Thought for the day: Don't tell students anything when you first display a prompt. Ask them what they notice and wonder, and let their ideas flow before you begin to focus their thinking on specific ideas!

Concepts: area (possibly polygons and perimeter)

Beginning

I notice four polygons with different colors. I wonder if the colors are important.
I notice that two of the polygons are made from whole-squares.
I notice that all of the polygons are made from whole-squares and half-squares.
I wonder if all four polygons have something in common.

Exploring

I notice that every polygon has a total of 4 whole squares inside of it. (They have areas of 4 square units).
I wonder how many polygons I can make that have areas of 4 square units.
I notice that I can make shapes using smaller parts of squares, too (like quarter-squares).
I wonder if the polygons have the same perimeters.
I wonder how long the slanted sides are.
I notice that the diagonal (from corner to corner) in a square is longer than a side.
I notice that I can use the edge of my grid paper as a “ruler” to measure the slanted sides.

Creating

As they notice and wonder, students may create and explore their own polygons. Many of their creations may flow out of things that they have wondered about. For example, they may

Create more polygons whose area is 4 square units. (Try for variety!)
Create a variety of polygons having some chosen fractional area (such as 2.5 square units).
Try to make their polygons look like recognizable objects (real or imaginary).
Create polygons using smaller pieces of squares (such as quarter-squares).
Create polygons that have the same area but as many different perimeters as possible.

Reflecting and Extending
I notice that polygons with the same area can look very different.
I notice that polygons with the same area can have different perimeters.
I wonder how may triangles (or other shapes like pentagons or hexagons) I can create that have areas of 4 square units.
I wonder what is the smallest (or largest) perimeter I can make for a polygon whose area is 4 square units.

Notes

A related Creative Math Prompt is part of an extended activity called "Area Challenge" on the Deep Math Projects page of this site. You can use this activity to learn more about the concepts in this prompt.

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!)

The diagonal of square is about 1.4 times as long as a side. Most students will probably estimate it at about 1.5 units. Students will learn methods to calculate the exact length in middle school or high school when they study the Pythagorean theorem.