An easier approach to remembering discriminant

JC Mathematics

I notice many students forget how discriminant works! I think I should inform A-level students that they need to know how it works and that is one thing they should not return to their O-level teachers. So I thought I share a bit on how to effectively, get it correct and also use it. At the same time, I hope to better the students’ understanding too.

Credits: learn-site.net
Credits: learn-site.net

Let us first look at the quadratic formula that is well engrained in our heads.

x= \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}

Now do we see a very familiar formula in there? Yes, its our discriminant formula, b^{2}-4ac!!! So how does it work, alongside with roots. Let’s look at what a root is next, a root is loosely put, a solution to the equation. I hope it is slowly making sense.

When the discriminant is negative, we have a \sqrt{-\mathrm{number}} which is imaginary, that means we have NO real roots!!

And when the discriminant is zero, we have \sqrt{0}=0 so we only have one single root, that is -\frac{b}{2a}.

BUT if we have a positive discriminant, than due to the \pm{ } there, we end up with two roots.

So I do hope this clears up the air as to how we can relate some algebra with roots. 🙂

This is a question worth looking at from A-levels which test us on discriminant.

Display of arithmetic skills

JC Mathematics

This is a video of an arithmetic challenge that took place in China in 2015. It is China against Japan, and the title is called the best brain of China. I shared a video on arithmetic previously, and a student mentioned that this only possible after years of training. So here is some kids showing us how math is done!

As you observe from the video, the participants are really young, as young as 9 years old! And the mental arithmetic skills exhibited here is truly insane. I shared this video to show students how “cool” is it to be able to do mental calculations really fast. At the same time, I hope to motivate students to brush up on their arithmetic skills.

From my experience, students with strong arithmetic backgrounds, are able to be more meticulous with their work. They can check their own work efficiently. Furthermore, it removes over-reliance on calculator. It pains me to see students do 2-6 with calculator. Just a few days back, one students got a question wrong and realised she calculated it correctly but keyed in wrongly in calculator. She trusted the calculator and went ahead. This is really a waste, and I notice this in many students.

If I do have kids, I will hope to train their arithmetic skills up. And that will begin by putting the calculator away.

The Modulus Sign #4

Secondary Math

I received this from a student of mine who was really confused with the transition from line 2 to line 3. Students should give it a thought before reading my explanation below.

Modulus of a Complex Number
Modulus of a Complex Number

Firstly, my student felt that |(2x-1)+2yi| \le |(x+1)+yi| was being squared on both sides to remove the modulus and the next step should be [(2x-1)+2yi]^{2} \le [(x+1)+yi]^{2}. She is very mistaken, as (2x-1)+2yi is a complex number, which is a vector, and not a mere scalar. So the transition from step 2 to step 3 was really an evaluation of the modulus. If we do not skip any steps, the following should clarify how we got from step 2 to step 3.

|(2x-1)+2yi| \le |(x+1)+yi|

\sqrt{(2x-1)^{2}+(2yi)^{2}} \le \sqrt{(x+1)^{2}+(yi)^{2}}

\sqrt{(2x-1)^{2}+(2yi)^{2}} \le \sqrt{(x+1)^{2}+(yi)^{2}}

(2x-1)^{2}+(2yi)^{2} \le (x+1)^{2}+(yi)^{2}

I do hope this raise awareness for students to treat modulus carefully and question themselves, when they are dealing with a scalar or vector.

The Modulus Sign #1

Secondary Math

A few weeks back, I conducted a weekend of lessons on the modulus sign (or absolute value sign) with all my JC2. Its to prepare them for the upcoming prelims and A-levels. Many students till today, are still having difficulties with the modulus sign. I thought I will start a little series, just like the integration to introduce some misconceptions.

So what does modulus do? We know that |-2|=2, so it pretty much makes everything negative, positive. Is that it? This is a very loose representation of modulus (absolute) function. I usually tell my students to think of modulus like magnitude. We are really interested in the magnitude to -2 here, and that will simply be 2. Why do I choose such a representation? Simply because it is the actual meaning!! Applying a modulus, is seeking the magnitude.

It so happen that -2 is a scalar quantity. And when we talk about |x+yi|=\sqrt{x^{2}+y^{2}}, we are applying pythagoras’ theorem here to find the magnitude. This is slightly different because x+yi is no longer a scalar, but a vector!

I hope clarifies your doubts about what exactly does modulus mean!

Partial Fractions made easy

JC Mathematics

Many of my students seem to be impressed with my method of doing partial fractions, and after I explain to them, I think they all can conclude that it is nothing more than kindergarten math tricks.

