Given a maths question, most of us will attempt solving it step by step, diligently, using formula and methods we have mastered.
However, there are times when a simpler solution can be done if we are able to see the logical side to the maths problem assigned.
Let me show an example.
Question:
2x = 24, find the value of x.
Solution: (mathematically)
Taking "log" on both sides, ==> x log 2 = 4 log 2 ==> x = 4 (log 2) / (log 2) = 4
Solution: (logically)
Comparing the values of their power, we get x = 4, since their base is the same (=2).
No working is needed!
Thus, maths does not actually just train us to do things systematically, it allows us to have a bit of mental freedom. This freedom is done in terms of the small little "twists" that make use of visual comparison or logical thinking (comparison).
Interesting approach to maths learning, right?
So many ways, mathematically and non-mathematically.
But best is the stretch to our mind to develop it to see things in many angles.
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Showing posts with label mental maths. Show all posts
Showing posts with label mental maths. Show all posts
Sunday, 16 November 2008
Sunday, 10 August 2008
Mental Multiplication | More Tips
With flexibility in the arrangement of numbers to be multiplied, mental multiplication can be easily done with numbers that seems to be big.
Example: 21 can be splitted into 20 + 1 = (2 x 10) + 1
With this splitting of "big" number, 21 becomes operations using only 2, 10 and 1. All these numbers are easy to handle mentally.
Mentally we need to double the number, attach a"0" behind, and add back the original number to the answer.
Let's do an example.
Example-A: 34 x 21
Mental computation is
34 x 2 ==> 68 ==> append "0" ==>680 ==> 680 + 34 = 714 (answer).
See it can be done easily!
Example-B: 34 x 15
Here, 15 can be splitted into 10 + 5.
But, 34 x 15 ==> 34 x (10 + 5) involves 34 x 5 which may be hard to handle.
What is a better way?
15 can be modified to 1.5 x 10. This looks better as 0.5 is "half" which is 1/2, an easier operation.
Thus 34 x 15 ==> 34 x 1.5 x 10 (All these uses simpler number of 2 and 10).
34 x 15 ==> 34 x ( 1 + half) x 10 ==> [ 34 + (34 / 2) ] x 10 = [34 + 17] x 10 = 510
Here we see that the "half" method works better than the "x5" method (mentally).
From the above 2 examples, we can see that multiplication of 2 seemingly big numbers can be done easily by modifying the numbers to simpler ones having 2, 10 and the likes. Coupled with simpler mathematical operations, these multiplication can be done mentally with ease.
Mental math technique can be created easily and effortlessly. What is needed is just a bit of creativity and basic math principle understanding.
Maths is interesting!
:-)
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Example: 21 can be splitted into 20 + 1 = (2 x 10) + 1
With this splitting of "big" number, 21 becomes operations using only 2, 10 and 1. All these numbers are easy to handle mentally.
Mentally we need to double the number, attach a"0" behind, and add back the original number to the answer.
Let's do an example.
Example-A: 34 x 21
Mental computation is
34 x 2 ==> 68 ==> append "0" ==>680 ==> 680 + 34 = 714 (answer).
See it can be done easily!
Example-B: 34 x 15
Here, 15 can be splitted into 10 + 5.
But, 34 x 15 ==> 34 x (10 + 5) involves 34 x 5 which may be hard to handle.
What is a better way?
15 can be modified to 1.5 x 10. This looks better as 0.5 is "half" which is 1/2, an easier operation.
Thus 34 x 15 ==> 34 x 1.5 x 10 (All these uses simpler number of 2 and 10).
34 x 15 ==> 34 x ( 1 + half) x 10 ==> [ 34 + (34 / 2) ] x 10 = [34 + 17] x 10 = 510
Here we see that the "half" method works better than the "x5" method (mentally).
From the above 2 examples, we can see that multiplication of 2 seemingly big numbers can be done easily by modifying the numbers to simpler ones having 2, 10 and the likes. Coupled with simpler mathematical operations, these multiplication can be done mentally with ease.
Mental math technique can be created easily and effortlessly. What is needed is just a bit of creativity and basic math principle understanding.
Maths is interesting!
:-)
.
Thursday, 7 August 2008
Mental Conversion of Fraction to Decimal - the Quick way
In maths, while doing division of numbers, we will normally come across the fraction, be it improper or proper form. And we may need to have the answer in decimal!
Mentally how do we get the answer fast and simple?
To have this quick conversion from fraction to decimal, we first need to memorise some simple basic fractions. This will speed up the process of mentally computing the conversions.
We can do mental division using any of the method presented in this blog (refer to links at the end of this post). But the answer is in fraction form.
Example is 40 / 7 = 5 5/7, a mixed numer format.
We like to have this 5/7 in decimal (sometimes).
You can see that these basic fraction/decimal conversions are not difficult to memorise.
In fact some of them are sub-product of others.
Example: 1/ 6 =(1/2) x (1/3) = 0.5 x 0.333 or 5 x 0.333 / 10
You can choose to memorise 1/2 and 1/3 and do simple mental multiplication later for 1/6.
Another example is 1/8. This can be achieved with (1/2) x (1/4).
After memorising the above basic numbers, we can straight away use them in the computation. Let's see an example.
Example: 42 / 5 ==> 8 2/5
We know that 8 2/5 = 8 + 2/5.
Mentally converting, 8 + 2/5 ==> 8 + (2 x 0.2) = 8.4 (the answer)
This is with the use of the memorised number of 1/5 = 0.2.
Another example: 37 / 8
Mental division produces 37 / 8 = [(8 x 4) / 8] + ( 5 / 8 ) = 4 + ( 5/8 )
We need only to convert the ( 5 / 8 ) into decimal.
5 / 8 = 5 x 1/8 = 5 x 0.125 ==> 0.625 using Mental Multiplication method (from left to right approach).
Therefore 37 / 8 = 4 + 0.625 = 4.625 done mentally!
From the above examples, we can see the effectiveness of the memorised fractions/decimals numbers simplifying the mental conversion process.
Practice with this method and it will get easier with times.
Links for mental division:
Mental Division
Mental Division - denominator approach
Mental Division - Inflation method
Mental Division - Deflation method
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Mentally how do we get the answer fast and simple?
