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.

H2 Maths Workshop - Headstart to JC

Dates in 2010: 7/1, 14/1, 21/1, 28/1, 4/2.

Outline of the content:
0. Introduction to the JC system and to H2 maths
1. Binomial expansion
2. Graphs of basic functions
3. Linear transformations of graphs
4. Inequalities
5. Applications of differentiation
6. Integration techniques

Positive feedback from the 2009 cohort:
1. Yes it is a good headstart, so students would not stumble during the first year in JC. It is thorough and detailed.

2. His explanations are detailed and in depth. He does not rush through the questions but rather spent more time on it in order for the students to understand better.

3. He will go through stuff that teachers in the school would not do so, in order to help us understand better.

Enquire or register with Rick or Jane at 63853133 now!

Workshop in Nov/Dec

Preparation workshop for JC 1 re-examination/consolidation

Lessons beginning 24 Nov 2009.

Objective: To help students prepare for the year-end re-exam as well as enable students to consolidate the year's learning.

7 lessons of 2.5 hours each.

Topics to be covered:
1. Binomial expansion
2. Inequalities
3. AP/GP
4. Method of differences
5. Recurrences
6. Mathematical induction
7. Equations of tangents and normals
8. Integration techniques
9. Differential equations
10. Graphs of rational functions
11. Vectors (involving lines)

Please speak to Rick or Jane @ 63853133.

JC 2 Last-lap Prep Workshop

JC 2 Last-lap Prep Workshop for H2 Maths

Starting 1 Oct 2009

Outline of 12 lessons each of duration 2.5 hrs:
Lessons 1 - 6: Focus is on selected 2008 A-level questions
Lessons 7 - 12: Focus is on selected school prelim questions 2009

Format:
Teacher highlights key concepts and skills relevant to a question.
Students apply them to practise on the question under time limit.
Teacher reviews the problem-solving approach, with students clarifying any doubt.

Please register with Rick or Jane @ 63853133.

JC 1 Promo Exam Prep Workshop

JC 1 Promo Exam Prep Workshop

@ Eton Education Centre

6 Lessons (2.5 hours each)

Dates: 10, 12, 17, 19, 24, 26 Sep '09

Content of focus:
1. Sequences and series
- AP/GP
- Summation and method of differences
- Recurrences, conjectures and mathematical induction

2. Applications of differentiation and integration
- Maxima and minima
- Rates of change
- Maclaurin's expansion
- Areas and volumes

3. Functions and graphing techniques
- Inverse and composite functions
- Graphs of rational functions
- Transformations of graphs (including the graph of the derivative function)

Encouraging words from students who attended our workshop:
1. It worked as a great revision. The questions were very relevant.

2. I am able to manage my time better and able to use the most effective approach to tackle a question.

3. The tutors made many things simpler.

4. The workshop covers a variety of topics, points out various question types and teaches exam skills.

5. It helps me to think fast under time constraint. Thus, I will be able to perform better under exam conditions.

6. It has really been able to bring up the main points of each topic and the various types of question that are possible to come out.

7. It was enjoyable. The teachers were very clear and sharp. The explanations were very good. I understood everything.

8. The things taught are comprehensive, tutors are not too fast in their explanations, concepts taught are easy to understand.

9. The tutors are friendly and approachable.

JC 2 Prelim Prep Workshop

  • Preparation classes for JC 2 preliminary examination (14 lessons of 2.5 hours each & 2 lessons of 3 hours each)
    Lesson plan:
    - 6 lessons to cover the first set of chosen school prelim papers 1 and 2.
    - 6 lessons to cover the second set of chosen school prelim papers 1 and 2.
    - 2 lessons (3 hours each) to take the internally set mock exam papers 1 and 2.
    - 2 lessons to review the solutions to the mock exam papers 1 and 2.

    Style of teaching and learning:
    - The teacher highlights key concepts and skills for a particular question.
    - The student attempts the question under time limit.
    - The class then discusses the problem-solving approach and clarifies any conceptual issues.
    Dates: Starting 11/7, 1/8

    We welcome new students to join the second half of this workshop (starting 1 Aug). To register immediately, please call Rick at 63853133.
  • Lesson 1 (11/7) Summary

    1. The table of specifications provides us with a clear idea of the topics assessed and their weightages. Sequences and series (including Maclaurin's expansion), calculus (i.e. differentiation, integration, differential equations), vectors and complex numbers, graphing techniques and functions are four large clusters in the pure maths syllabus.

