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:

 Base Height Points on perimeter Points inside Shape area

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.

 4.- Draw ten different quadrilaterals, five concave and five convex, on the electronic board. For each one write down the number of grid points in its perimeter, the number of points inside the shape and the area in your exercise book. Work out the area of these shapes using Pick's theorem and check for each shape that the area you calculate is the same as the area given on the electronic board. 5.- At home investigate the following: Does Pick's theorem work for any geometric plane shape? What happens when the vertices are not located on the grid's points? Josep Mª Navarro Canut Spanish Ministry of Education. Year 2001 