GREATER CHALLENGES I 

3rd year of secondary education 


1. THE AREA OF A TRIANGLE ON A
SQUARE GRID 

In
the first window there is a square
grid
with a triangle drawn on it. For practical purposes vertex A
can't move and B
can only move along the horizontal line at the bottom of the grid (using the
"base"
arrows). Point C
can be moved anywhere on the grid
(by changing the "height"
and "right"
values) except for the horizontal line at the bottom of the grid, as we can't
form a triangle from a segment. The vertices are always located on one of the
grid's points. 


1. Form
different triangles on the electronic board and complete the following table
in your exercise book:
NOTE:
The
unit of area is represented by the white square. The unit of length is
obviously the side of this square. The grid points which form part
of the perimeter of the shape are those which are located on any of its sides.
You will have to look at this carefully. 


2. THE AREA
OF A PARALLELOGRAM ON A SQUARE GRID 

In
the following window you can see different types of parallelograms.
To get different parallelograms just change the "height"
and "base"
values (either one of them or both of them). The vertices can be moved
following the same restrictions
as in the window above. 


2. Repeat
activity 1 above, this time with different parallelograms on the electronic board. 3. Use
the results from the last two activities to try and find a relation between
the number of points on the
perimeter, the
number of points inside the shape
and the area of the corresponding shape. If you manage to find the relation
you will actually have discovered "Pick's theorem". (CLUE:
if you take the result of multiplying one of the amounts by a certain quantity
and add it to the other amount you will get a quantity which differs from the
actual area of the shape by a small whole number). 

3. PICK'S THEOREM 

There
are no parameters
in this window and this time you can move any of the vertices
of the parallelogram to any point on the square grid. If you managed to find
the correct answer to the previous exercise you should be able to draw any
kind of quadrilateral and find out its area
without having to use traditional
geometric formulae.
If any two sides cross or if we make a triangle our results will not be
correct, as
our shape will not be a quadrilateral. 
























Josep Mª Navarro Canut 


Spanish Ministry of Education. Year 2001 



Except where otherwise noted, this work is licensed under a Creative Common License