Stars

Star 




We all know how a star looks like. The interior of a star is actually a pentagon. Let us now look at the vertices of the star. What is the sum of all the angles at the vertices?

We can generalise from there: Given a 2k+1 regular polygon, k>1, extend its edges to form a "star" pattern. What is the sum of the angles at the vertices?

Geometric properties of a circle

quad 


Consider the first picture, suppose the quadilateral as shown, with one vertex at the centre of the circle. Then we all know that the angle at the centre is twice the angle at the top vertex. Now suppose we have a quadilateral as shown in the second picture. How can we prove the converse? i.e. there exists a circle such that 3 of the corners cut the circle, and the centre of the circle coincides with the remaining vertex?

Relative velocity

This is a pretty interesting question. I've heard from my student that none of his classmates could solve it. Even some of his teachers had problems with it. It did make me think for a while, but once you have the correct diagram, it should not be too difficult.


Suppose a plane flies North at 100 km/h. The wind seems to be blowing at a bearing of 030(degrees). If the plane reduces its speed to 50 km/h, the wind seems to be blowing at a bearing of 050. What is the actual speed of the wind?

zero divisors

This is a common mistake students make. For instance,

x^2-x=0.

Some will divide both sides by x, and then obtain x=1. But this will result in a loss of a solution to the equation. Weird things start to happen when we divide through by a root (or zero). Perhaps we confuse students in the following case:

sinx =cosx, hence tan x =1.

The student will be confused. Why is it in the latter case we can divide both sides by cosx, yet in the former we cannot divide both sides by x?

Hence I think this example will show to students what is going on.

Why all numbers are equal to zero.

Let x=0. Dividing both sides by x, we have 1=0. So multiplying both sides by 2, we get 2=0, similarly 3=0,4=0 etc, hence all numbers are equal to zero.

Let the class figure out the mistakes themselves.

TAFN

The case of the missing 10 cents

Suppose Ali, Bala, and John buy a ruler that costs $2.50. They each have $1, and they pool a total of $3 to purchase the latter. So the change is 50 cents. Now, if we give each boy 10 cents back (since we cannot split 50 cents evenly among the 3 boys), we would have 20 cents left. But by doing so, each boy would have contributed 90 cents. In all, they contribute 90 X 3 = $2.70. If I add back the leftover 20 cents, we would have

$2.70 + $0.20=$2.90.

So where did the 10 cents go?

Angle bisectors in triangles

Figure 1 


Geometry is indeed a fascinating topic. However, many students tend to have alot of misconceptions. One type of fallacy is to assume some property of a figure by looking at the diagram. The diagram, sadly is not drawn to scale. Hence even though a picture is worth a thousand words, but it takes a mathematician to explain it completely.

Take for instance figure 1. Given that the cevian AD bisects the angle at D, is it true that it also bisects the length AC?

Some students I've taught felt that it is true. But of course, it is not. A bit of simple mathematics will prove otherwise. I'll leave this to the reader to show the following:

If AD bisects the angle at A, then the ratio AD:DC=AB:BC.

TAFN

H1, H2, H3 syllabus fo A level mathematics

As you'd have known by now, A level syallbus for Mathematics has been revamped. There is no longer any mechanics in A level math. In short, H1 Syallbus is for people who only need math for certain areas of application besides engineering. H2 is mainly concerned with science and engineering applications, while H3 is for research and varsity mathematics.

The outlines of the new syllabus can be found here:
H1
H2
H3

Interestingly, H3 syllabus consists of Geometry, ODE, Graph Theory and Combinatorics. Geometry is similar to the NUS course "Geometry: From ancient to modern", where one learns Ceva's and Menelaus' theorem. Furthermore, H3 exam questions will consist of proving statements given.

H2 syallbus is, in my opinion, Maths C plus a quarter of Further Maths. There is no longer any mechanics section, and even the statistics portion is reduced. Most notably, one is no longer expected to derive continous random variables, and t-test involving difference of 2 sample means is no longer included. The strange thing about the syallbus is that partial fractions is not mentioned at all. However, one would still need to use techniques involving the latter to do other topics such as series and integration.

The use of graphing calculators reduces the workload of the student. There will not be much questions on curve sketching, but rather more on questions on the behaviour of a function. For instance, finding asymptotes. Finding the inverse of 3x3 matrices is done by the calculator as well, and hence solving a system of 3 linear equations would not require one to use row operations anymore.

