TECHNICAL MATHEMATICS
EXAMINATION GUIDELINES
GRADE 12
2021
TABLE OF CONTENTS
Page | |
1. INTRODUCTION | 3 |
2. ASSESSMENT IN GRADE 12 4 2.1 Format of question papers for Grade 12 4 2.2 Weighting of cognitive levels 5 | 4 4 5 |
3. ELABORATION OF CONTENT FOR GRADE 12 (CAPS) | 6 |
4. ACCEPTABLE REASONS: EUCLIDEAN GEOMETRY | 11 |
5. INFORMATION SHEET | 14 |
6. GUIDELINES FOR MARKING | 16 |
7. CONCLUSION | 16 |
PAPER | TOPICS | Weighting of content areas | DURATION | TOTAL | DATE | MARKING |
1 | Number Systems (binary and complex numbers) Algebra (expressions, equations and inequalities including nature of roots, exponents, surds and logarithms) | 50 ± 3 | 3 hours | 150 | November | External |
Functions and graphs | 35 ± 3 | |||||
Finance, growth and decay | 15 ± 3 | |||||
Differential calculus and integration | 50 ± 3 | |||||
2 | Analytical Geometry | 25 ± 3 | 3 hours | 150 | November | External |
Euclidean Geometry | 40 ± 3 | |||||
Trigonometry | 50 ± 3 | |||||
Mensuration, circles, angles and angular movement | 35 ± 3 |
COGNITIVE LEVEL | DESCRIPTION OF SKILLS TO BE DEMONSTRATED | WEIGHTING (plus, minus) | APPROXIMATE NUMBER OF MARKS IN A 150-MARK PAPER |
Knowledge | Recall Identification of correct formula on the information sheet (no changing of the subject) Use of mathematical facts Appropriate use of mathematical vocabulary Algorithms Estimation and appropriate rounding of numbers Definitions Properties of functions | (25 ± 2 )% | 34 to 40 marks |
Routine procedures | Perform well-known procedures Simple applications and calculations which might involve few steps Derivation from given information may be involved Identification and use (after changing the subject) of correct formula Generally similar to those encountered in class | (45 ± 2 )% | 64 to 70 marks |
Complex procedures | Problems involve complex calculations and/or higher-order reasoning There is often not an obvious route to the solution Problems need not be based on a real-life context Could involve making significant connections between different representations Require conceptual understanding Learners are expected to solve problems by integrating different topics | (20 ± 2 )% | 27 to 33 marks |
Problem solving | Non-routine problems (which are not necessarily difficult) Problems are mainly unfamiliar Higher order reasoning and processes are involved Might require the ability to break the problem down into its constituent parts Interpreting and extrapolating from solutions obtained by solving problems based in unfamiliar contexts. | (10 ± 2 )% | 12 to 18 marks |
THEOREM STATEMENT | ACCEPTABLE REASON(S) |
LINES | |
The adjacent angles on a straight line are supplementary. | ∠s on a str line |
If the adjacent angles are supplementary, the outer arms of these angles form a straight line. | adj ∠s supp |
The adjacent angles in a revolution add up to 360°. | ∠s round a pt OR ∠s in a rev |
Vertically opposite angles are equal. | vert opp ∠s = |
If AB || CD, then the alternate angles are equal. | alt ∠s; AB || CD |
If AB || CD, then the corresponding angles are equal. | corresp ∠s; AB || CD |
If AB || CD, then the co-interior angles are supplementary. | co-int ∠s; AB || CD |
If the alternate angles between two lines are equal, then the lines are parallel. | alt ∠s = |
If the corresponding angles between two lines are equal, then the lines are parallel. | corresp ∠s = |
If the co-interior angles between two lines are supplementary, then the lines are parallel. | co-int ∠s supp |
TRIANGLES | |
The interior angles of a triangle are supplementary. | ∠ sum in Δ OR sum of ∠s in Δ OR Int ∠s Δ |
The exterior angle of a triangle is equal to the sum of the interior opposite angles. | ext ∠ of Δ |
The angles opposite the equal sides in an isosceles triangle are equal. | ∠s opp equal sides |
The sides opposite the equal angles in an isosceles triangle are equal. | sides opp equal ∠s |
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. | Pythagoras OR Theorem of Pythagoras |
If the square of the longest side in a triangle is equal to the sum of the squares of the other two sides then the triangle is right-angled. | Converse Pythagoras OR Converse Theorem of Pythagoras |
If three sides of one triangle are respectively equal to three sides of another triangle, the triangles are congruent. | SSS |
If two sides and an included angle of one triangle are respectively equal to two sides and an included angle of another triangle, the triangles are congruent. | SAS OR S∠S |
If two angles and one side of one triangle are respectively equal to two angles and the corresponding side in another triangle, the triangles are congruent. | AAS OR ∠∠S |
If in two right-angled triangles, the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and one side of the other, the triangles are congruent | RHS OR 90°HS |
The line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half the length of the third side | Midpt Theorem |
THEOREM STATEMENT | ACCEPTABLE REASON(S) |
The line drawn from the midpoint of one side of a triangle, parallel to another side, bisects the third side. | line through midpt || to 2nd side |
A line drawn parallel to one side of a triangle divides the other two sides proportionally. | line || one side of Δ OR prop theorem; name || lines |
