sec 23 syllabus (2023): mathematics - University of Malta · 2020-03-09 · SEC 23 SYLLABUS (2023): MATHEMATICS Page 2 of 196 Introduction This syllabus is based on the curriculum

SEC 23 Syllabus Mathematics

2023

MATSEC Examinations Board

SEC 23 SYLLABUS (2023): MATHEMATICS

Page 1 of 196

Table of Contents

Introduction ............................................................................................................................. 2

List of Subject Foci .................................................................................................................... 5

List of Learning Outcomes .......................................................................................................... 5

Programme Level Descriptors ..................................................................................................... 6

Learning Outcomes and Assessment Criteria ................................................................................. 9

Scheme of Assessment ............................................................................................................ 65

School candidates ................................................................................................................. 65

Private Candidates ................................................................................................................ 67

General notes ......................................................................................................................... 68

Results ............................................................................................................................... 68

School Candidates ............................................................................................................. 68

Private Candidates ............................................................................................................. 68

Table of Formulae ............................................................................................................... 69

Coursework Modes .................................................................................................................. 70

Coursework Mode 1: Investigating .......................................................................................... 71

Marking Criteria – Investigating ........................................................................................... 75

Coursework Mode 2: Modelling ............................................................................................... 77

Marking Criteria – Modelling ................................................................................................ 82

Coursework Mode 3: Solving .................................................................................................. 85

Marking Criteria – Solving ................................................................................................... 93

Coursework Mode 4: Maths Trail ............................................................................................. 95

Coursework Mode 5: Research Project .................................................................................. 110

Marking Criteria – Research Project ................................................................................... 113

Coursework Mode 6: Non-Calculator Test .............................................................................. 115

Specimen Assessments: Controlled Paper MQF 1-2 .................................................................... 123

Specimen Assessments: Marking Scheme for Controlled Paper MQF 1-2 .................................... 132

Specimen Assessments: Controlled Paper MQF 2-3 .................................................................... 137

Specimen Assessments: Marking Scheme for Controlled Paper MQF 2-3 .................................... 146

Specimen Assessments: Controlled Paper MQF 3-3* .................................................................. 150

Specimen Assessments: Marking Scheme for Controlled Paper MQF 3-3* .................................. 160

Specimen Assessments: Private Candidates Controlled Paper MQF 1-2 ......................................... 165

Specimen Assessments: Marking Scheme for Private Candidates Controlled Paper MQF 1-2 ......... 177

Specimen Assessments: Private Candidates Controlled Paper MQF 2-3 ......................................... 180

Specimen Assessments: Marking Scheme for Private Candidates Controlled Paper MQF 2-3 ......... 193


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Introduction

This syllabus is based on the curriculum principles outlined in The National Curriculum Framework for All (NCF) which

was translated into law in 2012 and designed using the Learning Outcomes Framework that identify what students

should know and be able to achieve by the end of their compulsory education.

As a learning outcomes-based syllabus, it addresses the holistic development of all learners and advocates a quality

education for all as part of a coherent strategy for lifelong learning. It ensures that all children can obtain the necessary

skills and attitudes to be future active citizens and to succeed at work and in society irrespective of socio-economic,

cultural, racial, ethnic, religious, gender and sexual status. This syllabus provides equitable opportunities for all

learners to achieve educational outcomes at the end of their schooling which will enable them to participate in lifelong

and adult learning, reduce the high incidence of early school leaving and ensure that all learners attain key twenty-

first century competences.

This programme also embeds learning outcomes related to cross-curricular themes, namely digital literacy; diversity;

entrepreneurship creativity and innovation; sustainable development; learning to learn and cooperative learning and

literacy. In this way students will be fully equipped with the skills, knowledge, attitudes and values needed to further

learning, work, life and citizenship.

What is Mathematics?

Mathematics is a body of knowledge – a product that consists of a series of rich, interconnected conceptual structures

made up of facts, techniques & skills, as well as concepts. At secondary school level these structures are built around

a small set of fundamental notions and their properties: numbers; constants, variables, and functions; two dimensional

and three dimensional shapes, and measures of physical quantities; data, and measures of the likelihood of events. In

the SEC Mathematics syllabus these conceptual structures are respectively known as the ‘Number’, ‘Algebra’, ‘Shape,

Space & Measures’ and ‘Data Handling & Chance’ strands.

Mathematics is also a way of thinking and a mode of enquiry – a creative process which helps us to discover and

understand the world of mathematics and the relationships between the mathematical objects that inhibit this world.

Finally, mathematics is a unique, rich, concise, precise, unambiguous, and universal language whose specialist

vocabulary, symbolism, syntax, and semantics allows us to communicate ideas that cannot easily be communicated

through other means. Moreover, as an internationally understood means of communication that goes beyond

linguistic, cultural, and social barriers it enables us to share our ideas and understanding with others across time and

space.

What does the study of Mathematics entail?

There are three issues that one has to consider when identifying what the study of mathematics involves. The first

issue deals with the nature of mathematics, the second deals with the aims of mathematics education, while the third

deals with the teaching approaches that should be used to help pupils learn mathematics.

The Teaching of Mathematics & the Nature of Mathematics

As seen in the previous section, SEC Mathematics is a body of knowledge that is made up of four interrelated, yet

distinct, strands. A brief outline of the knowledge pupils should acquire in each strand by the time they leave

compulsory schooling is as follows:

Number: Pupils should develop number sense and computational fluency by understanding the way numbers are

represented and the quantities they represent. They should also develop the ability to calculate mentally and on paper,

make estimates and approximations, and check the reasonableness of their results.


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Algebra: Pupils should recognise patterns and relationships in mathematics and in the real world. They should also

develop the ability to use symbols, notation, graphs and diagrams to represent and communicate mathematical

relationships and concepts.

Shapes, Space and Measures: Pupils should acquire knowledge of the geometrical properties of 2-D and 3-D shapes.

They should also develop spatial awareness as well as the ability to recognise the geometrical properties of everyday

objects. Pupils should know and understand systems of measurement.

Data Handling & Chance: Pupils should be able to gather, organise and analyse data. They should also be able to

present data in tables, charts and a variety of graphs. Pupils should be able to use data to estimate the likelihood of

an event occurring.

Given that mathematics is not only a body of knowledge, but also a way of thinking and a mode of enquiry, as well as

a language, the study of mathematics necessarily entails learning the processes, as well as the vocabulary, symbols,

syntax, and semantics of the subject. Thus when helping pupils acquire the programme learning outcomes (see List of

Learning Outcomes below on pg.5) SEC Mathematics teachers should give appropriate attention to all product,

process, and language facets of the subject in all four strands of the subject. Thus, they need to emphasize not only

the facts, skills & techniques, and concepts of the subject but also the processes of mathematics as well as its

vocabulary, symbols, associated rules of writing, and meanings.

The processes of mathematics can be classified under three very broad categories:

(a) Understanding & Doing strategies: simplifying, analysing, observing, pattern spotting, connecting, planning,

organizing, designing, devising, implementing, trialling & improving, calculating, estimating, approximating, reviewing,

checking, interpreting, and, reflecting.

(b) Reasoning & Deducing strategies: generalising, specialising, abstracting, conjecturing, predicting, testing, verifying,

justifying, proving, disproving, and, concluding.

(c) Communicating strategies: defining, verbalising, talking, describing, discussing, questioning, interpreting,

explaining, recording, and, presenting.

The emphasis on product, process, and language aspects of the subject should permeate all aspects of the teaching

and learning including both components of its assessment - controlled assessment & coursework.

The Teaching of Mathematics & the Aims of Mathematics Education

SEC Mathematics teachers should ensure that pupils are acquainted with the two main views of why mathematics

should be taught: purist views that look on mathematics as a subject to learn in its own right, one which focuses on:

the study of patterns, the relationships between mathematical objects, the logical and deductive reasoning required

when doing mathematics, and, the rich, interconnected conceptual structures that make up the subject, and,

utilitarian views that look on mathematics as a subject that is a useful tool that can solve problems in a wide range of

contexts (Chambers & Timlin, 2013).

These views are associated with the following two aspects of mathematics:

(i) Utilitarian Aspect: Mathematics is useful. It equips learners with the necessary knowledge to help them understand

and interact with the world around them. Moreover, it forms the basis of science, technology, architecture,

engineering, commerce, industry and banking. It is also increasingly being used in the medical and biological sciences,

economics and geography. This pervasiveness makes mathematics one of the most important subjects in the school

curriculum.


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(ii) Aesthetic Aspect: Mathematics is a beautiful subject with an evolving body of knowledge that is characterised by

its order, precision, conciseness and logic. Mathematics is also characterised by a search for pattern and relationships

within and between mathematical objects. Indeed, mathematicians seek patterns in numbers, in space and in motion

and such random processes as changes in weather. As such mathematics should offer learners intellectual challenge,

excitement, satisfaction, and wonder.

The Teaching of Mathematics & the Teaching & Learning Approaches

There is no single best way of teaching mathematics. Different approaches – including teaching through exposition,

teaching through discovery, and teaching through investigation – should be used as each one of these approaches

targets different types of learning and different learning outcomes. However, irrespective of which of these

approaches is adopted, it is important that all teaching and learning activities should be student-centred rather than

teacher-centred.

Thus, SEC Mathematics teachers should ensure that pupils:

Appreciate the beauty and elegance of Mathematics;

Recognise that Mathematics has always been an important part of human activity and has contributed greatly

to man’s advancement in a wide variety of areas;

Develop a spirit of curiosity and other personal qualities such as confidence and perseverance that enables

them to explore the world of Mathematics with profit;

Develop an ability to work independently and collaboratively when doing Mathematics;

Develop a positive attitude towards Mathematics and its teaching and learning;

Are provided with ample opportunities to develop a deep understanding of the concepts within each of the

four strands of the subject;

Understand and appreciate the place and purpose of mathematics as a subject in its own right;

Apply mathematical knowledge and understanding to solve a wide selection of ‘pure mathematics’ problems

arising from purely abstract considerations;

Understand and appreciate the place and purpose of mathematics in their own lives and in society;

Apply mathematical knowledge and understanding to solve several real-life ‘applied mathematics’ problems

arising from situations in their own lives as well as in society;

Appreciate the interdependence of the different strands of Mathematics;

Develop the ability to use Mathematics across the curriculum;

Think and communicate mathematically - precisely, logically, and creatively;

Make effective, creative, and efficient use of appropriate technology in Mathematics; and,

Acquire a secure foundation for the further study of Mathematics.


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How is SEC Mathematics related to candidate’s lives, to Malta, and/or to the world?

The world as we know it would be quite different were it not for the contributions that mathematics has made to a

number of fields of knowledge including science, technology, engineering, architecture, commerce, and art amongst

others. Mathematics has helped man to measure; to calculate; to approximate; to estimate; to solve problems; to

develop logical thinking and reasoning; to conjecture; to justify; to verify; to structure and organise; to seek patterns;

to simplify; to generalise; to abstract; to predict outcomes; to assess risks; and, to process and communicate

information. The assessment criteria in SEC Mathematics (see section Learning Outcomes and Assessment Criteria

below on p.9) have been specifically designed to help pupils acquire all of these abilities, mastery of which will enable

them in their efforts to become lifelong learners as well as active participants in the world around them. Moreover,

the Extension assessment criteria have been purposely designed to enable candidates to continue studying

mathematics at the post-secondary level if they so wish. It is strongly recommended that those pupils who wish to

continue studying STEM subjects at higher levels not only learn all of the SEC Mathematics ‘Extension’ assessment

criteria but obtain a grade ‘1’, ‘2’ or ‘3’ in the subject.

List of Subject Foci

Number – The number system

Number – Numerical calculations

Algebra – Fundamentals of algebra

Algebra – Graphs

Shape, Space and Measure – Measurement

Shape, Space and Measure – Lines, angles and shapes

Shape, Space and Measure – Constructions and loci

Shape, Space and Measure – Transformations

Data Handling and Chance – Statistics

Data Handling and Chance – Probability

List of Learning Outcomes

At the end of the programme, I can demonstrate an understanding of:

LO 1. The structure of the number system and the relationship between numbers.

LO 2. A variety of calculation procedures (mental methods, pen and paper methods, and assistive technology

methods) and associated skills.

LO 3. Patterns, sequences, algebraic expressions, formulae, equations and inequalities.

LO 4. Graphical representations of algebraic functions.

LO 5. Forms of measurement dealing with angles, length, area, volume, capacity, mass and time.

LO 6. The properties of lines, segments, parallel lines, perpendicular lines, transversals, angles, 2D and 3D shapes.

LO 7. The construction of angles, lines, triangles, circles, others polygons and loci in 2D.

LO 8. Position and movement of shapes in a plane.

LO 9. Measures of central tendency, measures of dispersion and graphical representation of data.

LO 10. The probabilities associated with certain, uncertain and impossible events.


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Programme Level Descriptors

This syllabus sets out the content and assessment arrangements for the award of Secondary Education Certificate in

Mathematics at MQF Level 1, 2 or 3. Level 3 is the highest level which can be obtained for this qualification.

Table 1 overleaf refers to the qualification levels on the Malta Qualifications Framework (MQF) with minor

modifications to reflect specific Mathematics descriptors. These are generic statements that describe the depth and

complexity of each MQF level of study and outline the knowledge, skills and competences required to achieve an

award at Level 1, 2 or 3 in Mathematics.

Knowledge involves the acquisition of basic, factual and theoretical information. Skills involve the application of the

acquired knowledge and understanding to different contexts. Competences indicate sufficiency of knowledge and

skills that enable someone to act in a wide variety of situations, such as whether one is competent to exercise skills

with or without supervision, autonomy or responsibility.


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MQF Level 1 MQF Level 2 MQF Level 3

Basic general mathematics knowledge

1. Acquires basic general knowledge of

mathematics related to the immediate

environment and expressed through a variety of

simple tools and context as an entry point to

lifelong learning;

2. Knows and understands the steps needed to

complete simple mathematical tasks and

activities in familiar environments;

3. Is aware and understands basic mathematics

tasks and instructions;

4. Understands basic mathematics textbooks.

Basic factual knowledge of the mathematics fields of

work or study.

1. Possess good knowledge of mathematics field of

work or study;

2. Is aware and interprets mathematics related

information and ideas;

3. Understands facts and procedures in the

application of basic mathematics related tasks and

instructions;

4. Selects and uses relevant mathematics knowledge

to accomplish specific actions for self and others.

Knowledge of facts, principles, processes and general

concepts in the mathematics field of work or study.

1. Understands the relevancy of theoretical

knowledge and information related to the

mathematics field of work or study;

2. Assesses, evaluates and interprets facts,

establishing basic principles and concepts in the

mathematics field of work or study;

3. Understands facts and procedures in the

application of more complex mathematics tasks

and instructions;

4. Selects and uses relevant mathematical knowledge

acquired on one’s own initiative to accomplish

specific actions for self and others.

Basic skills required to carry out simple mathematics

related tasks.

1. Has the ability to apply basic mathematics

knowledge and carry out a limited range of

simple tasks;

2. Has basic repetitive communication skills to

complete well defined routine mathematics

related tasks and identifies whether actions have

been accomplished;

3. Follows instructions and be aware of

consequences of basic actions for self and others.

Basic cognitive and practical skills required to use

relevant information to carry out tasks and to solve

routine problems using simple rules and tools.

1. Has the ability to demonstrate a range of skills by

carrying out a range of complex mathematics

related tasks within a specified field of work or

study;

2. Communicates basic mathematics related

information;

3. Ensures tasks are carried out effectively.

A range of cognitive and practical skills required to

accomplish mathematics related tasks and solve

problems by selecting and applying basic methods,

tools, materials and information.

1. Demonstrates a range of developed mathematics

skills to carry out more than one complex task

effectively and in unfamiliar and unpredictable

contexts;

2. Communicates more complex mathematics

information;

3. Solves basic mathematics related problems by

applying basic methods, tools, materials and

information given in a restricted learning

environment.


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MQF Level 1 MQF Level 2 MQF Level 3

Work out or study under direct supervision in a

structured context.

1. Applies basic mathematics related knowledge

and skills to do simple, repetitive and familiar

tasks;

2. Participates in and takes basic responsibility for

the action of simple mathematics tasks;

3. Activities are carried out under guidance and

within simple defined timeframes;

4. Acquires and applies basic mathematics key

competences at this level.

Work or study under supervision with some autonomy.

1. Applies factual mathematical knowledge and

practical skills to do some structured tasks;

2. Ensures one acts pro-actively;

3. Carries out mathematics related activities under

limited supervision and with limited responsibility

in a quality controlled context;

4. Acquires and applies basic mathematics key

competences at this level.

Take responsibility for completion of mathematics

related tasks in work or study and adapt own behaviour

to circumstances in solving mathematics problems.

1. Applies knowledge and skills to do some

mathematics tasks systematically;

2. Adapts own behaviour to circumstances in solving

mathematics related problems by participating

pro-actively in structured learning environments;

3. Uses own initiative with established responsibility

and autonomy, but is supervised in quality

controlled learning environments, normally in a

mathematics environment;

4. Acquires key mathematics competences at this

level as a basis for lifelong learning.


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Learning Outcomes and Assessment Criteria

Subject Focus: Number – The number system

Learning Outcome 1:

(Coursework and

Controlled)

At the end of the programme, I can demonstrate an understanding of the structure of the number system and the relationship between

numbers.

Assessment Criteria (MQF 1) Assessment Criteria (MQF 2) Assessment Criteria (MQF 3)

1.1a Read whole numbers to one billion in figures and words.

1.1b Write whole numbers to one billion in figures and words.

1.1c Order whole numbers to one billion in figures and words.

1.1d Identify whole numbers on a number line.

1.1e Identify simple fractions on a number line.

E.g.𝟏

𝟐,

𝟏

𝟒,

𝟑

𝟓,

𝟓

𝟖,

𝟕

𝟏𝟎.

1.1f Identify mixed numbers on a number line.

1.1g Identify decimals on a number line.

1.1h Recognise the place value of any digit in a whole number up to one billion.


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1.1i Compare whole numbers up to one billion using symbols such as <, > or =.

1.1j Identify odd and even numbers.

1.1k Count forward and backwards in 1s, 2s, 3s, 4s, 5s, 10s and 100s starting from any whole number.

1.1l Count forward and backwards in steps of 25 (to and from any multiple of 25) and 50 (to and from any multiple of 50).

1.1m Recall the first ten multiples of all numbers from 2 to 10.

1.1n List the first five multiples of any whole number up to and including 100.

1.2n Write multiples of numbers using power notation.

E.g. 2 x 2 x 2 x 2 = 24; 3 x 3 x 5 x 5 x 5 x 5 = 32 x 54.

1.1o Identify common multiples of two numbers. 1.2o Identify the common multiples of three numbers.

1.1p Identify the least common multiple (LCM) of two numbers.

1.2p Identify the least common multiple (LCM) of three numbers.

1.1q Identify factors of any two-digit number.

1.2q Identify all factors of any two-digit number.

E.g. factors of 24 are 1,2,3,4,6,8,12,24.

1.2r Identify all the common factors of up to three numbers.

1.3r Identify the highest common factor (HCF) of up to three numbers.

1.1s Work out the square of a number.


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1.1t Generate the first ten square numbers. 1.2t Deduce that squares and square roots are inverses of each other.

1.1u Work out the cube of a number.

1.1v Generate the first five cube numbers. 1.2v Deduce that cubes and cube roots are inverses of each other.

1.1w Define what a prime number is. 1.2w Express any integer as a product of prime factors.

1.1x Identify prime numbers up to 100.

1.3y Find the square root of large numbers using prime factorisation.

1.2z Find the LCM of up to 3 numbers not greater than 20.

E.g. Find the LCM of 4 and 6.

1.3z Find the LCM of up to 3 numbers using prime factorisation.

1.3aa Find the HCF of up to 3 numbers using prime factorisation.

1.1ab Express tenths, hundredths and thousandths using decimal notation.

1.1ac Recognise the place value of tenths, hundredths and thousandths written in decimal form.

1.1ad Associate 0.5 with one half, 0.25 with one quarter and 0.75 with three quarters.


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1.1ae Count forwards and backwards in steps of 0.1, 0.2, 0.25 and 0.5.

1.2af Recognise that particular fractions have specific recurring decimal patterns.

E.g. 1

3= 0.333 … = 0. 3̇

1.1ag Recognise equivalent fractions.

1.1ah Generate equivalent fractions of proper and/or improper fractions.

1.1ai Compare fractions using symbols such as <, > or =.

1.1aj Compare mixed numbers using symbols such as <, > or =.

1.1ak Compare decimals using symbols such as <, > or =.

1.1al Write fractions in order.

1.1am Write mixed numbers in order.

1.1an Write decimals in order.

1.1ao Associate a simple fraction which has a denominator which is a factor of 100 to a decimal fraction.


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1.1ap Associate 25% with a quarter.

1.1aq Associate 50% with one half.

1.1ar Associate 75% with three quarters.

1.1as Recognise the relationship between fractions and decimals.

1.2as Recognise the relationship between fractions and percentages.

1.2at Recognise the relationship between decimals and percentages.

1.1au State one number lying between two given decimal numbers.

1.2au State one fraction lying between two given fractions.

1.1av Associate a directed number to a real life situation such as temperature, floor levels and debt.

1.1aw Represent a directed number on a number line.

1.2ax Convert a number ≥1 in ordinary form to standard form.

1.3ax Convert a number <1 in ordinary form to standard form.

1.2ay Convert a number ≥1 in standard form to ordinary form.

1.3ay Convert a number <1 in standard form to ordinary form.

1.3az Find the reciprocal of a number.


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1.1ba Use appropriate mathematical terminology such as number line, fraction, percentage, etc.

1.2ba Use appropriate mathematical terminology such as square roots, cubes, improper fractions, prime numbers, Least Common Multiple, etc.

1.3ba Use appropriate mathematical terminology such as standard form, reciprocal, inequalities, highest common factor, etc.

1.1bb Use assistive technology (e.g. tablets and computers) and other learning resources (e.g. number frames, Cuisenaire rods, interlocking cubes, base 10 blocks, fraction wall) to learn about numbers and their properties.

1.2bb Use assistive technology (e.g. tablets and computers and calculators) and other learning resources (e.g. Cuisenaire rods, interlocking cubes, base 10 blocks) to learn about numbers and their properties.

1.3bb Use assistive technology (e.g. tablets and computers and calculators) and other learning resources to learn about numbers and their properties.

1.1bc Use appropriate mathematical processes to work on tasks and/or activities that are related to mathematical content at this level and which involve one or more modes of assessment such as solving, investigating, modelling, maths trails, and research projects.



1.1bd Work on tasks and/or activities, including word-problems, that are related to mathematical content at this level.




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Subject Focus: Number – Numerical Calculations

Learning Outcome 2:

(Coursework and

Controlled)

At the end of the programme, I can demonstrate an understanding of a variety of calculation procedures (mental methods, pen and paper

methods, and assistive technology methods) and associated skills.


2.1a Add by using the nearest multiple of 10, 100 or 1000 then adjusting.

E.g. 14 + 8 = 14 + 10 - 2 = 22.

2.1b Subtract by using the nearest multiple of 10, 100 or 1000 then adjusting.

E.g. 17 - 9 = 17 - 10+1 = 8.

2.1c Add numbers that involve four digits or less using column addition and/or a calculator.

2.1d Subtract numbers that involve four digits or less using column subtraction and/or a calculator.

2.1e Work through situations involving addition and/or subtraction.

2.1f Recognise that division is the inverse of multiplication.

E.g. If 7×8 = 56 then 56÷8=7 and vice versa.

2.1g Multiply any integer by 10, 100, or 1000.

2.1h Divide any integer by 10, 100, or 1000.


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2.1i Recognise unit fractions.

2.1j Find unit fractions of shapes, numbers, and/or quantities.

2.1k Associate unit fractions with the unitary method.

E.g. how much is 𝟏

𝟖 of 32? Therefore, how much is

𝟓

𝟖 of

32?

2.1l Recognise the relationship between division and fractions.

E.g. 𝟐

𝟑 means 2 ÷ 3 and vice versa.

2.1m Find remainders after division.

2.1n Express the remainder after division as a fraction.

2.1o Express the remainder after division as a decimal rounded up to two decimal places.

2.1p Work through situations involving addition, subtraction, multiplication, and/or division.

2.1q State, without the use of a calculator, the square root of squares up to 100 and/or the cube root of cubes up to 125.

2.2q Work out calculations involving powers and roots without the use of a calculator.

E.g. √1600, 90², 20³, ³√8000


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2.1r Find the square root of any number using a calculator.

2.2r Find the cube root of any number using a calculator.

2.1s Write tenths and/or hundredths in decimal form and vice versa.

2.1t Add numbers up to three decimal places

2.1u Subtract numbers up to three decimal places.

2.1v Add and/or subtract numbers using column methods using numbers up to three decimal places.

2.1w Derive mentally decimals that total 1 or 10.

E.g. 1 = 0.2 + 0.8; 10 = 1.7 + 0.1 + 8.2

2.1x Use pen and paper methods for either one or more of:

ThHTU × U (E.g. 2176 x 3)

HTU × TU (E.g. 804 x 16)

U.t × U (E.g. 7.9 x 8)

TU.t × U (E.g. 26.7 x 3)

U.th × U (E.g. 7.61 x 5)

2.1y Use pen and paper methods for either one or more of:

TU ÷ U (E.g. 32 ÷ 4)

HTU ÷ U (E.g. 114 ÷ 3)

HTU ÷ TU (E.g. 156 ÷ 12)

U.t ÷ U (E.g. 8.4 ÷ 7)

U.th ÷ U (E.g. 6.48 ÷ 4)


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2.1z Use pen and paper methods for multiplication and/or division by 10 and 100 including decimals.

2.2z Use pen and paper methods for multiplication and/or division of numbers by up to 2 digits.

HTU x U.t (E.g. 175 × 1.4)

TU.t x U.t (E.g. 18.6 × 2.7)

U.t ÷ U.t (E.g. 2.4 ÷ 0.6)

Limited to single digit decimal divisor in the case of division.

2.1aa Derive doubles and/or halves of numbers up to 1 d.p.

E.g. double 7.6 is 15.2; half of 13 is 6.5.

2.2ab Use the four operations (addition, subtraction, multiplication and/or division) with directed numbers.

2.1ac Evaluate expressions involving whole numbers, one or more of the four operations, and brackets.

E.g. find the value of 4 + (2 x 3); find the value of (15 - 3) ÷ (2 + 1).

2.2ac Use the BIDMAS rule with integers.

2.1ad Find fractions of numbers, shapes and/or quantities.

2.1ae Reduce a fraction to its simplest form.

2.1af Change an improper fraction into a mixed number and / or vice versa.

2.2af Change mixed numbers into decimals and /or vice versa.

2.1ag Interpret scales involving whole numbers. 2.2ag Interpret scales involving decimals.


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2.2ah Add two fractions (including mixed numbers) with different denominators using equivalent fractions.

2.2ai Subtract two fractions (including mixed numbers) with different denominators using equivalent fractions.

2.2aj Multiply two fractions (including mixed numbers).

2.2ak Divide two fractions (including mixed numbers).

2.2al Work through situations involving the addition, subtraction, multiplication and / or division of fractions (including mixed numbers).

2.2am Find percentages of quantities.

2.2an Convert percentages to fractions and/or decimals

and/or vice versa.

2.2ao Express a quantity as a percentage of another.

2.2ap Find percentage increase and/or percentage decrease.

E.g. cost and selling price; discounts; profit and loss;

(excluding percentage error).

2.3ap Work through situations involving percentage increase and/or decrease

E.g. cost and selling price; discounts; profit and loss; and

percentage error.


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2.3aq Work out reverse percentage calculations.

E.g. the cost of a T.V. set including VAT at 18% is €472. Calculate the cost excluding VAT.

2.3ar Work through situations involving successive percentage changes

2.2as Work out the simple interest using the simple interest formula.

2.3as Work out the principal, the rate, the time, or the amount using the simple interest formula.

2.3at Work out the compound interest, the appreciation and/or depreciation, working step by step.

2.3au Work out the compound interest, the appreciation and/or depreciation, using the formula.

𝑨 = 𝑷 (𝟏 ±𝒓

𝟏𝟎𝟎)

𝒏

where A – final amount; P – principal initial balance; r – interest rate; n - number of times interest applied per time period.

2.3av Work out the number of repayments needed to repay a loan.

2.1aw Convert euro to cent and/or vice versa. 2.2aw Convert from one currency to another using published exchange rates.

2.3aw Convert from one currency to another using published exchange rates.

Including the use of buying and/or selling rates.

2.1ax Work through situations that involve finding the correct change.


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2.1ay Work through simple situations involving special offers.

E.g. find the total cost of two items when the cheaper item is sold at half price.

2.2ay Work through simple situations involving directed numbers, personal and household finance.

E.g. pocket money accrued, how much it will cost to prepare a meal, which item is the best buy when items come in various sizes e.g. oil in one litre bottles vs oil in two litre bottles.

2.3ay Work through complex situations involving personal and household finance.

E.g. earnings, simple interest, income tax and VAT.

Excluding loans and compound interest.

2.2az Write ratios in their simplest form (including decimal numbers and numbers with different units).

2.2ba Find one quantity of a ratio given the other.

2.2bb Divide a quantity in a given ratio.

2.3bc Work through complex situations involving ratios.

Excluding area and volume.

E.g. A mixture of 900 g contains sugar and flour in the ratio 1:2. How much sugar should be added to change the ratio of the mixture to 2:3?

