2.1Rationale
The Higher course is geared to the needs of students of above average mathematical ability. Among the students taking the course are those who will proceed with their study of advanced mathematics not only for the Leaving Certificate but also at third level; some are the mathematicians of the next generation. However, not all students taking the course are future specialists or even future users of academic mathematics. Moreover, when they start to study the material, some are only beginning to be able to deal with abstract concepts.
A balance must be struck, therefore, between challenging the most able students and encouraging those who are developing a little more slowly. Provision must be made not only for the academic student of the future, but also for the citizen of a society in which mathematics appears in, and is applied to, everyday life. The course therefore focuses on material that underlies academic mathematical studies, ensuring that students have a chance to develop their mathematical abilities and interests to a high level; but it also covers the more practical and obviously applicable topics that students are meeting in their lives outside school.
For the target group, particular emphasis can be placed on the development of powers of abstraction and generalisation and on an introduction to the idea of proof--hence giving students a feeling for the great mathematical concepts that span many centuries and cultures. Problem-solving can be addressed in both mathematical and applied contexts. Alongside this, adequate attention must be paid to the acquisition and consolidation of fundamental skills, in the absence of which the students' development and progress will be handicapped.
2.2Aims
In the light of the general aims of mathematics education listed in section 1.2, the specific aims are that the Higher course will provide students with the following:
- a firm understanding of mathematical concepts and relationships;
- confidence and competence in basic skills;
- the ability to formulate and solve problems;
- an introduction to the idea of proof and to the role of logical argument in building up a mathematical system;
- a developing appreciation of the power and beauty of mathematics and of the manner in which it provides a useful and efficient system for the formulation and solution of problems.
2.3Assessment objectives
The assessment objectives are objectives A, B, C, D (dealing with knowledge, understanding and application), G (dealing with psychomotor skills) and H (dealing with communication). These objectives should be interpreted in the context of the aims of the Higher course as formulated above.
2.4Content
Knowledge of the content of the primary curriculum is assumed, but many concepts and skills are revisited for treatment at greater depth and at a greater level of difficulty or abstraction.
It is assumed that calculators and mathematical tablesare available for appropriate use.
Sets
1. Listing of elements of a set. Membership of a set defined by a rule. Universe, subsets. Null set (empty set). Equality of sets.
2. Venn diagrams.
3. Set operations: intersection, union, difference, complement. Set operations extended to three sets.
4. Commutative property and associative property for intersection and union; failure of commutativity and associativity for difference; necessity of brackets for the nonassociative operation of difference. Distributive property of union over intersection and of intersection over union; necessity of brackets.
Number systems
1. The set N of natural numbers. Order (<, , >, ). Place value. Sets of divisors. Pairs of factors. Prime numbers. Sets of multiples. Lowest common multiple. Highest common factor. Cardinal number of a set.
The operations of addition, subtraction, multiplication and division in N. Meaning of an for a, nN, n 0. Estimation leading to approximate answers.
2. The set Z of integers. Order (<, , >, ).
The operations of addition, subtraction, multiplication and division in Z. Use of the number line to illustrate addition, subtraction and multiplication. Meaning of an for aZ, nN, n 0. Estimation leading to approximate answers.
3. The set Q of rational numbers. Decimals, fractions, percentages. Decimals and fractions plotted on the number line.
Rational numbers expressed as decimals. Terminating decimals expressed as fractions.
The operations of addition, subtraction, multiplication and division in Q. Rounding off. Significant figures for integer values only. Estimation leading to approximate answers.
Ratio and proportion.
Not envisaged as examination terminology.
4. Meaning ap of where a, pQ.
Rules for indices (where a, b, p, qQ and a, b 0):
Square roots, reciprocals: understanding and computation.
Scientific notation: non-zero positive rationals expressed in the form a × 10n, where nZ and 1 a < 10.
5. The set R of real numbers: every point on the number line represents a real number. Order (<, , >, ).
Addition, subtraction and multiplication applied to where aQ, bQ+.
The set of irrational numbers R \ Q.
6. Commutative and associative properties for addition and multiplication; failure of commutativity and associativity for subtraction and division; distributive property of multiplication over addition.
Priority of operations.
Not envisaged as examination terminology.
Applied arithmetic and measure
1. Bills. Profit and loss. Percentage profit. Percentage discount. Tax. Annual interest. Compound interest (interest added at regular intervals to a maximum of three; formula not required). Value added tax (VAT).
2. SI units of length (m), area (m2), volume (m3), mass (kg), and time (s). Multiples and submultiples. Twenty-four hour clock, transport timetables. Relationship between average speed, distance and time.
3. Perimeter.
Area: square, rectangle, triangle.
Surface area and volume of rectangular solids (i.e. solids with uniform rectangular cross-section).
Length of circumference of circle = . Length of diameter
Use of formulae for length of circumference of circle (2 r) and for area of disc (i.e. area of region enclosed by circle, r2). Use of formulae for curved surface area and volume of cylinder (2 rh,r2h), right circular cone and sphere
Application to problems, including use of the Theorem of Pythagoras.
