mathematical explorations classroom-ready activities Finding Pi with Archimedes s Exhaustion Method salessandra King Studying historical puzzles can give students a perspective on the usefulness of math as a tool and on the creative aspects of problem solving. Pi Day offers a wonderful opportunity for such inquiry. My students and I have enjoyed retracing and applying some parts of Archimedes s method of exhaustion to measure the ratio of the circumference of a circle to its diameter. Archimedes lived from 287 BCE until 212 BCE and was one of the greatest mathematicians of all times, as well as an extraordinarily creative problem solver. He formulated how to approximate the circumference of a circle using a sequence of inscribed and circumscribed regular polygons whose perimeters converged to the circumference (see figs. 1a b). His system was a precursor to calculus, specifically the concept of limit, which we briefly discussed at the end of our lesson. The main objective of this exploration is to show students how formulas came into existence and how they describe clear mathematical relationships. Middle school students are gen- Edited by Barbara Zorin, drbzorin@ gmail.com, MATHBonesPro, and carrie Fink, cfink@methow.org, Liberty Bell Junior-Senior High School, Winthrop, Washington. Submit manuscripts through http://mtms.msubmit.net. erally familiar with the formula C = pd that connects the circumference of any circle to its diameter through the constant p. However, they may not always understand its real meaning or may not know how this formula was discovered. As students explore the process that led to the discovery of C = pd, they will develop a better appreciation of the nature of mathematics and of the profound connections between the various areas of this discipline. Studying math in context gives students the opportunity to participate in real-life problem solving as well as recognize the usefulness of mathematics. This activity can also offer a glimpse into the intellectual development of some fascinating ideas, a closer look at Archimedes s work, and the occasion to think and communicate mathematically about higher-order mathematical concepts. Ask students to research some aspect of Archimedes s life and the mathematical issues he confronted. Students can then share what they learned with the class. This is generally a wellreceived activity and a good introduction to the history of mathematics. To calculate the value of pi, my students used a method similar to Archimedes s, although it did not involve applying the great mathematician s precise geometric ratios, concepts well beyond the geom- Fig. 1 These diagrams illustrate Archimedes s ideas. (a) (b) etry knowledge possessed by middle school students. Instead, my students used a more experimental approach by directly measuring the lengths of the sides of the polygons using rulers and yardsticks. Activity sheet 1 provides 116 MatheMatics teaching in the Middle school Vol. 19, No. 2, September 2013 Copyright 2013 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
guidelines on how to replicate this process in a classroom. After sharing some information about Archimedes and breaking into groups, students drew a circle using a string of fixed length (the radius) tied to a clasp at the center. To make students calculations easier, we chose radii equal to 1/2 foot, 1/2 yard, or 1/2 meter so that the length of the circumference would (theoretically) equal pi feet, yards, or meters, respectively. Then the students inscribed a regular hexagon. First, they chose a point on the circumference. Then, with the compass open and using the same radius, they pointed the compass at their chosen point and drew a semicircle that cut the circumference at two more points. The students used their rulers and connected each point on the circumference with the center of the circle. They then extended the segment to reach the circumference on the other side of the center, producing three more vertices. They next connected the vertices by drawing a regular hexagon. The students took turns measuring the six sides of the polygon and recorded and added their data, determining the perimeter of the polygon and an estimate of the length of the circumference. Then, the averages of the measurements found by each student in a group were added; the group mean their first approximation of p was then calculated. Each student used table 1 to track results, and each group collected the data from each member in table 2. The students then doubled the number of sides of the polygon to obtain a dodecagon. They produced this shape by drawing (or constructing, for the more advanced students) the perpendicular bisector of each side. (This line intersects the circumference at one point, a vertex of the inscribed regular dodecagon.) Once all the vertices were created and connected, the Table 1 Each student used a grid to track and organize the inscribed polygon data. Type of Polygon Hexagon Dodecagon 24-gon No. of Sides Length of Sides students again measured the perimeter of the dodecagon using the same procedure outlined above. Some groups doubled the number of sides to construct and find the perimeter of a 24- gon (see fig. 2). Using Archimedes s method in its entirety, the students could also construct circumscribed polygons. This part of the activity, and the corresponding approximation of p, is described in activity sheet 2. Activity sheet 3 invites students to think in depth about Archimedes s procedure for approximating p and to ponder its creativity and meaning. I found that students realized that as the number of sides of the inscribed and circumscribed polygons increases, Archimedes s procedure would produce a lower and an upper limit for the value of p. Students concluded that as the number of sides increases indefinitely, the perimeter of the polygons approaches the circumference, and their area, the area of the circle. The Reflections section can also help assess students learning. In our Average Length Perimeter Value of p Table 2 Each group s inscribed polygon data were also contained in a table. Type of Polygon No. of Sides Perimeters Hexagon Dodecagon 24-gon Average Perimeter Value of p Fig. 2 Some groups doubled the number of sides to construct and find the perimeter of a 24-gon. case, we discussed the activity in class as a group, thus providing a good gauge of the students understanding. It also helped clarify their thoughts and ideas. Each student could then write their answers and submit them either individually or as a group. To promote this activity, students descriptions and Vol. 19, No. 2, September 2013 Mathematics Teaching in the Middle School 117
