Rationale
The Foundation course is intended to equip students with the knowledge and techniques required in everyday life and in various kinds of employment. It is also intended to lay the groundwork for students who proceed to further education and training in areas in which specialist mathematics is not required. It should therefore provide students withthe mathematical tools needed in their daily life and work and (where relevant) continuingstudy; but it should do so in a context designed to build the students' confidence, theirunderstanding and enjoyment of mathematics, and their recognition of its role in theworld about them. Hence, material is chosen for its intrinsic interest and immediate applicability as well as its usefulness beyond school.
The course is designed for students who have had only very limited acquaintance with abstract mathematics. Basic knowledge is maintained and enhanced by being approached in an exploratory and reflective manner - availing of students' increasing maturity- rather than by simply repeating work done in the Junior Cycle. Concreteness is provided in particular by extensive use of the calculator; this serves as an investigative tool as well as an object of study and a readily available resource. By means of such a developmental and constructive approach, the ground is prepared for students' advance to abstract concepts via a multiplicity of carefully graded examples. Computational work is balanced by emphasis on the visual and spatial.
For the target group, particular emphasis can be given to aims concerned with the use of mathematics in everyday life and work - especially as regards intelligent and proficient use of calculators- and with the recognition of mathematics in the environment.
2.2 Aims
In the light of the aims of mathematics education listed in Section 1.2, the aims of the Foundation course are:
- development of students' understanding of mathematical knowledge and techniques required in everyday life and employment;
- particular emphasis on meaningfulness of mathematical concepts;
- acquisition of mathematical knowledge that is of immediate applicability and usefulness;
- introduction of the students to mathematical abstraction;
- maintenance and enhancement of students' basic mathematical knowledge and skills;
- encouragement of accurate and efficient use of the calculator;
- promotion of students' confidence in working with mathematics.
Assessment Objectives
The assessment objectives are the objectives (a), (b), (c), (d) and (e) listed in Section 1.3. These objectives should be interpreted in the context of the statement of the aims of the Foundation course. Knowledge of the content of the Junior Certificate Foundation course will be assumed.
Structure and Content
The syllabus is presented without options. It is therefore envisaged that students would study the entire course.
Content
Number systems
Revision of the following, using calculator for all relevant aspects:
- Development of the systems N of natural numbers, Z of integers, Q of rational numbers and R of real numbers. The operations of addition, multiplication, subtraction and division. Representation of numbers on a line. Inequalities. Decimals. Powers and roots. Scientific notation.
- Factors, multiples, prime numbers in N. Prime factorisation.
- Use of brackets. Conventions as to the order of precedence of operations.
Arithmetic
Use of calculator for all relevant operations in the following:
- Approximation and error; rounding off. Relative error, percentage error, tolerance. Very large and very small numbers on the calculator. Limits to accuracy of calculators.
- Substitution in formulae.
Main stages of calculation should be shown - Proportion. Percentage. Averages. Average rates of change (with respect to time).
- Compound interest and depreciation formulae.
A = P ( 1 100r-) °
P = N l l - + r100)n
Formula provided in examinations; natural number. - Value added tax (VAT). Rates. Income tax (including PRSI); emergency tax; tax tables.
- Domestic bills and charges.
- Currency transactions, including commission.
- Costing. Materials and labour. Wastage.
- Metric system. Change of units. Everyday Imperial units.
Conversion factors provided for Imperial units.
Areas and Volumes
Use of calculator for all relevant operations in the following:
- Plane figures: disc, triangle, rectangle, square, H-figure, parallelogram, trapezium. Solid figures: right cone, rectangular block, cylinder, sphere, right prism.
See Appendix. Questions will be confined to the variables given in the formulae ("engineer's handbook" approach). - Use of Simpson's Rule to approximate area.
Algebra
- Consideration of the following, using calculator where relevant:
i)x + a = b;)
ii) ax = b; ) a, b, cQ
iii) ax + b = c;)
iv) ax + b = cx; a, b, c Z
v) ax+ b = c x + d; a , b , c , d Z
vi) ax + by = c; ) a, b, c, d, e, dx + ey = f; ) f Z
Cases with unique solutions only. - Problems giving rise to equations of type (i) - (vi)
vii) x2 = a; a ~ Q+
viii) x 2 + a = b ; b - a > O,a , b ~ Q
ix) ax2 = b; a, b ~ Q+
x) ax2 + b = c; a > 0 , (c-b) > 0,
xi) ax 2 + b x + c = 0 ; a > 0 , b2 ~ 4 a c , a,b, c s Z a, b, c ~ Z
Use of formula (provided in examinations) - Consideration of the inequalities:
i) x + a > b ; x + a < b ; )
ii) ax > b;ax < b;)
iii) a x + b > c ; a x + b < c ; ) a,b, c s Z
iv) x + a ~ b ; x + a b ; )
v) a x ~ b ; a x b ; )
vi) a x + b ~ c ; a x + b c ; )
Statistics and probability
- Fundamental Principle of Counting: if one task can be accomplished in x different ways, and following this a second task can be accomplished in y different ways, then the first task followed by the second task can be accomplished in xy different ways.
