Ratios and the Golden Ratio

Title: Ratios and the Golden Ratio

Audience: High School Geometry

Length of Lesson: two 50-minute periods

Sources:

http://hs.houstonisd.org/debakeyhs/Lessons/ratioprocedures.html

http://www.thirteen.org/edonline/nttidb/lessons/dn/golddn.html

I. Performance or learner outcomes

The student will be able to:

                    - Calculate the Golden Ratio
                        - Find examples of the Golden Ratio by measuring parts of the body
                      - Find other natural examples of the Golden Ratio
                      - Apply knowledge of these ratios to draw a body to scale

II. Overview

Teacher will initiate a brief review of ratios. Students will explore ratios of body measurements and their relation to the Golden Ratio and Fibonacci Numbers. They will hunt for other examples of Golden Ratios. They will apply their knowledge of the human body to draw a sketch using the ratios.

III. Resources, materials and supplies needed

3x5 index cards

measuring tape (fully flexible to measure around bodies)

or alternatively, string and meter sticks

calculators

overhead projector and blank transparencies

large sheets of paper for each student

Polaroid cameras

Basket with papers containing various heights

IV. Supplementary materials, handouts.

Student Instructions/ Data Sheet, see Appendix A

V. Standards

TEKS

§111.34.(c) Geometric patterns: knowledge and skills and

performance descriptions. The student identifies, analyzes, and describes patterns that emerge from two- and three-dimensional geometric figures.

§111.34 (f) Similarity and the geometry of shape: knowledge and skills and performance descriptions. The student applies the concepts of similarity to justify properties of figures and solve problems.

Engage

Teacher does

Let’s review ratios.

Pass out 3x5 index cards and a ruler.

Today we are going to learn about proportions of the human body.

Let students try out some of these examples.

Probing Questions

If the lengths of a right triangle ABC are 3,4,5 and we have discovered a similar triangle DEF with the ratio of ABC to DEF being 1/25 what are the lengths of the sides of DEF?

Can anyone tell me the ratio of the longer side to the shorter side of the index cards?

If I wanted to shrink the index card by 60%, what would the new measurements be? Would the ratio of the sides be the same?

Has anyone ever noticed specific properties about the measurements of the human body?

Student Does

75, 100, 125

1.666666

1.8x3

yes, the ratio would still be the same.

Various answers: possibly that the length of forearm= length of foot, arm span= height, etc.

Explore

Pass out page where they can fill in measurements of their bodies. They should be in groups of two, helping to measure each other.

Each student does a variety of measurements of their bodies and fills out the sheet. (see attachment).

Explain

Teacher does

Tell the students this ratio is called the Golden Ratio or phi. It is prevalent in nature and art. It is an irrational number (1+ sqrt5)/2 or 1.61803399. The Golden Ratio is thought to be very aesthetically pleasing.

Probing Questions

Can you make any conjectures or come up with some theories about these ratios?

What have we been learning about in the past few days?

Can I have a volunteer come to the board and write the Fibonacci Numbers?

Now, looking at these numbers can we find any thing close to the ratio of body proportions we found?

Lets think back to the 3x5 note card. Doesn’t this ratio seem close to the Golden Ratio? Do you think this is on purpose? Explore board game rectangles, etc at home to see how frequently the Golden Ratio is used.

Student Does

One student from each pair puts the results of their ratio calculations on the overhead.

Students each calculate the averages.

Hopefully, the students will find that the ratios are all near 1.6.

The Fibonacci Numbers.

1,1,2,3,5,8,13,21…

class helps out and the sequence continues.

Students have a few minutes to calculate ratios, and hopefully will realize that the ratio of the (n+1)th number over the nth number approaches the ratio as n gets larger.

On transparency, students calculate ratios of consecutive Fibonacci Numbers and they are shown to move progressively towards phi.

Extend

Students measure and make a list and sketch of any items they find that are in the golden ratio using measuring tape, etc.

Polaroid cameras can be used to take pictures of items. These pictures are later shown to the class and they explain them.

Evaluate

If time does not allow in class, this should be continued for homework. Using paper pencil, ruler and some imagination, consider the golden ratios found in the human body. Draw a body to scale. A basket will be passed around, giving heights of people. The assignment is to use these ratios to approximate a body shape/size. Students must scale down the body size to fit on the piece of paper. Therefore, two ratios will be applied. The drawing should include the ratio of the actual height of the person to the scaled down model.
Appendix A

Name _________________________ Date _____________________

Take the following measurements

B= the height from top of head to bottom of feet

N= Navel height from the floor to the naval of each person in the group

F= the length of an index finger

K= the distance form the big knuckle in the "middle" of the index finger to the finger tip

L= the length of a leg from hip joint to the floor

H= the distance from the hip joint to the knee of the same leg measured above

A= the length of an arm from shoulder to the fingertips

E= the distance from the elbow to the fingertips of the same arm above

C= the distance from the top of the head to the chin

Y= the distance from the center of the eyes to the chin

M= the circumference of the head and call it M.

I= the circumference of the neck and call it I.

_____________