in the form y = mx + c.
C3.6 Parallel lines
Find the gradient and equation of a straight
line parallel to a given line.
Chapter 9
Time
3 hours
C1.14 Using a calculator
1 Use a calculator efficiently.
2 Enter values appropriately on a calculator.
3 Interpret the calculator display
appropriately.
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C1.15 Time
1 Calculate with time: seconds (s), minutes
(min), hours (h), days, weeks, months, years,
including the relationship between units.
2 Calculate times in terms of the 24-hour
and 12-hour clock.
3 Read clocks and timetables.
Block 14: Total time 14 hours
Subject area
Approx. time
allocation
Learning objectives
Vocabulary
Mathematical investigations
and ICT
Chapter 20
Symmetry
4 hours
C4.5 Symmetry
Recognise line symmetry and order of
rotational symmetry in two dimensions.
line symmetry; order of
rotational symmetry;
rotational symmetry
Triangle count, Page 308
Students investigate the
number of triangles formed
when a larger triangle is
divided according to two
different rules.
C7.1 Transformations
Recognise, describe and draw the following
transformations:
1 reflection of a shape in a vertical or
horizontal line.
2 rotation of a shape about the origin,
vertices or midpoints of edges of the
shape, through multiples of 90°.
3 enlargement of a shape from a centre by a
scale factor.
x
4 translation of a shape by a vector
.
y
anticlockwise; clockwise;
enlargement; reflection;
rotation; scale factor;
transformation; translation;
vector; vertex
ICT activity, Page 309
Students use a geometry
package to explore
enlargements.
Chapter 26
10 hours
Transformations
()
Notes
Students need to know that
horizontal lines are in the form
y = a and vertical lines are in
the form x = b.
Suggested Scheme of Work: Core
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Resources in Cambridge IGCSE Core Mathematics Fifth Edition
Block 15: Total time 14 hours
Subject area
Approx. time
allocation
Learning objectives
Vocabulary
Mathematical investigations
and ICT
Notes
Chapter 15
Graphs in
practical
situations
5 hours
C2.9 Graphs in practical situations
1 Use and interpret graphs in practical
situations including travel graphs and
conversion graphs.
2 Draw graphs from given data.
conversion graph; distance;
distance–time graph; gradient;
speed; time; travel graph
Chapter 16
Graphs of
functions
6 hours
C2.10 Graphs of functions
1 Construct tables of values, and draw,
recognise and interpret graphs for
functions of the forms:
• ax + b
• ± x 2 + ax + b
• a (x ≠ 0)
x
where a and b are integer constants.
2 Solve associated equations graphically,
including finding and interpreting roots by
graphical methods.
equations; intersection;
linear function; reciprocal
function; root; simultaneous
linear function; simultaneous
quadratic function; symmetry
ICT activity 2, Page 155
Students use graphing
software to find the solutions
to quadratic equations
and equations involving
reciprocals.
Block 13 must be completed
first.
cubic sequence; linear
sequence; nth term; quadratic
sequence; sequence; square
numbers; term-to-term rule
House of cards, Page 153
Students can explore the
sequences produced from
building houses of cards.
Students should use
differences to help them find
rules for the nth term.
Blocks 8 and 11 must be
completed first.
C2.11 Sketching curves
Recognise, sketch and interpret graphs of the
following functions:
a linear
b quadratic.
3 hours
C2.7 Sequences
1 Continue a given number sequence or
pattern.
2 Recognise patterns in sequences,
including the term-to-term rule, and
relationships between different sequences.
3 Find and use the nth term of sequences:
a linear
b simple quadratic
c simple cubic.
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Suggested Scheme of Work: Core
Chapter 14
Sequences
This Scheme of Work has been devised to follow a logical route through the textbook for students following the Extended syllabus and using the Core
and Extended textbook. Its aim is for students to complete the course by the end of the second term in the second year of study; this will then allow
time for revision and preparation for examinations. The chapters have been divided into fourteen blocks each with 15 hours of teaching time; this
roughly equates to four weeks’ work, depending upon individual timetables. The timings are generous to allow for some flexibility in this area.
If necessary, the blocks can be interchanged to allow for local conditions, preferences, etc. Where prior knowledge is required before starting a block, this
is listed in the ‘Notes’ column in the Scheme of Work; please read this carefully to ensure necessary learning has taken place before attempting the work.
Similarly, the order in which each chapter is completed can be rearranged within each block if resources or timetabling dictates but, once again, some
care needs to be taken to ensure the necessary prior learning has taken place.
Learning objectives included in the schemes of work below are reproduced from the Cambridge IGCSETM and IGCSE (9–1) Mathematics syllabuses
(0580/0980) for examination from 2025. This Cambridge International copyright material is reproduced under licence and remains the intellectual
property of Cambridge Assessment International Education.
Resources in Cambridge IGCSE Core and Extended Mathematics Fifth Edition
Block 1: Total time 15 hours
Subject area
Approx. time Learning objectives
allocation
Chapter 1
Number and
language
4 hours
E1.1 Types of number
Identify and use:
• natural numbers
• integers (positive, zero and negative)
• prime numbers
• square numbers
• cube numbers
• common factors
• common multiples
• rational and irrational numbers
• reciprocals.
E1.3 Powers and roots
Calculate with the following:
• squares
• square roots
• cubes
• cube roots
• other powers and roots of numbers.
