Caution on Mixed Number

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Fractions are a necessary part of maths.
They come in many forms; improper, proper and mixed number.


Though improper and proper forms are direct in its presentation and interpretation, mixed number form may pose a potential mistake for young learners.

Example:
   
Is this 2 + (3/4) or 2 x (3/4) ?

Caution has to be taken to stress it as 2 + (3/4).

Some students have taken it to mean 2 pieces of (3/4) !
Dangerous isn't it.

But rest assure.
If you understand the language of maths and its "grammar", all will be well and interesting.

:-)

Using Units to Deduce Maths Formula

There are times when we cannot remember some simple formula for a maths application.

Or we have doubts to the some maths working  especially when many parameters got involved.

I ave a simple tip.

Look at the units for the numerical item.

Example:

To calculate distance travelled by a vehicle, given the speed it goes  and time taken,

we look at the speed's units.

Unit:  m / s

What does it tell?

Yes, it gave an indirect answer that speed = distance / time.

Thus if time is given, we are able to know that we just need to multiple speed by time in order to retain only the distance.

(m / s) x s  =  m (only)  ==> Distance

The above allow us to use units to deduce the working (and formula).

Hence, we should not overlook the power of knowing units.

It is simply disappointing to sometimes see people missing out on writing the units for certain parameters. Maths loses its value simply by ignoring this step.

Therefore treasure this little but powerful "units".

:-)
Maths is interesting!

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Number of Answers | Common mistake

Maths can be tricky when you are not careful.
This is not to frighten you, though.

This post is just to remind you of the wonderful aspect of maths in covering all areas.

Below is an example of what I meant.

Let's take the quadratic eqaution solving as a starting point

x² = 5x

x  = 5x  /  x   = 5  (Answer)

At first, this looks pretty fine. The answer, when substituted back, produces match of equation.

But this is actually not complete.

Those doing quadratic equation will know 2nd order (x2) equation evaluates to 2 answsers.
The answers may be the same though.

Now, if we approach it using another method, let's see the different.

x² - 5x = 0

==> x (x - 5) = 0   , after factorising

==>  x = 0  and  (x - 5) = 0
==>  x = 0  and   x = 5

There are two answers now.

We had the x = 5 initially, but what about this new x = 0.

We have missed out on the x = 0 with the first mehtod. It looks OK then.

What happen?
It may be due to lack of experience handling this form of maths question.

The concept in solving quadratic equation is actually not limited to second order.
The hidden message is depending on the order, the number of answers will follow suit.

What I meant is :
2nd order gives 2 answers,
3rd order gives 3 answers,
4th order gives 4 answers, etc.

It is this verry message that maths learner should capture. Otherwise you will be tricked to give only one answer which leads you to "mistakes" of being incomplete.

I agree that this is tricky, but within reasonable argument.
If a student practice hard (and smart), he will not fall prey to this type of simple math problem.

Do not get con again.

Enjoy maths. It's fun and interesting.

:D

Tips on Avoiding Mistakes (Unit writing)

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Maths involves many traps.

Any one of this traps will make the solution looks odd or even to the extent of wrong answer.

What are this traps ?

Mathematical operators, symbols, units, transferring of numbers, size of the written symbols, decimal points are some of the examples of traps contributing to the error.

Here I would like to mention about "unit".

In maths, calculation of items are aplenty. One of them is the study of speed.

In the topic of speed, students are dealing with three basic elements.
They are the distance, time and their ratio (speed).

All these three elements have different units all to themselve.

Distance ==  metre
Time ==  second
Speed == metre / sec

There are variations of the above.
km, mintues, hours, km / h, m / min, etc

Do you now see the danger?

If you are dealing with so many units in one maths question, what are the chance of making mistakes?
If you are careful, the chance is low, but it does not mean zero.

You still have to be careful.

How to avoid having mistakes due to this undesired slip?

One tip is to write down the units in the working steps.
Do not leave the numerical answer (in the working) without any unit indicated.

Make clear the item of interest, whether it is distance or time by reflecting the unit besides the number.

Example:  5 km,  40 sec.

A complete maths example will push the message across, thus ....