    2. It is important to pick the easiest question to begin with in order to build up confidence, rather than attempting the exam in question sequence.

    3. System of linear equations:
    - To find n unknowns, we need to form n equations using the information from the question.

    - Then, we use TI-GC to solve, either through RREF or APPS.

    4. Inequalities:
    - If an expression has a denominator (say 2-x), a solution has to be excluded, i.e. x \ne 2.

    - A useful strategy is to multiply both sides either by the square of the denominator or simply the denominator (when it is always positive, e.g.
    x^2 + 1).

    - Factorise, as early as possible.

    - At times, students may be asked to deduce the solution of a similar inequality by means of a suitable substitution. Common substitutions are
    -x, \frac{1}{x}, \text{ln}x, e^x, |x| and combinations of them.

    5. Graphing of basic functions:
    - A hyperbola has this standard representation: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 . The graph will have oblique asymptotes given by y = \pm \frac{b}{a}x , which intersect at the origin.
    Note: An ellipse has the equation given by \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (notice the positive sign) and it should not be confused with a hyperbola. 

    - Sometimes, the hyperbola may be expressed as \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 (where h, k > 0), which we saw in our question. The graph will have oblique asymptotes given by y - k = \pm\frac{b}{a}(x - h), which intersect at (h, k). Put simply, the asymptotes have been shifted to the right by h units and up by k units.

    - We may need to carry out some form of investigation when the equation of a curve has an unknown, e.g. y = \text{ln}(x + k) that we saw in our question. Because k is a positive integer, we will investigate, with GC, the behaviour when k = 1, 2, 3, ... .

    6. Transformations of graphs:
    - The use of a table helps us to plan for the sketching of a transformed graph. Read this article.

    - Reciprocal graph: read this article.

    - Graph of the derivative function: read this article. To sketch such a graph, it is necessary to take gradients at suitable locations as we move along the curve of y = \text{f}(x).

  • Lesson 2 (16/7) Summary

    1. Permutations and combinations: read this article.

    2. Probability:
    - When a question involves stages, like the one we saw during lesson, drawing a probability tree would be most helpful (see below).

    - Recall that: \text{P}(A \cup B) = \text{P}(A) + \text{P}(B) - \text{P}(A \cap B).
    If A, B are mutually exclusive events, then \text{P}(A \cap B) = 0 and so \text{P}(A \cup B) = \text{P}(A) + \text{P}(B).
    If A, B are independent events, then \text{P}(A \cap B) = \text{P}(A) \times \text{P}(B).

    3. Maclaurin's expansion:
    - One type of question involves binomial expansion.
    Recall that: (a + x)^n = a^n\left(1 + \frac{x}{a}\right)^n.

    - Another type of question involves trigonometric approximations.
    Recall that: For small x, sin x \approx x, tan x \approx, cos x \approx 1 - \frac{x^2}{2}. Note that x may be replaced by kx in our approximations.

    - The third type involves the use of standard series, which can be found in formulae list MF15.

    - The fourth type involves repeated differentiation.
    Recall that: y = \text{f}(0) + \text{f}'(0)x + \frac{\text{f}''(0)}{2!}x^2 + ... + \frac{\text{f}^{(n)}(0)}{n!}x^n + ... . Be careful in your differentiation steps and use suitable substitutions where appropriate (e.g. sin 2x = y^2 - 9 in the question we saw) to make your expression less complex for subsequent differentiation.

    4. Functions:
    - To find the inverse function, we express x in terms of y.

    - Domain of the inverse \text{f}^{-1} = Range of \text{f}.

    - Range of the inverse \text{f}^{-1} = Domain of \text{f}.

    - Condition for the composite function (e.g. \text{f} \circ \text{g}) to exist: Range of the first function (e.g. g) \subseteq Domain of the second function (e.g. f).

    - If the above condition fails, a common question would ask one to find a restriction of the domain of the first function (e.g. g) to force its range to become equal to the domain of the second function (e.g. f).

    - Domain of the composite function (e.g. \text{f} \circ \text{g}) = Domain of the first function (e.g. g).

