Thought for the day: Help students learn to feel comfortable sharing their "noticings" and "wonderings" by accepting all ideas without comment at first. Record their ideas and discuss them later.
Concepts: polygons; finding angles by reasoning (without measuring); interior angles in polygons; rotations (possibly concave polygons)
Examples of noticing and wondering
I notice three regular heptagons joined along their sides.
I notice two sides between the sides where the heptagons are joined.
I notice that the heptagons join to form a 17-sided polygon (a 17-gon).
I notice that the 17-gon is concave. (It has "indentations.")
I notice that the heptagons are "bending". (They might make a ring if I keep joining more of them!)
I wonder if the heptagons will match up perfectly to form a ring when I keep joining more of them.
I wonder how many heptagons it will take to make a ring.
I wonder what would happen if I left only one side between the sides where the heptagons are joined.
I wonder if I could make rings from other regular polygons.
I wonder what angle is formed by joining the centers of the three heptagons. (See the picture below.)
Young students can explore rings of polygons by making cut-outs of regular polygons and fitting them together. (They can use pattern blocks for hexagons.) Ask them to predict what will happen before they make the rings.
This prompt is perfect for students who have just learned about sums of angles in polygons. (It also helps if they know about vertical and supplementary angles.) They can apply their new knowledge to calculate angles and make predictions. In many cases, the polygons will in fact match up perfectly to make rings, but it depends on how many sides you leave between the sides that you are joining. In the case of the heptagon pattern above, it takes 14 heptagons to make the full ring. (Some students may wonder if the number of polygons needed is always twice the number of sides.)
Students may also like to explore the polygons formed by the inner and outer edges of the ring (as well as the new regular polygon created by joining the centers or the midpoints of the sides of the original polygons).