To have this quick conversion from fraction to decimal, we first need to memorise some simple basic fractions. This will speed up the process of mentally computing the conversions.
We can do mental division using any of the method presented in this blog (refer to links at the end of this post). But the answer is in fraction form.
Example is 40 / 7 = 5 5/7, a mixed numer format.
We like to have this 5/7 in decimal (sometimes).
- Firstly, we need to know (memorise) the below basic fractions/decimal numbers:
1/2 = 5
1/3 = 0.333
1/4 = 0.25
1/5 = 0.2
1/6 = 0.1667
1/7 = 0.1429
1/8 = 0.125
1/9 = 0.111
1/10 = 0.1
You can see that these basic fraction/decimal conversions are not difficult to memorise.
In fact some of them are sub-product of others.
Example: 1/ 6 =(1/2) x (1/3) = 0.5 x 0.333 or 5 x 0.333 / 10
You can choose to memorise 1/2 and 1/3 and do simple mental multiplication later for 1/6.
Another example is 1/8. This can be achieved with (1/2) x (1/4).
After memorising the above basic numbers, we can straight away use them in the computation. Let's see an example.
Example: 42 / 5 ==> 8 2/5
We know that 8 2/5 = 8 + 2/5.
Mentally converting, 8 + 2/5 ==> 8 + (2 x 0.2) = 8.4 (the answer)
This is with the use of the memorised number of 1/5 = 0.2.
Another example: 37 / 8
Mental division produces 37 / 8 = [(8 x 4) / 8] + ( 5 / 8 ) = 4 + ( 5/8 )
We need only to convert the ( 5 / 8 ) into decimal.
5 / 8 = 5 x 1/8 = 5 x 0.125 ==> 0.625 using Mental Multiplication method (from left to right approach).
Therefore 37 / 8 = 4 + 0.625 = 4.625 done mentally!
From the above examples, we can see the effectiveness of the memorised fractions/decimals numbers simplifying the mental conversion process.
Practice with this method and it will get easier with times.
Links for mental division:
Mental Division
Mental Division - denominator approach
Mental Division - Inflation method
Mental Division - Deflation method
.
Tuesday, 5 August 2008
Can It Be Divided By 11?
To be able to decide whether a number can be divided by 11, we need to understand this multiply by 11 concept first.
Click here to review the x 11 mental concept.
It is simply to do the reverse process of the mental multiplication by 11.
Example: Can the number 352 be divided by 11?
Looking at the centre digit, we can deduce that it is the addition of the 2 extreme digits.
As such the number 352 can be divided by 11.
Example: How about 3751 ?
Again, looking at the digit 3 and 7, we can see that their subtraction produces 7-3 = 4.
For the other 2 digits of 5 and 1, their subtraction also results in 5- 1 = 4.
This means that the number 3751 can be divided by 11 (based on the mental multiplication by 11).
Example: How about 542 ?
The centre digit should be 5 - 2 = 3 to be divisible by 11.
Therefore the number 542 can be quickly deduced not able to be divided by 11.
Check other divisibility concept by clicking the link here.
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Click here to review the x 11 mental concept.
It is simply to do the reverse process of the mental multiplication by 11.
Example: Can the number 352 be divided by 11?
Looking at the centre digit, we can deduce that it is the addition of the 2 extreme digits.
As such the number 352 can be divided by 11.
Example: How about 3751 ?
Again, looking at the digit 3 and 7, we can see that their subtraction produces 7-3 = 4.
For the other 2 digits of 5 and 1, their subtraction also results in 5- 1 = 4.
This means that the number 3751 can be divided by 11 (based on the mental multiplication by 11).
Example: How about 542 ?
The centre digit should be 5 - 2 = 3 to be divisible by 11.
Therefore the number 542 can be quickly deduced not able to be divided by 11.
Check other divisibility concept by clicking the link here.
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Sunday, 3 August 2008
Mental Logarithm Easily Computed
In our daily life, we encounter many maths computing which we do mentally. Some are simple addition and subtraction. Some involves more complex operation. But with breaking-down strategies, we are abale to simplify this mental process to simpler steps for the mind to handle.
Here I will discuss on the mental computation with maths operation like the indices and logarithm, that we may come also in some daily situations.
Below site an example. Read and enjoy it.
Example: 10x = 8
To solve this example, we need to convert it into the logarithm form to move forward.
log 10x = log 8 ---(A)
Applying the Laws of Logarithm, log Yx ==> x log Y
Expression (A) above, become x (1) = x = log 8
However, we are stuck here! What is the answer for log 8? And mentally?
How to carry on with mental logarithm computation?
Mental logarithm computation can be easily processed if we can simply memorise a few number of the basic logarithm.
The basic numbers that we need to remember and suffice for common mental logarithm usage are:
With the 3 basic logarithm numbers known, we can list out the first few log to see its usefulness:
With the above list, most of the logarithm numbers can be easily computed mentally.
But do note, however, that logarithm of prime number cannot be done using above method.
Examples are log 11, log 17, etc.
However, we should not be deterred from this as they are the minority and the list above serves to cover most of the logarithm solutions which can be easily done mentally.
:)
.
Here I will discuss on the mental computation with maths operation like the indices and logarithm, that we may come also in some daily situations.
Below site an example. Read and enjoy it.
Example: 10x = 8
To solve this example, we need to convert it into the logarithm form to move forward.
log 10x = log 8 ---(A)
Applying the Laws of Logarithm, log Yx ==> x log Y
Expression (A) above, become x (1) = x = log 8
However, we are stuck here! What is the answer for log 8? And mentally?
How to carry on with mental logarithm computation?
Mental logarithm computation can be easily processed if we can simply memorise a few number of the basic logarithm.
The basic numbers that we need to remember and suffice for common mental logarithm usage are:
- log 2 = 0.301
- log 3 = 0.477
- log 7 = 0.845
With the 3 basic logarithm numbers known, we can list out the first few log to see its usefulness:
- log 1 = 0
- log 2 = 0.301
- log 3 = 0.477
- log 4 = log (2 x 2) = log 2 + log 2 = 0.301 + 0.301 = 0.602
- log 5 = log (10 / 2) = log 10 - log 2 = 1 - 0.301 = 0.699
- log 6 = log (2 x 3) = log 2 + log 3 = 0.301 + 0.477 = 0.778
- log 7 = 0.845
- log 8 = log (2 x 2 x 2) = log 2 + log 2 + log 2 = 0.301 x 3 = 0.903
- log 9 = log (3 x 3) = log 3 + log 3 = 0.477 + 0.477 = 0.954
- log 10 = 1
With the above list, most of the logarithm numbers can be easily computed mentally.
