Lesson Four – Pentagons and Pentagrams
The regular pentagon with a pentagram inside it offers students the opportunity to discover many instances of the Golden Ratio. The images below depict some of the golden relationships found in this figure.
The regular pentagon with a pentagram inside it offers students the opportunity to discover many instances of the Golden Ratio. The images below depict some of the golden relationships found in this figure.
A suggestion for this lesson is to provide students with copies of the pentagram-inscribed pentagon (or, better yet, to have them build the model using Geometer’s Sketchpad, Geogebra, or a coding platform like Scratch), and have them find as many golden relationships as possible. Providing too much information up front (other than the fact that this figure contains a wealth of golden relationships) may impose too much direction upon the students' exploration. It is better to see what they find on their own.
Once again, this form of student investigation provides the teacher with time to circulate, conduct formative assessment, and offer descriptive feedback. Here are some things to look for:
Toward the end of the session, student findings should be shared in order for all present to see the true extent to which the pentagon/pentagram is based on the Golden Ratio.
Once again, this form of student investigation provides the teacher with time to circulate, conduct formative assessment, and offer descriptive feedback. Here are some things to look for:
- There are many Golden Triangles within the pentagon/pentagram. Seeing as the students will have become very familiar with them in Lesson Three, it will be interesting to see if they recognize them in a new context.
- Students will probably look for the Golden Ratio among the lengths of line segments in the figure. However, the Divine Proportion also exists among the area measurements of various sections of the pentagon/pentagram. Will they think to look there?
- Many golden pairs of line segments are repeated in the figure, but oriented differently in various places. Will the students record them as unique applications of the Golden Ratio, or will they generalize one pair to other occurrences in the figure?
Toward the end of the session, student findings should be shared in order for all present to see the true extent to which the pentagon/pentagram is based on the Golden Ratio.
Reference
Jarvis, D. (2007). Mathematics and visual arts: Exploring the golden ratio. Mathematics Teaching in the Middle School, 12(8), 467-473. Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/231093806?accountid=15115
Jarvis, D. (2007). Mathematics and visual arts: Exploring the golden ratio. Mathematics Teaching in the Middle School, 12(8), 467-473. Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/231093806?accountid=15115