# Hard Task of Teaching “Easy” Things

September 2, 2015

What could be easier to teach and to learn than the fact that the length of the diameter is twice the length of the radius and thus the length of the radius equals half of the length of the diameter. You could see this relation clearly in the picture of a circle.  To switch from one to another you could simply multiply or divide by 2.  Nothing to it.  So obvious. And yet, the link between radius and diameter seems to be the source of great confusion for Robert.  And thus for me.  It is very hard to teach obvious facts and  apparent connections.  There is really nothing to explain and not much to memorize.

It is much easier to teach and to learn other, more complex formulas. How to calculate areas of rectangles, triangles, and even trapezoids, not to mention areas of circles.

Today, Robert was dividing circles into six, eight, or twelve congruent sectors, cutting them out, and using them to build figures that resembled rectangles.

Judging by his sly smile, Robert noticed the fact that as the parts of the circle decreased in size but increased in the number, they could be arranged in a shape that more and more looked like a rectangle. (To illustrate the concept you can look at  http://www.mathsteacher.com.au/year8/ch12_area/07_circle/circle.htm )

I am not sure if he just guessed or sort of understood that the length of the rectangle was approaching half of the circumference while the width was moving toward  the length of the radius.  Nonetheless, we both arrived to the formula for area of the circle, which Robert later applied a few times.

Robert almost automatically calculated areas of the circles when he was given their radii.

When, however, the problem demanded that Robert find the area of the circle with known diameter, Robert hesitated for quite a while, then closed his workbook and said, “Tomorrow.”

Just to put this post in the context of our studying together, I need to add, that we devoted most of the time today not to areas of the circles, but to the clearer pronunciation of CVC words.  That is a real struggle.

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