Example :
Alan travelled at a speed of 60 km / h for 2 h. After that, he slowed down by 20 km / h and travelled the last quarter of the journey at this new speed. How long did he take to travel?

Working:
60  x20 = 120
120 / 3 = 40
60 - 20 = 40
40 / 40 = 1
2 + 1 = 3
Answer: 3 hrs.

What is your comment on the working?

I personally feel uncomfortable.  What about you?

The danger in that sort of working is the lack of showing the actual item in the calculation.
It does not allow a good way for checking after completing the worksheet (if many maths problems are within).

Clearly writing the units will, at least, make checking later an easier task.

It also allows the marker (teacher) a clearer picture instead of guessing what you intend to show.

Along the way, during the working, you will also have a lesser chance of getting confuse as the items are listed with the proper message (through the units).

So are you convince proper unit presentation is worth the while?

A pointer for your thoughts.....

Cheers  :-D

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Percentage Increase in Perimeter

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Percentage is a nice and mystery word in maths.

Why do I say that?

Look at the example below:

If the perimeter has increased by 30%, does the length also increases by the same amount?

The answer is obviously YES.

Next,

If the perimeter is increased by 30%, does the area covered by it also increases by the same amount?

???  The answer needs some pondering, right?

Answer to this:
If the perimeter is increased by 30%, the length and width will both increase by 30%.
This makes the area increase by 2(30%) + (30% x 30%) = ?

(I will explain this maths calculation in a later  post.)
For now, let's concentrate on the maths operation.

What do you get from 30% x 30%?
30% = 0.3
Thus 30% x 30% = 0.3 x 0.3 = 0.09 = 9%

This is a potential mathematical  mistake.
Error:  30% x 30% = 900% !

So, increase in area becomes 60% + 9% = 69%

Interesting how the mind works.

If the mind is not clear when doing maths, common mistakes do occur.
With more practice, however, this form of mistakes will be lesser.

Hence, be careful when dealing with parameter such as perimeter, length and AREA.
Know their relation and be aware of the "catch" when this type of maths question is being asked.

Do not fall for the maths trick.

:-)
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Proper Way Of Writing Maths Expression

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Maths expression tells certain message. When it is not written properly, or written in such a way that it causes wrong interpretation, then you will expect marks to be deducted.

Examples:

1)     y = cos (A + B)
2)    g = x + log K
3)    y / x + 2

Let's look at the above examples one by one.

Example 1:
If the brackets are taken out, y = cos A + B.
Does it also mean B + cos A?

Example 2:
If the sequence is swapped, y = log K + x
Does it mean y = log (K + x)?

Example 3:
Is the denominator just x or (x + 2)?
Or is the correct expression 2 + (y /x) ?

From the above 3 maths expressions, you will observe and sense that something will go wrong when you did not write "properly".

This need practice and does need some "maths" sense to go along with the practice.
You need to know the different form of expression and its implications.

Questions like:
- one term or two terms in the desired expression?
- which is the actual denominator?
- will anyone mis-interpret the logging of term?
- If the words or symbols are too small, will they be able to see clearly?

To save time and marks, write with the reader or marker at heart.
Write as though they are reading them.
Think and write like they will be.

Maths is afterall, a language that has to be shared and used to solve certain objectives.
Do write clearly and appropriately.

The practice and skill mastered will do you and everyone one good.
Strive to make less unnecessary mistakes and reduce the chance of your marks being subtracted off through improper writing.

Cheers!   ^.^

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Tricky Angles | Be Aware!

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Geometry in maths can means dealing with angles from a square or a rectangle.

Normally the question is to determine an unknown angle given some shape and angles.

However, mistakes can happen when basic knowledge of relationship between angles and  shapes are not proper understood.

Here, I will stress on the square and rectangular matters. This is basic but can pose a tricky problem to the unwarys. Poor thing.....

Let's look at the diagram below.

Here, if sides M and N are the same, that is, if the box is a square,  angle A will be 45 degree.

This is so since the corner where angle A lies is 90 degree divided EQUALLY by half due to the diagonal lines reaching to the opposite side. (symmetrical sides).

