Principles of Mathematics

Principles of Learning Mathematics - Know Your Limits

May 11th, 2008

While learning mathematics, we are always exposed to a few methods of solving a particular maths questions. The methods are taught in order to give us flexibility to select an apropriate technique to ”attack” any maths problems.

Is it good then to master all the techniques taught?

The answer is to know your limits.

If you are stressed up learning so many methods, let go of the ones that you find uncomfortable with. Master the one that seems to be the best and easiest to you.

Stay on with this chosen technique of solving problems, and apply it to similar maths questions. This is the first step in the principles of correct mathematics learning.

However, do not be compacent and stay stuck! Having more methods to solve a certain maths problems is always a better and sensible course of action any maths learners should aim for.

After mastering the first selected method, move on and try using another technique that was taught. Practice till it becomes comfortable and an easy tool to use.

In mathematics, flexibility is the norm. Questions are always varied, and thus, solution has to follow suit. It is this nature of solving mathematics that makes a good maths learners achieve much in his later life.

Knowing one’s limit is thus important especially when dealing with mathematics. Focus on one method first as too many at a go will only mess up the curious and greedy mind.

Understand that maths problem can be solved through the use of any one method. Even if the steps are more, it is still a way to obtain the answer. Slowly after mastering the selected method, you can explore another method that can shorten the solving process.

At least you are now more relax as you can fall back to the first method if the new technique cannot be comprehended finally.

However, having said that, do review the maths syllabus. Some syllabus do spell out that the students have to master a few methods to solve a certain type of maths questions. For that matter, you are left with no choice but to deal with them accordingly. Seek for help if necessary, and do not simply give up!

Finally a message that I like to share.  “Although practice makes perfect, good practice is the one that makes perfect ultimately!”

Choose the correct principles of learning and you will not go wrong. Know your mental limits.

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Solving Maths Develops Plan Foward Capability

May 9th, 2008

As in any case of solving problems, solving maths also requires certain strategy and procedures. Performing the mathematical steps to realise the result need certain skills.

One of this skill is the ability to see the “path” to the result. This ability, however, is obtained when we are able to plan what to do and reveal the intermediate goals.

Solving maths is like a mini-warfare where the enemy is the result that has to be obtained. Every steps that we take to achieve our goals has to be planned for.

We need to think ahead in each steps of computation and be capable of using whatever tools available to clear the obstacles lying in front of us.

Solving maths, thus, is a good platform for anyone to gear themselves for a future of planning, or to have better planning capability.

The steps required to solve a maths problem develops one to be able to plan forward. The steps taken at every maths operation serves a certain purpose to simplify mathematical expressions or eliminate unknowns.

Studying and doing maths is therefore a necessary part of human development in that it allows the learners to develop their mind to solve real-life problems through proper planning.

So to developing minds, cheers to maths!

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Why Do We Study Quadratic Equation?

May 8th, 2008

In maths class, we are hammered with expressions after expressions of quadratic equations. We are taught how to solve for its roots. We are taught all the necessary methods or mathematical techniques to handle quadratic equations.

But after all these, what is the purpose?

This is the question many students of maths studies ask.

Do we need this “quadratic” knowledge in working life?

See the diagrams and photos below. They will enlighten you.

The communication dish is parabolic in shape. Parabolic is the equivalent to quadratic mathematically. Engineers need to understand quadratic equation to dsign this beautiful profile.

This wok is designed using quadratic expression. With this, food can be fried to our liking!

Without quadratic equation, who knows how a wok would look like.

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Here you see that eye-glass lens are constructed with curves matching that of the quadratic equation.

Light is thus controlled to give good image to our eyes.

Quadratic equations to the rescue, right?

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Other examples are:

1)  Distance travelled given by the quadratic equation  s = ut + (1/2) a t²

2) Electrical characterisitcs of a MOSFET (Transistor device) 
                         i = k [(Vg - Vt)VD - (1/2)Vd²]

So now do you still wonder why you study quadratic equations?

Maths do have a purpose in our daily life. Rest assure that you are studying maths for a good cause.

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Merit of Grouping in Maths Solving

May 7th, 2008

For those who do maths at above elementary level, you will encounter many terms involved in the already many steps to solve a mathematical question.

Example is the solution of Partial fraction, that is highly needed in calculus.

One of the steps needed is the comparing of coefficients to extract out equations to determine the numerators in the individual terms of the partial fractions.

An example is shown below.

x - 3 = A x² + B x (x + 1) + C (x - 2)

How do we solve for the unknown A, B and C?

One useful technique is to do “grouping” of relevant terms.

This is a simple yet powerful method that make the process of solving maths less confusing as it serves to gather common or liked term in the same boundaries.

What I mean is ….

From the above example, we can rewrite them as,

x - 3 = A x² + Bx (x + 1) + C (x - 2) 
==>  x - 3 = A x² + Bx² + Bx + Cx - 2C
==> x - 3 = (A + B)x² +(B + C)x - 2C

Here, you can see that the coefficient of the terms can be equated nicely to be :-

x² term:  (A + B) = 0

x term:  (B + C) = 1

Constant:  2C = 3

From herer A, B and C can be easily determined comfortably.