Vocabulary
Mathematical
investigations and ICT
Notes
cube number; cube root;
factor; highest common
factor; integer; irrational
number; lowest common
multiple; multiple; natural
number; negative number;
positive number; power;
prime factor; prime
number; rational number;
reciprocal; square
number; square root
Mystic Rose, Pages
102–104
This fully worked example
takes students through
the process of carrying
out a mathematical
investigation and the
value of systematic
working. Students should
work through the problem
and then compare their
methods with the worked
solution.
Primes and squares,
Page 104
This is an investigation
into which prime numbers
can be written as the sum
of two squares.
This chapter covers the different types of
number and vocabulary that students need
to be familiar with.
In Exercise 1.6 (Page 8), students need to
recall some work from Lower Secondary
including Pythagoras’ theorem and the
formula for the circumference and area of
a circle.
The chapter covers non-calculator work as
well as giving the students the opportunity
to practise using their calculator to find
powers and roots.
Suggested Scheme of Work: Extended
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Subject area
Approx. time Learning objectives
allocation
Chapter 2
Accuracy
5 hours
E1.9 Estimation
1 Round values to a specified degree of
accuracy.
2 Make estimates for calculations
involving numbers, quantities and
measurements.
3 Round answers to a reasonable
degree of accuracy in the context of a
given problem.
Vocabulary
Mathematical
investigations and ICT
This chapter involves rounding to powers of
10, decimal places and significant figures.
It also includes using an appropriate degree
of accuracy and estimation. It is important
for students to use estimation as a means of
checking their calculations.
In exercise 2.4 on Pages 16–17, they need to
find area and volume of simple compound
2D and 3D shapes.
The chapter ends with finding upper and
lower bounds and then calculating with upper
and lower bounds.
In exercise 2.7 on Pages 21–22, they need to
use the formula for area of a circle and the
formulas for density and speed.
Remind students to round any inexact
answers to 3 s.f. Also when working with
angles, give inexact angles correct to 1 d.p. see Block 11 Chapter 25.
accuracy; decimal place;
estimate; lower bound;
rounding; significant
figure; upper bound
E1.10 Limits of accuracy
1 Give upper and lower bounds for data
rounded to a specified accuracy.
2 Find upper and lower bounds of the
results of calculations which have
used data rounded to a specified
accuracy.
Chapter 3
Calculations and
order
4 hours
E1.5 Ordering
Order quantities by magnitude and
demonstrate familiarity with the symbols
=, ≠, >, <, ⩾, ⩽.
E2.6 Inequalities
1 Represent and interpret inequalities,
including on a number line.
Football leagues, Page 104
Students use systematic
working to investigate
how many games there
are in total when t teams
play each other twice.
57
This chapter focuses on ordering decimals
and fractions, and order of operations with
integers.
E1.6 is split into two parts. In Block 2,
Chapter 4 students will learn:
E1.6 Use the four operations for
calculations with INTEGERS, fractions
and decimals, including correct ordering of
operations and use of brackets.
E2.6 is split into three parts.
In Block 5, Chapter 13 Pages 148–150,
students will learn:
E2.6.2 Construct, solve and interpret linear
inequalities.
In Block 5, Chapter 14, Pages 153–156
students will learn
E2.6.3 Represent and interpret linear
inequalities in two variables graphically.
E2.6.4 List inequalities that define a given
region.
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Suggested Scheme of Work: Extended
E1.6 The four operations
Use the four operations for calculations
with integers, fractions and decimals,
including correct ordering of operations
and use of brackets.
addition; division; indices;
inequality; multiplication;
order of operations;
subtraction
Notes
Approx. time Learning objectives
allocation
Chapter 35
Mean, median,
mode and range
3 hours
E9.2 Interpreting statistical data
1 Read, interpret and draw inferences
from tables and statistical diagrams.
2 Compare sets of data using tables,
graphs and statistical measures.
3 Appreciate restrictions on drawing
conclusions from given data.
E9.3 Averages and measures of spread
1 Calculate the mean, median, mode,
quartiles, range and interquartile
range for individual data and
distinguish between the purposes for
which these are used.
2 Calculate an estimate of the mean
for grouped discrete or grouped
continuous data.
3 Identify the modal class from a
grouped frequency distribution.
Vocabulary
Mathematical
investigations and ICT
Notes
average; discrete data;
frequency; grouped
frequency table; mean;
median; modal class;
mode; range
Reading ages, Pages
545–546
In this investigation
students find out
how reading ages
are determined and
investigate the reading
age of newspaper articles.
Students learn about measures of spread
and types of average. They learn to
calculate averages for raw, frequency and
grouped data and how to determine which
average is the most suitable for a given
data set.
Suggested Scheme of Work: Extended
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Resources in Cambridge IGCSE Core and Extended Mathematics Fifth Edition
Block 2: Total time 15 hours
Subject area
Approx. time Learning objectives
allocation
Chapter 4
Integers,
fractions,
decimals and
percentages
6 hours
E1.4 Fractions, decimals and
percentages
1 Use the language and notation of the
following in appropriate contexts:
• proper fractions
• improper fractions
• mixed numbers
• decimals
• percentages.
2 Recognise equivalence and convert
between these forms.
Vocabulary
Mathematical
investigations and ICT
Notes
decimal; denominator;
equivalent fraction;
fraction; improper
fraction; mixed number;
numerator; order of
operations; percentage;
proper fraction; recurring
decimal; simplest form
Hidden treasure, Pages
286–287
Students explore an
algorithm to work out
which contestant in a
game show will win the
hidden treasures.
In this chapter E.1.6 is revisited.
In Block 1, Chapter 3, Pages 25–30 students
about ordering decimals and fractions, and
order of operations with integers.
