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Sunday, 31 August 2008
Inverse Matrix | What is it used for?
If you are good with algebra, you will discover that this inverse matrix way of handling the solution of simultaneous equations is similar, except that they are done as a group, collectively.
The answers to the unknown variables are obtained at one go with this Inverse matrix method.
However, what do you need to know in order to use this Inverse matrix solving?
You have to understand:
1) Convertion of simultaneous equations into a set of matrices
2) Determinant and technique to get its numerical value
3) Minor of the individual elements within the matrix
4) Co-factor of this determinant formed with this set of matrices
5) Transpose of matrix
6) Adjoint matrix obtained with the co-factors and transposed matrix
7) Formula to relate determinant with the adjoint matrix ==> Inverse matrix
8) Matrices multiplication
The list looks amazingly long for matrix novice, but, DO NOT FEAR!
Matrices consist of numbers only, and simple mathematical operations, nothing abstract.
(The details are not presented here for fear that you will leave this site.)
Slowly research into the above terms and see for yourself that they are "friends" and not "foes".
Happy start to matrices and its application.
Monday, 25 August 2008
Graphical Art (Jelly Fish)
A jelly fish drawn out of trigonometrical expressions. Wondering what is maths about...
Yes, on the interesting part, maths is lively, as seen above.
Animals come alive with maths.....
Graphical Art (Volcano)
Saturday, 23 August 2008
Differences between Determinants and Matrices
Both contain numbers within. But ......
- determinants are bounded by two straights lines whereas matrices are by square braces
- determinant resulted in a single numerical value, whereas matrices are sets of numbers grouped within the braces
- determinant can be extracted from matrix, but not the other way round
- there are inverse matrix but not inverse determinant
- a scalar multiplier affects only a single row or single column of a determinant, but affects all the numbers within a matrix
The differences are aplenty. But those listed above are the least any maths students or person should know.
Why create such topics to excite maths learners?
Answer: So that maths will be made interesting and challenging (... the informal reply)
With open mindedness, maths is a very fantastic subject to learn and train for. It carries many hard works and beautiful thinking. Thoughts that are simple yet abstract at times.
Any more fun? ..... You are the answer!
Wednesday, 20 August 2008
Cramer and Sarrus | How are they related to maths
Who are they?
In which area of maths are they involved?
For those of you who is doing matrices, you will be familiar with their rules, I suppose.
They are famous for creating techniques used in the solution of simultaneous equations.
How so?
Cramer's Rule is a method of solving unknowns in simultaneous equations using division of various determinants.
Rule of Sarrus is a simple way of determining the numerical value of 3rd order determinant.
The two rules, therefore, complement each other to form an easy way to solve simultaneous equations.
However do note that students of these topics use to mix up the 2 rules.
When question states the use of Cramer's and Sarrus rule, they tend to get confuse. THe common notion is that Rule of Sarrus is another method to solve simultaneous equations which is not so.
It is only a technique to evaluate the determinant and get its value. It cannot solve the unknowns in the set of equations.
This is a good problem to get confuse. Why?
Getting confuse means a thinking brain and learning starts to take place. When there is no confusion, there can be two happenings.
- blank or "completely do not understand"
- "completely understood without questions"
Which is which does not matter. What is important is not to disappoint Cramer and Sarrus!
Work hard folks!
Tuesday, 19 August 2008
A Good Quadratic Solver
Even though you may not use it to solve math problems, you can take this Quadratic solver as a guide to check your answers.
It is a user-friendly tool that can give you the answer easily. Try it!
This is one of the easily available math tools that make maths learning exciting and and interesting. See for yourself.
Quadratic Solver
This solver is provided by Algebra.Com:
Log X Is Just A Number
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 converted the term into a number, and that was the mistake not captured.
Being careful with every steps taken in solving maths questions is a discipline one can treasure. This is one of the interesting part of doing maths!
"+" Its Many Interpretations
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:
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:
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:
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 also subjected to many interpretations. Thus everyone needs to be aware of its matching to the specific applications.
Mental Alertness In Maths Calculation
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.
Math Challenge 9
Add five to it.
Multiply the total by two.
Subtract the original number twice.
Is the answer "TEN"?
Thinker: How is it done?
Math Challenge 8
(Do not tell anyone).
Multiply the number by three.
Add six to it.
Divide the total by three.
Subtract the result by the original number that you kept secret.
Is the final answer "TWO"?
Thinker: How is it done? ==> Use Algebra
How To Draw Best Fitting Straight Line
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?
How To Get Gradient and Intercept from Two Points
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! :)
Monday, 18 August 2008
How To Reduce Stress In Learning 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 getting on your nerves!
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 for the moment, and coming back for in-depth understanding after mastering the necesary.
(NOTE: You still have to fully understand the topic, the approach explained here is to "buy" time to release some stress.)
This way, learning maths will not be that stressful and will become fun, since the basics can be managed comfortably with understanding.
Ways To Improve Logical Learning Part Of Maths
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).
logaY = X <==> aX = 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 ?
Benefits Of Knowing How To Solve Quadratic Equations
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.
- 3 cos2 A + 2cosA + 4 = 0
- 2 (log Y)2 + 2(logY) + 3 = 0
- 4x + 3(2x) - 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?
- Stare at the given expression
- Identify the terms that matches the x2 format.
- Identify the other two terms through the "x" format and pure number format.
- After re-writing 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 quadratic 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.
Indices & Logarithm - Less Mistakes with Proper Writing
Many mistakes and confusions were made due to improper writing of the above two items that are very much key to mathematics.
Two of the topics that are commonly populated with writing errors are Indices and Logarithm.
Indices: ax = Y
Logarithm: lognA = K
From the above two examples, it is apparent that writing is crucial to give correct meaning to mathematics.
Writing therefore is an important skill while doing these two maths topics.
Discipline is needed here to maintain consistency.
What do I mean?
Let me give an example.
log2 xy = log2x + log2y
The above is a correct expression written with proper size and positioning.
If writing discipline is not maintained, it may end up as:
log2 xy = log2x + log2y (size and position error)
Do you notice the change in meaning when size changes?
The subscript "2" representing "base 2" is conveyed as "2 times y"!
This is the result of a simple "slip" of the hand in writing the size of the "2".
Can this mistake be allowed?
I guess NO!
Why run the risk of marks being deducted or effort wasted in getting wrong answers?
Write properly is the only answer to less mistakes, if not at all.
If ax = Y is written as ax = Y, how do we expect to get correct answers?
Here, you realise that maths learning is not just about mathematical concepts, human discipline and strength of concentration matters also.
Being able to write clearly and presentable is a noble capability that we should be proud to demonstrate to anyone. Strive to be focused and discipline when dealing with maths.
Happy writing! :P
Know Your Limits While Learning Maths
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 complacent 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.
Solving Maths Develops Plan Foward Capability
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!
Why Do We Study Quadratic Equation?
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 design 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.
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?
Other examples are:
1) Distance travelled given by the quadratic equation s = ut + (1/2) a t2
2) Electrical characteristics 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.
Merit of Grouping in Maths Solving
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 10n5n/2 / 20n/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 )n5n/2 / (2 x 2 x 5)n/4
==> 5(n + n/2 - n/4) 2n- n/4 - n/4
==> 55/4n 2n/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 .....
Indices | Big and Small concept
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!
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.
Tips To Reduce Errors Doing Simultaneous Equations
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.
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.
Our Brain Is Tuned For Addition Versus Subtraction
Do you make more mistakes doing subtraction than doing addition?
If the answers to the above questions are all YES, you are not alone.
We are "programmed" since young to add more than subtract.
Do you count things starting from 1 up or down from 100?
Does the page number of books starts from 1 up or down from the last number?
Does the date on a calendar month starts from 1 up ?
Day in day out, we are bombarded by addition of numbers through counting up. It is not accidental though. It is due to the fact that we do not know the maximum in those counts.