However, if the side M and N are not equal in length, then angle A WILL NOT be 45 degree. It will depends on the ratio of side M and N.

Note this message and unnecessary mistake can be avoided.

Sometime it is to test the logical thinkng through maths, by not telling you angle A is 45 degree but stating that the box is a square.

This type of maths problem will require you to calculate another angle but using angle A which is not given.

It is tricky but good to have. Your brain will be stretched to make it "flexible" for future use.

Maths is good in this sense as it twists our mind and makes our life interesting!

Work hard as well as smart.

For more examples on avoiding unnecessary mistakes, visit this time calculation post.

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Misconception of Percentage

Maths is interesting!

 

It catches you nicely when you are not aware of its full implication.

 

Many of its techniques and concepts has wide boundaries.

 

Any maths learners has to understand its fundamentals and basic concepts in order to "escape" being caught with wrong usage.

 

Percentage is one area I wish to mention in this post.

 

You may find that percentage or percent is a simple term in maths.

 

Besides being a ratio, it is a comparative element.

 

It tells how big the target is to the original, or how small it is.

 

Misconception of Percentage:

 

The numerator is always smaller than the denominator (which is the total).

 

Is it true?

 

A definite NO!

 

Reason:

If a ratio has its numerator less than the denominator ==>  The numerator is relatively smaller in size than the total.

If the numerator is larger than the denominator ==> The numerator is bigger in size than the total.

An example can illustrate the concept.

Example:
If a costume is now priced at 90% of its original, $100, it means that the price is not only $90.
It is lesser than the original, since the ratio is 90 / 100 or 0.9.

If the costume is newly priced at 120% of its original, $100, it means that the price is now at $120!
A price value more than the original.
It is a practical real-life number.

• Percentage can be more than 100%.
• The numerator can be more than the denominator.

Understanding this concept will serve anyone good. Things in life goes up as well as down. Percentage reflects this sense through its number.

Interesting? You bet.

:)  Happy maths learning.
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Is Maths Really Interesting?

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One question those who detest maths will ask is "Is maths really interesting?".

It is a very subjective question.

Everyone has likes and dislikes.

However, in the case of maths, it is the gain versus the lack.

Maths is a necessary life skill to have.

Knowing it makes a whole lot of different.

It will speed up your solving to some daily questions.

"What is the time needed if I drive at 60 km/h for a distance of 90km?"
"What is the area of the metal sheet needed to cover this pillar?"

We are weak in maths due to many reasons.
If you do not arrest these reasons, or reduces the obstacles to it, you will always fear maths.
This will create a mental block to your maths learning.

Practice and practice to reveal your weakness. Learn through mistakes.
You will feel the confidence of handling maths problems after that phase.

Like what Mark Twain said "Action speaks louder than words".

I would like to tweet it in the context of maths.

Instead of pure saying that you cannot do maths or you hate maths, practice (action) on it.
You will feel the difference.
You will get the hang of doing maths.
You will realise that maths is not that difficult.
You will find that it is your mindset that is the block, not maths!

Practice.

Do it.

Practice speaks louder than words.

Maths is interesting.
That will be your final conclusion if you take action and do hands-on practice.

Cheers! :D

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Careful With The Use of 2 and Z in Maths

Mistakes are normal in maths. This is during the learning stages.

It will not be acceptable during tests and examinations. This everyone knows!

But human do make mistakes. This is a fact. Therefore, we have to know our weakness and avoid moving towards it (them).

A very common mistake made while doing maths is to use the variable "Z". From the writing, you can see that it is very similar to the number "2".

During examination, we are tensed up and our mind may not see what the eyes received.

2Z may end up as 22 finally. This "22" will then be used for computation and of course results in a BIG shock!

Thus knowing this danger, avoid using variables that are close to number in writing, unless stated by the maths question itself.

Do not choose a similar looking variable and end up with disappointment.

"b" and "6", "l" and "1" or "S" and "5". These are dangerous combination.

Thus look carefully when dealing with these numbers and variables.

Do not cause unnecessary mistakes. Save your effort to deal with better challenging thinking.

:)

Time Calculation May Not Be Easy

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Counting from 1 to 100 is normal for anyone. Just increment by 1. It's that simple.

