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; (possibly concave polygons, algebraic expressions and equations, centers and angles of rotation)
I notice three regular heptagons joined along their sides.
I notice that there are two sides between the pair of sides where the heptagons are joined.
I notice that the heptagons are "bending.”
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 notice that the heptagons join so that the outer edges form a 42-sided polygon (a 42-gon).
I notice that the 42-gon is concave. (It has "indentations.")
I notice that the heptagons join so that the inner edges form a concave 28-gon.
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 if I should still leave two sides between each pair of adjoining sides.
I wonder what angle is formed by joining the centers of the three heptagons. (See the picture below.)
As they notice and wonder, students may create and explore their own drawings like the one in the prompt. Many of their creations may flow out of things that they have wondered about. For example, they may
Finish drawing the ring of 14 heptagons.
Try to draw rings of heptagons with 1 or 3 sides between pairs of adjoining sides.
Create rings from other regular polygons.
Create and explore more concave polygons from the inner and outer edges of polygon rings.
Reflecting and Extending
I notice that the number of heptagons in the ring is twice the number of sides in the heptagon.
I wonder if this always happens.
I wonder if I can use algebra to predict which types of polygons form rings (and for which numbers of sides between adjoining sides).
I notice that I can create an adjoining heptagon by rotating a heptagon 180° about the midpoint of one of its sides.
I wonder if there are other centers of rotation (with other angles) that I could use to create adjoining heptagons.
I wonder if these centers of rotation form a pattern.
I wonder if I could create interesting patterns by overlapping rings of polygons.
I wonder what will happen if I keep joining polygons even when they don’t form rings.
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. This is the case for an n-gon if n is odd and if you leave (n–3)/2 sides between adjoining sides. (For an interesting challenge, try to prove this algebraically!) It is possible to make more than one type of ring from many polgyons! For example you can make a 6-polygon ring or an 18-polygon ring from a nonagon.)
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).