n5-course-spec-mathematics

National 5 Mathematics Course Overview

• Course code: C847 75
• Course assessment code: X847 75
• SCQF: level 5 (24 SCQF credit points)
• Valid from: session 2023–24
• Provides detailed information about the course and course assessment for consistent and transparent assessment.
• Describes the structure of the course and the course assessment in terms of the skills, knowledge, and understanding that are assessed.
• This document is for teachers and lecturers; contains all the mandatory information needed to deliver the course.
• Edition: May 2023, version 3.0

Course Rationale

• National Courses reflect Curriculum for Excellence values, purposes, and principles.
• Offers flexibility, more time for learning, more focus on skills and applying learning, and scope for personalisation and choice.
• Every course provides opportunities for candidates to develop breadth, challenge, and application.
• Focus and balance of assessment is tailored to each subject area.
• Mathematics engages learners of all ages, interests, and abilities.
• Learning mathematics develops logical reasoning, analysis, problem-solving skills, creativity, and the ability to think in abstract ways.
• It uses a universal language of numbers and symbols, which allows us to communicate ideas in a concise, unambiguous, and rigorous way.
• The course develops important mathematical techniques critical to successful progression beyond National 5 in Mathematics and many other curriculum areas.
• The skills, knowledge, and understanding in the course also support learning in technology, health and wellbeing, science, and social studies.

Purpose and Aims

• Using mathematics enables us to model real-life situations and make connections and informed predictions.
• It equips us with the skills we need to interpret and analyse information, simplify and solve problems, assess risk, and make informed decisions.
• The course aims to:
  • Motivate and challenge candidates by enabling them to select and apply mathematical techniques in a variety of mathematical and real-life situations.
  • Develop confidence in the subject and a positive attitude towards further study in mathematics.
  • Develop skills in manipulation of abstract terms to generalise and to solve problems.
  • Allow candidates to interpret, communicate, and manage information in mathematical form: skills which are vital to scientific and technological research and development.
  • Develop candidates’ skills in using mathematical language and in exploring mathematical ideas.
  • Develop skills relevant to learning, life, and work in an engaging and enjoyable way.

Who is this course for?

• This course is suitable for learners who have achieved the fourth level of learning across the mathematics experiences and outcomes in the broad general education, or who have attained the National 4 Mathematics course, or who have equivalent qualifications or experience.
• This course is particularly suitable for learners who wish to develop mathematical techniques for use in further study of mathematics or other curriculum areas, or in workplaces.

Course Content

• Throughout this course, candidates acquire and apply operational skills necessary for developing mathematical ideas through symbolic representation and diagrams.
• They select and apply mathematical techniques and develop their understanding of the interdependencies within mathematics.
• Candidates develop mathematical reasoning skills and gain experience in making informed decisions.

Skills, Knowledge, and Understanding

• The following provides a broad overview of the subject skills, knowledge, and understanding developed in the course:
  • Understand and use mathematical concepts and relationships
  • Select and apply numerical skills
  • Select and apply skills in algebra, geometry, trigonometry, and statistics
  • Use mathematical models
  • Use mathematical reasoning skills to interpret information, to select a strategy to solve a problem, and to communicate solutions

Skills, Knowledge, and Understanding for the Course Assessment

• The following provides details of skills, knowledge, and understanding sampled in the course assessment.

Numerical Skills

• Working with surds
  • Simplification
  • Rationalising denominators
• Simplifying expressions using the laws of indices
  • Multiplication and division using positive and negative indices including fractions
  • (ab)^m = a^m b^m
  • (a^n)^m = a^{mn}
  • a^m / a^n = a^{m/n}
• Calculations using scientific notation
• Rounding
  • To a given number of significant figures
• Working with reverse percentages
  • Use reverse percentages to calculate an original quantity
• Working with appreciation/depreciation
  • Appreciation including compound interest
  • Depreciation
• Working with fractions
  • Operations and combinations of operations on fractions including mixed numbers (addition, subtraction, multiplication, division)