There isn’t a name to this method of doing partial fraction per se, so let us call it “plus minus zero”! The trick is to understand that what follows after every polynomial of normal is a mere plus minus zero. so we can go ahead and do whatever we want to it; be it, “+2x – 2x” or “+2-2”. And after you learn how to see what to choose to plus and minus it, everything really boils down to some really straightforward cancellation

For example, when I see things like \frac{x}{x+2}, I will think of introducing -2+2. So we have \frac{x+2-2}{x+2} = \frac{x+2}{x+2} - \frac{2}{x+2} = 1 - \frac{2}{x+2}

Or when we have \frac{x^{2}}{x+1}, I will think of -1+1. Then \frac{x^{2}-1+1}{x+1} = \frac{(x+1)(x-1)}{x+1} + \frac{1}{x+1} = x - 1 + \frac{1}{x+1}

This is a really neat and good technique to long division method, cos I don’t like to do it haha. But this takes some skills, as most students will agree. But most of my J2s are so used to seeing me do it, that they have developed this skill too. 🙂

Confusion on when to put ± sign

JC Mathematics, Secondary Math

The \pm sign definitely can cause a lot of confusion for students. For starters,

|x|=\pm{x}
x^{2}={|x|}^2

and the list goes on.

So my focus today is to explain when to put \pm while dealing with \sqrt{ }. First, lets recognise a few things.

x=\sqrt{9}: This gives us x=3

x^{2}=9: This gives us x=\pm{3}

Yes, the square-root can be raised to a square for the x, but notice the difference in answer in both cases. The former is a +3 while the latter has -3 too. This is because the square-root is always a positive square-root and does not return \pm{ }. On the other hand, x^{2}={|x|}^2 and the |x|=\pm{x} which causes the \pm{ }.

So next time, please be alert when you introduce the \pm{ }.

Interpreting GP essay questions

JC General Paper

Writing a GP essay is always a daunting task if you are not prepared or worse, do not know how to start. Here is a step-by-step guide on how to tackle a GP essay question.

1. Understand the question

A lot of times, students make the fatal mistake of misinterpreting the question and thus end up writing off-topic or hijacking the question i.e. changing the question. A GP essay question becomes very straightforward if you can identify its main components:

  • opinion
  • context
  • task

The opinion gives you the discussion topic. The context tells you the time frame and/or society from which your examples should come from and finally, the task tells you what you need to do.

Example 1

Unlike the Arts, such as writing or music, Mathematics lacks the capacity for creativity. How far do you agree with this statement? (2013)

In the first example, the question is only made up of two components: opinion and task. There is no context, which means that your examples can come from anywhere and any time period.

Looking at the opinion, the question is hinting at us to compare the Arts’ and Mathematics’ ability to foster creativity. You then need to decide whether to agree to a large or small extent if ‘Unlike the Arts, Mathematics lacks the capacity for creativity’. This is your task.

Example 2

How important is it for people in your society to retain a sense of tradition? (2010)

In the second example, we see all three components: opinion, context and task. Looking again at the opinion, the question wants us to discuss the importance of tradition, however the context limits us to using examples from our society only i.e. Singapore. Finally, the task wants you to decide if preserving tradition is very or not so important.

2. Plan your essay

Now that you have understood what the question wants you to do, you can proceed to plan how to write your essay. As the saying goes, ‘If you fail to plan, you plan to fail’ so planning is VERY important. Planning gives your essay a clear direction. Moreover, you are less likely to deviate from the topic or suddenly get stumped in the middle wondering how on earth to continue.

You should spend a maximum of 10 minutes planning. First, brainstorm quickly for ideas, counter-arguments and relevant examples. And by quickly I mean anything that comes to mind (I usually brainstorm using a mind map but any way is fine as long as it is fast and easy for you to understand).

Image taken from: one.garazx.cu.cc

Second, select the strongest points which support your stand and the weaker points belonging to your counter-argument. Doing so will help make your argument sound more convincing. Typically, the ratio is 3 strong points : 1-2 counter argument(s).

Finally, plan your paragraphs. A basic GP essay layout includes a/an:

  • Introduction
  • Body: this may consist of 3-4 paragraphs depending on how many points you want to argue
  • Conclusion

For the essay body, order your points and counter-arguments in a way that allows you to present the most convincing argument.

3. Write your essay

Once you have finished planning, start writing your essay and stick to your plan. Do not add or change anything unless it is absolutely necessary because this will eat into your time.

In summary, it only takes three steps to kick-off writing a GP essay. Pull out past year papers and try your hand at planning some of the essay questions. With practice, one should get familiar with the process in no time.

Classical Mathematical Fallacies #2

JC Mathematics

This is a classical mathematical fallacies. At hindsight, everything looks perfectly fine.

"Ï€=3" Mathematical fallacy
“Ï€=3” Mathematical fallacy

This is a rather difficult one as the error is actually hidden in the second last line! The mistake here is that we conveniently removed the power, but overlooked that there will be a \pm{ }. If we do account for it, we will have the original result.

Classical Mathematical Fallacies #1

JC Mathematics

This is a classical mathematical fallacies. At hindsight, everything looks perfectly fine.

"2=1" Mathematical Fallacy
“2=1” Mathematical Fallacy

So did you manage to spot the error? It is at line 5, where we divide both sides of the equation by a-b. Now this is mathematical alright, but in this case, it is not! notice that a-b=0 on the account that we started with a=b. And we need to be reminded that we cannot divide anything by zero.