## The Golden Student

### Kevin HollandSanta Rosa District Schools

#### Description

This is an enrichment activity for the enhancement of the study of ratios and data collection. Students are introduced to the golden section in mathematics and use this ratio to determine if their bodies are -golden- through a group investigation.

#### Objectives

Understands that numbers can be represented in a variety of equivalent forms using integers, fractions, decimals, and percents, scientific notation, exponents, radicals, absolute value, or logarithms.

Adds, subtracts, multiplies, and divides real numbers, including square roots and exponents using appropriate methods of computing (mental mathematics, paper-and-pencil, calculator).

Relates the concepts of measurement to similarity and proportionality in real-world situations.

represents and applies geometric properties and relationships to solve real-world and mathematical problems including ratio, proportion, and properties of right triangle trigonometry.

Interprets data that has been collected, organized, and displayed in charts, tables, plots.

#### Materials

-Copy of the video Disney’s DONALD DUCK IN MATHMAGIC LAND, which can usually be obtained through your district media services or from one of the local elementary/middle school media centers.
-Measuring tape for each cooperative group
-Calculator for each cooperative group
-Worksheet for each student
-Poster board and markers
-Pictures of ancient Roman architecture, statues, or objects in nature which might have the -golden ratio- look. (Optional)

#### Preparations

1. Obtain a copy of the video -Donald Duck in Mathmagic Land-.
2. Obtain a measuring tape for each group of students you will have.
3. Obtain VCR/Television for the class period.
4. Copy worksheets for all students.
5. Teach a lesson on ratios and equivalent forms before the activity.

#### Procedures

**This lesson activity is a modification from an activity used by Michael Serra’s book DISCOVERING GEOMETRY: AN INDUCTIVE APPROACH (Key Curriculum Press, 1993). Michael Serra has some great information and extensions for this activity.

Prior Knowledge: The teacher will have taught a basic understanding of ratios, including equivalent forms of ratios such as fractions to decimals. Review how to construct a histogram (bar graph) with the class. The concepts of proportionality and similarity could also be introduced.

1. Hook the students with the question of -Which of you are -Golden Students? Today, we will find out who is golden.-
2. Show the video DONALD DUCK IN MATHMAGIC LAND up to the point where Donald is trying to fit into the golden rectangle.
3. Explain that as the students observed from the video, there are many things in nature which have proportions of the golden ratio. Divide students into cooperative groups to collect data from body measurements to form ratios. Students will analyze these ratios to determine which student is the closest to being golden.
4. Divide the class into groups of 3 or 4 depending upon the number of measuring tapes available. Have the students measure and record the different body lengths given on the worksheet. Round to the nearest ˝ inch or centimeter.
5. When they have completed all measurements, direct students to convert these ratios into their equivalent decimal forms using the calculator. Round to the nearest thousandth.
6. Tell your class that the golden ratio is equivalent to 1.618. As a class, decide a reasonable range of golden (ie: 1.568 to 1.668). Have the students circle the ratios on their lists which are -golden-.
7. Decide as a class, which student(s) are golden. This is usually the person with the most ratios circled by his or her name.
8. Continue the lesson by having the students make a class histogram/bar graph of a body ratio.
9. Have the groups collect ratios from all of the students and make a large class histogram using the poster board and markers. Discuss the characteristics of a good graph such as title, axis names, and scales.
10. Have students present their poster board in front of the class or display the poster boards around the room.
11. Summarize the activity by revisiting ratios, equivalent forms, and the golden ratio.
An added enhancement would be to show the pictures of various items that might be golden.
12. Reward the class by showing the end of the video.

#### Assessments

Observe group participation and collect student work. The students can be graded by the given assessment rubric at the bottom of the worksheet. A group grade for the histogram poster may also be given.

#### Extensions

Have students research and report on one of the following related topics: Golden Ratio, Golden Rectangle, Fibonacci numbers, architecture, mathematics in art, etc.