Algebraic skills

• Working with algebraic expressions involving expansion of brackets
  • (a + b)(c + d)
  • (ax + b)(cx + d)
  • (ax^2 + bx + c)(dx + e)
• Factorising an algebraic expression
  • Common factor
  • Difference of squares
  • Trinomials with unitary and non-unitary coefficient
  • Combinations of the above
• Completing the square in a quadratic expression with unitary coefficient
  • Writing quadratics of the form x^2 + bx + c in the form (x + p)^2 + q where p, q ∈ ℝ
• Reducing an algebraic fraction to its simplest form
  • Where a, b are of the form (mx + p) or nx + q and b ≠ 0
• Applying the four operations to algebraic fractions
  • \frac{a}{b} * \frac{c}{d}, where a, b, c, d can be simple constants, variables or expressions; * can be add, subtract, multiply or divide; and b, d ≠ 0
• Determining the equation of a straight line
  • Use the formula y - b = m(x - a) or equivalent to find the equation of a straight line, given two points or one point and the gradient of the line
  • Use functional notation, f(x)
  • Identify gradient and y-intercept from various forms of the equation of a straight line
• Working with linear equations and inequations
  • Where numerical coefficients are rational numbers
  • Where numerical solutions are rational numbers
• Working with simultaneous equations
  • Construct from text
  • Graphical solution
  • Algebraic solution
• Changing the subject of a formula
  • Linear formula
  • Formula involving a simple square or square root
• Recognise and determine the equation of a quadratic function from its graph
  • Equations of the form y = kx^2 and y = k(x + p)^2 + q where k, p, q ∈ ℝ
• Sketching a quadratic function
  • Equations of the form y = ax^2 + bx + c where a, b, c ∈ ℝ
  • Equations of the form y = k(x + p)(x + q) where k, p, q ∈ ℝ
• Identifying features of a quadratic function
  • Identify:
    • The nature and coordinates of the turning point
    • The equation of the axis of symmetry of a quadratic of the form y = k(x + p)^2 + q where k, p, q ∈ ℝ
• Solving a quadratic equation
  • Solving from factorised form
  • Solving having factorised first
  • Graphical treatment
• Solving a quadratic equation using the quadratic formula
  • Solving using the quadratic formula
• Using the discriminant to determine the number of roots
  • Know and use the discriminant
  • Determine the number and describe the nature of roots using the language ‘two real and distinct roots’, ‘one repeated real root’, ‘two equal real roots’ and ‘no real roots’

Geometric skills

• Determining the gradient of a straight line, given two points
  • m = \frac{y2 - y1}{x2 - x1}
• Circle geometry
  • Calculating the length of an arc
  • Calculating the area of a sector
• Calculating the volume of a standard solid
  • Sphere, cone, pyramid
• Applying Pythagoras’ theorem
  • Using Pythagoras’ theorem in complex situations including converse and three dimensions
• Applying the properties of shapes to determine an angle involving at least two steps
  • Quadrilaterals/triangles/polygons/circles
  • Relationship in a circle between the centre, chord and perpendicular bisector
• Using similarity
  • Interrelationship of scale — length, area and volume
• Working with two-dimensional vectors
  • Adding or subtracting two-dimensional vectors using directed line segments
• Working with three-dimensional coordinates
  • Determining coordinates of a point from a diagram representing a three-dimensional object
• Using vector components
  • Adding or subtracting two- or three-dimensional vectors using components
• Calculating the magnitude of a vector
  • Magnitude of a two- or three-dimensional vector

Trigonometric skills

• Working with the graphs of trigonometric functions
  • Basic graphs
  • Amplitude
  • Vertical translation
  • Multiple angle
  • Phase angle
• Working with trigonometric relationships in degrees
  • Sine, cosine and tangent of angles from 0° to 360°
  • Period
  • Related angles
  • Solve basic equations
  • Use the identities \cos^2 x + \sin^2 x = 1 and \tan x = \frac{\sin x}{\cos x}
• Calculating the area of a triangle using trigonometry
  • Area = \frac{1}{2}ab\sin C
• Using the sine and cosine rules to find a side or angle in a triangle
  • Sine rule for side and angle
  • Cosine rule for side and angle
• Using bearings with trigonometry
  • To find a distance or direction

