Students will be able to approximate the Golden Ratio

Golden Ratios

15 minutes

I have this written on the board:

“What down your observations? What strikes you as interesting in these videos?”

As students enter the room, I welcome them to the “Golden Movie Theatre” and give them their “Golden Movie Ticket” which has the prompts on the board written down. I show students selected segments from ViHart’s video series on the Golden Ratio. She currently has three videos available and I show clips from the first two, which discuss the Fibonacci sequence and show the patterns from fruit and flowers. These videos confirm their discoveries from the days before. The third video is especially helpful, because she discusses the universal connection between plants with Fibonacci and non-Fibonacci leaves and petals. Essentially, our discussion mixes biology and mathematics and points out that plants will grow however they need to in order to get the most sunlight possible. The angle spread between leaves tends to maximize a plants exposure to sunlight. This might change with the orientation and climate of the plant, but that is essentially the connection. She briefly mentions Lucas Numbers (2,1,3, 4, 7,…) and that is the launching point for our investigation for the day.

Remember that Fibonacci numbers are named after Fibonacci. Lucas numbers are also attributed to their creator. Today you will create your number sequence and explore the results of your pattern. The basic rule is that your pattern needs to be recursive and each number should be the sum of the two numbers before it. However you can start with *any *number. When you are happy the pattern sequence you have set up, come get your official number sequence log from me. We want to record your invention and share it with the class.

30 minutes

This is a fun activity because students take ownership of their pattern. They often start with fun types of numbers. Its especially fun when students pick decimals and large numbers. They name their patterns and then fill out the official number log of our class. This is listed as the “student numbers” document. They fill out their numbers and then write out pairs of consecutive numbers as both fractions and ratios. I ask them to write their observations on the back of the sheet. If they have time, I ask them to make a scatter plot of the terms, where the x axis represents the number pair (1^{st} pair, 2^{nd} pair, etc) and the y axis represents the decimal approximations (with increments of about .05 and a range of 0-1.7). This is a fun chance to spiral back to lines of best fit and other data content from the common core curriculum. Students are curious about each others patterns and are excited to share their patterns and table results. I always know things are going well when a pair of students notices how “crazy” it is that their decimal results are about the same.

15 minutes

This summary is more about their number discoveries then the Golden Ratio. We mention Phi and I use various images of spirals and references of structures to emphasize the wide spread recognition of this ratio, but the Golden Ratio is not something that students can fully comprehend in a single lesson or set of lessons. It is an enormously rich topic that we are simply introducing.

The summary starts with a number sequence share. I encourage students to simply describe the sequence that they created. We popcorn around the room and share a sequence of patterns. They great thing is that they seem to have nothing in common except for the general recursive nature of the sequence. After they share, I ask:

“What did all of these patterns have in common?”

“Each number is the sum of the two before it."

“What else?”

Of course this is an unfair question, but its meant simply to remind that there is more here than they might have thought. That is when I show a table as a sample. We talk about the decimal results as we read the table. Do these decimals seems to be approaching a certain value?”

I like to help students understand this pattern by setting up a quick scatter plot (something they are familiar with in the 8^{th} grade common core standards). We think about the line of best fit and set it at about 1.6. Students are surprised to find that *all *of the patterns hover around this line. We talk about this and share that the Fibonacci sequence and *every* sequence built like it approaches the same number, phi. I usually follow this by discussing the premise of the golden ratio with a simple line diagram cut where the ratio of the segments corresponds to the golden ratio.

“The golden ratio is often associated with the golden spiral, which you can draw through the golden rectangles formed by tiling the Fibonacci numbers. You tiled the Fibonacci numbers the other day. I will return them to you and give you a chance to draw the golden spiral through the rectangles in the tiling pattern.”

If nothing else, students enjoy the rich connections through all the facets of the lessons in this series. There is a level of mystery around these numbers that one can’t help get excited about. There is an undeniable pleasant beauty to the shapes that exhibit the golden ratio and spiral. Students draw the spirals easier on the floors they tiled in their previous lesson and are now aware that there is world of beauty in ratios.