Therefore, we are accustomed to addition and will do better with this maths operation than with subtraction.
Try doing this:
- y = 44 + 36
- y = 44 - 36
- y = 60 + 53
- y = 60 - ( - 53)
We can do the above simple maths questions easily.
But, honestly, did you take a longer time to compute the subtraction?
There is a slight mental block compared to addition, right? Mine does.
There is nothing wrong with this. What I like to bring to your attention is the fact that knowing addition is the brain's preference, we can minimise maths error by aiming for addition than subtraction, if possible.
If we are to do a maths operation y = 55 - (3 + 6), do the (3 + 6) first, followed by the subtraction.
Avoid doing y = 55 - 3 - 6, with 2 subtractions involved. Although mathematically it is correct, why take the risk?
It's the same for calculating y = 53x + 16x than doing y = 53x - (- 16x).
Did you get the message? Understand how your brain works and tweet the mathematical process for a less risky computation.
Principles of Learning (Mathematics)
Before we start learning anything useful, we need to question ourselves over its purpose.
If the answer to it is sufficient enough for you to strike on, you are in the correct path to a better future.
However, what is the mental approach to learning, before actual content acquisition?
Is the ownership of learning clear?
These are the questions you have to answer before you set out on the bright journey of learning.
The key issue here is OWNERSHIP of learning.
Besides learning the essential principles of mathematics, the crucial principles of learning has to be handled well too. Do note that both go hand-in-hand.
Principle of learning takes into account the idea that learning is the sole ownership of the learner.
The teacher is, just after all, a catalyst to speed up the learning process. He is there to address any questions that may be harder to solve or missing links that may unknowing been left out during the course of lesson delivery.
But to have a fruitful learning outcome, you must know the fact that you are the ultimate target of the process. You have to constantly remind yourself that "I am the one learning".
Whether can the teacher teach well or explain properly, the learning still goes back to you. (Do not blame anyone for failure to learn!). It becomes an excuse to deviate from the proper.
So, the principles of learning has to be clearly understood before principles of mathematics can be effectively captured.
Mathematics learning will then be a wonderful learning journey.
Wonderful here does not mean easy-going though.
Without decent struggle in learning and thinking, retention of knowledge will not last long. This is a well-known fact! (Struggle here means the mental processing of knowledge, past and current).
Study hard (and smart), and know that you are the final gem that any teacher would like to polish, provided you start off with the correct mindset.
Cheers to learning and cheers to you, the mathematics gem.
Maths Is Interesting!
Analysis of Maths Error
This is possible, provided the learners pick up the error made and understand them through detailed analysis.
I came across an error that puzzled me for quite some time.
The error made: Inverting y = 2x + (1/x) became 1/y = (1/2x) - x
What actually happen in the thinking during the math manipulation?
Understanding the thinking will definitely help in resolving the issue and prevent further mistakes made.
Here, I believe the student:
- confused "flipping" cards with the basic concept of maths (flipping the left y does not equate to individually flipping the right mathematical terms)
- confused the property of indices ( 1/a = a-1) with this operation, resulting in the last term having a shocking "-" appearing after the inversion of denominator to numerator placement.
- lack practices to enhance the algebraic manipulation during the learning phase. Knowledge is not retained.
Analysing maths error is not an easy task if done alone, and without guidance.
The thinking part is an abstract art of the mind. If there is no communication to reveal the mental procedure in solving the math question, a skilled guess has to be made especially by those teaching maths.
If guidance is not available, textbook will be the next best alternative to answer for the mistake encountered. Go back to basic to dig out more understanding.
The approach is to question and question till satisfied. It will cover most aspect of the learning objectives.
However, do note that identifying and analysing maths error is a great learning process.
It serves to filter out current bad learning habits and replaces the gap with proper systematic steps. This is of much importance especially in maths where every steps count.
Therefore, do enjoy "debugging" your maths error. It is fun and not meaningless!
Errors do tell many stories. Just be friendly to them.
Linear | Angular Speed
- Linear
- Angular
Linear speed is defined as the distance travelled for a given time.
Angular speed is defined as the angle covered for a given time.
Are they proportional ?
Or are they related in any sense?
Let us take 2 pendulums hung on a slim rotating rod for analysis.
If the 2 pendulums (A and B) rotate one full cycle, the time taken by them is the same.
They covered the same amount of angular distance (360 degree) within the same amount of time.
This showed that they have exactly the SAME angular speed.
But is the Linear speed similar?
The length of the 2 circumferences travelled by the individual pendulums are not the same.
The linear length or distance is therefore NOT the same.
Length = 2 x (pi) x radius.
They took the same time to complete one full cycle, though.
The linear speed is thus DIFFERENT, having travelled different length for the same amount of time.
In this case, the angular speed is the SAME whereas the linear speed is different. Pendulum A has a higher linear speed compared with pendulum B.
Can the angular speed be different and linear speed made to be the same?
If the same slim rod is used, the answer is NO.
But if they are held by different rods, like that of the traditional analogue clock, the answer is YES.
For the linear speeds to be the same, pendulum A has to take a longer time to complete one cycle compared to pendulum B timing.
In this case, their linear speeds are the SAME while the angular speed will be DIFFERENT. This is so, since, same angular coverage but different time taken.
Why are we talking so much about this?
This is an "old" principle that was applied to the famous grandfather's clock. The setting of the pendulum position along the swinging rod (or string) is the key to the time accuracy of the clock. Ancient people has used this understanding to produce something useful for their daily needs, and this is mathematics!
Wonderful isn't it? :)
How To Estimate Numerical Answer
It called for logically thinking and is done on a case-by-case basis.
Take for example, 302 x 11.
What is the estimated product of the 2 numbers?
We can choose for it to be 300 x 10, which gives 3000.
But how about being more accurate in our estimation?
300 x 11 gives 3300 !
We can use "11" instead of closing in to "10" of the first instance.
Logical thinking at play here...
But why choose "11" instead of the "10"?
The answer is simple.
Since the first number is 300, multiplying it with "11" is still within any person's ability. Therefore using "11" as part of the estimation is closer to the actual answer.
Now, how about 314 x 11?
Here, the answer estimated will be 310 x 10 = 3100, to be practical.
310 x 11 will be harder to estimate. Therefore "10" is chosen in preference to "11" as in the case of the previous example. (We are not talking about mental maths, but maths for "normal" people.)
Estimation, thus, calls for a fast but accurate production of numerical answer. It involves thinking and choosing numbers that are easy to handle.
Estimating answers can be fun as shown. It can be a challenging game where students aim to be first to give the most accurate answer.
Have fun estimating! :-) It gets more interesting as you go on.....
Volume and Surface Area | Geometrical Relationship ?
However, have you wondered what is the relationship between them?
If we want to maintain the volume of an box but reduce its surface area, is it possible?
Or does the volume ALWAYS increases with increase in its surface area?
How about reduction in surface area? Will the volume also reduce?
Let us throw some numbers into an example to figure out the answers.
For simplicity, let us use a simple box. (Diagram 1)
<== Diagram 1 Box of diagram 1 has a : Volume = (2 x 8 ) x 4 = 64 cubic units
Surface area = 2 (2 x 8 ) + 2(4 x 2) + 2(8 x 4) = 112 square units
Let modify the box (Diagram 2) to another dimension and see its geometric parameters.
<== Diagram 2 Volume = ( 4 x 4 ) x 4 = 64 cubic units
Surface area = (4 x 4) x 6 sides = 96 square units
Note, the volume remained but....
the surface area has reduced!
This saves material, right?
How about another dimension (Diagram 3)?
<== Diagram 3 ( A tall box!) Volume = (2 x 2 ) x 16 = 64 cubic units
Surface area = 2(2x2) + 4(2x 16) = 136 square units
What now?
The volume again remained,
but the surface area has increased!
What is the conclusion from here?
It is observed that althought the volume of the box has not changed, their surface areas has changed. The direction of change( increase or decrease), however, is on a case by case basis.