Statistical skills

• Comparing data sets using statistics
  • Compare data sets using calculated/determined:
    • Interquartile range
    • Standard deviation
• Forming a linear model from a given set of data
  • Determine the equation of a best-fitting straight line on a scattergraph and use it to estimate y given x

Reasoning skills

• Interpreting a situation where mathematics can be used and identifying a strategy
  • Can be attached to any operational skills to require analysis of a situation
• Explaining a solution and relating it to context
  • Can be attached to any operational skills to require explanation of the solution given
• Skills, knowledge, and understanding included in the course are appropriate to the SCQF level of the course.
• The SCQF level descriptors give further information on characteristics and expected performance at each SCQF level (www.scqf.org.uk).

Skills for Learning, Skills for Life and Skills for Work

• This course helps candidates to develop broad, generic skills. These skills are based on SQA’s Skills Framework: Skills for Learning, Skills for Life and Skills for Work and draw from the following main skills areas:
  • Numeracy
    • Number processes
    • Money, time and measurement
    • Information handling
  • Thinking skills
    • Applying
    • Analysing and evaluating
• These skills must be built into the course where there are appropriate opportunities and the level should be appropriate to the level of the course.
• Further information on building in skills for learning, skills for life and skills for work is given in the course support notes.

Course assessment

• Course assessment is based on the information provided in this document.
• The course assessment meets the key purposes and aims of the course by addressing:
  • Breadth — drawing on knowledge and skills from across the course
  • Challenge — requiring greater depth or extension of knowledge and/or skills
  • Application — requiring application of knowledge and/or skills in practical or theoretical contexts as appropriate
• This enables candidates to:
  • Demonstrate mathematical operational skills
  • Integrate mathematical operational skills developed throughout the course
  • Demonstrate mathematical reasoning skills
  • Apply numerical calculation skills without the use of a calculator to demonstrate an underlying grasp of mathematical processes

Course assessment structure

Question paper 1 (non-calculator)

• 40 marks
• The purpose of this question paper is to allow candidates to demonstrate the application of mathematical skills, knowledge, and understanding from across the course. A calculator cannot be used.
• This question paper gives candidates an opportunity to apply numerical, algebraic, geometric, trigonometric, statistical and reasoning skills.
• These skills are those in which the candidate is required to demonstrate an understanding of the underlying processes.
• They involve the ability to use numerical skills within mathematical contexts in cases where a calculator may compromise the assessment of this understanding.
• This question paper has 40 marks out of a total of 90 marks. It consists of short-answer and extended-response questions.
• Setting, conducting and marking question paper 1 (non-calculator)
  • This question paper is set and marked by SQA and conducted in centres under conditions specified for external examinations by SQA.
  • Candidates complete this in 1 hour.

Question paper 2

• 50 marks
• The purpose of this question paper is to assess mathematical skills. A calculator may be used.
• This question paper gives candidates an opportunity to apply numerical, algebraic, geometric, trigonometric, statistical and reasoning skills.
• These skills are those which may be facilitated by the use of a calculator, allowing more opportunity for application.
• This question paper has 50 marks out of a total of 90 marks. It consists of short-answer and extended-response questions.
• Setting, conducting and marking question paper 2
  • This question paper is set and marked by SQA and conducted in centres under conditions specified for external examinations by SQA.
  • Candidates complete this in 1 hour and 30 minutes.
• Specimen question papers for National 5 courses are published on SQA’s website. These illustrate the standard, structure, and requirements of the question papers candidates sit. The specimen papers also include marking instructions.

Grading

• A candidate’s overall grade is determined by their performance across the course assessment.
• The course assessment is graded A–D on the basis of the total mark for all course assessment components.

Grade description for C

• For the award of grade C, candidates will typically have demonstrated successful performance in relation to the skills, knowledge, and understanding for the course.

