Mastering Mathematics Smartly
by Wee Wen Shih

A unique self-help website that provides comprehensive coverage of mathematics at A-level & beyond, written in a student-friendly style.

Pre-course resources for the keen learner

For those who are keen to start already, here are 3 e-books and a weblink for your reference. Have a very merry christmas and a happy new year!

Week 1

  • Course intro slides.

  • Chapter 1 slides.

  • Chapter 1 main points:

    1. A definition consists of defined and undefined terms. An undefined term could be a geometric term, a general mathematics term or a logical term.

    2. A proposition is a statement that is either true or false, but not both.

    3. A proposition that is assumed to be true is an axiom or a postulation. A proposition that can be proved to be true is a theorem.

    4. A proposition "p \rightarrow q" has 3 other variations, i.e. its converse "q \rightarrow p", its contrapositive "\sim q \rightarrow \sim p" and its inverse "\sim p \rightarrow \sim q". Note that "\sim p" refers to "not-p" and "\sim q" refers to "not-q".

    5. To prove that "p \rightarrow q" is true, we start with "\sim q" and proceed to show that the conclusion is "\sim p". Then, we apply the law of contrapositive and claim that "p \rightarrow q" is indeed true.

    6. Recall the various forms of the implication statement at the end of Chapter 1 slides.

    7. Do the questions in the DSM 101 specimen paper, as a form of practice and consolidation. Write down the converse, contrapositive and inverse for all "if-then" statements. We will check solutions briefly during the next lesson on Wednesday, 14 Jan '09.

    8. For the exercise, please submit your own work for Q8 and Q9 to me on Friday, 16 Jan '09.

Week 2

  • Specimen paper Q1(a)(i):

    Proposition: If I bring you to the zoo, then it is sunny.
    Converse: If it is sunny, then I bring you to the zoo.
    Contrapositive: If it is not sunny, then I do not bring you to the zoo.
    Inverse: If I do not bring you to the zoo, then it is not sunny.

    Q1(a)(iii):

    Proposition: If a>0, then ab>0.
    Converse: If ab>0, then a>0.
    Contrapositive: If ab \leq 0, then a \leq 0.
    Inverse: If a \leq 0, then ab \leq 0.

  • Chapter 2 slides.
    Please take note of two errors on slide page 7.
    (i) No, since R does not lie on the line segment PR.

    (iv) No, since R does not lie on the ray RP.

  • Chapter 2 main points:

    1. Corresponding angles postulation:
    (a) two corresponding angles are equal \rightarrow two lines are parallel
    (b) two lines are parallel \rightarrow two corresponding angles are equal

    2. Alternate angles theorem:
    (a) two alternate interior angles are equal \rightarrow two lines are parallel
    (b) two lines are parallel \rightarrow two alternate interior angles are equal

    3. Sum of interior angles of a triangle = 180^{\circ}.

    4. Exterior angle is the sum of two interior opposite angles (can you prove it?).

    5. Attempt questions 5, 6, 7 and 8 and submit your work next Friday (23 Jan '09). We could start preliminary work on exercise 2 at the next lesson on Friday :)

    6. Next lesson on Friday (16 Jan '09) will be held at TR65.

Week 3

We had you to familiarise the use of Geometer Sketchpad. Have a very happy 'niu' year!

Week 4

  • We will have lesson on Wednesday at TR62, 9.30am.

  • The set of slides on chapter 4 is put up in advance, for your reference.

  • Main points:

    1. Both triangles are congruent: both have identical shape and size.

    2. Both triangles are similar: both have identical shape and one triangle is a scaled version of the other.

    3. Congruency tests: SSS, SAS (included angle is located between 2 sides), ASA (included side is located between 2 angles), AAS (corresponding side is adjacent to an angle).

    4. Similarity tests: AA, SSS, SAS (included angle is located between 2 sides).

    5. A median is constructed by joining a vertex to the mid-point of a side of the triangle, opposite to the vertex (see example 1).

    6. An altitude (or height) is constructed by dropping a perpendicular from a vertex to a side of the triangle, opposite to the vertex (see example 2).

    7. Homework questions to be completed and submitted for marking: Q4, Q5, Q7, Q9, Q10, Exam 05/06 Q5(a) [ please check Blackboard ].

Week 5

Plan for lessons:

1. Complete the proof of Pythagorean Theorem and start work on Exercise 4.

2. Clarify any doubt in preparation of Quiz 1.

Main points:

  •  When we wish to prove that sides have same lengths (Ex. 4 Q4), we essentially show congruency of triangles.