This concluded that there is no relationship between volume and surface area of any object.
Therefore don't be tricked into reckoning that surface area increase will cause a definite increase in volume. Likewise for reduction also.
Hope you gain much from here.
Geometry is exciting and mysterious at the same time, right?
For relationship comparsion between another set of geometrical parameters, perimeter and area, click here.
Simple Flipping of Math Terms
Many weird answers can come out when done wrongly.
Look at an example of changing numerator to denominator of terms in an equation.
Example: Express y in terms of x for the equation (1 / y) = 5 + x
If the answer is simply to "flip" numerator to denominator,
getting y = (1/5) + (1/x), did the student learn his lesson?
Yes, he applied the concept of x / y = a / c ==> y / x = c / a neatly, but......
something went amiss!
First loophole: Student did not understand the meaning of "="
Second loophole: Student did not understand the meaning of summation
* Right Concept
First item:
The "=" symbol means whatever collectively on the left equals whatever collectively is on the right side.
Second item:
The "+" symbol means terms are added up to form a bigger piece of "something".
This bigger piece of "something" therefore operates as a whole, and not as individual terms.
The correct answer, hereby, should be, following the above 2 concepts,
(1 / y) = 5 + x ===> y = 1 / (5 + x)
The whole of "5 + x" acts as a collectively bigger piece of "something", and is flipped together as a single piece.
Note, however, here the word "flipped" has many "stories" within it.
Like to hear the "stories"? Read on....
Flipping is an abbreviation for the cross multiplication operation.
Mathematically speaking, it is
1/y = 5 + x ==> Multiply both side by "y"==> (1/y) * y = (5 + x) * y
==> 1 = (5 + x) * y, the "y" is brought over.
Divide both side by "(5 + x)" ==> 1 / (5 + x) = [(5 + x) * y] / (5 + x)
==> 1 / (5 + x) = y .The correct answer.
Finally, at long last, we obtain the outcome of the "flipping".
You can see from the above that if you understand the math principle, you can actually do simple flipping of numerator to denominator, which those ignorant will not be able to comprehend.
They will simply follow the flipping and will get flopped at the end.
Understand what you are doing. Equations in algebra can reveal your ignorance!
Math Challenge 7
2 0 2 0 ?
This is not that difficult if you have attempt the challenge in this link.
Math Challenge 6
It is not advanced math that we are talking about.
Just elementary level is good enough to solve this thinker.
Maths Challenge 5
Find the numerical value for the power (x) of both bases such that,
1233 = 12 x + 33 x
Sunday, 17 August 2008
How To Study Math Wisely
But, whichever method one selects and sticks with, an important learning concept is reflected in the Pareto Ratio.
What is this all-important "Pareto Ratio" about?
Pareto ratio is simply a 80 /20 ratio. Though it is simple, its impact to many things are fantastic, and is the deciding element that constitutes wise and effective learning.
What do I mean?
Pareto ratio actually allows us to understand that 80% of the work are done by only 20% of the people.
And 80% of knowledge can be mastered from 20% of the learning material!
This is good news, isn't it?
Yes, it is true. But where is this 20% of the key learning material?
In math textbooks, this important 20% normally lies at the beginning of a new topic or chapter. It consists of the principles and concepts portion within the complete chapter.
Here, knowledge is "told". After this portion, worked examples follow up to demonstrate the application of the concepts and principles of the new topic. At the end of this worked examples and explanation, comes the exercise part for student to practice.
Take a look at any math book on hand. The flow is almost the same with slight deviation to make it outstanding.
The worked examples and exercises constitute the 80% of the learning material that only serves to enhance knowledge. They are important though and should not be ignored!
The message here is that to really master math, focus should be placed at the front 20% of the notes. Understand deeply the concepts and principles within it, and then move on to apply the learned knowledge to test for comprehension of the materials. This is learning math wisely.
Some math learner do otherwise by jumping directly (and impatiently) into the exercise even before a decent level of understanding sets in. This is academic disaster! Avoid it to save time and frustration.
Thus, to learn wise, aim for the 20% and you are effective for the other 80%.
Wonderful right? :)
Mathematics Graphics Helps In Better Understanding
Though the questions and explanations may come in the normal English or any equivalent first language, mathematics still consists of symbols, abbreviations, variables and mathematical operators, besides the numerous expressions or equations.
Learning and teaching mathematics, thus, become a chore for those "newbies" as well as those who detest the subject.
Graphics to represent mathematics meaning is therefore a welcome move for them. Although not all mathematical expression can be presented in graphics, it is a better option in term of knowledge retention.
Take an example of solving the simultaneous equations.
Example 1:
Solve y = 2x - 4 and y = -x + 6
To the mathematics learners, this may seem daunting even after explanation. They may not be able to visualise mentally what really is these 2 equations doing, not to say, solving them.
However, if the question is shown in graphics or converted likewise, the understanding will immediately be clear.
Compare the above "worded" question with the "graphical" question shown below.
From the graphical interpretation of the mathematical question, it is easier then for the students to understand what is needed. The equations are here clearly shown as lines that intersect at a common point. See the advantage now?
Let's see another example.
Example 2:
The all-famous Pythagorean Theorem is presented as x2 + y2 = z2.
But this is a simple mathematical expression or formula, and that's all (to them).
What actually does it means? Besides pure memorising, what else?
If this Theorem is presented in the graphical form (below) , you will see the difference.
Here the idea that the "squaring" of the sides are related through the Theorem will be easily accepted. The message of the Pythagorean Theorem as areas of various squares will be captured and locked mentally for understanding and retention with ease.
Why is graphical form a better technique of learning and teaching?
This has to do with our brain. It is well known that our right hemisphere of the brain is for image, and the left hemisphere is for logical matters including studying for mathematics.
This graphical learning and teaching, therefore, goes in line with well-known and accepted method of linking right and left brain for better comprehension.
For mathematics learners, it is hence a merit to be able to convert most mathematical knowledge to images, if possible (although it is understandable that not all is possible).
For mathematics teachers, use graphics if possible to save your breath.
Happy imaging !
Who Says The Compass Triangle Is Right-Angled?
Look at this post for details on what I am talking (or rather writing) about.
However, the question is how do we know it is truly a right-angled triangle?
The answer is "Go back to basic and prove it is!".... (or it is not)
This ability to prove will enhance your understanding of its principles.
Let the spirit of the mathematician starts flourishing ...
A diagram to simplify explanation is shown herein.
Our target: To prove that Angle C is 90 degree.
Two key items exist here. They are:
1) Length DA = DC & Length DC = DB,
since they are the radius of the circle
2) Angle C1 = Angle A & Angle C2 = Angle B,
due to the facts in item 1) above.
These 2 key points will ensure that angle C is 90 degree (or right-angle).
Moving on ...
Angle d2 = 1800 - Angle C2 - Angle B = 1800 - 2 x (Angle C2)
Angle d1 = 1800 - Angle d2
Which makes Angle d1 = 1800 - [ 1800 - 2 x (Angle C2) ]
This produces an important observation: Angle d1 = 2 x (Angle C2).
Also Angle A + Angle C1 + Angle d1= 1800, which also means,
===> [ 2 x Angle C1 ] + Angle d1= 1800.
Re-writing the above gives [ 2 x Angle C1 ] + 2 x (Angle C2) = 1800
Factorising the above equation yields:
2 x (Angle C1 + Angle C2) = 1800 ---(1)
Hang on, we are coming to the end ...
(Our target) Angle C = Angle C1 + Angle C2
From equation (1),
we can show that Angle C = Angle C1 + Angle C2 = 900!
Therefore, it is true that
Compass Triangle has Angle C as right-angle.
Hurray!! Eureka!!
Fall back to maths basic if in doubt. The skill picked up will serve you good in the application of maths theory.
How To Draw Right-Angled Triangle Using Geometric Compass
One way is to make use of a protractor.