Fig. 3 The variation in the readings and measurement errors often surprised students. (a) extension activities (available online). In closing, my students responded enthusiastically to this exploration, which provided a very intuitive and logical introduction to basic concepts of calculus, such as that of limit. This approach not only places mathematics in context and highlights the connections and the creativity involved in complex problem solving but also offers a glimpse of higher-order thinking. It also helps demystify some advanced mathematical concepts. BIBLIOGRAPHY Boyer, Carl. 1949. The Concept of the Calculus. Wakefield, MA: Hafner. Dunham, William. 1990. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley.. 2010. Great Thinkers, Great Theorems (The Great Courses). Chantilly, VA: The Teaching Company. Edwards, C. Henry. 1979. The Historical Development of the Calculus. New York: Springer-Verlag. Heath, Thomas. 1981. A History of Greek Mathematics. New York: Dover. reflections could be used in a morning assembly, school newspaper, class blog, or newsletter. The experimental, hands-on approach that we used to find the perimeter of the various polygons by directly measuring the sides with rulers and yardsticks presented a few challenges and corresponding learning opportunities. Students found out that they needed to be precise when drawing. Some groups decided to practice the constructions a few times on separate sheets of paper; others asked the most meticulous student to complete the drawings. (b) A more important intellectual hurdle was the presence of measurement errors, as middle school students are often surprised by the variation in the readings (see fig. 3) and by the notion of significant digits. Students, especially at the middle school level, are often exposed only to problems with precise numerical answers and do not expect to find errors in a math problem. These complexities led to interesting conversations and to a desire to understand how wrong our finding of p was or how far our estimated value was from the generally accepted value, which we studied with the Alessandra King, alessandra.king@ holton-arms.edu, teaches at the Holton- Arms School in Bethesda, Maryland. She has taught mathematics and physics at the middle school and high school levels in Europe, Asia, and America, and is interested in the cultural and historical development of mathematical concepts and their applications. Download a free app for your smartphone, then scan this tag to access an activity sheet extension for Mathematical Explorations at http://www.nctm.org/mtms054. 118 Mathematics Teaching in the Middle School Vol. 19, No. 2, September 2013
activity sheet 1 INSCRIBING POLYGONS We are going to use a method similar to that of Archimedes to estimate the value of p, a mathematical constant that represents the ratio of the circumference to the diameter for any circle. Archimedes inscribed and circumscribed polygons of increasing number of sides to a circumference. Materials: A copy of this activity sheet Poster board with the drawing of a circle of diameter 1 foot, 1 yard, or 1 meter Compass Paper and pencil Ruler or yardstick Calculator Part 1: Inscribing a Hexagon within a Circle 1. Using a compass a. Mark any point on the circumference as your starting point. b. Open your compass, using the same radius of your circle. c. Set the compass point on your chosen point and draw a semicircle that intersects the circumference in 2 more points. You should now have 3 points on the circumference. d. Using your ruler, connect each of these points with the center of the circle and extend each segment to intersect the circumference on the other side of your starting points. You should now have 6 points on the circumference. e. Connect each point with its adjacent vertices to produce a regular hexagon. 2. Using a protractor OR a. Draw a diameter. The 2 points where the diameter intersects the circumference are 2 vertices of the hexagon. b. Measure an angle of 60 degrees at the center and mark that point. c. Connect that point to the center and extend the segment both ways so that it intersects the circumference in 2 points. These are 2 more vertices of the hexagon. d. Measure an angle of 120 degrees at the center and mark that point. e. Connect that point to the center and extend the segment both ways so that it intersects the circumference in 2 points. These are the last 2 vertices of the hexagon. f. Connect the 6 vertices to build a regular hexagon.
activity sheet 1 (continued) Part 2: Find the First Approximation of p 1. Measure the length of each side of the hexagon and report your data in the table below. 2. Find the average length of the hexagon sides and add that information to the table. 3. Find the perimeter of the hexagon and add it to the table. 4. Calculate the approximate value of p, using the hexagon perimeter as an approximation for the circumference of the circle. Inscribed Polygon Data Type of Polygon No. of Sides Length of Sides Average Length Perimeter Value of p Hexagon 6 Dodecagon 12 24-gon 24 5. Explain why this will be an underestimate of p. Part 3: Inscribe a Dodecagon First, you will learn to construct a perpendicular bisector of a line segment, which is the perpendicular line that divides the segment in half. 1. Using a compass a. Open your compass a little longer than half the line segment you want to bisect. b. Place the compass point on 1 of the endpoints of the segment and draw a semicircle. c. Without changing the radius, draw another semicircle, centering it on the other endpoint of the segment. d. The two semicircles intersect at 2 points. Draw the line that connects these 2 points: This line is the perpendicular bisector of the segment.
