A. Math Archives - Page 6 of 8

Sec 4 A. Math Reflections 02/02/13 – Higher Derivative

Posted on February 5, 2013February 24, 2016 by admin

Today I taught them the Higher Derivatives, using displacement, velocity and acceleration as real-life examples. I chose not to introduce the 2nd derivative as a means to test whether a turning point is a maximum or minimum, as I preferred to do that when teaching the application of differentiation in tangents and normals at a point in a curve.

Points to note:

(1) students must not write the 2nd derivative as dy^2/dx^2;

(2) d^2y/dx^2 is not the same as (dy/dx)^2;

(3) d^2y/dx^2 is the same as f”(x) or f^2(x);

(4) there may be product rule, quotient rule or chain rule involved when finding the higher derivatives.

Rgds,

Ilyasa, M.Ed, PGDE, ex-MOE Math and Physics teacher (hp: 97860411)

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A. Math

Sec 4 A. Math Reflections 26/01/13 – Quotient Rule

Posted on January 29, 2013February 22, 2016 by admin

I introduced Quotient Rule to the class last week so today we continued with it. Quotient Rule is actually very easy to use and should not be replaced by the Product Rule, even though it’s very convenient for students to write a fraction as a product of two terms.

What students find difficult is in simplifying the resulting expression, which often consists of fractions within fractions, with some terms having a square root sign over it. Of course there is a ‘trick’ to get the factorisation done quickly and correctly, but it’s too cumbersome for me to demonstrate here. More importantly, this difficulty that students often face highlights the point that I’ve posted earlier, that Sec 1 and Sec 2 Algebra is VERY IMPORTANT.

For the next lesson, I’ll be going into the Second Derivative as well as Application of Differentiation to curves, tangents and normal. We may be covering the Chain Rule if we have time.

Rgds,

Ilyasa, M.Ed, PGDE, ex-MOE Math and Physics teacher (hp: 97860411)

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A. Math, A. Math Tips

Sec 4 A. Math Reflections sat1415 19/01/2013 – Differentiation

Posted on January 21, 2013February 22, 2016 by admin

This was our third lesson in Jan. We did the Product Rule for Differentiation after spending the last two sessions doing the Basic Rule and the Power Rule (a subset of the chain rule).

I told students to differentiate directly, in line, and not use the “let u = f(x) and v = g(x)” which is longer, more confusing and can cause more careless mistakes.

Now they are able to perform dy/dx = f(x)g'(x) + g(x)f'(x) quite quickly without introducing new variables such u and v which can be quite clumsy. Most of the time, finding dy/dx is part of a larger question, so it is unwise to make this part too long.

I ended the lesson with an intro to the Quotient Rule. Again, I don’t recommend the use of u’s and v’s and instead differentiate directly in line. We’ll be practising the Quotient Rule in the next session.

Again I tell students that Calculus is almost one-third of the A. Math syllabus, so if you want to get A1 you have to master Differentiation and Integration, which, in my opinion (and students hate it every time I say it), are amongst the easiest of topics in A.Math.

However, Integration at the A-Levels (H2 Math) can be quite difficult. But we’ll cross the bridge when we come to it. : )

Ilyasa, M.Ed, PGDE, ex-MOE teacher

(hp: 97860411)

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For our latest timetable, click here =>

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A. Math, Sec Math

Reflections on Sec 4 A. Math class (9/12/2012) …

Posted on December 9, 2012December 15, 2012 by admin

Two new students joined us ytdy, one from Monfort Sec and one from SCGS. Apparently, I tutored the latter’s cousin a few yrs ago, all the way from Sec 2 to JC2 in Math and Physics, and who is now a first year undergrad; just learnt that he scored 2 A’s and 2 B’s in his A-levels (he might have told me about it); anyway, that proves my point, that you can come from a ‘neighbourhood’ sec sch and then go to a ‘low-ranked’ JC and still do well at the A-Levels. Contrast this with the two ex-IP students that I helped this year to re-take their A-Levels. So students out there pls wake up; no one owes you good grades.

The three tutees agreed to revise some sec 3 topics instead of me teaching them a Sec 4 topic. So for this class (Sat 2.15 to 3.45pm), I will only start teaching Differentiation in Jan 2013. So ydty two of them covered Indices, Surds and Log while the remaining one wanted to revise Trigonometry. Every time, I find joy in proving to students that Logarithms is a VERY EASY topic. Once you understand what a logarithm is, everything about it becomes very easy (I’ve posted another article on Log; pls do a search on it, under A Math study tips I think).

Trigo is a much harder topic, especially the proving of some Trigonometric Identities. However, there are heuristics to use in solving the latter, and these techniques work 95% of the time. I like ‘proving’ qns because there is no answer to find, and students normally dislike such qns precisely because there is no answer to find. But I can’t blame them; imagine spending 6 yrs of your life in pri sch only learning how to find answers, so students become obsessed with finding a numerical answer, and eventually get defeated by qns that ask them to prove something already known.

