3.1Rationale
The Ordinary course is geared to the needs of students of average mathematical ability. Typically, when such students come in to second level schools, some are only beginning to be able to deal with abstract ideas and some are not yet ready to do so. However, many of them may eventually go on to use and apply mathematics--perhaps even quite advanced mathematics--in their future careers, and all of them will meet the subject to a greater or lesser degree in their daily lives.
The Ordinary course, therefore, must start where these students are, offering mathematics that is meaningful and accessible to them at their present stage of development. It should also provide for the gradual introduction of more abstract ideas, leading the students towards the use of academic mathematics in the context of further study. The course therefore pays considerable attention to consolidating the foundation laid at primary level and to addressing practical topics; but it also covers aspects of the traditional mathematical areas of algebra, geometry, trigonometry and functions.
For the target group, particular emphasis can be placed on the development of mathematics as a body of knowledge and skills that makes sense and that can be used in many different ways--hence, as an efficient system for the solution of problems and provision of answers. 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.
3.2Aims
In the light of the general aims of mathematics education listed in section 1.2, the specific aims are that the Ordinary course will provide students with the following:
- an understanding of mathematical concepts and of their relationships;
- confidence and competence in basic skills;
- the ability to solve problems;
- an introduction to the idea of logical argument;
- appreciation both of the intrinsic interest of mathematics and of its usefulness and efficiency for formulating and solving problems.
3.3Assessment 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 Ordinary course as formulated above.
3.4Content
The content of the primary curriculum is taken as a prerequisite, but many concepts and skills are revisited for treatment at greater depth and at a greater level of difficulty or, ultimately, of 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.
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 anfor 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.
The operations of addition, subtraction, multiplication and division in Q. Rounding off. Estimation leading to approximate answers.
Ratio and proportion.
Not envisaged as examination terminology.
4. Rules for indices (where aQ, m, nN, m 0, n 0):
aman =am+n
am=amn, m > nan
(am)n=amn
Meaning of
Square roots, reciprocals: understanding and computation.
Scientific notation: non-zero positive rationals expressed in the form a × 10n, where nN and 1 a < 10.
5. The set R of real numbers: the idea that every point on the number line represents a real number. Order (<, , >, ).
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.
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) and sphere
Application to problems.
( , ). 4 2 4 3
3 rr
aa 12 0 , .
Not envisaged as examination terminology.
Percentage profit based on cost price or selling price (relevant one to be specified in examination questions).
Multiples and submultiples: mm, cm, km, cm2, hectare, km2, cm3, g, tonne, minute, hour. Use of "litre".
Problems may include compound figures made out of those specified above.
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 the form: (ax + b)(cx + d) (ax + b)(cx2+ dx + e) where a, b, c, d, eZ.
Addition and subtraction of expressions of the form ax + b ± ... ± dx + ecf where a, b, c, d, e, fZ.
3. Use of the distributive law 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+ bxx2 + bx + c where a, b, cZ.
Difference of two squares. Simple examples.
4. Formation and interpretation of number sentences leading to the solution of first degree equations in one variable.
First degree equations in two variables, with coefficients elements of Z and solutions also elements of Z. Problems and their solutions.
Quadratic equations of the form x2+ bx + c = 0 where b, cZ and x2+ bx + c is factorisable. Solution of simple problems leading to quadratic equations.
Examples: (2x + 3) + (4x 2) (3x + 2y) (x + 3y 4) (5x2+ 7x 2) + (2x2 x 7)
Examples: 3(x + 4) 5(2x + 3) + 2(x + 3) + 2(5x 6) y(2x + 1)
Examples: (2x 3)(5x + 4) (x 4)(x2 5x 11)
Example: 8xy 4y
Examples: x2 y2
x2 16 9 y2
5. Solution of equations of the form ax + b ± ... ± dx + e = gcf h where a, b, c, d, e, f, g, hZ.
6. Solution of linear inequalities in one variable, of forms such as ax + bc, where a, b, cZ, xZ.
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.
Mean and mode.
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.
Examples: 2x 1 9 10 2x >2
Harder examples than those expected at primary level.
Practical approach, for example using drawings.
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.
"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).
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.
Ruler allowed.
"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.
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.
Construction: To divide a line segment into three equal parts.
Circle: centre, arc, chord, tangent, segment, sector, radius, diameter, semicircle. Cyclic quadrilateral.
Theorem: An angle subtended by a diameter at the circumference is a right angle.
Theorem: The sum of opposite angles of a cyclic quadrilateral is 180°.
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.
Intuitive approach using drawings.
3. Coordinate geometry:
Coordinating the plane.
Coordinates of images of points under translation, axial symmetry in the x or y axis and central symmetry in the origin.
Using two points to get the midpoint, distance, slope.
Equation of a line in the form y y1= m(x x1).
Intersection of a line with x and y axes (using algebraic methods).
Trigonometry
1. Cosine, sine and tangent of angles less than 90°. Values of these ratios for integer values of angle. Value of angle (to nearest degree), given value of sin, cos, tan.
2. Solution of right-angled triangle problems of a simple nature involving heights and distances, including use of the Theorem of Pythagoras.
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 solutions of equations of the type f (x) = 0.
3. Graphing solution sets on the number line for linear inequalities in one variable.
4. Graphical treatment of solution of first degree simultaneous equations in two variables.
Formulae provided in examinations. Same scale to be used on each axis in diagrams.
Example: 2x + 1 < 5, xR