    - To find the range of the composite function (e.g. \text{f} \circ \text{g}), follow these steps:
    (1) Sketch the graphs of f and g, corresponding to their respective domains.

    (2) Find the range of g from the graph of g.

    (3) Map the range of g onto the x-axis of the graph of f.

    (4) Find the range of f corresponding to the map, which is essentially the range of the composite \text{f} \circ \text{g}.

    - The geometrical relationship between the graph of \text{f} and \text{f}^{-1} is a reflection of each other in the line y = x. Note that any point
    (x, y) on the graph of \text{f} becomes (y, x) on the graph of \text{f}^{-1}. Also, a horizontal asymptote y = a present in the graph of \text{f} becomes a vertical asymptote x = a in the graph of \text{f}^{-1}. 

    Sometimes, y = x may be expressed as
    y = \text{f} \circ \text{f}^{-1}(x), where the domain is the domain of \text{f}^{-1} which is also the range of \text{f} 
    or
    y = \text{f}^{-1} \circ \text{f}(x), where the domain is the domain of \text{f} which is also the range of \text{f}^{-1}.
    In either case, one should draw the line y = x based on the domain. 
  • Lesson 3 (18/7) Summary

    1. Integration and its applications:
    - In the question, we saw the need to find the area of a region by evaluating the integral by means of integration by a given substitution. Read this article for details.

    - Be sure to know what is a chord (i.e. a straight line joining two points) and what is an arc (i.e. a curved line).

    - Be sure to know how to use GC to evaluate an integral via the command fnInt, e.g. \int_{0}^{\sqrt{6} - 2} \frac{1}{(x+2)^2 (x^2 + 4x + 1)} \text{d}x will be entered in GC like this:


    Note that the GC produces an error message when we are not careful with our entry.

    - Read this article for more content about solids of revolution, especially the skill of expressing x in terms of y.

    2. Binomial distribution:
    - If X \sim \text{B}(n, p), we recall that \text{E}(X) = np and \text{Var}(X) = np(1 - p).

    - In the question, we can use the algebraic approach (by taking ln on both sides) or GC (by creating a table of values) to solve an inequality involving probability, i.e. \text{P}(X = 0) < 0.08.

    - Suppose that X follows any distribution with mean \mu and variance \sigma^2. Central Limit Theorem states that for large n, X_1 + X_2 + ... + X_n \sim \text{N}(n\mu, n\sigma^2).

    3. Poisson distribution:
    - The question we solved was about the demand and supply of vehicles. The demand was modelled as a Poisson random variable.

    - Sometimes, it is necessary for us to define random variables clearly at the start of our working, e.g. let V represent the number of vans demanded per day.

    - It is important for us to read the question carefully so that we may be able to interpret the sentences correctly in the mathematical form. In the question, the sentence "probability that not all requests can be met" would be interpreted as \text{P}(\text{demand} > \text{supply}).

    - When evaluating a conditional probability, it may be necessary for us to consider different cases. In the question, we saw that
    \text{P}(\{(V \ge 1) \text{and} (C \ge 2)\} \cap (V + C = 4)) = \text{P}(V = 1) \times \text{P}(C = 3) + \text{P}(V = 2) \times \text{P}(C = 2), by using multiplication and addition principles.

    - When a Poisson (discrete) variable is approximated to become a Normal (continuous) variable for \lambda > 10, continuity correction has to be applied.

    We use rounding up and rounding down concepts for continuity correction. Let's suppose we have to evaluate \text{P}(22 < X < 30). A useful strategy is to rewrite the probability as \text{P}(23 \leq X \leq 29) first. Then we decide what number we will need for it to be rounded up to 23 and what number we will require for it to be rounded down to 29. This gives us the probability \text{P}(22.5 < X < 29.5).
  • Lesson 4 (23/7) Summary

    1. Equations of tangents and normals:
    - Given a pair of parametric equations x = \text{f}(t) and y = \text{g}(t), we will need to carry out parametric differentiation to find the derivative of y with respect to x: \frac{\text{d}y}{\text{d}x} = \frac{\frac{\text{d}y}{\text{d}t}}{\frac{\text{d}x}{\text{d}t}}.

    - If gradient of tangent =m, then gradient of normal = -\frac{1}{m}.

    - Equation of tangent/nomal with gradient k at the point where t = a is: y - \text{g}(a) = k[x - \text{f}(a)].