TAFN

Applications of maths

I just started tutoring and my student is quite inquisitive. For example, he likes to ask what's the use of learning this math concept etc. I find this very encouraging, and I would like to share some application of the topics. But somehow, it's not as easy as I thought.

Take for instance, today I was going through functions. He asks me, so what's the use of learning inverse functions? I was quite stumped for a moment. Obviously, all these things are very important and apply to all sorts of situations. But to think up of one on the spot can be a daunting task.

But later on I realise I've been thinking in the wrong direction. Each concept has its purpose and applies to many areas of our lives. But not every concept can be substatiated by giving a real life example. Most math concepts are more likely to be used as a tool, or rather, a tool of a tool, to approach a problem. For instance, learning about what are injective functions or surjective functions seem rather meaningless for a first-time learner, but it is important in the study of graphs. For instance, we use such properties to interept the behaviour of a graph. And graphs are important in empirical data in many fields of study: be it in physical sciences, economics, engineering, statistics etc.

So that is what I mean by somethings being a tool of a tool. Concepts that seem to be meaningless are actually the foundation to build our "weapons" to handle real life problems. If weapons are indeed used to handle problems, then mathematical concepts we learn from 'O' and 'A' levels serve as the parts of a bullet.

However, I do lament that not enough real life examples are given to students to cultivate their interest in maths. Indeed, there are areas where applications immediately follow theory. For example, any student who has learnt differentiation knows how it applies to optimization problems. But there are many other areas left unexplored. For instance, how does complex numbers apply to real life situations?

If I put myself in a teacher's shoe, I do understand the predicament. It is often difficult to talk about an application without going in-depth into other fields. In other words, one would be inviting alot of "bombastic" terms and terminology in order to discuss an application. Alternatively, one can always just remark that such a concept is applied in a specific area of engineering say, but no one will truly be satisfied with this answer.

I still feel optimistic since the world is so vast, surely it is possible to collect some solid yet simple (to understand) examples to present to our students. Maybe in the event that it is truly difficult to find a simple example, it is best to quote a area of study as above, and guide the student in the right direction to find out more about it himself.

Perhaps it is due to my ignorance that I have not come across any good example of how complex numbers are used to real life. The example I used is geometry. I explained to my student about geometry and transformations. Then I compared cartesian geometry and looking at the same shape on a Argand Diagram. By comparison, and by applying some of the concepts he learnt earlier, he was astonished to see that transformations can be done so easily on the argand diagram, just by mere multiplication of a complex number to the orignal function. And I left it to his imagination in what disciplines such knowledge would be important.

If anyone has any good examples in any area of mathematics that we learn in 'O'/'A' level, please share with me. I would like to accumulate a wealth of information to help other teachers as well. Ultimately, the students will benefit the most.

Sudoku

I've been thinking of 2 math questions recently. Let me just post it here.

The first one is an interesting question. We have been bombarded with Sudoku recently, you know, the addictive puzzle which we need to fill up 9 latin sqaures, with the additional restriction that every row and column cannot have a repeated number. I've been trying to compute the total number of puzzles that can be generated. It turns out that it is quite hard. Moreover, I found on a site that, according to this fellow, the total number of puzzles that can be generated is:

6,670,903,752,021,072,936,960

Note that this number is the number of puzzles which can produce a unique solution, not the number of solutions to Sudoku, which is obviously less. (perhaps someone can come up with a number?)

One more thing, it appears that Solving Sudoku puzzles is an NP problem. That is, no efficient algorithm can be used to compute an arbitrary Sudoku Puzzle.

All this information can be found here.

The second math issue is a really fundamental one, and it was pointed out to me by Jasper. Consider this trigonemetric identity:

cot (x)=1/tan(x)

Sounds familiar right? But did you notice something wrong with it? It appears that this isn't really the definition for cotangent fuction. It lies in the fact that when we subtistitute x=pi/2, or in fact take x to be any odd multiple of pi/2, we get something funny:

cot(pi/2)=0 yet, 1/tan(pi/2) is not defined. Note that tan(pi/2) is undefined. The left and right limits do not match. Although 1/tan(x) as x tends to pi/2 yields zero, this is just the limit and is not an equality. Hence this definition is false!!

The true definition for cot (x) should be

cot(x)=cos(x)/sin(x)

TAFN