This objective is revisited to include a
greater focus on non-calculator methods
when working with larger integers and
calculations with fractions.
E1.6 The four operations
Use the four operations for calculations
with integers, fractions and decimals,
including correct ordering of operations
and use of brackets.
Chapter 11
Algebraic
representation
and manipulation
9 hours
E2.2 Algebraic manipulation
1 Simplify expressions by collecting
like terms.
2 Expand products of algebraic
expressions.
3 Factorise by extracting common
factors.
E2.3 Algebraic fractions
1 Manipulate algebraic fractions.
2 Factorise and simplify rational
expressions.
59
E2.5 Equations
7 Change the subject of formulas.
algebraic fraction;
bracket; expand;
expression; factorise;
formula; quadratic
expression; subject
Chapter 11 is split between Block 2 and
Block 5. In this first section, there is a
focus on expanding brackets, simple
factorisation, substitution into formulas and
changing the subject of a simple formula.
The rest of E2.2 is covered in Block 5
4 Factorise expressions of the form:
• ax + bx + kay + kby
• a 2 x 2 − b 2y 2
• a 2 + 2ab + b2
• ax 2 + bx + c
• ax 3 + bx 2 + cx
5 Complete the square for expressions in
the form ax 2 + bx + c.
The rest of E2.5 on constructing and
solving equations is covered in Block 5,
Chapter 13.
Changing the subject of more complicated
formulas is covered later in Block 5,
Chapter 11.
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Suggested Scheme of Work: Extended
E2.1 Introduction to algebra
1 Know that letters can be used to
represent generalised numbers.
2 Substitute numbers into expressions
and formulas.
Block 3: Total time 15 hours
Subject area
Approx. time Learning objectives
allocation
Vocabulary
Mathematical
investigations and ICT
Notes
Chapter 26
Measures
6 hours
E5.1 Units of measure
Use metric units of mass, length,
area, volume and capacity in practical
situations and convert quantities into
larger or smaller units.
area; capacity; centimetre;
gram; kilogram;
kilometre; length; litre;
mass; metre; millilitre;
millimetre; volume
Metal trays, Page 388
This is an investigation
into a maximum box for
the same surface area.
This chapter focuses on units and
conversions.
Chapter 22
Geometrical
vocabulary and
construction
5 hours
E4.1 Geometrical terms
1 Use and interpret the geometrical
terms:
• point
• vertex
• line
• plane
• parallel
• perpendicular
• perpendicular bisector
• bearing
• right angle
• acute, obtuse and reflex angles
• interior and exterior angles
• similar
• congruent
• scale factor.
2 Use and interpret the vocabulary of:
• triangles
• special quadrilaterals
• polygons
• nets
• solids.
3 Use and interpret the vocabulary of a
circle.
acute; bearing; centre;
circle; circumference;
cone; congruent;
construction; cube;
cuboid; cylinder;
decagon; diameter; edge;
equilateral triangle;
exterior angle; face;
frustum; hemisphere;
hexagon; interior angle;
irregular polygon;
isosceles triangle; kite;
line; net; obtuse and
reflex angles; octagon;
parallel; parallelogram;
pentagon; perpendicular;
perpendicular bisector;
plane; point; polygon;
prism; pyramid;
quadrilateral; radius
(plural radii); rectangle;
regular polygon; rhombus;
right angle; rightangled triangle; scale
factor; scalene triangle;
similar; solid shape;
sphere; square; surface;
trapezium; vertex
Fountain borders, Page
345
This investigation looks at
the number of tiles needs
to border different sized
fountains. Students need
to work systematically to
solve the problem.
This chapter is an introduction to
geometrical vocabulary and properties
of shapes. It also covers constructions of
triangles and scale drawings.
E4.3 is split between Block 3, Chapter 22
and Block 9, Chapter 28 with the following
covered in Chapter 28:
E4.3 Scale drawings
2 Use and interpret three-figure bearings.
E4.2 Geometrical constructions
1 Measure and draw lines and angles.
2 Construct a triangle, given the
lengths of all sides, using a ruler and
pair of compasses only.
3 Draw, use and interpret nets.
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E4.3 Scale drawings
1 Draw and interpret scale drawings.
Suggested Scheme of Work: Extended
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Subject area
Approx. time Learning objectives
allocation
Chapter 16
Proportion
4 hours
E2.8 Proportion
Express direct and inverse proportion
in algebraic terms and use this form of
expression to find unknown quantities.
Vocabulary
Mathematical
investigations and ICT
Notes
direct proportion; inverse
proportion; proportion;
variation
Modelling: Stretching a
spring, Page 255
This is practical
investigation exploring
how the extension of a
spring varies with the
mass suspended from it.
This chapter focuses on different types of
proportion.
Suggested Scheme of Work: Extended
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Block 4: Total time 15 hours
Subject area
Approx. time Learning objectives
allocation
Chapter 36
Collecting,
displaying and
interpreting data
8 hours
E9.1 Classifying statistical data
Classify and tabulate statistical data.
E9.2 Interpreting statistical data
1 Read, interpret and draw inferences
from tables and statistical diagrams.
2 Compare sets of data using tables,
graphs and statistical measures.
3 Appreciate restrictions on drawing
conclusions from given data.
E9.4 Statistical charts and diagrams
Draw and interpret:
a bar charts
b pie charts
c pictograms
d stem-and-leaf diagrams
e simple frequency distributions
E9.5 Scatter diagrams
1 Draw and interpret scatter diagrams.
2 Understand what is meant by positive,
negative and zero correlation.