2.2bd Draw scale drawings.

E.g. 2D plans.

2.3bd Draw scale drawings involving bearings; angles of elevation and depression.

2.2be Interpret scale drawings.

E.g. 2D plans and maps.

2.3be Interpret scale drawings.

E.g. bearings; angles of elevation and depression.

2.2bf Interpret simple proportion using ratio notation.

E.g. what is the value of □ in: 1:3 = □:6?


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2.1bg Work through simple situations that involve direct proportion using the unitary method (including price, distance, time, mass, and capacity).

2.3bg Work through situations that involve direct and / or inverse proportion.

2.2bh Use the rules for multiplying and dividing integer powers of numbers (positive indices)

E.g. 78 x 72 = 710, 65 ÷ 62 = 63, (23)5 = 215.

2.3bh Use the rules for multiplying and dividing integer powers of numbers (positive, negative and zero indices)

E.g. 78 x 7-5 ÷ 7 = 72, 65 ÷ 6-2 = 67,(23)-5 = 2-15, 90 = 1

2.3bi Use the rules for multiplying and dividing fractional powers of numbers (positive, negative indices).

2.2bj Evaluate numbers given in index form (positive indices).

E.g. 34 = 81; 23 x 52 = 200.

2.3bj Evaluate numbers given in index form (positive, negative, zero and fractional indices).

E.g. 70 = 1; 2-1 = ½; 8-2/3 = ¼.

2.3bk Apply the four rules on numbers in standard form.

2.1bl Give a rough approximation of calculations involving two-step situations requiring any of the four operations.

E.g. what is the approximate change received after buying three yoghurts costing 26c each, when paying with a €1 coin? Change ≈ 100 – (25x3) = 25c.

2.2bl Give a rough approximation of calculations requiring any of the four operations.

E.g. 37 x 5.8 ≈ 40 x 6 = 240.

2.3bl Give a rough approximation of the answer of any given calculation.

E.g. √80 ≈ 9; 1.8³ ≈ 8.

2.1bm Round any whole number to the nearest powers of ten.


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2.1bn Round up or down decimal numbers to the nearest whole number, depending on the context.

E.g. how many 50 seater buses are needed to transport 130 students? 130 ÷ 50 = 2.6 rounded up to 3 buses; E.g. how many yoghurts can one buy with €10 if each yoghurt costs €1.15? €10 ÷ €1.15 = 8.695 rounded down to 8 yoghurts.

2.1bo Round a decimal number with two decimal places to the nearest tenth or to the nearest whole number.

2.2bo Round any decimal number up to a given number of decimal places.

2.3bo Round any number to a given number of significant figures.

2.1bp Use appropriate mathematical terminology such as rough estimate, nearest 10, add, subtract, improper fraction, proportion, etc

2.2bp Use appropriate mathematical terminology such as, decimal places, square root, equivalent fractions, etc.

2.3bp Use appropriate mathematical terminology such as simple interest, scale drawing, inverse proportion, significant figures, etc.

2.1bq Use assistive technology (e.g. tablets and computers) and other resources (e.g. Cuisenaire rods, Unifix cubes, base 10 blocks, fraction wall, euro coins and notes) appropriate to this level to calculate and to learn about numerical calculations.

2.2bq Use assistive technology (e.g. tablets, computers and calculators) and other resources (e.g. Cuisenaire rods, Unifix cubes, base 10 blocks) appropriate to this level to calculate and to learn about numerical calculations.

2.3bq Use assistive technology (e.g. tablets, computers and calculators) and other resources appropriate to this level to calculate and to learn about numerical calculations.

2.1br Use appropriate mathematical processes to work on tasks and/or activities that are related to mathematical content at this level and which involve one or more modes of assessment such as solving, investigating, modelling, maths trails, and research projects.



2.1bs Work on tasks and/or activities, including word-problems, that are related to mathematical content at this level.




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Subject Focus: Algebra – Fundamentals of Algebra

Learning Outcome 3:

(Coursework and

Controlled)

At the end of the programme, I can demonstrate an understanding of patterns, sequences, algebraic expressions, formulae, equations and

inequalities.


3.1a Recognise pictorial patterns, linear number sequences and square number sequences.

3.1b Extend pictorial patterns, linear number sequences and square number sequences.

3.1c Write a sequence given the first term and the rule.

3.1d Tabulate the terms corresponding to the first few stages of a pictorial pattern.

3.2d Determine the terms in the next stages of a pictorial pattern.

3.3e Generate the terms of a sequence given the nth term.

3.3f Describe the nth term of a linear sequence using an algebraic expression.

E.g. –3n – 1

3.1g Evaluate linear expressions by substituting pictures with positive numbers.

E.g. Evaluate + 3 x , given that =6 and = 2.

3.2g Evaluate linear expressions by substituting directed numbers.

3.3g Evaluate non-linear expressions by substituting directed numbers.


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3.2h Represent unknowns, variables and constants in algebraic expressions using letters.

E.g. If x represents the number of blue boxes and y represents the number of red boxes, give an expression for the total number of boxes. Given that a blue box costs 2 euro and a red box costs 3 euro, write an expression to represent the total cost of buying x blue boxes and y red boxes.

3.2i Simplify linear algebraic expressions by collecting like terms.

3.3i Simplify non-linear algebraic expressions by collecting like terms.

3.2j Simplify algebraic expressions by multiplying linear terms.

E.g. −4 × 5b; −2a × −3b and p × (−3p).

3.2k Expand a linear algebraic expression by multiplying a number over a bracket

E.g. Expand: −2(a + 3); 3(x + 4) + 2(x − 1); 2(1 − x) – 3(x + 2).

3.3k Expand a non-linear algebraic expression by multiplying a single term over a bracket.

E.g. Expand 2x(3x + 4y) − x(x − y).

3.3l Expand two brackets of the form

(ax ± b)(cx ± d ) and (ax ± b)2

3.3m Factorise expressions by using the common factor method.

3.3n Factorise quadratic expressions involving trinomials and/or difference of two squares.


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3.3o Simplify algebraic fractions with numerical denominators.

3.3p Use the four operations on algebraic fractions with numerical denominators.

3.3q Simplify algebraic fractions with linear algebraic denominators.

E.g. Simplify: 4

2𝑥 + 4;

3.3r Add algebraic fractions with linear and quadratic algebraic denominators.

3.3s Subtract algebraic fractions with linear and quadratic algebraic denominators.

3.3t Use the rules for multiplying and dividing integer

powers (positive, negative and zero indices).

E.g. 2a8 × 3a-5 = 6a3; 6y5 ÷ 3y-2 = 2y7; (2c3)5 = 32c15;

x0 = 1; x-2 = 1

𝑥2.

3.2u Change the subject of a formula that uses one or two operations.

3.3u Change the subject of the formula that includes

squares and square roots.

3.3v Change the subject of the formula when the same subject letter occurs more than once.

3.2w Form an equation involving unknown and whole numbers on both sides.

3.3w Form linear equations involving an unknown and integers or fractions on both sides.


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3.1x Work through an equation where there are

pictures instead of numbers.

E.g. If ⎕ + 6 =16, then ⎕ = 10 E.g. If ⎕ + ⎕ =24, then ⎕ = 12

3.2x Solve an equation involving unknown and whole numbers on both sides using the balancing method or other methods.

3.2y Solve simple linear equations involving brackets.

E.g. Solve for x: 4(x – 1) = 2x + 6.

3.3y Solve linear equations involving fractions with numerical denominators.

3.2z Work through situations leading to the solution of linear equations in one unknown.

E.g. mystery numbers, geometric shapes, etc.

3.3aa Solve two simultaneous linear equations

algebraically.

3.3ab Solve two simultaneous equations, one linear

and one quadratic, algebraically.

3.3ac Solve quadratic equations by factorisation.

3.3ad Solve quadratic equations by using the formula.

3.3ae Solve equations involving algebraic fractions with linear algebraic denominators.

3.3af Solve equations involving algebraic fractions with quadratic algebraic denominators.

3.3ag Find approximate solutions to equations for which there is not a simple method of solution using the trial and improvement method.

E.g. Solve for x: x3 − 2x = 100.


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3.3ah Solve simple equations using indices when x is a power.

E.g. Solve for x: 2x = 8; 9x = 3x+1.

3.3ai Represent a simple linear inequality on a number

line.

E.g. represent x ≤ 9 on a number line.

3.3aj Solve simple linear inequalities in one variable with one inequality sign.

E.g. 2x + 5 > 7.

3.3ak Work through situations that use direct and

inverse variation (y ∝ xn for n = ±1, ±2, ±3, ±𝟏

𝟐, ±

𝟏

𝟑).

3.1al Use appropriate mathematical terminology such as sequence, square numbers, etc.

3.2al Use appropriate mathematical terminology such as term, sequence, evaluate, simplify, expand, subject of the formula, solve the equation, number machine, etc.

3.3al Use appropriate mathematical terminology such as sequence, nth term, simplify, factorise, difference of two squares, index form, subject of the formula, trial and improvement, simultaneous equations, etc.

3.1am Use assistive technology (e.g. tablets and computers and other resources (e.g. equation balance) appropriate to the level to learn about the fundamentals of algebra.



3.1an Use appropriate mathematical processes to work on tasks and/or activities that are related to mathematical content at this level and which involve one or more modes of assessment such as solving, investigating, modelling, maths trails, and research projects.



3.1ao Work on tasks and/or activities, including word-problems, that are related to mathematical content at this level.




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Subject Focus: Algebra – Graphs

Learning Outcome 4:

(Coursework and

Controlled)

At the end of the programme, I can demonstrate an understanding of graphical representations of algebraic functions.


4.1a Read coordinates from a grid in the first quadrant. 4.2a Read coordinates from a grid in all four quadrants.

4.1b Plot points on a grid in the first quadrant given their coordinates.

4.2b Plot points on a grid in all four quadrants given their coordinates.

4.2c Write the coordinates of a set of points for

equations of the form y = ± mx ± c.

4.2d Construct tables of values for linear functions. 4.3d Construct tables of values for quadratic functions.

4.3e Construct tables of values for cubic and reciprocal functions.

4.2f Plot the graph of a linear function from a table of values.

4.3f Plot the graph of a quadratic function from a table of values

4.3g Plot the graph of a cubic function and a reciprocal function from a table of values.


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4.2h Explain what the gradient of a line represents.

4.2i Recognise that parallel lines have equal gradients.

4.2j Recognise that the y-intercept represents the point where a line cuts the y axis.

4.2k Recognise that for the equation y = mx + c the value of m determines the gradient of the graph and the value of c determines the y-intercept.

4.2l Find the gradient of a line from the coordinates of any two points on the line.

4.2m Write the equation of a straight line given the gradient and the y-intercept.

4.3m Write the equation of a straight line given a pair of coordinates or the line graph.

4.2n Verify whether a line passes through a point.

4.2o Use straight line graphs to find the value of one coordinate given the other.

4.2p Interpret straight line graphs arising from real life

situations.

E.g. conversion graphs, distance–time graph, etc.

4.3p Interpret non-linear graphs arising from real life situations.


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4.3q Compare information from two or more straight line graphs concerning real life situations.

4.3r Interpret rates of change in linear graphs.

E.g. speed, distance and time.

4.3s Solve two simultaneous linear equations graphically.

4.3t Solve two simultaneous equations, one linear and one quadratic, graphically.

4.3u Solve two simultaneous equations, one linear and one cubic, graphically.

4.3v Solve two simultaneous equations, one linear and one reciprocal, graphically.

4.3w Identify the maximum value or the minimum value from a quadratic graph.

4.3x Find the value of a coordinate given the other from a quadratic graph.

4.3y Solve quadratic equations graphically.

4.1z Use appropriate mathematical terminology such as coordinates, quadrant.

4.2z Use appropriate mathematical terminology such as gradient, intercept, conversion graph.

4.3z Use appropriate mathematical terminology such as

plot, straight-line graph, speed, etc.


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4.1aa Use assistive technology (e.g. tablets and computers) and other resources appropriate to the level to learn about graphical representations of algebraic functions.



4.1ab Use appropriate mathematical processes to work on tasks and/or activities that are related to mathematical content at this level and which involve one or more modes of assessment such as solving, investigating, modelling, maths trails, and research projects.



4.1ac Work on tasks and/or activities, including word-problems, that are related to mathematical content at this level.




Page 33 of 196

Subject Focus: Shape, Space and Measure – Measurement

Learning Outcome 5: (Coursework and

Controlled)

At the end of the programme, I can demonstrate an understanding of forms of measurement dealing with angles, length, area, volume, capacity, mass and time.


5.1a Label the eight compass points.

5.3b Interpret three figure bearings.

5.3c Interpret diagrams involving three figure bearings.

5.3d Draw diagrams involving three figure bearings.

5.1e Illustrate that a whole turn is the same as 4 right angles while half a whole turn is the same as 2 right angles.

5.1f Define an angle as a measure of turn.

5.1g Illustrate that an angle is a measure of turn.

5.1h Estimate angles up to 1800. 5.2h Estimate angles up to 3600.


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5.1i Sketch angles up to 1800.

5.2i Sketch angles up to 3600.

5.1j Measure angles up to 1800 with a protractor. 5.2j Measure angles up to 3600 with a protractor.

5.1k Draw angles up to 1800 with a protractor. 5.2k Draw angles up to 3600 with a protractor.

5.1l Distinguish between acute, right and obtuse angles.

5.2l Distinguish between right, acute, obtuse, and reflex angles.

5.3m State Pythagoras’ Theorem.

5.3n Apply Pythagoras’ Theorem in 2D shapes to find the length of missing sides.

5.3o Apply Pythagoras’ Theorem in 3D shapes to find the length of missing sides.

5.3p State the Converse of Pythagoras’ Theorem.

5.3r Apply the Converse of Pythagoras’ Theorem to check whether a triangle is right-angled.

5.3s Recognise that a triangle whose sides are in the ratio of 3:4:5 or in the ratio of 5:12:13 is a right-angled triangle.


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5.3t Define the trigonometric ratios (sine, cosine and tangent) as the ratios of sides in a right-angled triangle.

5.3u Find unknown lengths and angles in right-angled triangles using the trigonometric ratios.

5.3v Interpret angles of elevation and depression.

5.3w Use trigonometric ratios in 2D situations involving angles of elevation/depression and/or bearings.

5.3x Use trigonometric ratios in 3D situations involving angles of elevation/depression and/or bearings.

5.3y Solve any triangle using the sine and/or the cosine rule.

5.1z Measure line segments.

5.1aa Define perimeter as the boundary of a 2D shape.

5.1ab Measure the perimeters of regular and/or irregular polygons.

5.1ac Calculate the perimeters of regular and/or irregular polygons.


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5.1ad Define the area as the amount of surface of a 2D shape.

5.1ae Use the metric units of area (square kilometres, square metres, square centimetres, square millimetres) including their abbreviations.

5.1af Recognise that 1 square unit is the area occupied by a square of side length 1 unit.

E.g. 1 square metre (1 m2) is equivalent to the area of a square of side 1 m.

5.2af Recognise that 1 cubic unit is the volume occupied by a cube of side length 1 unit.

E.g. 1 cubic centimetre (1 cm3) is equivalent to the volume of a cube of side 1 cm.

5.1ag Express metric measures of area using decimal

notation.

5.2ah Convert between larger and smaller metric units of mass (kg, g) length (km, m, cm, mm) and capacity (l, ml).

5.3ah Convert between larger and smaller metric units of area and volume.

5.1ai Find the areas of irregular and/or regular shapes by counting unit squares on a grid.

5.1aj Work out the area of squares and/or rectangles using the formula: Length x breadth.

5.2aj Derive the formula to find the area of a square and/or rectangle.

5.1ak Work out the area of a right angled triangle by considering it as half a rectangle.

5.2ak Derive formulae to find the area of a parallelogram (bh), a triangle (bh/2) and/or a trapezium (h/2(a+b)).

5.3ak Derive the formula A = ½ ab Sin C for the area of acute-angled and obtuse-angled triangles.

5.2al Use formulae to find the area of a parallelogram, a triangle and/or a trapezium.

5.3al Find the area of acute-angled and obtuse-angled triangles using A = ½ ab Sin C.

5.1am Work out the area of compound shapes that are made up of squares and rectangles

5.2am Calculate the area of compound shapes that include triangles, parallelograms and/or trapezia.


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5.2an Define the notion of 𝜋 as the ratio of the circumference to the diameter.

5.2ao Calculate the circumference of a circle using the appropriate formulae.

5.2ap Calculate the area of a circle using the appropriate formula.

5.2aq Define the surface area of a solid shape as the total amount of surface of the shape.

5.2ar Calculate the surface area of cubes and cuboids.

5.2as Define the volume of a solid shape as the amount of space that it occupies.

5.2at Calculate the volume of cubes and cuboids using the appropriate formulae.

5.2au Calculate the volume of compound shapes made of cubes and cuboids.

5.3av Calculate the length of an arc of a circle.

5.3aw Calculate the area of a sector and/or segment of a circle.


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5.3ax Calculate the area of compound shapes that include sectors and/or segments of circles.

5.3ay Derive the formulae for the curved surface area and/or total surface area of a cylinder.

5.3az Calculate the curved surface area and/or total surface area of a cylinder and/or prism.

5.3ba Derive the formulae for the volume of a prism and/or a cylinder using the cross-sectional area and height.

5.3bb Calculate the volume of a prism and/or cylinder.

5.3bc Calculate the surface area of: a square based right-pyramid, right circular cone and sphere.

5.3bd Calculate the volume of a: square-based right-pyramid, frustum of a square-based right-pyramid, right circular cone, frustum of a right circular cone, sphere.

5.3be Calculate the surface area and/or volume of compound 3D shapes involving cubes, cuboids, cylinders and/or prisms.

5.3bf Find missing dimensions of 3D shapes.

E.g. A cube has a volume of 1000cm3. Calculate the length of the side of the cube.


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5.3bg Give appropriate upper and lower bounds for measurement data to a specified accuracy.

E.g. measured lengths and masses.

5.3bh Obtain appropriate upper and lower bounds to solutions of simple calculations.

5.1bi Convert between larger and smaller units of time (hours, minutes, seconds).

5.2bj Convert between larger and smaller units of time (year, days, hours, minutes, seconds).

5.1bk Read time to the hour/half hour/quarter hour using terms ‘o’clock’, ‘half past’, ‘quarter past’, and ‘quarter to’.

5.1bl Write time to the hour/half hour/quarter hour using terms ‘o’clock’, ‘half past’, ‘quarter past’, and ‘quarter to’.

5.1bm Read, the 12-hour clock (analogue and digital) to 1 minute also using the terms ‘past’ and ‘to’.

5.1bn Write the 12-hour clock (analogue and digital) to 1 minute also using the terms ‘past’ and ‘to’.

5.1bo Read the 24-hour clock (analogue and digital).

5.1bp Convert 12-hour to 24-hour clock times and vice versa.

5.1bq Interpret a time table, a timeline and/or a calendar.

5.1br Workout the duration of a time interval, the starting time and/or the finishing time.


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5.1bs Estimate time using seconds, minutes and/or hours.

5.2bs Determine time intervals in days, hours and/or minutes.

5.1bt Measure time using seconds, minutes and/or hours.

5.1bu Work through situations involving addition and subtraction of time given in hours and minutes.

5.2bu Work through situations involving addition and subtraction of time given in days, hours, minutes and seconds.

5.3bu Interpret time zones.

5.1bv Use appropriate mathematical terminology such as perimeter, area, squares, rectangles, cm2, compass points, acute angle, obtuse angle, right angle, hours, minutes, seconds, as o'clock, half past, quarter to, timeline, etc.

5.2bv Use appropriate mathematical terminology such as triangle, parallelogram, trapezium, cuboid, volume, acute angle, obtuse angle, reflex angle, protractor, analogue, time interval, calendar, etc.

5.3bv Use appropriate mathematical terminology such as segment, sector, prism, bearings, angle of depression, sine, cosine, tangent etc.

5.1bw Use assistive technology (e.g. tablets, computers and calculators) and other resources (e.g. plastic money, cardboard clocks, 2D and 3D plastic shapes, measuring instruments) appropriate to this level to learn about measures.

5.2bw Use assistive technology (e.g. tablets,

computers, and calculators) and other resources (e.g.

plastic money, cardboard clocks, 2D and 3D plastic

shapes, measuring instruments) appropriate to this

level to learn about measures.

5.3bw Use assistive technology (e.g. tablets,

computers, and calculators) and other resources

appropriate to this level to learn about measures.

5.1bx Use appropriate mathematical processes to work on tasks and/or activities that are related to mathematical content at this level and which involve one or more modes of assessment such as solving, investigating, modelling, maths trails, and research projects.



5.1by Work on tasks and/or activities, including word-problems, that are related to mathematical content at this level.




Page 41 of 196

Subject Focus: Shape, Space and Measures – Lines, angles and shapes.

Learning Outcome 6: (Coursework and

Controlled)

At the end of the programme, I can demonstrate an understanding of the properties of lines, segments, parallel lines, perpendicular lines, transversals, angles, 2D and 3D shapes.


6.1a Identify examples of horizontal and/or vertical lines.

6.2a Explain the difference between a line and a line segment.

6.1b Draw examples of horizontal and/or vertical lines.

6.1c Draw examples of parallel lines.

6.1d Draw examples of perpendicular lines.

6.2e Explain what a transversal is in relation to a set of parallel lines.

6.2f Identify examples of transversals drawn across a set of parallel lines.

6.2g Draw examples of transversals across a set of parallel lines.

6.2h Identify vertically opposite angles within a pair of intersecting lines.

6.2i Identify alternate angles within sets of parallel lines and transversals.


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6.2j Identify corresponding angles within sets of parallel lines and transversals.

6.2k Identify interior angles within sets of parallel lines and transversals.

6.2l Work out the size of missing angles in situations involving vertically opposite angles.

6.2m Work out the size of missing angles in situations involving alternate angles.

6.2n Work out the size of missing angles in situations involving corresponding angles.

6.2o Work out the size of missing angles in situations involving interior angles within sets of parallel lines and transversals.

6.1p State that angles around a point add up to 360°.

6.1q Work out the size of missing angles in diagrams showing angles at a point.

6.1r Deduce that the angles on a straight line add up to 180°.

6.1s Work out the size of missing angles in diagrams showing angles on a straight line.


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6.1t Classify a triangle according to the length of its sides and/or the size of its angles.

6.1u State that the sum of the interior angles of a triangle is 180°.

6.3u Prove that the sum of interior angles of a triangle is 180˚.

6.1v Work out the size of missing interior angles in triangles.

6.2v Use the properties of one or more types of triangles (i.e. equilateral, isosceles, scalene, and/or right-angled triangles).

6.2w Use the result that the exterior angle of a triangle is equal to the sum of the interior angles at the other two vertices.

6.3w Prove that the exterior angle of a triangle is equal to the sum of the interior angles at the other two vertices.

6.3x Explain the concept of congruency.

6.3y Identify congruent shapes.

6.3z Prove that two triangles are congruent using either one or other of the following properties of congruent triangles: SSS; SAS; ASA; RHS.

6.3aa Use the fact that two triangles are congruent to find the lengths of missing sides and/or angles in one or the other.

6.3ab Explain the concept of similarity.


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6.3ac Identify similar shapes.

6.3ad Prove that two triangles are similar using either one or other of the following properties of similar triangles: (i) the angles of one are respectively equal to the angles of the other (AAA); (ii) the ratio of the three sides of the first triangle is equal to the ratio of the corresponding sides of the other; (iii) one angle of one triangle is equal to one angle of the other triangle and the sides about these equal angles are in the same ratio.

6.3ae Find missing elements (length and/or angle) in triangles and other flat shapes using similarity.

6.3af Find missing elements (area and/or volume) in 2D and/or 3D shapes using similarity.

6.1ag Describe the properties of a square and/or rectangle.

6.2ag Describe the properties of a rhombus, parallelogram, trapezium and/or kite.

6.2ah Recognise that any 2D shape with four sides is a quadrilateral.

6.2ai Classify a quadrilateral according to its properties.

Limited to: square, rectangle, rhombus, parallelogram, trapezium and/or kite.

.

6.2aj State that the sum of the interior angles of a quadrilateral is 360°.

6.3aj Prove that the sum of interior angles of a quadrilateral is 360˚.


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6.2ak Work out the size of missing interior angles of a quadrilateral.

6.2al Use the properties of one or more types of quadrilaterals (squares, rectangles, rhombuses, parallelograms, trapeziums, kites, and/or any other quadrilateral).

6.1am Name a polygon using the number of sides (from 3 to 10 sides).

6.1an Classify polygons using the number of sides.

6.1ao Identify between a regular and an irregular polygon.

6.2ao Define a regular polygon as a polygon with all sides equal and all interior angles equal.

6.3ap Describe the symmetrical properties of a regular polygon in terms of reflective and/or rotational symmetry.

6.3aq Derive the sum of the interior angles of any polygon.

6.3ar Derive the sum of the exterior angles of any polygon.

6.3as Find missing interior angles using the sum of the interior angles of a polygon.


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6.3at Find missing exterior angles using the sum of the exterior angles of a polygon.

6.1au Identify one or more of the main components of a circle (i.e. centre, radius, diameter, and circumference).

6.3au Identify one or more of the additional main components of a circle (chord, tangent, arc, sector, and segment).

6.1av Name one or more of the main components of a circle (i.e. centre, radius, diameter, and circumference).

6.3av Name one or more of the additional main components of a circle (chord, tangent, arc, sector, and segment).

6.1aw Draw one or more of the main components of a circle (i.e. centre, radius, diameter, and circumference).

6.3aw Draw one or more of the additional main components of a circle (chord, tangent, arc, sector, and segment).

6.3ax Apply Circle Theorem 1: The angle in a semicircle is a right angle.

6.3ay Apply Circle Theorem 2: The angle which an arc of a circle subtends at the centre is twice that which it subtends at any other point on the remaining part of the circumference.

6.3az Apply Circle Theorem 3: Angles in the same segment of a circle are equal.

6.3ba Apply Circle Theorem 4: The opposite angles of a cyclic quadrilateral are supplementary.

6.3bb Apply Circle Theorem 5: The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

6.3bc Apply Circle Theorem 6: The angle between the radius and the tangent at the point of contact is a right angle.


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6.3bd Apply Circle Theorem 7: Equal chords are equidistant from the centre.

6.3be Apply Circle Theorem 8: Chords which are equidistant from the centre of a circle are equal.

6.3bf Apply Circle Theorem 9: The perpendicular bisector of a chord passes through the centre.

6.3bg Apply Circle Theorem 10: A straight line drawn from the centre of a circle to bisect a chord is at right angles to the chord.

6.3bh Apply Circle Theorem 11: If two tangents are drawn to a circle from a point outside the circle, then:

(a) the tangents are equal in length; (b) the angle between the tangents is bisected by the line joining the point of intersection of the tangents to the centre; and, (c) this line also bisects the angle between the radii drawn to the points of contact.

6.3bi Apply Circle Theorem 12: If a straight line touches a circle, and from the point of contact a chord is drawn, the angle which the chord makes with the tangent is equal to the angle in the alternate segment.

6.1bj Recognise a simple 3D shape (i.e. cylinder, cone, sphere, triangular prism and square-based right pyramid).

6.1bk Name a simple 3D shape (i.e. cylinder, cone, sphere, triangular prism and square-based right pyramid).


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6.1bl Recognise a simple 3D shape (i.e. cube, cuboid, cylinder, cone, sphere, triangular prism and square-based right pyramid) from a 2D drawing.

6.2bm Draw the plan of a simple 3D shape (i.e. cube and/or cuboid).

6.3bm Draw the plan of a simple 3D shape (i.e. cylinder, cone, triangular prism, and square-based right pyramid from a 2D drawing).

6.2bn Draw the front elevation of a simple 3D shape (i.e. cube, cuboid).

6.3bn Draw the front elevation of a simple 3D shape (i.e. cylinder, cone, triangular prism, and square-based right pyramid from a 2D drawing).

6.2bo Draw the side elevation of a simple 3D shape (i.e. cube, cuboid).

6.3bo Draw the side elevation of a simple 3D shape (i.e., cylinder, cone, triangular prism, and square-based right pyramid from a 2D drawing).

6.1bp Identify the faces, edges and/or vertices of simple 3D shape (i.e. cube, cuboid, triangular prism, and pyramid) from a 2D drawing and/or a 3D model.

6.1bq Identify the flat and curved surfaces of the cylinder, the cone and the sphere from a 2D drawing and/or a 3D model.

6.1br Count the faces, edges and/or vertices of simple 3D shape (i.e. cube, cuboid, triangular prism, and pyramid) from a 2D drawing and/or a 3D model.

6.1bs Count the flat and curved surfaces of the cylinder, the cone and the sphere from a 2D drawing and/or a 3D model.

6.1bt Identify a net as being either that of a closed cube, or that of an open cube, or of neither.

6.2bt Identify a net as being either that of a cuboid, triangular prism, square-based right pyramid, or of neither.

6.3bt Draw the net of a cube, a cuboid, a triangular prism, and/or a square-based right pyramid.


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6.1bu Use appropriate mathematical terminology such as scalene triangle, isosceles triangle, equilateral triangle, right-angled triangle, regular polygon, pentagon, hexagon, horizontal, vertical, perpendicular lines, circle, parallel lines, cylinder, cone, triangular prism, vertices, edges, faces, etc.

6.2bu Use appropriate mathematical terminology such as rhombus, trapezium, kite, vertically opposite, alternate angles, corresponding angles, interior angles, transversal line, centre, circumference, radius, diameter, net of a solid shape, square based pyramid, etc.