    2. Maxima and minima:
    - Many questions start by asking students to show that a certain expression holds. Read the problem statement carefully to look out for information to form useful relationship between variables.

    - To determine the stationary value(s), we set the derivative to zero.

    - To determine the nature of stationary value(s), we use either the first derivative or the second derivative test. The first derivative test is useful especially when it is difficult to differentiate the expression the second time.

    - First derivative test:
    \frac{\text{d}y}{\text{d}x}>0 when x = a^-, \frac{\text{d}y}{\text{d}x}=0 when x = a, \frac{\text{d}y}{\text{d}x}<0 when x = a^+: x = a gives maximum y.

    \frac{\text{d}y}{\text{d}x}<0 when x = a^-, \frac{\text{d}y}{\text{d}x}=0 when x = a, \frac{\text{d}y}{\text{d}x}>0 when x = a^+: x = a gives minimum y.

    \frac{\text{d}y}{\text{d}x}>0 when x = a^-, \frac{\text{d}y}{\text{d}x}=0 when x = a, \frac{\text{d}y}{\text{d}x}>0 when x = a^+: x = a gives stationary inflexion point.

    \frac{\text{d}y}{\text{d}x}<0 when x = a^-, \frac{\text{d}y}{\text{d}x}=0 when x = a, \frac{\text{d}y}{\text{d}x}<0 when x = a^+: x = a gives stationary inflexion point.

    - Second derivative test:
    \frac{\text{d}^2y}{\text{d}x^2} < 0 when x = a: x = a gives maximum y.

    \frac{\text{d}^2y}{\text{d}x^2} > 0 when x = a: x = a gives minimum y.

    \frac{\text{d}^2y}{\text{d}x^2} = 0 when x = a: no conclusion can be made, use the first derivative test instead.

    3. Complex numbers:
    - One type of question asks students to solve equations involving complex numbers. One useful strategy is to compare real and imaginary parts.

    - Another type of question asks students to solve z^n = r\text{e}^{\text{i}\theta}. Therefore, z = r^{\frac{1}{n}}\text{e}^{\text{i}\left(\frac{\theta + 2k\pi}{n}\right)} where k = 0, 1, 2, ...,n-1.

    - A third type of question asks students to sketch loci on an Argand diagram and find their intersection(s). The three standard loci are:
    (a) circle of the form |z - c| = r (centre at the point represented by c and radius r);

    (b) half-line of the form arg(z - c) = \theta (a line, beginning at the excluded point c, makes an angle \theta with the horizontal);

    (c) perpendicular bisector of the form |z - c_1| = |z - c_2| (a line, passing through the mid-point of the points represented by c_1 and
    c_2, which is perpendicular to the line segment joining the points c_1 and c_2).

    To find the intersection(s) between a circle and a line passing through the centre (of the circle), we add or subtract (to the x- and y-coordinates of the centre) the horizontal or vertical component of the radius (of the circle). In the example below:

    we see that A and B are intersection points between the two loci; it is easy to see (from the diagram) that A has coordinates (1 - \sqrt{2}\:cos\:\theta, 1 + \sqrt{2}\:sin\:\theta) and B has coordinates (1 + \sqrt{2}\:cos\:\theta, 1 - \sqrt{2}\:sin\:\theta).
  • Lesson 5 (25/7) Summary

    1. Vectors:
    - We can grasp all of the concepts in this topic by appreciating the importance of ratio theorem, scalar product and vector product.

    - Read more about applications of the scalar product.

    - Read more about applications of the vector product.

    - Read more about lines & planes and geometric interpretations.

    - We can expand scalar products like the way we do in algebra, for example:
    \textbf{a}.(\textbf{b} - \textbf{a}) = \textbf{a}.\textbf{b} - \textbf{a}.\textbf{a}
    = \textbf{a}.\textbf{b} - |\textbf{a}|^2.

    - When the line lies in the plane, the equation of the line must satisfy the equation of the plane. For example, consider
    l : \textbf{r} = \textbf{a} + \lambda\textbf{d} and \pi : \textbf{r}.\textbf{n} = p. Then we need to show that (\textbf{a} + \lambda\textbf{d}).\textbf{n} = p, regardless of the value of \lambda.

