Wednesday, 30 July 2008
Another sister method of Inflation method is the so-called Deflation Method.
In this Deflation method of Mental Maths division, we need to scale down to a "better" number.
Examples of "better"number can be 2 and 10.
But do note that both number has to be even for the reduction or deflation.
Let us do an example to illustrate the case.
164 / 16
You will notice that 16 can be scaled down to 8 by dividing the denominator 16 by 2.
The numerator has to follow suit too.
Therefore 164 / 2 = 82.
164 / 16 ==> 82 / 8
Using the numerator approach mental division method,
82 / 8 ==> ( 80 + 2) / 8
==> (80 / 8) + ( 2 /8 ) ==> 10 2/8
==> 10 1/4 ANSWER.
These cute mental steps for the division are not complicated.
They make use of "better" number to perform tasks that the mind can handle, that is, use small manageable number to do the maths operation.
It applies the saying "Keep it short and simple!".
With simplicity, the mind will be clearer.
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.
- 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
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.
The followings rules illustrate the point:
1) Odd + Odd = Even number
2) Odd x Odd = Odd number
3) Even + Even = Even number
4) Even x Even = Even number
5) Even x Odd = Even number
6) Even + Odd = Odd number
Though they are basic mathematical concepts, they may trick the mind when one is not careful.
It is a form of self-checking too.
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.
You can read on to find out more...
To illustrate, let us do an example of 40 / 7
Break up as much as possible the original dividend 40 in term of 7 ==> (7 X 5) + 5
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
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
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)
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
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).
Principle: (a - b)2 = a2 - 2ab + b2
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.
Concept: Dealing with 10's is simpler than dealing with non-10's.
Principle: (a + b)2 = a2 + 2ab + b2
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.
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 .
It possess the ability to make any number that multiplies with it looks special.
Example: 123 x 1001
Answer becomes 123123.
Look at the pattern. The original number 123 repeats itself !
This property of number 1001 can be used to do maths trick.
We know that 1001 can be splitted into its prime number 7, 11, and 13.
With this knowledge,
- we can ask someone to come out with a 3-digit number (e.g. 135),
- ask him to repeat the number beside the original (135135)
- ask him to divide the number by 13,
- ask him to divide it again by 11, and lastly
- ask him to divide it by 7.
Without him telling you the answer, tell him your answer (135).
He will be very surprised!
You can customise the number of digits to other quantity also.
Why does the trick works?
By repeating the number besides the original, we are actually doing a 1001 multiplication.
abc(1001) = abc (100 + 1) = abc000 + abc = abcabc.
And by doing a division of 13, followed by 11, and 7, we are trying to do a division of 1001.
Therefore abc (1001) / 1001 gives us back the original abc.
Simple maths operations and its number theory can create interesting games and tricks that looks challenging at first. But with creativity and a bit of thinking, the answer to the maths puzzles can be easily realized.
By the way, this 1001 can be taken as binary ! (Not wrong)
It can then be used for other trick.
How can the number 1001 be 9?
Answer: By changing its base from 2 to 10!
Tricky? It is limited by our imagination.
The combination of these items forms a meaning of either a maths relationship or problem.
The same variable can also be written in many ways.
Examples of written forms (with y as the variable):
- numerator (above the dividing line) ==> y / a
- denominator (below the dividing line) ==> a / y
- power of a base (small size font and superscript) ==> ay
- base in logarithm format (small size font and subscript) ==> log y a
From the above few examples, you will realise that writing the mathematical expression can be a bit tricky when its form is not correctly presented.
Confusion may arise when they are not written properly.
Examples of confusion:-
5y being written 5 y.
Is the meaning still the same?
log5A being written as log5A.
Mathematical meaning of the relation changes!
5a(y + 6) written as 5(ay + 6).
Maths expression has been modified !
Why did the confusion or error comes about?
It boils down to the attitude in writing. If one did not write properly, especially in maths, the whole meaning of the expression is lost or misinterpreted.
In maths, the meaning of the relationship between the symbols, variables and operators resides in its rightful presentation.
It is different from the English language where a spelling error can still be recognized with correctness in meaning (not all though, if error is too extensive).
Maths trains a person to write properly and with a certain discipline, in order to retain the original meaning of the maths expression.
Good writing skill is, therefore, developed as a by-product of learning maths.
Clear presentation steps are also enhanced during maths question solving.
From the benefits of the above, it can be seen that learning maths is a very great activity in that, it is, not only,the mathematical content acquired, other skills are also indirectly picked up.
So can we deduce that a good mathematician can write nicely? Your guess....
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.
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.
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.
Here is one simple mathematical manipulation that will amaze your friends.
To pre-release what is going on.....
The trick is by performing some mathematical operations to an unknown number, and asking for the digit in the one's placing after the final step, you can proudly inform the audience or friend the number(answer) of that final step with confidence.
Step 1: Ask someone to say a three-digit number (the digit in the hundred's has to be different from the digit in the one's)
Step 2: Reverse these digits making the one's becoming the hundred's.
Step 3: Subtract the resulting smaller number from the bigger number.
Step 4: Ask the same person for the last digit.
i) To get the digit in the hundred's position, subtract the digit requested for in step 4 from nine.
ii) The centre digit is always nine after all the mathematical operations.