3 Draw by eye, interpret and use a
straight line of best fit.
E9.7 Histograms
1 Draw and interpret histograms.
2 Calculate with frequency density.
Vocabulary
Mathematical
investigations and ICT
Notes
bar chart; class width;
composite bar chart;
correlation; dual bar
chart; frequency density;
grouped frequency table;
histogram; line of best
fit; pictogram; pie chart;
scatter diagram; stem and
leaf; tally table; two-way
table
Heights and percentiles,
Pages 544–545
Students interpret an
unfamiliar percentile
chart showing heights
versus ages. The activity
introduces the idea of
percentiles which they
will meet again in Block
11, Chapter 37.
This chapter focuses on the collection,
display and interpretation of data.
Suggested Scheme of Work: Extended
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Subject area
Approx. time Learning objectives
allocation
Chapter 7
Indices, standard
form and surds
7 hours
E1.7 Indices I
1 Understand and use indices (positive,
zero, negative, and fractional).
2 Understand and use the rules of
indices.
E1.8 Standard form
1 Use the standard form A × 10 n where
n is a positive or negative integer, and
1 ⩽ A < 10.
2 Convert numbers into and out of
standard form.
3 Calculate with values in standard
form.
Vocabulary
Mathematical
investigations and ICT
Notes
equation; exponential;
index; laws of indices;
powers; standard form;
surds
Towers of Hanoi,
Pages 440–441
Students investigate the
classic problem of the
Towers of Hanoi.
The rule for the number
of moves to move n discs
is 2 n − 1.
Students learn the laws of indices, standard
form and surds. Students following the
extension syllabus need to carry out
calculations involving standard form
without the use of a calculator.
E1.18 Surds
1 Understand and use surds, including
simplifying expressions.
2 Rationalise the denominator.
E2.4 Indices II
1 Understand and use indices (positive,
zero, negative and fractional).
2 Understand and use the rules of
indices.
Suggested Scheme of Work: Extended
63
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Block 5: Total time 15 hours
Subject area
Approx. time Learning objectives
allocation
Vocabulary
Mathematical
investigations and ICT
Notes
Chapter 12
Algebraic indices
3 hours
E2.4 Indices II
1 Understand and use indices (positive,
zero, negative and fractional).
2 Understand and use the rules of
indices.
index; indices
Chequered boards,
Page 254
This is an investigation
into the total number of
black and white squares
on an m by n chequered
board. It is a variation of
the problem ‘How many
square are there on a
chess board?’
Block 4 must be completed first.
Chapter 11
Algebraic
representation
and manipulation
4 hours
E2.2 Algebraic manipulation
4 Factorise expressions of the form:
• ax + bx + kay + kby
• a 2 x 2 − b 2y 2
• a 2 + 2ab + b2
• ax 2 + bx + c
• ax 3 + bx 2 + cx
algebraic fraction;
bracket; expand;
expression; factorise;
formula; quadratic
expression; subject
E2.3 Algebraic fractions
1 Manipulate algebraic fractions.
2 Factorise and simplify rational
expressions.
E2.5 Equations
7 Change the subject of formulas.
Chapter 11 is also covered in Block 2 and
the material in Block 2 must be covered
first.
In Block 5, there is a focus on more
complicated algebraic manipulation:
factorising quadratic equations, algebraic
fractions and changing the subject of more
complicated formulas.
The below part of E2.2 is covered in Block 2:
1 Simplify expressions by collecting like
terms.
2 Expand products of algebraic
expressions.
3 Factorise by extracting common factors.
The rest of E2.2 is covered in Block 5,
Chapter 13:
5 Complete the square for expressions in
the form ax 2 + bx + c.
The rest of E2.5 on constructing up and
solving equations is also covered in Chapter
13 which is the next chapter in this block.
Suggested Scheme of Work: Extended
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Approx. time Learning objectives
allocation
Chapter 13
Equations and
inequalities
5 hours
E2.2 Algebraic manipulation
1 Simplify expressions by collecting
like terms.
2 Expand products of algebraic
expressions.
3 Factorise by extracting common
factors.
4 Factorise expressions of the form:
• ax + bx + kay + kby
• a 2 x 2 − b 2y 2
• a 2 + 2ab + b2
• ax 2 + bx + c
• ax 3 + bx 2 + cx.
5 Complete the square for expressions
in the form ax 2 + bx + c.
Vocabulary
Mathematical
investigations and ICT
Notes
This chapter focuses on setting up and
solving a variety of equations: linear,
simultaneous and quadratic. It also covers
solving linear inequalities.
The rest of E2.5 is covered in Chapter 11
7 Change the subject of formulas.
E2.6 is split into three parts.
In Block 1, Chapter 3, students will learn:
1 Represent and interpret inequalities,
including on a number line.
In Block 5, Chapter 14, students will learn to:
3 Represent linear inequalities in two
variables graphically.
4 List inequalities that define a given
region.
completing the square;
elimination; inequality;
linear equation; quadratic
equation; quadratic
formula; simultaneous
equation; substitution
E2.6 Inequalities
2 Construct, solve and interpret linear
inequalities.
Chapter 14
Graphing
inequalities and
regions
3 hours
E2.6 Inequalities
3 Represent linear inequalities in two
variables graphically.
4 List inequalities that define a given
region.
inequality; region
ICT activity, Page 287
Students can use a
graphing package to
explore inequalities and
regions.
65
E2.6 is split into three parts.
In Block 1, Chapter 3, students will learn:
1 Represent and interpret inequalities,
including on a number line.
