Formulas (Understanding & Application)Formulas are very important. They make solving of problems easier and faster.Its very important to understand the formula and to apply them at correct...
Formulas are very important. They make solving of problems easier and faster.Its very important to understand the formula and to apply them at correct place.For example, let us take simple expansion formula(expansion of square of a binomial expression) i.e. (a+b)2. We know the formula is :- (a+b)2 = a2 + 2.a.b + b2 . Now in order To apply the formula, We must know that the letters/alphabets ‘a’ and ‘b’ represent the two terms of the binomial expression and understand the formula like this à the square of a binomial(containing two terms) expression = square of first + 2 times the product of the two terms + square of the second term. Let us apply this formula Eg. 1: Let us find the square of a binomial expression 3ax+4xy (3ax+4xy)2 = (3ax)2+2.(3ax).(4xy)+(4xy)2= 9a2x2+24ax2y+ 16y2 Hence, (3ax+ 4xy)2 = 9a2x2+ 24ax2y+ 16y2 ← Answer Eg2. (a+b+c)2 [ Square of a trinomial(containing 3 terms) expression ] Let us take 1st term as (a+b) and 2nd term as c (a+b+c)2=(a+b)2+2(a+b).c+ c2= a2+2.a.b+b2+2a.c+2b.c+ c2 = a2+b2+c2+2ab+abc+2ac Hence,(a+b+c)2 = a2+b2+c2+2ab+abc+2ac ← Answer
Formulas are very powerful, because they give you a way to solve a problem. Let's say that you need to buy new carpet for your room. You need to know the area, right? Length times width is area. Without this, you have no carpet!
I finished my school about 50years back. Applied arithmetic used to be the focus and we learnt some of the basic formulae during school which would make the life easy for someone who has just completed Grade 10. With the introduction of calculators and later by computers, the emphases shifted on the advanced sciences even in the junior classes. It has both good and bad effects, good for those who want to pursue higher studies, bad for those who will not find an opportunity to apply their knowledge of the relatively advance concepts for the rest of their life. Its a dilemma for the educators to draw the line. However, I feel that teachers can do a lot in making students understand the simple formulae so that these can be applied by them in practice.
One thing in this respect must also be understood by students, as a Chinese proverb say: "A teacher can take you to the door, he cannot open the door for you".
Teachers can only explain as to how a formula can be applied, he is not there when you need to apply it in a real life situation.
Agreed... However, at schools... We're somehow taught the exact opposite, unfortunately, and are forced to focus on the 'how' rather than the 'why', and therefore fail when attempting to apply knowledge outside of the classroom, or even when dealing with word problems.
It's kinda sad actually :(
We can use this formula [(a+b)^2=a^2+2*a*b+b^2] to find the squares of numbers such as 63,102,59,97 etc. easily. Let us find the square of 63 :
Example 1 :-
63^2=(60+3)^2= 60^2+2*60*3+3^2= 3600+360+9=3969 <--Answer
Example 2 :-
(97)^2=(100-3)^2=100^2-2*100*3+3^2= 10000+600+9= 9409 <--Answer
Understanding of formulae makes maths interesting. It can help you to do mental arithmetic as can be seen in the following example:
We know that A^2 - B^2 = (A+B)(A-B) hence A^2 = (A+B)(A-B)+B^2
Now if we have to take the square of 23 then:
23^2 = (23+3)(23-3)+3^2 = 26*20+9 = 520+9 = 529
And one who is good in metal arithmetic can find out squares of 2 digit numbers without using a calculator as illustrated above. The trick is to add or subtract a number to the original number so that you have a zero in the unit place in one of the two numbers i.e. A+B or A-B. In the example I have chosen to add 3 to 23 to get A-B equal to 20 to simplify the multiplication that can be done without using calculator.