Unknown number : 741
Reversed number: 147
Subtracted number: 741 - 147 = 5 9 4
Digit in one's position: 4
Digit in hundred's position: 9 - 4 = 5
Centre digit: Always 9
Answer without knowing the final subtracted number but only knowing the last digit 4
==> 5 9 4
Try out this maths trick. It will guarantee you fun!
Sunday, 27 July 2008
They are purposely made that way, though.
What's the reason?
The answer has to be explained through geometry in mathematics.
For a circle, the distance of its centre point to the outer edge is always a constant.
Comparing this to other type of shapes, for example, rectangle or hexagon, these other shapes do not have this characteristics.
If someone happens to close the lid of the manhole at a tilted angle, the shapes other than the circular one, has a possibility to drop through the manhole.
The circular-shaped lid will never has a chance to slide through and enter the manhole at all!
The constant radius of a circle ensured that there will not be any gaps in all possible condition between the lid and the manhole opening.
With this reason, the manhole are made circular in shape.
This theory relates geometry to problem-solving of real-world matters.
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.
This post is about the way mental addition is done and its benefit.
In school, we are always taught to add numbers starting from the right-sided digit, then move on to the left-sided digit. This is the conventional way of doing addition.
But this has a demerit in that the first mention of a number to the answer starts from the right. We normally says a number starting from the LEFT!
On paper, this right to left method of addition is fine, but to do it mentally, it is a bit tricky.
For mental addition, we need to start addition from the left digit (as opposed to the conventional right side).
Let's start with an example.
1 2 +
We do the 4 + 1 = 5 first.
(Left side first)
Later do 5 + 2 = 7
Answer is 57.
(This is correct based on the traditional right to left addition also)
We can mentally compute this and say out the number directly as we have started with the left digit addition first which coincide with the left side saying.
Try another examle:
1 3 4
4 2 3+
The answer is 557 ! Easy?
Before we can finalised the number to a particular digit, look at the next digit and check if the addition of that 2 numbers in that digit position will be more than 10.
If the added sum of the next digit is more than 10, we need to add 1 to the current digit.
See next example.
Example (next level up):
2 3 6
1 2 5 +
The left digit final sum will be 2 + 1 = 3, and will not change since the addition of the next digit numbers will not be more than 10, therefore no carry over of 1.
Next, we do the addition to the centre digits which gives us 3 + 2 = 5, BUT look at the right digit summation.
It is 6 + 5 = 11 which is more than 10.
This will affect the centre digit final number by adding 1 to the initial 5.
Centre number answer ==> 5 + 1= 6.
Lastly the right digit number will be 1 (taking only the last number in the 6+ 5 = 11 answer).
Answer therefore = 3 6 1.
Mental maths has the advantage that the answer format matches the presentation format, which save time in flipping the final answer as done using the traditional or conventional method of right to left sequence of addition.
(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.
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.
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!
I personally find it interesting enough for me to share it in this post. You can create deviation of this to excite people.
Here it goes.....
The magic number is 1089.
That is the final result of doing a series of mathematical operations using a random number (integer only) that anyone can call out.
1) Start with any number (integer). Let's say 321
2) Reverse the integer. ==> 123
3) Subtract the second number from the original first number. 321 - 123 = 198
4) Note this subtracted answer and reverse it. ==> 891
5) Add the number in step 3 to number in step 4. ===> 198+ 891 = 1089
NOTE: Which ever number you start with, the final numerical answer is always 1089.
You can write this number 1089 on a piece of paper and put it inside a bottle or box on a table.
Identify a target person and request him to shout out any number that comes to his mind.
Let him perform the series of mathematics operation with your guidance. (He should get the final number as 1089.)
Tell him to retrieve the paper inside the bottle or can on the table.
He will be surprised (and impressed!) to see the same answer 1089 written on that piece of paper.
Try this maths trick to amuse yourself and others.
Saturday, 26 July 2008
In English, they are words that spell the same forward and reverse.
In maths , they are number likes 121, 232, 1221, 12321, 11211, etc
We can design our number palindromes using basic mathematical operation:
Answer: 6 2 6 is a number palindromes
Another palindromes in mathematics:
It is an interesting series of number that I like to present in this post.
It begins with the number 1.
12 = 1
112 = 121
1112 = 12321
11112 = 1234321
111112 = 123454321
Is it interesting?
Maths can create many wonderful patterns if we border to hunt for them.
Swimming is a common activity that many people does.
What is its relationship to maths?
The followings are some of its relationships:-
1) Speed of swim (measurement of distance and time)
2) Surface area of palm (area measurement of odd shape)
3) Kicking angle of the legs (trigonometry, angle)
4) Body profile - streamlining ( shape , geometry sector in maths)
5) Rhythm of the stroke (sequence, counting, pattern sector)
6) Breathing (volume of air required, space measurement)
7) Force of stroke (analysis sector, measurement)
8.) Friction of body to water (analysis sector)
The above are items we can discuss during maths lesson or during learning of maths with emphasis of practical application in real life.
This will make maths education more meaningful and enticing to learners. Use as many relationships to maths terms and concepts to bring in linkages so as to better retain the topics learned.
Hard facts are difficult to memorise.
Therefore, with bonding to real-life applications, retention of concepts can be more easily retained. Have fun with maths.
Maths is seemingly getting more interesting now, right?
Click here to have an overview of Sectors of Maths.
