Did you make simple maths error especially after a certain period of performing maths calculations?

Our mind has to have the strength and stamina to stay active and alert to have 0% error.

This is not a maths problem. Rather it is a psychological problem.

How then are we to minimise, if not eliminate, these simple maths errors that we hate to make?

Below I list a simple trick that I often used and advise maths learners.

- Before starting to do any maths assignment, or test, or examination,  scan through the whole paper to scout out the simple and difficult questions.

- Start off with the complex questions or those that you are not that confident with. You need the attention to “attack” those questions while your brain is still alert. (Note the time factor too, otherwise, you may consume too much time for them leaving little time for the simple ones!)

- After completing the complex questions, you will be more relax mentally to solve the other questions. And given the simplicity, less error will be made since you have scanned through the paper initially and has better confidence in dealing with them.

This is a proposal that balance mental alertness with solving math questions.

Maths solving can be taxing, but with proper strategy, you can lessen the anxiety and, in fact, after more practice, like and enjoy maths.

We use them easily in many maths expressions. In normal usage, they are maths operator. But there are some instances where their meaning deviates and is important that the maths learners are aware of.

This deviation to the normal meaning happens to allow maatching to the application.

Let’s take a simple example of the “+” operation.

Case 1:

H + H + O = H₂O

This is the addition of chemical elements with a different results from normal maaths.

The interpretation differs to suit the chemical equation. It is valid. But not for true maaths!

Case 2:

A + A = A

Why is it not A + A = 2A?

This is an instance of the Boolean Algebra. In boolean operation, there is no “2″, being in base 2.

In boolean or digital operation, the “A” can mean a “High” logic or “1″.

Therefore, a “High” added to “High” will still give a “High” logic (in the electrical sense).

Thus A + A = A> Still a valid add (+) operation in this digital sense.

Case 3:

a++

In this case, the operation is actually an abbreviation of a programming language.

What it really does is to replace the result of a mathematical operation of a + 1 back to itself.

A location called “a” is provided (in hardware) and the results of the operation “a + 1″ is placed back to the “a” location. This is again a valid “+” process with a different meaning to the normal maths operation.

 The 3 cases highlighted serves to let maths learners know that the symbol “+”, though looks simple, is als subjected to many interpretations. Thus everyone needs to be aware of its matching to the specific applications.

The mistake inter-mixed the principles of algebra with logarithm.

The maths question is to solve the value of x given the expression:

2+ log (5x - 1)  = log 3x

The expression, after transferring the “log (5x -1)” to the right side of the “=”, became

2 = log 3x  -  log (-5x + 1) !

Spotted the mistake done ?

Why was the “log (-5x + 1)” in that form?

The correct expression should be 2 = log 3x - log (5x - 1).

What actually went inside the student’s mind was confusion between algebra and logarithm. He did not understand the concept of “logging” the (5x -1).

Log X is always a number!

Similarly log (5x - 1) is also a number.

Therefore log (5x - 1) moves as a number, same as in moving algebraic term.

If we have 2 + (x-a) = y,       re-arranging the expression,    gives us 2 = y - ( x-a).

The term “x-a” is taken as a whole, with change in the sign of (x -a)  and not including that of the individual internal “a” and “x”. This is basic algebra.

Moving log (5x - 1) is the same. Being a number, it operates equivalent to the algebraic manipulation.

The log (5x -1) is thus taken as a whole and sign change affects only the term as a whole. It does not affect the individual internal “5x” and “-1″!

Part of learning maths is following rules and principles.

The mistake made by the student was a reflection of correct algebraic change, but in the wrong sense. “Log” had convert the term into a number, and that was the mistake not captured.

The data given are mostly random values that may not be lying on a straight line.

Therefore the technique of drawing this straight line on the graph becomes a crucial skill as it may results in producing inaccurate answers or outcomes.

An example is presented here for discussion.

Diagram 1:  Various random co-ordinates scattered over the sheet

How do we plot a suitable straight line from these few scattered co-ordinates?

Many “funny” ways exist that reflects poor understanding of the purpose for straight line plotting.