Grade description for A

• For the award of grade A, candidates will typically have demonstrated a consistently high level of performance in relation to the skills, knowledge, and understanding for the course.

Equality and inclusion

• This course is designed to be as fair and as accessible as possible with no unnecessary barriers to learning or assessment.
• For guidance on assessment arrangements for disabled candidates and/or those with additional support needs, please follow the link to the assessment arrangements web page: www.sqa.org.uk/assessmentarrangements.

Further information

• The following reference documents provide useful information and background.
  • National 5 Mathematics subject page
  • Assessment arrangements web page
  • Building the Curriculum 3–5
  • Design Principles for National Courses
  • Guide to Assessment
  • SCQF Framework and SCQF level descriptors
  • SCQF Handbook
  • SQA Skills Framework: Skills for Learning, Skills for Life and Skills for Work
  • Coursework Authenticity: A Guide for Teachers and Lecturers
  • Educational Research Reports
  • SQA Guidelines on e-assessment for Schools
  • SQA e-assessment web page

Appendix 1: course support notes

• These support notes are not mandatory. They provide advice and guidance to teachers and lecturers on approaches to delivering the course. They should be read in conjunction with this course specification and the specimen question paper.

Approaches to learning and teaching

• The purpose of this section is to provide general advice and guidance on approaches to learning and teaching across the course.
• The overall aim of the course is to develop a range of mathematical operational and reasoning skills that can be used to solve mathematical and real-life problems. Approaches to learning and teaching should be engaging, with opportunities for personalisation and choice built in where possible.
• A rich and supportive learning environment should be provided to enable candidates to achieve the best they can. This could include learning and teaching approaches such as:
  • Investigative or project-based tasks such as investigating the graphs of quadratic functions, perhaps using calculators or other technologies
  • A mix of collaborative and independent tasks which engage candidates, eg identifying gradient and y-intercept values from various forms of the equation of a straight line
  • Using materials available from service providers and authorities, eg working with real-life plans and drawings, using trigonometric skills to calculate line lengths and angle sizes
  • Problem-solving and critical thinking
  • Explaining thinking and presenting strategies and solutions to others — candidates may be provided with information which could be used to solve a problem, eg using simultaneous equations, and could then discuss their strategies in groups
  • Effective use of questioning and discussion to encourage more candidates to explain their thinking and to determine their understanding of fundamental concepts
  • Making links across the curriculum to encourage the transfer of skills, knowledge and understanding such as in science, technology, social subjects and health and wellbeing, eg liaising with physics on applications of appropriate formulae, such as F = ma, v = u + at, and s = ut + \frac{1}{2}at^2 — there should be a shared understanding across curriculum areas regarding approaches to changing the subject of a formula
  • Using technology to provide richer learning experiences and to develop confidence
• Developing mathematical skills is an active and productive process, building on candidates’ current knowledge, understanding and capabilities. Existing knowledge should form the starting point for any learning and teaching situation, with new knowledge being linked to existing knowledge and built on. Presenting candidates with an investigative or practical task is a useful way of allowing them to appreciate how a new idea relates to their existing knowledge and understanding.
• Questions can be used to ascertain a candidate’s level of understanding and provide a basis for consolidation or remediation where necessary. Examples of probing questions include:
  • How did you decide what to do?
  • How did you approach exploring and solving this task or problem?
  • Could this task or problem have been solved in a different way? If yes, what would you have done differently?
• As candidates develop concepts in mathematics, they will benefit from continual reinforcement and consolidation to build a foundation for progression.
• Throughout learning and teaching, candidates should be encouraged to:
  • Process numbers without using a calculator
  • Practise and apply the skills associated with mental calculations wherever possible
  • Develop and improve their skills in completing written and mental calculations in order to develop fluency and efficiency
• The use of a calculator should complement these skills, not replace them.