Use in examples. - Discrete probability: simple cases. For equally likely outcomes, probability = (number of outcomes of interest)/ (number of possible outcomes).
Examples including coin tossing, dice throwing, birthday distribution, card drawing (one or two cards), and sex distribution. - Statistics: graphical and tabular representation of statistical data; grouped and ungrouped frequency distributions. Mean; cumulative frequencies and cumulative frequency graph; median; weighted mean. Concept of dispersion; standard deviation of ungrouped array of not more than ten numbers.
Emphasis on use of calculator. Median obtained from array or cumulative frequency graph; finding median from histogram excluded (but histogram itself included).
Trigonometry
- Sine, cosine and tangent as ratios in a right-angled triangle.
- Solving for an unknown in a right-angled triangle.
Problems to include diagrams.
4. Interpretation of graphs in following cases:
Case 1:
cases in which information is available only at plotted points
Case 2:
continuous graphs:
- distance/time
- speed/time
- depth of liquid/time
- conversion of units
Examples:
currency fluctuations
inflation
employment/unemployment
temperature
temperature chart (medical)
pollen count
lead levels
smog
Interpretation to include: given range of values of one variable, estimate from the graph the corresponding range of values of the other.
Geometry
- Co-ordinate geometry:
Distance between two points.
Slope of a line through two points. Perpendicular lines.
Midpoint of a line segment.
Equation of line: y = mx + c.
Parallel lines.
Formulae will be given in examinations.
Obtaining equation of line, given slope and one point or given two points. - Geometrical results- knowledge of the following and use in numerical examples:
Proofs excluded.
(a) Vertically opposite angles are equal;
"Equal" means equal in measure.
(b) When a transversal cuts two parallel lines, the corresponding angles are equal, and the alternate angles are equal;
(c) Opposite sides and angles of a parallelogram are equal;
(d) The sum of the angles of a triangle is 180°;
(e) The base angles of an isosceles triangle are equal;
(f) The angle on a (straight) line is 180°;
(g) The Theorem of Pythagoras;
(h) The angle in a semicircle is a right angle. - Constructions:
(a) To draw a perpendicular from a given point an a line;
(b) To draw a perpendicular from a point at the end of a line segment;
(c) To draw a perpendicular to a given line from a point not on the line;
(d) To construct an angle of 60°;
(e) To construct an angle equal to a given angle;
(f) To draw a line parallel to a given line through a point;
(g) To construct a parallelogram (given sufficient data);
(h) To draw the circumscribed circle of a given triangle;
(i) To draw the inscribed circle of a given triangle;
(j) To draw the tangent to a circle at a given point on the circle. - Enlargements:
Main emphasis on construction.
Enlargement of a rectilinear figure by the ray method. Centre of enlargement. Scale factor k. Two cases to be considered:
k > 1, k Q (enlargement): 0 < k < 1, k Q (reduction).
A triangle abc with centre of enlargement a, enlarged by a scale factor k, gives an image triangle ab'c' with bc parallel to b'c'.
Object length, image length, calculation of scale factor.
Finding the centre of enlargement.
A rectilinear region when enlarged by a scale factor k has its area multiplied by a factor k2. - Repeating patterns. Identification of axial symmetry, planes in of symmetry, central symmetry, and rotational symmetry in given figures. Patterns in different cultures.
Co-ordinate treatment not included.
3. Assessment
It is envisaged that, at present, the courses would be assessed by means of final written examinations.
The following principles would apply:
a. The status and standing of the Leaving Certificate would be maintained.
b. Candidates would be able to demonstrate what they know rather than what they do not know.
c. Examinations would build candidates' confidence that they can do mathematics, rather than undermining the confidence of those who attempt them.
Note
Restriction at present to assessment by formal written examination has governed the specification of assessment objectives (for those for the Foundation course, see Section 2.3); they have been limited to a subset of the general objectives (see Section 1.3). In the future, it may be possible to introduce a coursework component. This would facilitate assessment of the other objectives, notably problem-solving, communicative and creative skills, and in particular of work done with the aid of computers.
4. Select Bibliography
Among the national and international literature consulted, the following three national reports are of particular relevance:
- Curriculum and Examinations Board. Mathematics Education: Primary and Junior Cycle Post-Primary. Dublin: Curriculum and Examinations Board, 1986.
- Mathematics Counts. Report of the Committee of Inquiry into the Teaching of Mathematics in Schools under the Chairmanship of Dr. W. H. Cockcroft. (The Cockcroft Report). London: HMSO, 1982.
- National Research Council. Everybody Counts: a Report to the Nation on the Future of Mathematics Education: Washington, D.C.: National Academy Press, 1989.