One example can be simply riding a bicycle.
In this activity, many things relating to maths can be cited.
Let's see the relation.
To start off, riding and getting the bicycle to move forward steadily, we need the momentum and speed (this is measurement).
How to balance properly while riding ?
(This is related to the EQUAL operator in algebra, left side equals right side).
How far to ride on?
(This is distance measurement and unit term, metres or kilometres?, exponential expression).
Objects can be counted, example, lamp-posts that pass by along the way.
(This is counting and pattern recognition ability).
Braking force and how much to apply?
(This is analysis sector of maths).
Curve turning during cycling is another task to relate to maths.
(This is the geometry sector).
The parts of the bicycle can also be related to maths.
- Diameter of the wheel
- The angle of the seat
- The gear ratio of the pedal
- The strength of the metal support
- The weight of the bicycle
How long can the bicycle last?
(This is related to analysis sector, and probability study)
The above are some examples of citing maths term to actual real-world activity and item. Maths is not an isolated subject that is abstract from real-life.
With practice, we can quote many examples of maths relation to daily activities.
For more details on the numerous Sectors of Maths, view this link.
For another example of maths relation to real-life case, view here.
One case application is a camping trip outdoor. In this activity, we can see the applications of the various sectors of the maths in the tasks involved. It allows the learning of mathematics to be attached to real-life issues.
In preparation for the outdoor trip, the below tasks are involves:
Planning a camping trip includes inventory control, number of members going (number usage sector), what is used to carry the items (measurement sector), and the schedule of tasks (relating and analysing sectors).
Packing of the items into various bags and containers (volume measurement sector) comes next.
The journey (time measurement sector) to the camp site is another relation to maths that can be discussed and emphasised.
The location or direction to the camp site (geometry and measurement sectors) is related to trigonometry, angles, and distance measurement.
Selecting and setting up the camp site (analyzing, problem-solving sectors and number usage sector) involves getting the correct number of members to do various tasks, like pitching the tent (geometry sector - tent layout), and driving the wooden pegs to secure the tent (measurement sector - how deep in).
Cooking for meals involves calculating the individual amount (volume) of food and procedures (fractions, analysing, problem-solving, pattern and measurement sectors).
All the above tasks involve maths relation sector as well, because they are directly related to real-life practical issues.
From this case application of a simple coutdoor trip, we can see that the learning and teaching of maths can be factored into the numerous activities within. This makes future maths lessons and topics easy to accept as maths takes on a different platform with the acknowledgement that maths is everywhere in our daily life and can solve real-life problems.
Maths is reallllly interesting!
First, we need to clarify the name of the terms, for fraction, before we proceed.
Fraction is written as n / d, where the "n" is known as the NUMERATOR and the "d" is the DENOMINATOR. Note: "d" is below the dividing line.
Explanation of fraction representation:1/5 means one part of a whole (which is divided into 5 equal parts).
2/4 means two parts of a whole that consists of 4 equal parts.
In ratio term, 2/4 means the same as 1/2 regardless of physical size.
(Thinking of the cake as an example may helps).
Rule for Addition of Fraction: a / b + c / b = (a + c) / b <== same denominator.
When the denominators are the same, we can simply add the numerators together keeping the denominator as it is. Example: 1/4 + 2/4 = (1 + 2) / 4 = 3 / 4. This is simple.
How about when the denominators are different?
Example: 1 / 4 + 1 / 2 = ?
Tips: Make the denominator for both terms the same so that the rule for fraction addition can be applied.
In this case we have different denominators "4" and "2". We noticed that "2" is related to "4" by 2 times.
Therefore, we can intentionally multiply the second term denominator by 2, which gives us 2 x 2 = 4, and which matches the first term denominator.
But take note, by multiplying the second term denominator by 2, we also need to multiply the second term numerator by the same amount, that is, 2 also.
This is to ensure that we did not change the meaning for the second term 1/2. By multiplying the second term by 2/2, which is 1, we did not change anything since anything multiplied by 1 is still the same as original (in this case, 1/2). Make sense?
So the math example of above,1 / 4 + 1 / 2 becomes 1 / 4 + [(1 x 2) / (2 x 2)] = 1 /4 + 2 / 4 = 3 / 4.
Another example:(2 / 5) + (1 / 10) = [(2 x 2) / (5 x 2)] + 1 /10 = (4 / 10) + (1 / 10) ==> same denominator now!Answer: ( 4 + 1 ) / 10 = 5 / 10 = 1 / 2.
So, to be able to apply the rule of fraction addition, we simply make the denominators for all the terms the same.
This rule of fraction addition applies equally to fraction subtraction also.
Common mistakes made during fraction addition
(s / b) + (s / c ) is not equal to s / ( b + c ) !
This is a very common mistake students make. Only common denominator allows us to add the numerators. Common numerators does not qualify for simply addition of denominators. We need to make the denominators the same before we can add the numerators.
For more information about learning fraction and its implication, click this link.
They may not be easy themselves, but the process to solve these maths questions will not be difficult when the other sectors of maths are handled well. This problem solving sector comes after the Analyzing sector, which builds up skill in deciding the apprioprate approach to a given maths question.
Problem solving is not done by memorising all the facts and steps in solving maths problems. If it is so, then maths education losses its meaning.
