The students are required to find the area of odd shapes on a grid. There are square units, easily seen half units and often difficult to see other units depending on the diagonal of the line. I try to emphasize thinking of the diagonals as halving some rectangle. By doing this, one can find the areas of parts of a shape and eventually add them up.
Every year I find this type of activity fulfilling and challenging. Fulfilling because it leads to a deeper understanding of the real nature of area (not always a formula to be solved). It also foments a lot of logical thinking and observational math. It is quite challenging for many students to actually "see" the parts of a whole in a figure. Many try to simply estimate ("it sorta looks like"). This type of estimation is ok, but I want them to use logic and clearly lay out their case for the area.
Here is an example of one such problem:
Here is how I go about solving for the area:
Can you see how I found the area?
(hint: it is : 8 1/2)