Double-click on the movie to PLAY.

Click once to PAUSE.

RELOAD the page or double-click to return to the START of the movie.

This movie displays the values of w and the area of the M-shape, as w changes.
Can you spot when the area is a mimimum? You might want to PAUSE the movie at various points, since the numbers change so rapidly (or use the arrow keys or the JAVA-M worksheet to show it frame-by-frame).

[The range of values taken by the area is fairly small, and we have therefore measured the area to several (4) decimal places. It can be challenging to compare such numbers: for example, some students might think that 0.917 is smaller than 0.9168. The minimum area (of about 0.9167) seems to occur when w lies somewhere in the range 0.162 to 0.171. It is worth asking students, "Roughly what fraction (of 1) is this?", since it turns out the the minimum occurs when w is a simple fraction, namely 1/6.

It might be worth drawing examples of the M-shape on squared paper, using 12 or 24 units instead of 1 for the sides of the bounding square. This results in much friendlier numbers.

On the next two pages we go through the same procedure with the N-shape. If you prefer, you might want to skip to the T-shape or P-shape pages, which are perhaps the most illuminating.