But do note, however, that logarithm of prime number cannot be done using above method.
Examples are log 11, log 17, etc.
However, we should not be deterred from this as they are the minority and the list above serves to cover most of the logarithm solutions which can be easily done mentally.
:)
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Wednesday, 30 July 2008
Inflation Method In Mental Maths
There are many methods to do mental math division. One method is presented here.
I call it the Inflation method.
The key concept is to multiply the number (denominator) to a "better" or simpler number to operate.
What is these "better" number ?
Examples of "better" number are 10, 20 , or 100 .
Let us do an example to see the process.
34 / 5 = ?
5 when multiplied by 2 gives 10 which is a "better" number.
Therefore to maintain the original math question, we need to multiply the numerator by 2 also.
(34 x 2) / (5 x 2) = 68 / 10 ==> 6.8 ANSWER.
For the Divide by 5 question:
Alternatively, we can short-cut the above step by multiplying the numerator by 2 and later shifting the decimal point of the answer one digit to the left.
NOTE:
- Choosing the multiplication of numerator by 2 is because of the "5" division number,
- Shifting the decimal point to one digit to the left is similar to dividing by 10.
- Denominator is not multiplied by 2 because the decimal point shifting cater for that step.
Example of Decimal Shifting method:
34 / 5==> Numerator: 34 x 2 = 68.0==> Shift decimal point to left: 6.80 (Final answer).
Another example of using Inflation method for Mental division: 26 / 25
Steps:
1) Denominator 25 x 4 = 100 ( a "better" number)
2) Numerator also x 4 ==> 26 x 4 = 104
Answer : 104 / 100 = 1.04 All done mentally!
Visit the Deflation Method in Mental Division for another alternative to solving the maths problem.
.
I call it the Inflation method.
The key concept is to multiply the number (denominator) to a "better" or simpler number to operate.
What is these "better" number ?
Examples of "better" number are 10, 20 , or 100 .
Let us do an example to see the process.
34 / 5 = ?
5 when multiplied by 2 gives 10 which is a "better" number.
Therefore to maintain the original math question, we need to multiply the numerator by 2 also.
(34 x 2) / (5 x 2) = 68 / 10 ==> 6.8 ANSWER.
For the Divide by 5 question:
Alternatively, we can short-cut the above step by multiplying the numerator by 2 and later shifting the decimal point of the answer one digit to the left.
NOTE:
- Choosing the multiplication of numerator by 2 is because of the "5" division number,
- Shifting the decimal point to one digit to the left is similar to dividing by 10.
- Denominator is not multiplied by 2 because the decimal point shifting cater for that step.
Example of Decimal Shifting method:
34 / 5==> Numerator: 34 x 2 = 68.0==> Shift decimal point to left: 6.80 (Final answer).
Another example of using Inflation method for Mental division: 26 / 25
Steps:
1) Denominator 25 x 4 = 100 ( a "better" number)
2) Numerator also x 4 ==> 26 x 4 = 104
Answer : 104 / 100 = 1.04 All done mentally!
Visit the Deflation Method in Mental Division for another alternative to solving the maths problem.
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Mental Division With Denominator Focus
There are numerous ways to solve and simplify a math problem.
A simple math division can also be done differently, mentally or otherwise.
It boils down to selecting the appropriate approach.
It also depends on one's preference in solving the math question, and one's confident level.
To review another mental division focusing on the numerator, click this link for information.
The mental methods focusing on the denominator are listed below.
Number divided by 4: y / 4 ==> Do y / 2 twice ==> y/2 and y/2
Number divided by 5: y / 5 ==> Do y times 2, followed by divide by 10. Actually 2y / 10 = y / 5
Number divided by 6: y / 6 ==> Do y / 3 followed by y / 2
Number divided by 8: y / 8 ==> Do y / 3 thrice since 2 x 2 x 2 = 8
Number divided by 9: y / 9 ==> Do y / 3 twice since 3 x 3 = 9
The strategy for this mental math division is to break down the denominator to its lowest factor and perform the operation a number of times.
You may based on the knowledge that dividing by, example, 2 is simpler than dividing straight from 8, which being a bigger number, is harder to handle.
The concept of this multiple simpler division makes mental division easier and less stressful.
If possible involve the use of the number 10 which everyone should find easy to manipulate in the mind.
:-)
.
A simple math division can also be done differently, mentally or otherwise.
It boils down to selecting the appropriate approach.
It also depends on one's preference in solving the math question, and one's confident level.
To review another mental division focusing on the numerator, click this link for information.
The mental methods focusing on the denominator are listed below.
Number divided by 4: y / 4 ==> Do y / 2 twice ==> y/2 and y/2
Number divided by 5: y / 5 ==> Do y times 2, followed by divide by 10. Actually 2y / 10 = y / 5
Number divided by 6: y / 6 ==> Do y / 3 followed by y / 2
Number divided by 8: y / 8 ==> Do y / 3 thrice since 2 x 2 x 2 = 8
Number divided by 9: y / 9 ==> Do y / 3 twice since 3 x 3 = 9
The strategy for this mental math division is to break down the denominator to its lowest factor and perform the operation a number of times.
You may based on the knowledge that dividing by, example, 2 is simpler than dividing straight from 8, which being a bigger number, is harder to handle.
The concept of this multiple simpler division makes mental division easier and less stressful.
If possible involve the use of the number 10 which everyone should find easy to manipulate in the mind.
:-)
.
How Is Mental Division Done
Ever wonder how is mental division done?
You can read on to find out more...
To illustrate, let us do an example of 40 / 7
Mentally solving:-
Step 1:
Break up as much as possible the original dividend 40 in term of 7 ==> (7 X 5) + 5
Step 2:
Divide the result in step 1 by original divisor 7, giving ==> (7 x 5 )/7 + 5/7 = 5 + 5/7
Remark: ( a + b ) / c ==> a / c + b / c
Answer : 5 5 / 7
-------------------------------------------------------------------**
One more example: 38 / 4
Mentally solving: 39 / 4 ==> (4 x 9) / 4 + 3/4 ==> 9 + 3/4 ===> 9 3/4
The idea is to convert the improper fraction to a mixed number.