Thus, grouping has the ability to simplify the thinking steps due to clarity as reflected above.

Another example is in Indices simplification.

Take the example of   10ⁿ5^n/2 / 20^n/4.

We cannot see any way to combine the different base number (10, 5 and 20) unless we split them into their lowest factors. (Don’t lost focus now, we are aiming for Grouping technique!)

10ⁿ = (5 x 2 )ⁿ
20ⁿ = (2 x 2 x 5)ⁿ

Rewriting to prepare for grouping,

(5 x 2 )ⁿ5^n/2   /   (2 x 2 x 5)^n/4

==> 5^{(n + n/2 - n/4)}   2^{n- n/4 - n/4}

^==> 5^5/4n   2^n/2

Here again, you can see the merit of grouping the common base number in order to perform the Indices operation.

Maths is “tricky” at times, but isn’t this to train our mind to stay active and flexible to counter any challenges put forward. It actually enhances our self-esteem and confidence to handle problems in real life.

Happy grouping …..  

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Indices | Big and Small concept

May 5th, 2008

In the study of indices, symbols are written with two sizes and in two different positions.  They have their own unique meanings.

a²  means   a times a,  or simply  a x a.

(a + b)²  means    (a + b) x (a + b).

(anything)³  means     (anything) x (anything) x (anything).

Therefore, from above examples, we can see that “anything” operated by a small number higher up above it means repetition by that number of times (defined by that little number).

Mistakes normally made:

(a + b)²  ==>  a² + b²            This is wrong!
a² - b²    ==> (a - b) ²        This is also incorrect!

Reason

The “2″ is a power and not a factor.  It means repetition.  And therefore cannot be factorised.  

The ”2″ is written above the normal line (called the base), and thus has “bigger” power than the base element.

The correct answer to the mistakes:
(a + b)² = ( a + b) ( a + b) = a² + 2ab + b²
a² - b² = ( a x a ) - ( b x b ) = (a - b ) ( a + b ).

To summarise, the “smaller” number (or letter) makes the “larger” base repeats the number of times indicated by that small number.

Principles of mathematics and its indices’ concept …..
Understand the principles and concepts, and you will be fine.

For more common mistakes in indices, click here.

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Tips To Reduce Errors Doing Simultaneous Equations

May 4th, 2008

Solving simultaneous equations involves many simple steps. The simple steps mostly include addition, subtraction and multiplication.

Though the mathematical operations are simple, mistakes made while solving simultaneous equations are aplenty. The errors are mostly “slip-of-the-mind” human errors.

To review the elimination method employed to solve simultaneous equations, please click here.

How then are we to reduce these careless errors?

Here, I propose 2 tips.

Tip one

Make use of addition instead of subtraction to eliminate the selected unknown.

Example:  ( to eliminate unknown “y”)

7x   +  2y = 11   — (A)
-5x +  2y = -1   — (B)

In normal doing, we perform (A) - (B) equation subtraction. But what is the risk?

The result may end up as 2x + 0 = 10 ! 

The correct answer should be 12x + 0 = 12.

Why the error? 

This is because our brain is use to addition more than subtraction. Therefore the “slip-of- the-mind” error happened.
Mentally doing 7x - (-5x) is harder to operate with.

Refer to this post to understand why our brain likes addition.

So how?

Negate the equation (B) so that we can perform addition.

(-5x + 2y) times (-1) = -1  times (-1)   
will become   5x - 2y  =  1   —-(C)

Rewriting the question,

7x + 2y = 11   —-(A)
5x - 2y = 1 —-(C)

This becomes simpler!
We now need to ADD the 2 equations.   (Instead of the risky subtraction).

(A) + (C) : 12x + 0 = 12   <== This is the correct result that we want, risk-free!

Message:  Negate the unknown variable of one equation and do ADDITION to remove it.

Tip  two

Avoid making the number (coefficient) bigger through multiplication.

Example:

9x + 2y = 13  — ( K )
  x - 4y = -7   — ( L )

Here, we have the option to remove either the “x” or the “y”.

Which to select depends on the proper selection of multiplication factor in order not to make the coefficient big.

Case 1:  Remove “x”.  
We need to multiply equation (L) by 9 to cause the first term (x) to be the same as that in equation (k), so as to eliminate the “x” unknown.

What happened?

Equation (L) became        9x - 36 y = - 63 !
Look at the coefficient of “y” ===> It became a GIANT!

Case 2:  Remove unknown “y”.
Multiply equation (k) by 2. This produces equation (k) as 18x + 4 y = 26.

This is still manageable.  And less error will occur since big number is harder to handle.

Message:   Seek to multiply coefficient such that the resultant equation has smaller number.

I have seen many maths students fumbling with these simple operations, and making many unnecessary mistakes.

I do hope that these two little tips will aid you and any maths learners of simultaneous equations to reduce errors.

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