If a line divides two sides of a triangle in the same proportion, then the line is parallel to the third side. | line divides two sides of Δ in prop |
If two triangles are equiangular, then the corresponding sides are in proportion (and consequently the triangles are similar). | ||| Δs OR equiangular Δs |
If the corresponding sides of two triangles are proportional, then the triangles are equiangular (and consequently the triangles are similar). | Sides of Δ in prop |
If triangles (or parallelograms) are on the same base (or on bases of equal length) and between the same parallel lines, then the triangles (or parallelograms) have equal areas. | same base; same height OR equal bases; equal height |
CIRCLES | |
The tangent to a circle is perpendicular to the radius/diameter of the circle at the point of contact. | tan ⊥ radius tan ⊥ diameter |
If a line is drawn perpendicular to a radius/diameter at the point where the radius/diameter meets the circle, then the line is a tangent to the circle. | line ⊥ radius OR converse tan ⊥ radius OR converse tan ⊥ diameter |
The line drawn from the centre of a circle to the midpoint of a chord is perpendicular to the chord. | line from centre to midpt of chord |
The line drawn from the centre of a circle perpendicular to a chord bisects the chord. | line from centre ⊥ to chord |
The perpendicular bisector of a chord passes through the centre of the circle; | perp bisector of chord |
The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circle (on the same side of the chord as the centre) | ∠ at centre = 2 ×∠ at circumference |
The angle subtended by the diameter at the circumference of the circle is 90°. | ∠s in semi- circle OR diameter subtends right angle |
If the angle subtended by a chord at the circumference of the circle is 90°, then the chord is a diameter. | chord subtends 90° OR converse ∠s in semi -circle |
Angles subtended by a chord of the circle, on the same side of the chord, are equal | ∠s in the same seg. |
If a line segment joining two points subtends equal angles at two points on the same side of the line segment, then the four points are concyclic. | line subtends equal ∠s OR converse ∠s in the same seg. |
Equal chords subtend equal angles at the circumference of the circle. | equal chords; equal ∠s |
Equal chords subtend equal angles at the centre of the circle. | equal chords; equal ∠s |
THEOREM STATEMENT | ACCEPTABLE REASON(S) |
Equal chords in equal circles subtend equal angles at the centre of the circles. | equal circles; equal chords; equal ∠s |
The opposite angles of a cyclic quadrilateral are supplementary | opp ∠s of cyclic quad |
If the opposite angles of a quadrilateral are supplementary then the quadrilateral is cyclic. | opp ∠s quad supp OR converse opp ∠s of cyclic quad |
The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. | ext ∠ of cyclic quad |
If the exterior angle of a quadrilateral is equal to the interior opposite angle of the quadrilateral, then the quadrilateral is cyclic. | ext ∠ = int opp ∠ OR converse ext ∠ of cyclic quad |
Two tangents drawn to a circle from the same point outside the circle are equal in length | Tans from common pt OR Tans from same pt |
The angle between the tangent to a circle and the chord drawn from the point of contact is equal to the angle in the alternate segment. | tan chord theorem |
If a line is drawn through the end-point of a chord, making with the chord an angle equal to an angle in the alternate segment, then the line is a tangent to the circle. | converse tan chord theorem OR ∠ between line and chord |
QUADRILATERALS | |
The interior angles of a quadrilateral add up to 360°. | sum of ∠s in quad |
The opposite sides of a parallelogram are parallel. | opp sides of ||m |
If the opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram. | opp sides of quad are || |
The opposite sides of a parallelogram are equal in length. | opp sides of ||m |
If the opposite sides of a quadrilateral are equal, then the quadrilateral is a parallelogram. | opp sides of quad are = OR converse opp sides of a parm |
The opposite angles of a parallelogram are equal. | opp ∠s of ||m |
If the opposite angles of a quadrilateral are equal then the quadrilateral is a parallelogram. | opp ∠s of quad are = OR converse opp angles of a parm |
The diagonals of a parallelogram bisect each other. | diag of ||m |
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. | diags of quad bisect each other OR converse diags of a parm |
If one pair of opposite sides of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram. | pair of opp sides = and || |
The diagonals of a parallelogram bisect its area. | diag bisect area of ||m |
The diagonals of a rhombus bisect at right angles. | diags of rhombus |
The diagonals of a rhombus bisect the interior angles. | diags of rhombus |
All four sides of a rhombus are equal in length. | sides of rhombus |
All four sides of a square are equal in length | sides of square |
The diagonals of a rectangle are equal in length. | diags of rect |
The diagonals of a kite intersect at right-angles. | diags of kite |
A diagonal of a kite bisects the other diagonal. | diag of kite |
A diagonal of a kite bisects the opposite angles | diag of kite |