In Block 5, Chapter 13, students will learn:
2 Construct, solve and interpret linear
inequalities.
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Suggested Scheme of Work: Extended
E2.5 Equations
1 Construct expressions, equations and
formulas.
2 Solve linear equations in one unknown.
3 Solve fractional equations with
numerical and linear algebraic
denominators.
4 Solve simultaneous linear equations
in two unknowns.
5 Solve simultaneous equations, involving
one linear and one non-linear.
6 Solve quadratic equations by
factorisation, completing the square
and by use of the quadratic formula.
Block 6: Total time 15 hours
Subject area
Approx. time Learning objectives
allocation
Chapter 27
Perimeter, area
and volume
15 hours
E5.2 Area and perimeter
Carry out calculations involving the
perimeter and area of a rectangle,
triangle, parallelogram and trapezium.
E5.3 Circles, arcs and sectors
1 Carry out calculations involving the
circumference and area of a circle.
2 Carry out calculations involving arc
length and sector area as fractions
of the circumference and area of a
circle.
E5.4 Surface area and volume
Carry out calculations and solve
problems involving the surface area and
volume of a:
• cuboid
• prism
• cylinder
• sphere
• pyramid
• cone.
E5.5 Compound shapes and parts of
shapes
1 Carry out calculations and solve
problems involving perimeters and
areas of:
• compound shapes
• parts of shapes.
2 Carry out calculations and solve
problems involving surface areas and
volumes of:
• compound solids
• parts of solids.
Vocabulary
Mathematical
investigations and ICT
Notes
arc; area; circumference;
compound shape;
cone; cuboid; cylinder;
diameter; frustrum;
parallelogram; perimeter;
prism; pyramid; radius;
rectangle; sector; sphere;
surface area; trapezium;
triangle; volume
Tennis balls, Pages
388–389
This is an investigation
into a packing problem
involving 12 tennis balls.
ICT activity, Page 389
This is an ICT
investigation in which
students find the
maximum volume cone
made from a sector with a
fixed radius.
Answers may need to be given in terms of π.
Formulas for
• area of a triangle
• area of a circle
• circumference of a circle
• curved surface area of a cylinder
• curved surface area of a cone
• surface area of a sphere
• volume of a sphere
• volume of a pyramid
• volume of a cone
• volume of a cylinder
• volume of a prism
will be given.
Suggested Scheme of Work: Extended
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Block 7: Total time 15 hours
Subject area
Approx. time Learning objectives
allocation
Chapter 25
Angle properties
11 hours
E4.6 Angles
1 Calculate unknown angles and
give simple explanations using the
following geometrical properties:
• sum of angles at a point = 360°
• sum of angles at a point on a
straight line = 180°
• vertically opposite angles are
equal
• angle sum of a triangle = 180° and
angle sum of a quadrilateral = 360°.
2 Calculate unknown angles and give
geometric explanations for angles
formed within parallel lines:
• corresponding angles are equal
• alternate angles are equal
• co-interior (supplementary)
angles sum to 180°.
3 Know and use angle properties of
regular polygons.
Mathematical
investigations and ICT
Notes
alternate angles;
centre; circumference;
corresponding angles;
cyclic quadrilateral;
exterior angle; interior
angle; parallel; polygon;
radius; segment; semicircle; supplementary;
tangent; vertically
opposite angles
ICT activity 2, Page 347
Students use a geometry
package to demonstrate
the circle theorems:
• angle subtended at
the centre of a circle
by an arc is twice the
size of the angle on
the circumference
subtended by the same
arc
• angles in the same
segment of a circle are
equal
• exterior angle of a
cyclic quadrilateral is
equal to the interior
opposite angle.
Block 6 must be completed first.
Angles properties include basic properties
of angles round a point and along a line,
angles on parallel lines, angles in polygons
and circle theorems.
Further circle theorems are covered in
Block 10, Chapter 24 Pages 318–320.
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Suggested Scheme of Work: Extended
E4.7 Circle theorems I
Calculate unknown angles and give
explanations using the following
geometrical properties of circles:
• angle in a semicircle = 90°
• angle between tangent and radius = 90°
• angle at the centre is twice the angle
at the circumference
• angles in the same segment are equal
• opposite angles of a cyclic
quadrilateral sum to 180°
(supplementary)
• alternate segment theorem.
Vocabulary
Approx. time Learning objectives
allocation
Chapter 6
Ratio and
proportion
4 hours
E1.11 Ratio and proportion
Understand and use ratio and
proportion to:
• give ratios in their simplest form
• divide a quantity in a given ratio
• use proportional reasoning and ratios
in context.
E1.12 Rates
1 Use common measures of rate.
2 Apply other measures of rate.
3 Solve problems involving average
speed.
Vocabulary
Mathematical
investigations and ICT
Notes
average speed; compound
measure; density; direct
proportion; inverse
proportion; population
density; pressure; rate;
ratio
ICT activity 2, Page 105
Students use a graphing
package to investigate
velocities at different
points of a 100 m sprint.
This chapter involves solving problems
involving direct and inverse proportion and
the use of compound measures.
Suggested Scheme of Work: Extended
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Block 8: Total time 15 hours
Subject area
Approx. time Learning objectives
allocation
Chapter 23
Similarity and
congruence
4 hours
Chapter 33
Probability
5 hours
Vocabulary
Notes
congruent; scale factor;
E4.4 Similarity
similar
1 Calculate lengths of similar shapes.
2 Use the relationships between lengths
and areas of similar shapes and
lengths, surface areas and volumes of
similar solids.