But counting time may be another story for young math students.

Why is this so?

There is the seconds, the mintes, and the hours to handle. They differ by the umber 60.

The "carry over" through addition is the number SIX!

1 min 30 sec added by 40 sec gives .......

1 min 70 sec?

Here the 70 sec includes 60 sec + 10 sec. You need to understand that for time, 60 sec means one whole minute.
Thus, 70 sec = 60 sec + 10 sec ===> 70 sec = 1 min + 10 sec.

Therefore, 1 min 30 sec add 40 sec gives ==> 2 min 10 sec.

Compare this to adding 30 by 40. Answer = 70. No more analysis.

Time base on 1 hour = 60 min, 1 min = 60 sec concept.

To test true understanding of addition (and subtraction), time is a good gauge and tool to assess learners.

Test it on young kids today to annoy them... *#^&@!
Cheers!

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Common Mistake Of Gaps and Length

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There are some maths questions that will be out to catch the careless learner.

A common one is that which require you to calculate the distance or length given the gap of items.

Look at the diagram below for an example.


Here, can you find the length from the left-most pillar to the 8th pillar, given the distance from start to 3rd pillar is 30m?

Solution: (Wrong slip-of-the-mind working)

Since the distance is 30m for 3rd pillar, answer to 8th pillar has to be 80m.

Seems to be right and logical. ==> Careful here!

Why?

Look at the step distance in between pillar. It is 30m / 2 = 15m.

As the gap between pillar from start to 8th pillar is only 7 gaps,
the actual correct distance is 7 x 15m = 105m.

Interestingly tricky question, right?
Be careful and alert for this "step" or "gap" maths problem.
Slamp down this carelessness, and mistake will eventually disappear (for this type).

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Simple Way to Master Indices Maths Question

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Have you seen many mistakes like the below?


Question 1:
3^x = 27

Solution:
3^x = 3⁴ ===> x = 4

Question 2:
9^x = 3³

Solution:
3^3x = 3³ ===> 3x = 3 ===> x = 1

Why the error?

A simple explanation is that the maths learner is not familiar with the basic multiplication of repeated numbers.

3 x 3 = 9
4 x 4 = 16
5 x 5 = 25
6 x 6 = 36
7 x 7 = 49

2 x 2 x 2 = 8
3 x 3 x 3 = 27

2 x 2 x 2 x 2 = 16

etc.....

Once you have mastered this basic repeated multiplications, you can rest assure that indices question will not be there to haunt you.

How about solving "x" in this 9^{x + 1} + 2(3^x) - 3 = 0 ?

I bet that if you understood the above criteria of learning indices, the equation can be easily solved for x (using quadratic formula as a hint).

All complex things start off with simple things.
Do you agree this applies to maths?

:D

Quadrant Identification for Trigonometrical Questions

For question regarding trigonometry, quadrant is one of the key parameter to obtain correct answers.

What is this quadrant about?
A complete cycle (360 degree) is divided into 4 quarters.
They are zones defined for specific trigonometric functions.

The first quarant (0 to 90 degree) gives positive sign for ALL trigonometric functions.
The second quarant (90 to 180 degree) allows only "sine" to have positive number.
For the thrid quarant (180 to 270 degree), "tangent" has positive number only.
Lastly, the fourth quarant (270 to 360 degree), "cosine" gives positive number only.

So, you can see that given a sign of a trigonometrical operation, the specific quadrant can be found or identified.

Example:

sin X = - 0.5 ===> Identifies quadrant as 3rd and 4th.

tan X = 0.2 ===> Identifies the 1st and 3rd quadrant.

This is simple, right?

However, do note the below example.
It causes a mistake that is common!

Example of potential error:

sin 2X = -0.5 ====> which quadrants ?

The answer is not that direct!

Why?

Now the math question is not on "X", but on "2X".

To identify the quadrant, you need to start off from the "2X", working as per normal.
But, after identifying the 2 quadrants, you have to compute the "2X" reference angle.
Using the reference angle, you have to obtain the 2 angles.
After which, you need to divide the angles obtained by 2.
The divided angles is then the final angles lying within the quadrants.