6.3bu Use appropriate mathematical terminology such as cyclic quadrilateral, supplementary, complementary, tangent of a circle, equidistant, chord, line, line segment, similar, congruent, hypotenuse, Pythagoras’ theorem, side elevation, front elevation, plan of a solid shape, triangular prism, etc.

6.1bv Use assistive technology and other resources (e.g. 2D and 3D plastic shapes) appropriate to this level to learn about lines, angles and shapes.

6.2bv Use assistive technology (and other resources (e.g. 2D and 3D plastic shapes) appropriate to this level to learn about lines, angles and shapes.

6.3bv Use assistive technology and other resources appropriate to this level to learn about lines, angles and shapes.

6.1bw Use appropriate mathematical processes to work on tasks and/or activities that are related to mathematical content at this level and which involve one or more modes of assessment such as solving, investigating, modelling, maths trails, and research projects.



6.1bx Work on tasks and/or activities, including word-

problems, that are related to mathematical content at

this level.



this level.



this level.


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Subject Focus: Shape, Space and Measures – Constructions and loci

Learning Outcome 7:

(Coursework and

Controlled)

At the end of the programme, I can demonstrate an understanding of the construction of angles, lines, triangles, circles, others polygons

and loci in 2D.


7.1a Construct a circle of a given radius using a pencil, a ruler, and a pair of compasses only.

7.2b Construct a 60° angle using a pencil, a straight-edge, and a pair of compasses only.

7.2c Construct a 90° angle using a pencil, a straight-edge, and a pair of compasses only.

7.2d Construct the perpendicular bisector of a line segment using a pencil, a straight-edge, and a pair of compasses only.

7.2e Construct the perpendicular from a point to a line using a pencil, a straight-edge, and a pair of compasses only.

7.2f Construct the perpendicular at a point on a line using a pencil, a straight-edge, and a pair of compasses only.

7.2g Construct the angle bisector of a pair of intersecting lines using a pencil, a straight-edge, and a pair of compasses only.


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7.2h Construct a triangle given the length of one side and two angles using a pencil, a ruler, and a protractor only.

7.2i Construct a triangle given the length of two sides and the included angle using a pencil, a ruler, and a protractor only.

7.2j Construct a triangle given the length of three sides using a pencil, a ruler, and a pair of compasses only.

7.3j Construct a triangle given various conditions using a pencil, a ruler and a pair of compasses only.

7.2k Construct a triangle containing a 60° angle using a ruler and a pair of compasses only.

7.2l Construct a right-angled triangle using a ruler and a pair of compasses only.

7.2m Construct a regular hexagon using a ruler and a pair of compasses only.

7.3m Construct a regular polygon using some of the following tools: a pencil, a ruler, a protractor, and a pair of compasses.

7.3n Explain what a locus of a point is.

7.3o Use a ruler and a pair of compasses only to construct the locus of a point which is at a fixed distance from a given point.


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7.3p Use a ruler and a pair of compasses only to construct the locus of a point which is equidistant from a straight line and/or line segment.

7.3q Use a ruler and a pair of compasses only to construct the locus of a point which is equidistant from two given points.

7.3r Use a ruler and a pair of compasses only to construct the locus of a point which is equidistant from two intersecting straight lines and/or line segments.

7.3s Work through situations involving intersecting loci and regions using appropriate loci constructions.

7.2t Use appropriate mathematical terminology such as construct, straight-edge, compasses, etc.

7.3t Use appropriate mathematical terminology such as angle bisector, locus, etc.

7.1u Use assistive technology (e.g. tablets, computers, mathematical instruments) and other resources appropriate to this level to learn about constructions.



7.1v Use appropriate mathematical processes to work on tasks and/or activities that are related to mathematical content at this level and which involve one or more modes of assessment such as solving, investigating, modelling, maths trails, and research projects.




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Subject Focus: Shape, Space and Measures - Transformations

Learning Outcome 8:

(Coursework and

Controlled)

At the end of the programme, I can demonstrate an understanding of position and movement of shapes in a plane.


8.1a Recognise reflective symmetry in regular polygons.

8.1b Identify lines of symmetry in 2D shapes and pictures.

E.g. flags and dominoes.

8.1c Draw lines of symmetry in 2D shapes and pictures.

E.g. flags and dominoes.

8.3c Identify planes of symmetry in simple 3D shapes.

8.1d Classify triangles using reflective symmetry. 8.2d Classify quadrilaterals (square, rectangle, parallelogram, trapezium, rhombus and kite) using reflective symmetry.

8.1e Complete symmetrical patterns given one and/or two lines of symmetry at right angles.

8.2e Draw reflections in the x-axis, y-axis, y = ±c and/or x = ±c.

8.3e Draw reflections in the line y = ±x.

8.3f Explain informally that reflections preserve length and angle.

8.2g Describe reflections in the x-axis, y-axis, y = ±c and/or x = ±c.

8.3g Describe reflections in the line y = ±x.


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8.1h Recognise clockwise and/or anticlockwise rotations in multiples of 45° based on the 8 compass points.

E.g. if you are facing South what direction will you face if you turn 90° clockwise?

8.2h State the order of rotational symmetry of a regular polygon.

8.2j State the order of rotational symmetry of 2D shapes and/or partly shaded 2D shapes.

E.g.

8.3j Complete the shading of a given 2D shape to obtain a shape of a required order of rotational symmetry.

E.g. Shade the least number of squares in this shape so that it has a rotational symmetry of order 4.

8.2k Draw the rotation of a simple shape about any vertex of the shape and/or about the origin by angles of 90° and/or 180° using transparencies or otherwise.

8.3k Find the centre of rotation by inspection and/or by construction.

Limited to angles of rotation of 90o and 180o.

8.2l Describe the rotation of a simple shape about any vertex and/or about the origin using angles of 90°and/or 180°.

8.3l Explain informally that rotations preserve length and angle.

8.1m Move a point right, left, up and/or down on a grid. 8.2m Draw translations using a column translation vector.

8.2n Describe translations using a column translation vector.

8.3n Explain informally that translations preserve length and angle.


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8.3o Describe enlargements of a simple shape by a scale factor about a centre of enlargement.

8.3p Draw the enlargement of a shape, given the centre of enlargement, using positive scale factors (integral and/or fractional).

8.3q Explain informally that enlargements preserve angle but not length.

8.3r Distinguish between reflection, rotation, translation and enlargement in terms of congruency and/or similarity.

8.3s Transform 2D shapes by a combination of reflections, rotations, translations and enlargements.

8.2t Create tessellating shapes.

8.2u Draw a tessellation.

8.1v Use appropriate mathematical terminology such as lines of symmetry, clockwise, left, right, rotation, etc.

8.2v Use appropriate mathematical terminology such as translation, reflection, rotational symmetry, etc.

8.3v Use appropriate mathematical terminology such as tessellation, scale factor, centre of rotation, enlargement, etc.

8.1w Use assistive technology (e.g. tablets and computers) and other resources (e.g. 2D and 3D plastic shapes) appropriate to this level to learn about transformation geometry.




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8.1x Use appropriate mathematical processes to work on tasks and/or activities that are related to mathematical content at this level and which involve one or more modes of assessment such as solving, investigating, modelling, maths trails, and research projects.



8.1y Work on tasks and/or activities, including word-problems, that are related to mathematical content at this level.




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Subject Focus: Data Handling and Chance - Statistics

Learning Outcome 9:

(Coursework and

Controlled)

At the end of the programme, I can demonstrate an understanding of measures of central tendency, measures of dispersion and graphical

representation of data.


9.1a Collect data. 9.3a Distinguish between discrete and continuous data.

9.1b Classify data according to given criteria.

E.g. Classify numbers according to whether they are odd or even.

9.3b Represent data in the form of an inequality.

E.g. Under 21 players (P) should be at least 16 years old but less than 21: 16 ≤ P < 21

9.1c Sort data.

E.g. Sort a deck of playing cards according to criteria selected by the learner (picture vs non picture cards).

9.1d Read a frequency table. 9.3d Read a cumulative frequency table and/or a cumulative frequency graph.

9.1e Interpret a frequency table. 9.3e Interpret a cumulative frequency table and/or a cumulative frequency graph.

9.3f Compare two cumulative frequency tables and/or cumulative frequency graphs.

9.1g Construct a frequency table using a tally column. 9.2g Construct a frequency table with grouped and/or ungrouped discrete data.

9.3g Construct a frequency table using grouped and/or ungrouped continuous data.

9.3h Construct a cumulative frequency table.


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9.3i Draw a cumulative frequency graph from a cumulative frequency table.

9.1j Read a bar chart and/or a bar-line graph. 9.2j Interpret line graphs and/or bar charts. 9.3j Interpret stacked and/or clustered bar charts.

Limited to a maximum of 3-section stacked/clustered bar charts.

9.kj Construct a bar chart and/or a bar-line graph. 9.2k Construct a bar chart using grouped and/or ungrouped discrete data from a frequency table.

9.3k Construct stacked and/or clustered bar charts.

Limited to a maximum of 3-section stacked/clustered bar charts.

9.1l Read a pictograph where the symbol represents a number of units.

9.1m Interpret a pictograph where the symbol represents a number of units.

9.1n Draw a pictograph where the symbol represents a number of units.

9.2o Interpret pie charts.

9.2p Construct pie charts.

9.3q Interpret a histogram with equal class intervals.

9.3r Construct a histogram with equal class intervals.

9.3s Interpret a histogram with unequal class intervals.


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9.3t Construct a histogram with unequal class intervals.

9.1u Interpret a given Carroll diagram.

9.1v Complete a Carroll diagram. 9.2v Construct a Carroll diagram.

9.2w Work through a situation involving data in tables and/or statistical diagrams.

9.2x Compare data from two tables and/or statistical diagrams.

9.1z Find the mean of a set of ungrouped data. 9.2z Find the mean of a set of ungrouped data from a frequency table.

9.3aa Find the mean of a set of grouped data from a frequency table.

9.1ab Work out the total amount given the mean and the number of items.

9.2ab Differentiate between the mean, median, mode of a set of ungrouped data.

9.2ac Work through situations involving the mean, mode, median and/or range.

9.2ad Decide when best to use each type of average using words like “outlier”.


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9.1ae Find the median of a set of ungrouped data. 9.2ae Find the median of a set of ungrouped data from a frequency table.

9.3af Find the class interval in which the median of a set of data lies from a grouped frequency table.

9.1ag Find the mode of a set of ungrouped data. 9.2ag Find the mode of a set of ungrouped data from a frequency table.

9.3ag Identify the modal class from a grouped frequency table.

9.1ah Find the range of a set of ungrouped data. 9.2ah Find the range of a set of ungrouped data from a frequency table.

9.3ah Find the range of a set of data from a grouped frequency table.

9.3ai Explain what the median, quartiles and/or the interquartile range of a set of data represent.

9.3aj Estimate the median, the quartiles and/or the interquartile range from a cumulative frequency graph.

9.3ak Interpret box-and-whisker plots.

9.3al Draw box-and-whisker plots.

9.1am Use appropriate mathematical terminology such as tally, pictograph, mean, Carroll diagram, etc.

9.2am Use appropriate mathematical terminology such as median, mode, data, bar chart, pie-chart, etc.

9.3am Use appropriate mathematical terminology such as discrete data, continuous data, histogram, stacked bar-chart, clustered bar-chart, etc.

9.1an Use assistive technology (e.g. tablets and computers) and other learning resources to learn about statistics.

9.2an Use assistive technology (e.g. tablets, computers and calculators) and other learning resources to learn about statistics.

9.3an Use assistive technology (e.g. tablets, computers and calculators) and other learning resources to learn about statistics.


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9.1ao Use appropriate mathematical processes to work on tasks and/or activities that are related to mathematical content at this level and which involve one or more modes of assessment such as solving, investigating, modelling, maths trails, and research projects.



9.1ap Work on tasks and/or activities, including word-problems, that are related to mathematical content at this level.




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10.1a Mention events that are certain to happen, and others that are impossible.

10.2b Identify events as certain, very likely, likely, equally likely (even chance), unlikely, very unlikely and/or impossible.

10.2c Estimate the probability of an event by experiment.

10.2d Determine the probability of an event.

E.g. the probability of getting 4 when throwing a die = 1/6; the probability of not getting a 4 when throwing a die = 5/6.

10.3d Determine the probability of an event from a frequency table.

10.2e Distinguish between experimental and theoretical probability.

10.2f Deduce that the probability of a certain event is 1 and the probability of an impossible event is 0.

Subject Focus: Data Handling and Chance - Probability

Learning Outcome 10:

(Coursework and

Controlled)

At the end of the programme, I can demonstrate an understanding of the probabilities associated with certain, uncertain and impossible

events.


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10.2g Mark the probability on a probability scale.

10.2h Identify the set of all possible outcomes of a single event.

E.g. Picking a counter from a bag containing two red counters (R1, R2) and one blue counter (B). The possible outcomes are: picking R1, picking R2 or picking B.

10.2i Identify the set of all possible outcomes of two independent events.

E.g. One bag contains two red counters (R1, R2) and a blue counter (B). Another bag contains a red counter (R3) and a green counter (G). One counter is picked from each bag. The possible outcomes are six: R1R3 ; R1G; R2R3; R2G; BR3; and BG.

10.2j Deduce that the probability of all mutually exclusive outcomes add up to 1.

10.3j Work out the probability of mutually exclusive events.

10.2k Construct a possibility space for two independent events.

10.2l Work out the probability of two independent events using a possibility space.

10.3m Distinguish between dependent and independent events.


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10.3n Construct a probability tree (tree diagram) for up to three independent and/or dependent events.

10.3o Use a probability tree (tree diagram) to work out the probability of up to three independent and/or dependent events.

10.2p Use appropriate mathematical terminology such as event, likely, even chance, certain, outcome, possibility space, etc.

10.3p Use appropriate mathematical terminology such as mutually exclusive, etc.

10.2q Use assistive technology (e.g. tablets, computers and calculators) and other learning resources to learn about probability.

10.3q Use assistive technology (e.g. tablets, computers and calculators) and other learning resources to learn about probability.

10.1r Use appropriate mathematical processes to work on tasks and/or activities that are related to mathematical content at this level and which involve one or more modes of assessment such as solving, investigating, modelling, maths trails, and research projects.



10.1r Work on tasks and/or activities, including word-problems, that are related to mathematical content at this level.




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Scheme of Assessment

School candidates

The assessment consists of:

Coursework: 30% of the total marks; comprising 3 assignments of equal weighting i.e. 10% each; set during the three-

year course programme.

Coursework assignments can be pegged at either of two categories:

A coursework assignment at MQF level categories 1-2 must identify assessment criteria from these two MQF

levels. The ACs are to be weighted within the assignment's scheme of work and marking scheme at a ratio of

40% at Level 1 and 60% at Level 2.

A coursework assignment at MQF level categories 1-2-3 must identify assessment criteria from each of Levels

1, 2, and 3. These ACs are to be weighted within the assignment’s scheme of work and marking scheme at a

ratio of 30% at each of Levels 1 and 2 and 40% at Level 3.

The mark for assignments at level categories 1-2 presented for a qualification at level categories 2-3 is to be

recalculated to 60% of the original mark. The mark stands in all other cases.

Controlled assessment: 70% of the total marks; comprising of a two-hour written exam; set at the end of the

programme and differentiated between three tiers:

a. MQF levels 1 and 2;

b. MQF levels 2 and 3;

c. MQF level 3 and level 3 Extension Assessment Criteria.

Candidates can obtain a level higher than Level 1 if they satisfy the examiners in both coursework and controlled

assessments, irrespective of the total marks obtained.

Part 1: Coursework

The coursework will be based on LO 1 – LO 10. An overview of the suggested coursework assignments is shown in the

table below:

Part 1: Coursework – Category Levels 1-2-3 (30 %)

Assignment 1

(10 %)

Assignment 2

(10 %)

Assignment 3

(10 %)

Assignment 4

(10 %)

Assignment 5

(10 %)

Investigating (8%)

+

Non-Calculator

Test (2%)

Modelling (8%)

+

Non-Calculator

Test (2%)

Solving (8%)

+

Non-Calculator

Test (2%)

Maths Trail (8%)

+

Non-Calculator

Test (2%)

Research Project (8%)

+

Non-Calculator

Test (2%)

Figure 1: Suggested Coursework Assignments for School Candidates

Candidates will be assessed through 3 assignments carried out during this three-year programme – 1

assignment in Year 9, 1 assignment in Year 10 and 1 assignment in Year 11.

Coursework Assignment 1 or 2 is mandatory and one can choose or repeat any of the 5 suggested Assignments

in completing their coursework part.


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All coursework assignments shall be marked out of 100 (20% from the Non-Calculator Test and 80% from the

other mode) according to guidelines and marking criteria available with this syllabus.

School candidates’ assignments, forming part of coursework, are to be available at the candidates’ school for

moderation purposes as indicated by the MATSEC Board.

Part 2: Controlled Assessment II

Part 2: Controlled Assessment II (2 hours) (70 %)

Paper consisting of questions at Level 1-2 OR

Paper consisting of questions at Level 2-3. OR

Paper consisting of questions at Level 3-3*(Extension Assessment Criteria).

Figure 2: Part 2 Controlled assessment for School Candidates

Controlled Assessment II:

o will cover all learning outcomes;

o be marked out of 100 and all questions are compulsory - answers are to be written on the examination

paper provided.

o Controlled Paper 1-2 will consist of 10 – 15 questions:

40% of the marks allotted will be based on Assessment Criteria from MQF 1





o Controlled Paper 3-3* will consist of 9 – 13 questions:


60% of the marks allotted will be based on Extension Assessment Criteria at MQF 3


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Private Candidates

Private candidates shall be assessed by means of two controlled assessments.

Part 1: Controlled Assessment I

The first controlled assessment will focus on the learning outcomes identified for school candidates’ coursework.

Learning outcomes with assessment criteria in the psychomotor domain can be assessed by asking questions in pen-

and-paper format seeking understanding of the activity.

Part 1: Controlled Assessment I (2 hours) (30 %)

Paper consisting of 4 – 8 questions at Level 1-2

OR

Paper consisting of 4 – 8 questions at Level 2-3.

Figure 4: Part 1 Controlled Assessment 1 for Private Candidates

Controlled Assessment I:

o will be marked out of 100 and all questions are compulsory - answers are to be written on the examination

paper provided;

o will include a balanced selection of questions which simulate the work done in the coursework modes, or

part of, outlined in the syllabus.

Part 2: Controlled Assessment II

Part 2: Controlled Assessment II (2 hours) (70 %)

Paper consisting of questions at Level 1-2 OR

Paper consisting of questions at Level 2-3. OR

Paper consisting of questions at Level 3-3*(Extension Assessment Criteria).

Figure 2: Part 2 Controlled assessment for School Candidates

Controlled Assessment II:

o will cover all learning outcomes;

o be marked out of 100 and all questions are compulsory - answers are to be written on the examination

paper provided.







o Controlled Paper 3-3* will consist of 9 – 13 questions:


60% of the marks allotted will be based on Extension Assessment Criteria at MQF 3


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General notes

Candidates who intend to further their study in Mathematics and Science subjects at Intermediate Level

or Advanced Level are STRONGLY advised to sit for Paper 2-3 and attempt questions in Section C.

Questions will be set in English and must be answered in English.

Candidates are expected to abide by the following principles of good mathematical practice:

inclusion of justifications in solutions whenever appropriate; inclusion of all appropriate steps in solutions to problems.

Questions will typically involve numerical calculations, approximations, estimations, data and graphical

interpretations, application of formulae, recall and applications of properties of shapes, recall and

applications of mathematical facts.

To answer these questions, particularly those involving numerical calculations, candidates are advised to

choose and use the more efficient techniques (mental and pencil-and-paper). They are expected to have a

range of strategies to aid mental calculations of unknown facts from facts that can be rapidly recalled.

Candidates are allowed to use mathematical instruments and scientific calculators with statistical

functions. Programmable calculators are not allowed.

Candidates are allowed to use transparencies for drawing transformations.

It is very important that candidates see and make connections between strands.

The overall weighting (±6%) in the controlled papers for each of the four strands is shown below:

Number Algebra

Shape, Space & Measures

Data Handling & Chance

Paper 2-3 20% 35% 30% 15%

Paper 1-2 35% 20% 35% 10%

Results

School Candidates

School Candidates sitting for Controlled Assessment II Level 1-2 may qualify for Grades 6, 7 or 8. The results for candidates who do not obtain at least a Grade 8 shall remain Unclassified (U). School Candidates sitting for Controlled Assessment II Level 2-3 may qualify for Grades 4, 5, 6 or 7. The results for candidates who do not obtain at least a Grade 7 shall remain Unclassified (U). School Candidates sitting for Controlled Assessment II Level 3-3* (Extension Assessment Criteria) may qualify for Grades 1, 2, 3, 4, or 5. The results for candidates who do not obtain at least a Grade 5 shall remain Unclassified (U).

Private Candidates Private Candidates sitting for Controlled Assessment I, Level 1-2, and Controlled Assessment II, Level 1-2, may qualify for Grades 6, 7 or 8. The results for candidates who do not obtain at least a Grade 8 shall remain Unclassified (U). Private Candidates sitting for Controlled Assessment I Level 2-3, and Controlled Assessment II, Level 2-3, may qualify for Grades 4, 5, 6 or 7. The results for candidates who do not obtain at least a Grade 7 shall remain Unclassified (U). Private Candidates sitting for Controlled Assessment I Level 2-3, and Controlled Assessment II, Level 3-3* (Extension Assessment Criteria), may qualify for Grades 1, 2, 3, 4 or 5. The results for candidates who do not obtain at least a Grade 5 shall remain Unclassified (U).


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Table of Formulae

A table of the formulae reproduced below will be provided for the use of the candidates. These formulae will

be provided for Paper 3 – 3* (Extension Assessment Criteria) only.

Area of a Triangle ½ 𝑎𝑏 sin 𝐶

Curved Surface Area of Right Circular Cone πrl

Surface Area of a Sphere 4𝜋 𝑟2

Volume of a Pyramid /Right Circular Cone 1

3𝑏𝑎𝑠𝑒 𝑎𝑟𝑒𝑎 × 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 ℎ𝑒𝑖𝑔ℎ𝑡

Volume of Sphere 4

3𝜋𝑟3

Solutions of ax2 + bx + c = 0 𝑥 = −𝑏 ± √𝑏2−4𝑎𝑐

2𝑎

Sine formula 𝑎

sin 𝐴=

𝑏

sin 𝐵=

𝑐

sin 𝐶

Cosine formula 𝑎2 = 𝑏2 + 𝑐2– 2𝑏𝑐 𝑐𝑜𝑠𝐴

Compound Interest / Appreciation & Depreciation 𝐴 = 𝑃 (1 ±𝑟

100)

𝑛


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Coursework Modes

This section presents sample assessments with respective marking schemes. Teachers may use these

guiding documents to develop an assignment based on one of the modes presented in this syllabus.

Otherwise, teachers may develop their own assignment and select an appropriate assessment tool other

than those listed in the syllabus as long as this assignment is sent to MATSEC for approval before being

given to students. A marking scheme is not a list of model answers, but a guideline of how expected

answers should be marked.

The following is a table of abbreviations, and the respective meanings, which are used in the sample marking schemes.

Abbreviation Meaning

M Method marks are awarded for knowing a correct method of

solution and attempting to apply it.

A Accuracy marks (Type A) given for correct answer only.

B Accuracy marks (Type B) are awarded for specific results or

statements independent of the method used.

f.t. follow through

e. e. o. o. each error or omission


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Coursework Mode 1: Investigating

Investigating

80 marks

internally-assessed

externally-moderated

Defining the Process of Investigating The process of ‘investigating’ involves exploring a given situation in order to understand its basic features; identifying various areas of interest; posing a number of questions aimed at deepening knowledge; answering a number of these questions by applying already learnt content and/or process knowledge; drawing suitable conclusions based on appropriate mathematical reasoning; and, finally, identifying avenues of further exploration. The Aim of Investigating Tasks The aim of investigating tasks is to provide candidates with opportunities that promote the development of independent mathematical thinking through the posing and solving of problems. It allows candidates to apply previously learnt knowledge as well as to make connections between different areas of mathematics. Moreover, the act of investigating promotes creative thinking, encourages systematic ways of working, as well as the development of initiative and flexibility in approach. In-depth study of a mathematical topic also promotes the development of valuable personal characteristics such as persistence and commitment. Types of Investigating Tasks In investigating tasks one has to choose either the goal, or the method, or both Thus, investigating tasks can be one of two types: (a) Given goal Choose method: tasks in which the pupils are only expected to choose a method as the goal is already determined in the statement of the problem Example 1: “Build a cuboid” Example 2: “Draw a compound shape made up of triangles, rectangles, and circles that has two lines of reflective symmetry” (b) Choose goal Choose method: tasks in which the pupils are expected to explore the situation and decide on a goal and method when there is no apparent goal in the statement of the problem (Example 1) or tasks in which pupils are required to search for a goal and method that are implicit in the statement of the problem (Example 2). Example 1: “Investigate the multiples of 9 between 9 to 999” Example 2: “Classify quadrilaterals according to their properties” Guidelines for Setting, Conducting, and Assessing Investigating Tasks The following guidelines on how to set, conduct and assess an investigating task are designed to ensure that investigating tasks are set in a systematic way (see section ‘(i) Steps’ below); candidates know exactly what they have to do when carrying out an investigating task (see section ‘(ii) Framework’ below); and, teachers know how they can carry out reliable and valid assessment (see section ‘Marking Criteria’ below). (i) Steps: The following steps related to the implementation of the investigating coursework in the classroom setting, should be followed:

An investigating coursework should consist of one investigating task which can be of the ‘given goal – choose method’ variety; or of the ‘choose goal – choose method’ variety; or a mixture of both. For summative assessment purposes, candidates are required to hand in an individual report. (for further details see section (ii) below).

Teachers should ensure that pupils have understood what the investigating task entails.

Candidates should draw up a short plan to demonstrate that they understood the task and have started thinking about how to conduct the task. A short paragraph paraphrasing the task and a number of initial exploratory questions should be provided.

The plan presented to the teacher who may provide feedback which must not be prescriptive but indicative. It is important that candidates are left with the freedom to choose what they feel is the most suitable way to work on the tasks.


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Once they receive feedback on their plans candidates may adjust their plan in line with the feedback received. They should then start working according to the timeline indicated by the teacher.

When teachers monitor the work done by the candidates they should ensure that any help offered should limit itself to the asking of questions aimed at making pupils think and reflect on what they are doing. It should not consist of the teachers telling pupils what to do.

Investigating tasks should be assessed according to the rubric given below in Marking Criteria section.

(ii) Framework: The report should be structured according to the following framework:

Section Details

Title This section should include:

The name of the candidate.

The title of the investigating coursework.

The date when the investigating coursework was given.

The date of handing in of the investigating coursework report.

Understanding

In this section the candidates should provide clear evidence that they have understood what the task is all about. Where applicable, they should:

Write a few sentences paraphrasing the task in their own words.

Identify and define the key terms in the statement of the task.

Indicate the goals they intend to achieve.

Planning In this section the candidates are expected to show that they have devised a plan that will lead to a successful solution of the investigating task. Where applicable, they should:

Identify a number of main areas of investigation.

Pose a series of questions for each of the identified main areas of the investigation.

Identify which of these questions they will attempt to answer.

Indicate which required and/or additional data and/or information has been collected and/or researched.

Identify alternative strategies for the purposes of comparing results of calculations and reasonableness of conclusions reached.

Enacting

In this section the candidates are required to demonstrate how they have carried out their plan and how they went about generating a solution. Where applicable, they should:

Present relevant and accurate calculations.

Show evidence that appropriate care and attention has been exercised when performing calculations (e.g. by showing all relevant working; by checking reasonableness of answer according to context; by including an approximation; by re-working calculation in reverse using the inverse operation; by stating level of precision used; etc.)

Indicate predictions made and how they were checked.

Explain how conclusions were reached from these predictions.

Concluding In this section the candidates are required to show that they have come up with an acceptable solution and have examined this solution in order to check that it is reasonable. Where applicable, they should:

Present a solution.

State a number of conclusions.

State to what extent conclusions are reasonable, valid, and complete.

Suggest a number of possible extensions to the investigation.


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Communicating Throughout all the sections, where applicable, the candidates are expected

to:

Present their work in a logical, clear, concise, and appealing fashion.

Use mathematical language that is correct.

Provide clear justifications using appropriate language.

Use appropriate forms of representation (notation, diagram, figure, table, graph, etc.) in each of the various parts of the task.

Include references.


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Investigating

exemplar

Two points on the surface of a solid

Investigate the mathematical connections between any two points

that lie on two different faces of a cuboid:

The report

Student is expected to write a report including the following 5 distinct sections.

1. Title

Title of the investigating coursework

Student’s name and class

Date of the coursework

Date of handing in of the report

2. Understanding

A few sentences paraphrasing the task in your own words.

Identification the key terms in the statement of the task.

Definition of the identified key statements.

Indication of the goals intended to be achieved.

3. Plan

Identification of the main area of the chosen investigation.

A number of questions for the area of the investigation.

Identification, if possible, of at least two strategies for the purpose of

comparing results of any calculations and conclusions reached.

4. Implementation

The relevant and accurate calculations in which:

i. All relevant working is shown.

ii. The reasonableness of the calculations is checked.

iii. The level of precision used is indicated.

An appropriate form of representation is used (e.g. notation, diagrams,

tables, etc.)