    When the line meets the plane, (\textbf{a} + \lambda\textbf{d}).\textbf{n} = p for one value of \lambda only.
  • Lessons 6 & 7 (30/7 & 1/8) Summary

    1. Normal distribution:
    - It is useful to use the symmetrical property of a normal distribution, i.e. \text{P}(X < \mu - k) = \text{P}(X > \mu + k).

    - It is important to appreciate the difference between X_1 + X_2 + ... + X_n (i.e. the sum of n observations of X) and nX (i.e. n times of X). Read this article for more details.

    - Reading and making sense of the key words in the problem statement is really an art of interpretation. Read this article for more details.

    - \text{B}(n, p) is approximated to become \text{Po}(np) when n is large and np < 5. No continuity correction is necessary.

    - Casio GC commands:
    (a) To find the normal probability \text{P}(a < X < b), the sequence of keys are MENU \to STAT \to DIST \to NORM \to Ncd.

    (b) To find x such that the normal probability \text{P}(X < x) = p (where p is some given probability), the sequence of keys are MENU \to STAT \to DIST \to NORM \to InvN. Make sure that the Tail option is set to Left.

    (c) To find the Poisson probability \text{P}(X \leq x), the sequence of keys are MENU \to STAT \to DIST \to POISN \to Pcd. 


    2. Differential equations:
    - Formulating a DE requires some skill in interpreting the problem statement. Read this article for more details.


    3. Recurrences:
    - Finding a general formula from a recurrence relation requires one to follow three logical steps repeatedly. Read this article for more details.


    4. Summation of series:
    - To show that \text{P}(k + 1) is true in MI for any summation, we apply the fact that \sum_{r=1}^{k+1}U_r = \sum_{r=1}^{k}U_r + U_{k+1}. To simplify further, it is a good strategy to factorise.

    - To evaluate a complex summation like \sum_{r=1}^{n}(U_r - V_r), we simplify it to \sum_{r=1}^{n}U_r - \sum_{r=1}^{n}V_r and tackle each sum separately.

    - We saw that as n \to \infty, an expression like \frac{n+3}{2^n} \to 0, because the denominator is going to be much larger than the numerator.

    - For take-home question IJC/P1/Q9(a): Sum of A.P. is S_n = \frac{n}{2}[2a + (n - 1)d]. We want to solve for n such that S_n \ge 80000.
    Answer for you to check: 74.


    5. System of linear equations:
    - Casio GC steps: Select EQUA from the main menu, choose F1, specify the number of unknowns, enter the values into the matrix and then select SOLV.


    6. Inequalities:
    - For take-home question HCI/P1/Q2: To solve the inequality, we need to use the first part and to multiply both sides by the square of the denominator. Remember that x \ne \pm\sqrt{5}.
    Answer for you to check: -\sqrt{5}<x\leq-1 or 1 \leq x < \sqrt{5}.


    7. Rate of change:
    - For take-home question HCI/P1/Q3: Apply the chain rule, i.e. \frac{\text{d}h}{\text{d}t} = \frac{\text{d}V}{\text{d}t} \times \frac{\text{d}h}{\text{d}V}.
    Answer for you to check: -\frac{2}{\pi}\:\text{cm\:s}^{-1}.

  • Lesson 8 (6/8) Summary

    1. Homework tips (p1, p2).


    2. Key takeaways from the lesson (p3, p4).

  • Lesson 9 (8/8) Summary

    1. Homework tips (p1, p2).

Ongoing & completed classes

  • Sequences & series (3 sessions of 2.5 hours each) [ Completed ]
    Specific skills to be covered: AP/GP, method of differences, recurrence relations, mathematical induction involving conjectures
    Dates: 12, 14 & 19 Mar '09
     
    - Personal summary for lesson 1 on 12 Mar (picture 1 and picture 2)

    - Personal summary for lesson 2 on 14 Mar (picture)

    For Q2(a), we note that \sum_{r = 0}^{N - 1} (2r + 1)\text{sin}(2r + 1)x = -\frac{\text{d}}{\text{d}x}\left[ \sum_{r = 0}^{N - 1} \text{cos}(2r + 1)x \right], since the question requires us to use calculus.

    Before we differentiate, we need to obtain the expression of \sum_{r = 0}^{N - 1} \text{cos}(2r + 1)x.