Simple isn't it?
Maths is non-threatening if you are able to see the trick.
In fact, it becomes interesting as a result of its challenging nature.
.
You can read on to find out more...
To illustrate, let us do an example of 40 / 7
Mentally solving:-
Step 1:
Break up as much as possible the original dividend 40 in term of 7 ==> (7 X 5) + 5
Step 2:
Divide the result in step 1 by original divisor 7, giving ==> (7 x 5 )/7 + 5/7 = 5 + 5/7
Remark: ( a + b ) / c ==> a / c + b / c
Answer : 5 5 / 7
-------------------------------------------------------------------**
One more example: 38 / 4
Mentally solving: 39 / 4 ==> (4 x 9) / 4 + 3/4 ==> 9 + 3/4 ===> 9 3/4
The idea is to convert the improper fraction to a mixed number.
Simple isn't it?
Maths is non-threatening if you are able to see the trick.
In fact, it becomes interesting as a result of its challenging nature.
.
Tuesday, 29 July 2008
Mental Multiplication of 2 digits by 2 digits
Mentally multiplying one digit by one digit is simple (normally memorising the answers).
How about mentally multiplying 2 digits by one digit? Maybe OK.
Principle of 2 digit by one digit mental multipication:
- Expand the 2 digit to include a 10's number.
Example: 45 ==> 40 + 5
Let's do an example: 45 x 6
Mental solution: (40 + 5) x 6 ==> (40 x 6) + (5 x 6) ==> 240 + 30 ===> 270 (ANSWER)
Mentally multiplying 2 digit by 2 digit ?
Can also be done using the same splitting math principle.
Example: 45 x 12
Mental Solution
Step 1: Split the 45 to (40 + 5)
Step 2: Perform 40 x 12 ==> 10 x 4 x 12 ==> 10 x 48 ==> 480
Step 3: Perform 5 x 12 ==> 60
Step 4: Add up the above 2 results, 480 + 60 = 540 (ANSWER)
Concept:
Split the original to the 10's number and use addition instead of direct multiplication.
Just remember that it is always easier to manage 10's multiplication and addition.
.
How about mentally multiplying 2 digits by one digit? Maybe OK.
Principle of 2 digit by one digit mental multipication:
- Expand the 2 digit to include a 10's number.
Example: 45 ==> 40 + 5
Let's do an example: 45 x 6
Mental solution: (40 + 5) x 6 ==> (40 x 6) + (5 x 6) ==> 240 + 30 ===> 270 (ANSWER)
Mentally multiplying 2 digit by 2 digit ?
Can also be done using the same splitting math principle.
Example: 45 x 12
Mental Solution
Step 1: Split the 45 to (40 + 5)
Step 2: Perform 40 x 12 ==> 10 x 4 x 12 ==> 10 x 48 ==> 480
Step 3: Perform 5 x 12 ==> 60
Step 4: Add up the above 2 results, 480 + 60 = 540 (ANSWER)
Concept:
Split the original to the 10's number and use addition instead of direct multiplication.
Just remember that it is always easier to manage 10's multiplication and addition.
.
Monday, 28 July 2008
Mental Squaring | (a - b) (a + b) approach
Of the 3 approaches in mental number squaring, this approach is more conceptual.
Have an overview of the 3 approaches here.
Principle: (a - b) (a + b) = a2 - b2
But note that "b" is the difference to make the original number go to a 10's.
Example: 24 ==> a = 24 (original number), b = 4 (to make the original 24 go to 20)
However, note also, the principle has caused a new term "-b2" to appear.
Therefore to maintain the original squaring, we need to offset the new term with a "+b2".
The new formula to do mental squaring by the (a - b)(a + b) approach is:
a2 = (a - b) (a + b) + b2
Example of usage: 342
The 342 = (34 - 4)(34 + 4) + 42 letting b = 4.
Mentally multiplying (30)(38) can be easy ==> 1140 (click this link for method)
Finally, adding the b2 = 42 = 16 ==> 1140 +16 = 1156 (ANSWER).
:-) Happy?
.
Have an overview of the 3 approaches here.
Principle: (a - b) (a + b) = a2 - b2
But note that "b" is the difference to make the original number go to a 10's.
Example: 24 ==> a = 24 (original number), b = 4 (to make the original 24 go to 20)
However, note also, the principle has caused a new term "-b2" to appear.
Therefore to maintain the original squaring, we need to offset the new term with a "+b2".
The new formula to do mental squaring by the (a - b)(a + b) approach is:
a2 = (a - b) (a + b) + b2
Example of usage: 342
The 342 = (34 - 4)(34 + 4) + 42 letting b = 4.
Mentally multiplying (30)(38) can be easy ==> 1140 (click this link for method)
Finally, adding the b2 = 42 = 16 ==> 1140 +16 = 1156 (ANSWER).
:-) Happy?
.
Mental Squaring using (a - b)^2 approach
In this (a - b)2 approach to mental number squaring, the concept is to split the original number to one having 10's. It is similar to the ( a + b)2 approach but differs in the expanded expression.
Principle: (a - b)2 = a2 - 2ab + b2
Example: 342
Step 1: Split 34 into 40 - 6
Step 2: Replace 342 by (40 - 6)2
Step 3: Expand Step 2. 402 - 2(40)(6) + 62
Mentally it is easy to do 402 = 1600.
Mentally it is also simple to do 62 = 36.
Adding the above 2 results mentally is also easy, 1600 + 36 = 1636, with the "00" aiding the process.
Next, we need to perform the centre term "2ab" ==> 2(40)(6) = 480
Subtracting this last maths operation result of 480 from the 1636 gives 1156 (ANSWER).
The last operation of subtracting is the obstacle in speed compared to the other approaches in Mental Number Squaring.
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Principle: (a - b)2 = a2 - 2ab + b2
Example: 342
Step 1: Split 34 into 40 - 6
Step 2: Replace 342 by (40 - 6)2
Step 3: Expand Step 2. 402 - 2(40)(6) + 62
Mentally it is easy to do 402 = 1600.
