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Education

Common sense for math exercises

 "Intelligence needs to be nurtured by guiding the students and not by solving the problems for them".

Diagram 1

MY CHILD Apoorva, unable to solve a math problem approached me. The problem was "If the diagonal of a square is doubled, its area becomes how many times?" Most naturally, I took out a piece of paper to solve the problem for her, but was surprisingly stopped by my friend Mr. Chandra Sekhar, who happened to be with us on the spot.

"What is the relationship between `side' and `diagonal' of the square?" he asked Apoorva.

"Exactly here is where I get confused. Is it `a = d√2' or `d = a√2?"

"If you are confused, go to basics. Into how many right-angled triangles can a square be divided?"

"Two, but in what way will that solve the problem?"

Diagram 2

"Wait a moment. In a right-angled triangle, which one is greater, diagonal or side?"

Apoorva thought for a while, and replied with a smile "diagonal", and started explaining it with diagram 1.

"If side be `a' and is `1' unit, then according to the Pythagoras theorem, d² = a² + a². Therefore, d =√a²+a².

We know that √2 = 1.41 units. As diagonal is 1.41 and a = 1, `d = a√a2' is correct but not `a = d√2'. She could clear the bottleneck on her own.

"Excellent! Now can you go ahead with solving the problem", encouraged Mr. Chandra Sekhar, who is a cricketer too. Apoorva started working on the problem.

"Let the side be `a'. Therefore, d = a√2. And `a' = (d/√2).

Area of the square is a² = (d/√2)². = d²/2.

Now the diagonal is doubled. Therefore, a = 2 d/√2. New area = (2d/√2)². =4d²/2. Thus, area has become 4d²/2 from d²/2, which means area has become four times", concluded Apoorva.

Diagram 3

"So far so good. Could you solve the problem without any mathematical calculations, whatsoever? The clue is to think laterally, club common sense with observation", asked Mr. Chandra Sekhar.

Apoorva started searching the surroundings and stopped looking at the floor, which had square tiles.

"Uncle, look at the floor", said she and started explaining. "Observe the tiles, this slant line is diagonal", and drew the diagram 2.

"If the diagonal is doubled, we get four squares", she said, while drawing another diagram 3 to prove her point.

"Thearea has become four times" concluded Apoorva triumphantly.

Throughouttheinnings, I was a dumb slip fielder, for, Chandra Sekhar pushed the ball for a perfect boundary through his square-cut, which brought a paradigm change in me.

"Intelligenceneedsto be nurtured by guiding the students and not by solving the problems for them".

K.GANESH