    To do so, we use the result of \sum_{r = 0}^{N} \text{cos}(2r + 1)x = \frac{1}{2}\text{sin}2(N + 1)x\:\text{cosec}x and substitute N with N - 1.

    So \sum_{r = 0}^{N - 1} \text{cos}(2r + 1)x = \frac{1}{2}\text{sin}(2N)x\:\text{cosec}x.

    Now we apply the product rule of differentiation (i.e. keep a differentiate b, then keep b differentiate a)  and recall that \frac{\text{d}}{\text{d}x} \left[ \text{cosec}x \right] = -\text{cot}x \:\text{cosec}x (which can be derived easily).

    - Personal summary for lesson 3 on 19 Mar (picture)

    Approach to solving consolidation Q2:

    1. Use graphic calculator to find the value of \alpha.

    2. For part (i), we start with the key phrase: If the sequence converges, as n \to \infty, x_n \to \infty and x_{n + 1} \to \infty.

    3. For part (ii), see the graphic calculator screenshots.

    4. For part (iii), consider x_{n + 1} - x_n = 5\text{ln}\:(x_n + 2) - x_n.
    Now \frac{x_{n + 1} - x_n}{5} = \text{ln}\:(x_n + 2) - \frac{x_n}{5}.
    We can then make good use of the given graph to conclude that \frac{x_{n + 1} - x_n}{5} > 0 when 0 < x_n < \alpha
    and that \frac{x_{n + 1} - x_n}{5} < 0 when x_n > \alpha.

    5. For part (iv), try to reason it out on your own.

    6. For the last part, use graphic calculator the way we did in part (ii), but change u(n) and u(nMin) to those given in the question.  

  • Graphing techniques (3 sessions of 2.5 hours each) [ Completed ]
    Specific skills to be covered: Graphs of rational functions and of y = \text{f}\:'(x), simple transformations (reflections, translations and scalings), transformations of the form y = \text{f}(|x|), y = \frac{1}{\text{f}(x)} and y^2 = \text{f}(x).
    Dates: 21, 26, 28 Mar '09

    - Personal summary for lesson 4 on 21 Mar (picture)

    1. The discriminant b^2 - 4 ac helps us to determine the condition for which there are stationary points on a curve. It is also applied to determine the condition for which the curve has no values.

    2. Approach to solving discussion Q2:

    (i) Asymptotes are: y = x + (2a + 1) and x = 1. 

    (ii) Find the values of x such that \frac{\text{d}y}{\text{d}x} = 0. You will need to carry out the second derivative test.

    (iv) First, note that the parabola has a line of symmetry x = -a, because it is given that point B is its minimum. To find k, use the fact that the parabola passes through the point A.

    The graph of y = \text{f}\,'(x) has the same shape as the one we drew for Q1.

    3. Approach to solving consolidation question:

    (i) b = 1, a = 4.

    (ii) Use long division to obtain the answer: y = x + 3.

    (iii) C does not lie between 0 and 4.

    (iv) Use the graphic calculator to sketch and check your drawing.

    (v) Include the graph of y^2 = 4 - x^2 (i.e. a circle) and find the number of intersections. Answer: 2 real roots.

    - Personal summary for lesson 5 on 26 Mar (picture)

    - Lesson 6 on 28 Mar

    1. Approach to solving discussion Q2:

    (i) Use long division and compare the quotient with the oblique asymptote to obtain p - q = 1. Then use the fact that \text{f}(0) = 5 to find the value of q. You should get the answers p = 2, q = 1.

    (ii) Use GC to sketch your graph of y = \text{f}(x) = \frac{x^2 + 2x + 5}{x + 1} and find its maximum and minimum turning points to determine the range of values of k. Answer: -4 < k < 4.

    Use GC to check your sketches for (a) and (b).