Mentally it is also simple to do 62 = 36.
Adding the above 2 results mentally is also easy, 1600 + 36 = 1636, with the "00" aiding the process.
Next, we need to perform the centre term "2ab" ==> 2(40)(6) = 480
Subtracting this last maths operation result of 480 from the 1636 gives 1156 (ANSWER).
The last operation of subtracting is the obstacle in speed compared to the other approaches in Mental Number Squaring.
.
Mental Squaring using (a + b)^2 approach
In this (a + b)2 approach to mental number squaring, the trick is to split the original number to one having 10's. After which, apply the (a + b)2 expansion to solve the multiplication.
Concept: Dealing with 10's is simpler than dealing with non-10's.
Principle: (a + b)2 = a2 + 2ab + b2
Example: 342
Step 1: Split the 34 into 30 + 4
Step 2: Replace the 342 by (30 + 4)2
Step 3: Expand Step 2. 302 + 2(30)(4) + 42
Mentally it is easy to do the 302 ==> 900.
Mentally it is also easy to do the 42 ==> 16.
It is also easy to mentally add up the above 2 results ==> 900 + 16 = 916.
Next, we need to multiply the centre term, which is the "2ab" part ==> 2(30)(4) ==> 240
Mentally we are able to add, the 916 to the last maths operation 240.
916 + 240 = 1156 (ANSWER).
Simple isn't it!
What we have done is to split the original number to a 10's and simplified the mental processing.
:-)
To see other ways to do number squaring mentally, click this link.
.
Concept: Dealing with 10's is simpler than dealing with non-10's.
Principle: (a + b)2 = a2 + 2ab + b2
Example: 342
Step 1: Split the 34 into 30 + 4
Step 2: Replace the 342 by (30 + 4)2
Step 3: Expand Step 2. 302 + 2(30)(4) + 42
Mentally it is easy to do the 302 ==> 900.
Mentally it is also easy to do the 42 ==> 16.
It is also easy to mentally add up the above 2 results ==> 900 + 16 = 916.
Next, we need to multiply the centre term, which is the "2ab" part ==> 2(30)(4) ==> 240
Mentally we are able to add, the 916 to the last maths operation 240.
916 + 240 = 1156 (ANSWER).
Simple isn't it!
What we have done is to split the original number to a 10's and simplified the mental processing.
:-)
To see other ways to do number squaring mentally, click this link.
.
Squaring Number Mentally
Squaring a number can be challenging with paper and pen. It is even so when done mentally using the conventional right-to-left method.
For small number, it may not be difficult. But how about big 2-digit numbers?
The squaring may pose a great task!
Try doing 242.
With the conventional method, it will take sometime and also with the answer starting from the one's (the undesired reverse presentation). And accident-prone too!
Here, I propose 3 simple approaches to do the number squaring:
1) Use the principle (a + b)2 = a2 + 2ab + b2
2) Use the principle (a - b) 2 = a2 - 2ab + b2
3) Use the principle (a - b) (a + b) = a2 - b2
For these 3 approaches, the catch is to split the original number to one containing 10's.
Example: 24 ==> 20 + 4
By splitting the original number to a simpler 10's, we can apply any of the 3 ways above mentally to solve the number square. E.g. 24 ==> a = 20, b = 4.
Merit and demerit of first two mental approaches:
- Can start straight away with the mental calculation but may be slowed down at the last part in the 2ab processing.
Merit and demerit of third mental approach:
- Simple and fast at the end processing part, but slow at the initial splitting .
.
For small number, it may not be difficult. But how about big 2-digit numbers?
The squaring may pose a great task!
Try doing 242.
With the conventional method, it will take sometime and also with the answer starting from the one's (the undesired reverse presentation). And accident-prone too!
Here, I propose 3 simple approaches to do the number squaring:
1) Use the principle (a + b)2 = a2 + 2ab + b2
2) Use the principle (a - b) 2 = a2 - 2ab + b2
3) Use the principle (a - b) (a + b) = a2 - b2
For these 3 approaches, the catch is to split the original number to one containing 10's.
Example: 24 ==> 20 + 4
By splitting the original number to a simpler 10's, we can apply any of the 3 ways above mentally to solve the number square. E.g. 24 ==> a = 20, b = 4.
Merit and demerit of first two mental approaches:
- Can start straight away with the mental calculation but may be slowed down at the last part in the 2ab processing.
Merit and demerit of third mental approach:
- Simple and fast at the end processing part, but slow at the initial splitting .
.
More Tips On Mental Multiplication
If you are longing for more tips on mental multiplication, you have arrived at the correct post!
In this post, you will learn to do ABC x 33 type of mental multiplication.
This type of mental multiplication calls for the A x 11 category of calculation.
To have a review of A x 11 type of multiplication , please visit this link.
Example: 45 x 33
This can be separated into 45 x 3 x 11.
Doing 45 x 3 is simple.
Step: 40 x 3 = 120. Next, 5 x 3 = 15. Add the 120 to 15 = 135.
(You can view this post for a quick review of mental multiplication in 2 digits by 1 digit. )
Next, we need to do the x 11 part.
135 x 11 = 1 4 8 5 directly and mentally.
How did we get the digit 4 and 8 in the final answer 1 4 8 5?
Look at the answer 1 4 8 5.
The digit 1 in the final answer is the original 1 of the 135.
The digit 4 is the addition of the 1 and 3 of 135.
The digit 8 is the addition of the 3 and 5 of 135.
The last digit 5 is the original 5 in the 135.
With consistent practice, mental multiplication can be fast and plain sailing. It saves you time and will also boost your confidence.
Pick this skill up by training your mind for it.
Reading about it will also enhance your knowledge in this field.
Enjoy yourself using mental mathematics.
:-)
.
In this post, you will learn to do ABC x 33 type of mental multiplication.
This type of mental multiplication calls for the A x 11 category of calculation.
To have a review of A x 11 type of multiplication , please visit this link.
Example: 45 x 33
This can be separated into 45 x 3 x 11.
Doing 45 x 3 is simple.
Step: 40 x 3 = 120. Next, 5 x 3 = 15. Add the 120 to 15 = 135.
(You can view this post for a quick review of mental multiplication in 2 digits by 1 digit. )
Next, we need to do the x 11 part.
135 x 11 = 1 4 8 5 directly and mentally.