5. Conclusion

A solution, if applicable to the chosen area of the investigation.

Any final conclusions that may have been reached.

Any references that may have been used in the task.

Suggestion of a number of possible extensions to the investigation.


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Marking Criteria – Investigating

MARKING CRITERIA – Investigating

Maximum 40 marks*

* The total mark should be doubled and added to the Non-Calculator Test mark.

Understanding (6 marks)

(0 – 1 marks) (2 – 3 marks) (4 – 6 marks)

Paraphrasing of task in own words tends to be poor or non-existent.

Paraphrasing of task in own words tends to be adequate but not comprehensive.

Paraphrasing of task in own words tends to be comprehensive.

Only a few or none of the key terms in the statement of the task are identified.

More than half but not nearly all of the key terms in the statement of the task are identified.

All or nearly all of the key terms in the statement of the task are identified.

Only a few or none of the key terms in the statement of the task are defined. Definitions tend to be sketchy, unclear, and/or imprecise.

More than half but not nearly all of the key terms in the statement of the task are defined. Definitions tend to be acceptable but may lack sufficient detail, clarity, and/or precision.

All or nearly all of the key terms in the statement of the task are defined. Definitions tend to be quite detailed, clear, and precise.

Uses a form of representation that is inappropriate shedding little, or no, significant information about the task. Most, if not all, of any introduced notation is unsatisfactory.

Uses a form of representation that while being appropriate was inaccurate and/or inefficient shedding only some information about the task. More than half, but not nearly all, of the introduced notation is satisfactory.

Uses a form of representation that sheds all the information about the problem helping to clarify the meaning of the task. All, or nearly all, of the introduced notation is.

Poorly stated goal showing inadequate understanding of the task. There is little or no indication of what the expected outcome will be.

Adequately stated goal showing some understanding of the task. Although there is an indication of an expected output it is not that clear what this output will be.

Well-stated goal showing full understanding of the task. The expected output is clearly indicated.

There is clear evidence that the candidate has not understood what the task is all about.

There is evidence that the candidate has understood some, but not all, aspects of the task.

There is considerable evidence that the candidate has understood, most, if not all, aspects of the task.

Planning (10 marks)


An insignificant number of main areas of investigation are identified.

A small but significant number of main areas of investigation are identified.

A large number of main areas of investigation are identified.

A series of questions is asked for only a few of the identified main areas of investigation. Each series tends to be poor.

A series of questions is asked for some but not nearly all of the identified main areas of investigation. Each series tends to be adequate but not extensive.

A series of questions is asked for all or nearly all of the identified main areas of investigation. Each series tends to be extensive.

Which questions are going to be answered is stated in a poor fashion. No, or unclear, reasons are given for the choices made.

Which questions are going to be answered is stated in a partially adequate fashion. A small number of incompletely clear and/or ambiguous reasons are given for the choices made.

Which questions are going to be answered is stated in a more than adequate fashion. A considerable number of clear and unambiguous reasons are given for the choices made.

There is no evidence that required data or information has been appropriately collected and/or researched.

There is some evidence that required data or information has been appropriately collected and/or researched.

There is considerable evidence that required data or information has been appropriately collected and/or researched.

No alternative strategies for the purpose of comparing results are identified.

A few alternative strategies for the purpose of comparing results are identified.

A significant number of alternative strategies for the purpose of comparing results are identified.


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Enacting (8 marks)


Appropriate methods to represent the information are evident in only a small part of the presented work.

Appropriate methods to represent the information are evident in most, but not nearly all, of the presented work.

Appropriate methods to represent the information are evident in all, or nearly all, of the presented work.

Relevant and accurate calculations are performed throughout only a minor part of the task.

Relevant and accurate calculations are performed throughout most, but not nearly all, of the task.

Relevant and accurate calculations are performed throughout all, or nearly all, of the task.

No, or an insignificant, number of predictions are made and checked.

A small, but inadequate, number of predictions are made and checked.

An adequate number of predictions are made and checked.

Sound conclusions are made for only a few of the stated predictions.

Sound conclusions are made for most, but not nearly all, of the stated predictions.

Sound conclusions are made for all, or nearly all, of the stated predictions.

Concluding (8 marks)


There is no proposed solution or is incomplete.

The proposed solution is complete but either is difficult to understand and/or not consistent with the statement of the original task.

The proposed solution is complete, easy to understand, and consistent with the statement of the original task.

The conclusions reached are clearly stated in just a few of the cases. There is little or no evidence of what has been learnt from these conclusions.

The conclusions reached are clearly stated in most but not nearly all cases. There is some evidence of what has been learnt from these conclusions.

The conclusions reached are clearly stated in all or nearly all cases. There is clear evidence of what has been learnt from these conclusions.

No statement explaining to what extent conclusions are reasonable, valid, and complete.

Statement explaining to what extent conclusions are reasonable, valid, and complete but this tends to be unclear and/or ambiguous.

Clear and unambiguous statement explaining to what extent conclusions are reasonable, valid, and complete.

There is little or no evidence that enough care and attention has been exercised when calculations have been made.

There is some evidence that care and attention has been exercised when calculations have been made.

There is clear evidence that sufficient care and attention has been exercised when calculations have been made.

There are just a few suggested extensions to the investigation. Only a negligible number of these suggestions tend to show mathematical insight as well as critical and creative thinking.

There are a small but significant number of suggested extensions to the investigation, some of which tend to show mathematical insight as well as critical and creative thinking.

There are a large number of suggested extensions to the investigation. The vast majority of these suggestions tend to show mathematical insight as well as critical and creative thinking.

Communicating (8 marks)


Only some of the work is presented in a logical, clear, concise, and appealing fashion.

Most but not nearly all of the work is presented in a logical, clear, concise, and appealing fashion.

All or nearly all of the work is presented in a logical, clear, concise, and appealing fashion.

Use of mathematical language is correct in only a minor part, or none, of the work.

Use of mathematical language is correct in more than half, but not nearly all, of the work.

Use of mathematical language is correct in all, or nearly all, of the work.

Very few, or no, appropriate forms of representation are included in the presented work. All, or nearly all, of those that are included lack key information and/or are placed too far away from accompanying text.

An inadequate number of appropriate forms of representation are included in the presented work. Most, but not nearly all, that are included lack key information and/or are placed too far away from accompanying text.

An adequate number of appropriate forms of representation are included in the presented work. None, or just a few, that are included lack key information and/or are placed too far away from accompanying text.

References are included in just a few cases where they are required. Only a small but inadequate number of these references tend to contain the required information (where appropriate).

References are included in most but not nearly all cases where they are required. An adequate number of these references tend to contain the required information (where appropriate).

References are included in all or nearly all cases where they are required. The vast majority of these references tend to contain the required information (where appropriate).


30% 30% 40%


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Coursework Mode 2: Modelling

Modelling

80 marks

internally-assessed


Defining the Process of Modelling The process of ‘mathematical modelling’ or, more simply, ‘modelling’ uses mathematics to solve relative complex, and, possibly, not well defined, real-life situations. Bliss and Libertini (2016) define ‘modelling’ as “a process that uses mathematics to represent, analyse, make predictions or otherwise provide insight into real-world problems” (p. 8). In a sense. the process of ‘modelling’ can be seen as being a special case of the process of ‘investigating’ where the situations one deals with are situations taken from the real world rather than from some imagined world or from abstract mathematics. However, the fact that when ‘modelling’ one is tied to a specific real world context and answers have to be reconciled with reality makes this process distinct from the process of ‘investigating’. The Aim of Modelling Tasks The aim of modelling tasks is to provide candidates with opportunities that allow them to understand and solve genuine, real-life problems by making use of the language of mathematics. Thus, mathematics is seen, not as an abstract subject separate from real life and which is important for its own sake, but as a valuable tool that is not only meaningful but also relevant – a subject that can actually make a difference in their day-to-day lives. By their very nature modelling tasks support the development of links between different areas of mathematics, foster independent mathematical thinking, encourage creative thinking and decision making, promote systematic ways of working, and, develop initiative and flexibility in approach. Types and Examples of Modelling Tasks In modelling tasks, pupils are given a real-life situation and asked to attain a specific goal. Thus they need to find a method that reaches that goal. Examples of modelling tasks are the following1:

i. Plan a school interclass handball tournament for your grade. ii. Help your local mini-market to decide whether to operate two check-out lines with one

cashier at each line or one check-out line with one cashier and one bagger. iii. Determine the correct duration of the yellow light in one of your local traffic lights. iv. Alter the parking slots in the school car park so as to maximise the number of cars that

can park in it. v. Examine the feasibility of a class trip to an attraction of your choice.

vi. Study the relationship between the mass, height, and shoe-size of your classmates. vii. Examine the relationship between a person’s height and a person’s length of step.

viii. Plan a month-long packed school lunch menu for yourself. ix. Determine the best place to locate first-aid stands, refreshment outlets, and mobile

toilets at the open air ‘Isle of MTV’ concert. x. Help your mother find a way how to hang pictures along a staircase so that they look

straight. Guidelines for Setting, Conducting, and Assessing Modelling Tasks The following guidelines on how to set, conduct and assess a modelling task are designed to ensure that modelling tasks are set in a systematic way (see section ‘(i) Steps’ below); candidates know exactly what they have to do when carrying out a modelling task (see section ‘(ii) Framework’ below); and, teachers know how they can carry out reliable and valid assessment (see section ‘Marking Criteria’ below).

(i) Steps: The following steps related to the implementation of the modelling coursework in the class, should be followed:

Each modelling coursework should consist of one modelling task. Thus, for summative assessment purposes, candidates are required to hand in one report which should consist of the proposed solution of the modelling task as well as a short write-up of the procedure used (for further details see section (ii) below).

1 Based on suggested examples in Swetz & Hartzler (1991) and Garfunkel & Montgomery (2016).


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Once a modelling task is set, teachers should ensure that pupils have understood what it entails by briefly going over the task with the pupils.

Every candidate should draw up a short plan for the task. Each plan should demonstrate that the candidate has understood the task and has started thinking about how to conduct the task. A short paragraph paraphrasing the task and a number of initial exploratory questions should be sufficient to satisfy these two requirements.

Every candidate presents this plan to the teacher. The teacher may provide feedback which must not be prescriptive but indicative it is important that candidates are left with the freedom to choose what they feel is the most suitable way to work on the task.

Once they receive feedback on their plan candidates can, if they so wish, adjust their plan in line with the feedback received. They should then start working according to the timeline indicated by the teacher.

When teachers monitor the work done by the candidates they should ensure that any help offered should limit itself to the asking of questions aimed at making pupils think and reflect on what they are doing. It should not consist of the teachers telling pupils what to do.

One report should be handed in by each candidate as outlined below in section (ii). (ii) Framework: The report should be structured according to the following framework:

Section Details


The title of the modelling coursework.

The name of the candidate.

The date when the modelling coursework was given.

The date of handing in of the modelling coursework report.

Understanding

In this section the candidates should provide clear evidence that they have understood what the task is all about. Where applicable, they should:

Write a few sentences paraphrasing the task in their own words.

Identify the key terms in the statement of the task.

Define the identified key statements.

Indicate the goals they intend to achieve.

Planning In this section the candidates are expected to show that they have devised a plan that will lead to a successful solution of the modelling task. Where applicable, they should:

Identify different ways the situation can be modelled.

Select one method that will be used to model the situation and give a suitable explanation.

Identify a number of assumptions that need to be made in order to simplify the problem to one that can be solved.

Identify a number of variables and/or parameters.

Identify any relationships between the variables and/or parameters.



Enacting

In this section the candidates are required to demonstrate how they have carried out their plan and how they went about generating a solution. Where applicable, they should:

Present relevant and accurate calculations.

Show evidence that appropriate care and attention has been exercised when performing calculations (e.g. by showing all relevant working; by checking reasonableness of answer according to context; by including an approximation; by re-working calculation in reverse using the inverse operation; by stating level of precision used; etc.)

Indicate predictions made and how they were checked.

Explain how conclusions were reached from these predictions.


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Concluding In this section the candidates are required to show that they have come up with an acceptable solution and have examined this solution in order to check that it is reasonable. Where applicable, they should:

Present a solution.

State a number of conclusions.

State to what extent the model is practical, reasonable, valid, and complete.

State the limitations of the model.

Identify possible refinements to the model.

Communicating Throughout all the sections the candidates are expected to show that they have taken the appropriate care in presenting their work. Where applicable, they should:

Present their work in a logical, clear, concise, and appealing fashion.

Use mathematical language that is correct.

Provide clear justifications using appropriate language.

Use appropriate forms of representation (notation, diagram, figure, table, graph, etc.) in each of the various parts of the task.

Include references.


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Modelling exemplar

Turning tables

Next year, the headmaster is considering furnishing the classrooms with

new chairs and trapezoidal (semi-hexagonal) tables. One student can sit

at each of the shorter sides of the tables and two students can sit at the

longer side. Two tables can be arranged:

- to seat 6 students;

- or eight students.

Model other ways of using similar tables to seat all the students in your class.

The report

Write a report that should comprise of the following 5 distinct sections. 1. Title

The title of the coursework

Your Name and class

The date of the coursework

The date that you handed in the report 2. Understanding

Write a few sentences paraphrasing the task in your own words.



Indicate the goals they intend to achieve. 3. Plan

Identify other different ways the situation can be modelled.

Select one method that will be used to model the situation and give a suitable explanation. (e.g. drawing, physical 2D/3D model, etc.)

Identify a number of assumptions that need to be made in order to simplify the problem to one that can be solved. (e.g. unlimited availability of chairs and tables)

Identify a number of variables and/or parameters. (size of room, number of students and number of tables used).




4. Implementation

Present relevant and accurate calculations in which you need to: i. Show all relevant working.

ii. Check the reasonableness of your calculations. iii. Indicate any the level of precision used.

Use an appropriate form of representation (e.g. notation, diagrams, tables, etc.) 5. Conclusion

Present a solution, if applicable to your chosen area.

State any final conclusions that you may have reached.

Include any references that you may have used in aiding you with your inquiry. Suggest a number of possible extensions to the problem.


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Modelling Exemplar

Plan a month-long packed school lunch menu

for yourself.

The report

Write a report on the given modelling task. This report should contain the following 5 sections. 1. Title

Your name and class

The date of the coursework

The date that you handed in the report


2. Understanding



Indicate the goal/s you intend to achieve.

3. Planning

Identify a number of assumptions that you need to make in order to simplify the problem to one that can be solved (e.g. budget)

Identify a number of variables (e.g. cost) and/or parameters (e.g. amount you wish to eat or the type of food you wish to eat)


Indicate which required and/or additional data and/or information you have collected and/or researched (e.g. cost of a loaf of bread or cost of ham slices).

4. Enacting

Present relevant and accurate calculations in which you need to: i. Show all relevant working.

ii. Check the reasonableness of your calculations. iii. Indicate any the level of precision used.

Present justifications were necessary.

Use appropriate forms of representation (e.g. notation, diagrams, tables, etc)

5. Concluding

Present a solution.


State to what extent your solution is practical, reasonable and valid.

State any limitations of your solution.

Identify possible refinements to your solution.

Include any references that you may have used to aid you in your task.


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Marking Criteria – Modelling

MARKING CRITERIA – Modelling

Maximum 40 marks*


Understanding (4 marks)

(0 – 1 marks) (2 – 3 marks) (4 marks)



Paraphrasing of task in own words tends to be comprehensive.

Only a few or none of the key terms in the statement of the task are identified.

More than half but not nearly all of the key terms in the statement of the task are identified

All or nearly all of the key terms in the statement of the task are identified.

Definitions tend to be sketchy, unclear, and/or imprecise.

Definitions tend to be acceptable but may lack sufficient detail, clarity, and/or precision.

Definitions tend to be quite detailed, clear, and precise.

Uses a form of representation that is inappropriate shedding little, or no, significant information about the task. Most, if not all, of any introduced notation tends to be unsatisfactory.

Uses a form of representation that while being appropriate was inaccurate and/or inefficient shedding only some information about the task. More than half, but not nearly all, of the introduced notation tends to be satisfactory.

Uses a form of representation that sheds all the information about the problem helping to clarify the meaning of the task. All, or nearly all, of the introduced notation tends to be satisfactory.






There is considerable evidence that the candidate has most, if not all, aspects of the task.

Planning (10 marks)


An insignificant number of ideas on how to model the given situation are identified.

A small, but non-negligible, number of ideas on how to model the given situation are identified.

A large number of ideas on how to model the given situation are identified.

Explanation given for the method used to model the situation tends to be unclear.

Explanation given for the method used to be model the situation tends to be reasonably clear though expressed somewhat inadequately.

Explanation given for the method used to be model the situation tends to be quite clear.

There is no statement specifying the simplified situation that the model will attempt to solve.

There is a statement specifying the simplified situation that the model will attempt to solve but it lacks sufficient clarity and/or is ambiguous.

There is a clear and unambiguous statement specifying the simplified situation that the model will attempt to solve.

None, or just a few, of the main assumptions made in the proposed model are identified. Most of these assumptions tend to be unreasonable. No justifications given for the choice of assumptions.

Most, but not nearly all, of the main assumptions made in the proposed model are identified. Nearly half of these assumptions tend to be unreasonable. A small number of justifications are given for the choice of assumptions but these tend to be unclear.

All, or nearly all, of the main assumptions made in the proposed model are identified. Most of these assumptions tend to be reasonable. A clear and comprehensive list of justifications is given for the choice of assumptions.

No, or just a few, important variables and/or parameters are identified. No explanation is given for their inclusion.

Most, but not nearly all, important variables and/or parameters are identified. Explanations for their inclusion are unclear and/or insufficient.

All, or nearly all, important variables and/or parameters are identified. Explanations for their inclusion are clear and comprehensive.


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There is no explanation on how the required data or information will be appropriately collected and/or researched.

There is an explanation on how the required data or information will be appropriately collected and/or researched but this tends to be unclear and/or ambiguous.

There is a clear and unambiguous explanation on how the required data or information will be appropriately collected and/or researched.

No alternative strategies for the purpose of comparing results are identified.

A few alternative strategies for the purpose of comparing results are identified. However, an explanation on how these will be used is unclear and/or ambiguous.

An adequate number of alternative strategies for the purpose of comparing results are identified. Explanation on how these will be used is clear and unambiguous.

Enacting (10 marks)


There is no evidence that required data or information has been appropriately collected and/or researched.

There is some evidence that required data or information has been appropriately collected and/or researched.

There is considerable evidence that required data or information has been appropriately collected and/or researched.

Appropriate methods to represent the information are evident in only a small part of the presented work.

Appropriate methods to represent the information are evident in most, but not nearly all, of the presented work.

Appropriate methods to represent the information are evident in all, or nearly all, of the presented work.

Meaningful mathematics is used in just a small part, or none, of the presented work.

Meaningful mathematics is used in most, but not nearly all, of the presented work.

Meaningful mathematics is used in all, or nearly all, of the presented work.

Relevant and accurate calculations are performed throughout only a minor part of the task.

Relevant and accurate calculations are performed throughout most, but not nearly all, of the task.

Relevant and accurate calculations are performed throughout all, or nearly all, of the task.

The model is either not presented or contains a significant number of errors.

The model is presented but contains a small number of errors, is difficult to understand, and/or is incomplete.

The model is presented and is mainly correct, clear to understand and basically complete.

No, or an insignificant, number of predictions are made and checked.

A small, but inadequate, number of predictions are made and checked.

An adequate number of predictions are made and checked.

Sound conclusions are made for only a few of the stated predictions.

Sound conclusions are made for most, but not nearly all, of the stated predictions.

Sound conclusions are made for all, or nearly all, of the stated predictions.

There is no evidence that a number of iterations have been made in effort to test and improve the model.

There is some evidence that a small number of iterations have been made in effort to test and improve the model.

There is clear evidence that a significant number of iterations have been made in an effort to test and improve the model.



There is no proposed solution or it is incomplete.

The proposed solution is complete but is either difficult to understand and/or not consistent with the statement of the original task.


The conclusions reached, if any, tend to be unclear.

The conclusions reached are clearly stated in most but not nearly all cases

The conclusions reached are clearly stated in all or nearly all cases.

There is no statement explaining to what extent model is practical.

There is a statement explaining to what extent model is practical but this tends to be unclear and/or ambiguous.

There is a clear and unambiguous statement explaining to what extent model is practical.

There is no concluding statement explaining why the model is reasonable given the initial conditions.

There is a concluding statement explaining why the model is reasonable given the initial conditions but this tends to be unclear and/or ambiguous.

There is a clear and unambiguous concluding statement explaining why the model is reasonable given the initial conditions.

There is no concluding statement explaining to what extent the model is valid given the initial conditions.

There is a concluding statement explaining to what extent the model is valid given the initial conditions but this tends to be unclear and/or ambiguous.

There is a clear and unambiguous concluding statement explaining to what extent the model is valid given the initial conditions.


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There is no concluding statement explaining to what extent the model is complete given the initial conditions.

There is a concluding statement explaining to what extent the model is complete given the initial conditions but this tends to be unclear and/or ambiguous.

There is a clear and unambiguous concluding statement explaining to what extent the model is complete given the initial conditions.

There is no statement explaining limitations of the model.

There is a statement explaining limitations of the model but this tends to be unclear and/or ambiguous.

There is a clear and unambiguous statement explaining limitations of the model.

There is no statement on how the model can be improved.

There is a statement on how the model can be improved but this tends to be unclear and/or ambiguous.

There is a clear and unambiguous statement explaining how the model can be improved.



Only some of the work is presented in a logical, clear, concise, and appealing fashion. Instances where this is not so are substantial.

Most, but not nearly all, of the work is presented in a logical, clear, concise, and appealing fashion. Instances where this is not so are not negligible but not substantial.

All, or nearly all, of the work is presented in a logical, clear, concise, and appealing fashion. Instances where this is not so are few and far between.

Correct spelling is used in some of the presented work. Significant disregard for common grammatical rules.

Correct spelling is used in most, but not nearly all, of the presented work. Some disregard for common grammatical rules.

Correct spelling is used in all, or nearly all, of the presented work. Little, or no, disregard for common grammatical rules.




Very few, or no, visual representations such as diagrams, graphs and/or tables are included in the presented work. All, or nearly all, of those that are included lack key information and/or are placed too far away from accompanying text.

An inadequate number of visual representations such as diagrams, graphs and/or tables are included in the presented work. Most, but not nearly all, that are included lack key information and/or are placed too far away from accompanying text.

An adequate number of visual representations such as diagrams, graphs and/or tables are included in the presented work. None, or just a few, that are included lack key information and/or are placed too far away from accompanying text.

References are included in just a few cases where they are required. Only a small, but inadequate, number of these references tend to contain the required information.

References are included in most, but not nearly all, cases where they are required. An adequate number of these references tend to contain the required information.

References are included in all, or nearly all, cases where they are required. The vast majority of these references tend to contain the required information.


30% 30% 40%


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Coursework Mode 3: Solving

Solving

80 marks

internally-assessed


Defining the Process of Solving The process of solving is a convergent activity (Evans, 1987; HMI, 1985; Frobisher, 1994) that requires pupils to reach a predetermined goal. It requires examining a defined situation in order to identify the stated goal; picking out the provided data and conditions; making a plan on how to proceed; implementing the plan by using the provided information as well as previously learnt content and/or process knowledge; arriving at a solution; and, finally, checking the accuracy and reasonableness of this solution. Therefore, a ‘solving task’ – a task that requires a process of solving – is a closed task that has a pre-established outcome. The Aim of Solving Tasks The aim of solving tasks is to provide pupils with a wide variety of opportunities where they can use mathematics as a tool in their efforts to solve problems. When solving problems using previously learnt content and/or process knowledge pupils understand that mathematics can be of great utility in a wide range of activities in everyday life including those related to the home and those related to the work place. They also discover that mathematics can be a rich source of pleasure and enjoyment when they solve problems that are challenging, engaging, and novel. Solving problems thus helps pupils appreciate that the mathematical knowledge that they learn is not an end in itself but a means to an end. Types and Examples of Solving Tasks Solving tasks are well-structured: they have a clear, well-formulated goal which the solver can reach by making use of the data and conditions set out in the task and by applying previously learnt mathematics. Although the problem might have multiple solutions the solution set is unique. Thus, in the context of working through solving tasks, the term ‘problem-solving’ will be taken to mean the process of arriving at the goal/s explicitly or implicitly indicated in the problem. Solving tasks can be either one of two types: routine and non-routine. Routine solving tasks are activities that indicate, explicitly or implicitly, the teacher-taught strategy the solver has to use in order to solve them. They are also content-specific being based on content found in either one or more of the strands of the mathematics curriculum. The emphasis in such tasks should be on helping pupils practise and consolidate the mathematical knowledge they have learnt so that it can be subsequently used to solve more challenging tasks. Thus, these tasks are useful in teaching for problem-solving (Foong, 2002) and should be the staple diet of most mathematics lessons. Routine solving tasks typically involve either exercises or word problems: Examples of exercises: (i) Which of the following are prime numbers? 3, 16, 19, 21, 55, 100. (ii) The line segment joining the centre of a circle to a point on the circumference is called a __. (iii) Work out 2 + 3 × 4 – 6 ÷ 2. (iv) Solve 2x + 5 = 9. (v) Find the circumference of a circle whose radius is 7 cm.

Examples of word problems: (i) ‘What is the sum of twelve and fifteen?’ instead of ’12 + 15 = ?’ (ii) ‘Find the value of the square root of 0.08234’ instead of ‘(0.08234)1/2 = ?’ (iii) Given that a 20 cm pizza serves one, how many should three 30 cm pizzas serve? (iv) The sides of a right-angled triangle are (x – 1) cm, 2x cm and (2x + 1) cm. What is x? (v) Choose any two whole numbers. Add them. Subtract the smaller number from the larger

one. Now add these two results. Try some other pairs of whole numbers. What can you notice?


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Non-routine solving tasks are activities that give little indication, if any, on how one should proceed but allow the pupils the freedom to develop and choose their own strategies as well as the liberty to implement those strategies. They are not meant to be tasks where pupils simply replicate the strategies they learnt from their teachers but tasks that should stimulate intellectual curiosity. The emphasis in such tasks is on helping pupils to develop problem-solving heuristics (e.g. draw a diagram, construct a table, work systematically, look for pattern, work backwards, consider simpler exemplars, guess & test) as well as general personal qualities (e.g. work independently, work collaboratively, participate in discussions, use imagination and flexibility of mind, engage in exploration, engage in experimentation). Thus, these tasks are useful in teaching about problem-solving (Foong, 2002) and are useful intermediate steps in the transition from solving routine tasks to doing investigative tasks. One should note that although pupils have freedom of action when it comes to choosing and implementing strategies, they should all converge to the same answer set. Non-routine solving tasks can be either one of four types: open-search problems, whimsical problems, mathematical puzzles, and mathematical games. a) Open-search problems are activities that do not indicate a working-out strategy in the

statement of the activity (Butts, 1980). Some typical phrases used in such problems include ‘For which ___ is ___’, ‘Find all ___’, ‘What___ if ___’, and ‘Prove that ___’, (Butts, 1980). Some examples are the following: (i) 5 is a not a factor of 4 × 3 × 2 × 1 but 6 is a factor of 5 × 4 × 3 × 2 × 1. For which

positive integers n (30 > n > 4) is n a factor of (n – 1) × (n – 2) × … × 4 × 3 × 2 × 1?

(ii) What values are possible for the area of quadrilateral ELCK if ABCD and EFGH are squares of side 24 cm and E is the centre of square ABCD? Give a reason for your answer.

(iii) ABCD is a rectangle. Prove that its diagonals AC and BD are equal.

b) Whimsical problems are mathematical problems that present a totally unrealistic and

fantastical context (Butts, 1980). The main aim of such problems is to arouse the intellectual curiosity of the pupils by presenting a situation that is far removed from their day-to-day reality. The following are two examples:

(i) An army platoon on the march through the jungle came to a river which was wide, deep, an infested with man-eating crocodiles. On the far bank they could see two young native boys with a boat. The boat could only hold the two boys or one soldier with his rifle and kit. How can the platoon cross the river? (Adapted from Bolt, 1982, Activity Number 24)

(ii) Two pirates, Captain Bluebird and Captain Jack Sparrow, were burying their booty on an island. Near the shoe were two large rocks and a lone palm tree. Bluebird started from one of the rocks and, walking along the line at right angles to the line joining the rock to the palm tree, paced off a distance equal to that between the rock and the palm tree. Jack Sparrow did a similar thing with respect to the other rock and palm tree. They then walked toward each other and buried the treasure halfway between. Two years later the pirates returned to the island to dig up their treasure but found that the palm tree was no longer there. How can they find the treasure? (Adapted from Butts, 1980)

c) Mathematical puzzles are mathematical problems with well-defined solutions that are

designed to test ingenuity and perseverance but which also try to entertain the solver. Although they do require knowledge of specific mathematics in order to solve, they are mainly reliant on such staples of mathematical thinking such as logical thinking, deductive thinking, inductive thinking, and, pattern spotting, as well as the solvers’ abilities to use a wide range of problem-solving strategies such as using the solution of an analogous problem, using trial & error, working backwards, drawing a diagram, dividing the task into


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smaller tasks and solving each one, etc., In short, mathematical puzzles are designed to test one’s general readiness to think outside of the box – i.e. lateral thinking. One can think of several ways of classifying mathematical puzzles. A simple and useful way of categorising them is to classify them according to different strands of the mathematics curriculum.

