How "Phi" Is My Face?     
by Suzanne Alejandre

Teacher Lesson Plan


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NCTM Standards:
Number and Operations:
      Understand numbers and relationships among numbers
      Understand and use ratios and proportions to represent quantitative relationships

Measurement:
      Understand processes of measurement
Geometry:
      Specify locations and describe spatial relationships using coordinate geometry

Objectives:
      Students use pixel measurements to calculate the ratios of different face measurements.
      Students determine how close a ratio is to the golden ratio or Phi.

Part I:

In the art world, if a face has certain measurements that are close to the golden ratio they are pleasing. This idea can be introduced to students in a variety of ways. The Web site, Phi: The Golden Number, has a lot of information on this topic.

There are many rectangles that could be used to illustrate this idea but the two that will be measured in this activity are illustrated here:



Top of head to chin.
Side to side.


Side to side at eye level.
Eye level to chin.

Students start the activity by opening the Java Applet Index page.

Note: It will open in a separate window. If you are displaying the page for students, arrange your browser windows so that the applet and the directions can be easily viewed. If students are working individually they should be encouraged to do this.

Ask the question:

Can you determine the "pixel" distance between two points?

Prompt students to notice the:

  • display of x (in pixels): How can you calculate how long a line segment is in pixels if you know the x coordinate of both endpoints of that line segment?

  • display of y (in pixels): How can you calculate how long a line segment is in pixels if you know the y coordinate of both endpoints of that line segment?

Students should be aware of the following:

If the line segment is horizontal, pay attention to the x-coordinates. If the line segment is vertical, pay attention to the y-coordinates.

Have the students work through the first example as a class, in groups, pairs or individually. Sample data is given for Annie's Face on the student page.

Part II:

Once students have the technique established, they can continue the activity by finding the measurements for all the photos and deciding which is the most perfect using the given criteria.

Assessment:

Ask students to generalize this method and write the steps needed to calculate the two different ratios.

Extension:

A fun project would be to have the students' photographs instead of strangers' photographs. If you or your students have access to a digital camera, take photographs of the students.

Jill Britton tried this activity and wrote some excellent directions to follow. Please view: How to Replace the Photo in This Applet with Your Own.

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