Thought for the day: Creative Math Prompts are great tools for differentiation, because students respond at their own level of thinking.

Concepts: polygons; segments; lengths and angles; parallel segments; (possibly tiling (tessellations), symmetry, area)

Beginning

I notice that the polygon has 6 sides. (It is a hexagon.)
I notice that it looks different than the hexagons I usually see.
I notice that the hexagon has three different side lengths.
I notice that opposite sides are the same length.
I wonder how the lengths compare to each other.
I wonder if I could find a shape like this in real life.
I notice that opposite sides are parallel.
I notice that I can split the hexagon into two quadrilaterals (or pentagons) that look the same.

Exploring

I wonder how many ways there are to split the hexagon into two congruent parts.
I wonder if I could make this shape from equilateral triangles.
I wonder if I can make other hexagons with just three different sides lengths.
I wonder
if I can make all 6 sides different lengths without changing the angles.
I notice
that the (interior) angles in the hexagon are all the same size (1-third of a circle, or 120°).
I wonder if the opposite sides have to be parallel whenever the hexagon's angles are all the same.
I notice that the hexagon looks the same if I turn it upside-down.
I wonder if this is always true when I make a hexagon with opposite sides parallel and congruent.

Creating

As they notice and wonder, students may create their own polygons satisfying various conditions. Many of their creations will flow from things that they have wondered about. For example, they may

Create other hexagons whose angles are all congruent but whose sides are not.
Try to create other kinds of polygons whose angles are all congruent but whose sides are not.
Put the hexagons together to create tessellations (by joining them to create patterns with no gaps or overlaps).
Create other patterned designs by combining their hexagons in different ways.

Reflecting and Extending

I notice a connection between parallel opposite sides and congruent opposite sides.
I wonder if the hexagons would fit together to make a "honeycomb" pattern.
I wonder if I can make heptagons or octagons whose angles are the same but whose sides are not.
I notice that it matters whether the polygon has an even or an odd number of sides.
I wonder if I can reverse the problem and create hexagons who sides are all congruent but whose angles are not.
If so, I wonder if their opposite sides will be parallel.
I wonder what the area of the hexagon is. (Some students might try to measure it in triangular units by asking how many equilateral triangles it would take to make it. Consider using pattern blocks!)

Notes

See Problem #4 in Problem Set 3 of the Intrepid Math series for a more structured problem on the same topic (or just for more information about it).

Pattern blocks are an excellent tool for exploring the ideas in this problem—for all ages, but especially for younger students.