Ilyasa

(1) A. Math study tips.

(2) Sec 4 A. Math tuition.

A. Math, A. Math Tips, JC Math (H2/H1), Metacognition, Pri Math, Pri Math Olympiad, Sec Math

Is metacognition part of the mathematics curriculum in Singapore?

Posted on November 20, 2012February 13, 2016 by admin

One of the aims of mathematics education in schools in Singapore is to enable students to acquire thinking and problem solving skills and to make effective use of these skills to formulate and solve problems (MOE, 2007).

The existing curriculum framework for mathematics designed by the Ministry of Education (MOE) lists metacognition as one of the components on which the development of mathematical problem solving ability depends.

According to the MOE (2007), metacognition can be defined as the realization of, and the ability to regulate one’s thinking processes, in particular the choosing and application of problem-solving strategies. The MOE believes that it is important to provide students with metacognitive experience in order to help them develop their problem solving abilities.

Ilyasa

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A. Math, Metacognition, Pri Math, Sec Math

Metacognition enhances learning

Posted on November 20, 2012February 13, 2016 by admin

Various studies have revealed that metacognition helps to enrich students’ learning in different domains. For example, it has the potential to increase students’ capacities for independent learning (Ganz & Ganz, 1990).

Research also shows that knowledge of metacognition, such as being familiar with one’s strengths and weaknesses and searching for ways to overcome the latter, contributes to more effective learning (Bransford, Brown, & Cocking, 1999). Research also suggests that metacognition improves one’s chances of success when it comes to completing activities that rely heavily on thinking processes (Garner & Alexander, 1989; Pressley & Ghatala, 1990).

Many studies in metacognition have concluded that those who have advanced metacognitive abilities are more adaptable and steadfast in problem solving (e.g., see Artzt & Armour-Thomas, 1992; Swanson, 1990). Studies have also shown that one’s ability to plan and monitor a problem-solving process requires several metacognitive skills such as regulation and evaluation of thought processes (Mayer, 1999), and the use of metacognitive skills has the potential to identify the more able students from the less able ones (Pellegrino, Chudowsky & Glaser, 2001).

In addition, research has shown that one’s individual and group learning skills can be improved through the acquisition of metacognitive competencies (White & Frederiksen, 2005). Recent studies have also revealed that students who often fail to choose appropriate strategies, monitor or regulate their work, or articulate their thought processes are more likely to perform poorly in mathematics (e.g., see Lucangeli & Cabrele, 2006; Carlson & Bloom, 2005).

Ilyasa

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TUITION CLASSES:

_______________________________________________________________

EDUCATIONAL SERVICES:

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By EX-MOE TEACHERS & EXPERIENCED TUTORS

@ BLK 644, BUKIT BATOK CENTRAL, #01-68. S(650644).

CALL 65694897 OR SMS 98530744 OR 97860411.

A. Math, A. Math Tips, Metacognition, Pri Math, Sec Math

Metacognition and mathematical problem solving

Posted on November 20, 2012February 13, 2016 by admin

Metacognition is a crucial element in problem-solving, which is itself a key component in mathematics learning. To monitor and regulate one’s cognitive processes in problem-solving, Polya (1945) describes a four-step method: first, one has to comprehend the problem by sub-dividing it into more manageable parts and recognize any given data, conditions and variables to be found; second, one devises or selects a strategy to find the connections between the known data and the unknowns to be found; third, one executes the plan, scanning, regulating and examining each step; and, finally, after obtaining the solution, one evaluates the results which may involve re-visiting the previously taken steps.

Building on Polya’s work, Schoenfeld (1987) describes effective mathematical problem-solving as being contingent on how one uses four types of knowledge/skills: (1) resource knowledge, which is knowledge about one’s abilities and cognitive processes including knowledge of how to perform tasks or procedures; (2) heuristics, which are specific problem-solving methods or strategies; (3) regulatory processes, which includes the organisation and selection of resources and strategies; and (4) beliefs, which includes perceptions of and assumptions about mathematics (Gama, 2004).

Ilyasa

_______________________________________

TUITION CLASSES:

_______________________________________________________________

EDUCATIONAL SERVICES:

______________________________________________________________

By EX-MOE TEACHERS & EXPERIENCED TUTORS

@ BLK 644, BUKIT BATOK CENTRAL, #01-68. S(650644).

CALL 65694897 OR SMS 98530744 OR 97860411.

A. Math, Headstart Classes, Holiday Classes

Sec 4 A. Math Headstart Programme – Differentiation

Posted on November 9, 2012February 13, 2016 by admin

For our latest timetable, click here =>

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ORIGINAL POST (OUTDATED):

The details of the 2-day, 6-hr programme are as follows:

Date/Time (choose one pair of sessions only):

(1) Sat 10 Nov (10 am to 1 pm) and Sun 11 Nov (10 am to 1 pm);

(2) Thurs 22 Nov (4.30 pm to 7.30 pm) and Fri 23 Nov (4.30 pm to 7.30 pm);

(3) Thurs 6 Dec (4.30 pm to 7.30 pm) and Fri 7 Dec (4.30 pm to 7.30 pm);

Location: Blk 627, Bukit Batok Central, 07-640.