The following are typical examples related to the ‘Number’ strand: (i) A boy has as many sisters as brothers, but each sister has only half as many sisters

as brothers. How many brothers and sisters are there in the family? (Kordemsky, 1992, Puzzle Number 45)

(ii) The following numbers can be added quite quickly without using a calculator. How? 328,645 491,221 816,304 117,586 671,355 508,779 183,696 882,414 (Kordemsky, 1992, Puzzle Number 43)

The following group of examples are related to the ‘Shape & Space’ strand: (i) The Girl Guides have built a tent. Preparing for its opening, the Guides decorate the tent on all four sides with garlands, electric bulbs, and small flags. There are 12 flags in all. At first they arrange the flags 4 to a side, as shown, but

then they see that the flags can be arranged 5 or even 6 to a side. How? (Adapted from Kordemsky, 1992, Puzzle Number 7)

(ii) Take away four matches to leave exactly four equilateral triangles all of the same size. (Adapted from Bolt, 1985, Activity 8)

Some stereo-typical ‘Algebra’ strand puzzles are: (i) Given two numbers, if we subtract half the smaller number from each number, the

result with the larger number is three times as large as the result with the smaller number. How many times is the larger number as large as the smaller number? (Kordemsky, 1992, Puzzle Number 229)

(ii) Solve in your head: 6751x + 3249y = 26751 3249x + 6751y = 23249

The following are examples of puzzles from the ‘Data Handling’ Strand: (i) Your friend made three dice:

The red die had the numbers 2, 4, 9 twice on its faces. The blue die had the numbers 3, 5, 7 twice on its faces. The yellow die had the numbers 1, 6, 8 twice on its faces. The total on the faces of each die was the same but your friend was confident that if she let an opponent choose a die first and roll it, she would be able to select a die which would give her a better chance of obtaining a higher score. Explain! (Bolt, 1982, Activity 139)

(ii) You are a prisoner sentenced to death. The Emperor offers you a chance to live by playing a simple game. He gives you 50 black marbles, 50 white marbles and 2 empty bowls. He then says, "Divide these 100 marbles into these 2 bowls. You can divide them any way you like as long as you use all the marbles. Then I will blindfold you and mix the bowls around. You then can choose one bowl and remove ONE marble. If the marble is WHITE you will live, but if the marble is BLACK... you will die." How do you divide the marbles up so that you have the greatest probability of choosing a WHITE marble?

(https://www.braingle.com/brainteasers/15446/life-or-death-the-emperors-proposition.html)

https://www.braingle.com/brainteasers/15446/life-or-death-the-emperors-proposition.html


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d) Mathematical games are multiplayer games that do not require players to know specific knowledge of mathematics in order to play but which can be won using winning strategies that can be identified using mathematical thinking. Mathematical games have the following characteristics: o involve two or more players that compete with each other (Gough, 1999; Oldfield,

1991a) making moves that are mainly based on choice rather than chance (Gough, 1991) and which influence in some way or other the subsequent moves of the other players (Gough, 1993);

o are turn-based (Gough, 1999); o are governed by a set of rules (Oldfield, 1991a); o have a clear underlying structure (Oldfield, 1991a) which can be analysed

mathematically; o normally have a distinct finishing point (Oldfield, 1991a); o have specific mathematical objectives (Oldfield, 1991a); and, o require some form of mathematical thinking (Gough, 1993).

Some examples of these different mathematical game genres are the following:

A game for developing strategies: ‘Nim’ This is a game for two players. 21 counters are placed in a heap. Players taken it in turns to remove 1, 2 or 3 counters from the heap. The player that takes the last counter is the winner. (Adapted from Oldfield, 1991a)

A game to stimulate mathematical discussion: ‘3 Dice Game’ This is a game for 2 players. Players have a number grid with numbers 1-30 on it.

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

Player A throws three dice. The numbers thrown may be combined in any way using the four rules, to get a number to cross off on the grid. For example, (3 x 5) - 4 = 11, so 11 could be crossed off with a throw of a 3, 4, and 5. Player A may cross off up to 2 different numbers from her one throw, but for each must explain to player B how the result is obtained, if player B challenges her. Player B then has a turn. The aim is to be the first person to cross off three adjacent numbers along a row (e.g. 3, 4, 5); along a column (e.g. 5, 15, 25); or along a diagonal (e.g. 7, 18, 29). (Adapted from Oldfield, 1991b).

A game for developing concepts: ‘Thirty-Six’ This is a game that is ideally played by 2-4 players. To determine order of play, each player throws a six-sided die in turn. The player who has thrown the smallest number then throws again and adds the new number to her first number, and continues throwing and adding until she wishes to stop. If her total exceeds 36 she is "over the top" and therefore out of the game. The person with the second lowest initial throw then does the same, right through to the player with the highest initial throw. The player whose total is nearest to (but not over) 36 is the winner. (Adapted from Oldfield, 1991d)

A game for reinforcing skills: ‘Pairs’ This is a game that is ideally played by two to four players. After the players have been informed of the pairing rule for their particular game (e.g. the product of the numbers on the two cards must exceed 50), a set of playing cards (without the K, Q, and J) are shuffled and placed face down on a table in five rows of eight cards each. Player A turns up one card, and then another, so that all players can see them clearly. If they make a pair, she keeps that pair of cards and repeats the process. When two cards are turned up which do not make a pair, they are turned face down again in their original places. Player B then has a turn. Play continues until all pairs have been found. The player with the most pairs wins the game. (Adapted from Oldfield, 1992)


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Guidelines for Setting, Conducting, and Assessing Solving Tasks The following guidelines on how to set, conduct, and assess a solving task are designed to ensure that solving tasks are set by the teachers in a systematic way (see section ‘(i) Steps’ below); candidates know exactly what they have to do when carrying out a solving task (see section ‘(ii) Framework’ below); and, teachers know how they can carry out reliable and valid assessment (see section ‘Marking Criteria’ below). (i) Steps: The following steps related to the implementation of the solving coursework, should be followed: ● A solving coursework should consist of one task of the non-routine variety and a similar

non-routine problem creation activity. In the former, pupils are expected to work on the task, either individually, or collaboratively, as indicated by the teacher, while in the latter, pupils are expected to work on their own to create and solve a similar non-routine task. For summative assessment purposes, each candidate is required to hand in an individual report. (for further details see section (ii) below).

● Teachers should make sure that pupils have understood what the solving task entails. ● Candidates should draw up a short plan for the assigned non-routine problem. The plan

should demonstrate that the candidates understood what they have to do in this part of the coursework and have started thinking about how to go about completing it. A short paragraph paraphrasing this part and indicating the way to proceed should be provided.

● The plan is presented to the teacher who may provide feedback which must not be prescriptive but indicative - it is important that pupils are left with the freedom to choose what they feel is the most suitable way to work on the tasks.

● Once they receive feedback on their plans pupils can, if they so wish, adjust their plan in line with the feedback received. They should then start working according to the timeline indicated by the teacher.

● When teachers monitor the work done by the candidates they should ensure that any help offered should limit itself to the asking of questions aimed at making pupils think and reflect on what they are doing. It should not consist of the teachers telling pupils what to do.

Section Details


● The title of the solving coursework.

● Name of the candidate.

● The date when the solving coursework was given.

● Date of handing in of the solving coursework.

Understanding In this section the candidates should provide clear evidence that they have understood what the solving coursework is all about. Where applicable, they should:

● Write a few sentences paraphrasing the task in their own words.

● Identify and define the key terms in the statement of the task.

● Indicate the goal/s they intend to achieve.

● Identify the given data, the required unknowns and the stated conditions (if any).

Planning In this section the candidates are expected to show a plan they devised and that will lead to a successful solution of the solving task. Where applicable, they should indicate:

● Whether they have seen the problem before in the same form or in a slightly different form.

● How they are going to use the information they know.

● The strategy they are going to adopt to arrive at the goal/s.

● Alternative solution strategies and give reasons for the chosen solution strategy.


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Implementing & Checking

In this section the candidates are expected to show how they have carried out their plan and how they went about generating a solution. Where applicable, they should:

● Present relevant and accurate calculations.

● Present justifications for each step.

● Show evidence that appropriate care and attention has been exercised when performing calculations (e.g. by showing all relevant working; by checking reasonableness of answer according to context; by including an approximation; by re-working calculation in reverse using the inverse operation; by stating level of precision used; etc.)

Concluding In this section the candidates are expected to show that they have come up with an acceptable solution and have examined this solution in order to check that it is reasonable. Where applicable, they should:

● Present a solution.

● State a number of conclusions.

● State to what extent conclusions are reasonable, valid, and complete.

Creating In this section the candidates are expected to create and solve a similar problem. They should:

● Create a non-routine problem similar to the one given in the task.

● Present a solution.

Communicating Throughout all the sections, where applicable, the candidates are expected to:

● Present their work in a logical, clear, concise, and appealing fashion.

● Use mathematical language that is correct.

● Provide clear justifications using appropriate language.

● Use appropriate forms of representation (notation, diagram, figure, table, graph, etc.) in each of the various parts of the task.

● Include references.


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References

Bolt, B. (1982). Mathematical Activities. A Resource Book for Teachers. Cambridge: Cambridge University Press.

Bolt, B. (1985). More Mathematical Activities. A Resource Book for Teachers. Cambridge: Cambridge University Press.

Butts, T. (1980). ‘Posing Problems Properly’. In: S. Krulik & R.E. Reys (Eds.) Problem Solving in School Mathematics. NCTM

1980 Yearbook. Reston, Virginia: National Council of Teachers of Mathematics

Evans, J. (1987). ‘Investigations: The state of the art.’ Mathematics in School, 16(1), 27-30.

Foong, F.P. (2002). ‘The role of problems to enhance pedagogical practices in the Singapore mathematics classroom’. The

Mathematics Educator, 6(2), 15-31.

Frobisher, L. (1994) ‘Problems, Investigations and an Investigative Approach’. In: A. Orton and G. Wain (Eds.) (1989). Issues in

Teaching Mathematics. London: Cassell.

Gough, J. (1991). ‘Mathematics Games That Really Teach Mathematics’. Prime Number, 6(1), 3-6.

Gough, J. (1993). ‘Playing Games to Learn Mathematics’. In: J. Mousely & M. Rice (Eds.) Mathematics: of Primary Importance.

Brunswick: Mathematical Association of Victoria, 1993: Proceedings of the 30th Annual Conference of the MAV, La Trobe

University, 2, 3 December 1993; 218-221.

Gough, J. (1999). ‘Playing (Mathematics) Games: When is a Game not a Game?’ Australian Primary Mathematics Classroom,

4 (2), 12-17.

HMI (1985). Mathematics from 5 to 16. HMI Series: Curriculum Matters No.3. London: HMSO.

Kordemsky, B.A. (1992). The Moscow Puzzles: 359 Mathematical Recreations. New York: Dover Publications.

Oldfield, B.J. (1991a). ‘Games in the Learning of Mathematics: 1: A Classification’. Mathematics in Schools, 20(1), 41-43.

Oldfield, B.J. (1991b). ‘‘Games in the Learning of Mathematics: 2: Games to simulate discussion’. Mathematics in Schools,

20(2), 7-9.

Oldfield, B.J. (1991c). ‘Games in the Learning of Mathematics: 3: Games for developing strategies’. Mathematics in Schools,

20(3), 16-18.

Oldfield, B.J. (1991d). ‘Games in the Learning of Mathematics: 4: Games for developing concepts’. Mathematics in Schools,

20(5), 36-39.

Oldfield, B.J. (1992). ‘Games in the Learning of Mathematics: 5: Games for reinforcement of skills’. Mathematics in Schools,

21(1), 7-13.


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Solving exemplar

What values are possible for the area of quadrilateral ELCK if ABCD and EFGH are squares of side 24 cm and E is the centre of square ABCD? Give a reason for your answer. The report Write a report on the given solving task. This report should contain the following 5 sections. 1. Title

Your name and class

The date when the solving coursework was given

The date that you handed in the report


2. Understanding


Identify the key terms in the statement of the task (e.g. square, etc.).


Identify the given data (e.g. squares of side 24 cm, etc.).

Identify the required unknowns.

Indicate the goals you intend to achieve.

3. Planning

Indicate whether you have seen the problem before in exactly the same form or in a slightly different form.

Indicate how you are going to use the information you know.

Indicate the strategy you are going to adopt to arrive at your goal/s.

4. Implementing & Checking

Present relevant and accurate calculations in which you need to: i. show all relevant working;

ii. check the reasonableness of your calculations; iii. indicate any the level of precision used.

Present justifications for your steps.

Use appropriate forms of representation (e.g. notation, diagrams, tables, etc.).

5. Conclusion

Present a solution.


Include any references that you may have used to aid you in your task.

6. Creating Present a similar non-routine problem together with its solution.


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Marking Criteria – Solving

MARKING CRITERIA – Solving Maximum 40 marks*

* The total mark should be doubled and added to the Non-Calculator Test mark. Understanding (6 marks)




Paraphrasing of task in own words tends to be clear and comprehensive.

Only a few, or none, of the key terms in the statement of the task are identified and defined.

More than half of the key terms in the statement of the task are identified and defined. Definitions are acceptable but may lack sufficient detail, clarity, and/or precision.

All, or nearly all, of the key terms in the statement of the task are identified and defined properly.




Only some, or none, of the data, unknown/s and conditions (if any) are identified.

More than half of the data, unknown/s and conditions (if any) are identified.

All, or nearly all, of the data, unknown/s and conditions (if any) are identified.



There is considerable evidence that the candidate has understood most, if not all, of the aspects of the task.

Planning (6 marks)


No indication that the task has been seen before in the same form or in a slightly different form.

Limited indication that the task has been seen before in the same form or in a slightly different form.

Clear indication of a recognition that the task has been seen before in exactly the same form or in a slightly different form.

Only some of the data and/or conditions are taken into consideration. Evidence that most of the essential ideas of the problem have been ignored and/or misunderstood.

Most, but not all, of the data and/or conditions are taken into consideration. Evidence that some essential ideas of the problem were ignored and/or misunderstood.

All of the data and/or conditions are taken into consideration. Evidence that none of the essential ideas of the problems have been ignored and/or misunderstood.

Selected solution strategy, if any, is clearly inappropriate and will lead to an incorrect result.

Selected solution strategy is adequate though inefficient and can lead to both a partially correct or a correct result.

Selected solution strategy is appropriate and efficient and will lead to a correct result.

No alternative solution strategies are identified.

One or two alternative solution strategies are identified.

More than two alternative solution strategies are identified.

Implementing & Checking (6 marks)


Solves correctly some of the parts of the problem, or none at all.

Solves correctly about half of the parts of the problem.

Solves correctly all, or nearly all, parts of the problem.

Only a few, or none, of the computations are performed correctly and completely.

About half of the computations are performed correctly and completely. Some steps are left out.

All, or nearly all, of the computations performed are completely and correctly. No steps are left out.

Justifications are provided for only some, or none, of the steps.

Justifications are provided for about half of the steps.

Justifications are provided for all, or nearly all, of the steps.

The reasoning used in the solution was mainly unclear. This made practically all of the solution difficult to understand.

The reasoning used in the solution was sometimes not clear. This made significant parts of the solution difficult to understand.

The reasoning used in the solution was clear. This made all, or nearly all, of the solution easy to understand.

There is no evidence that an attempt to check the solution has been made.

There is some evidence that an attempt to check the solution has been made though this is only partially successful.

There is clear evidence that a successful attempt to check the solution has been made.


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There is no evidence that an attempt to examine the reasonableness of the result in the light of the essential information given in the statement of the problem and/or other relevant knowledge of the context has been made.

There is some evidence that an attempt to examine the reasonableness of the result in the light of the essential information given in the statement of the problem and/or other relevant knowledge of the context has been made. Some of the reasoning used in this examination has been given.

There is clear evidence that an attempt to examine the reasonableness of the result in the light of the essential information given in the statement of the problem and/or other relevant knowledge of the context has been made. The full reasoning used in this examination has been given.



There is no proposed solution or is incomplete.

The proposed solution is complete but is either difficult to understand and/or not consistent with the original task statement.


The conclusions reached are clearly stated in just a few of the cases. There is little, or no, evidence of what has been learnt from these conclusions.

The conclusions reached are clearly stated in most but not nearly all cases. There is some evidence of what has been learnt from these conclusions.

The conclusions reached are clearly stated in all or nearly all cases. There is clear evidence of what has been learnt from these conclusions.

Creating (10 marks)


Similar non-routine problem and its solution not/poorly presented.

An incomplete similar non-routine problem presented. Solution is either missing, incomplete or incorrect.

An appropriate similar non-routine problem created and a good logical solution presented.



Only some of the work is presented in a logical, clear, concise, and appealing fashion.

Most of the work is presented in a logical, clear, concise, and appealing fashion.





Justifications badly articulated/not articulated at all, with most of the language used being inappropriate and/or incorrect.

Justifications are poorly articulated with nearly half of the language used being inappropriate and/or incorrect.

Justifications are well-articulated with all, or nearly all, of the language used being appropriate and correct.

Very few, or no, appropriate forms of representation are included in the presented work. All, or nearly all, of those that are included lack key information and/or are placed too far away from accompanying text.

An inadequate number of appropriate forms of representation are included in the presented work. Most, but not nearly all, that are included lack key information and/or are placed too far away from accompanying text.

An adequate number of appropriate forms of representation are included in the presented work. None, or just a few, that are included lack key information and/or are placed too far away from accompanying text.

References included in just a few cases where required. Only a small but inadequate number of these references tend to contain the required information.

References are included in most but not nearly all cases where required. An adequate number of these references tend to contain the required information.

References are included in all or nearly all cases where required. The vast majority of these references tend to contain the required information.


30% 30% 40%


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Coursework Mode 4: Maths Trail

Maths Trail

80 marks

internally-assessed

externally-

moderated

Defining the Maths Trail A maths trail is an organised route involving a number of different tasks where pupils look for clues, spot mathematical designs and patterns, take measurements using appropriate instruments, make estimations, draw sketches, work out calculations, find solutions, solve problems, etc.

Maths trails can be conducted in various environments such as: the boundaries of the school or outside the school which could be another building like a museum or a supermarket or any other outside space like parks and playgrounds.

The aim of Maths trails The aim of Maths trails is to help pupils appreciate the use of mathematics in environments outside the classroom thus bringing into greater focus the utilitarian aspect of mathematics. It helps pupils make the necessary connections between the various topics learnt in class and links classroom-based exercises with real-life contexts.

Guidelines for Setting, Conducting, and Assessing a Maths Trail The Maths Trail assignment can be set at either MQF Level 1-2, (40% at Level 1 – 60% at Level 2), or at MQF Level 1-2-3 (30% at Level 1 – 30% at Level 2 – 40% at Level 3). It will consist of 6 to 10 mathematical tasks designed by the teacher and may take the form of a set of handouts/booklet. These are set in various locations in the designated site of the trail, targeting a number of learning outcomes covered in class. Each task is made of a number of parts, some of which are marked with an asterisk (*) meaning that these will be filled on site.

The assignment will be conducted in two phases: The first phase takes place on site where pupils may work either individually, or collaboratively, as indicated by the teacher. They will be supplied with all the required measuring instruments like compasses, thermometers, clinometers, pedometers, stop watches, trundle wheels and measuring tapes as required. Pupils will collect and write down all the required data at each stage of the trail.

In the second phase, each pupil, working individually, uses the data gathered on site to complete the tasks. The pupil’s work is then collected by the teacher for marking using a marking scheme. Each pupil is required to hand in this assignment on an individual basis.

Marking Scheme Guidelines

Types of Marks

Method Marks

Method marks (denoted by M) are awarded for knowing a correct method of solution and attempting to apply it. Method marks cannot be lost for non-method related mistakes. They can only be awarded if the method used would have led to the correct answer had a non-method mistake not been made. Unless otherwise stated, any valid method, even if not specified in the marking scheme, is to be accepted and marked accordingly.

Accuracy Marks

There are two types of Accuracy marks (denoted by A or B):

i) A marks are accuracy marks given for correct answer only.

Incorrect answers, even though nearly correct, score no marks.

Accuracy marks are also awarded for incorrect answers which are correctly followed through (denoted by A f.t.) from a previous incorrect answer, provided that f.t. is indicated in the marking scheme.

ii) B marks are accuracy marks awarded for specific results or statements independent of the method used.

Note: No Method marks (M) or Accuracy marks (A) are awarded when a wrong method leads to a correct answer.

The global mark for the Maths Trail is 80. If the total mark of all the tasks is different, it should be proportionately recalculated out of 80. This mark will be added to the mark obtained in the Non-Calculator Test for a final mark of the assignment out of 100.


Page 96 of 196

Maths Trail

exemplar tasks:

The following are a

number of Maths

Trail exemplar

tasks. When a

Maths Trail set of

tasks is formulated,

it should follow the

30:30:40 ratios.

These 10 exemplar

tasks follow these

ratios when taken

as a complete set

and the marks add

up to 100. In this

case, since the

total mark adds up

to 100, the mark

obtained by the

candidate should

be recalculated out

of 80.

Task 1: The school building

a) *Use your compass and mark with an arrow on the map below the North direction of

your school from the point marked X.

b) *Name a part of your school (e.g. library, Head of school office), which is due East of

point X.

c) Use your map to measure the bearing of the blue flag from point A.

Ans: _____________

d) Calculate the bearing of point A from the blue flag.

Ans: _____________

e) *Measure the distance on the map and the actual distance between the points P and

Q

Ans: _______ ; ________

f) Use your answer in e) to determine the scale of the map in the form 1 : n

Ans: _________


Page 97 of 196

Cement Sand Aggregate

Task 2: Concrete mixture

You are in the design and technology lab. Here you can find a cylindrical bucket, some

cement, sand and aggregate.

A worker needs to fill the bucket completely with a concrete mixture of cement, sand and

aggregate in the ratio 1 : 2 : 3.

For this task you will be needing a measuring tape, a measuring jug and bathroom scales.

a)

i) * Measure the required bucket dimensions you will need to calculate the capacity of

the bucket.

Ans: ___________________________________

ii) * Calculate the capacity of the bucket in Litres.

Ans: _________Litres

iii) *Determine how much cement, sand and aggregate in Litres you will need to fill the

bucket completely with the correct concrete mixture.

Ans: Cement: _________ Sand: __________ Aggregate: __________

b) *Use the measuring jug to mix the exact mixture of concrete to completely fill the

bucket. Weigh the filled bucket. Give your answer in kg.

Ans: ___________

c) Cement, sand and aggregate come in 10 Litre bags which cost €6, €2.50 and €2.30

respectively. 1.5 m3 of concrete mixture is needed for a project. Calculate the cost of

the concrete mixture.

Ans: € ___________


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Task 3: How many?

You are in the Mathematics activities room. The big box container is completely filled with

an assortment of coloured beads. Your task is to estimate the number of beads by their

colour without counting the whole lot.

a) *Fill the small box completely with a sample of the beads.

*Count and write down the number of each coloured bead.

Ans: Red = ______; Blue = _______; Green = _______

b) *Measure the length, breadth and height of both containers.

Ans: Big box: L = ____ , B = ____ , H = ____.

Small box: L = ____ , B = ____ , H = ____.

c) Calculate the capacity of the two boxes.

Ans: Big box: ________

Small box: ________

d) Calculate the approximate total amount of each coloured bead in the big container.

Ans: Red = ______; Blue = _______; Green = _______

e) One bead is picked at random from the big box. What is the probability of choosing:

i) a blue bead?

Ans: ________

ii) a red or a green bead?

Ans: ________


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Task 4: Classroom Plan

Your task is to draw a scale drawing of the

plan of the classroom.

a) *Draw a sketch of the plan of the classroom below. Make sure to include all desks,

windows, doors and other furniture. Do not include chairs.

Measure the necessary dimensions of the room, desks, furniture, doors and windows.

Measure also the position of each desk, door, window and furniture in the classroom.

Write down these measurements on your sketch.

b) Choose an appropriate scale and make a scale drawing of your classroom on a blank

page.


Page 100 of 196

Task 5: A flight of stairs

a) *Measure the length of the railing along this flight of stairs in metres.

Ans: ____________

b) *Count the number of steps and measure the vertical height of each step.

Ans: ________ , _________.

c) Calculate the vertical height of the whole flight of steps.

Ans: ________

d) Mark the distances AC and BC on the diagram below and calculate the horizontal

distance AB, giving your answer correct to 2 significant figures.

Ans: _________

e) Calculate the angle marked x.

Ans: ________

x

A B

C


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Task 6: How high is the building?

You are facing the backside of the school.

a) *Measure the height of each pupil in the group and fill in the table below.

b) *Each pupil takes turns in measuring the angle of elevation of the top of the building

using the clinometer making sure that each pupil is at a different distance from the

building. Each time note the distance from the wall.

c) *Complete the table below.

Height of the pupil Distance from the wall Angle of elevation

d) Complete the diagrams below with the recorded measurements, one diagram for each

pupil. For each diagram calculate the height of the wall.

e) Calculate the mean of your four answers in part (d).

Ans: ________

______ cm

____ cm

Ans:

_________

______cm

____ cm

Ans:

_________

______ cm

____ cm

Ans:

________

______ cm

____ cm

Ans:

_________


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Task 7: The Running Track

For this question assume that the outer and inner perimeter of the school running track

is made up of semi-circles and straight edges as shown by the bold black line.

a) *Measure the inner diameter d and the length L of the straight edge. You

may need to use a cone so that you take your measurements in parts.

Ans: _________, _________

b) *Measure the width W of the running track.

Ans: ________

c) Calculate the outer diameter D, of the semi-circles in the running track.

Ans: ________

d) Use your measurements to find the Area of the running track.

Ans: ________

D

L

d

W


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e) *Use the stopwatch to measure the time it takes for each one of the pupils in your

group to run along the straight edge l of the running track. Calculate the speed of each

pupil in m/s. Enter your results in the table below.

Student Time in seconds Speed in m/s

1

2

3

4

f) Use your data to calculate the mean running speed of your group.

Ans: _________

g) Write down the range of speed correct to 1d.p. for your group in the form:

_______ ≤ x ≤ ________


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Task 8: Façade of School Building

You are now facing the façade of your school.

a) *Make an estimate of the percentage area of

the façade taken by windows and doors.

Ans: ______

b) *Measure the length and breadth of one of the eight windows and calculate the total

area of all the windows.

Ans: L = _____ m; B = _____ m; Total area =______ m2

c) *Measure the length of the building façade.

Ans: ______ m

d) Count the number of courses (filati) of the façade and measure the height of one

course. Hence calculate the height of the building.

Ans: Height =_________ m

e) Calculate the area of the façade.

Ans: ______ m2

f) Calculate the actual percentage area of the façade being taken by the windows.

Ans: ________


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Task 9: The Sun

You are in the Science lab. Read the information on the chart about the sun and answer the

questions beneath.

Complete:

a) *The sun has the shape of a ________________ and has a diameter _________ times

that of the Earth.

b) *The radius of the sun is _________________ km.

c) *The sun has a mass _____________ times that of the Earth and accounts for ________%

of the mass of the whole solar system.

d) * _______% of the sun’s mass is hydrogen, _______% is helium and _______% consist of

other heavier elements.

e) Use the formula 𝑉 =4

3𝜋𝑟3 to calculate the volume of the sun. Give your answer in

standard form in m3.

Ans: ______________km3

f) Given that Density = Mass ÷ volume, and the mass of the earth is 5.972 × 1024 kg,

calculate the Density of the sun in g/cm3.

Ans: ___________g/cm3

The Sun

The Sun is the star at the centre of the Solar System. It is a nearly perfect

sphere of hot plasma. It is by far the most important source of energy for life

on Earth. Its diameter is about 1.39 million kilometres or 109 times that of

Earth, and its mass is about 330,000 times that of Earth. It accounts for

about 99.86% of the total mass of the Solar System. Roughly three quarters

of the Sun's mass consists of hydrogen (~73%); a quarter is

mostly helium with much smaller quantities of heavier elements,

including oxygen, carbon, neon, and iron.

https://en.wikipedia.org/wiki/Sun

https://en.wikipedia.org/wiki/Star

https://en.wikipedia.org/wiki/Solar_System

https://en.wikipedia.org/wiki/Plasma_(physics)

https://en.wikipedia.org/wiki/Energy

https://en.wikipedia.org/wiki/Solar_mass

https://en.wikipedia.org/wiki/Hydrogen

https://en.wikipedia.org/wiki/Helium

https://en.wikipedia.org/wiki/Oxygen

https://en.wikipedia.org/wiki/Carbon

https://en.wikipedia.org/wiki/Neon

https://en.wikipedia.org/wiki/Iron

https://en.wikipedia.org/wiki/Sun


Page 106 of 196

Task 10: The simple pendulum

You are in the physics lab. Look at the simple pendulum on

the work bench.

a) *Measure the length L of the string from the top to the

middle of the bob in metres.

Ans: L = ______ m

b) *Displace the bob at a small angle (≈5°-10°) and use the

stopwatch to measure the time for 20 oscillations in

seconds. (an oscillation is the movement of the bob from

one side to the other and back)

Ans: _______ seconds

c) Calculate the time T for 1 oscillation

Ans: T =_______ seconds

d) The time for one oscillation T depends on the length of the string L and the acceleration

due to gravity g. It can be found using the formula:

𝑇 = 2𝜋√𝐿

𝑔

Make g the subject of the formula.

Ans: g = _____________

e) Use your answer in part (d) and the values for L and T to calculate the acceleration due

to gravity g. Give your answer correct to 1 decimal place.

