### Summary

This lesson teaches students about the golden ratio and Fibonacci numbers. The lesson focuses on finding the golden ratio in art, nature, and common objects, as well as in their own skeletal structure. Students take measurements and use calculations to identify examples of the golden ratio both inside and outside the classroom.

### Essential Question(s)

Are there common mathematical patterns expressed in nature?

### Snapshot

**Engage**

Students examine various images that seemingly have nothing in common, but upon further investigation, patterns they all share are revealed.

**Explore**

Students discuss general size relationships in humans, then they measure each other looking for these relationships in their own bodies.

**Explain**

Students analyze the measurement data collected in the Explore phase, looking for patterns and relationships.

**Extend**

Students search for other common objects in the classroom that exhibit the golden ratio.

**Evaluate**

Students find three objects in their homes that exhibit the golden ratio. They take pictures or draw sketches of these objects.

### Materials

"Are We Golden?" teacher slides (attached)

Student Data Collection Excel spreadsheet (attached)

Student Data Collection PDF handout (attached)

Teacher data collection spreadsheet (attached)

Variety of measurement tools: rulers, meter sticks, tape measures, etc.

Rectangular objects to be measured, such as playing cards, cereal boxes, plasticID cards (the same shape and size as a credit card), or business cards. Make sure to also include objects that do not exhibit the golden ratio.

### Engage

Introduce the lesson and present the attached teacher slideshow. Show students a variety of images, beginning with DaVinci’s "The Mona Lisa" and "Vitruvian Man" on slide four. Ask them to discuss what they believe the images have in common. Be sure to note students' responses as you progress through each set of images on **slides 5-6**.

Next, transition to **slide 7** and conduct the "Rectangle Pageant." There are four rectangles shown (A, B, C, and D). Ask students to vote on the rectangle they believe is the most pleasing to their eye.

Show students the definition of the golden ratio on **slide 9**. Offer your own explanation, then show students **slides 10–12**, which show the same images they saw earlier but with diagrams and overlays illustrating the golden ratio. Discuss these items and their relationship to the golden ratio.

Next, discuss how this ratio not only can be found and has been purposely included in man-made objects but that it also shows up in nature. Transition to **slide 13** and show the images from nature with diagrams indicating the golden ratio.

### Explore

Prior to allowing students to begin the Explore portion of the lesson, lead a whole-class discussion asking the following questions:

*Are all of you aware that the length of your foot is relatively the same length as your forearm?*(You can demonstrate this by removing your shoe and holding it up against your arm between your elbow joint and your wrist.)*Do any of you know any other comparisons like these?*

Share responses and allow students to verify them if needed. Next, tell them that today’s activity will involve examining measurements of their bodies that express another relationship.

Have students select a partner with whom they will feel comfortable measuring and being measured.

Students will work in groups of two and in teams of two groups (so a total of four students per team). Students have two options for recording measurements and ratios. If the technology is available, they can record them on the attached Student Data Collection Excel file (this file can also be opened and used in Google Sheets if preferred). If you prefer for students to record the data manually, you will also find PDF and Word versions attached. Each pair of students should select the measuring tools they think they will need to accomplish the required measurements.

Have students complete all required measurements with their partner and record them on the team’s Student Data Collection file, either in digital or paper form. Measurements should be recorded in centimeters and each ratio of measurements should be expressed as a decimal.

### Explain

Once all groups have completed their measurements, recorded all ratios, and expressed them as a decimal, have students use a calculator to find the average of each of the columns on their worksheet. Have one member from each team should report the averages for each column while you collect the data for the class.

During this time, monitor and manage the reporting of the averages either on the board or in a spreadsheet. Once reporting is complete, initiate and engage a whole-class discussion regarding the findings.

This discussion should revolve around the idea that bones demonstrate linear growth and that the skeletal system overall grows at the same proportional rate. For example, our right arm does not decide to grow for a few years while everything else remains the same. Our skeleton grows at a continual rate. While the rate is not constant throughout our life (e.g., we have growth spurts and periods of slower growth), our body's growth is proportional. Finally, ask students to reflect on the Engage phase of this lesson: Ask students, "Do you think the fact that our own bodies reflect the golden ratio influences what we determine is beautiful or pleasing?" Refer to the rectangle from the rectangle pageant. Be explicit in pointing out that many of the relationships in their bodies have the same ratio as the rectangle they all thought was visually appealing at the beginning of this lesson. It is also the same ratio as many of the pictures of art and architecture they looked at in the slides.

If time permits, this short video contains animations that demonstrate where the golden ratio and the Fibonacci sequence are found in nature.

### Extend

Students should rejoin their original partners and work together to measure and identify at least three rectangular objects in the room that exhibit the golden ratio. Encourage students to identify objects that have not already been demonstrated, such as smartphones. The rectangular objects do not have to be movable. They could be cabinets, tiles, a window, a door, or whatever they see in the room they think exhibits the golden ratio.

In their notebooks, have students record the objects they identified and their measurements, along with an explanation of whether or not they exhibited the golden ratio.

### Evaluate

Ask students to locate three objects at home that they believe exhibit the golden ratio and photograph them with a digital camera or smartphone. If access to a digital camera is not available, encourage students to sketch and describe what they found. They should measure each item and record the measurements in order to confirm that the object is an example of the golden ratio. Have students email you their photos or turn in their sketches. Be sure they include the measurements in the body of the email or written on the sketches.

Finally, have students use a Two-Minute Paper strategy using the following prompt: Explain to a friend who was not in our class today what the golden ratio and share a few examples of where it can be found in nature and in man-made objects.

### Resources

Free Math Apps—used by over 100 Million Students & Teachers Worldwide. (n.d.). Retrieved from https://www.geogebra.org/?lang=en

K20 Center. (n.d.). Two-Minute Paper. Strategies. https://learn.k20center.ou.edu/strategy/d9908066f654727934df7bf4f506cf73

Reeder, S. (2007, October). Are we golden? Mathematics Teaching in the Middle School. 3(13), 150-155. Retrieved from http://www.shastacoe.org/uploaded/scmp2/are_we_golden_copy_x.pdf

TheBITK. (2013, April 05). Nature by Numbers - By Cristobal Vila - A display of the Fibonacci Sequence/Spiral. Retrieved from https://www.youtube.com/watch?v=BaXjWXXwQTk

K20 Center. (n.d.). GeoGebra. Tech Tools. https://learn.k20center.ou.edu/tech-tool/2352