  • Vectors (3 lessons of 2.5 hours each) [ Completed ]
    Specific skills to be covered: ratio theorem, points and lines, lines and planes, applications of scalar and vector products
    Dates: 2, 4 & 9 Apr '09

    - Personal summary for lesson 1 on 2 Apr (picture)

    Approach to solving discussion Q2:

    (i) First, \overrightarrow{OC} = \begin{pmatrix} 4 + \lambda \\ 3 + \lambda \\ 7 \end{pmatrix} for some \lambda. Then consider \overrightarrow{CA}.\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} = 0 to find \lambda.
    Answer: \overrightarrow{OC} = \begin{pmatrix} -8 \\ -9 \\ 7 \end{pmatrix}

    (ii) Consider \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}.\begin{pmatrix} q \\ 0 \\ 2 \end{pmatrix} = \sqrt{2}\sqrt{q^2 + 4}\:\text{cos}\frac{\pi}{3} to find q.
    Answer: q = \pm \frac{2}{\sqrt{3}}

    Approach to solving consolidation question:

    (a)(i) Draw a diagram. Then use the fact that AO and AB are perpendicular.

    (ii) Use the ratio theorem.

    (iii) You will need to use (i) and (ii) to find the length. Answer: \frac{77}{25}

    (b)(i) The direction vector of the line l is parallel to \overrightarrow{AB} \times \overrightarrow{AC}.

    (ii) Straightforward. Answer: \overrightarrow{OP} = \begin{pmatrix} 4 \\ 0 \\ -4 \end{pmatrix} \text{or} \begin{pmatrix} -2 \\ 4 \\ -2 \end{pmatrix}

    - Personal summary for lesson 2 on 4 Apr (picture)

    - Personal summary for lesson 3 on 9 Apr (picture)

  • Complex numbers (4 lessons of 2.5 hours each) [ Completed ]
    Specific skills to be covered: Modulus and argument, equations involving complex numbers, Argand diagrams
    Dates: 11, 16, 18 & 23 Apr '09

  • Preparation workship for JC 2 mid-year exam  (5 lessons of 2.5 hours each) [ Completed ]
    Specific skills to be covered: Applications of differentiation, applications of integration, differential equations, composite and inverse functions, graphing techniques, sequences and series, vectors, complex numbers, probability and statistics
    Dates: 25, 30 Apr, 2, 7 & 9 May '09

    Answers to the first question on 'Lines and planes':
    (i) Point of intersection: (1, -2, 3)

    (iii) \overrightarrow{ON} = \left(\begin{matrix}-5\\10\\6\end{matrix}\right)

    (iv) \overrightarrow{OQ} = \left(\begin{matrix}-20\\13\\-36\end{matrix}\right) 

    These five lessons are part of Module 4 of Eton's H2 Mathematics Mastery of Topics Series.

  • Preparation workshops for JC 2 mid-year examination (6 lessons of 2.5 hours each) [ Completed ]
    Specific concepts and skills to be covered:
       1. Applications of differentiation 
       2. Applications of integration
       3. Differential equations
       4. Composite and inverse functions 
       5. Graphing techniques
       6. Sequences and series 
       7. Vectors
       8. Complex numbers 
       9. Probability and statistics
    Dates: 23/5, 28/5, 30/5, 4/6, 6/6, 9/6
  • Preparation workshops for JC 1 mid-year examination (5 lessons of 2.5 hours each)
    Specific concepts and skills to be covered:
       1. Binomial expansion
       2. Inequalities
       3. Composite and inverse functions
       4. Graphing techniques
       5. Sequences and series
       6. Differentiation techniques
       7. Applications of differentiation 
    Dates: Starting 11/6

Module 1 (completed)

Are you still struggling with JC 1 topics, even though you may have cleared your promo exams and have been promoted to JC 2?

Come early-Feb '09, I plan to run the 'mastery of topics' series of lessons @ Eton (where I teach) for you to overcome your areas of difficulty once and for all.

Details of lessons:

  • Our main objective is to enable our students to acquire a strong competency of the key mathematical problem-solving skills, after attending the series of focused and intense lessons.
  • 10 lessons (2.5 hrs each) on 5 Feb, 7 Feb, 12 Feb, 14 Feb, 19 Feb, 21 Feb, 26 Feb, 28 Feb, 5 Mar, 7 Mar '09.
  • Classes on Thursdays: 7.15 - 9.45pm. Classes on Saturdays: 9.30am - 12pm. 
  • Topics include: applications of differentiation, integration techniques, areas and volumes, Maclaurin's expansion, differential equations and functions.
  • Lesson materials will be provided.
  • Lessons will focus on having students to summarise key concepts and to practise typical examination questions under time contraint, with comprehensive review from the teacher. As such, lessons will be focused and intense. Lesson 10 will be a total consolidation session of all content covered in previous lessons. 
  • Further support may be provided through the teacher's website, e.g. more learning resources, clarifications on school work.

