Lesson Two - The Golden Rectangle
Activate student thinking by presenting Tony Rubino’s Golden Ratio Abstract from Lesson One. Ask students to recall that the Golden Ratio was found by dividing the height of the artwork by its width. Tell them that rectangles that conform to the Divine Proportion in this manner are known as Golden Rectangles. If the 3 interior nested rectangles were not explored during Lesson One, have students predict, then confirm (through measurement and calculation), whether they are golden (Hint: they are).
Inform students that the Golden Rectangle is quite prevalent in our world, and that they probably see versions of it several times a day without knowing it. As evidence, offer the National Geographic logo. The iconic yellow rectangle is so well known that it is often appears, and is recognized, without the accompaniment of the magazine’s title. Students may assume that the outside of the logo is golden. When this hypothesis is tested, they will find that it is incorrect. It is the interior rectangle of the logo that is golden. Students can discover this by working with measurements taken on the logo, or the teacher can show them the video below.
Activate student thinking by presenting Tony Rubino’s Golden Ratio Abstract from Lesson One. Ask students to recall that the Golden Ratio was found by dividing the height of the artwork by its width. Tell them that rectangles that conform to the Divine Proportion in this manner are known as Golden Rectangles. If the 3 interior nested rectangles were not explored during Lesson One, have students predict, then confirm (through measurement and calculation), whether they are golden (Hint: they are).
Inform students that the Golden Rectangle is quite prevalent in our world, and that they probably see versions of it several times a day without knowing it. As evidence, offer the National Geographic logo. The iconic yellow rectangle is so well known that it is often appears, and is recognized, without the accompaniment of the magazine’s title. Students may assume that the outside of the logo is golden. When this hypothesis is tested, they will find that it is incorrect. It is the interior rectangle of the logo that is golden. Students can discover this by working with measurements taken on the logo, or the teacher can show them the video below.
Send the students on a scavenger hunt for Golden Rectangles. Encourage them to look for samples in the immediate environment (i.e., the classroom and school) and online. The use of the National Geographic logo may cause some students to wonder if other corporate logos feature Golden Rectangles. Provide pairs/groups with a copy of the National Geographic logo (or another Golden Rectangle) so that they will be able to locate potential Golden Rectangles more effectively through comparison. Allow them to use a variety of tools (e.g., measuring tapes, metre sticks, Geometer’s Sketchpad, Scratch 2.0) to determine which rectangles are golden and which ones exhibit different ratios. As the students investigate, circulate among the groups to ensure that they are applying the Golden Ratio accurately. This is an excellent opportunity for formative assessment of students’ understanding of ratio and proportion.
After an appropriate amount of time, gather the students together to discuss their findings. If they have discovered many Golden (and non-golden) Rectangles, it may be necessary to limit informal presentations to one of each per group. It is perfectly acceptable if a group has not found a Golden Rectangle. The investigation into non-golden samples still requires the students to work with ratios and proportions, and is therefore a worthy exercise in the development of mathematical understanding. It should be made very clear during this consolidation discussion that the learning goal is actually bigger than one specific ratio, and that it concerns deepening student understanding of ratio and proportion. While it would be nice for students to come across Golden Rectangles in this lesson, they still have equal opportunity for mathematical growth without finding one. However, this unit does explore the aesthetics of the Golden Ratio, and so a few Golden Rectangles should be present in this discussion. Have some ready in case the students do not find any. As the image to the right shows, the covers of many novels are Golden Rectangles. Have a variety of novels (some golden, some not) ready in case they are needed for the post-investigation discussion. It would be interesting to have golden and non-golden versions of the same novel on hand to discuss whether a person’s interest in reading is influenced by the aesthetics of the books’ shapes.
After an appropriate amount of time, gather the students together to discuss their findings. If they have discovered many Golden (and non-golden) Rectangles, it may be necessary to limit informal presentations to one of each per group. It is perfectly acceptable if a group has not found a Golden Rectangle. The investigation into non-golden samples still requires the students to work with ratios and proportions, and is therefore a worthy exercise in the development of mathematical understanding. It should be made very clear during this consolidation discussion that the learning goal is actually bigger than one specific ratio, and that it concerns deepening student understanding of ratio and proportion. While it would be nice for students to come across Golden Rectangles in this lesson, they still have equal opportunity for mathematical growth without finding one. However, this unit does explore the aesthetics of the Golden Ratio, and so a few Golden Rectangles should be present in this discussion. Have some ready in case the students do not find any. As the image to the right shows, the covers of many novels are Golden Rectangles. Have a variety of novels (some golden, some not) ready in case they are needed for the post-investigation discussion. It would be interesting to have golden and non-golden versions of the same novel on hand to discuss whether a person’s interest in reading is influenced by the aesthetics of the books’ shapes.
References
The Outsiders cover [Electronic image]. (1988). Retrieved from https://www.goodreads.com/book/show/231804.The_Outsiders
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
National Geographic logo [Electronic image}. (n.d.). Retrieved from http://www.nationalgeographic.com
Rubino, T. (2013, October 2). Golden ratio abstract [Electronic image]. Retrieved from https://fineartamerica.com/featured/golden-ratio-abstract-tony-rubino.html
The Outsiders cover [Electronic image]. (1988). Retrieved from https://www.goodreads.com/book/show/231804.The_Outsiders
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
National Geographic logo [Electronic image}. (n.d.). Retrieved from http://www.nationalgeographic.com
Rubino, T. (2013, October 2). Golden ratio abstract [Electronic image]. Retrieved from https://fineartamerica.com/featured/golden-ratio-abstract-tony-rubino.html