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Topic: hero's formula

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Subject:   RE: hero's formula
Author: Mathman
Date: Dec 7 2006
On Nov  8 2006, gcoston wrote:
> I am a first-year teacher, and I'm trying to come-up with an
> interactive way to present heron's theorem.  I know that I can
> always use a java site (which I'm having some difficulty finding
> quickly), but I think if I can get them out of their seats with an
> activity - they might retain it a bit longer.  Does anyone have any
> suggestions?

First of all, you have not indicated what sort of student you are trying to
reach, so I must make a point to begin with:  Why do people try to teach more
advanced mathematics to those who have great difficulty, and then ask how they
might do it the more easily than is usually done?  Is there no wonder on its own
merit of the development itself and then its application?

That said, how familiar are you with the interrelationships between the sides of
a triangle and the incircle, circumcircle, and three excircles for example?
There are also the trig half-angle formulas to consider. ...and so on.  There
is room for much study and development there to expand and relate to the idea of
the semi-perimiter and it uses.  As with other such studies, much that
inhibits learning is not what appears at the time, but is more likely that which
was not fully understood earlier.  For example, if not familiar with a
difference of squares in factoring, it is then difficult for them to follow a
development using that, but keeping that on the back-burner while really
looking at the development itself.

For example, in the half-angle formulas, can they readily see that a^2 -
(b-c)^2 = (a+b-c)(a-b+c) ?
Can they see readily that this can be expressed as (2s - 2c)(2s - 2b) ...and so
on?  If they don't get too excited about that, or have difficulty after plugging
in values to do an actual calculation, it is difficult to see how else they can
become more interested.

Unfortunatley, to my knowledge at least, there is no simpler way to visualise
Heron's formula without looking at the larger picture, having them see the
triangle and related circles, and subsequent division of the sides as tangents
to those circles.  If there was, I'd have grabbed onto it myself. However, I do
know that a more intensive study, as I've suggested, can help tie things
together.  It's sort of like wandering through the woods and looking about.  It
helps in better understanding of the particular by stoppping now and then and
turning over the rocks and seeing a bigger picture.


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