Section 6.3 Instructor Notes for Chapter 6
This chapter provides optional opportunities to explore topics outside the standard curriculum and to observe the connections between geometry, algebra, number, and operations. Observing patterns, making conjectures, and justification is encouraged throughout. For prospective teachers, the sections Section 6.1 and Section 6.2 offer an opportunity to review the greatest common factor and least common multiple of integers and to see that these concepts play a role outside of operations on fractions. The choice to use GCF instead of GCD was intentional as I wanted to emphasize the term factor here.
Subsection 6.3.1 Notes for Section 6.1
In this section, we begin by showing how a star polygon is constructed. For an inclass activity, the instructor may want to provide handouts with the templates and colored pencils. Small transparent straight edges can aid students in drawing these designs. GeoGebra interactives are provided and may be used in lieu of hand sketches.
Five-dot and six-dot star polygons are used to illustrate this process and to introduce the notion of connected components. In graph theory, the term connected component refers to a subset of vertices (points) that one can reach by traveling along edges. In fact, each star polygon is an example of a graph with \(n\) vertices and \(n\) edges and an instructor could use this section to introduce students to the field of graph theory. Star polygon \(\{_{3}^{6}\}\) shows that a star "polygon" sometimes ends up simply being an asterisk-like intersection of line segments.
Star polygons drawn on equally spaced dots around a circle will have rotational and reflectional symmetry. If you notice student work lacking symmetry, it is a good idea to help the student focus on skipping the same number and connecting dots. Although some students may want to sketch segments in a different order like 1 to \(k\text{,}\) 2 to \(k+1\text{,}\) 3 to \(k+2\text{,}\) etc., they are encouraged to sketch one connected component at a time possibly without lifting their pencil. Drawing each connected component in a different color will make it easier to make and verify conjectures.
Once the students have drawn a few star polygons, they will be invited to predict other star polygon designs based on the number of dots and the skip number. Give the students time to figure these out on their own through experimentation and discussion. I sometimes suggest other \(n,k\) pairs that might confirm or contradict their claims as a way of helping them refine and support their conjectures.
Key terms: star polygon, connected component, greatest common factor.