How did we get the digit 4 and 8 in the final answer 1 4 8 5?
Look at the answer 1 4 8 5.
The digit 1 in the final answer is the original 1 of the 135.
The digit 4 is the addition of the 1 and 3 of 135.
The digit 8 is the addition of the 3 and 5 of 135.
The last digit 5 is the original 5 in the 135.
With consistent practice, mental multiplication can be fast and plain sailing. It saves you time and will also boost your confidence.
Pick this skill up by training your mind for it.
Reading about it will also enhance your knowledge in this field.
Enjoy yourself using mental mathematics.
:-)
.
Mental Multiplication (2 by 1 digit), the Fast Way
Mental multiplication of two-digit number by a one-digit number can be done easily if you know the technique.
The key concept is to simplify the numbers before solving for the numerical answer.
Let me show you the steps.
Example: 32 x 6
The number 32 can be broken up into 30 + 2.
Therefore, we can do 30 x 6 first. 30 x 6 = 180
Next, we do 2 x 6 = 12.
Finally, just add up the two multiplied numbers 180 + 12 ===> 192 (ANSWER)
Here, what we have done is to obtain a simpler 30 for multiplication instead of a 32. After which addition is perform. Addition is always deemed to be easier to handle mentally compared to multiplication.
Another example:
53 x 7 ===> (50 + 3) x 7
50 x 7 = 350
03 x 7 = 021
Add up 350 + 021 = 371 (answer)
From the above two examples, you can see that by splitting the number (in the main question) into simpler manageable numbers, you can mentally perform the multiplication easily, thus, faster and accurately.
For read more on mental maths through simplification, you may like to visit this post by clicking here.
.
The key concept is to simplify the numbers before solving for the numerical answer.
Let me show you the steps.
Example: 32 x 6
The number 32 can be broken up into 30 + 2.
Therefore, we can do 30 x 6 first. 30 x 6 = 180
Next, we do 2 x 6 = 12.
Finally, just add up the two multiplied numbers 180 + 12 ===> 192 (ANSWER)
Here, what we have done is to obtain a simpler 30 for multiplication instead of a 32. After which addition is perform. Addition is always deemed to be easier to handle mentally compared to multiplication.
Another example:
53 x 7 ===> (50 + 3) x 7
50 x 7 = 350
03 x 7 = 021
Add up 350 + 021 = 371 (answer)
From the above two examples, you can see that by splitting the number (in the main question) into simpler manageable numbers, you can mentally perform the multiplication easily, thus, faster and accurately.
For read more on mental maths through simplification, you may like to visit this post by clicking here.
.
Sunday, 27 July 2008
Mental Subtraction - The Fast Way
Mental Subtraction is similar to Mental Addition with a bit of deviation.
The steps to compute the answer to the subtraction is still the unconventional left-to-right approach. This allows the answer to the mental subtraction to be recalled straight from the left digit.
We can do an example using the left-to-right approach.
5 4 3
3 2 2 -
------
We start from left digit by doing 5 - 3 = 2, next, the centre digit 4 - 2 = 2,
and finallyright digit 3 - 2 = 1==> Answer is 221.
You can get this with the conventional right-to-left approach but need to flip the final answer to get 221.
How about this subtraction which is a bit more challenging?
4 0 0
3 1 6 -
-------
This calls for a bit of deviation and creativity.
Mental maths operation lies in simplification to achieve speed. In the above last question, we can actually simplify the 316 by splitting it to 300 + 10 + 6.
This makes the mental subtraction easier in that the first subtraction deals with a simplier number of 300.
Later the answer is followed by subtracting 10 and lastly subtracting 6.
400 - 300 = 100
100 - 10 = 90
90 - 6 = 84
That is the final answer !
See how amazingly simple mental subtraction can be when simplified.
Another example of mental subtraction (with deviation)
4 0 0
2 9 4 -
------
For this question, we note that 294 is very close to 300.
So we mentally subtract using the 300 ==> 400 - 300 = 100
But since we over-subtract by 6, we return 6 to the above subtracted answer
===> 100 + 6 = 106.
That it, the final answer done mentally!
With practice using some creativity, mental maths can be fun and fast, and impressive too.
See another method of simplification for mental maths at this link.
.
The steps to compute the answer to the subtraction is still the unconventional left-to-right approach. This allows the answer to the mental subtraction to be recalled straight from the left digit.
We can do an example using the left-to-right approach.
5 4 3
3 2 2 -
------
We start from left digit by doing 5 - 3 = 2, next, the centre digit 4 - 2 = 2,
and finallyright digit 3 - 2 = 1==> Answer is 221.
You can get this with the conventional right-to-left approach but need to flip the final answer to get 221.
How about this subtraction which is a bit more challenging?
4 0 0
3 1 6 -
-------
This calls for a bit of deviation and creativity.
Mental maths operation lies in simplification to achieve speed. In the above last question, we can actually simplify the 316 by splitting it to 300 + 10 + 6.
This makes the mental subtraction easier in that the first subtraction deals with a simplier number of 300.
Later the answer is followed by subtracting 10 and lastly subtracting 6.
400 - 300 = 100
100 - 10 = 90
90 - 6 = 84
That is the final answer !
See how amazingly simple mental subtraction can be when simplified.
Another example of mental subtraction (with deviation)
4 0 0
2 9 4 -
------
For this question, we note that 294 is very close to 300.
So we mentally subtract using the 300 ==> 400 - 300 = 100
But since we over-subtract by 6, we return 6 to the above subtracted answer
===> 100 + 6 = 106.
That it, the final answer done mentally!
With practice using some creativity, mental maths can be fun and fast, and impressive too.
See another method of simplification for mental maths at this link.
.
Solving Mental Math Questions through Simplification
Math questions can be solved mentally through many ways or methods.
(This is provided no specific method is spelled out in the math question itself).
Selection of the method to be used depends on our liking. If we found one method suitable for us, it will make the math solution easy and encouraging.
Different people has different preference.
Therefore choosing one method over another is not wrong.
There are 2 main approaches to solving mental math question.
- Solving directly which may be a faster way but care has to be taken as it may be complex and involves many variables and maths relation.
- Solving by simplification.
Let me cite an example (the focus of this post).