Confused? Never mind. See the numerical solution below.....

Solution:
2X = sin^-1 (0.5) = 30⁰
This is the reference angle used to compute the actual answers.

Final answers are (Quad 3)= 180 + 30 = 210⁰
and (Quad 4) = 360- 30 = 330⁰.

Common mistake is to obtain reference "2X" angle and straight away divide it by 2.
Using this newly found "X", you proceed to identify the angles of the quadrant identified using the "2X". THIS IS INCORRECT!

Do not confuse double angle with single angle.
When the problem is "2X", solve all the way using the "2X" first until reaching the end.
After which, you then divide the angles by 2 to get to the final answers.

Maths is simple if you follow the rules accordingly.
If you mess up double angle with single angle while solving, you just literally mess up the workings.

Maths forces you to follow rules set out. It punishes only if you do not obey orders.

Maths is interesting isn't it? Never expect that maths can police your behaviour while practicing it, right?

:-)

Careless Algebraic Mistake

There are times when simple algebraic operations are confused by introducing trigonometric functions or logarithmic terms.

Example:

13 = 7 - 3x

This can be easily computed to be
13 - 7 = - 3x
==> 6 = - 3x
==> x = -2

But how about 13 = 7 - 3 tan X ?

Solution: 13 = 4 tan X ==> tan X = 13 / 4 , ..... and got into hot soup!

Why?

A careless mistake has been made.

When tan X was substituted into the original equation, the eyes refused to acknowledge this "complicated" tan X.
The eyes can only see the simpler "7 - 3" and thus compute it to be (7 - 3) = 4!

This caused the 7 - 3 tan X to be 4 tan X, which is WRONG.

The correct mathematical process of solving should maintain.

Therefore,

13 = 7 - 3 tan X
==> 13 - 7 = - 3 tan X
==> 6 = - 3 tan X
==> tan X = -2
.......

Maths is not that complicated when you follow the rules closely, even when the terms have changed into a seemingly complex expression / term.

By following what you have known with simple expression / term, any challenging equation can be easily solved.

This is the power of learning maths properly.
Being discipline in the way you handle maths is the key.

With a discipline mind, maths becomes fun , .. and interesting.

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Logarithm | Common Mistake

A common mistake occurs normally during simplification to a single logarithm term.

Question like,

" Simplify log X - log Y + log Z into a single term "

catches many students who are careless.

What is the error or mistake made?
- Doing the solving at one go when not familiar with the logarithmic rules
- Sign interpretation

Wrong answer given: log X/(YZ)

Correct answer: log XZ/Y

"log Z" is commonly taken to follow the previous log term, which is, "- log Y ".
Since "- log Y " causes the "Y" to be a denominator, "Z" is also taken to be a denominator too!

This is a mis-cue. A mental slip, mathematically.

Advise:
Look at the sign carefully before jumping to conclusion.
Go slow in the combination to a single log term.

Remember the idiom: "Slow and steady wins the race"

You can apply this to log simplification when you are new to it.

Cheers! :-D

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Complex Number | Common Mistakes (2)

Complex number consists of 2 parameters.

They are the modulus (length) and argument (direction).

Argand diagram is the pictoral form of representing this complex number.
In the Argand diagram, quadrants define the position of the "complex" line.

Z = a + ib (click for information) is the general form of writing the complex number.
"a" and "b" will define its polar counterparts, modulus and argument.

Having 4 quadrants in the Argand diagram means having 4 combinations of "a" with "b".

They are:
1) Z = a + ib
2) Z = -a + ib
3) Z = -a - ib
4) Z = a - ib

The first case lies in the first quadrant (Q1).
The second case lies in Q2.
Third case lies in Q3, and
fourth case in Q4.

Therefore knowing the sign of "a" and "b" let you know which quadrant the complex line lies.
And this is where the mistake lies!

The argument is always computed wrongly for the Q2, Q3 and Q4.
Only the positive sign of "a"and "b" is taken to get the value of the angle (argument).

Example of error:
Z = 5 + i5 ==> Argument = +45⁰ (Q1)

However, Z = -5 + i5 ==> Also taken as + 45⁰ forcing it to lie in Q1 (wrong!).
It should lie in Q2 since now the real term is negative.

