Solutions for Mathematics Enrichment

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Progress to Higher Mathematics

A book designed to aid progression towards the SQA Higher Mathematics examination.

Progress to Higher Mathematics

Written by a team of highly experienced mathematics teachers, Progress to Higher Mathematics provides carefully selected materials which will aid progression towards the Higher Mathematics examination offered by the Scottish Qualifications Authority.

Designed for use either as a precursor to the Higher syllabus or as introductory exercises throughout the teaching of the course, Progress to Higher Mathematics will help students to:

• build on previous knowledge and skills gained at National 5, Standard Grade or Intermediate 2
• consolidate the essential skills necessary for Higher Mathematics
• enhance algebraic expertise
• utilize graphicacy as a means of improving mathematical understanding
• develop the geometrical ideas needed for a thorough understanding of analytical geometry

A sample page from each chapter can be seen in the Book Gallery.

Reviews of Progress to Higher Mathematics appear in:

• The Mathematical Gazette (March 2009)
• Mathematics in School (November 2008)
• The Scottish Mathematical Council Journal 37 (March 2008)

• Mathematical Gazette review (PDF)
• Mathematics in School review (PDF)
• SMC Journal review (PDF)
• Book Gallery
• Order
• Order form (PDF)

Learning outcomes

A Checklist of Learning Outcomes for pupil use is available as a free PDF document. Teachers may freely print or photocopy the checklist for distribution to students. The document may be viewed or downloaded using the link:

• Checklist of Learning Outcomes (PDF)

Preface

We have written this book in response to the experience of teachers in Scotland who are familiar with the difficulties students face as they make the transition from Standard Grade or Intermediate 2 to Higher Mathematics: specifically a lack of mathematical skills which makes some concepts inaccessible. Progress to Higher Mathematics aims to make the transition smoother and more motivating.

Every year SQA examiners and the Inspectorate comment on the need to improve skills among Higher Mathematics candidates. In many schools, the relaxation of the age and stage regulations makes it possible to teach the Mathematics curriculum with greater flexibility and provide the time to consolidate the essential skills necessary for Higher Mathematics. For teachers in these schools, this book should be a welcome resource.

Progress to Higher Mathematics provides a series of exercises which can be used either as a precursor to the Higher syllabus or as introductory material throughout the course. The approach emphasises graphicacy since it is recognised that graphics play a key role for the modern learner. Basic algebraic skills are also stressed. Collectively the exercises will help students gain a thorough grounding in the essential skills required for success at Higher Mathematics.

We have designed the units to suit students with any starting level of competence, from basic to well beyond Intermediate. The range of exercises should enable students of all abilities to practise the techniques essential to improving their performance at Higher. Each exercise has clearly defined learning outcomes, allowing it to be used individually or as part of a complete unit. Within each exercise the questions are graded and cross-referenced to related material elsewhere in the book. Extension materials are indicated by a snowflake symbol.

Contents

Progress to Higher Mathematics contains 53 exercises organised into nine units, as shown below, where an asterisk * indicates an extension topic.

Answers are given to all the exercises and there is a two-page index.

1. Preliminaries
   1. Expanding brackets
   2. Linear equations
   3. Simultaneous equations
   4. Expressions with fractions
   5. Surds
   6. Indices
   7. The angles 30°, 45° and 60°
2. Solving equations
   1. Equations with fractions
   2. Quadratic equations
   3. General quadratic equations
   4. Quadratic equations resulting from a substitution
   5. Equations of the form k(x − p)² = q
   6. * Cubic equations
3. Lines and circles
   1. The gradient and equation of a straight line
   2. Lines parallel to the axes
   3. Parallel lines
   4. Midpoint and length of a line
   5. Collinear points
   6. The angle between a line and the x-axis
   7. Circles
   8. The distance between circles
4. Graph sketching
   1. Parabolas of the form y = k(x − a)(x − b)
   2. Parabolas of the form y = k(x − p)² + q
   3. * Cubic curves
5. Equations of curves
   1. The parabola equation y = kx² + q
   2. The parabola equation y = k(x − a)(x − b)
   3. The turning point of the parabola y = k(x − p)² + q
   4. The parabola equation y = k(x − p)² + q
   5. The equation of a parabola
   6. * Changing the form of a parabola equation
   7. * The cubic curve equation y = k(x − a)(x − b)(x − c)
6. Intersecting lines and curves
   1. The intersection of two straight lines
   2. The intersection of a straight line and a parabola
   3. The intersection of two parabolas
   4. * The intersection of a straight line and a cubic curve
7. Trigonometric graphs
   1. Basic trigonometric graphs
   2. Graphs of the form y = p sin qx° or y = p cos qx°
   3. Graphs of the form y = p sin x° + s or y = p cos x° + s
   4. Graphs of the form y = p sin(x + r)° or y = p cos(x + r)°
   5. Graphs of the form y = sin(x + r)° + s or y = cos(x + r)° + s
   6. * Graphs of the form y = p sin(x + r)° + s or y = p cos(x + r)° + s
8. Trigonometry
   1. Basic trigonometric equations
   2. Harder trigonometric equations
   3. Angles related to 30°, 45° and 60°
   4. Equations involving 30°, 45°, 60° and related angles
   5. Trigonometric ratios of the same angle
   6. The identities sin² x + cos² x ≡ 1 and tan x ≡ sin x/cos x
   7. Quadratic trigonometric equations
   8. * Further trigonometric graphs and equations
9. Algebra
   1. Functional notation
   2. Constructing expressions
   3. Forming and solving equations
   4. * Surds and indices

© A K Jobbings 2004–2017