Ans: _________

L


Page 107 of 196

Maths Trail

exemplar tasks

suggested marking

scheme

Task Workings and Solutions Additional guidance Marks

1. (a) Arrow from X showing correct direction Allow 10° error B1

(b) Hallway o.e. B1

(c) 009° Allow 5° error B2

(d) 180 + 9 = 189° Allow 5° error M1 A1

(e) 12 cm , 12 m Accept any units B1 B1

(f) 12 : 1200 = 1 : 100 M1 A1

(Total: 10 marks)

2. (a)i Radius = 10.3 cm or Diameter=20.6 cm Height =14.8 cm

Allow 0.5 cm error Both correct

B1

(a)ii 𝑉 = 𝜋𝑟2ℎ 𝑉 = 𝜋 × 10.32 × 14.8 𝑉 = 4932.7 cm3

𝑉 = 5 Litres

M1 A1 f.t.

(b) Cement: 5 ÷ 6 = 0.833 ≈ 0.8 𝐿

Sand: 0.833 × 2 = 1.666 ≈ 1.7 𝐿 Aggregate: 0.833 … × 3 = 2.5 𝐿

All correct M1 A1 f.t

(c) 8.32 kg Accept 7.3 to 9.3 kg B2

(d) 1.5 m3 = 1500 Litres Cement: 1500 ÷ 6 ÷ 10 × 6 = €150 Sand: 1500 ÷ 6 × 2 ÷ 10 × 2.5 = €125 Aggregate: 1500 ÷ 6 × 3 ÷ 10 × 2.3 = €172.50 Total = €447.50

B1

M1 A1

(Total: 10 marks)

3. (a) Red = x beads Blue = y beads Green = z beads

Numerical values of

x, y and z

All correct

B1

(b) 30 cm; 20 cm; 15 cm 9 cm; 5 cm; 5 cm

All correct B1

(c) Big box: 9000 cm3

Small box: 225 cm3 For using correct formula Both answers correct

M1

A1

(d) 9000 ÷ 225 = 40

Red: value of 40 × x Blue: value of 40 × y Green: value of 40 × z

M1 A1 B1

B1

B1

(e)i Value of

𝑥

𝑥+𝑦+𝑧

B1

(e)ii Value of

𝑥+𝑦

𝑥+𝑦+𝑧

B1

(Total: 11 marks)


Page 108 of 196

4. (a) Correct sketch and correct measurements

2 marks for correct sketch and 2 marks for correct measurements.

Allow 1 cm error for desks, doors, windows and furniture.

Allow 2 cm error for the dimensions of the room.

B4

(b) Correct scale drawing Appropriate scale used Correct size of room Correct size of desks Correct distances between desks Correct sizes of door and windows Correct position of door and windows

B1 B1 B1 B1 B1 B1

(Total: 10 marks)

5. (a) 291.5 cm Allow 0.5 cm error B1

(b) 10 steps ; 15 cm Allow 0.5 cm error B1

(c) 150 cm A1 f.t.

(d) Uses Pythagoras: 250 cm M2 A1 f.t

(e) sin-1 150

291.5= 30.9° Allow 5° error M1 A1 f.t

(Total: 8 marks)

6. (a) Correct heights of pupils B1

(b) Correct angles of elevation B1

(c) Table is complete B1

(d) h = 𝑑 × tan 𝜃

H = h + t = 8.5 m

𝜃 - angle of elevation. d - distance from the wall. t - height of the pupil. 1 mark for each correct answer.

Allow 0.3 m error each time.

B4

(e) Add heights and dividing by 4 = 8.5 m Allow 0.3 m error M1 A1

(Total: 9 marks)

7. (a) d = 73 m; L = 84.39 m Allow 0.1 m errors B1 B1

(b) W = 9.76 m Allow 5 cm error B1

(c) D = 73 + 9.76 × 2 = 92.52 m Allow 0.2 m errors M1 A1 f.t.

(d) Radii = 46.26 m and 36.5 m Area between circles

= π × 46.262 - π × 36.52

Area of two rectangles = 84.39 × 9.76 × 2 Total area = 4184 m2

Allow 100 m2 errors

B1

M1

M1

A1

(e) Correct times Finding the speeds by dividing 84.39 m by respective time in seconds

B1

M1

(f) Total ÷ number of pupils Correct answer

B1

(g) Minimum speed x Maximum speed Values of min & max B1

(Total: 13 marks)


Page 109 of 196

8. (a) Any percentage in the range 2% - 10% B1

(b) L = 1 m; B = 0.5 m; Total area = 1 × 0.5 × 8 = 4 m2

Allow 0.02 m error B1 M1 A1 f.t.

(c) 8.5 m Allow 0.05 m error B1

(d) 11 m (44 courses @ 25 cm each) Allow 0.5 m error M1 A1

(e) 11 × 8.5 = 93.5 m2 Allow 5 m2 error B1

(f) 4 ÷ 93.5 × 100 = 4.3% Allow 0.3 error M1 A1 f.t.

(Total: 10 marks)

9. (a) Sphere ; 109 Both correct B1

(b) 695 000 km or 6.95 × 105 km Accept 0.695 million B1

(c) 330 000 ; 99.86% Both correct B1

(d) 73% ; 25% ; 2% All correct B1

(e) V = 4 ÷ 3 × π × (695 000 × 1000)3

= 1.40618682 × 1027 m3

M1 A1

(f) Mass = 5.972 × 1024 × 330 000 000 = 1.997076 × 1033 g Volume = 1.40618682 × 1027 × 1003

= 1.40618682 × 1033 cm3

Density = Mass/Volume = 1.401

1.4 or more accurate

M1

M1

M1 A1 f.t.

(Total: 10 marks)

10. (a) 0.5 m B1

(b) 28.4 seconds M1 A1

(c) 1.42 seconds M1 A1

(d) 𝑇2 =

4𝜋2𝐿

𝑔

𝑔 =4𝜋2𝐿

𝑇2

M1

M1

(e)

𝑔 =4 × 𝜋2 × 0.5

1.422= 9.8

M1 A1

(Total: 9 marks)


Page 110 of 196

Coursework Mode 5: Research Project

Research Project

80 marks

internally-assessed


Defining the Research Project

A research project is a planned detailed research study carried out over a period of weeks not

exceeding two months that aims to increase and widen pupils’ understanding and knowledge of

one or more elements of the mathematics syllabus. Through research project work pupils not

only learn new concepts, skills, and facts but can also gain a better understanding of other

aspects of mathematics that are not normally covered in depth in the classroom such as its

history, its links with other subjects in the curriculum, and, particularly, its utility and application

in the world around them. This helps pupils to appreciate the true beauty of the subject and

challenges them to think beyond the limits of the classroom. They also develop the competencies

essential for life-long learning such as the ability to formulate essential questions, conduct

research, evaluate, synthesize results, and communicate those results to others while reflecting

on the strengths of their work and the ways in which they can improve. Participating in research

project work will also help them develop the communication skills necessary for success in the

21st-century such as reading, writing, discussing, interviewing, and other creative skills. The

widespread availability of data and information, especially via information and communications

technology, not only provides access to various kinds of authentic materials to work with but also

helps pupils develop essential technology-related skills.

In order to create a sense of learner ownership of the learning process and to help develop self-

assurance, pupils should be given the opportunity to choose their own topic (Educators’ Guide

for Pedagogy and Assessment, Learning Outcomes Framework, 2017) from a list of suggestions

given by the teacher. The suggested topics should be linked to the learning outcomes covered by

the class teacher during that year. Guidelines on how the projects should be conducted, as well

as parameters regarding time frames, length of project, amount of work carried out in class and

at home, and whether the project can be carried out individually or in groups, should be provided

by the teacher.

Characteristics of Research Projects to be used in the coursework

● When compiling a list of suggested projects teachers should take into consideration the pupils’

interests and hobbies such as sports, music, drama, art, travel, games, etc. This will ensure

learner ownership. Ideally, the suggested titles should be the result of a discussion between the

teacher and the pupils.

● The amount of work conducted by pupils on a project should be between 10 to 15 hours. Of

these, between 3 to 5 hours should be work conducted in class. Ideally these should be spread

out over the duration of the project which should not exceed two months. Thus, the teacher

should dedicate an adequate number of lessons during this period to let the students work on

their research project in class. The role of the teacher should be to offer guidance and monitor

the pupils’ progress.

● Research Projects may be partly carried out in groups of two to four students, thus fostering

healthy collaboration. However, each pupil should present a separate report containing his/her

own individual and distinctive contribution. For example, when carrying out research, pupils may

gather the data together as a group, but then each pupil should analyse the data on his/her own

and present a different individual report.


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The Process of doing the Research Project:

● Interpreting the Task: interpretation of the task by analysing what information is available and

what data can be collected and from where.

● Line of enquiry: narrowing down the research area to focus on one or more particular aspects

that connect to the students’ interests and abilities.

● Planning: choosing an appropriate strategy to carry out the project including the material and

equipment needed, a plan to carry out the project, and a timeline to complete it.

● Recording: finding a meaningful form of recording and organising the information and data

gathered.

● Reporting: drawing a conclusion and finalizing the end product. This should include a written

report.

Guidelines for each section of the report

Section Details

Title

This section should include:

● Name of pupil.

● The title of the Research Project.

● The date when the Research Project was given.

● Date of handing in of the Research Project.

Abstract & Introduction

This section should be brief and should indicate clearly what the project is all about. Where applicable, it should include:

● A short abstract (2-3 sentences) about what the Research Project is about.

● A list of the major objectives of the project.

● The issues that are relevant to the project.

● The set of issues that the pupils will be focusing on and the rationale why they are deciding to focus on those particular issues which could include reasons why they relate to their interests.

● A clear statement of the mathematical topics involved.

Gathering of Data and Information

In this section, the pupil should give a brief outline of the methodology used to gather the information. Where applicable, it should include:

● A description of the methods and intermediate steps used to obtain the information.

● A list of the resources used to gather data and information.

● A list of the sources and references used.

Discussion and Development

This is the main part of the report. In it the pupil should present his research based on the data and information gathered. It provides clear answers to all issues identified in the introduction. It includes all the appropriate diagrams, figures, equations, graphs, and illustrations as needed. It addresses any other interesting observations noticed while working through the project.

Where applicable, this section should include the following: ● All observations, measurements, data and/or information presented

in an organised form. ● Any calculations made including all the necessary intermediate

steps. ● Graphical and/or tabular representations used to display data. ● Any interesting connections, patterns, generalizations etc. noted. ● Clear references to appropriate mathematical ideas.


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Conclusion

In this concluding part of the report the pupils should summarise their work and give indications of possible extensions to their work.

This section should include: ● A summary of the most important findings. ● If applicable, the identification of interesting aspects related to the

issues considered which could be the scope of further research.

Non-exhaustive suggested ideas for a Research Project

1. “Adolescents are spending too much time playing computer games.” Conduct research to check whether this is true or false.

2. Who was Pythagoras? Conduct research about his famous theorem and its proof. 3. Draw a scale drawing of your house. Make separate drawings of each floor of the house.

Make an enlarged copy of your room to show the exact position and dimensions of the furniture of the room.

4. Make a budget plan for a 7-day holiday abroad for a family or group of persons. Include all expenses related to travel, accommodation, food, excursions and other activities.

5. What is a tessellation? Give examples of various tessellations. Research the use of tessellations in artistic representations and/or other real world situations.

6. Conduct research on the Roman number system. Compare and contrast the Roman number system and the Hindu-Arabic number system we are using now. Include examples of the use of the Roman number system in the local context.

7. How long does it take your classmates to travel from home to school and from school to home? Conduct research to compare and contrast the different travelling times.

8. Research how is Music related to Mathematics. 9. Find about old Maltese units of length, area, capacity and mass that were used in everyday

life. 10. Conduct research on three of the following devices for mathematical calculation:

o Abacus o Napier’s bones o Nomograms o The Slide rule o Logarithms and Four figure tables o Mechanical calculators


Page 113 of 196

Marking Criteria – Research Project

The assessment mark is based solely on the report presented by the student. In assigning a final mark out of 80, the following

Marking Criteria are to be used, considering each of the four main criteria.

MARKING CRITERIA – Research Project

Maximum 40 marks*


Abstract & Introduction (4 marks)

(0 – 1 marks) (2 – 3 marks) (4 marks)

The abstract is poor and does not summarise clearly what the project is about.

The abstract is simple and partly summarises what the project is about.

The abstract is good and clearly summarises what the project is about.

The list of objectives is poor and incomplete.

The list of objectives is adequate and incomplete.

The list of objectives is adequate and complete.

None or very few questions/queries are identified. None of these issues are relevant.

A few questions/queries are identified. Most of these issues are relevant.

A good number of questions/queries are identified. All of these issues are relevant.

No or poor indication of the issues being focused on with no reasons given.

Limited indication of the issues being focused on with adequate reasons given.

A clear indication of the issues being focused on with adequate reasons given.

No statement of the mathematical topics involved.

An incomplete statement of the mathematical topics involved.

A complete statement of the mathematical topics involved.

Gathering of Information and Data (8 marks)


Report exhibits little or no ability at making a general plan and of selecting appropriate methods that allow poor handling of the task.

Report exhibits some ability at making a general plan and of selecting appropriate methods that allow adequate but unexceptional handling of the task.

Report exhibits a clear ability at making a general plan and of selecting appropriate methods that allow exceptional handling of the task.

There is no evidence that required data or information has been appropriately researched.

There is some evidence that required data or information has been appropriately researched.

There is considerable evidence that required data or information has been appropriately researched.

No references are included. References are included in some cases. References are included in all or nearly all cases where they are required.

Discussion and Development (20 marks)


Observations, measurements, data and/or information is incomplete or disorganised.

Observations, measurements, data and/or information is partially complete and organised.

Observations, measurements, data and/or information is complete and very well organised.

Calculations are either not present and/or do not include any necessary intermediate steps and/or are inaccurate.

Calculations are present but a number of necessary intermediate steps are missing and/or are partly accurate.

Calculations are present, accurate and include all necessary intermediate steps.

Use of mathematical language is poor and incorrect.

Use of mathematical language is adequate and partly correct.

Use of mathematical language is good and mostly correct.

Graphical and/or tabular representations are either not present or incorrect.

Mathematical representations using a pictorial, tabular or graphical methods are present but are partially correct.

Clear and correct mathematical representations using a mixture of pictorial, tabular, graphical and algebraic methods.

No connections, patterns and generalization are obtained and mathematical ideas are basic.

Sufficient connections, patterns and generalization are obtained and some of the mathematical ideas are appropriate.

A substantial amount of connections, patterns and generalizations are obtained and the mathematical ideas are appropriate.

No reference to appropriate mathematical ideas.

Some reference to appropriate mathematical ideas.

Substantial reference to appropriate mathematical ideas.


Page 114 of 196

Only some or none of the work is presented in a logical, clear, concise, and appealing fashion.

About half of all of the work is presented in a logical, clear, concise, and appealing fashion.


Conclusion (8 marks)


No summary of the main points. A sufficient summary including some of the main points.

A proper summary including all or nearly all of the main points.

The conclusions reached are not stated and/or are inaccurate. There is little or no evidence of what has been learnt from these conclusions.

The conclusions reached are clearly stated in some of the cases. There is some evidence of what has been learnt from these conclusions.

The conclusions reached are clearly stated in all or nearly all of the cases. There is clear evidence of what has been learnt from these conclusions.

Ideas for further research lack clarity and relevance.

Ideas for further research are partly clear and/or relevant.

Ideas for further research are clear and relevant.


30% 30% 40%


Page 115 of 196

Coursework Mode 6: Non-Calculator Test

Non-Calculator Test

20 marks

internally-assessed


The Non-Calculator Test

a) The Non-Calculator Test consists of 20 questions of 1 mark each, which have to be answered

in 20 minutes. It can be pegged at either of two categories:

A Test at MQF level categories 1-2 must include:

o 8 questions based on assessment criteria from MQF 1 and;

o 12 questions based on assessment criteria from MQF 2.

A Test at MQF level categories 1-2-3 must include questions include:

o 6 questions based on assessment criteria from MQF 1;

o 6 questions based on assessment criteria from MQF 2 and:

o 8 questions based on assessment criteria from MQF 3.

b) The Non-Calculator Test should cover at least 5 Learning Outcomes.

c) Calculator use is not allowed during the Test.

d) The questions in a Non-Calculator Test:

o should involve numerical and/or algebraic computations that can either be done quickly

by inspection, mentally or using paper and pencil methods;

o should be such that calculator use might facilitate the working of the questions set;

o are all compulsory.

e) The Test should be marked out of 20. Answers are to be written on the examination paper

provided and sufficient space for rough work should be provided in the Test paper.

f) No partial marks should be allotted or awarded when marking.


Page 116 of 196

Non-Calculator

Test exemplar

(Level 1 – 2)

Attempt ALL questions Time: 20 minutes

Write your answers in the space available on the test paper.

The use of calculators is not allowed.

This paper carries a total of 20 marks.

QUESTIONS AND ANSWERS

ALL QUESTIONS CARRY ONE MARK

SPACE FOR ROUGH WORK

(IF NECESSARY)

1.

Write 2

5 as a decimal.

Ans _______________

2.

Write down all the factors of 76.

Ans _______________

3.

Work out the area of the triangle below.

Ans _______________

4.

Write in order, largest first:

3

5

2

9

2

3

Ans _______________

5.

Workout the value of the angle marked 𝑥.

Ans _______________


Page 117 of 196

6.

What is the cube root of 64?

Ans____________

7.

The angles in this shape are all right angles. Calculate

the area of the shape.

Ans _______________

8.

What is the fifth term of the following sequence?

100, 81, 64, 49, ____

Ans _______________

9.

Calculate the value of = 10 − 4𝑥 when 𝑥 = −1

2.

Ans _______________

10.

A car was bought for €5800 last year. This year it costs

5% less. What is the cost of the car this year?

Ans_______________

11.

The radius of a circular roundabout is 5 m.

What is the circumference of the roundabout correct to

the nearest metre?

Ans _______________

12.

What is the mean of the following set of numbers?

13.7, 15.8, 12.3 and 14.2

Ans _______________

20 cm

8 cm

2.5 cm

12 cm


Page 118 of 196

13.

Which of the following shows all the prime numbers

between 40 and 50?

(A) 41, 43, 45

(B) 41 and 47

(C) 43, 47 and 49

(D) 41, 43 and 47

Ans _______________

14.

Work out: 31

4− 2

2

3

Ans _______________

15.

€600 is shared between George and Joe in the ratio 2 : 3.

How much money does Joe receive?

Ans _______________

16.

Work out the value of z.

Ans _______________

17.

The equation of a straight line is 𝑦 = 4𝑥 − 5.

One point on the line has coordinates (1.5, 𝑑). What is the value of 𝑑?

Ans _______________

18.

A number is chosen at random from the set:

{7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}.

What is the probability of getting a common factor of 2

and 3?

Ans _______________


Page 119 of 196

19.

Workout the volume of a reservoir in the form of a

cuboid with length 5 m, width 3 m and height 4.5 m

Ans _______________

20.

A point P (3, -2) is translated onto point Q (-1, 5) by

the translation vector (𝑎𝑏

).

Write down the values of 𝑎 and 𝑏.

Ans 𝑎 = ____ ; 𝑏 = _____


Page 120 of 196

Non-Calculator

Test exemplar

(Level 1 – 2 – 3)

Attempt ALL questions Time: 20 minutes

Write your answers in the space available on the test paper.

The use of calculators is not allowed.


QUESTIONS AND ANSWERS

ALL QUESTIONS CARRY ONE MARK

SPACE FOR ROUGH WORK

(IF NECESSARY)

1.

Write 0.72 as a fraction in the lowest terms.

Ans _______________

2.

What is the fifth term of the following sequence?

100, 98, 94, 88, ____

Ans _______________

3.

A regular pentagon has sides of length 11.9 cm. How

long is its perimeter?

Ans _______________

4.

Write in order, smallest first:

1

3

2

5

2

7

Ans _______________

5.

Write down all the prime numbers which lie between

25 and 35.

Ans _______________

6.

Kevin goes to sleep at 22:45 on Saturday. At what

time does he wake up on Sunday if he sleeps exactly

7 hours and 25 minutes?

Ans _______________

7.

Give a reason why it is impossible to construct a

triangle with sides 3.7 cm, 4.8 cm and 8.9 cm.

Ans _____________________________


Page 121 of 196

8.

How many circular rings of radius 14 mm can be

made from a thin wire 88 cm long?

Ans _______________

9.

Find the value of 3𝑥−2

12−𝑥 when 𝑥 = −2.

Ans _______________

10.

A point P (3, -2) is translated onto point Q (-1, 5) by

translation vector (𝑎𝑏

).

Write down the values of 𝑎 and 𝑏.

Ans 𝑎 = ____ ; 𝑏 = _____

11.

A line cuts the y-axis at (0, 2) and has gradient 6.

One point on the line has coordinates (5, d).

What is the value of d?

Ans _______________

12.

The mean of 4 numbers is 20. Three of these

numbers are 18, 23 and 24. What is the fourth

number.

Ans _______________

13.

Given that 𝑎 = 2.8 × 10−1 and 𝑏 = 2 × 10−3, find the

value of 𝑎

𝑏 in standard form.

Ans _______________

14.

Work out the size of the interior angle of a regular

hexagon.

Ans _______________


Page 122 of 196

15.

A motorcycle is moving at a speed of 10 km/hr.

How long does it take in minutes for the motorcycle

to cover 1.5 km?

Ans _______________

16.

Find the value of: 13.6 × 13.7 + 13.7 × 13.4

13.7

Ans _______________

17.

Find the area of rectangle ABCD, given that the

diagonal is 5.5 m long and side CD is 4.4 m long.

Ans _______________

18.

Two unbiased dice are tossed. What is the probability

of getting a 6 on both dice?

Ans _______________

19.

What is the highest common factor (HCF) of 24, 40

and 96?

Ans _______________

20.

A vase is priced €99 during a sale when a 10%

discount is offered.

Find the original price before the sale.

Ans _______________

A B

C D


Page 123 of 196

Specimen Assessments

Specimen Assessments: Controlled Paper MQF 1-2

MATRICULATION AND SECONDARY EDUCATION CERTIFICATE

EXAMINATIONS BOARD

SECONDARY EDUCATION CERTIFICATE LEVEL

SAMPLE CONTROLLED PAPER

SUBJECT: Mathematics

PAPER NUMBER: Level 1 – 2

DATE:

TIME: 2 hours

Answer ALL questions.

Write your answers in the space available on the examination paper.

Show clearly all the necessary steps, explanations and construction lines in your working.

Unless otherwise stated, diagrams are drawn to scale.

The use of non-programmable electronic calculators with statistical functions and mathematical

instruments is allowed.




Page 124 of 196

1. Liam, Sandra and Jillian find the following recipe for 12 scones on the internet:

400 g flour 170 g butter

70 g sugar 1 egg

2 teaspoons baking powder 250 ml milk

½ teaspoon salt

a) Underline the best estimate for the total weight of all the ingredients?

893 ½ g 1 kg 1500 g 2.5 kg

(3)

b) Liam uses 1 litre of milk to make a number of scones. How much sugar does he

use?

(2)

c) Sandra makes 24 scones. How much flour does she use?

(2)

d) Jillian uses 105 g of sugar. How many scones does she make?

(2)

(Total: 9 marks)

2. a) Enter the missing values to complete the following sequence:

7, 7.25, _____, 7.75, _____, _____, 8.5.

(3)

b) i. Write the number seven hundred and fifty-four thousand and fifty-eight in

figures.

ii. Calculate the difference between the two values of the digit “5” in the number

seven hundred and fifty-four thousand and fifty-eight.

(4)

c) Use < , > or = to compare the following pairs:

(3)

i. 1

4

1

3 ii.

3

4 0.75 iii. 0.5 0.4

d) Write a number that lies between 0.5 and 0.75.

(1)


Page 125 of 196

(Total: 11 marks)

3. Work out the size of the angles marked x, y and z.

(6)

(Total: 6 marks)

4. a) Complete the shape to make it symmetrical about the dotted line.

(2)

b) i. Reflect shape A in the horizontal dotted line. Label the image B.

ii. Now reflect A and B in the vertical dotted line.

(3)

c) i. Move point A: 5 right and 2 up. Label it A1.

ii. Move point B: 9 left and 3 down. Label it B2.

(2) A

B

z

x

79° y

66°

125°

Diagram not to scale

A


Page 126 of 196

(Total: 7 marks)

5. a) Sort the following numbers in the Carroll diagram:

1, 4, 7, 8, 12, 16, 27, 36, 64

Square

numbers

Not square

numbers

Cube

numbers

Not cube

numbers

(3)

b) From the following set of numbers,

2, 5, 9, 13, 17, 19, 26, 30:

i. choose THREE numbers that add up to 60.

ii. calculate the mean of all the odd numbers.

(1)

(3)

(Total: 7 marks)

6. a) Martin and Claire are playing a game. They roll two fair dice, each numbered 1 to

6 and the score is the product of the two numbers.

Show all the possible outcomes in the possibility space below:

1st Dice

1 2 3 4 5 6

2nd

Dice

1 1 3

2

3 9

4 24

5

6 12

(2)

b) What is the probability that the score is:

i. 15?

ii. a factor of 80?

iii. a multiple of 3?

(1)

(2)


Page 127 of 196

(2)

c) Which TWO scores have the greatest probability?

(1)

d) List all the different possible scores.

(2)

e) Which score has a probability of

1

12?

(2)

(Total: 12 marks)

7. The side AC of a scalene triangle ABC is x cm long.

Side BC is 3 cm longer than side AC.

(1)

a) Write an expression for the length of side BC in terms of x.

b) Side AB is 4 cm shorter than side AC. Write an expression for the length of side

AB in terms of x.

(1)

c) The perimeter of the triangle is 26 cm. Write an equation in terms of x and solve

it.

(3)

d) Write down the lengths, of the three sides of the triangle ABC.

AC = ______ cm AB = _______ cm BC = _____ cm

(2)

A

B C


Page 128 of 196

(Total: 7 marks)

8. 271 school children and 11 teachers are going on a school trip. Coaches that carry 51

passengers cost €80 and minivans that carry 14 passengers cost €30. For the school trip, the

school cannot spend more than €475 for transport.

a) i. If only coaches are used, how many coaches will be completely full?

ii. How many passengers will remain?

(2)

(2)

b) Show that the school cannot afford to hire another coach.

(2)

c) i. If the school hires minivans to carry those who remain, how many minivans

are needed?

(2)

ii. Calculate the total cost of the transport.

(2)

(Total: 10 marks)

9. Chantelle went on holiday in Sweden and Norway. Before departing from Malta, she bought

6000 Swedish Krona (SEK) and 5000 Norwegian Krone (NOK). The exchange rates, at the

time, were 10.4769 SEK and 9.6122 NOK, both for one Euro.

a) Find the value of 1 NOK in Euro cent, to the nearest cent.

(2)

b) Calculate the total amount, to the nearest Euro, that Chantelle paid when buying

the foreign currency notes.

(3)

c) When she was in Sweden, Chantelle bought a dress that cost €329.00. What is the

price that Chantelle paid in SEK?

(2)


Page 129 of 196

(Total: 7 marks)

10. The compound shape in the diagram is made up of a semicircle, a trapezium and a rectangle.

a) What is the radius of the semicircle?

(1)

b) Calculate the area of the semicircle. (Use 𝜋 = 22

7 )

(2)

c) Calculate the area of the trapezium.

(2)

d) Calculate the total area of the compound shape.

(2)

12 cm

7 cm

3 cm

2.4

cm

Diagram not to scale


Page 130 of 196

(Total: 7 marks)

11. a) Complete the table for the graph of the equation 𝑦 = 2𝑥 − 1.

x -1 0 1 2 3

y

(2)

b) Draw the graph of 𝑦 = 2𝑥 − 1

(2)

c) Use your graph to find the value of:

i. y when x = 2.6 ii. x when y = -4

(1,

1)

d) Draw a second straight line graph that passes through the two points (2, -4) and

(-1, 5).

(1)

e) Write down the y-intercept of this graph.

(1)

f) Write down the coordinates of the point of intersection of the two graphs.

(1)

-2 -1 O 1 2 3

-4

-2

2

4

6

x

y

0


Page 131 of 196

(Total: 9 marks)

12. a) In this question, use ruler and compasses only.

i. Construct a circle of radius 4 cm.

ii. Construct a regular hexagon with its vertices on the circumference of the

circle. Label the vertices of the hexagon A, B, C, D, E and F.

iii. Draw and measure the diameter AD and the chord CE.

(1)

(2)

(2)

b) Take accurate measurements to calculate the area of the hexagon.

(3)

(Total: 8 marks)

END OF PAPER


Page 132 of 196

Specimen Assessments: Marking Scheme for Controlled Paper MQF 1-2


EXAMINATIONS BOARD


MARKING SCHEME FOR SAMPLE CONTROLLED PAPER



DATE:

TIME: 2 hours

Question

No. Workings and Solutions Additional guidance Marks

1. a) Approximate weight of milk: 250 g

Approximate weight of egg: 60 g

Approximate weight of baking powder and salt: 10 g

Total weight = 400 + 70 + 170 + 250 + 60 + 10 =

960 g ≈ 1 kg

For approximations

Converting g to kg

M1

M1 A1

b) (1000 ÷ 250) × 70 = 280 g M1 A1

c) 24 scones mean double ingredients

Double flour = 800 g

M1

A1

d) 12×105

70 = 18 scones

M1

A1

(Total: 9 marks)

2. a) 7.5 ; 8 ; 8.25 B3

b) a) 754,058 or 754058 B1

b) 50,000 – 50

= 49,950

Correct 50,000 & 50 B1 B1

B1

c) i.