Example: Calculate MENTALLY 24 x 5
In mental math, multiplying using 5 is known to be more difficult than using 10.
We also know that 10 = 5 x 2, so let's use multiply by 10 first.
Step 1:
24 X 10 = 240.This is not the answer since it is x10, but since 5 = 10 / 2, we divide the 240 by 2.We again know that divide by 2 is also simple.
Step 2:
Divide the answer in step 1 by 2, 240 / 2 = 120(This is the final answer for 24 x 5)
What we have done is to use 2 simple numbers, that is 10 and later 2, to perform multiplication using 5 which may be mentally difficult to do.
We have converted a more difficult direct mental solving into 2 simple math steps that is less prone to error and fast.
The message here is we can choose to solve mental math questions using something that we are comfortable with by using the simplification method.
You see the nice part about choosing the right strategy?
You wouldn't get old mentally practicing mental maths!
:-)
.
(This is provided no specific method is spelled out in the math question itself).
Selection of the method to be used depends on our liking. If we found one method suitable for us, it will make the math solution easy and encouraging.
Different people has different preference.
Therefore choosing one method over another is not wrong.
There are 2 main approaches to solving mental math question.
- Solving directly which may be a faster way but care has to be taken as it may be complex and involves many variables and maths relation.
- Solving by simplification.
Let me cite an example (the focus of this post).
Example: Calculate MENTALLY 24 x 5
In mental math, multiplying using 5 is known to be more difficult than using 10.
We also know that 10 = 5 x 2, so let's use multiply by 10 first.
Step 1:
24 X 10 = 240.This is not the answer since it is x10, but since 5 = 10 / 2, we divide the 240 by 2.We again know that divide by 2 is also simple.
Step 2:
Divide the answer in step 1 by 2, 240 / 2 = 120(This is the final answer for 24 x 5)
What we have done is to use 2 simple numbers, that is 10 and later 2, to perform multiplication using 5 which may be mentally difficult to do.
We have converted a more difficult direct mental solving into 2 simple math steps that is less prone to error and fast.
The message here is we can choose to solve mental math questions using something that we are comfortable with by using the simplification method.
You see the nice part about choosing the right strategy?
You wouldn't get old mentally practicing mental maths!
:-)
.
Friday, 25 July 2008
Improve Your Mind With Some Simple Math Games
Mathematics has long been recognized for it's mental benefits.
Working with numbers improves your concentration, memory, focus, problem solving skills, and general clarity of thought.
To enjoy these benefits, you don't have to indulge in any complicated formulas. All you need is a few minutes daily practice playing some simple math games.
And before you rush out to buy the latest Xbox console and software, realize that numbers are all around you...
Look at the clock on your computer. Usually it's located in the lower right-hand corner of your screen (or use any clock to tell the time).
The 24-hour format works best. On my computer right now, the time is 15:38.
There are all kinds of creative games you can play with this. Here are ten to get you started:
#1 Add the single digits together from left to right:1 + 5 + 3 + 8 = (say "6... 9... 17")
#2 Add the single digits together from right to left:8 + 3 + 5 + 1 = (say "11... 16.. 17")
#3 Add the inner and outer digits together, then add the resulting pairs together:1 + 8 = 95 + 3 = 8and so 9 + 8 = 17
#4 Add the single digits on either side of the colon, and multiply the results:1 + 5 = 6 and 3 + 8 = 116 x 11 = 66
#5 Subtract the single digits on either side of the colon, and multiply the results:1 from 5 = 43 from 8 = 54 x 5 = 20
#6 Multiply the single digits on either side of the colon, and multiply the results:1 x 5 = 5 and 3 x 8 = 245 x 24 = 120
#7 Add the two-digit numbers either side of the colon:Add 15 to 38 to get 53
#8 Subtract the two-digit numbers either side of the colon:Subtract 15 from 38 to get 23
#9 Divide the two-digit numbers either side of the colon:Divide 15 into 38 to get 2 remainder 8
#10 Feeling brave? Multiply the two-digit numbers either side of the colon:Multiply 15 by 38 to get... 570
You can repeat the above exercises as many times a day as you like.
Try them anytime you have a spare minute, like when you're placed on hold in a telephone queue.
You may not turn into a mathematical genius, but you'll certainly keep your brain in gear!
ABOUT THE AUTHOR: Murdo Macleod is co-author of the popular "Fun With Figures" mental math course, which shows anyone aged between 8 and 80 the easy way to do impressive mental calculations. Visit the website today for more details at: http://www.FunWithFigures.com
Source: www.ArticleTrader.com
http://www.articletrader.com/self-improvement/improve-your-mind-with-some-simple-math-games.html
.
Working with numbers improves your concentration, memory, focus, problem solving skills, and general clarity of thought.
To enjoy these benefits, you don't have to indulge in any complicated formulas. All you need is a few minutes daily practice playing some simple math games.
And before you rush out to buy the latest Xbox console and software, realize that numbers are all around you...
Look at the clock on your computer. Usually it's located in the lower right-hand corner of your screen (or use any clock to tell the time).
The 24-hour format works best. On my computer right now, the time is 15:38.
There are all kinds of creative games you can play with this. Here are ten to get you started:
#1 Add the single digits together from left to right:1 + 5 + 3 + 8 = (say "6... 9... 17")
#2 Add the single digits together from right to left:8 + 3 + 5 + 1 = (say "11... 16.. 17")
#3 Add the inner and outer digits together, then add the resulting pairs together:1 + 8 = 95 + 3 = 8and so 9 + 8 = 17
#4 Add the single digits on either side of the colon, and multiply the results:1 + 5 = 6 and 3 + 8 = 116 x 11 = 66
#5 Subtract the single digits on either side of the colon, and multiply the results:1 from 5 = 43 from 8 = 54 x 5 = 20
#6 Multiply the single digits on either side of the colon, and multiply the results:1 x 5 = 5 and 3 x 8 = 245 x 24 = 120
#7 Add the two-digit numbers either side of the colon:Add 15 to 38 to get 53
#8 Subtract the two-digit numbers either side of the colon:Subtract 15 from 38 to get 23
#9 Divide the two-digit numbers either side of the colon:Divide 15 into 38 to get 2 remainder 8
#10 Feeling brave? Multiply the two-digit numbers either side of the colon:Multiply 15 by 38 to get... 570
You can repeat the above exercises as many times a day as you like.