Another way is to use a try square.
But what if you have only a geometric compass, how do you do it?
(Not the compass that give you directional bearings of north, south, east and west!
It is the one that let us draw circle with.)
Don't worry. The trick is simple!
The steps are :
- Draw a (light) circle to serve as boundary
- Draw a line across the diameter of this circle to serve as one side of the triangle
- From the 2 ends of the line drawn in step 2, produce 2 more lines, joining them at any point along the circle boundary.
- Smile! You have created a wonderful right-angled triangle true to target.
See the diagram 1 beside for an overview of the steps used.
Diagram 1 ===>
Try out this method and see for yourself!
Proof using a protractor or try square at angle C to verify the 900.
Enjoy yourself! Geometry is fun! :)
Exciting Area and Perimeter in Geometry
I suppose, you are also good at calculating their numerical value.
However, in the midst of doing the many calculations and heavily involved with different funny shapes, the fundamentals of geometry have to be maintained. The principles are always the main issue while studying mathematics. Take a thinker below.
Is perimeter related to area in any way?
Does the perimeter expands with increase in area?
Does the area shrink with decrease in perimeter?
These questions serve to address understanding of these 2 key geometrical parameters.
If you are able to answer them correctly, give yourself a pat.
Otherwise, give yourself a slap!
Let's look further into them, if you have any queries.
The 2 diagrams to the left has the SAME area, but look at their perimeter. One is longer than the other even when the areas are the same.
Amazing right?
Now, look at the 2 diagrams on the left. These 2 shapes has the SAME perimeter, but how about their areas?
The areas are highly different!
Exciting area and perimeter relation, right?
Do note, however, that the above 2 cases are just to illustrate concepts of area and perimeter in geometry. There are also normal cases where increase in area results in increase in perimeter, and likewise for decreasing condition.
Stick to principles and understanding is assured. Cheers!
Why Is Net Important In Geometry?
The height factor of the geometrical outlook is presented in the 3-dimensional view.
But does it really present the object with a complete understanding of its shape?
There may be assumptions made if the angle of view is not appropriate.
The assumptions may be wrong!
Let me cite one example below.
From looking at the shape presented on the left, can you tell whether the shape is that of a tetrahedron or pyramid?
Rather difficult isn't it?
It is almost impossible to tell!
Therefore, NET comes in useful in this aspect.
So, what is this net about?
Net is a different way of presenting shape. It utilises only 2-dimensional view. It is actually a 3-D object "flattened" into 2-D perspective.
The usefulness of net can be seen from the answer it presents below using the above object as reference.
Here, if the net of the object is shown, as compared to its 3-D version, the message is clearly presented.
If the object is a tetrahedron, then the net is that of the left.
If the object is a pyramid, then the net is that of the right.
Thus, net in geometry serves its purpose in showing a clearer idea of shape when the 3-D perspective is not sufficient to reveal all details.
One common application of net is in ORIGAMI, an interesting paper folding activity liked by many.
How Fast Can Spiderman Get Freedom? | A tricky question
You need to read the question, understand it, and think carefully about it.
Do not be afraid, though.
It is only applicable for fun or amusement.
I have one here. * Dare to attempt... ? *
Spiderman fell into a deep and slippery oil tunnel dugg vertically into the ground. To get out into the city, he has to, obviously, climb out of it. However, the tunnel is 20 metres deep, and he can only manage to move up 3 metres per hour due to back injury. But as the wall of the tunnel is too slippery, any attempt to move 3 metres will result in a slip back of 2 metres. This make spiderman's rate of climb to be effectively pegged at 1 metre per hour only. How long will spiderman then take to reach out of the tunnel?
NOTE: It seems like spiderman needs take 20 hours to achieve this feat. This is due to the tunnel height of 20 metres and his rate of upward climb being at 1 metre / hour. This is wrong! The assumption has to be properly thought over. Spiderman will be happy as it will take lesser time for him to reach freedom.
....... think it over, if you surrender and urgently need to see spiderman .......
the answer is ==>
Spiderman will be over the edge of the tunnel at the " see below " hours.
As he can only progress at 1 metre / hour, it meant that at the 2th hour, he will only start at a 2 metre height from bottom. At the 18th hour, he will start at 18 metres from bottom. From there on, he just need to cover another 2 metres to be over the edge of the tunnel (before falling back 2 metres). Therefore he needs only 18 hours plus another 2/3 hour to be happy. Got it?
Principle of Speed Calculation
The question asked about the time taken when two persons or cars travelling at different speed and opposite direction meet each other.
A quick way to solve this question for its time is to apply the below:
Time taken = Total distance between the two objects / Total sum of the two speeds
However, what if we forgot this so-called formula?
As Albert Einstein once said "Education is what is left after all else is forgotten".
This is very true in learning, especially for maths.
If you forgot the formula to a question and without understanding its concepts and principles, do you think you can solve that question?
Yes, only by accident, if you are lucky.
Understanding the principles is very important as it enables you to solve any question starting from basic if all else has been "returned to the teacher".
Let us look at an example.
If 2 cars travelling at the speed of 30 km/h and 50 km/h meet at a distance X km away from the starting point of A, what is the time they will meet if they start off at the same time? The distance between them is 650 km apart.
Quick way to solve: T = 650 / (30 + 50) = 650 / 80 = 8.125 hrs (from start time).
From first principle:
Let the distance they meet be X km (from A point).
Since the time they meet is the same from A and B starting points,
X / 30 = (650 - X) / 50 ==> To find the value of X
80X = 1500 times 13 = 19500 ===> X = 19500 / 80 = 240.375 km
Time taken (T) for them to meet = X / 30 = 8.125 hrs (from start time).
Look here:
T = 19500/ (30 x80) hrs or
T = (19500/30) / 80 = 650 / 80 hrs
The "650" above (in bold) isn't it equal to the distance apart between the 2 cars?
The "80" above (in bold) isn't it equal to the total sum of the 2 speeds?
The working from first principle produces the SAME answer as the quick method!
But the major difference is that the solution from the Quick way has no long-lasting impact on learning as compared to the latter technique, which bank on understanding of principles of speed, time and distance.
Therefore, which way do you prefer (for longer retention of knowledge)?
Maths Game - "Reversity"
You will find it wonderful and importantly, it is free and requires only a piece of paper and a pencil to go with it.
This activity is not new. I have seen many with deviation.
Here I will share with you the concept.
CONCEPT: The target ( a list of number from 1 to 10, or similar list) is to be produced using a set of pre-defined numbers, or a certain number only. Any mathematical operation can be utilised to achieve the target.
To produce a list of number from 0 to 10 using ALL the numbers 1, 2, 3 and 4, with any of the mathematics operators, + , -, /, x, ) ( and exponential.
Here it goes...
0 = 4 - 3 - ( 2 - 1 )
1 = (4 - 2 )/ (3 - 1 )
2 = (4 + 21) / 3
3 = 41 + 2 -3
4 = 4 x 2 - 3 - 1
5 = 4 x 2 - 31
6 = 4 + 2 / 13
7 = 4 + 3 / 12
8 = 4 x 2 / 13
9 = 4 x 2 + 13
10 = 4 x 2 + 3 - 1
This is a simple maths game, but it involves the application of the basic principle of mathematics to achieve the targeted number for a given set of constraints.
"Reversity" - a reversal of the mathematical procedure.
Another tougher Reversity game is to have only the number 4 used.
Any deviation can be created to generate different levels of difficulties. Explore it!
Keep your mind in reverse gear ....... ! s r e e h C (reversed)
Number Game To Make Maths Fun
You can play games with maths. With a bit of creativity, you can create lots of fun.
I just encountered one word game initiated by a nice member at the social site BlogCatalog .
The game started with a word (has to be more than 5 letters). The next member has to continue with another word but has to start it using the last letter of the previous word. Sort of fun going especially when you see the people whacking at the comment space to participate.