Integrating skills

• Integrating with other operational skills
  • Skills, knowledge, and understanding may be integrated with other operational skills, for example:
    • Expressions could be combined with equations
    • Gradients could be combined with the equation of a straight line
    • Indices could be combined with fractions
    • Completing the square could be combined with sketching a quadratic function
    • Related angles could be combined with sine and cosine rules
    • Vectors could be combined with Pythagoras’ theorem for vectors at right angles
    • Area of a triangle could be combined with area of sector
• Integrating with reasoning skills
  • Skills, knowledge, and understanding may be integrated with reasoning skills, for example:
    • Expressions could be derived from a mathematical problem before simplification
    • Compound solids could be broken down into simple solids to enable a volume to be calculated — this could be enclosed in a design problem
    • An algebraic fraction could be derived from a mathematical situation before simplifying
    • Accuracy in rounding answers could be required to suit a given situation
    • Simultaneous equations could be derived from a problem before solving
    • A quadratic equation and graph could be used in a context-based problem
    • A problem involving the use of Pythagoras’ theorem could be set in a real-life context
    • A problem involving the use of similarity could be set in a real-life context
    • Sine and cosine rules could be used in a problem situation involving bearings
    • Vectors could be used in a context-based problem
    • A problem involving the use of line of best fit could be set in a scientific context

Useful websites

The table below lists organisations that may provide resources suitable for the National 5 Mathematics course.

The above resources were correct at the time of publication and may be subject to change.

Preparing for course assessment

• The course assessment focuses on breadth, challenge and application. Candidates draw on and extend the skills they have learned during the course. These are assessed through two question papers: one non-calculator and a second paper in which a calculator may be used.
• In preparation for the course assessment, candidates should be given the opportunity to:
  • Analyse a range of real-life problems and situations involving mathematics
  • Select and adapt appropriate mathematical skills
  • Apply mathematical skills with and without the use of a calculator
  • Determine solutions
  • Explain solutions and/or relate them to context
  • Present mathematical information appropriately
• The question papers assess a selection of knowledge and skills acquired in the course and provide opportunities to apply skills in a wide range of situations, some of which may be new to the candidate.
• Prior to the course assessment, candidates may benefit from responding to short-answer questions, multiple-choice questions and extended-answer questions.

Developing skills for learning, skills for life and skills for work

• Course planners should identify opportunities throughout the course for candidates to develop skills for learning, skills for life and skills for work.
• Candidates should be aware of the skills they are developing and teachers and lecturers can provide advice on opportunities to practise and improve them.
• SQA does not formally assess skills for learning, skills for life and skills for work. There may also be opportunities to develop additional skills depending on approaches being used to deliver the course in each centre. This is for individual teachers and lecturers to manage.
• Significant opportunities to develop the skills for learning, skills for life and skills for work are described in the table below.

• During the course there are opportunities for candidates to develop their literacy skills and employability skills. Literacy skills are particularly important as these skills allow candidates to access, engage in and understand their learning, and to communicate their thoughts, ideas and opinions. This course provides candidates with the opportunity to develop their literacy skills by analysing real-life contexts and communicating their thinking by presenting mathematical information in a variety of ways. This could include the use of numbers, formulae, diagrams, graphs, symbols and words.
• Employability skills are the personal qualities, skills, knowledge, understanding, and attitudes required in changing economic environments. The mathematical operational and reasoning skills developed in this course aim to enable candidates to confidently respond to the mathematical situations that can arise in the workplace. It aims to provide candidates with the opportunity to analyse a situation, decide which mathematical strategies to apply, work through those strategies effectively, and make informed decisions based on the results.
• Additional skills for learning, skills for life and skills for work may also be developed during this course. These opportunities may vary and are at the discretion of the centre.

Appendix 2: skills, knowledge and understanding with suggested learning and teaching contexts

Examples of learning and teaching contexts that could be used for the course can be found below. The first two columns are identical to the tables of ‘Skills, knowledge and understanding for the course assessment’ in this course specification. The third column gives suggested learning and teaching contexts. These provide examples of where the skills could be used in individual activities or pieces of work.

| Algebraic skills | Explanation | Suggested learning and teaching contexts