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Constructing Numerical Expressions
Before giving pre-constructed numerical expressions, it is better to train students in constructing them.
One way would be to tell them a math story which involves shopping and then ask students to construct expressions to summarise the story. There could be several ways of writing the expressions and the class can discuss as to which of them are equivalent and which are not. Then the class could simplify and check which is the correct expression. In case of expressions which give the wrong answers, the class could discuss what was wrong in the way the expressions were constructed.
A sample story - I went to a shop with a friend on my birthday with Rs 100. I bought 4 pencils each costing Rs 5 and both had a drink for Rs 20. Since it was my birthday, the casher told me that I get Rs 10 as a bonus. How much did I have with me when I left the shop?
Evaluating Numerical Expressions
Evaluating expressions involving several numbers & several operations is one of the basic tasks in arithmetic. But many expressions may yield different results depending on the sequence in which the operations are evaluated
If we take 3 + 8 + 7 + 4 or 3*8*7*4 we can perform the additions & multiplications in any order that we want and the result would be the same. Hence the order of addition or multiplication can be changed. As per the fundamental laws of arithmetic, they are ?associative".
But take a simple math problem, which recently became a viral sensation in the Internet for reasons which are not really connected with math! What is the value of the expression 8 ÷ 2(2 + 2)?
It can have 2 different values depending on the order in which the operations are performed!
8 ÷ [2(2 + 2)] which implies 2 + 2 = 4. 2 * 4 = 8 & 8 ÷ 8 = 1 OR
[8 ÷ 2]*(2 + 2) which implies 2 + 2 = 4. 8 ÷ 2 = 4. 4 * 4 = 16
Which is correct? It has confused even the technical staff who edit the magazine Popular Mechanics.
BODMAS
In the 17th century itself, mathematicians realized that, an expression which has a mix of different operations, could yield different results, depending on the order in which these operations were performed. Hence, they agreed that these operations should be performed as per an assigned priority. In India we call it the BODMAS rule. In western countries it is known as PEMDAS.
A math expression can have several operations in it. In arithmetic, the following are considered the normal operations. They are given along with a letter of the alphabet which is a short form for that operation.
B - Brackets O - Order (Exponentiation) D - Division M - Multiplication A - Addition S - Subtraction
Of the above, only "O" may need some explanation. It refers to operations like or 21/4. This refers to the exponentiation operation. The reason it is called O is not very clear.
BODMAS rules also take into account the fact that A & M are commutative in that the order does not matter, whereas S & D are not commutative.
The priority allotted to the operations is given below.
B has the highest priority
O has the next highest priority
M & D have equal priority, but their priority is next to O
A & S have equal priority, but their priority is next to M & D
But statement 3 & 4 are not absolutely true.
As in the case of the problem we started the article, there are cases where the order in which the operations are done, can give different results.
But, mathematicians had already realized these exceptions and included an additional clause while applying BODMAS. The unfortunate aspect is that most people are not aware of this additional clause!
The additional clause says that in case operations of equal priority, like M & D or A & S, occur together, the operations should be carried out in the "left to right" direction
Applying this additional clause, we can see that the correct result is the 2nd one.
First the bracket (2 +2) is worked out as 4.
Then from left to right 8 ÷ 2 = 4.
Then 4 * 4 = 16
The table below summarizes whatever we have discussed.
BODMAS is a convention
BODMAS is not a math concept which can be logically deduced. It is just a convention commonly accepted by mathematicians all over the world to ensure that the result of multiple operations yields the same result.
It is like the use of the symbol "X" for multiplication and "+" for addition. It is just a convenience, to ensure clarity in communication.
To give a non-math example, it is like the direction of driving a car. When we are in a particular country, we adopt their convention! In the US we drive on the right side of the road, while in India we drive on the left side of the road! Imagine the dangers of not following this convention!
Collectively we can call these the "Information" aspect of math. They are neither concepts or skills. They have to be learnt from external sources and remembered by repeated use.
That this problem has become viral on the Internet is a sign that even in today's technological world, fundamentals of math have still not been clearly understood.
BODMAS in Real Life
While we learn BODMAS, we also need to keep in mind that is possibly of more use in mechanical calculations than to humans.
When humans express their ideas through an expression that needs to be evaluated, the order in which the operations need to be done is fairly clear to other humans, with appropriate use of parentheses, as the very purpose of the math language is clear communication.
Many expressions which regularly appear on the NET and which yield multiple solutions, are actually constructed by humans with the intention of confusing. In reality BODMAS rules are intuitive and real life expressions rarely confusing.
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