Problem solving sector is aim at creating skill in solving problems using principle and facts related to the given problem. With this problem solving skill picked up through maths education, we will be ready to face any problem not even related to maths!
Problem solving ability involves exploring relationship between all the factual sectors (fractions, geometry, patterns, number usage, and measurement) and analyzing through the process to obtain a decent answer. This mathematical solving skill will be used to expand on other non-mathematical skill like language and presentation know-how.
What can be done to boost up Problem Solving skill in maths education?
Below are some of my suggestions:
1) Case studies on certain unworkable projects. Ask the learners to figure out the problems and come up with as many possible solutions to the problems.
2) Set a goal and allow the maths learners to determine the process to achieve the goal. (e.g. building a table with 2 uneven legs, with maths calculations to support solution)
Problem solving sector encompasses all the skills needed to handle maths quesions. It integrates both the thinking and factual part in maths education.
For children new to formal maths learning, the various factual maths sectors have to be mastered before they can proceed to the thinking part, which requires a certain amount of maturity and self-discipline.
To read other sectors related to this sectors and an overview, visit Sectors of Maths.
This maths sector digs out all the factual part in complement with the relating sector skill to enhance or push up the learning level of maths solving and applications.
This analyzing skill helps to determine whether all the facts obtained through linkage are relevant to the solving process and whether they leads to the final solution.
Given a maths problem, the ability to analyze will determine the appropriate steps and maths solving tools to be used. If we can select the optimum method to solve a maths problem, we can solve the problem in a faster manner since less steps are used.
This is useful, for example, in computer porgramming where length of the software programme may decide the solving speed.
Sometimes, we are presented with alot of information to a problem.
The ability to analyze will allow us to dig out suitable facts to incorporate into the solving process.
As such, this analyzing sector is very closely related to the problem solving sector, which is the most important part of maths solving.
How then can we enhance this Analyzing skill in this sector of maths?
My suggestions are listed below:
1) Ask though provoking questions that may not have a direct answer. This forces the maths learners to think and analyze.
2) Do case studies on simple projects to find out their operating principle (if possible get some maths calculations done to prove the method)
3) Purposely create simple errors in work process to start them thinking.
Note: The outcome of the analysis is not important at this stage since the focus is on the analysis skill. Let them learn to think.
If this analyzing sector is carried out nicely, it can be very enjoyable and creates confidence boosting effect. The maths lessons will then be meaningful with positive results. This sector is to have a breakthrough in the thinking habit of the math learners which will greatly benefits anyone learning maths.
To read about other sectors of maths, and their relations to the maths education proper, you can check Sectors of Maths.
Relating sector is one sector that gathers and associates the factual knowledge for math applications.
Relating sector links up hard mathematical facts and skills into useful problem solving of real-life issues.
Only through relating, a person can see a bigger picture or knows the constraints to the problem and the way to solve the problems.
Relating is not limited to math only, but can also be cross-discipline. It can be relating topics in the arts, music, or science, and even feng-shui. This wide relation of math to other areas gives a strong foundation to the use of math for analysis.
How to strengthen this relation sector in maths education?
Some examples are presented below:
1) Say targeted work process from mathematical point of view, to reflect the unknown linkage of mathematical factual parts to the work (e.g. "pour 2 litres of water into the tank to reach the desired water level", instead of saying, "pour 2 pails of water....")
2) Quote and link as many factual parts (geometry, fractions, number usage, measurement, and patterns) into an activity (cooking, drawing, etc)
3) Discuss directly on issues related to math and others (to arouse interest of the vast dimension in math applications to real world problems or needs).
The final objective is to rope in relation of math terms to real-life applications, and allow maths learning to be less abstract. The main objective is to break down all possible barriers to maths learning and ease acceptance of maths lessons.
To read an overview of Sectors of Maths, you can click this link here.
Numbers can be represented in various ways; fractions, whole number, decimal, percentage and many others. For fraction itself, they are again sub-divided into improper fraction, proper fraction and mixed number.
But actually what is the significance of fractions and whole number in math education?
Fractions represent connections to a whole item or system. It shows the relationship between a parent part to a smaller part.
It presents to math learners an idea of sharing and size of the smaller part in relation to the whole (parent) part. Only through understanding the meaning of fractions can one link up mathematical expressions and their variables.
This is more so in algebra manipulation, where division (denominator)and multiplication (numerator) are involved.
Addition and subtraction of fractions can also be easily done when the math learners know the meaning of the physical division.
Suggestions of activities to enhance knowledge in Fractions & Whole numbers sector in math education are listed below:
1) Cutting up a cake into equal pieces
2) Playing with LEGO blocks to form objects
3) Card games where distribution of card from a pile is involved
4) Dough playing where a piece is separated from the original whole
5) Paper craft where cutting shapes to form objects are involved.
These examples are to expose the math students of using fractions to represent real world division of items. With knowledge obtained here and complementing it with other sectors of math, a stronger understanding of math can be builded upon.
You can read an overview of other sectors of math by clicking this link .
The title serves to implies measurement of anything using distance for mental effect.
Why is measurement important to math education?
Firstly, measurement involves planning of what to measure and how to measure.
Secondly, it involves the selection of a suitable unit of measurement and tools of measurement.
These are skills needed for good math analysis and math question / problem solving. The logical and systematical steps involved in measurement can be directly applied to solving math questions.