3 Solve problems and give simple
explanations involving similarity.
ICT activity 1, Pages
346–347
Students use a geometry
package to investigate the
ratio of corresponding
sides in similar triangles.
It is important that students have a sound
grasp of similarity before they tackle
trigonometry in Block 9.
E8.1 Introduction to probability
1 Understand and use the probability
scale from 0 to 1.
2 Understand and use probability
notation.
3 Calculate the probability of a single
event.
4 Understand that the probability
of an event not occurring = 1 – the
probability of the event occurring.
event; expected
frequency; outcome;
probability scale; relative
frequency; Venn diagram
Students need to study Chapter 33 before
ICT activity: Buffon’s
they study probability further in Block 14,
needle experiment, Page
Chapter 34.
503
Buffon’s needle is a classic
probability experiment
used to produce an
estimate for π.
cubic; exponential
sequence; linear sequence;
nth term; quadratic; termto-term rule
House of cards, Page 254
Students can explore the
sequences produced from
building houses of cards.
E8.2 Relative and expected frequencies
1 Understand relative frequency as an
estimate of probability.
2 Calculate expected frequencies.
Chapter 15
Sequences
6 hours
E2.7 Sequences
1 Continue a given number sequence or
pattern.
2 Recognise patterns in sequences,
including the term-to-term rule,
and relationships between different
sequences.
3 Find and use the nth term of
sequences.
Includes subscript notation and linear,
quadratic, cubic and exponential sequences
and simple combinations of these.
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Mathematical
investigations and ICT
Block 9: Total time 15 hours
Subject area
Approx. time Learning objectives
allocation
Chapter 28
Bearings
2 hours
E4.3 Scale drawings
2 Use and interpret three-figure
bearings.
three-figure bearings
Chapter 29
Trigonometry
8 hours
E6.1 Pythagoras’ theorem
Know and use Pythagoras’ theorem.
adjacent; cosine;
depression; elevation;
hypotenuse; opposite;
Pythagoras’ theorem; sine;
tangent
E6.2 Right-angled triangles
1 Know and use the sine, cosine and
tangent ratios for acute angles in
calculations involving sides and
angles of a right-angled triangle.
2 Solve problems in two dimensions
using Pythagoras’ theorem and
trigonometry.
4 Carry out calculations involving
angles of elevation and depression.
E6.3 Exact trigonometric values
Know the exact values of:
1 sin x and cos x for x = 0°, 30°, 45°, 60°
and 90°
2 tan x for x = 0°, 30°, 45°, 60°.
E6.4 Trigonometric functions
1 Recognise, sketch and interpret the
graphs for 0° ⩽ x ⩽ 360°:
• y = sin x
• y = cos x
• y = tan x.
2 Solve trigonometric equations
involving sin x, cos x or tan x, for
0° ⩽ x ⩽ 360°.
Vocabulary
Mathematical
investigations and ICT
Notes
Block 2 must be completed first.
E4.3 is split into two parts and scale
drawings are covered in Block 3, Chapter 22:
1 Draw and interpret scale drawings.
ICT activity, Page 441
Students can explore the
use of a graphing package
to solve trigonometric
equations.
Angles will be given in degrees. Answers
should be written in degrees and decimals
to one decimal place.
Part of E6.2 is covered in Block 13,
Chapter 30:
3 Know that the perpendicular distance
from a point to a line is the shortest
distance to the line.
Suggested Scheme of Work: Extended
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allocation
Chapter 5
Further
percentages
5 hours
E1.13 Percentages
1 Calculate a given percentage of a
quantity.
2 Express one quantity as a percentage
of another.
3 Calculate percentage increase or
decrease.
5 Calculate using reverse percentages.
Vocabulary
percentage; percentage
increase / decrease;
reverse percentage
Mathematical
investigations and ICT
Notes
Part of E1.13 is covered in Block 12,
Chapter 8:
4 Calculate with simple and compound
interest.
Suggested Scheme of Work: Extended
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Block 10: Total time 15 hours
Subject area
Approx. time Learning objectives
allocation
Chapter 21
Straight-line
graphs
12 hours
E3.1 Coordinates
Use and interpret Cartesian coordinates
in two dimensions.
E3.2 Drawing linear graphs
Draw straight-line graphs for linear
equations.
E3.3 Gradient of linear graphs
1 Find the gradient of a straight line.
2 Calculate the gradient of a straight
line from the coordinates of two
points on it.
E3.4 Length and midpoint
1 Calculate the length of a line
segment.
2 Find the coordinates of the midpoint
of a line segment.
E3.5 Equations of linear graphs
Interpret and obtain the equation of a
straight-line graph.
E3.6 Parallel lines
Find the gradient and equation of a
straight line parallel to a given line.
E3.7 Perpendicular lines
Find the gradient and equation of a
straight line perpendicular to a given line.
Vocabulary
Mathematical
investigations and ICT
Notes
axes; bisector;
coordinates; gradient;
intercept; midpoint;
origin; parallel;
perpendicular; segment
Plane trails, Pages
285–286
Students investigate the
number of crossing points
between the vapour trails
from p planes.
This chapter includes finding the equation
of the perpendicular bisector.
Suggested Scheme of Work: Extended
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allocation
Chapter 24
Symmetry
3 hours
E4.5 Symmetry
1 Recognise line symmetry and order
of rotational symmetry in two
dimensions.
2 Recognise symmetry properties
of prisms, cylinders, pyramids and
cones.
