4.1Rationale
The Foundation course is geared to the needs of students who are unready for or unsuited by the mathematics of the Ordinary course. Some are not yet at a developmental stage at which they can deal with abstract concepts; some may have encountered difficulties in adjusting to post-primary school and may need a particularly gradual introduction to second level work; some have learning styles that essentially do not match the traditional approach of post-primary schools. Many of the students may still be uncomfortable with material presented in the later stages of the primary curriculum. Nonetheless, they need to learn to cope with mathematics in everyday life and perhaps in further study.
The Foundation course, therefore, must help the students to construct a clearer knowledge of, and to develop improved skills in, basic mathematics, and to develop an awareness of its usefulness. Appropriate new material should also be introduced, so that the students can feel that they are making a fresh start and are progressing. The course therefore pays great attention to consolidating the foundation laid at primary level and to addressing practical issues; but it also covers new topics and lays a foundation for progress to more traditional study in the areas of algebra, geometry and functions. An appeal is made to different interests and learning styles, for example by paying attention to visual and spatial as well as numerical aspects.
For the target group, particular emphasis can be placed on promoting students' confidence in themselves (confidence that they can do mathematics) and in the subject (confidence that mathematics makes sense). Thus, attention must be paid to the acquisition and consolidation of fundamental skills, as indicated above; and concepts should be embedded in meaningful contexts. Many opportunities can thus be presented for students to achieve success.
4.2Aims
In the light of the general aims of mathematics education listed in section 1.2, the specific aims are that the Foundation course will provide students with the following:
- an understanding of basic mathematical concepts and relationships;
- confidence and competence in basic skills;
- the ability to solve simple problems;
- experience of following clear arguments and of citing evidence to support their own ideas;
- appreciation of mathematics both as an enjoyable activity through which they experience success and as a useful body of knowledge and skills.
4.3Assessment 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 Foundation course as formulated above.
4.4Content
The content of the primary curriculum is taken as a prerequisite, but many concepts and skills are revisited for revision and for treatment at a greater depth or level of difficulty.
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 and union (for two sets only), complement.
4. Commutative property for intersection and union.
Number systems
1. The set N of natural numbers. Order (<, , >, ). Idea of place value. Sets of multiples. Lowest common multiple.
The operations of addition, subtraction, multiplication and division in N where the answer is in N. Meaning of an for a,nN, n 0. Evaluation of expressions containing at most one level of brackets. Estimation leading to approximate answers.
2. The set Z of integers. Positional order on the number line.
The operation of addition in Z.
3. The set Q+ of positive rational numbers.
Fractions: emphasis on fractions having 2, 3, 4, 7, 8, 16, 5, 10, 100 and 1000 as denominators. Equivalent fractions. The operations of addition, subtraction and multiplication in Q+. Estimation leading to approximate answers. Fractions expressed as decimals; for computations without a calculator, computation for fractions with the above denominators excluding 3, 7 and 16.
Decimals: place value. The operations of addition, subtraction, multiplication and division. Rounding off to not more than three decimal places. Estimation leading to approximate answers.
Percentage: fraction to percentage. Suitable fractions and decimals expressed as percentages.
Equivalence of fractions, decimals and percentages.
Not envisaged as examination terminology.
Examples: 2 + 7 (4 1) 6 + 10 × 3 3(14 5) (7 + 2)
Example: 32 100
; 32%
Example: 42 100
; 0.42; 42%
4. Squares and square roots.
5. Commutative property.
Priority of operations.
Applied arithmetic and measure
1. Bills: shopping; electricity, telephone, gas, etc. Value added tax (VAT). Applications to meter readings and to fixed and variable charges. Percentage profit: to calculate selling price when given the cost price and the percentage profit or loss; to calculate the percentage profit or loss when given the cost and selling prices. Percentage discount. Compound interest for not more than three years. Calculating income tax.
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. Calculating distance from a map. Use of scales on drawings.
4. Perimeter.
Area: square, rectangle, triangle.
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), for area of disc (i.e. area of region enclosed by circle, r2). Use of formula for volume of cylinder ( r2h).