Way one: 
Connecting up the points in sequence  ==>  Results in non-straight line end-to-end.

Way two:
Connecting the extreme two co-ordinates ==> Results in unbalanced straight line (diagram 2)

Diagram 2:   Uneven gaps between points and line drawn

Here you can clearly see that there are 2 points that are above the straight line by a certain gap. The spread of the random points are not evenly balanced about the straight line drawn.

Let’s see a better way to fit the line to these randomly scattered points. (Diagram 3)

Diagram 3

In diagram 3, you can see that the scattered points are evenly balanced across the length of the straight line drawn. The gaps of the crosses are almost the same through.

This is termed the “Best fitting straight line“.

Therefore in plotting a straight line, it is the evenness that counts. The spread of the points has to be balanced so that the individual errors of the given data to the line can be minimised. The line drawn is then one that will produce a more accurate results. 

Never imagine that so much thoughts are needed just to plot a straight line graph, right?

In maths, to obtain the equation of a line from two given co-ordinates, we inevitably think of graph plotting. This is one good way to obtain the answer by finding the gradient and intercept.

Let’s take an example.

2 points:   (1, 5)  and (3, 11) are given. What is the straight line expression?

By plotting these two points on a graph, we can easily determine the gradient and intercept, and then the mathematical expression for the equation.

Diag: Graph with 2 points.

From the graph, to determine the gradient, we can check:

• increase of the vertical unit with reference from the 2 points as 11 - 5 = 6 units, and
• increase of the horizontal unit from the 2 points as 3 - 1 = 2 units,

Gradient = change in vertical / change in horizontal = 6 / 2 = 3

The next item is the intercept, and directly from the graph, it showed the value to be 2.

Therefore, the straight line expression comes to    y = 3 x + 2.

This graphical method is OK, simple and easy to do.

But is this the only way to get the straight line expression from 2 points given?

No, there is at least one other method. Don’t forget maths is exciting and amazing, if one wishes it to be.

What is the other way?   The answer is the use of simultaneous equations!

How so?

Note, given 2 co-ordinates and having to find 2 unknowns satisfies the basic requirement to set up 2 equations for simultaneous solving.

We know the general straight line equation to be y = mx + c.

Therefore with the known co-ordinates,

11 = m(3) + c   —–(A)

5 = m(1)  + c    —–(B)

By elimination method, (A) - (B),  gives,

6 = m(2)   ===>  this gives m = 6 / 2 = 3  (The gradient!)

With  m = 3 found, let’s put back into equation (B),

5 = (3)(1) + c  ===>  c = 5 - 3 = 2  (The intercept!)

Thus, the straight line equation is y = 3 x + 2.   This is the same as the one obtained with graphical method.

Therefore, from the simultaneous way, we can still obtain the expression from the 2 given co-ordinates, without plotting the graph.

Either which way is fine.

What is interesting is that once you master the principles of maths, you can be flexible to choose the method that you like and still arrive at an appropriate answer.

Enjoy maths!  

“How to remember?”

“Why teach so fast?”

“Any simpler method?”

These are sampled questions that express the stress and anxiety that many students faced.

Though some of the components of teaching and learning can be simplified, they take time to implement with due consideration for many other factors like curriculum, key learning objectives, and quality.

What then can we do to lessen the stress that keep on coming?

The answer is to stay focus and be aware of the key objectives of the topic.

If the maths topic is about logarithm and its various laws, the objectives should be to apply the logarithmic laws to solve any related maths questions.

Do not ask questions like

“Why must I write the base with a smaller font size?” and

“Why do I need to add the individual logs to form a logarithmic product?”.

Asking questions is definitely good for learning, but in these instances, the focus is not to dig out the history part of logarithmic studies. The main aim is understanding the application of the laws in order to apply and solve problems.  It is simply just that!

Trying to find out more than what the maths syllabus calls for within the short time frame is tantamount to facing learning disaster.

Just know that analysing too much into a subject or topic is equivalent to paralysing the learning mind. Stress, thus, appears and starts making friends.