Improvise this concept to maths :
Maths Game
Make use of any 3 different maths operators to generate the number 18.
Rule of the game: The last maths operator has to be the first for the next person to continue, and no numbers in the operation are to be the same.
Start: 18 ==> 40 / 2 - 5 + 3 ==> .......
Happy thinking :D
Maths Concept and Its Implications
Most maths learners do face this sort of situation while learning maths. It is a learning phase in education.
What is therefore the message behind this phase?
In this phase, you will discover that purely understanding the steps or procedures in "attacking" a mathematics problem may not cover all aspect of the topic involved. Extra (hidden) elements are also mixed into the learning to form complete comprehension of the mathematics topics.
What normally is the strategy maths teacher use to teach, especially in a classroom setting, is to "drill" students into mastering a topic, practising till the idea goes into the head. It is an effective way to transfer knowledge, but with limitation.
Students are left, sometimes, to figure out the underlying concepts themselves. This is so due to many factors, examples like time resources, ability of the class to understand details, teaching style, and assumptions on the part of students and teachers.
All these contributed to gray areas in the learning of maths, thus producing gaps in the knowledge acquiring process.
What is a better way to handle maths in a classroom learning environment?
Do not blame. Take learning to be a personal challenge to be guided by the teacher. There are always too many factors in class to handle, and some aspect of learning and teaching will definitely be missed out, especially when different students has different aptitude towards maths.
After all, you, the maths student, is the one who is learning and aiming to be better, not the teacher!
What you can do with in maths learning is the understanding of maths concept as opposed to ability to solve rigid questions step-by-step. (You can solve many similar types of maths problems but still do not know its concepts - this is not true learning).
Let me cite a few examples to illustrate the "concept" element.
Example 1:
Finding the area of a given triangle with known height and base.
Formula is Area = (1/2) x Height x Base length.
Yes, you can calculate the area from the formula and given information.
If we give you another triangle with the SAME information but slanted in angles. What is the approach?
For those who did not master the geometrical area concept, they will start off with the calculation as usual, and obtaining an answer that is correct in number.
For those who had mastered the underlying area concept, they will know that the area calculation is not necessary, and the answer will be the same as previous. This is so since the area formula spells out clearly that angles of the triangle has no relation with the area computation. (Even if they showed the workings, they reflect it to get marks for the steps).
The two styles of learning, therefore, clearly, showed the strength in the latter case as it allows you to grasp an overall understanding of area. You are not mechanically drilled into rigid step-by-step solving. You are elevated into "flexible" application of concept at a higher learning platform when you practise the latter technique. You are able to link with a bigger picture in mind.
Example 2:
Solve 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = ?
If you understand the concept of "Multiply", you will solve the question efficiently.
(Solution: 8 x 8 = 64 , short and sweet)
You can, however, solve this the rigid way by adding up all the "8"s also, but, is this the proper method when you already know of a better technique?
Example 3:
Simplify log3 X + log3 Y into a single logarithmic term.
You can solve this by applying the logarithmic Product formula to get an answer of log3 XY.
But what if the question changes to 1 + log3 X ?
Without understanding the Product concept, you will be stuck!
Product concept in logarithm calls for terms to be expressed in the "log" form for proper combination later.
(Solution: 1 has to be changed to log3 3 to be able to continue).
Therefore ways has to be found or linked to previous knowledge to capture another "log" term for the "1" in the new maths question to simplify the problem.
The above three examples, hopefully, demonstrated the importance of comprehending concepts (the extra hidden element) to reap effective learning. This addresses the issue of solving ANY mathematics questions regardless of how they are asked and how they are twisted.
The implication of studying with mastery of concepts in mind has far-reaching advantages in your future endeavours as it allows you to have the "flexible" approach to problem-solving in an effectiveness manner.
How Many Decimal Places To Use?
"How many decimal places have I to use for this computing?".
This is not a simple question to answer.
Though the teacher may simply say "Use 3 decimal places" as a yardstick or to standardise marking or solutions, what is the underlying importance of selecting the correct amount of decimal places?
Many a times, the number of decimal places to be used in any mathematical calculation is not significant at elementary maths level. What is required at that education level is the understanding of the maths working or steps in solving the maths questions.
However, as we progress up the educational ladder, the importance or precision of the answer becomes significant.
Let us look at one example and its impact.
Example A: 2.143 has 2 parts.
- 1st part is the integral portion of "2"
- 2nd part is the 3 decimal places for "143" (after the decimal point).
Now let's divide the number 4 by the number in example A, and see the effect to the answer when we select various number of decimal places.
1) 4 / 2.143 = 1.867
2) 4 / 2.14 = 1.869
3) 4 / 2.1 = 1.905
What do you see in these results?
Not much changes right?
This is because the number divided "4" is too near or of the same unit as the divisor.
Let's change the number to be divided to 40.
1) 40 / 2.143 = 18.67
2) 40 / 2.143 = 18.69
3) 40 / 2.1 = 19.05
Do you notice that the difference got bigger?
What is the impact when the number to be divided increases to 400?
The differences will be drastic now!
400/ 2.143 = 186.7 and 400 /2.1 = 190.5, a difference of 3.8!
So what's the rationale in deciding the number of decimal places?
We need to understand the relative scale of all the numbers involved in the computation of the maths questions.
When the numbers involved are of the same scale, the number of decimal places is not an issue. However, when numbers are relatively bigger, more decimal places are required to have better precision to reflect an accurate answer.
In the last division, it may be a difference of 3.8 metre of length or $3.8 difference in monetary change!
Maths learning, thus, not only involves pure manual calculation, but also involves the logical interpretation of the steps and the impact of the numbers and their decimal placement.
The logical thinking part in maths education does places a significant benefit to anyone mastering maths in his future life, as he is able to decide and use the appropriate method and techniques to achieve results.
Pyramid - An Ancient Shape
Geometry has lasted a long time ago.
The Egyptian Pyramids of Giza in Egypt is a famous example.
What is so special about the shape of this pyramid?
It differs from other shape in that, besides the base, all its other surfaces end up in a vertex, or pointed tip at one end.
Actually this pyramid is closely related to the prism. In fact, pyramid is part of the prism, and therefore is mathematically linked.
The volume of a prism is given as
"Area of Base x Height".
The volume of the pyramid is thus given as one-third that of the prism.
In other words, volume of pyramid = (1/3) x (Area of Base) x Height.
For more information about this volume relationship and why is it one-third, you may read Volume of Common Solids.
To better understand pyramid, test yourself with the question below.
Look at the diagram below and tell which volume is the biggest.
Note that they have the same base area and height, but has different slanting angles.
Any ideas? Read on for the answer...
The answer is that they are all equal in volume.
This is because the angle of the slanted edge does not play a part in the computation the pyramid volume. The formula tells the "story".
After understanding the concept of pyramid, do you have a better admiration for the "Great Pyramids" of Egypt?
I sure do! :)
Mastering Maths Using PDCA Cycle
However, there may be cases of the results, after applying these learning steps, not meeting expectation. No marked improvement is seen after all the hard work.
What happened? What went wrong?
These are the very questions that this group of maths learners would ask.
The answer lies in the flow of the process. The execution of the learning cycle is important in reaping the desired targets. Random application of the learning steps on an ad-hoc basis may be the reason for the confusion in mastering the maths lessons.
To master maths, or any subject for that matter, we need a solid learning model. A model whereby we can follow through step-by-step, and also without any confusion. This learning model has also to be simple and specific.
One such model of learning is the PDCA model.
What is this PDCA about?
P stands for Plan.
D stands for Do.
C stands for Check.
A stands for Action.
This PDCA, therefore governs our steps in the learning process.
Let us dig more into this model.
Here, what we meant, is to plan for the topic to be mastered, the resources needed and the time frame, plus the desired outcome. This stage sets the goals where we need to achieve to consider mastery of the targeted maths topics.