Statistics and data handling
1. Collecting and recording data. Tabulating data. Drawing and interpreting pictograms, bar-charts, pie-charts (angles to be multiples of 30° and 45°). Drawing and interpreting trend graphs. Relationships expressed by sketching such graphs and by tables of data; interpretation of such sketches and tables.
2. Discrete array expressed as a frequency table.
Mean and mode.
Not envisaged as examination terminology.
Percentage profit based on cost price or selling price (relevant one to be specified in examination questions). Also: percentage increase, e.g. 5% increase in attendance at a match.
Multiples and submultiples: mm, cm, km, g, cm2, km2, cm3, minute, hour. Use of "litre". Students should be familiar with everyday use of "weight".
See Guidelines for Teachers.
Algebra
1. Formulae, idea of an unknown, idea of a variable.
Evaluation of expressions of forms such as ax + by and a(x + y) where a, b, x, yN; evaluation of quadratic expressions of the form x2 + ax + b where a, b, xN.
2. Use of associative and distributive properties to simplify expressions of forms such as: a(x ± b) + c(x ± d) x(x ± a) + b(x ± c) where a, b, c, d, xN.
3. Solution of first degree equations in one variable where the solution is a natural number.
Relations, functions and graphs
1. Couples. Use of arrow diagrams to illustrate relations.
2. Plotting points. Joining points to form a line.
3. Drawing the graph of forms such as y = ax + b for a specified range of values of x, where a, bN. Simple interpretation of the graph.
Geometry
1. Synthetic geometry:
Preliminary concepts: The plane. Line ab, line segment [ab], |ab| as the length of the line segment [ab]. Angle; naming an angle with three letters. Straight angle. Angle measure; | abc| as the measure of abc. Acute, right and obtuse angles. Parallel lines; perpendicular lines. Vertically opposite angles. Triangle (scalene, isosceles, equilateral), quadrilateral (convex), parallelogram, rectangle, square.
Informal treatment (see Guidelines for Teachers).
Examples: Find the value of 3x + 7y and of 6(x + y) for given values of x and y. Find the value of x2 + 5x +7 when x = 4.
Examples: 3(x 2) + 2(x + 1) x(x + 1) + 2(x + 2) (see Guidelines forTeachers).
Examples: Solve 3x + 4 = 19. Solve 4(x 1) = 12.
Example: "is greater than"
Example: Draw the graph of y = 3x + 5 from x = 1 to x = 6.
Practical, intuitive approach, for example using drawings and paper-folding.
For constructions, the use of compasses, set squares, protractor, and straight-edge are allowed unless otherwise specified.
Use of geometrical instruments--ruler, compasses, set squares and protractor--to measure the length of a given line segment, the size of a given angle and the perimeter of a given square or rectangle.
Construction: To construct a line segment of given length.
Construction: To construct a triangle when given:
- the lengths of three sides;
- the lengths of two sides and the measure of the included angle;
- the length of a base and the measures of the base angles.
"Fact": A straight angle measures 180°.
"Fact": Vertically opposite angles are equal in measure.
"Fact": The measure of the three angles of a triangle sum to 180°.
Construction: To construct a right-angled triangle, given sufficient data.
"Fact" (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.
Construction: To construct a rectangle of given measurements.
"Fact": A diagonal bisects the area of a rectangle.
Construction: To draw a line through a point parallel to a given line.
Construction: To divide a line segment into two or three equal parts.
Construction: To bisect an angle without using a protractor.
Meaning of distance from a point to a line.
Meaning of base and corresponding perpendicular height of a parallelogram.
"Fact": The area of a parallelogram = |base| × (corresponding) perpendicular height.
2. Transformation geometry:
Central symmetry, axial symmetry.
Use of instruments to construct the image (rectilinear figures only) under (i) axial symmetry and (ii) central symmetry.
Ruler allowed.
Ruler allowed.
For interpretation of the use of the word "fact", see Guidelines for Teachers.
Ruler allowed.
Verification by finding the areas of the squares on the three sides or otherwise.
Ruler allowed.
Verification by paper-cutting or otherwise.
Intuitive approach using drawings.