# Parametric Equation

One nice question I shared with my students during our revision lesson.

One question on parametric equation I shared with my students during our revision lesson. Most of them knew how to sketch the graph using the GC but none of them understand that it is also important to know the orientation of the curve, i.e. how the x, y coordinate moves as t varies from – pi to pi.

Without this concept of orientation, they could not find the coordinates of point Q.

It is hence important to understand how parametric equations work rather than just memorising the procedure to get the final graph, which is lacking in various schools that my students come from.

Mastery through understanding takes time and it can be a painful process. However once you get it, it will stay with you.

# A Level Math Preparation Tip #3: Complex numbers

Complex numbers usually come as 2 questions, 1 on the loci and 1 on algebraic properties involving Cartesian, Polar or Exponential form.

One common type of question is on the properties of argument and modulus when we multiply, divide or raise a complex number to a certain power. Recall:

$\displaystyle \arg ({{z}_{1}}{{z}_{2}})=\arg ({{z}_{1}})+\arg ({{z}_{2}})$

$\displaystyle \arg \left( {\frac{{{{z}_{1}}}}{{{{z}_{2}}}}} \right)=\arg ({{z}_{1}})-\arg ({{z}_{2}})$

$\displaystyle \arg ({{z}_{1}}^{n})=n\arg ({{z}_{1}})$

$\displaystyle \left| {{{z}_{1}}{{z}_{2}}} \right|=\left| {{{z}_{1}}} \right|\left| {{{z}_{2}}} \right|$

$\displaystyle \frac{{\left| {{{z}_{1}}} \right|}}{{\left| {{{z}_{2}}} \right|}}=\frac{{\left| {{{z}_{1}}} \right|}}{{\left| {{{z}_{2}}} \right|}}$

$\displaystyle \left| {{{z}_{1}}^{n}} \right|={{\left| {{{z}_{1}}} \right|}^{n}}$

Next is the finding roots of a polynomial with unknown coefficients.

One common “not smart” approach student is to write “since -2+i is a root, the conjugate -2-i is also a root”. Followed by factorising the polynomial into linear factors and compare coefficient to find “a” and “b” and then the roots.

It is easier to follow the instructions given the question. A root means when you sub z = -2+i into the polynomial you will get 0. Then compare coefficient (real part = 0, imaginary part = 0) to get “a” and “b”. This method is faster as once I know the values of “a” and “b”, I can use GC “polynomial root finder” to search for the 3 remaining roots (usually it is quite nice).

Also if the question did not mention “a” and “b” are real, it is not right to say -2-i is a root.

# 1. YEARLY – TOPICAL – YEARLY (YTY) approach

In the final stage of the preparation you should be spending most of your available time doing revision. Yes breaks are important but you would prefer to study hard now and enjoy a true happy break after A levels knowing you have done well.

Step 1:

You probably need to set aside 3 + 3 + 3 hours a week on Mathematics. The two rounds of 3 hours is to try one year’s past year A level paper 1 and paper 2, each paper doing it without any breaks.

Step 2:

Mark your script, circle the questions that you could not do and spend the remaining 3 hours doing topical revision and do corrections for the questions answered wrongly. You may want to seek help from your teacher, your tutor or your friends.

Step 3:

Repeat this 3 + 3 + 3 cycle with another set of A level papers, hopefully you can apply the lessons learned from the mistakes and perform better in the next set of papers.

# 2. Study the trends of the exam papers (there is IMO)

In one of the past years’ examination markers’ report, it was mentioned that students may not do well if they try to predict questions based on past year papers.

There is some truth in this statement as every year there may be a couple of questions that are “unusual”, which serve as a benchmark to separate the A students from the rest.

However this doesn’t mean the entire paper is going to be different. In my opinion there is a core structure of how the paper is set according to the syllabus set by MOE or SEAB.

# 3. Never give up!!

Remember A level is one sitting 100%. If you have performed badly for prelims and done well for A levels, your certificate will only reflect the A grade you obtained. Therefore there is always the chance that your grades will jump up significantly. However if you choose to give up then nothing can be done. I recalled one of my students who never scored more than 20 marks for all the test / exams since JC 1. However by following some tip I gave, he never give up and managed to pass the subject during A levels. This is already a very big improvement, to jump by at least 25 marks which is equivalent to 4 or 5 grades.

# 4. Never be too confident!

Scoring well for Mathematics requires consistent practice, you can get “rusty” if you stop practicing for a period of time. You would not want to waste all the efforts of scoring A in all school assessment only to not obtain it at the only important moment. I had a student who scored high As throughout the two years in JC yet scoring B in A level. Therefore if you have been scoring As, please maintain it.

Good luck to all students taking A levels!

Mr Ang

http://alevelmathtuition.sg

ang.alevelmath@gmail.com

94385929

# Review: Edusnap Part 2 (Not so great afterall)

I have reviewed Edusnap probably 2 weeks ago and I found the app to be noble in idea on how students are able to receive academic help from kind souls anytime. Edusnap was also featured in the news a couple of days ago.

However being one of the more active people offering help to people in need, I realised that this app has a serious downside: People abuse it to get answers to questions without any / minimal effort on their own.

This app, used in the hands of the wrong people have made them more lazy and reliant on others.

I took a snapshot of one of the requestor:

I may be wrong but it seems to me that this person is simply asking people to do his / her homework so that he / she can submit in time tomorrow.

Not just that : The creators of Edusnap mentioned that the helpers should offer hints and not spoon feed the solutions. However there seems to be no moderator around to deter such behaviour.

It may be a “hype” using technology but is it really helpful to all? Or is it merely a form of entertainment compared to handwriting and wrecking one’s brain to find the solutions on his / her own?

# Question that is not well thought out.

My student had difficulty showing that 3 < A < 3.75 and I was stuck for a couple of seconds too. Reason? The question did not state clearly how many rectangles are required. The answer is 2 rectangles, but if a student used 3 or more rectangles or 2 rectangles that are different from what the setter has in mind, then they would probably not get the values 3 and 3.75.

The setter did not consider enough when setting this question.

# Review: Edusnap

I came across this application through random google search and decided to download it onto my android phone to test it out.

Basically the idea is you have a question on any subject, you can just take a snapshot of the question and just *scream* for help. Any kind souls with the answers will reply back taking a snapshot of their solutions. Others may post comments on the questions of the solutions provided.

As far as I know, you cannot access to the poster’s or answer provider’s information. This includes their email address even if they want to share. This actually makes it hard for tutors trying to promote themselves by offering free help.

It is pretty organised in the sense that you can attach your question according to level and topics. Students  who need some reference just for knowledge sake can come to the menu for easy access.

Overall my experience with this app is pretty positive, just that sometimes it crashes with my phone. Another area of improvement is that whenever you go back to the question page it does not stay at your previous position and you have to search for it all over again. Since this is free, I thought it is good for students to post questions and train their mastery skills through providing solutions. However I suspect some parents uses this platform to find answers for their kids instead. I have seen adult posters asking primary school math questions.

This is not encouraged as I believe one way to learn is to let the kid’s hands get dirty.

# PJC’s unique syllabus

A new JC 1 student from PJC joined one of my classes today and I realise she doesn’t fit into any of the lessons I am conducting. Reason? PJC is adopting a syllabus that is different from the majority of the JCs or IPs.

PJC chose to begin with vectors, which is voted to be one of the 2 HARDEST topic for Pure Math section. Most schools would start with either 1) Graphing Techniques or 2) Sequence and series. I am not sure what signal PJC is trying to give their students but the students will be struggling in school due to the difficulty level of the topic, plus they will find it very hard to fit into tuition classes because the rest are doing something else.