Try them anytime you have a spare minute, like when you're placed on hold in a telephone queue.
You may not turn into a mathematical genius, but you'll certainly keep your brain in gear!
ABOUT THE AUTHOR: Murdo Macleod is co-author of the popular "Fun With Figures" mental math course, which shows anyone aged between 8 and 80 the easy way to do impressive mental calculations. Visit the website today for more details at: http://www.FunWithFigures.com
Source: www.ArticleTrader.com
http://www.articletrader.com/self-improvement/improve-your-mind-with-some-simple-math-games.html
.
Thursday, 24 July 2008
How To Easily Multiply By 11 Mentally (Video)
The maths video posted here demonstrates a very interesting technique to easily handle multiplication of a number by eleven.
View it to understand what I am talking about.
.
View it to understand what I am talking about.
.
Brain Development With Abacus Mental Math
Children have an amazing ability to learn, but their vast brain potential is not always nurtured to the fullest extent.
With the proper guidance and tools, children as young as 4 or 5 are capable of mastering mathematical skills and calculating ability that will yield benefits that last a lifetime.
“Learning Mathematics with the Abacus” is a set of books offering simple, enjoyable instructions for using the abacus, an ancient calculating device that provides modern children with valuable mental stimulation and proficiency in mathematics.
Scientific analyses indicate abacus training can improve a child's ability to:
concentrate, visualize, memorize, observe, and process information.
But how can it accomplish all that, and so much more?
Our brain has two hemispheres, the left brain and the right brain.
About 95% of our children use only the left brain, which provides the ability to analyze information concerning languages and sound.
But the right side of the brain, which is focused on thinking, creativity and integration of information, needs to be used and stimulated as well.
Learning to use the abacus can help develop this right brain/left brain integration.
When children use both hands to move abacus beads in arithmetic calculations, it stimulates cells in both the right and left sides of the brain.
This results in quick, balanced whole brain development, leading to greater mental capacity.
Using the abacus, a child can do all arithmetic calculations up to 10 digits without relying on an electronic calculator.
Nurtureminds publishes "Learning Mathematics with the Abacus" as well as abacus and mental arithmetic reference materials.
We offer three books with simple step-by-step instructions that make learning the abacus enjoyable.
Beginners use the Learning Mathematics with the Abacus Year 1 textbook and activity book to start adding and subtracting numbers up to 100. They begin by identifying the different parts of the abacus, holding and using it correctly, mastering the right fingering technique in moving the beads, and learning to visualize as they calculate.
The Learning Mathematics with the Abacus Year 2 textbook focuses on addition and subtraction of numbers up to 1,000, and develops addition and multiplication skills.
These books have been used by tens of thousands of students in Malaysia and many other nations. They are regarded as the best abacus learning books for children on the market.
Activities in these books have been carefully designed and structured by our panel of academicians, curriculum specialists and instructional designers to ensure that pupils not only learn mathematics effectively, but also develop the ability to perform mental calculations.
With these books, you can help your child achieve more than just math skills. You can boost your child's confidence, provide a sense of achievement, promote intuitive thinking, enhance problem-solving capability, enhance creativity, and improve concentration and mental endurance.
Find out more on why our books are becoming increasingly popular in many countries like Malaysia, the United States, the United Kingdom, Australia, Canada, India, Singapore and elsewhere. They have become valuable teaching tools in schools, tuition centers and community centers, and are used by homeschooling parents around the world.
By: NurtureMinds
Article Directory: http://www.articledashboard.com
www.NurtureMinds.com/ is owned by Dhaval Shrimankar, a private tutor based in Fairfax VA.
With the proper guidance and tools, children as young as 4 or 5 are capable of mastering mathematical skills and calculating ability that will yield benefits that last a lifetime.
“Learning Mathematics with the Abacus” is a set of books offering simple, enjoyable instructions for using the abacus, an ancient calculating device that provides modern children with valuable mental stimulation and proficiency in mathematics.
Scientific analyses indicate abacus training can improve a child's ability to:
concentrate, visualize, memorize, observe, and process information.
But how can it accomplish all that, and so much more?
Our brain has two hemispheres, the left brain and the right brain.
About 95% of our children use only the left brain, which provides the ability to analyze information concerning languages and sound.
But the right side of the brain, which is focused on thinking, creativity and integration of information, needs to be used and stimulated as well.
Learning to use the abacus can help develop this right brain/left brain integration.
When children use both hands to move abacus beads in arithmetic calculations, it stimulates cells in both the right and left sides of the brain.
This results in quick, balanced whole brain development, leading to greater mental capacity.
Using the abacus, a child can do all arithmetic calculations up to 10 digits without relying on an electronic calculator.
Nurtureminds publishes "Learning Mathematics with the Abacus" as well as abacus and mental arithmetic reference materials.
We offer three books with simple step-by-step instructions that make learning the abacus enjoyable.
Beginners use the Learning Mathematics with the Abacus Year 1 textbook and activity book to start adding and subtracting numbers up to 100. They begin by identifying the different parts of the abacus, holding and using it correctly, mastering the right fingering technique in moving the beads, and learning to visualize as they calculate.
The Learning Mathematics with the Abacus Year 2 textbook focuses on addition and subtraction of numbers up to 1,000, and develops addition and multiplication skills.
These books have been used by tens of thousands of students in Malaysia and many other nations. They are regarded as the best abacus learning books for children on the market.
Activities in these books have been carefully designed and structured by our panel of academicians, curriculum specialists and instructional designers to ensure that pupils not only learn mathematics effectively, but also develop the ability to perform mental calculations.
With these books, you can help your child achieve more than just math skills. You can boost your child's confidence, provide a sense of achievement, promote intuitive thinking, enhance problem-solving capability, enhance creativity, and improve concentration and mental endurance.
Find out more on why our books are becoming increasingly popular in many countries like Malaysia, the United States, the United Kingdom, Australia, Canada, India, Singapore and elsewhere. They have become valuable teaching tools in schools, tuition centers and community centers, and are used by homeschooling parents around the world.
By: NurtureMinds
Article Directory: http://www.articledashboard.com
www.NurtureMinds.com/ is owned by Dhaval Shrimankar, a private tutor based in Fairfax VA.
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