Measurement item in this post means anything from length, time, angle, colour parameter, volume, weight and other physical properties. It is a way for math learners to relate measured data to the real world.
This measurement sector in math education allows learners to have a perspective of objects and their dimension.
They will be able to (as examples):
a) feel for themselves how hot is the water (temperature),
b) how big is the bench (length),
c) how long must they wait (time),
d) how deep must they dig a well (depth),
e) how fast must they run (speed), and
f) how big is the petrol tank (capacity, volume)
To enhance the skills in measurement sector of math education, below are some suggestions:
1) Involves the learners with daily chores like preparing materials for cooking (volumes learning)
2) Preparing a dinning table for food (layout planning given size of plates)
3) Time keeping for some activities like watching television
4) Taking and recording weight of people for a period of time (for weight control understanding)5) Building a project, example, chair or table (length measurements)
6) Troubleshooting an electronic circuit (electrical parameter measurement)
7) Playing games that involves comparing sizes (like fitting wooden pegs into targets)
8.) Discussing methods of measurement and the tools or equipment needed
9) Art and craft exercises where cutting and placement are involved
10) Checking for tolerance in items measured ( small scale unit )
In the above activities, type of measurement tools are part of the learning objectives.
What are the tools used for measuring?
Is it a slide ruler, try-square, protractor, string, container, electronic meter or a weighting machine?
Let them know the tools available for the measurement and what measurement units these tools give (centimetre, litre, amperes, degree, etc)
All these are part of the important math skills that we, as parent or math teacher, can train them for the sole purpose of mastering math.
They will see the significant of math in those applications mentioned above. And I wish to say this math sector is the easiest to pick up as it is directly linked to many daily activities. Students will fine this exercise fun and enjoyable.
For an overview of other related math sectors, click Numerous Sectors of Math for more details.
What is meant by Geometry sector and why is it important?
Geometry involves the ability to visualise actual shapes, angles, positions and spatial diemension of objects.
It also covers the ability to create the drawings in all possible angular view.
The students in this geometry sector must be able to relate math analysis to actual object creations, like the 3-dimensional view.
They must be skillful enough to describe the appearance of the object or solution with math tools, and relate them to the world physically.
It is with this ability that engineers create wonderful structures like the Great Wall of China or the Tokyo Tower., or the famous Pyramid of Egypt.
But, note that this skill complements the other sectors of math to achieve a complete math education.
Examples of Geometry sector ability:
a) Relating buildings to height calculation
b) Relating the Egyptian Great Pyramid to gradient or slope parameter
c) Drawing a perspective drawing of a room (angle dimensioning)
d) Creating a 3-D picture (physical visualising)
e) Tailoring of clothing (matching of different shapes and dimensions)
What are the activities needed to enhance this geometry aspect of math education?
Examples of activities that can help in developing the skill in geometry sector:
1) Playing buliding toys (e.g. LEGO, wooden blocks) to create physical objects
2) Playing Jigsaw puzzles that requires matching shapes
3) Doing art and craft exercises that need figuring out shapes or curves and bends for objects
4) Playing doughs ( 3-D objct creation and dimensioning)
5) Playing slides (example of gradient) or see-saw (pivot point application in real world)
6) See a movie in a 3-D theatre (space dimensioning effect)
7) Discuss about real life examples in relation to shapes (example water ripple to sinewave in trigonometry)
8.) Building a project given some pre-defined objects like ball, box, stick and objects of various shapes.
9) Packing things ( example: tidying up the bookself with proper arangement of books)
10) Cooking to relate the term "volume" of items in a pot or pan.
Take the chance to let the students explore with shapes and relate them to math terms used in geometry, example, gradient, height, volume and angles.
Use hand-on practical assignments to aid math understanding and better impression of the geometry learning.
In summary, there are alot of ways to enable math students learn geometry. It is the relationship between real world applications that they need to be aware before geometry in math education beomes meaningful.
A brief overview of the other math sectors can be viewed by clicking at Numerous Sectors of Math.
Pattern mentioned here refers to the numerical or pictorial sequence that tests one's intelligence.
This patterns sector in math is important because it trains learners in relating things with some logic or reason, and predicting future events.
It also trains the brain to plan base on previous data available. As such, recognizing pattern is challenging and can easily create interest in math.
Examples of pattern recognizing and its application are:
a) sorting items from smallest to biggest (production line)
b) sorting items from shortest to longest (production line)
c) music rhythm (3 beat, 4 beat)
d) image processing, image recognition and image recovery (from the group of pixels identified )e) error correction in data transmission (from the sequence of received data)
What can be done to enhance this Pattern sector of math education?
A few suggestions are listed below for reference:
1) Figuring out Jigsaw puzzles through recognizing and matching colour and shape properties
2) Listening to music and identifying the beats (anticipating skill)
3) Doing up decorative accessories, like necklace, to be creative in pattern sequencing
4) Playing Number sequence quiz (example: 3, 5, 8, ___ , 17, 23)
5) Checking Calendar date and weeks information (a fixed pattern of days)
6) Talking about real world application related to pattern issue (link to actual examples to let learners know of usefulness of math)
Pattern recognizing is important in math learning as it trains the mind to handle real world issue.
Through learning this pattern sector, math students will have an advantage when solving math questions, since they are able to plan the steps of the solution given the available information.