E4.8 Circle theorems II
Use the following symmetry properties
of circles:
• equal chords are equidistant from the
centre
• the perpendicular bisector of a chord
passes through the centre
• tangents from an external point are
equal in length.
Vocabulary
Mathematical
investigations and ICT
Notes
bisector; centre;
tangent; chord; cone;
cylinder; equidistant;
line symmetry; order of
rotational symmetry;
perpendicular; prism;
pyramid; rotational
symmetry
Students should study Chapter 24 before
Tiled walls Page 346
they study Block 11, Chapter 32.
Students can investigate
the number of spacers
(T shaped or + shaped)
used to separate the tiles
in different tiling patterns.
Suggested Scheme of Work: Extended
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Block 11: Total time 15 hours
Subject area
Approx. time Learning objectives
allocation
Vocabulary
Mathematical
investigations and ICT
Notes
Chapter 32
Transformations
7 hours
E7.1 Transformations
Recognise, describe and draw the
following transformations:
1 reflection of a shape in a straight line.
2 rotation of a shape about a given
centre through multiples of 90°.
3 enlargement of a shape from a centre
by a scale factor.
x
4 translation of a shape by a vector
.
y
enlargement; reflection;
rotation; scale factor;
translation
Triangle count, Pages
476–477
Students investigate
the number of triangles
formed when a larger
triangle is divided
according to two different
rules.
Blocks 1 and 4 must be completed first.
ICT activity, Page 546
Students gather height
data from students in
their class and draw a
cumulative frequency
diagram of the results.
The material on box - and - whisker plots
(Pages 539–540) is extension material only.
()
Chapter 37
Cumulative
frequency
4 hours
E9.6 Cumulative frequency diagrams
1 Draw and interpret cumulative
frequency tables and diagrams.
2 Estimate and interpret the
median, percentiles, quartiles and
interquartile range from cumulative
frequency diagrams.
cumulative frequency;
cumulative frequency
diagram; interquartile
range; median;
percentiles; quartiles
Chapter 20
Functions
4 hours
E2.13 Functions
1 Understand functions, domain and
range and use function notation.
2 Understand and find inverse
functions f−1(x).
3 Form composite functions as defined
by gf(x) = g(f(x)).
composite; domain;
function; inverse;
mapping; range
Students are introduced to function using
mappings and mapping diagrams.
Suggested Scheme of Work: Extended
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Block 12: Total time 15 hours
Subject area
Approx. time Learning objectives
allocation
Chapter 8
Money and
finance
4 hours
E1.13 Percentages
1 Calculate a given percentage of a
quantity.
2 Express one quantity as a percentage
of another.
3 Calculate percentage increase or
decrease.
4 Calculate with simple and compound
interest.
5 Calculate using reverse percentages.
Vocabulary
Mathematical
investigations and ICT
Notes
compound interest;
cost price; currency
conversion; deposit;
depreciation; discount;
earnings; exponential
decay; exponential
growth; profit and loss;
selling price; simple
interest
ICT activity 1, Page 105
In this activity students
investigate how the share
price of their chosen
company changes over
time.
Blocks 2 and 9 must be completed first.
12-hour clock; 24-hour
clock; distance; speed;
time
A painted cube, Page 475
Students investigate how
many faces of small cubes
making up a larger cube
are painted when the
outside of the larger cube
is painted.
Ensure students understand that say 1.25
hours is not 1 hour 25 minutes.
Students may need to solve problems
involving different time zones.
E1.14 Using a calculator
1 Use a calculator efficiently.
2 Enter values appropriately on a
calculator.
3 Interpret the calculator display
appropriately.
E1.16 Money
1 Calculate with money.
2 Convert from one currency to
another.
Chapter 9
Time
1.5 hours
E1.14 Using a calculator
1 Use a calculator efficiently.
2 Enter values appropriately on a
calculator.
3 Interpret the calculator display
appropriately.
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E1.15 Time
1 Calculate with time: seconds (s),
minutes (min), hours (h), days,
weeks, months, years, including the
relationship between units.
2 Calculate times in terms of the 24hour and 12-hour clock.
3 Read clocks and timetables.
Suggested Scheme of Work: Extended
E1.17 Exponential growth and decay
Use exponential growth and decay.
Approx. time Learning objectives
allocation
Chapter 31
Vectors
3.5 hours
E7.2 Vectors in two dimensions
1 Describe a translation using a vector
x represented by
, AB or a.
y
2 Add and subtract vectors.
3 Multiply a vector by a scalar.
()
Vocabulary
Mathematical
investigations and ICT
Notes
collinear; magnitude;
parallel; position vector;
scalar; vector
ICT activity, Page 477
Students use a geometry
package to explore the
addition, subtraction, and
multiplication of vectors.
Modulus bars |a| are used to show the
magnitude of a vector.
complement; element;
empty set; intersection;
set; subset; union;
universal set; Venn
diagram
Numbered balls, Page 440
Students investigate
number sequences using
the rule:
If the last term was even,
divide by 2 to find the
next term
If the last term was odd,
add 1 to find the next
term.
Venn diagrams will be limited to two or
three sets only.
E7.3 Magnitude of a vector
x
Calculate the magnitude of a vector
y
as x 2 + y 2.
()
E7.4 Vector geometry
1 Represent vectors by directed line
segments.
2 Use position vectors.
3 Use the sum and difference of two or
more vectors to express given vectors
in terms of two coplanar vectors.
4 Use vectors to reason and to solve
geometric problems.