Remove the unnecessary questions that blocks the main objectives and move along with the requirements. Pick up the extras only if you have the time and in a comfortable and relaxed mood.

Why cause undue stress and anxiety upon yourself if the key matters in maths learning is not treated properly? Contain the scope and stay focus!

Do not dwell on things that are not the main issue and causing mental blockage. Accept what is taught with the knowledge that they are meant for application.

This way, learning maths will not be that stressful and will become fun, since the basics can be managed comfortably with understanding.

Each of us has one particular dominant style. Knowing which one will thus serve us good. However, the learning style itself is still not enough, we need to also know the types of intelligence we possess.

The learning styles are used to gather information and ideas, through the senses, for the brain. But how do we process the information captured afterwards depends much on the intelligences that we also have.

In maths learning, what we need is the logical / mathematical intelligence.

This lets us do all sort of computations and comparisons that maths requires.

An example is the conversion of an expression in logarithmic form to its index form (and vice versa).

log_aY = X <==> a^X = Y

Though, this conversion seems simple enough, it is found that many students still have difficulties in converting the above.  Do they lack this logical intelligence?

The answer is “maybe” and “maybe not”.

They may have possess this logical intelligence, but has not yet enhanced it.

One way to offset this weakness in the logical comparison part of intelligence is to introduce the visual intelligence into maths learning.

What is this visual intelligence?

This is the “appearance” aspect of information processing. Here, colour, shapes, and the likes are linked up to the information.

Why introduce the visual intelligence?

The main reason is to bridge up the right and left brain, a theory that is now well-known and practiced throughout the learning and teaching communities.

Now, let’s see how we can solve the logarithmic and index conversion through the use of the visual intelligence.

The examples above showed two very interesting ways to enhance the learning. 

The first one is to replace and “beautify” the symbols with pictures or graphics to arouse retention ability.

The second one is to make use of colours to strengthen the symbols and their placements.

By seeing pictures and colours, although they are no way close to any maths topics, the learning of maths is greatly improved especially for the logical comparison portion.

Therefore, if possible, introduce as many “visuals” into maths as you can. They will make maths learning a totally new experience. Take care of intelligences and intelligences will take care of you.

Inspired ? 

The generic quadratic equation is  y = a x² + bx + c.

We know there are various methods to handle the quadratic equations.

There are:

• Factoring
• Completing the Square
• Quadratic Formula
• Graphical

Mastering all these techniques allow anyone studying maths to have the flexibility of choosing a better or suitable method that fits the nature of the question.

However, do note that if there is problem learning all these techniques at one go, click here to get some pointers.

What is the benefit?

Many maths questions are actually quadratic in expression. They may not appear so, but, on closer look, they are.

Examples:

• 3 cos² A + 2cosA + 4 = 0
• 2 (log Y)² + 2(logY) + 3 = 0
• 4^x + 3(2^x) - 5 = 0
• 5x^-2 - 7x^-1 - 6 = 0

Being able to handle the generic quadratic equation solving means having the potential to solve numerous other types of quadratic equations as listed above.

What is the obstacle if you still cannot map the quadratic solving method to the other types of expressions?

Tips: 

1. Stare at the given expression
2. Identify the terms that matches the x² format.
3. Identify the other two terms through the “x” format and pure number format.
4. After re-writting the questions in the generic quadratic form, apply any of the method to solve this quadratic equation.

And that’s all.

Simple isn’t it?

Thus, mastering any one method of handling quadratic equation allows anyone to solve many other types of quadratic equations. Therefore, it is worth the time and effort to know solving these type of mathematical expression.

Bonus information:

Let’s look into this example  

The first term can be modified to 5(x^-1)².
The second term can be modified to 3(x^-1).
The last term will be obviously the pure number “2″.

Selecting the use of quadractic formula, we can say that a = 5, b= -3 and c = -2.

Next, just apply the quadratic formula and you are close to the two answers (roots) of the equation,

x^-1 = -4/10 or 1.   Clear?  

If not, read again… Our brain needs some mental exercise at times.