Execution takes place here. It requires the application of the resources needed to meet all the objectives set forth in the Planning stage. Practicing of the maths questions, analysis of the steps and understanding of the mathematical concepts are captured in this learning stage. All scope within the topics has to be covered, preferably, here.
This is a critical step in the PDCA model. This is where we check for correct understanding of the mathematical concepts and procedures in getting the answers. Model answers to the given maths problems can be a reference here to confirm the solutions. In this checking stage, the question "Did the DO stage results in me getting the correct steps and answers?" is the key element to complete this part of the learning cycle. If there is any discrepancies in the solutions or answers, the next stage in the PDCA will be executed.
Action here is what the words literally means. Take action to rectify any hiccups in the understanding of the mathematical concepts and tools. Analysis of the problems involved is the main focus here. "What went wrong?" and "Why it happened?" are the type of questions to be addressed in this stage. The analysis has to be as detailed as possible in order to cover all areas including linkages to other maths concepts learnt previously. (A good source for information is our textbooks). ACTION stage is value-adding and accumulative in the learning process.
* After identifying the mistakes made, go back to the PLAN stage to revise the objectives for the second round or cycle.
Follow these 4 simple steps in the PDCA model to control your flow of thoughts while learning. It is a systematic approach to learning. This ensures that your learning will not jump at random doing things you deemed proper but which may not be the case.
PDCA is a recurring process which halts only when mastery of the maths topics occurred. All questions are then answered correctly with ease.
Give the PDCA learning model a try! Happy practicing with it.


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Guitar Chords For Beginners – Get Started Immediately With This Easy Guitar Chord
Learning to play guitar is an exciting adventure. The guitar can sound single notes or chords. I want to talk to you about guitar chords for beginners today. We need to take care of a couple of things first so you understand things later.
First, a chord is made up of three or more notes that are played together. If you are playing a two note combination, that is called a diad and is different. It takes three notes to make a chord and the notes are played with the fingers of the left hand.
A chord is made up of a root note, a third above the root and a fifth above the root. If you are starting a chord at the "C" note, it is a "C" chord. Therefore the notes in a "C" chord are the root "C", the third "E" and the fifth "G". We get that by going up from the root alphabetically like this: "C", "d", "E", "f", "G". You can find the notes in any chord this way.
One of the best guitar chords for beginners is the C major and is played like this:
1st string is open
2nd string, 1st fret. This is a "C" and is an octave (8 notes above) the root "C". Played with the index (# 1) finger
3rd string is open. This is the fifth, or "G"
4th string, 2nd fret. This is the third, or "E". Played with the middle (# 2) finger.
5th string, 3rd fret. This is the root "C". Played with the ring (# 3) finger.
6th string is not sounded. Although it is an "E", it does not sound good with the chord, so is silenced.
Now draw the pick across the strings from the 5th string downward and listen to the sounds and how they blend together. The notes are what we call harmonic. They sound nice together. You have just played a C major chord. See how easy guitar chords for beginners are? A little practice and you'll play it perfect every time.
The other two guitar chords for beginners in the key of C major are the "F" and the "G". The F major has the notes F, A and C. The G major chord has the notes G, B, and D. Notice how all the chords contain the notes of the key of C?
Play the F chord like this:
1st string, 1st fret, index finger (F)
2nd string 1st fret, index finger (C). You can kind of lay your index finger across these strings.
3rd string, 2nd fret, middle finger (A)
4th string, 3rd fret, ring finger (F)
5th and 6th strings are not played.
Play the G chord like this:
1st string, 3rd fret, ring finger (G)
2nd string, open (B)
3rd string, open (G)
4th string, open (D)
5th string, 2nd fret (B)
6th string, 3rd fret (G)
As you see, the G chord actually spans three octaves of the G and two octaves of the B.

subset: commoncrawl


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Pine dressers may also be found in ads. From circulars to trade journals, you could possibly find a wide option of dressers on sale. Each advertisement usually consisted of brand, price and manufacturer's details. This way, you are completely secured in the event of a mishap. Some advertisements may require you to visit the supplier directly. This is a means of securing their integrity, while making a sale face-to-face. Regardless your choice, pine dresser devices will supply you with years of wonderful use.
A recliner is a sort of chair that could be pushed back at an angle to flatten the backrest. It makes seating a lot more relaxing, like a makeshift bed that enables the user to lie down a bit. This type of chair is actually inviting to remain on after a challenging day's work. It actually is quite relaxing and relaxing. Recliners have ended up being so popular now that almost everyone wants to have one for his home. It really is not only extremely practical, it also beautifies a residence.
Take a look at the cost of a single couch which of a futon bunk bed. You'll be astonished to see the distinction on your own. So, now you shall likewise agree that this bunk bed can be referred to as an inexpensive futon bunk bed. Proceeding, in regards to the space it occupies and the location that would be had to position a sofa and bunk bed in a space, this cheap futon bunk bed likewise ratings higher, as you require just the floor location of a bunk bed to position this two in one in your room.
Ranges of corner sofas include types such as the corner sofa bed due to their severe area performance. Corner sofa beds can conveniently be made into an extra bed by easily connecting it to other products of furnishings. A corner sofa bed can be a terrific selection for a small room since it will certainly make the room appear larger. An L setup couch set lends a more advanced, graceful aim to a living room, while offering comfort and style. The corner sofa sits in an L setup, a look that numerous consider elegant and graceful. The excellent thing about armless corner sofas is that they can be integrated with other sofas to produce an even higher sectional sofa with additional seating.
Top Five Steps To Better Couches
Yet another option is getting a memory foam futon bed. These are similar to the standard memory foam beds but without the added cost. You can find these online too and will be astonished at how they feel. Some individuals get the very best sleep they've ever carried these types of memory foam beds. There are properties in the foam that disperses weight so that pressure points are not developed and disperses body temperature better so that you do not get too warm resting.
Basically, a futon serve as a sofa during the day and is transformed into a sleeping area during the night. Some of the futons have wooden frames that can be folded and put away to conserve area. The futon sofa has cushions that are typically stuffed with cotton batting which can be found in a wide range of colors, sizes and fabrics. Those used to sleeping and sitting on a futon sofa rave about the great back support it provides. Another plus point of futon bed mattress and cushions is that they can be conveniently moved outdoors to air out. Sometimes, the futon sofa might need some assembly. The most crucial thing to do while putting together the sofa is to take your time; it is best to commit a whole afternoon to assemble it. It is suggested to take an inventory of the parts of the sofa before beginning. And, you must read the guidelines not when, but twice for clearness. If utilizing power devices while putting together the sofa, beware as these devices are so strong that they might over-tighten or strip out the hardware used to hold the frame together.
Solutions To Improve Your Custom Leather Couches
Every lady needs a jewelry armoire to arrange her things. The finest component is you can easily find a jewelry armoire in numerous sizes today. You can pick from the huge armoires which can function as a stylish furniture item in one corner of your space or you can choose a wall mounted device that might also function as a mirror.
As soon as you have chosen you want a transitional sofa set, you will want to know how to look after and care for it. Simply as holds true with other furnishings in your house, transitional furnishings requires the very same quantity of attention. If you have pets in your home, you surely will not allow them to lift the furniture. The very same puts on children, as they have to clean themselves up when they comeback from play, prior to they pile up on your cherished sofa. About the other furnishings in your house, you would not wish to feel you have bought a piece with an unique d cor or one that is a straight-out mismatch. Buy one that blends with the interiors of your home and matches the general plan of the other furnishings in the house.
Good Work! Your Couches Is About To Start Being Relevant
Needing price cut sectional sofas at thrift establishments or garage sales is the right step, however not the fastest or hassle-free methods of purchasing. The items offered here might be slightly utilized that needs a haul, however will certainly save your cash than spending for a brand name new sofa from a department store. There are lots of outlet shopping malls offering sectional sofas at price cut rates, however such shopping malls are constantly found on the outskirts of the town as the suppliers consider it more affordable. This suggests that it is a huge drive from the city for the buyers, however deserves a drive as they are available in much better cheap rates than from the department store and itsmiddle-man.