Class size: Max 6 students.

Investment amt: $150 total for both days (6 hrs).

Topics: Calculus – Differentiation & Its Applications

Tutor: Mr Ilyasa; M.Ed (NIE), PGDE (NIE), BSc (NUS), A-Level (RJC); ex-sch teacher, full-time tutor (8 years) of PSLE, O and A Level Math and Physics.

To book a place in the programme, sms to or call Mr Ilyasa at 97860411.

A. Math, A. Math Tips

A. Math Reflections & Study Tips

Posted on October 10, 2012February 13, 2016 by admin

(1) Simultaneous Equations

Generally an easy topic, but beware of certain kinds of questions such as:

(a) Questions involving reciprocals of x and y, such as

Solve 3(1/x) + 1/y = 1; 1/(x^2) + 1/(y^2) = 5

Ans: x = 1, y = -1/2 or x = -5/2, y = 5/11

Do not make common denominators and cross-multiply; the better technique is to let p = 1/x and q = 1/y.

(b) Questions involving coefficient matrices that are singular (determinant = 0), such as

Given that x and y satisfy the simultaneous equations mx + (m-1)y = 10 and (m-2)x + 3my = 20,

(i) if the equations have no unique solution, find the values of m; (ans: 1/2, -2)

(ii) if the equations have no solution, find the value of m.(ans: 1/2)

(adapted from Additional Math, EPB Panpac, p. 13)

(2) Indices, Surds & Logarithms

Generally an easy topic, except for students who do not understand or memorize the laws of indices and logarithms properly. I find that quite a number of weak students do not know the meaning of logarithm, thus not appreciating and enjoying the topic. Weak students do not realize that while Indices is concerned with the answer or expression obtained when a power is applied to a base number, Logarithms is about the power itself; the power that is needed to be applied to a base to give a certain number.

For eg, why is lg1000 = 3? ‘Evaluate lg1000’ or ‘What is lg1000?’ is the same as asking, “What is the Power that must be applied to the number 10 (the base) to obtain 1000?” Since 10^3 = 1000, therefore the answer is 3.

Weak students almost always make one or more of the following mistakes, thinking that (i) logA x logB = logA + logB, (ii) log(A + B) = logA + logB, (iii) logAB = logA x logB, (iv) (logA)^n = nlogA, (v) logAB^n = nlogAB, (vi) (a^m)^n = a^(m+n). All these are wrong.

How to be good in Indices and Logarithms? READ AND UNDERSTAND THE LAWS OF INDICES AND LOGARITHMS CAREFULLY. Yes, there is such a thing as READING MATH, not just practising Math.

(3) Quadratic Functions & Equations

Some of the points to note are:

(1) Understand that alpha and beta by themselves also satisfy the quadratic eqn because they are the roots of the equation! So it’s not just abt finding the sum and product of roots;

You must be able to solve questions like:

If α is the root of the equation x^2 = 2x – 1, show that α^4 – α^2 = 2α – 2.

(2) Understand that the discriminant (b^2 – 4ac) is less than or equal to zero when the question involves the phrase “for which the function is never positive or never negative”;

(3) Understand that sometimes you are required to solve an inequality involving the discriminant but at other times the inequality involves the function itself;

(4) Understand how to complete the square to determine the maximum or minimum value of a quadratic function;

(5) Recognise that “real and distinct roots”, “real and equal roots”, and “no real roots” have other names that express the same meaning.

TO BE CONTINUED ……….. (by Mr Ilyasa)

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A. Math, Pure Physics, Sec Math

Sec 4 Physics, A. Maths One-to-One Tuition

Posted on October 6, 2012February 13, 2016 by admin

For our latest timetable, click here =>

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ORIGINAL POST (OUTDATED):

Objective: Intensive Revision or Crash Course for O-Levels 2013

Focus: Understanding Key Concepts, Problem-Solving Techniques, Difficult Topics

DETAILS:

Available Time Slots:

Mon 4pm – 7pm

Thur 1.30pm – 4.30pm, 8pm – 10pm

Fri 6pm – 8pm

Sat 9am – 11am

Investment Amt: $80 per hr

Tutor: Mr Ilyasa; M.Ed (NIE), PGDE (NIE), BSc (NUS), A-Level (RJC); ex-sch teacher, full-time tutor (8 years) of PSLE, O and A Level Math and Physics.

Location: Blk 627 Bukit Batok Central #07-640

Average Class Size: 2-6 students

For appointment, call or sms to 97860411. Thank you.