Algebra
1. Meaning of variable, constant, term, expression, coefficient.
Evaluation of expressions.
2. Addition and subtraction of simple algebraic expressions of forms such as: (ax + by + c) ± ... ± (dx + ey + f) (ax2 + bx + c) ± ... ± (dx2 + ex + f) where a, b, c, d, e, fZ.
Use of the associative and distributive property to simplify such expressions as: a(bx + cy + d) + ... + e(fx + gy + h) a(bx2+ cx + d) ax(bx2+ c) where a, b, c, d, e, f, g, hZ.
Multiplication of expressions of forms such as: (ax + b)(cx + d) (ax + b)(cx2+ dx + e) where a, b, c, d, eZ.
( , ). 4 2 4 3
3 rr
( , ) rlr h 1 3
Multiples and submultiples: mm, cm, km, cm2, hectare, km2, cm3, g, tonne, minute, hour. Use of "litre".
Derivation and use of the relevant formulae for perimeter, area and volume.
Problems may include compound figures made out of those specified above.
Examples: (2x + 3) + (4x 2) (3x + 2y) (x + 3y 4) (5x2+ 7x 2) + (2x2 x 7)
Examples: 3(x + 4) 5(2x + 3) + 2(5x 6) 5(3x2 4x + 8)
Examples: (2x 3) (5x + 4) (x 4) (x2 5x 11)
Percentage profit based on cost price or selling price (relevant one to be specified in examination questions).
Division of expressions of forms such as: (ax2 + bx + c) ÷ (ex + f) (ax3 + bx2+ cx + d) ÷ (ex + f) where a, b, c, d, e, fZ.
Rearrangement of formulae.
Addition and subtraction of expressions of the form: ax + b ± ... ± dx + ecf where a, b, c, d, e, fZ, a ± pbx + c qx + r where a, b, c, p, q, rZ.
3. Use of the distributive property in the factorising of expressions such as: abxy + ay where a, bZ, sx ty + tx sy where s, t, x, y are variable.
Factorisation of quadratic expressions of the form: ax2 + bxax2 + bx + c where a, b, cZ.
Difference of two squares of the form a2x2 b2y2, where a, bN.
4. Formation and interpretation of number sentences leading to the solution of first degree equations in one variable.
First degree equations in two variables. Problems and their solutions.
Quadratic equations of the form ax2 + bx + c = 0. Solution using factors and/or the formula for real roots only. Problems and their solutions.
5. Equations of the form: ax + b ± ... ± dx + e = gcfh where a, b, c, d, e, f, g, hZ. a ± ... ± p = dbx + cqx + re where a, b, c, p, q, r, d, eZ. Problems and their solutions.
6. Solution of linear inequalities in one variable, of forms such as: ax + bcabx + c < d where a, b, c, dZ.
Examples: (2x2 + 11x + 15) ÷ (x + 3) (6x2 + x 12) ÷ (3x 4) (6x3 x2 33x 28) ÷ (3x + 4)
Example: 8xy 4y
Examples: 2x 1 9 1 2x 1 < 11 3 > 2x 7 > 5
Statistics
1. Collecting and recording data. Tabulating data. Drawing and interpreting bar-charts, pie-charts, and trend graphs.
2. Discrete array expressed as a frequency table. Drawing and interpreting histograms.
Mean and mode. Mean of a grouped frequency distribution.
Cumulative frequency. Ogive, median, interquartile range.
Geometry
1. Synthetic geometry:
Preliminary concepts: The plane. Subsets of the plane: line ab, line segment [ab], half line [ab; collinear points. |ab|as the length of the line segment [ab]. Half-planes. Angle; naming an angle with three letters. Straight angle. Angle measure; |abc| as the measure of abc. Acute, right, obtuse, and reflex angles. Parallel lines; perpendicular lines. Vertically opposite, alternate and corresponding angles. Triangle (scalene, isosceles, equilateral), quadrilateral (convex), rhombus, parallelogram, rectangle, square, circle. Concept of area in relation to these figures.
"Fact": A straight angle measures 180°.
*Theorem: Vertically opposite angles are equal in measure.
"Fact": Alternate angles are equal in measure when formed by two parallel lines intersecting a third line.
"Fact": Corresponding angles are equal in measure when formed by two parallel lines intersecting a third line.
"Fact": In the diagram below:
(a) If |abc|= |bcf |then line L || line M (b) If |abc|= |dcg|then line L || line M
(i.e. the converses of the two previous "facts" are true).
In the case of the theorems marked with an asterisk (*), formal proofs may be examined; in the case of other results stated, proofs will not be examined.
For constructions, the use of compasses, set squares, protractor, and straight-edge are allowed unless otherwise specified.
See Guidelines for Teachers.
For interpretation of the use of the word "fact", see Guidelines for Teachers.
*Theorem: The measures of the three angles of a triangle sum to 180°.
*Theorem: An exterior angle of a triangle equals the sum of the two interior opposite angles in measure.
Construction: To construct a triangle, given sufficient data.