Beds are crucial bedroom custom leather couches that ought to be decorated in such a way that it looks welcoming and warm. Embellishing your bed with accessories like draperies or with canopy bed hangings can make your room appearance elegant. Headboards can also provide a touch of enrichment to your bedroom. The size of the headboard ought to be in proportion to the size of your bedroom. Smaller sized headboards are better if your bedroom has a low ceiling, while a bedroom with a high ceiling offers you the scope to try out bigger ones. There are also some special headboards offered like the bookcase headboard. Meant to be a functional furniture accessory, a bookcase headboard has cubbies that can save books and photos, eliminating the requirement for a bedside table and conserving space.
A bulky well constructed alternative can be quite costly, while on the other hand a trendier area conserving style can be less costly and much better matched to contemporary homes. These are leather recliners with a push back style when you lean back, while the bulkier designs will have a lever for the same operation. For those who love everything elegant, the very best recliners to buy are those that come as speciality chairs and have a massager too in the backrest in addition to footrest. Though costly, these are a true blessing.
How Consumers Fall In Love With Couches
With memory foam being introduced to the futon mattress the game for the futon has actually entirely altered. Because of the elasticity of memory foam it can quickly be folded up exactly like a futon mattress does. So now days you do not have to utilize the plain and rock tough 2 inch thick cloth mattress. Memory foam is a standard polyurethane foam that has actually had particular chemicals contributed to it to make it Visco elastic. Visco-elastic foam is on other name for memory foam. Visco-elastic foam reacts to temperature level and pressure from your body. If you've ever laid on memory foam then you know that when you first lay on it it feels a little cold and a little firm but after a moment or two the foam begins to warm and soften allowing you to slowly sink a little down into it.
Just Where Can You Obtain The Best Couches?
Sleepovers have actually become treasured time invested specifically preteen age and parents always scuffle to produce the best accommodation for their little visitors. Normally sleeping bag suffices for children to snuggle chat and invest awesome time with their little friends how ever the introduction of air beds change all that. The current choice of Air Mattress is becoming the more appealing choice for sleepover offered their versatility and convenience. Kids like Air Beds they sleep even more than one, One even more reason children are cooler to air beds is they are comfortable for family games and become comfortable seats around a fireplace or tables for late night snack. Parents seem to choose the concept of investing even more time with their children sleep over friends when there is a more comfortable environment and Raised Air Beds seem to be best platform for a board or card games and late night story informing.
Sofas set the style of the room they are in. Convenience and style of your sofa constantly depends upon what material you choose. Softer microfiber materials, like plush or chenille, includes comfort to your sofa while crisp and smooth materials have a neater and tidier look. View as the general decor of the room prior to choosing in between a modern or a modern microfiber sofa. Threats such as untidy children, eating on your sofa, or a light colored material, makes sofa covers a good idea. Having an extra cover could save you money and humiliation rather than needing to change your sofa after a while. Covers are not that pricey and could be convenient if you change the color design of the room and simply have to get a new matching sofa cover rather of buying an entire new sofa.
Another alternative is the Flip-Over microfiber sofa beds. They are very simple to transform as they don't feature a mattress that is different from the sofa. This is why they cost less than other forms of sofa beds. The real seat of the sofa ends up being the sleeping location. They are terrific for kids too as they are very close to the ground. You will not have to stress over them falling out of bed in the middle of the night. See to it you check out all of these various types of microfiber sofa beds. That means you can get the one you will certainly be able to gain from the most. There are many things you want to look at. Initially, what size is the sofa bed? It is going to depend upon the size of couch that you have space for. Most of us have slept on thin mattresses that leave us with the bars under it bothering us.
Practical Ways The Specialists Utilize Custom Leather Couches
Egyptian cotton sheets are the most preferred choice today. Egyptian cotton sheets are equipment washable. Egyptian cotton sheets costs vary with their thread count, the higher the thread count the higher the rate.
Speaking of comfort, kindly do yourself and your guests a favor and recognize that the products used in covering convertible sofas make a huge difference in how they feel. Nobody suches as trying to sleep on a surface area that gets too hot throughout the night. Materials like vinyl have the tendency to trap one's body heat and moisture, and can be very lame to try to sleep on unless effectively covered. Bed mattress covers or perhaps a spare blanket under the bottom sheet can solve this problem. Naturally, choosing fabric or upholstered coverings are similarly subject to issues like scratchy or rough structures. Again, the correct covering can solve most of these issues. The best thing to do is take note of these sorts of things when you are selecting the ideal sofa bed for you. A little diligence at the front end will certainly settle handsomely in the long run. Convertible sofas are distinct, stylish, economical, convenient and comfy, and with a little shopping around, you can discover the best one for your house.
Convertible sofas are heavy to move, however offer enough variety that they are entirely worth it. Having the ability to have both a sofa and a bed all in one is a sweet alternative for lots of in today's world of house living and smaller houses. When I first heard about these furnishings marvels of the modern world, I honestly didn't identified what to make of it. I knew that a convertible is a kind of automobile. I get it that there is no roofing and all that. I also identified what a sofa is. A couch, right? However somehow, putting the 2 words together just didn't register anything in my head. Convertible didn't refer to something not having a top, however to something that can "transform" from one thing to another. See the light coming on?
Ottoman sleepers are available in a number of various designs and colors, although all them can be put into any among the three major classifications of sofa bed. The most common kind of sleeper ottomans are the ones that can be converted into a bed when you pull the extension from under the cushions. You likewise have the kind that allows you to adjust the back of it for even more support and upright seating along with 180 degrees down for putting down to sleep. There are likewise the sectional kinds of sleepers that are modular types that can be upright, set or even organized for propping your feet. You can even make use of a chaise to lounge or even sleep on, while others sit upright on the ottoman. The ottoman likewise can be pushed beside the chaise to develop a larger area for resting. By doing this you and your guest can hang out and relax while in comfort and design.
Simple Steps To Replacing Couches
If you are one of those who desire more from everything in life, you can choose recliner sofas. This is possible due to the fact that these sofas are designed in such a way that your legs recline while the chair part of the sofa supports your back and neck really firmly yet conveniently. These are popular amongst those who love seeing TV while sitting conveniently on their couch. Lying down conveniently after a difficult day at work is better than sitting on a sofa. Some special features like headrests and foot rests are likewise readily available in modern recliners, which make them utmost in convenience.
A sofa is an useful addition to any house. Nevertheless, searching for one is oftentimes a difficulty for a great deal of individuals. This is due to the fact that the incorrect option of sofa for your house can generate not only a physical catastrophe however likewise a space catastrophe. This is especially real if you have a little living room. Advantage that there are a great deal of corner sofas to pick from. They are the best options for a tight space. This is not to mention yet that leather sofas of this kind are in the picture too. Meaning, space and beauty comes in one with an option of a leather corner sofa.
For an actual stylish combining to improve your living-room, you can buy red sectional sofas with a dark frame matching with dark or chocolate brown drapes and red of carpet. You can likewise establish a big mirror in your living-room exposing your creativity in the form of red and black glass paints on the frames of the mirror. The remarkable thing about red coloring product is that it shines energy. Just at the one time red color too gives an actual welcoming experience.
Sleeper sofas are of wonderful use, specifically in small residences that do not have enough space to accommodate a huge-sized bed. A sleeping sofa serves several purposes, i. e, you can utilize it as a seating to recline, sleep or simply sit and relax. The structure of a sofa bed is such that a bed is connected to the sofa and it can be pulled out whenever required. Folding sofas are of enormous utility considering that they can be dislodged and rolled up even in a little storage space. Sleeper sofas have actually become essentials in lots of families where guests pay frequent visits. Studio houses are geared up with sleeper sofas therefore increasing the readily available space, which would have otherwise been occupied by large beds. Above all, sleeper sofas provide extreme convenience while sleeping.