Chapter 10
Set notation and
Venn diagrams
6 hours
E1.2 Sets
Understand and use set language,
notation and Venn diagrams to describe
sets and represent relationships between
sets.
Definition of sets
e.g.
A = {x: x is a natural number}
B = {(x, y): y = mx + c}
C = {x: a ⩽ x ⩽ b}
D = {a, b, c, …}
Suggested Scheme of Work: Extended
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Block 13: Total time 15 hours
Approx. time Learning objectives
allocation
Chapter 17
Graphs in
practical
situations
8 hours
E2.9 Graphs in practical situations
1 Use and interpret graphs in practical
situations including travel graphs and
conversion graphs.
2 Draw graphs from given data.
3 Apply the idea of rate of change to
simple kinematics involving distance–
time and speed–time graphs,
acceleration and deceleration.
4 Calculate distance travelled as area
under a speed–time graph.
acceleration; conversion
graph; deceleration;
distance; distance–time
graph; speed; speed–time
graph; time; travel graph
Block 9 must be completed first.
Chapter 30
Further
trigonometry
7 hours
E6.2 Right-angled triangles
3 Know that the perpendicular distance
from a point to a line is the shortest
distance to the line.
adjacent; cosine; cosine
rule; depression;
elevation; hypotenuse;
opposite; perpendicular;
plane; Pythagoras’
theorem; sine; sine rule;
tangent
It is important that students are confident
with earlier wok on Pythagoras’ theorem
and right-angled trigonometry before they
study this chapter.
E6.2 has been split over two chapters:
Block 9, Chapter 29:
1 Know and use the sine, cosine and
tangent ratios for acute angles in
calculations involving sides and angles
of a right-angled triangle.
2 Solve problems in two dimensions using
Pythagoras’ theorem and trigonometry.
4 Carry out calculations involving angles
of elevation and depression.
Bear in mind that inexact answers should
be given to 3 s.f. and angles to 1 d.p.
E6.3 Exact trigonometric values
Know the exact values of:
1 sin x and cos x for x = 0°, 30°, 45°, 60°
and 90°.
2 tan x for x = 0°, 30°, 45°, 60°.
E6.5 Non-right-angled triangles
1 Use the sine and cosine rules in
calculations involving lengths and
angles for any triangle.
2 Use the formula
1
area of triangle = ab sin C.
2
E6.6 Pythagoras’ theorem and
trigonometry in 3D
Carry out calculations and solve
problems in three dimensions using
Pythagoras’ theorem and trigonometry,
including calculating the angle between a
line and a plane.
Vocabulary
Mathematical
investigations and ICT
Notes
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Subject area
Block 14: Total time 15 hours
Subject area
Approx. time Learning objectives
allocation
Chapter 18
Graphs of
functions
6 hours
E2.10 Graphs of functions
1 Construct tables of values, and draw,
recognise and interpret graphs for
functions of the following forms:
• axn (includes sums of no more
than three of these)
• abx + c.
1
1
Where, n = −2, −1, − , 0, , 1, 2, 3; a
2
2
and c are rational numbers; and b is a
positive integer.
2 Solve associated equations
graphically, including finding and
interpreting roots by graphical
methods.
3 Draw and interpret graphs
representing exponential growth and
decay problems.
Vocabulary
Mathematical
investigations and ICT
Notes
asymptote; cubic function;
decay; exponential
function; growth;
intersection; linear
function; quadratic
function; reciprocal
function; simultaneous
equations; symmetry;
turning point
ICT activity, Page 255
Students use a graphics
calculator or graphing
package to explore the
graphs of exponential
functions and use
the graphs to solve
exponential equations.
Blocks 8 and 11 must be completed first.
Students learn to estimate gradients of
curves by drawing tangents.
The rest of E2.12 is covered in Chapter 19:
2 Use the derivatives of functions of the
form axn , where a is a rational constant
and n is a positive integer or zero, and
simple sums of not more than three of
these.
3 Apply differentiation to gradients and
stationary points (turning points).
4 Discriminate between maxima and
minima by any method.
E2.12 Differentiation
1 Estimate gradients of curves by
drawing tangents.
Chapter 19
Differentiation
and the gradient
function
5 hours
E2.12 Differentiation
2 Use the derivatives of functions of
the form axn , where a is a rational
constant and n is a positive integer
or zero, and simple sums of not more
than three of these.
3 Apply differentiation to gradients and
stationary points (turning points).
4 Discriminate between maxima and
minima by any method.
derivative; differentiation;
gradient; maximum
(maxima); minimum
(minima); stationary
point; tangent
It is essential that students have covered the
work from Chapter 18 first.
E2.12 is also covered in Chapter 18:
1 Estimate gradients of curves by drawing
tangents.
Suggested Scheme of Work: Extended
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allocation
Chapter 34
Further
probability
4 hours
E8.3 Probability of combined events
Calculate the probability of combined
events using, where appropriate:
• sample space diagrams
• Venn diagrams
• tree diagrams.
E8.4 Conditional probability
Calculate conditional probability using
Venn diagrams, tree diagrams and tables.
Vocabulary
Mathematical
investigations and ICT
Notes
conditional probability;
event; outcome;
probability; sample space
diagram; tree diagram;
Venn diagram
Probability drop, Page 501 Students study combined events and
conditional probability.
An investigation into
Pascal’s triangle.
Dice sum, Page 502
Students should already
know the possible scores
when two six-sided dice
are rolled together. This
activity extends the idea
to m-and n-sided dice.
Suggested Scheme of Work: Extended
79
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applying knowledge from previous topics to
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