From the available information in the math questions, they can anticipate or deduce the next step to take, applying the skill learned in recognizing number pattern sequence.
It is not the math content that is the focus, but the skill in planning forward that is important.
Thus this factual part of the math education is critical and prepares the learners for future math challenges.
To read an overview of this math topic, refer to Numerous Sectors of Math for details.
It is the basic foundation where math revolves.
Without number, what is maths?
In this post, let me explain the details and its involvement in math education.
Number usage means the ability to use numbers and "play" with it in any situations that they apply. It also involves recognising the type of numbers for different usage or applications.
It also means knowing the relationships between all the entities involving numbers.
Number Usage implies the ability to count correctly numbers in ascending as well as descending order.
The skill to count at multiples are also part of number usage. Number bonding is also a part of this sector.
Adding, subtracting and understanding any matters related to numbers are defined in this sector.
How to groom learners in this Number Usage sector?
There are many ways (of course!). I have listed some examples below:
1) Do dot-to-dot drawings to reveal the profile of a hidden diagram (counting skill)
2) Play number puzzles (recognizing number pattern)
3) Reading date on calendar (recognizing multiples; week and months)
4) Count steps while climbing staircase (counting skill)
5) Count with real objets to expose learners to real life relationship between math and reality (number relation)
6) Play board games like "snake and ladder" to enhance number sequence (counting, relation skill)
7) Play maths card game (example UNO) (number relation skill)
8.) Count whatever comes to sight (counting skill)
-picking and counting seashells at the seaside,
-count beads while playing,
-counting money by buying things and counting the monetary change
The list is not exhaustive but up to one's imagination.
Do take the chance to expose the usage of numbers to the learners for the sake of letting them relate number to real applications and see meaning in numbers.
This will help them later in their math education.
Refer to Sectors of Maths for a complete review.
In order to handle maths well, students and parents, besides teachers has to know and understand these sectors which I have categorised under 2 parts.
What is these parts / sectors that I am referring to? Read on ....
1) Thinking part:
- Relating sector
- Analyzing sector
- problem solving sector
2) Factual part:
- number usage sector
- patterns sector
- geometry sector
- distance measurement sector
- fractions & whole number sector
These are the basic maths sectors that I believe control most of our maths applications.
Different sectors will need different set of skills to manage the learning and teaching.
Identify the approach to learning (and teaching) maths pertaining to which sector, and I can say that half the war is conquered!
This post will serve as the beginning of a series of posts to discuss in-depth the particular maths sector and what should be done on that sectors.
The auditorical, visual and kinesthetic style of learning are the basic ones.
In the classroom environment, if class management is not handle well by the teaching staff, noise level will definite override the vocal volume of the teacher. Students with preference for the auditory style will suffer as a result.
Though visual type of students do have less impact from the noise generated, they are exposed to only one type of delivery.
To reap full benefits of learning, it is known that we have to use more senses. Thus, abacus comes into the picture.
Abacus is a maths tool that consist of beads.
These beads represent numbers. And by placing these beads in a certain pattern, we can have different numbers represented.
What benefit does it have on maths education?
The physical moving of the beads to their respective position is an action that impacts the brain with the other senses (visual or hearing) aiding it. It is a multi-sensory learning skill.
Visually, students are also able to identify the numbers and have a better understanding of the counts and their relation to physical items. In other words, they find meaning linking numbers to real objects.
This method, however, applies better with young learners where their inquisitive mind are still absorbing anything interesting.
No students like it.
During exam revision period, many maths students are tensed up preparing for the challenge.
There are many ways and strategies to revise.
Working in groups?
Or alone at a secluded location isolated from mankind?
Revised one day before?
Or one month before?
Spotting potential questions or study all?
Actually there is no one way to revise.
The ultimate strategy is still to one's liking. As long as one is comfortable with the method, try it.
If it does not work, explore other method or strategy.
But do not keep on trying.
You know why!
Exam will be over by then, right?
Basically, I would suggest some pointers (as a maths teacher) to overcome this maths exam anxiety:
1) Do more maths questions to recap knowledge acquired
2) Work as a group to leverage on each others' strengths and understanding (depending on one's learning style, may not always suit everyone)
3) Participate more in class quiz or activities led by maths teacher
4) Do not stay late into the night for revision one day before the actual exam
5) Stay fit physically by doing exercise to keep the mind clear
6) Be proactive in learning with positive confidence attitude
The above are pointers that you may follow.
But, the truly best way to learn maths is continuous practice and reading up during normal maths lesson time.
It is especially beneficial if you can read the maths topics taught within one day to keep the newly acquired knowledge fresh.
DO NOT LEAVE everything till the LAST MINUTE!
This is bad for your maths learning. The human brain needs time to digest information over a period of time, and squeezing everything into a short period of time is not only challenging but putting your result at stake.
Why risk it? You do not deserve this!
With proper learning schedule and planning, you CAN enjoy maths and do well in the exam.
Maths anxiety is an undesirable situation anyone learning maths should avoid.
It slows down the progress of maths studies. When you have it, you tend to mentally avoid maths, and anything relating to maths. Some examples are the counting and organising numbers which to any layman are basic daily task. Yet those of you who have "captured" this anxiety will avoid them.
