This task involves visualising a shape as it changes. Thus it has a strong dynamic/visual component. However, it also lends itself to an algebraic approach, as well as giving students an opportunity to think about decimals and fractions.

After considering the M-shape (which has 3 'legs'), we look at an N-shape (2 legs), a T-shape (1 leg) and a P shape (1 leg).

For the M-shape, we present two movies that show the shape as w increases from 0 to 1/3. the second movie displays the numerical values of w and of the area of the M-shape as these change. At both extremes the shape reduces to a square (of area 1 square unit). So our primary question is,
For what value of w is the area of the M-shape a minimum?

We repeat the process for the N-, T- and P-shapes. It turns out that there is a nice, elegant rule for the number of legs the shape has and the value of w when the area is a minimum. [This rule emerges from a related task we developed on the ICCAMS project.]

The relationship between w and the perimeter (and the number of legs) is simpler than the area relationship, and we don't consider it explicitly. It is an interesting relation, though, especially for the T- and P-shapes.

On the NEXT PAGE we see what happens to the M-shape as w changes.

a high-res pdf file of this new GEOaa-zz task