1

4<

1

3

ii. 3

4= 0.75

iii. 0.5 > 0.4

B1

B1

B1

d) Any number between 0.5 and 0.75 B1

(Total:11 marks)

3. 𝑥 = 180° − 79° − 66° = 35°

𝑥 = 180° − 79° = 101°

𝑧 = 360° − 66° − 125° = 169°

M1 A1

M1 A1

M1 A1

(Total: 6 marks)


Page 133 of 196

4. a)

1 mark for each two

correct squares

B2

b) i.

ii.

1 mark for each

correct reflection

B1

B2

c) i.

ii.

B1

B1

(Total: 7 marks)

5. a)

Square

numbers

Not square

numbers

Cube

numbers

1

64

8

27

Not cube

numbers

4

16

36

7

12

(-1 e.e.o.o.)

B3

b) i. 13, 17 and 30

ii. (5 + 9 + 13 + 17 + 19) ÷ 5 = 12.6

Correct odd numbers

For adding and

dividing

B1

M1

M1

A1

(Total: 7 marks)

6. a)

1st Dice

1 2 3 4 5 6

2nd

Dice

1 1 2 3 4 5 6

2 2 4 6 8 10 12

3 3 6 9 12 15 18

4 4 8 12 16 20 24

5 5 10 15 20 25 30

6 6 12 18 24 30 36

deduct 1 mark for

every three incorrect

entries

B2

A

B1

B

A1

A

B


Page 134 of 196

b) i. 1

18

ii. factors of 80: 1, 2, 4, 5, 2, 4, 8, 10, 4, 8, 16,

20, 5, 10 and 20.

= 15

36=

5

12

iii. multiples of 3: 3, 6, 6, 12, 3, 6, 9, 12, 15, 18,

12, 24, 15, 30, 6, 12, 18, 24, 30 and 36.

= 20

36=

5

9

Or equivalent

For identifying

factors of 80

Or equivalent

For identifying

multiples of 3

Or equivalent

B1

M1

A1

M1

A1

c) 6 and 12 B1

d) Different scores: 1, 2, 3, 4, 5, 6, 8,10,12,9, 15, 18,

16, 20, 24, 25, 30 and 36.

18

For identifying the

scores

M1

A1

e) 1

12=

3

36 and there are three 4’s.

The score is 4

M1

A1

(Total: 12 marks)

7. a) 𝑥 + 3 B1

b) 𝑥 − 4 B1

c) 𝑥 + 𝑥 + 3 + 𝑥 − 4 = 26 3𝑥 = 27 𝑥 = 9

or equivalent B1

M1

A1

d) 9 cm ; 5 cm ; 12 cm B2

(Total: 7 marks)

8. a) i. 282 ÷ 51 = 5.52…

5 Coaches

ii. 282 ÷ 51 = 5 remainder 27

27 passengers

M1

A1

M1

A1

b) 80 × 6 = 480

€480 is more than €475

M1

A1

c) i. 27 ÷ 14 = 1.92…

2 minivans

M1

A1

ii. 80 × 5 + 30 × 2

= €460

M1

A1

(Total: 10 marks)

9. a) 1 ÷ 9.6122 = €0.104…

= 10 cent

M1

A1

b) (6000 ÷ 10.4769) + (5000 ÷ 9.6122)

= 1092.8607…

= €1093

M1

M1

A1

c) 329.00 × 10.4769

= 3446.90 SEK

M1

A1

(Total: 7 marks)


Page 135 of 196

10. a) 3.5 cm or 3 ½ cm B1

b) A = 𝜋𝑟2

2

A = 22

7× (3.5)2 ÷ 2

= 19.25 cm2

M1

A1

c) 𝐴 =

ℎ(𝑎 + 𝑏)

2

𝐴 =3(12 + 7)

2

= 28.5 cm2

M1

A1

d) Area of rectangle = 28.8

Total area = 28.8 + 19.25 + 28.5

= 76.55 cm2

M1

A1

(Total: 7 marks)

11. a)

x -1 0 1 2 3

y -3 -1 1 3 5

(-1 e.e.o.o.)

B2

b)

Correct plotting of

points

Correct line

B1

B1

c) i. 4.2

ii. -1.5

B1

B1

d)

B1

e) 2 B1

f) (0.6, 0.2) 1 mark for the x-coordinate and 1 mark for the y-coordinate

B1

(Total: 9 marks)


Page 136 of 196

12. a) i.

ii.

iii. AD = 8 cm; CE = 6.9 cm

Circle of radius 4 cm

Correct construction

of arcs and hexagon

CE 0.2 cm

B1

B2

B1 B1

b) Perpendicular distance between two parallel sides of

trapezium = 3.45 cm Area of hexagon = 2 × 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡𝑟𝑎𝑝𝑒𝑧𝑖𝑢𝑚

= 2 ×1

2× 3.45 × (4 + 8) = 41.4 cm2

M1

M1

A1(ft)

(Total: 8 marks)


Page 137 of 196

Specimen Assessments: Controlled Paper MQF 2-3


EXAMINATIONS BOARD





DATE:

TIME: 2 hours










Page 138 of 196

1. a)

b)

i. What is the least common multiple of 4, 6 and 8.

ii. Use equivalent fractions to put 5

8 ,

5

6 and

3

4 in ascending order.

The planet Saturn has a mass of 5.7 × 1026 kg and the planet Jupiter has a mass of

1.9 × 1027 kg.

i. How much is Jupiter heavier than Saturn? Write your answer in standard form.

ii. The mass of Saturn is 𝑎

𝑏 the mass of Jupiter. Write down the fraction

𝑎

𝑏 where a

and b are whole numbers.

(2)

(2)

(2)

(2)

(Total: 8 marks)

2. The side AC of a scalene triangle ABC is x cm long.

Side BC is 3 cm longer than side AC.

a) Write an expression for the length of side BC in terms of x.

(1)

b) Side AB is 4 cm shorter than side AC. Write an expression for the length of side AB

in terms of x.

(1)

c) The perimeter of the triangle is 26 cm. Write an equation in terms of x and solve it.

(3)

A

B C


Page 139 of 196

d) Write down the lengths, in cm, of each of the three sides of the triangle ABC.

AC = AB = BC =

(2)

(Total: 7 marks)

3. 271 school children and 11 teachers are going on a school trip. Coaches that carry 51 passengers

cost €80 and minivans that carry 14 passengers cost €30. For the school trip, the school cannot

spend more than €475 for transport.

a) i. If only coaches are used, how many coaches will be completely full?

ii. How many passengers will remain?

(2)

(2)

b) Show that the school cannot afford to hire another coach.

(2)

c) i. If the school hires minivans to carry those who remain, how many minivans are

needed?

(2)

ii. Calculate the total cost of the transport.

(2)

(Total: 10 marks)

4. Chantelle went on holiday in Sweden and Norway. Before departing from Malta, she bought

6000 Swedish Krona (SEK) and 5000 Norwegian Krone (NOK). The exchange rates, at the time,

were 10.4769 SEK and 9.6122 NOK, both for one Euro.

a) Find the value of 1 NOK in Euro cent, to the nearest cent.

(2)

b) Calculate the total amount, to the nearest Euro, that Chantelle paid when buying the

foreign currency notes.

(3)


Page 140 of 196

c) While in Sweden, Chantelle lost her mobile phone, which she had bought in Malta

for €350.00. She bought another one, of the same model, for 5000 SEK. Calculate

the percentage difference, in the price of the new mobile phone, over its cost in

Malta. Give your answer correct to 2 decimal places.

(4)

(Total: 9 marks)

5. Bernard would like to visit friends in Canada and looks for possible flights offered by different

airlines. The following are the times of flights for each itinerary, local time. (Local time means

the time of the city from which the plane is leaving or at which it is arriving). Toronto is 6 hours

and London is 1 hour, behind Central European Time. Malta, Munich and Vienna are in the Central

European Time zone.

Departure Time Arrival Time Departure Time Arrival Time

A Malta 16:50 Munich 19:20 Munich 11:50 Toronto 14:40

B Malta 07:20 Vienna 09:35 Vienna 10:30 Toronto 13:55

C Malta 07:15 London 10:20 London ? Toronto 15:05

a) Calculate the departure time from London.

(2)

b) The flight from London to Toronto takes 7 hours 10 minutes. How long is the waiting

time at London airport?

(1)

c) Work out the total time that elapses, between leaving Malta and arriving in Toronto,

for itineraries A and B. Give your answer in hours and minutes.

A:

B:

(5)


Page 141 of 196

d) What is the difference in total duration of the journey between the itinerary offered

by airline B and that offered by airline C? Give your answer in hours and minutes.

(2)

(Total: 10 marks)

6. The stacked bar chart below shows the number of different staff employed with a firm in the

years 2016, 2017 and 2018.

a) What fraction of the total staff in 2016 consisted of Clerical staff?

(2)

b) Calculate the increase in Manual staff between 2017 and 2018.

(1)


Page 142 of 196

c) A report in the company magazine stated that, for each clerk, there were more

manual and executive workers in 2017, than there were in 2018.

Show that this report is correct.

(3)

(Total: 6 marks)

7. EDOC is a straight line. DC is a diameter to the circle centre O. The line ABC is the hypotenuse

of the right angled triangle AEC.

a) What is the size of angle DBC? Give a reason for your answer.

(1)

b) Show that ABDE is a cyclic quadrilateral.

(3)

c) Angle ACE = 30°. Find the size of angle BOD. Give a reason for your answer.

(2)

(Total: 6 marks)

A

B

C O D E


Page 143 of 196

8.

a) Use ruler and compasses only to construct a triangle ABC such that

BC = 10 cm, angle ABC = 45° and angle ACB = 60°.

(4)

b) Construct the perpendicular bisectors of lines AB and BC.

Label the point O where these two perpendicular bisectors meet. (2)

c) Draw the locus of the point which is 5.2 cm from O. (2)

(Total: 8 marks)

9.

a) Mark a point C to complete the rectangle OABC. Draw the rectangle OABC. (1)

b) Draw and label the reflection of OABC in the line x = 4, to form rectangle O1A B1C1. (2)

c) Rotate rectangle OABC 90 anticlockwise about the origin to form OA2B2C2. Draw and

label rectangle OA2B2C2. (4)

(Total: 7 marks)


Page 144 of 196

10. The following distance - time graph shows the journey of a man who went for a walk.

a) How far did he travel altogether?

(2)

b) Calculate his average speed in km/h.

(3)

c) For how long did he stop?

(1)

d) What does part E represent?

(1)

e) In which stage did he travel fastest? Give a reason for your answer.

(2)

(Total: 9 marks)

11. John has a sum of €235 that is made up of 5 euro notes and 20 euro notes. He has twelve more

5 euro notes than 20 euro notes.

a) Let x be the number of 5 euro notes.

Let y be the number of 20 euro notes.

Write down TWO equations in terms of x and y to represent the above information.

(2)

b) Find the number of 5 euro notes and the number of 20 euro notes that John has.

(6)

(Total: 8 marks)


Page 145 of 196

END OF PAPER

12. Alex conducted a survey on the number of people travelling in cars as they passed through

a road.

Alex’s Morning Survey

Number of people

in a car

Frequency

1 144

2 70

3 16

4 10

a)

How many cars were surveyed?

(1)

b) How many people passed through the road while travelling in these cars?

(2)

c) What is the mean number of people per car?

(2)

d) What is the probability that a car that passes through the road has 3 or more people

travelling in it?

(2)

e) After doing some calculations Alex stated:

“It is equally likely for cars to have less than 3 people travelling in them as for cars

to have 3 or more people travelling in them”.

i. Is Alex correct?

ii. Explain your reasoning.

(1)

(4)

(Total: 12 marks)


Page 146 of 196

Specimen Assessments: Marking Scheme for Controlled Paper MQF 2-3


EXAMINATIONS BOARD





DATE:

TIME: 2 hours

Question

No. Workings and Solutions Additional guidance Marks

1. a) i. Multiples of 4: 4, 8, 12, 16, 20, 24

Multiples of 6: 6, 12, 18, 24

Multiples of 8: 8, 16, 24

LCM = 24

M1

A1

ii. 5

8=

15

24 ;

5

6=

20

24 ;

3

4=

18

24

∴ 5

8 ,

3

4 ,

5

6

Equivalent fractions

with a same common

denominator

Correct order

M1

A1

b) i. 1.9 × 1027 ̶ 5.7 × 1026 = 1.33 × 1027 M1 A1

ii. 5.7 × 1026

1.9 × 1027=

3

10

M1 A1

(Total: 8 marks)

2. a) 𝑥 + 3 B1

b) 𝑥 − 4 B1

c) 𝑥 + 𝑥 + 3 + 𝑥 − 4 = 26 3𝑥 = 27 𝑥 = 9

or equivalent B1

M1

A1

d) 9 cm; 5 cm; 12 cm B2

(Total: 7 marks)

3. a) i. 282 ÷ 51 = 5.52…

5 coaches

ii. 282 ÷ 51 = 5 remainder 27

27 passengers

M1

A1

M1

A1

b) 80 × 6 = 480

€480 is more than €475

M1

A1

c) i. 27 ÷ 14 = 1.92…

2 minivans

M1

A1

ii. (80 × 5) + (30 × 2)

= €460

M1

A1

(Total: 10 marks)


Page 147 of 196

4. a) 1 ÷ 9.6122 = €0.104…

= 10 cents

M1

A1

b) (6000 ÷ 10.4769) + (5000 ÷ 9.6122)

= 1092.8607…

= €1093

M1

M1

A1

c) 5000 SEK = 5000 ÷ 10.4769

= €477.2404…

% Difference = 477.2404… − 350

350 𝑥 100

= 36.35%

M1

A1

M1

A1 (f.t.)

(Total: 9 marks)

5. a) Arrival in Toronto (London Time):

15:05 + 5 hrs = 20:05

Departure from London:

20:05 – 7:10 = 12:55

M1

A1

b) Waiting time in London:

12:55 – 10:20 = 2 hours 35 minutes

A1

c) Airline A:

Arrival in Toronto (Central European time) is

14:40 + 6 hrs = 20:40

Time between 16:50 and 20:40 (next day)

= 7 hrs 10 min + 20 hrs 40 min

= 27 hrs 50 min

Airline B:

Arrival in Toronto (Central European time) is

13:55 + 6 hrs = 19:55

Time between 07:20 and 19:55

= 12 hours 35 minutes

M1

M1

A1

M1

A1

d) Airline C:

Arrival time in Toronto (Central European

time): 15:05 + 6 hrs = 21:05

Total time elapsed:

21:05 – 07:15 = 13 hrs 50 min

Difference:

13 hrs 50 min – 12 hrs 35 min

= 1 hour 15 minutes

M1

A1

(Total: 10 marks)

6. a) 20 clerks out of 74

10

37

M1

A1

b) 112 – 90 = 22 B1

c) 2017: 96 ÷ 30 = 3.2

2018: 122 ÷ 44 = 2.77

3.2 > 2.77

M1

M1

A1

(Total: 6 marks)


Page 148 of 196

7. a) 90° Angle in a semicircle B1

b) ABD = 180° - DBC = 180° - 90° = 90°

(angles on a straight line)

AED + ABD = 90 ° + 90° = 180°

Since opposite angles are supplementary,

ABDE is a cyclic Quadrilateral

M1

M1

M1

c) BOD = 30° × 2 = 60°

Angle at the centre is twice the angle on the

circumference.

B1

B1

(Total: 6 marks)

8. a)

b)

c)

correct line BC

correct B (45°)

correct C (60°)

correct perpendicular

bisectors

correct locus: Circle

correct size and

position of circle

B1

B2

B1

B1 B1

B1

B1

(total: 8 marks)

9. a)

b)

c)

Correct rectangle

OABC

Correct reflection

Correct labels

Correct rotation

Correct labels

B1

B1

B1

B2

B2

(Total: 7 marks)

10. a) 4.5 × 2 = 9 km M1 A1

b) Total time = 4.5 hrs

Av. Speed = 9.0 ÷ 4.5 = 2 km/h

M1

M1 A1

c) 30 minutes B1

d) Journey back to starting position B1

e) E

Gradient is steepest

(Accept working out

all speeds)

B1

B1

(Total: 9 marks)

C

A1

C1

O1

B1

O2

A2

B2

C2


Page 149 of 196

11. a) 5𝑥 + 20𝑦 = 235 𝑥 − 𝑦 = 12

Or any other

equivalent equations

B1

B1

b) 5𝑥 + 20𝑦 = 235 eq (1)

5𝑥 − 5𝑦 = 60 eq (2)

eq (1) – eq (2) gives 25𝑦 = 175

𝑦 =175

25= 7

𝑥 − 7 = 12 𝑥 = 19

Multiplying by 5

Subtracting

Substituting

M1

M1

M1 A1

M1

A1

(Total: 8 marks)

12. a) 144 + 70 + 16 + 10 = 240 B1

b) (144 × 1) + (70 × 2) + (16 × 3) + (10 × 4)= 372

M1

A1

c) 372 ÷ 240 = 1.55 M1 A1(ft)

d) 26

240 =

13

120

Or equivalent M1 A1

e) 𝑃(3 𝑜𝑟 𝑚𝑜𝑟𝑒 𝑝𝑎𝑠𝑠𝑒𝑛𝑔𝑒𝑟𝑠) =

13

120= 0.1083̇

𝐿𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 3 𝑝𝑎𝑠𝑠𝑒𝑛𝑔𝑒𝑟𝑠 = 144 + 70 = 214

𝑃(𝐿𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 3 𝑝𝑎𝑠𝑠𝑒𝑛𝑔𝑒𝑟𝑠) =214

240= 0.8916̇

0.8916̇ ≠ 0.1083̇

Alex is incorrect

M1

M1

M1

M1

A1

(Total: 12 marks)


Page 150 of 196

Specimen Assessments: Controlled Paper MQF 3-3*


EXAMINATIONS BOARD




PAPER NUMBER: Level 3 – 3*

DATE:

TIME: 2 hours









Useful information:

Area of a Triangle ½ 𝑎𝑏 sin 𝐶

Curved Surface Area of Right Circular Cone πrl

Surface Area of a Sphere 4𝜋 𝑟2

Volume of a Pyramid /Right Circular Cone 1

3 𝑏𝑎𝑠𝑒 𝑎𝑟𝑒𝑎 𝑥 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 ℎ𝑒𝑖𝑔ℎ𝑡

Volume of Sphere 4

3𝜋𝑟3

Solutions of ax2 + bx + c = 0 𝑥 = −𝑏 ± √𝑏2−4𝑎𝑐

2𝑎

Sine formula 𝑎

sin 𝐴=

𝑏

sin 𝐵=

𝑐

sin 𝐶

Cosine formula 𝑎2 = 𝑏2 + 𝑐2– 2𝑏𝑐 𝑐𝑜𝑠𝐴

Compound Interest / Appreciation & Depreciation 𝐴 = 𝑃 (1 ±𝑟

100)

𝑛


Page 151 of 196

1) EDOC is a straight line. DC is a diameter to the circle centre O. The line ABC is the hypotenuse of the

right-angled triangle AEC.

a) Show that ABDE is a cyclic quadrilateral.

(3)

b) Angle ACE = 30°. Find the size of angle BOD. Give a reason for your answer.

(2)

(Total: 5 marks)

A

B

C O D E Diagram not drawn to scale


Page 152 of 196

2) In this question, use ruler and compasses only.

a) Construct a triangle ABC in the space below such that BC = 10 cm, angle ABC = 45° and angle

ACB = 60°.

(4)

b) Construct the perpendicular bisectors of lines AB and BC. Label the point O where these two

perpendicular bisectors meet.

(2)

c) Draw the locus of the point which is 5.2 cm from O.

(1)

(Total: 7 marks)

3)

a) In a supermarket, the price of a 0.75 litre bottle of sparkling water was €0.55. It is now being

sold in a six-pack of 1 litre bottles at €5.28 per pack.

i. Calculate the percentage increase in the cost per litre of sparkling water.

(4)

ii. The supermarket sold 203 six-packs of 1 litre bottles and made 12% profit. How much

did these bottles cost to the supermarket?

(4)


Page 153 of 196

b) In the first 5 years, the price of a new luxury car will decrease by 10% each year. The price

will then decrease by 8% each year. How much will the car cost in 7 years’ time if the price of

a new car today is €72 000?

(4)

(Total: 12 marks)

4)

a) Draw and label the reflection of OABC in the line x = 4, to form rectangle O1A B1C1.

(2)

b) Rotate rectangle OABC 90 anticlockwise about the origin to form OA2B2C2. Draw and label

rectangle OA2B2C2.

(4)

(Total: 6 marks)

𝑥 O

A

B

C

𝑦


Page 154 of 196

5)

a) Solve each of the following TWO inequalities [P] and [Q].

[P] 2𝑥 + 1 < 7 [Q] 1 − 5𝑥 ≤ 𝑥 + 4

(4)

b) Show the solutions of the two inequalities [P] and [Q] on the same number line below.

(2)

c) Write down the largest integer that satisfies both the inequalities [P] and [Q].

(1)

(Total: 7 marks)

6)

a) Solve the following equation giving your answers correct to 2 decimal places:

2𝑥2 + 5𝑥 − 1 = 0

(4)

b) Solve the equation: 6

2𝑥−1−

3

𝑥+1= 1.

(4)

(Total: 8 marks)

-1 0 1 2 3 4 -2


Page 155 of 196

7) The stacked bar chart below shows the staff employed by a firm in the years 2016, 2017 and 2018.

a) What fraction of the total staff in 2016 consisted of Clerical staff?

(2)

b) A report in the company magazine stated that, for each clerk, there were more manual and

executive workers in 2017, than there were in 2018.

Show that this report is correct.

(3)

(Total: 5 marks)


Page 156 of 196

8) When it does not rain, the probability that Nathan goes fishing is 6

7. When it does rain, the probability

that Nathan goes fishing is 1

5. The chance that it will rain tomorrow is 30%.

a) Complete the tree diagram writing the probabilities on the branches.

(2)

b) Calculate the probability that tomorrow, Nathan will not go fishing.

(5)

c) Last year Nathan went fishing 22 times and on 3 occasions he caught more than 2 kg of fish.

What is the probability that tomorrow, Nathan will catch more than 2 kg of fish?

(3)

(Total: 10 marks)

9)

a)

i. Use prime factors to find Highest Common Factor of 128 and 72.

(4)

Rain

No Rain

Goes fishing

Does not go fishing

Does not go fishing

Goes fishing


Page 157 of 196

ii. Factorise completely the expression: 128𝑦𝑥2 − 72𝑦.

(2)

b) Solve the equation: 252𝑥 = 125(𝑥+3).

(3)

(Total: 9 marks)

10) Traffic police use two speed cameras, set up at two points, A and B on a main road, 0.5 km apart

(correct to 1 d.p.). The speed limit is 70 km/h.

Malcolm takes 25 seconds (correct to the nearest second) to drive from point A to point B and he

is fined for over speeding.

a) Write down the lower and the upper bound of the distance AB and of the time taken by

Malcolm to cover this distance.

(2)

b) Malcolm says that he was travelling under the speed limit. Show how Malcolm could be right.

(3)

(Total: 5 marks)

11)

a) Neglecting air resistance, the distance (d) covered by a falling object is directly proportional to

the square of the time (t) it has been falling.

If an object falls 19.56 m in 2 seconds, determine the distance it will fall in 6 seconds.

(4)


Page 158 of 196

b) The full capacity of a cone is 2.5 litres. The cone is filled with water exactly half way up.

Calculate the volume of the empty space in the cone.

(4)

(Total: 8 marks)

12) A sphere is cut in half to form two identical hemispheres. The total surface area of the two

hemispheres together is 192 cm2.

a) Show that the surface area of the sphere is 128 cm2.

(3)

b) Points A, B and C form a triangle on a plane where AC = 125 m, BC = 320 m and AB = 431 m.

B is on a bearing of 083° from C.

i. Calculate the size of angle ACB.

(2)

ii. Calculate the bearing of C from A.

(2)

(Total: 7 marks)

A

B

C

Diagram not drawn to scale


Page 159 of 196

13)

a) Complete the table of the graph 𝑦 = 2𝑥2 − 5𝑥.

(3)

b) Draw the graph of 𝑦 = 2𝑥2 − 5𝑥 on the grid below.

(2)

c) On the same grid above draw a suitable straight-line graph and use both graphs to solve the

equation: 𝑥2 − 3𝑥 + 1 = 0.

(6)

(Total: 11 marks)

END OF PAPER

𝑥 -0.5 0 0.5 1 1.5 2 2.5 3

𝑦

𝑥

𝑦

0


Page 160 of 196

Specimen Assessments: Marking Scheme for Controlled Paper MQF 3-3*


EXAMINATIONS BOARD




PAPER NUMBER: Level 3 – 3*

DATE:

TIME: 2 hours

Question

No. Workings and Solutions

Additional

guidance Marks

1) a)

DBC = 90° (angle in a semicircle)

ABD = 180° - 90° = 90° (angles on a straight line)

AED + ABD = 90 °+ 90° = 180°

Since opposite angles are supplementary,

ABDE is a cyclic Quadrilateral

B1

M1

M1

b) BOD = 30° × 2 = 60°

Angle at the centre is twice the angle on the

circumference.

B1

B1

(Total: 5 marks)

2) a)

b)

c)

correct line BC

correct B (45°)

correct C (60°)

correct

perpendicular

bisectors

correct locus:

Circle

correct radius

and position of

circle

B1

B1

B1

B1 B1

B1

B1

(Total: 7 marks)

A

B

O

C


Page 161 of 196

3) a) i. 0.55 ÷ 0.75 = 0.73̇

5.28 ÷ 6 = 0.88

% 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 = 0.88 − 0.73̇

0.73̇ × 100 = 20

ii. 100% + 12% = 112% = 1.12

203 × 5.28 ÷ 1.12 = € 957

M1

M1

M1 A1 (f.t.)

M1 A1

M1 A1

b) 100% − 10% = 90% = 0.9 and

100% − 8% = 92% = 0.82

72 000 × 0.95 × 0.922 = €35 985

Both correct

M1 for correct

use of power.

M1 for

multiplying.

M1 M1 M1 A1

(Total: 12 marks)

4)

a)

b)

Correct

reflection

Correct labels

(all correct)

Correct rotation

Correct labels

(1 mark for

each two

correct labels)

B1

B1

B2

B2

(Total: 6 marks)

5) a)

b)

c)

[P] 2𝑥 < 6

𝑥 < 3

[Q] −3 ≤ 6𝑥

−1

2≤ 𝑥

2

Correct arrows

and endpoints

M1

A1

M1

A1

B1 B1

B1

(Total: 7 marks)

6) a) 𝑥 =

−5 ± √52 − 4 × 2 × (−1)

2 × 2

𝑥 =−5 ± √33

4

𝑥 = 0.19 𝑎𝑛𝑑 − 2.69

1 mark for each

correct value

M1

M1

A2

-12 -10 -8 -6 -4 -2 O 2 4 6 8 10 12 14

-6

-4

-2

2

4

6

8

10

12

𝑥 O

A

B

C

𝑦

C1

B1

O1

A2

B2

C2

-1 0 1 2 3 4 -2


Page 162 of 196

b) 6(𝑥 + 1) − 3(2𝑥 − 1) = (2𝑥 − 1)(𝑥 + 1)

6𝑥 + 6 − 6𝑥 + 3 = 2𝑥2 + 2𝑥 − 𝑥 − 1

2𝑥2 + 𝑥 − 10 = 0

(2𝑥 + 5)(𝑥 − 2) = 0

𝑥 = −5

2 𝑜𝑟 2

Removing

denominators

Simplifying and

equating to

zero

Factorising

Both values correct

M1

M1

M1

A1

(Total: 8 marks)

7) a) 20 clerks out of 74

10

37

M1

A1

b) 2017: 96 ÷ 30 = 3.2

2018: 122 ÷ 44 = 2.77

3.2 > 2.77

M1

M1

A1

(Total: 5 marks)

8) a)

Correct first

branch

Correct second

branch

B1

B1

b) 3

10×

4

5=

6

25

7

10×

1

7=

1

10

6

25+

1

10=

17

50

B1

B1

M1 M1A1(f.t)

c) 1 −

17

50=

33

50

3

22×

33

50=

9

100

M1

M1A1(f.t)

(Total: 10 marks)

3

10

7

10

1

5

4

5

6

7

1

7

Goes

fishing

Does not

go fishing

Goes

fishing

Does not

go fishing

Rain

No Rain


Page 163 of 196

9) a) i. 128 = 27

72 = 23×32

HCF = 23 = 8

ii. 8𝑦(16𝑥2 − 9)

8𝑦(4𝑥 − 3)(4𝑥 + 3)

M1

M1

M1 A1

M1

A1

b) (52)2𝑥 = (5)3(𝑥+3)

4𝑥 = 3(𝑥 + 3)

4𝑥 = 3𝑥 + 9

𝑥 = 9

M1

M1

A1

(Total: 9 marks)

10) a) i. 0.45 ≤ 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 ≤ 0.55

24.5 ≤ 𝑡𝑖𝑚𝑒 ≤ 25.5

Both values correct

Both values correct

B1

B1

b) 𝑇𝑖𝑚𝑒 (𝑢𝑝𝑝𝑒𝑟 𝑏𝑜𝑢𝑛𝑑) 𝑖𝑛 ℎ𝑜𝑢𝑟𝑠 = 25.5 ÷ 60 ÷ 60

𝑆𝑝𝑒𝑒𝑑(𝑙𝑜𝑤𝑒𝑟 𝑏𝑜𝑢𝑛𝑑) = 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒(𝑙𝑜𝑤𝑒𝑟 𝑏𝑜𝑢𝑛𝑑)

𝑇𝑖𝑚𝑒(𝑢𝑝𝑝𝑒𝑟 𝑏𝑜𝑢𝑛𝑑)

𝑆𝑝𝑒𝑒𝑑(𝑙𝑜𝑤𝑒𝑟 𝑏𝑜𝑢𝑛𝑑) = 0.45

25.5÷60÷60= 63.5 𝑘𝑚/ℎ

M1

M1

A1 (f.t.)