It is detrimental and is spiral in effect. Losing confidence in solving maths questions sets in, and will "strengthen" this weakness and gives assurance that you cannot do maths. Frightening isn't it?
Remove or minimise this anxiety!
How to overcome or minimise it?
First, you have to be aware that this maths anxiety, from professional studies, comes from past experience when dealing with maths.
An example can be that very first maths lesson with a horrible, selfish and uncaring maths teacher.
But the good news is that past experience can be overridden with new positive experience, this maths anxiety is curable.
Replace this undesirable maths anxiety with something positive and stimulating. Something that you can grasp and have fun with, but is still related to maths. Example is the Sudoku electronic maths player.
Positive replacements can also be to clarify maths question till a satisfactory answer is obtained. Speak your mind for the sake of your maths education and the sake of learning.
Do not fear failure or shyness to obtain an answer.
Seek out as much answers to any obstacles as possible.
With more solution obtained, confidence to solve maths problems will be boosted. This will help lower the level of this fearful maths anxiety.
Knowing there is problem and not taking action is detrimental to your maths learning. Take active action, and not passive, being-prompted type of action.
Learning mindset has to change towards maths, which will in turn arouse interest in the subject.
Think positively to motivate yourself.
Boosting up your self-image is a way to reduce, if not remove, this maths anxiety.
Expose to as many avenues or alternatives as possible to enhance yourself in this area.
Be aware that learning through failure is also a very important process of education, especially with maths where the answer is a clear-cut right or wrong answer.
Let the past go, and embrace new changes. This new change in learning attitude is a mind-opener to having better result in maths. Do not fear maths.
Relax and love maths to overcome its anxiety.
Maths is InteRestinG !
Friday, 25 July 2008
It showed the interesting side of playing with maths.
Maths when dealt with on the light side can be very stimulating for actual studies.
Do take a look at this fun video. Sure to amuse you!
And will inspire you to like maths, if you have yet to.
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
I happened to come across an interesting math story that I like to share.
It about a young German boy who did math in a unique and creative way, by simply using simple principles.
I hope that with this inspiring story, anyone reading this story will find math a very amazing creation of mankind.
Here's the story ...
There was a boy in a class studying math with, of course, a math teacher. This boy's name is Carl Friedrich Gauss (1777 - 1855). One day this math teacher presented a challenging mathematical problem to the class where Gauss is in.
The math problem is to add up all the numbers starting from 1 and ending with 100.
Every students picked up a piece of paper and started to add up the numbers one after another from number 1 onwards.
Within a short span of time, while his fellow students were still struggling, Gauss went forward to the teacher and submitted his answer.
That action surprised not only his math teacher but the whole class. But that is not all.....
The interesting thing is that his answer is correct.
How did he do that so fast?
He came out a different way of analysing the mathematical problem. Instead of the normal way of adding the first numbers onwards, Gauss looked at the problem with a different angle.
What he did was to split the range of number from 1 to 100 into two equal halves, 1 to 50 and 51 to 100. He noticed that if he flipped the last half to start from 100, and adding it the two ranges together, he will get something stunting.
He discovered that by adding the first pair, 1 + 100, he got an answer of 101. For the second pair, 2 + 99, he again got the same answer 101.
This answer of 101 was still valid for the rest of the number pair addition. And since there were 50 pairs of numbers, the final total is 101 x 50 which gave Gauss an answer of 5050.
The way he perceived and analyzed the mathematical problem surprised everyone.
From this story, you can see that math is a very interesting subject that tests the limitation of human mind. With different approaches, math solving can achieve a new dimension completely different from convention. This shows that math can be fun and exciting if we choose it to be.
You can beat any calculator and turn into a human computing machine- the wonderful calculator!
Thursday, 24 July 2008
In school, we are made to sit still facing the whiteboard or slides to learn numbers, mathematical expressions and numerous maths equations.
With that attention span will be shortened and focus lost in the midst.
The teacher will get angry with us for not paying any attention.
Is this typical of your maths learning journey also?
Day after day, we are bombarded by maths worksheets, maths lessons, and tutorials...
Is this type of learning proper?
This question is more so, especially when it is the enforced type like done in the army, or worst, the prison!.
There are however, merits as well as demerits in this enforced training style.
Merits of learning maths using enforced discipline:
- Fast coverage of syllabus
- All students learn at the same pace (or slight deviation)
- Attention span are trained to be longer
- Concentration enhanced, the list goes on..
Demerits of the enforced discipline learning:
- Learn for the sake of learning
- Forget after the exam or test
- Learning not internalised
- Low self-esteem
- Guided thinking and no freedom to explore at own pace ....
From the above points, is learning maths through enforced discipline suitable?
You can guess that the answer is a case-by-case issue.
It depends on the character of the teaching, his teaching style, the attitude of the students, the confidence level of the maths learners, the complexity of the maths lesson, the amount of maths tutorials and homework, time frame, expectation level, etc.
It is a complicated issue of matching the style of the teaching staff to the students.
The teacher has to be aware of the demerits of strict discipline in "hijacking" the minds of the students and you, the students has to understand the learning situation of the classroom. It is therefore a give-and-take process of learning maths.
Education is not all about results as do maths education. It is sometimes the hidden soft skills, like tolerance, time management, ability to focus, and appreciation, that is all the more important, and learning maths is just a journey to building a person.
