Lesson plan based on Cycladic Figure

Conduct an experiment to determine the frequency of pulling various parts of an ancient Greek sculpture out of a hat.

Skills and Focus: Geometry, Calculation

Subject Area: Mathematics

Thematic Connection: Counting and Calculating, Identifying Patterns

Grade Level: Secondary School

Time Needed: 90 minutes

Objectives

• Conduct a statistical experiment and evaluate its results.

• Introduce basic statistical methods.

Instructional Materials Needed

Cycladic figure

chart

Scissors

Paper and pencils

Shennan, S. Quantifying Archaeology, 2nd ed. Iowa City: University of Iowa Press, 1997.

Activity:

Step 1: Have students divide the Cycladic figure into 12 simple geometric forms (rectangles, triangle, ovals). Have each student cut up one figure according to these divisions.

Step 2: Record the number of rectangles, triangles, and ovals that result. Then distribute the chart.

Step 3: Put all the parts into a hat. Discuss the chances of pulling a head or a thigh out randomly (1/12, 1/6 and so on). Ask students to speculate whether the odds would change if you put parts from several figures into the hat.

Step 4: Pull out ten items from the hat, one at a time, and record them. After pulling a piece out, replace it so the odds of pulling it or another type out do not change. Tally the results and compare them to the predictions made before. If the predictions were wrong, why might they be wrong? Try the same experiment again and compare results.

Step 5: Use the c2 test to determine whether or not the results obtained are statistically significant.

Determine the c2 values for each type, with Oi = obtained results and Ei = estimated results. Estimated results are indicated on the chart. Thus if two heads were pulled out of the hat in ten tries the value would be (2-.825)2 2 = 0.69. Adding up the values for each category results in a c2 value for the trial. Now students must determine the degrees of freedom (n) in order to establish significance.

where k is the number of categories being considered.

Thus n would equal 7 if one were deciding on significance according to body parts, but would equal 2 if one were establishing significance with respect to shape. If a table of c2 values is available, have students compare the results to those on the table. If not, the c2 value (at the 0.05 confidence level) for n = 2 is 5.99 and for n = 7 is 14.06. If the c2 value is higher than this then the null hypothesis (that the distribution of samples removed is proportional to their proportions in the population as a whole) is correct. If not, then the null hypothesis is void, and there is some bias affecting the results.

Step 6: Discuss the results.

Critical Thinking Ask students to

• explain what c2 represents.

• explain why the c2 value would be different for different degrees of freedom.

Goals

This activity meets Illinois State Goal 6: Demonstrate and apply a knowledge and sense of numbers, including basic arithmetic operations, number patterns, ratios, and proportions.

This activity meets Illinois State Goal 10: Collect, organize, and analyze data using statistical methods to predict results and interpret uncertainty and change in practical applications.

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