Last however not the least, modern desks, dressers and credenzas that you purchase should be great quality. These days you have a range of option in products made use of to make these items. From different types of wood to metals each of these have their own benefits and drawbacks. So decide after doing an extensive study on the quality of the furniture. Online stores are the best location to purchase modern credenzas, desks and dressers as they provide you a wide range of items to choose from. You can also purchase furniture from around the world from these stores.
Furniture that is commonly made use of to show the art material such as exclusive silver and brass posts, earthenware and tableware is known as Furniture Dresser. In older days this furniture piece was thought about simply a side table; and it periodically include drawers. However, with the growing awareness people have become more innovative and creative to decorate their rooms. Dressers own a special location for the people who like to have furniture with functions.
Orthopedic pillows are said to be the finest pillow for neck pain. How do you go about finding the finest pillow for YOUR neck pain?
Other fabrics that may be made use of for sofas are cotton, wool, jute, and linen. These, however, are all high maintenance as they stain quickly and are not capable of handling excess wear and tear. Also bear in mind to obtain guidelines on how to clean the material that you have actually picked, and methods to safeguard it, so that it lasts as long as possible. As long as you know how to make the most of your sofa and have it suit any interior area, you are bound to have a good looking piece of furniture that can be the star of your room.
It appears simple while purchasing upholstered furniture for your house. We often tend to forget particular points to inspect while selecting then furniture or rather we forget to make a precautionary examine the sturdiness of the furniture. Furniture is made to offer comfort with appropriate engineering and a well structured frame to help support the spine. Contemporary pieces tend to have clean lines, and are normally made with modern materials such as MDF, board and foam. While this may sound uninspiring, select wisely and you can expect a hard-wearing piece of furniture. While modern furniture may include a speck of antique delight in your room. While purchasing a sofa for your room proportions matter a lot. Before in fact stepping in the sofa shop you have to first analyze the theme of your living room. Look for the average measurement of your room, if the other furniture are enough large, you must stay clear of purchasing big and large sofas. Finally, before you buy, you must constantly inspect that the piece will fit with your doors, specifically if your house has a slim hall or tilted corridor.
Half Truths About Couches That You Should Ignore
After working doggedly throughout the day, everybody will certainly need a safe place to rest his/her tired body. No one would ever before like to be interrupted when resting but unfortunately some bed we lay might be a trouble. Some are smaller sized in size with weaker joints such that when one turns at the middle of the evening; some squeaky sound is produced thus distracting your rest. Picture oversleeping such a bed with someone who will turn every now and then? Save yourself from unneeded disturbances by buying bunk beds. It is sturdy, well repaired and strong to produce any noise when one turns. Some bed are also small such that they cant support the entire mattress or ones body thus too awkward but for a platform bed, it has a broad area to accommodate any amount of weight or lot of people.


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The Dr. Gesundheit Clown Therapy Show mixes physical comedy, eccentric dance, comic juggling, wisecracks & wordplay, acrobatics, and ingenious audience participation. Volunteers become the Dr.'s partners in classic vaudeville and commedia routines.
"Craig's shows entertain all ages by mixing expert juggling, wisecracks & wordplay, and energetic clown routines. This clown character is not the heavily made-up sort that frightens small children. Yet his routines preserve the strengths of traditional clowning. He falls, he flips, and his timing is so good he can get a laugh by slowly raising an eyebrow."
-Rick Kinnebrew
Niles Public Library
Wednesday, July 31, 2019 - 4:30pm - 5:30pm
Event Type:
Family/All Ages
Meeting Room


text: Sections View Full Chapter Figures Tables Videos Full Chapter Figures Tables Videos Supplementary Content ++ The contribution of genetic variation to health and disease has become increasingly clear as a result of our burgeoning understanding of the human genome. Genomic medicine uses an understanding of the individual's genetic makeup to enhance the quality of medical care, including presymptomatic identification and prevention, determination of susceptibility to disease, selection of treatments, and individual clinical care based on genotype. Most diseases fall somewhere on a continuum from single-gene disorders, disorders with a major gene effect, disorders with complex traits, and those with a predominantly environmental causation. The majority of genetic disorders are multisystemic in their manifestations. Most children with special health care needs have disorders that are caused by mutations in single genes or variations in chromosomes or that are heavily influenced by genetic factors. Genetics can no longer be considered as only the identification of rare syndromes and inborn errors of metabolism. Care for the child with a genetic disorder involves five major activities—suspicion, diagnosis, management, genetic counseling, and advocacy. These activities require a partnership between the pediatrician, one or more specialists or specialty teams, the family, and schools and other support services. No physician can hope to recognize and personally manage all genetic disorders, many of which are quite rare. Treatment for genetic conditions is advancing rapidly, encompassing much more than the supportive care previously available. ++ Suspicion of a genetic condition must rest with the primary care physician. Situations suggesting a genetic disorder are outlined in Table 172-1. Single organ diseases or birth defects may also be genetic or have a major genetic component. Neuropsychiatric conditions such as autism spectrum disorder are frequently genetic in causation. The American College of Medical Genetics has published a detailed guide to indications for genetic referral.1 ++Table Graphic Jump LocationTable 172-1. Conditions Suggesting a Genetic DisorderView Table|Favorite Table|Download (.pdf)Table 172-1. Conditions Suggesting a Genetic DisorderFamily history of a genetic disorderAbnormal prenatal diagnostic testingAbnormal newborn screening resultMultiple congenital anomaliesUnusual facial features and minor skeletal abnormalitiesDisorders of growth, prenatally or postnatallyDevelopmental delay or regressionChronic multisystem dysfunctionAcute life-threatening eventPerinatal or neonatal death ++ Once a genetic condition is suspected, a diagnostic evaluation should be undertaken. Consultation with a geneticist or other specialist is usually valuable to focus the testing. There are no generalized screening tests for genetic disease. As for any clinical diagnosis, a detailed history, family history, and physical examination are the initial steps. The family history or specific biochemical or chromosomal testing may supplement diagnostic information. Mutation analysis (DNA testing) requires a definite clinical or biochemical diagnosis or a suspicion of a specific diagnosis.1 Comparative genomics uses a microarray of DNA segments to evaluate the patient against a known normal panel.2 Although this testing may be useful in some situations, the amount of genetic variability in the population often makes interpretation of results difficult. Exclusion of more easily diagnosed conditions ... GET ACCESS TO THIS RESOURCE Sign In Username Error: Please enter User Name Password Error: Please enter Password Forgot Username? Forgot Password? Sign in via OpenAthens Sign in via Shibboleth Get Free Access Through Your Institution Contact your institution's library to ask if they subscribe to McGraw-Hill Medical Products. What is MyAccess? Create a FREE MyAccess profile to: Use this site remotely Bookmark your favorite content Track your self-assessment progress and more! GET ACCESS TO THIS RESOURCE Subscription Options Pay Per View Timed Access to all of AccessPediatrics 24 Hour $34.95 (USD) Buy Now 48 Hour $54.95 (USD) Buy Now Best Value AccessPediatrics Full Site: One-Year Individual Subscription $595 USD Buy Now View All Subscription Options


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Innova Disc
Portable Economy Discatcher
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The DISCatcher Sport is easily portable. Set it up at home for putting practice or a game of HORSE with a friend. Bring it to camping trips to play disc golf just about anywhere. It will really improve your short game, where the game is won or lost
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- Portable Economy Discatcher
- Galvanized and powder coated construction
- 18 chain assembly
- Light at only 40 lbs
- Collapses to 6 pieces
- Not meant for permanent outdoor installation
- Chains - Durable yellow powder coated finish with pre-attached chain set
- Basket - Electroplate galvanized finish
- 2 Poles - Two sections with an electroplate galvanized finish
- Base - Our simple electroplate galvanized, four spoke
- System is wide enough and strong enough to stabilize the target, yet is still lightweight