Places are limited. To avoid disappointment, you may indicate your interest in this series of lessons early

- via an online form, or
- by calling 63853133 (speak to Rick or Jenny) to leave your name and contact information.

Eton Education Centre Private Limited is conveniently located opposite to Hougang Mall.
Nearest NEL MRT station: Hougang.
Nearest bus interchange: Hougang Central.

Module 1 lessons' log

  • Lesson 1's personal summary

  • Lesson 2's personal summary

  • Lesson 3's personal summary

  • Lesson 4's personal summary

  • Lesson 5's personal summary

  • Lesson 6's personal summary and worked solutions (Q4, Q5) to discussion questions

  • Lesson 7's personal summary (very brief)

  • Lesson 8's personal summary (questions only)

    Some important points:

    - Functions and standard graphs (parabolas, rectangular hyperbolas, circles, ellipses, exponential and logarithmic graphs) are always related.

    - Although f_1(x) = \left|\frac{2x+1}{x-3} \right|, the rule can be rewritten as -\frac{2x+1}{x-3} (thus removing the modulus sign) because the graph of y = \frac{2x+1}{x-3} is reflected in the x-axis when we sketched the graph of y = \left|\frac{2x+1}{x-3}\right|.

    - The point (x, y), the vertical asymptote x = a and the horizontal asymptote y = b on the graph of any function become the point (y, x), the horizontal asymptote y = a and the vertical asymptote x = b respectively on the graph of its inverse. This is due to the reflection in the line y = x.

  • Lesson 9's personal summary (questions only)

    Note the error in (iv): Find the range of hg (not gh).

    Important points:

    - The range of any composite function can be obtained rather easily from graph sketches.

    - The domain must be taken into consideration when a graph is sketched.

  • Lesson 10 - total consolidation (short quiz)

Doing a personal summary

The summary that a set of notes provides may not meet the specific, unique needs of a learner. I strongly encourage every student to answer the following questions when preparing a personal summary that will work for him/her:

1. What do you think are the key concepts, results and relationships for this topic? Refer to your notes.

2. What kinds of skill are you often assessed on? Refer to your notes, daily work, tests and exams.

3. What kinds of difficulty do you often face? Refer to your daily work, tests and exams.

4. What kinds of mistake you you often commit? Refer to your daily work, tests and exams.

If possible, select carefully a number of typical questions (e.g. from your notes, tutorials, test papers, reference books, etc.) in which you'll be able to apply the bulk of your mathematical understanding.

For incoming JC 1 students of 2009

Waiting for your O-level results and wishing to stay ahead in H2 mathematics when you start JC 1?

Come mid-Jan '09, I plan to do a SECOND 'headstart' workshop @ Eton (where I teach) for you to stay ahead in your JC academic career.

Details of lessons:

  • Our main objective is to enable our students to acquire a basic competency of some H2-level mathematics topics, by facilitating their ability to build on prior knowledge of O-level additional mathematics so as to internalise new mathematical skills. 
  • 4 lessons (2.5 hrs each) on 22, 24, 29, 31 Jan '09.
  • Classes on Thursdays: 7.15 - 9.45pm. Classes on Saturdays: 9.30am - 12pm.
  • During these lessons, I will connect the new H2 mathematics content with what you have already learnt at O-level. Read this document which provides of details what I actually do in class.
  • Content covered: partial fractions, binomial expansion, inequalities, trigonometry, small angle approximations, graphing techniques, differentiation and integration.

Places are limited. To avoid disappointment, you may indicate your interest in this workshop early

- via an online form, or
- by calling 63853133 (speak to Rick or Jenny) to leave your name and contact information.

Eton Education Centre Private Limited is conveniently located opposite to Hougang Mall.
Nearest NEL MRT station: Hougang.
Nearest bus interchange: Hougang Central.

This workshop has ended its run. Thanks to all students who attended the two workshops that were held.

Some positive students' feedback (thanks too!):

1. It helps me to understand the links between H2 maths and Add. maths.

2. It helps me to approach the questions with strategies.

3. It helps me to have more confidence in taking H2 maths and not feel that it is a very complex subject.

4. He is extremely detailed, has a good pace of teaching and takes effort to list out important points.