Ratios and Rates

NZ Level 6 (NZC) Level 1 (NCEA)

Golden Ratio (Investigation)

Lesson

The Golden Ratio can be found everywhere! It’s in architecture, art and even in nature. In this investigation we will investigate the evidence of the Golden Ratio in the human body.

- To find ratios in daily life.
- To practice with decimal ratios.
- To practice creating equations from ratios.

- Internet
- Computer printer
- Calculator
- Measuring tape

- Measure the following lengths on your own body.
- Total height head to toe
- Bellybutton to feet
- Waist to floor
- Top of head to waist
- Tip of nose to chin
- Middle of eye to the tip of nose
- Length of the lips
- Width of the nose
- Top of head to chin
- Top of head to the middle of eye
- Pupil of eye to lip
- Outside distance between the eyes
- Hairline to the pupil of eye
- Lips to the chin
- Wrist to mid elbow
- Tip of finger to wrist

- Find the ratio of $A:B$
`A`:`B`for each of the $9$9 pairs of measurements. Each pair should result in a ratio very close to the Golden Ratio. For example, to create the first ratio the measurements you obtained would be plugged into the ratio total height head to toe : bellybutton to feet. - Convert each of the ratios you just found into a decimal. Use your calculator if necessary.

- Compare all of the ratio decimals you just found. Take the average of these ratios and use it to guess what the golden ratio is.
- Once you have made your guess, read about the Golden Ratio here.
- Was your guess close?
- Now that you know what the Golden Ratio is, use it to find the total height of a person who has a length of 114 centimetres (45 inches) from their bellybutton to their feet.
- What would be the length from bellybutton to feet for a person whose total height is 198cm (78 inches)?

- Work with friends! Have your friends complete the same measurements as you did and compute the ratios in decimal form. Take the average of all of your ratios and their ratios. Is it closer to the actual value of the Golden Ratio?
- Look up pictures of your favorite celebrities. Find a clear photograph just of their face and print it out as large as you can. Measure the following distances on their face:

- Find the ratio of $A:B$
`A`:`B`for each of the $6$6 pairs as done previously with your own measurements. Convert the ratios you find into decimals. Take the average of these ratios combined with the ratios found in your own measurements. Is this average number closer to the Golden Ratio?

If you want to read more about the many places where you can find the Golden Ratio check out this link.

Apply direct and inverse relationships with linear proportions

Apply numeric reasoning in solving problems