Let's not take maths learning lightly, but treat it with respect and enjoy the meaningful process.
The results they produced can be awesome.
With simple addition and multiplication, sometimes we see shocking maths answer.
But all can be explained if our principles to mathematics is concrete.
Below presents the interesting mathematics operations and their results:
1) (4 x 9) + (4 + 9) = 49
2) (5 x 9) + (5 + 9) = 59
See that by using the same digit, with the above maths operation, we can get the answer without thinking! But do note that one of the digit has to be 9.
To see more of such fun, click here.
But there is a group of quadratic equations that has a certain format that can be easily factorised without much computation.
They, therefore, deserve some attention!
They are the so-called "Special Factored Products" in this post.
Special Factored Products
- x2 + 2xy + y2 = (x + y)2
- x2 - 2xy + y2 = (x - y)2
- x2 - y2 = ( x + y )( x - y )
NOTE: The centre term of the quadratic equation is the product of the square root of the two others and 2.
**They must ONLY be in these special form before we can proceed to apply the quick factorisation method**.
If you can be familiar with their quadratic equation format, you can translate them to their factorised form straight away without even computing !
Let's look at some example to enhance understanding of this important maths principles.
Example 1: Factorise x2 + 4x + 4
Applying the first factorisation format, the "y" can be replaced by 2 since y2 = 22.
Therefore the factored answer = (x + 2)2.
Example 2: Factorise 4a2 - 4a + 1
Applying the second factorisation format since now the maths sign for the term with power of 1 (that is the "a" term) is "minus", we can replace the "y" by 1 since 12 is 1.
How about the first term 4a2 which looks different now?
Using the knowledge of Indices, we know that 4a2 = 22a2 which becomes (2a)2.
We can use the understanding to replace the "x" term in the factorised form (x - y) 2.
Thus, the factorised form of 4a2 - 4a + 1 = (2a - 1)2.
Common mathematical mistakes made:
1) Forget to cater for the coefficient in the "x2" term. Take care of the number too!
2) The minus sign ONLY applies to the term with the power of 1 in the original unfactorised form. The term with pure number is still "+"!
If the quadratic equation has the minus sign in the number term, we cannot apply the second factorised form.
Example 3: Factorise 4a2 - 9
It is very obvious that this maths equation format is the third type.
Here we need to split the two terms into the "squared" format to convert the quadratic equation to the factorised product form.
Look at the first maths term. 4a2 = (2a)2 as in Example 2.
This is used to replace the "x" in the special factored product form (x - y)2.
The second term can be easily obtained to be 3 since 9 = 32 and the 3 will therefore replace the y term.
Quick factorising 4a2 - 9 ==> (2a - 3)(2a + 3). Simple?
It is good practice to remember this 3 special factored products and their conversion as I find that it saves time and many maths questions appear to be in this form.
Learners are not able to recognise this easy 2-term quadratic relation.
To be able to recognise this maths format, practice till the trick catches on ==> instinctive awareness!
Maths will be interesting if sufficient effort is put in. This goes to any thing in life too!
:-) :) :P
This is so when numbers are multiplied by eleven.
For x 11, the process revolves round the fact that "any number" x 1 = "the same number".
Let me explain.
Y x 11 = Y x (10 + 1) = 10Y + Y.
25 x 11 = 25 x (10 + 1) = 250 + 25 = 275
Now, look carefully!
The final multiplied result of 275 has its digit "2" taken from the 10-th digit of the original "25", and the digit 5 of the multiplied (product) result is taken from the 1-th digit of the original 25.
What then is in the centre digit of the multiplied result (275)?
It is the maths addition of the 2 end digits of the original 25 , which is "2 + 5" = 7!
Got the picture?
This is due to the fact that the "x10" shifts the 25 left by one digit. Thus retaining the number 2 on its leftmost digit at the multiplied end product.
Multiplying 25 x 11 = a three digit number. This three digit number is obtained by inserting into the centre of the final product result, the addition of the two numbers 2 and 5 (the original 25). OK?
32 x 11 ==> 3, (3 + 2), 2 ==> 352 (answer).
Maths is fun and interesting if you understand the underlining tricks !
For other exciting details about this topics, refer to this link.
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.
Article Directory: http://www.articledashboard.com
www.NurtureMinds.com/ is owned by Dhaval Shrimankar, a private tutor based in Fairfax VA.
The "Completing the Square" method of solving quadratic equation is demonstrated here. Through this video, any steps missed out can be replayed.
Make full advantage of this video to master this method as it is a sure way to solve ANY quadratic equations.
In the below video clip, you will learn the method to solve quadratic equations of this nature.
For more ways of solving quadratic equations, you may click this link to view the video.
and using only the basic math operators, +, -, /, x, and ( ),
figure out the operators place of insertion for the set of numbers to get a final answer of 500.
Not that difficult if you have patience.....
Enjoy doing. Cheers!
I do not mean that it is hard to learn.
What I meant is the wide variety of methods used to solve them.
There are 5 basic ways to solve quadratic equations.
This video lists out the methods although it did not show you the steps. Good to be aware of the methods in solving quadratic maths equation so as to have a better choice upon doing. Some methods are simpler than others. While others are tedious.
The video here shows you the method to apply using factoring to solve the maths equation.
Factoring is, however, not a fool-proof method to solve ANY quadratic equations. It is simple though.