Relating Two Polygons
Homage to Smudge's Perimeters
More or Less Grids
Homage to Smudge's Perimeters
More or Less Grids
Applets Relating Two Polygons

Two Polygons:
Numbers of sides can be varied between 3 and 20; Graphs of perimeters and areas can be displayed; Uses the signature of a shape: the ratio of the square of the SemiPerimeter to the Area Applet Based on and developed from an idea of Dan Meyer 

Two Rectangles:
Sizes of rectangles can be varied; Graphs of perimeters and areas can be displayed; Uses the signature of a shape: ratio of the square of the SemiPerimeter to the Area; Applet 
Homage to Prof Smudge's Perimeters
Inspired by a task about comparing the perimeters of two polygons, this applet was developed in order to exploit variations. It asks for which path, the black or the green, has the longer path length.
The subpaths can be flipped in various ways using buttons. It also supports thinking about which has the largest area (area below the axis is deemed to be negative), which is nontrivial. Applet & Notes 
More or Less Grids

How few cells can you change from the original for each of the 8 parts of the table?
Which is the most challenging and what makes it challenging? What principles about perimeter and area did you encounter? Applet See More or Less and Bi–Quality Tables 