  • When we wish to show that lines are perpendicular (Ex. 4 Q5), we do the same as the above.

  • The main approach of the proof of Pythagoras' Theorem is to (1) identify similar triangles and then to (2) compare the corresponding sides of these triangles.

  • Please submit your work on Exercise 4 on Wednesday, 11 February '09.

Week 6

Plan for lessons:

1. Deliver Chapter 5 content. Please refer to the set of slides.

2. Start work on Exercise 5.

Main points:

1. The mid-point theorem starts with a line segment connecting two mid-points. Its proof requires ideas from earlier topics.

2. The intercept theorem starts with a line segment, intersecting two sides, that is parallel to the third side. Its proof requires ideas from earlier topics.

3. In the discussion question, we saw that the angle bisector coincides with the median and altitude in the case of an isosceles triangle.

4. Attempt Q4, Q5, Q10, Q11, Q12 + one other question (to be confirmed) and submit your work for marking on 18 Feb '09.

Week 7

Plan for lessons:

1. Return Quiz 1 scripts.

2. Cover content of chapter 6. Slides for your reference.

3. Key concepts:
(i) convex & concave polygons;

(ii) sum of interior and external angles of a polygon;

(iii) properties & classification of quadrilaterals;

(iv) more proofs (to be completed when term resumes)

Main points:

  • When we study the properties of quadrilaterals, we need to distinguish between the basic property and the derived properties. For example, the basic property for a parallelogram is that 'opposite sides are parallel', and a derived property (because it can be obtained from the basic property) is that 'opposite angles are equal'.

  •  Please attempt exercise 6 questions 1 - 8 during the break. Have a restful vacation!

  • Approach to solving Q11 in Exercise 5:

    - Construct a diagonal DB.

    - Consider triangle ADB. Construct a line through M that is parallel to AB, which will cut DB at P. By Intercept Theorem, P is the mid-point of DB.

    - Consider triangle BCD. Construct a line through N that is parallel to CD, which will cut DB at P. By Intercept Theorem, P is the mid-point of DB.

    - The statements above thus answer the hint given in the question. The rest of the proof follows the approach given in the lecture.

Week 9

Plan for lessons:

  • Complete the rest of chapter 6.
  • Deliver part of chapter 7.
  • Address learning issues in preparation of quiz 2.

Week 10

Plan for lessons:

  • Deliver chapter 7.

  • In the lesson, we proved several theorems involving areas. This picture gives us a overview of the relationships.

  • During the tutorial, we went through Q4, Q9, Q10. This picture provides the approach to solving Q9 (by the method of isolating different parts of the figure in order to see more clearly).

Week 11

Plan for lessons:

  • Complete Chapter 7.

    - Nets are useful to help us visualise lateral areas of solids.

    - It is important to distinguish between height (h) and slant height (l) in the case of a cone. Relationship: l^2 = h^2 + r^2.

    - Proportional reasoning is a useful method to help us derive various results (e.g. arc length, area of sector, lateral area of a cone).

    - To find the area of a regular polygon, we break it up into congruent triangles (of which we are familiar with the area).
    Note: In any regular polygon, the centre is the point that is equidistant (i.e. of equal distance) from each vertex or corner. Thus, in a regular hexagon, each triangle is equilateral; and in a regular n-gon, each triangle is isoceles.

    - We start with the area of a regular polygon to derive the area of a circle, by considering n \to \infty (i.e. n approaches infinity).

    - Homework questions to attempt: Q8, Q9, Q10, Q12, Q16. 

  • Review Quiz 2.

Week 12

Plan for lessons:

  • Cover Chapter 8.

    Main points:

    1. Tessellations are produced by means of translations, rotations and reflections.

    2. To describe a translation, one needs to specify the translation vector.

    3. To describe a rotation, one needs to specify the centre of rotation, the angle of rotation and the direction of rotation. Some special angles of rotation are 45 degrees, 90 degrees, 180 degrees and 270 degrees.

    4. To describe a reflection, one needs to specify the line of reflection.

    5. A modified figure can tessellate if the transformation that changes its shape is again applied to repeat it.

    6. We cannot tessellate a regular pentagon because the size of its interior angle does not divide 360 degrees evenly. Using the same principle, it is possible for quadrilaterals, triangles and regular hexagons to tessellate.

  • Useful link on motion geometry and tessellation.

  • Useful link on regular tessellations.

  • Interesting link on the soccer ball.

  • Picture of bee hive.

  • Picture of an elaborately-tiled wall in a mosque in Afghanistan.