(Total: 5 marks)

11) a) 𝑑 = 𝑘𝑡2

19.56 = 𝑘(2)2 𝑘 = 19.56 ÷ 4 = 4.89

𝑑 = 4.89𝑡2 = 4.89 × 62 = 176.04 𝑚

M1

M1

M1 A1

b) Volume scale factor = 23 = 8

Volume of small cone = 2.5 ÷ 8 = 0.3125 𝑙

Volume of frustum = 2.5 – 0.3125 = 2.1875 𝑙

Accept 2.2 l

or more

accurate

M1

M1

M1 A1

(Total: 8 marks)

12) a) 2×(S.A.) = 2×(2πr2 + πr2)

192 = 6πr2 = 192

πr2 = 32

Surface area of sphere = 4 πr2 = 128 cm2

Formulating

M1

M1

A1

ii) i. 𝐶𝑜𝑠 𝐴�̂�𝐵 =

𝑎2+𝑏2−𝑐2

2𝑎𝑏

𝐶𝑜𝑠 𝐴�̂�𝐵 =3202 + 1252 − 4312

2 × 125 × 320= −0.8467

𝐴�̂�𝐵 = 𝑐𝑜𝑠−1 − 0.8467

= 148°

ii. 360° − (148° − 83°) = 295°

M1

M1 A1

B1

(Total: 7 marks)


Page 164 of 196

13) a)

𝑥 -0.5 0 0.5 1 1.5 2 2.5 3

𝑦 3 0 -2 -3 -3 -2 0 3

-1 e.e.o.o.

B3

Correct plotting

Correct curve

B1

B1

b) 2(𝑥2 − 3𝑥 + 1) = 0

2𝑥2 − 6𝑥 + 2 = 0

2𝑥2 − 6𝑥 + 2 + 𝑥 − 2 = 𝑥 − 2

2𝑥2 − 5𝑥 = 𝑥 − 2

𝑦 = 𝑥 − 2

𝑥 = 2.6 𝑎𝑛𝑑 0.4

Multiplying by 2

Adding 𝑥 − 2 on

both sides of

equation.

Simplifying.

Finding

equation of

straight line.

Drawing of

straight line.

𝑥 values of

points of

intersection.

0.1

M1

M1

M1

M1

B1 B1

(Total: 11 marks)


Page 165 of 196

Specimen Assessments: Private Candidates Controlled Paper MQF 1-2


EXAMINATIONS BOARD


SAMPLE PRIVATE CANDIDATES CONTROLLED PAPER



DATE:

TIME: 2 hours










Page 166 of 196

1. In a solving task, Samuel is trying to cut 12 identical rectangles exactly in half so that the two parts

are also identical. He already did two.

a) What is the order of rotational symmetry of the first two shapes?

(1)

b) Help Samuel cut the remaining 6 rectangles exactly in half but in different ways. Note that

and count as the same cut.

(14)

(Total: 15 marks)

2. Paul was working on a solving task involving reflective symmetry. He was presented with the

following scenario, where a digital clock was resting on a cabinet facing a mirror as shown in the

diagram. His friend Martin is standing behind the clock and facing the mirror.

Martin


Page 167 of 196

The following is the time of the digital clock as seen in the mirror.

a) Write the time as is seen on the digital clock.

(2)

b) As part of the creation activity Paul prepared the following images and was going to draw their

reflection in the mirror line. Help Paul draw the reflected images.

(8)

(Total: 10 marks)


Page 168 of 196

3. Maria, Lara and Amy’s teacher takes her class to do a maths trail at the Chinese Garden. The

three students are placed in the same group and need to do some tasks together.

a) The students find this entryway shown below:

Maria drew the shape of the entryway on her copybook as shown below.

i. Draw all the lines of symmetry of this shape.

(2)

ii. How many half circles make up the perimeter of the shape? _________

(1)


Page 169 of 196

iii. Maria measures the length d and finds that it is 1.05 m long. Find the perimeter of this

shape.

(3)

b) For the second task, the three students find the space shown in the picture below:

i. Name the TWO solid shapes:

A:________________________________ (1)

B:________________________________ (1)

ii. Name the TWO outlined flat shapes:

C:________________________________ (1)

D:________________________________ (1)

A

B

C

D

A

B

C

D


Page 170 of 196

iii. Lara uses footsteps to find the length and breadth of this rectangular area. The length of

one footstep is 42 cm.

She counts 8 footsteps to walk along the length. Calculate the length in metres.

(2)

She counts 5 footsteps to walk along the width. Calculate the width in metres.

(2)

iv. Hence find the area in m2 of this rectangular space.

(2)

v. The rectangular space is covered with rectangular tiles each measuring 15 cm by 8 cm.

Estimate how many tiles are needed to cover the rectangular space. Give your answer to

the nearest whole number.

(4)


Page 171 of 196

c) Amy wants to find the height of a tree in the garden. She moves away from the tree and looks

at its top, as shown in the diagram.

i. What is the size of angle ABO?

(1)

ii. What kind of triangle is ∆ ABO?

(1)

iii. How tall do you think Amy is? (Circle the right answer)

0.16 m 1.6 m 1600 cm

(1)

iv. Maria uses a trundle wheel and finds that the distance between Amy and the tree is

2.38 m. Calculate the height of the tree.

(2)

(Total: 25 marks)

A

O

B

45o


Page 172 of 196

4. Ryan, Stephanie, Mark and Lisa go to different schools. They carried out a survey to find the average

number of books carried to school by a secondary school student. The tables below show the data

they collected from their schools.

Ryan:

Number of

books 1 2 3 4 5 Total

Tally IIII IIII

I

IIII IIII

IIII IIII

I

IIII IIII

I IIII III

Frequency 11 5

Stephanie:

Number of


Tally IIII III IIII IIII

IIII IIII II IIII IIII IIII I

Frequency 22

Mark:

Number of


Tally IIII IIII

III

IIII IIII

IIII II IIII III IIII II II

47

Frequency

Lisa:

a) Complete the tables above.

(4)

b) Fill in the following frequency table showing all the data gathered by the four students from

the four schools together.

Number of


Frequency 180

(2)

Number of


Tally IIII I IIII

IIII II

IIII IIII

I IIII II

Frequency 6 12 2


Page 173 of 196

c) Draw a pie chart below, using the frequency table in part (b), to illustrate all the data.

(5)

d) Now answer the following question:

What is the average number of books carried to school by a secondary school student?

(2)

(Total: 13 marks)


Page 174 of 196

5. Ana was doing a research project on “How geometry

influences logo design”. She came across this design

showing two interlocking equilateral triangles. It has a

rotational symmetry of order 6. The distance from the

centre to the tips of the triangles is 7 cm.

Ana decided to construct the design herself. Use a well

sharpened pencil, a ruler, and a pair of compasses only to

help Ana construct the design in the construction area below.

Use the centre of the construction area as the centre

of your construction.

Start with a circle of radius 7 cm.

You need to figure out what else needs to be done to

complete the design.

When finished, outline the design in bold.

Leave any construction lines and arcs visible.

(Total: 12 marks)


Page 175 of 196

6. a) Joseph’s teacher has given each pupil in his class the following set of numbers and symbols on

pieces of card:

1 3 2 7 1 + + + +

She then showed them how to create numbers from 10 to 22 by using some of the cards and at

least one of the + cards. For instance:

She created 10 by using the ‘3’, ‘7’ and ‘+’ cards: 3 + 7

She created 12 by using the ‘2’, ‘3’, ‘7’ and ‘+’ cards: 2 + 3 + 7

She created 14 by using the ‘1’, ‘1’, ‘3’ and ‘+’ cards: 1 3 + 1

The teacher displayed these results in the table below. She then told the pupils that it is possible

to create all numbers up to 22 in this manner and asked them to complete the table. Help Joseph

complete the table by filling in the rest of the table.

(10)

b) The teacher now asked the pupils to set the ‘+’ cards aside and begin creating 5 digit numbers

by placing all the other cards near each other. As an example she created the number 13712

as follows:

The teacher now challenged the pupils to write:

i. the largest possible 5-digit odd number: (1)

ii. the largest possible 5-digit even number: (1)

iii. the smallest possible 5-digit odd number: (1)

iv. the smallest possible 5-digit even number: (1)

Number Sum of numbers Number Sum of numbers

10 3 + 7 17

11 18

12 2 + 3 + 7 19

13 20

14 13 + 1 21

15 22

16

1 3 7 1 2


Page 176 of 196

c) The teacher now asked pupils to use a ‘+’ symbol card and the ‘1’, ‘1’, ‘2’, ‘3’, and ‘7’ cards to

create a sum involving a 2-digit number and a 3-digit number. As an example she created:

2 1 1 + 7 3

She now added the two numbers 211 and 73 together to give an answer of 284.

i. She now challenged the class to find a sum of a 3-digit number and 2-digit number that

added up to 248. Help Joseph complete this task by filling in the following:

+ = 248

(3)

ii. She now challenged the class to find the largest 3-digit number and 2-digit number sum.

Help Joseph complete this task by filling in the following:

+ =

(4)

iii. Finally, the teacher challenged the class to find the smallest 3-digit number and 2-digit

number sum. Help Joseph complete this task by filling in the following:

+ =

(4)

(Total: 25 marks)

END OF PAPER


Page 177 of 196

Specimen Assessments: Marking Scheme for Private Candidates Controlled Paper MQF 1-2


EXAMINATIONS BOARD


MARKING SCHEME FOR SAMPLE PRIVATE CANDIDATES

CONTROLLED PAPER



DATE:

TIME: 2 hours

Que. Marks Additional Guidance

1. a) 2 B1

15

b) 2 mark each for the first 4 correct shapes; 3 marks each

for the remaining 2 correct shapes.

B8

B6

2. a) 13:56 B2

10

1 mark for each two correct digits

b)

B2

B2

B2

B2

1 mark for correct

orientation of image

1 mark for correct

position of the image

3.

a)

i.

B2

1 mark given

if 2 lines of

symmetry

are marked.

ii. 4 B1

iii. Circumference = 𝜋d

Perimeter = 2 × 𝜋 × 1.05 = 6.6 m

M1

M1 A1


Page 178 of 196

b)

i.

A: Cube

B: Cuboid

B1

B1

25

ii. C: Trapezium

D: Hexagon

B1

B1

iii. L = 0.42 m x 8

= 3.36 m

W = 0.42 m x 5

= 2.1 m

M1

A1

M1

A1

iv. Area = 3.36 × 2.1

= 7.1 m2 (or more accurate)

M1

A1 f.t.

v. Area of 1 tile = 0.15 × 0.08

= 0.012m2 Number of tiles = 7.056 ÷ 0.012

= 588 tiles

M1

A1

M1

A1 f.t.

c)

i.

45° B1

ii. Isosceles or right-angled triangle B1

iii. 1.6m B1

iv. Height = 1.6m + 2.38m

= 3.98m

M1

A1 f.t.

4. a) Ryan:

No. 1 2 3 4 5 Total

Freq. 11 21 11 5 3 51

Stephanie:

Mark

Lisa

No. 1 2 3 4 5 Total

Freq. 8 22 10 5 1 46

No. 1 2 3 4 5 Total

Freq. 13 17 8 7 2 47

No. 1 2 3 4 5 Total

Freq. 6 12 11 5 2 36

B1

B1

B1

B1

13

b)

No. 1 2 3 4 5 Total

Freq. 38 72 40 22 8 180

B2

Award one mark for each 3 correct entries

c)

Correct pie chart drawn

No. 1 2 3 4 5

Angle 76° 144° 80° 44° 16°

B3

B2

Deduct one mark for each incorrect angle

d) 2 books B2

5.

Excellent Good Average Poor

Symmetry 3 2 1 0

Dimensions 3 2 1 0

Completeness 3 2 1 0

Neatness 3 2 1 0

12


Page 179 of 196

6. a) Sum adding to 11: 7 + 3 + 1

Sum adding to 13: 7 + 3 + 2 or 7 + 3 + 1 + 1

Sum adding to 15: 13 + 2 or 13 + 1 + 1

Sum adding to 16: 13 + 2 + 1

Sum adding to 17: 13 + 2 + 1 + 1

Sum adding to 18: 17 + 1

Sum adding to 19: 17 + 2 or 17 + 1 + 1

Sum adding to 20: 17 + 3 or 17 + 2 + 1

Sum adding to 21: 17 + 3 + 1 or 17 + 2 + 1 + 1

Sum adding to 22: 17 + 3 + 1 + 1 or 21 + 1

B1

B1

B1

B1

B1

B1

B1

B1

B1

B1

25

b) i) Largest possible five digit odd number: 73211 B1

ii) Largest possible five digit even number: 73112 B1

iii) Smallest possible five digit odd number: 11237 B1

iv) Smallest possible five digit even number: 11372 B1

c) i) 211 + 37 B3

ii) 731 + 21 or 721 + 31

Largest 3-digit & 2-digit number sum: 752

B3

B1

(for getting

the digits

correct)

(for getting

the sum

correct)

iii) 113 + 27 or 123 + 17 or 117 + 23 or 127 + 13

Smallest 3-digit & 2-digit number sum:

B3

B1

(for getting

the digits

correct)

(for getting

the sum

correct)


Page 180 of 196

Specimen Assessments: Private Candidates Controlled Paper MQF 2-3


EXAMINATIONS BOARD


SAMPLE PRIVATE CANDIDATES CONTROLLED PAPER



DATE:

TIME: 2 hours










Page 181 of 196

1. Lara is doing a survey for an online radio station. She gave this questionnaire to a number of

randomly selected persons.

Lara draws the following stacked bar chart to show her results.

(I) Which type of radio programme do you prefer most?

Music shows Talk shows News and weather Sports

(II) What is your age group?

Under 30 30 – 50 Over 50

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

105

110

115

120

125

130

135

140

145

150

Music shows Talk shows News andweather

Sports

Over 50

30 - 50

Under 30


Page 182 of 196

a)

i. Complete this two-way table.

Music shows Talk shows News and

weather Sports Total

Under 30 33 18 22 55 128

30 - 50 15 35 40 55 145

Over 50 8 30 35 31 104

Total 56 97 141 377

(2)

ii. Draw the missing bar for Talk shows in the stacked bar chart above. (2)

b) Answer the following questions:

i. How many persons between 30 and 50 years old prefer Music shows or Sports

programmes?

(1)

ii. What percentage of the persons taking part in the questionnaire prefer Talk shows?

(2)

iii. Lara says that: “One out of thirteen persons over 50 years old prefer Music shows.” Show

that Lara is correct.

(2)

iv. What fraction of the persons 50 years old or younger prefer Sports?

(2)

v. What is the probability that a 24-year-old person prefers News and weather?

(2)

c) Suggest ONE way of improving the survey.

(1)

(Total: 14 marks)


Page 183 of 196

2. Jasmine was given a solving task which included a diagram showing two concentric circles, (two

circles that have a common centre O) and line AB which is a chord of the outer circle touching the

inner circle at Q.

a) What can you tell Jasmine about point Q?

(1)

b) Jasmine decided to draw the line of symmetry of the diagram. Draw this line on the diagram

above.

(1)

c) What is the relationship between OQ, QB and OB?

(2)

d) Show that the area between the circles is π × (QB)2.

(5)

e) Given that AB = 50 cm, calculate the area between the two concentric circles.

(2)

(Total: 11 marks)

A Q

O

B


Page 184 of 196

3. Brian conducts a probability experiment using an app which simulates an 8-sector spinner.

The following is a screen shot of the results after 50 spins.

https://www.nctm.org/adjustablespinner/

a) Explain why the theoretical probability for all the colours is the same.

(1)

b) Give a reason why the experimental probabilities are not equal to the theoretical probabilities.

(2)

c) What is the theoretical probability that the spinner stops on either yellow or red?

(2)

d) In this case:

i. what is the experimental probability that the spinner stops on either yellow or red?

(1)


Page 185 of 196

ii. what is the experimental probability that the spinner does not stop on either yellow or red?

(1)

e) Would you say that the app is fair or biased? Give a reason for your answer.

(2)

f) What should Brian do so that he can safely conclude whether the app is fair or not?

(2)

(Total: 11 marks)

4. Alan was given a solving task where he had to find the sum of the terms from the 23rd to the 124th

term of a linear sequence that starts as follows:

8, 11, 14, 17, …

While working on the task, he noted that the nth term of this sequence is:

nth term = 3𝑛 + 5

a) Help Alan find the 23rd term and the 124th term of this sequence.

(4)

b) Alan thought that he should find how many terms there from the 23rd to the 124th term. Find the

number of terms for Alan.

(2)


Page 186 of 196

c) Alan noticed something about the following totals. What did Alan notice?

23rd term + 124th term, 24th term + 123rd term, 25th term + 122nd term. …

(3)

d) Hence or otherwise, work out the sum of all the terms from the 23rd to the 124th.

(4)

(Total: 13 marks)

5. Maria’s teacher has asked the class to investigate the factors of numbers. She instructed the class

to choose a set of consecutive whole numbers between 1 and 100 and to find the factors of each

number in this set. Maria chose from 16 to 26 and first found the factors of 15. She wrote the results

in the second column of a table:

a) i. Help Maria complete the second column in the table above by finding the factors of the

remaining numbers.

(5)

ii. Which numbers have just two factors? _____________________

(1)

iii. Which numbers have an odd number of factors? ______________________

(1)

Number Factors

Sum of factors

(apart from number

itself)

16 1, 2, 4, 8, and 16 15

17

18

19

20

21

22

23

24

25

26


Page 187 of 196

The teacher now asked the pupils to explore the factors of the numbers they had selected. Maria

decided to investigate further the number of factors. In order to do so she decided to take a look at

whole numbers in the range from 27 to 50.

b) Help Maria by finding:

i. TWO other numbers in the range from 27 to 50 that have just two factors:

_______________________ (2)

ii. TWO other numbers in the range from 27 to 50 that have an odd number of factors:

_______________________ (2)

Maria now decided to draw some conclusions from what she found.

c) Help Maria draw up these conclusions by answering the following questions.

i. How would you describe the numbers that have only two factors?

______________________________________ (1)

ii. How would you describe the numbers that have an odd number of actors?

______________________________________ (1)

The teacher then asked the pupils to add the factors obtained in the second column but not to add

the number itself. As an example she used the number 12. She first found the factors of 12:

Factors of 12: 1, 2, 3, 4, 6, and 12.

She then added the factors of 12 apart from 12 itself:

Sum of factors of 12 apart from 12 itself: 1 + 2 + 3 + 4 + 6 = 16.

d) Help Maria complete the third column in the table above. The first one has been done for you.

(5)

Maria’s teacher then asked the pupils to repeat the process on each of the numbers in the third

column. To continue with her example, the teacher took 16, which was the sum of the factors of

12 (except 12 itself), and found its factors:

Factors of 16: 1, 2, 4, 8, and 16.

Sum of factors of 16 apart from 16 itself: 1 + 2 + 4 + 8 = 15.

She then repeated the process on the new number which was 15. She continued repeating the

process on each subsequent new sum of factors (except for the number itself) until she got 1 as

the sum of factors (except for the number itself). Her work is as follows:

Factors of 15: 1, 3, 5, and 15.

Sum of factors of 15 apart from 15 itself: 1 + 3 + 5 = 9

Factors of 9: 1, 3, and 9.

Sum of factors of 9 apart from 9 itself: 4

Factors of 4: 1, 2, and 4.

Sum of factors of 4 apart 4 itself: 3

Factors of 3: 1, and 3.

Sum of factors of 3 apart 3 itself: 1


Page 188 of 196

The teacher then wrote the result as a chain of 7 numbers:

12 16 15 9 4 3 1

She then challenged the class to find another chain of 7 numbers or a longer one. Maria decided to

try to find such a chain by considering the numbers from 16 to 20.

e) Help Maria do the challenge by finding the chains for the numbers 16 to 20. Write down your

answers in the spaces provided underneath:

Chain for 16: _______________________________________________________________ (1)

Chain for 17: _______________________________________________________________ (1)

Chain for 18: _______________________________________________________________ (1)

Chain for 19: _______________________________________________________________ (1)

Chain for 20: _______________________________________________________________ (1)

f) Maria then thought that if she multiplied 12 by 2, the chain for the resulting number, 24, could be

longer than that for 12. So she decided to find the chain for 24.

i. Was Maria correct? ________________ (1)

ii. Give a reason for your answer.

______________________________________________________________________________

____________________________________________________________________________ (1)

(Total: 25 marks)


Page 189 of 196

6. Maria, Lara and Amy’s teacher takes her class to do a maths trail at San Anton Gardens. The three

students are placed in the same group and need to do some tasks together.

a) In the first task the students have to calculate the height of this monument in the garden.

Maria moves to a point X, 6 m away from the monument and measures the angle of elevation

from point X to the top of the monument.

i. What is the angle of elevation shown on the clinometer in the diagram above?

(1)

ii. Mark the angle of elevation, and the distance of Maria away from the monument on the given

diagram.

(2)

The clinometer measures the angle of

elevation.


Page 190 of 196

iii. Maria’s height is 1.6 m. Calculate the height of the monument, giving your answer correct to

1 d.p.

(3)

iv. Maria now moves closer to the monument such that she is 4 m away from it. Maria checks the

angle of elevation again using the clinometer. What is the angle of elevation now? Give your

answer correct to the nearest degree.

(2)

b) For the second task, the group finds this circular pond in the same garden. The pond is

surrounded by a hedge.

i. The length of one footstep is 42 cm. Calculate the circumference of the outer part of the hedge

in metres.

(1)

The group wants to find the

circumference of the pond.

Lara walks round the hedge of the pond.

She counts 35 foot steps to go round the

hedge.


Page 191 of 196

ii. Calculate the diameter of the hedge.

(2)

iii. Lara and seven other classmates measure the width of the hedge at a number of places around

the pond. They get the following widths:

54 cm 49 cm 57 cm 50 cm

53 cm 48 cm 51 cm 34 cm

Calculate the mean width of the hedge.

(2)

Calculate the median width of the hedge.

(1)

Which of the calculated mean or median is the better estimate of the width of the hedge?

Give a reason.

(1)

iv. Calculate the diameter of the pond without the hedge.

(1)


Page 192 of 196

v. Hence calculate the capacity of the pond if it is 70 cm deep.

(2)

vi. To fill in this pond, the gardeners use a hose-pipe from which water comes out at a rate of 0.005

m3 per second. Find how many minutes does it take to fill this pond.

(3)

c) For the third task, the students find this clock at the entrance of San Anton Palace. The clock

shows the time of their arrival on site.

ii. How many degrees would the minute hand turn in the time that is left?

(3)

(Total: 25 marks)

END OF PAPER

i. How much time is left for the maths trail if the

students have to meet their teacher at the

garden gate at 1:00 pm? Give your answer

to the nearest 5 minutes.

(1)


Page 193 of 196

Specimen Assessments: Marking Scheme for Private Candidates Controlled Paper MQF 2-3


EXAMINATIONS BOARD


MARKING SCHEME FOR SAMPLE PRIVATE CANDIDATES

CONTROLLED PAPER



DATE:

TIME: 2 hours

Que. Marks Additional Guidance

1. a) i)

ii) clustered bar at 18, 53 and 83

Music

shows

Talk

shows

News

and

weather

Sports Total

Under

30 33 55

30 -

50 40 145

Over

50 8 104

Total 56 83 97 141

B2

B2 14

-1 each

error or

omission

(e.e.o.o.)

-1 e.e.o.o.

(b) i) 15 + 55 = 70 B1

ii) 83

377× 100 = 22% M1 A1

Or more

accurate

iii) 8

104=

1

13 M1 A1

iv) 55 + 55 = 110 and 128 + 145 = 273

110

273

M1

A1

v) 22

128=

11

64

M1

A1

c) Use 4 or more class intervals

or

Add other types of programmes

A1

Or any other valid statement

2. a) It is the mid-point of AB B1

11

or equivalent

b) A vertical line through the centre seen in diagram. B1

c) OQ2+ QB2 = OB2 B2

d) Area required = 𝜋(𝑂𝐵)2 − 𝜋(𝑂𝑄)2

Area = 𝜋(𝑂𝐵2 − 𝑂𝑄2) (I)

Since OQB = 90°, by Pythagoras’ theorem,

𝑄𝐵2 = 𝑂𝐵2 − 𝑂𝑄2

And substituting 𝑄𝐵2 in (I)

Area = 𝜋 × (𝑄𝐵)2

B1

B1

M1

M1

A1

Formulating

Factorising

Identifying theorem

Using theorem correctly


Page 194 of 196

e) Now, 𝑄𝐵 =𝐴𝐵

2=

50

2= 25 cm

Therefore, Area = 𝜋 × 252

= 1963.5 cm2

M1

A1

3. a) Angle at the centre of all sectors is the same. B1

11

Or any other valid reason

b) Theoretical probability is what one expects to happen, but

it isn’t always what actually happens during an experiment B2

c) 12.5 × 2 = 25% M1 A1

d) i) 10 + 12 = 22%

ii) 100 – 22 = 78%

B1

B1

e) Fair because for most of the outcomes, the experimental

probability is very close to the theoretical probability

and/or

Biased because there are two outcomes where there is a

big difference between the experimental and the

theoretical probabilities.

B2

Or any other valid reason

f) Increase the number of spins.

or

For a fair spinner the more spins, the closer you get to the

theoretical probability.

B2

4. a) 23rd term = 3 × 23 + 5

= 74

M1

A1

13

Using nth term

124th term = 3 × 124 + 5

= 377

M1

A1

b) Number of terms = 124 – 23 +1

= 102

M1

A1

c) Finding the terms and adding

451

They all add up to the same number

M1

M1

A1

d) There are 102÷2

= 51 pairs

Sum required = 451 × 51

= 23 001

M1

A1

M1

A1

5. a) i. Factors of 17: 1 & 17

Factors of 18: 1, 2, 3, 6, 9, & 18

Factors of 19: 1 & 19

Factors of 20: 1, 2, 4, 5, 10, & 20

Factors of 21: 1, 3, 7, & 21

Factors of 22: 1, 2, 11, & 22

Factors of 23: 1 & 23

Factors of 24: 1, 2, 3, 4, 6, 8, 12, & 24

Factors of 25: 1, 5, 25

Factors of 26: 1, 2, 13, & 26

B5

25

Award 1 mark for each two correct sets

of factors

ii. 17, 19, & 23 B1

iii. 16 & 25 B1

b) i. Any two prime numbers from 29, 31, 37, 41, 43, & 47.

B1

B1

ii. 36 & 49 B1 B1

c) i. Prime numbers B1

ii. Square numbers B1

d) i. Sum of factors of 17 (excluding 17): 1

Sum of factors of 18 (excluding 18): 21 B5

Award one mark for every two correct sums


Page 195 of 196

Sum of factors of 19 (excluding 19): 1








e) Chain for 16: 16 15 9 4 3 1 (from example

worked out by teacher)

Chain for 17: 17 1

Chain for 18: 18 21 11 1 (can use column 3 in

table to write down factors of 18 and 21)

Chain for 19: 19 1

Chain for 20: 20 22 14 10 8 7 1

B1

B1

B1

B1

B1

f) i. No

ii. Because the chain for 24: 24 36 55 17 1

B1

B1

6. a) i. 25° B1

25

ii.

B2 f.t.

iii. Tan 25° = 𝑥

6

x = 6 × tan 25°

x = 2.798

Add Maria’s height: 2.798 + 1.6

Height of Monument 4.4 m

M1

A1

A1

iv. Tan ө = 2.798

4

Angle of elevation = 35°

M1

A1

b)

i. 42 × 35 = 1470 cm

= 14.7 m

B1

ii. Diameter = 𝐶

𝜋 =

14.7

𝜋

= 4.68 m

M1

A1 f.t.

iii.

Mean = 54+49+57+50+53+48+51+34

8 = 49.5 cm

Median: 34 48 49 50 51 53 54 57

(50 + 51) ÷ 2 = 50.5 cm

The median gives a more reliable value of the average

width of the hedge because 34 is an outlier and will

affect the mean.

M1 A1

B1

B1

iv. Diameter of pond:

4.68 - 1.01 = 3.67 m

B1

v. Capacity = 𝜋r2h

25°

6m


Page 196 of 196

= 𝜋 × 1.8352 × 0.7

= 7.4 m3 (or more accurate)

M1

A1

vi. 7.41

0.005 = 1482 seconds

1482 ÷ 60 = 24.7 minutes

M1 A1

A1

c. i. Time now: 11:20

Time left: 1hr 40 mins B1

ii. 1 hr = 360°

40 mins = 240°

Total = 600°

M1

M1

A1

Documents

sec 23 syllabus (2023): mathematics - University of Malta · 2020-03-09 · SEC 23 SYLLABUS (2023): MATHEMATICS Page 2 of 196 Introduction This syllabus is based on the curriculum