Samples of noticing and wondering
I notice that all four shapes are polygons.
I notice that two of the shapes are made of four whole squares.
I notice that all of the polygons are built from squares and half-squares.
I notice that the green polygon has a greater perimeter than the pink square.
I notice that all of the polygons have an area of four square units.
I wonder if the polygons all have something in common.
I wonder if the polygons are shown in a special order.
I wonder how many different polygons I can make by joining four squares.
I wonder how many different polygons I can make that have an area of four square units.
I wonder how long I can make the perimeter of a polygon that has an area of four square units.
A related Creative Math Prompt is part of an extended activity called "Area Challenge" on the Deep Math Projects page of this site.
This prompt is focused on area, but it can also lead to great discussions about perimeter—or even about the meaning of the word polygon, especially for younger students. For example, when my students were trying to create more polygons of area 4, I once had a student who asked if this was a polygon:
Eventually, the class decided that it was not, because many of the sides cross each other. (This fits the usual grade-school definition of a polygon. However, sometimes mathematicians use a definition that allows the sides to cross. For them, this would be polygon!)