Values of "a" and "b" are not the only parameters needed to find the angle.
Their signs are equally important!

Advice:
Think of the Argand diagram representation before the definition of the angle.
This will ensure that the angle is correctly calculate later on since you have an idea which quadrant the complex line should lies then.

Doing the complex number and its conversion from rectangular form to polar form properly will make you happy and like maths. Proper thinking process will path you into a good habit that leads to confidence in maths.

Maths is Interesting! And fun ...

:D, Smile.

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Complex Number | Common Mistake (1)

Multiplication of complex numbers remains the same as done for normal algebraic operation.

However, due to complex number having 2 terms, namely, real and imaginary terms, care has to be taken for the "i"unit.

This is specially so when multiplication of conjugate is involved.

A popular mistake made while doing this form of multiplication is:

(3 + i2)(3 - i2) = 3² + (i2)²

What is wrong?

The concept of conjugate and its multiplication states that:
(a + ib)(a - ib) = a² + b²

The "i" symbol is NOT reflection in the final outcome!

Only the "a" and the "b", the numerical part, are extracted out for computation.

Taking the "i" into account will cause the sign of the last term (i2) to be incorrect.
This is because i² = -1.

Therefore, regardless of the sign in the multiplicands, just pull out the numerical part in the complex number and use them for calculation, that is, the 3 and 2 in the example above.

The correct answer, thus, is (3 + i2)(3 - i2) = 3² + 2².

Looking carefully at the application of the formula, you will notice that this is a simple and easy technique to do conjugate multiplication.

Message: "Touch me not" i said.

Mastery takes place when we do not repeat mistakes.

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Simple Factor Multiplication

Multiplying is simple.

What is 4 x 3?
Answer is 4 x 3 = 12.

Simple?
Sure it is.

How about y(y - 1)?
Answer is y² - y.

Again simple? Sure.

But how about (y + 1)(2y + 3)?
Many of you may find this simple and basic.

But you may still come across some who did not grasp this factor multiplication.
Mistake still occur for this maths operation involving factors.

What is the mistake commonly seen?

(y + 1)(2y + 3) is given as (y)(2y) + (1)(3).
First term multiply by first term, second one multiply with the second one. That's all.
This is incorrect mathematically.

This is a misconception of what multiplication does.

Let me explain.
(y + 1)(2y + 3) can be interpreted as (y)(2y + 3) plus (1)(2y + 3).
This is key to this form of maths operation.

The second term (2y + 3) is multiplied by the first term "y" of the first factor (y + 1).
(2y + 3) is next multiplied by the second term "1" of the first factor.
The result of these two operations are then added up, since it is y add 1 (as reflected in the first factor).

The correct answer is then:
(y + 1)(2y + 3)
= (y)(2y) + (y)(3) + (1)(2y) + (1)(3)
= 2y² + 3y + 2y + 3
= 2y² + 5y + 3

Learn from the mistake, and do not repeat it.
This is the basic concept in learning from mistakes. They are our teacher.

Remember, maths is interesting!
A twist can be destructive or constructive.
That is where maths is special and challenging.

Cheers! :-)
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Indices | Interesting Mistakes (3)

If you are not careful with the base and power in a mathematics index expression, simple mistakes will occur. These simple mistakes will expose the weakness in your basic understanding of indices concepts and principles.

But, although, mistakes do occur, rest assure that learning from them is a good thing to have in the learning process.

What is the common mistake student normally make when doing indices?

Here is an example:

y = 9^{x + 1}

y = (3²)^{x + 1}

y = 3^{2x + 1}

You see the error here?

Yes, the power to the base 3 is wrongly done.

It should be 2(x + 1) = 2x + 2. The last term was left out of the multiplication.

A good way to prevent this mistake, or slip-of-the-mind error, is to use parentheses.

Using parentheses at the (x + 1) index will visually group up the "target" for multiplication by 2.

The correct answer: y = 3^{2x + 2}

Take note of this simple maths error and you are on the way to a happy maths learning journey.

Cheers! Making maths interesting goes a long way.....

:-)