Experiment: Not for Reading.
Let us take two arm-length measuring $12$ units and $5$ units, hinged at one end $C$. Let the two arms be inclined at an angle $X$ to each other. The two arms named $AC$ and $BC$ can be moved, varying the angle $X$, bringing arms closer or farther. Experiment making more and more triangles, changing $X$. How many triangles can be made with arms $AC$ and $BC$?
Shall we try to make triangles with larger third side $AB$, denoting the measure of the third side as $c$? Also experiment making triangles with smaller and smaller third side $AB$.
What happens of the triangle $ABC$ when we attempt to make the third side $AB$ of maximum length? Shall we attempt to formulate our finding? The teacher may please wait till the class, one student supplementing and correcting the statement of the other and so on...... concludes the observation that the length of the third side in a triangle is ............ sum of the lengths of the other two sides.
We continue the experiment. Shall we investigate, what we do to find a triangle with the smallest third side? What the third side must be like? It would not be difficult to encourage the class to formulate a result as to what the third side must be greater than.
In terms of the usual notations, denoting $BC$ as $a$ and $AC$ as $b$, the first result is \[ c < a + b. \]
Analogously, we can obtain $a < b + c$ and $b < a+c$. But that implies that $c > a - b$. Interpreting geometrically we are getting the second result worked out above.
Is it not wonderful!! It gives us an opportunity, how algebraic representations facilitates the study of Geometry; and reveals facts not so obvious.
II
Examine the following. Take two points $A$ and $B$ in a plane. Supposing, there are no barriers in the plane. How do you travel starting from $A $ towards $B$? How about travelling via a third point $C$? Would it be as economical distance-wise or time-wise? Is there a situation that it will not make any difference, even if we travel via $C$? Formulate your finding in the form of a result. Compare the result with the finding of the preceding discussion and the result there.
We pick up our earlier experiment once again. May we eye upon the space (area) enclosed by the triangle. Once again we begin with a collapsed triangle $ABC$ with $AC$ upon $CB$ enclosing a zero area.
We separate $AC$ and $CB$, making an angle $X$, and enclosing some area. Increase $X$ further, area increases further.
As you repeat `increase $X$ further', in a chorus, most oftenly, the class joins `area increases'. They realise to have erred only when $X$ nearing $180^o$, they see the area vanishing to zero. Repeat the experiment. Some in the class, growing wiser, would say in between `area decreases'. Repeat the experiment, moving slow ... and in between preparing to ask, ``stop me when the area starts decreasing''.
I must take a break to share my experience, when during this experiment in the class; at this stage, a little village girl from Orissa jumps up to ask; ``Is it not turning when the angle is $90^o$''. She is able to see the truth that me and the entire faculty at Institute of Mathematics in Bhubaneshwar found impossible to prove to her at her age of $8-9$; and feel guilty to ask her to wait for another $4-5$ years.
Bandita, the little girl expects $90^o$ to be the turning point and not $80^o$ or $100^o$. The truth is not $89^o$ or $91^o$, either? Why? Mathematics depends on faith in `Order' in nature. The nature is proclaiming this `Order' to everyone, everywhere, all the while. We can expose the students to this invitation of the Nature and help them in discovering the `Order' for themselves.
Let us take two arm-length measuring $12$ units and $5$ units, hinged at one end $C$. Let the two arms be inclined at an angle $X$ to each other. The two arms named $AC$ and $BC$ can be moved, varying the angle $X$, bringing arms closer or farther. Experiment making more and more triangles, changing $X$. How many triangles can be made with arms $AC$ and $BC$?
What happens of the triangle $ABC$ when we attempt to make the third side $AB$ of maximum length? Shall we attempt to formulate our finding? The teacher may please wait till the class, one student supplementing and correcting the statement of the other and so on...... concludes the observation that the length of the third side in a triangle is ............ sum of the lengths of the other two sides.
We continue the experiment. Shall we investigate, what we do to find a triangle with the smallest third side? What the third side must be like? It would not be difficult to encourage the class to formulate a result as to what the third side must be greater than.
In terms of the usual notations, denoting $BC$ as $a$ and $AC$ as $b$, the first result is \[ c < a + b. \]
Analogously, we can obtain $a < b + c$ and $b < a+c$. But that implies that $c > a - b$. Interpreting geometrically we are getting the second result worked out above.
Is it not wonderful!! It gives us an opportunity, how algebraic representations facilitates the study of Geometry; and reveals facts not so obvious.
II
Examine the following. Take two points $A$ and $B$ in a plane. Supposing, there are no barriers in the plane. How do you travel starting from $A $ towards $B$? How about travelling via a third point $C$? Would it be as economical distance-wise or time-wise? Is there a situation that it will not make any difference, even if we travel via $C$? Formulate your finding in the form of a result. Compare the result with the finding of the preceding discussion and the result there.
III
We separate $AC$ and $CB$, making an angle $X$, and enclosing some area. Increase $X$ further, area increases further.
I must take a break to share my experience, when during this experiment in the class; at this stage, a little village girl from Orissa jumps up to ask; ``Is it not turning when the angle is $90^o$''. She is able to see the truth that me and the entire faculty at Institute of Mathematics in Bhubaneshwar found impossible to prove to her at her age of $8-9$; and feel guilty to ask her to wait for another $4-5$ years.
Bandita, the little girl expects $90^o$ to be the turning point and not $80^o$ or $100^o$. The truth is not $89^o$ or $91^o$, either? Why? Mathematics depends on faith in `Order' in nature. The nature is proclaiming this `Order' to everyone, everywhere, all the while. We can expose the students to this invitation of the Nature and help them in discovering the `Order' for themselves.
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