Exploring Pi using the computer in middle school mathematics

School Science and Mathematics, Nov 1996 by Pyzdrowski, Laura, Holtan, Boyd

The National Council of Teachers of Mathematics (NCTM) in its Curriculum and Evaluation Standards for School Mathematics encourages a middle school curriculum that integrates technology. It recommends that students should be able to identify and use functional relationships and make connections among seemingly diverse concepts and topics. In this activity of exploring the derivation of Pi, students take a constructivist role by collecting data and making conjectures. Using the data they construct graphs and tables and discourse about appropriate algebraic representations. The computer is used as an instructional aid enabling the students to view the data in a variety of forms. They are encouraged to communicate about the connections among the various representations.

The National Council of Teachers of Mathematics (NCTM) encourages a curriculum that integrates technology (1989). One recommendation in the 5-8 standards is that every classroom have atleast one computer and that additional computers should be available for individual, small-group, and whole-class use. The necessity to memorize algorithms has decreased while the skills and understanding required to think critically and use technology have increased. The Council recommends that teachers increase the attention given to the communication skills of students. Students must be encouraged to read, write, listen to, and discuss mathematics. Students need to be able to identify and use functional relationships as well as to develop and use tables, graphs, and rules and make connections among the three. Problem solving must play a major role in the classroom. Real life situations should drive the mathematics when appropriate, and open ended questions and extensions to the original problem are essential.

An activity that encompasses many of these recommendations is one in which students study Pi by using a computer graphing utility as an aid. The materials required for the initial activity are circular regions such as jar lids and coffee cans, yam, a sheet of paper large enough to accommodate a graph of the data collected in the activity, scissors, tape, markers, and a yard stick or meter stick. To begin the activity, students collect data by recording the circumference and the diameter of various circular objects. On the large paper, students should draw axes and identify the yaxis as the circumference and the x-axis as the diameter. They should then place a circular region on the x-axis beginning at the origin and mark its diameter on the x-axis. The circumference of this circular region is measured by wrapping yam once around the circular object and cutting it to this length. The students should then fasten, with tape, the circumference yarn vertically on the paper beginning at the diameter marking on the x-axis. They should proceed to measure the diameter and circumference of several other circular regions in the same manner. Figure 1 illustrates the yarn graph. Although time may be taken for the class to discuss the important parts of a graph, the focus of this step in the activity is to record and visualize the data.

The yam graph is a good first step in the activity because it allows the students to conjecture that a linear relationship seems to occur. Noticing this relationship is important. Later, when the computer graphing utility is used, the line of best fit will be plotted.

Next the students in the class make a table using the measurements for the diameter (D) and circumference (C). Table 1 illustrates a sample recording of measurements. The teacher should encourage discussions among the students concerning the accuracy of the measurements. The teacher may allow the students to decide the acceptable accuracy based on the materials being used. These first parts of the activity may be done in small groups so that the class may have several charts and tables to compare for discussions. Encourage the students to communicate within groups about mathematical ideas and then communicate the group findings to the class. Measure as many circular regions as possible to allow for more data. Students should look at the data in their tables and conjecture about relationships between the circumference and the diameter as the circle size changes. Once the students have conjectured that the ratio of the circumference and the diameter of all circular regions seems to be constant, the class is ready to proceed to the next section of the activity.

Depending on resources students may work in pairs at computers, or the teacher may use a computer for class presentation. Using a computer graphing utility such as DERIVE, plot the class data of related diameter and circumference recordings. Figure 2 shows the DERIVE plot of the data found in Table 1. Once again the students should notice that the points seem to form a straight line. Use this opportunity to have the class discuss the points that do not fit into the linear relationship. Discuss where errors may have originated. Additionally, check for data entry errors and accuracy.