Meaning of congruent triangles.
"Fact": Two triangles are congruent if they satisfy any one of the following four conditions:
- three sides in one equal in measure to three sides in the other (SSS);
- two sides and the included angle in one equal in measure, respectively, to two sides and the included angle in the other (SAS);
- two angles and a side in one equal in measure, respectively, to two angles and a corresponding side in the other (ASA);
- a right angle, hypotenuse and a side equal in measure, respectively, in each (RHS).
Construction: To bisect an angle without using a protractor.
*Theorem: If two sides of a triangle are equal in measure, then the angles opposite these sides are equal in measure.
Converse: If a triangle has two angles equal in measure, then the sides opposite these angles are equal in measure (i.e. the triangle is isosceles).
"Fact": If in a triangle two sides are of unequal length, then the angles opposite these sides are unequal in measure and the larger angle is opposite the longer side.
"Fact": Any two sides of a triangle are together greater in measure than the third side.
"Fact": The area of any rectilinear figure is equal to the sum of the areas of any two non-overlapping rectilinear figures of which it is composed.
*Theorem: Opposite sides and opposite angles of a parallelogram are respectively equal in measure.
*Theorem: A diagonal bisects the area of a parallelogram.
Theorem: The diagonals of a parallelogram bisect each other.
Meaning of distance from a point to a line.
Ruler allowed.
Meaning of base and corresponding perpendicular height of a triangle and a parallelogram.
"Fact": The area of a rectangle = length × breadth.
Theorem: The area of a triangle = |base| × (corresponding) perpendicular height.
Theorem: The area of a parallelogram = |base| × (corresponding) perpendicular height.
Construction: To construct the perpendicular bisector of a line segment without using a protractor or set square.
Theorem: Any point on the perpendicular bisector of a line segment [ab] is equidistant from a and b.
Converse: Any point equidistant from two points a and b lies on the perpendicular bisector of the line segment [ab].
Circle: centre, arc, chord, tangent, segment, sector, radius, diameter, semicircle. Cyclic quadrilateral.
Construction: To construct the circumcircle of a triangle.
Theorem: Any point on the bisector of an angle is equidistant from the half lines forming the angle.
Construction: To construct the incircle of a triangle.
*Theorem: The measure of the angle at the centre of the circle is twice the measure of the angle at the circumference, standing on the same arc.
Deduction: All angles at the circumference on the same arc are equal in measure. Deduction: An angle subtended by a diameter at the circumference is a right angle. Deduction: The sum of opposite angles of a cyclic quadrilateral is 180°.
Theorem: If a line passes through a point t on a circle and is perpendicular to the diameter at t, then the line is a tangent to the circle at t.
Converse: The tangent at any point of a circle is perpendicular to the diameter drawn to the point of contact.
*Theorem: A line through the centre of a circle perpendicular to a chord bisects the chord.
Theorem: A line drawn parallel to one side of a triangle divides the other two sides in the same ratio.
Deduction: In the diagram above:
ab =
acaxay
Construction: To divide a line segment into a given number of equal parts.
*Theorem: If two triangles are equiangular, the lengths of corresponding sides are in proportion.
*Theorem (Theorem of Pythagoras): In a right-angled triangle, the square of the length of the side opposite to the right angle is equal to the sum of the squares of the lengths of the other two sides.
Converse of the Theorem of Pythagoras: If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle has a right angle and this is opposite the longest side.
2. Transformation geometry:
Translation, central symmetry, axial symmetry. Translation and central symmetry map a line onto a parallel line. Axis and centre of symmetry. Rotation.
3. Coordinate geometry:
Coordinating the plane.
Coordinates of images of points under translation, axial symmetry and central symmetry.
Distance. Midpoint.
Slope of a line. Parallel and perpendicular lines.
Equation of a line in the forms: ax + by + c = 0 y = mx + cy y1 = m(x x1) where a, b, c, m, x1, y1 Q.
Intersection of lines.
Intuitive approach using drawings.
Trigonometry
1. Cosine, sine and tangent of angles between 0° and 360° (inclusive). Functions of 30°, 45° and 60° in surd form, derived from suitable triangles.
2. Solution of right-angled triangles and triangles requiring applications of the sine rule. Relevant problems.
Use of formulae 1 2ab sin C, 1 2bc sin A, 1 2ca sin B for finding area.
Functions and graphs
1. Concept of a function. Couples, domain, codomain, range.
2. Use of function notation:
f (x) =
f : x
y =
Drawing graphs of functions f : xf (x), where f (x) is of the form ax + b or ax2+ bx + c, where a, b, cZ, xR.
Using the graphs to estimate the (range of) value(s) of x for which f (x) = k, where kR.
3. Maximum and minimum values of quadratic functions estimated from graphs.
4. Graphing solution sets on the number line for linear inequalities in one variable.
5. Graphical treatment of solution of first degree simultaneous equations in two variables.
Proof of sine rule not required.
Proof of formulae not required.
Solution of quadratic inequalities is excluded, but students may be asked to read off a range of values for which a function is (say) negative.