activity sheet 1 (continued) e. Now, construct a perpendicular bisector for each side of your inscribed hexagon. f. Each perpendicular bisector intersects the circumference at a point. You now have 12 points on the circumference: the 6 vertices of the inscribed hexagon and the 6 intersections with the bisectors. Connect the 12 vertices to construct a regular dodecagon. 2. Using midpoints and the diameter OR a. Find the midpoint of 3 consecutive sides of the hexagon and connect them to the center of the circumference. b. Extend these 3 segments both ways, drawing diameters. The 2 endpoints of each diameter make 6 vertices. You should now have 12 points on the circumference: the 6 vertices of the hexagon and the 6 new points. c. Connect the 12 vertices to build the dodecagon. Part 4: Find the Second Approximation of p 1. Repeat the steps in part 2 for the dodecagon for a new approximation of p and complete the dodecagon row of the table. Part 5: Comparing Estimations 1. Repeat part 3 to construct a regular 24-gon in the circumference. Using the same method, fill in the 24-gon row of the table.
activity sheet 2 CIRCUMSCRIBING POLYGONS Part 1: Circumscribe a Hexagon 1. Before circumscribing a hexagon, you need to construct a tangent line as described below. a. Draw a diameter of the circle and extend this line beyond the circumference. b. Open the compass to a measure smaller than the radius of the circle. Then, setting its point where the extended diameter intersects the circumference, draw a circle. c. This circle will intersect the extended diameter at 2 points. Using these 2 points as the endpoints of a segment, construct the perpendicular bisector to the segment. This perpendicular bisector is the tangent line to the circle. 2. Follow the instructions in activity sheet 1 to construct an inscribed regular hexagon but do not join the vertices. 3. Instead, extend the 3 diameters beyond the edge of the circle and construct the tangent lines at each of the points in which the diameters intersect the circumference. The 6 tangent lines intersect one another at the vertices of the circumscribed hexagon. Part 2: First Approximation of p 1. Measure the length of each of the sides of the hexagon, and report your data in the table below. Circumscribed Polygon Data Type of Polygon No. of Sides Length of Sides Average Length Perimeter Value of p Hexagon 6 Dodecagon 12 24-gon 24 2. Find the average length of the hexagon sides and write it in the same table. 3. Find the perimeter of the hexagon and add that information in the table as well. 4. Calculate the approximate value of p using the hexagon perimeter as an approximation for the circumference of the circle. 5. Explain why this will be an overestimate of p.
activity sheet 2 (continued) Part 3: Circumscribe a Dodecagon 1. Using the circumscribed hexagon that you have, draw 3 lines connecting 2 opposite vertices that pass through the center of the circle. 2. Each of these lines intersects the circumference in 2 points. There are 6 points. Construct the tangent lines to the circle through each of these points. You now have 12 tangent lines: 6 from the hexagon and 6 new ones. These tangent lines intersect and describe a regular dodecagon. Part 4: Find Your Second Approximation of p 1. Repeat part 2 for the dodecagon and complete the dodecagon row of the table. Part 5: Construct a Regular 24-gon 1. Repeat part 3 to construct a regular 24-gon and complete the 24-gon row of the table. REFLECTIONS 1. a. Explain why successive approximations of p from inscribed polygons of increasing numbers of sides increase in value each time. b. Explain why successive approximations of p from circumscribed polygons of increasing numbers of sides decrease in value each time. 2. Explain why an approximation of p from an inscribed regular polygon of any number of sides will be less than that from a circumscribed regular polygon of the same number of sides. 3. Could the approximation from circumscribed polygons get to any number, no matter how small, simply by increasing the number of sides enough times? Could the approximation from inscribed polygons get to any number, no matter how large, simply by increasing the number of sides enough times? Explain your answer. If not, do the perimeters get closer to a particular number? 4. In your words, summarize Archimedes s method to find the value of the number p. What do you think of this method? 5. Mathematically, we can say that as the number of sides of the inscribed polygons increases, the perimeter of the polygon provides a lower limit for the value of p. What do you think this means? 6. Mathematically, we can say that as the number of sides of the circumscribed polygons increases, the perimeter of the polygon provides an upper limit for the value of p. What do you think this means? 7. What are the lower and upper limits for your approximation of p? 8. Extra credit: Show that the hexagon constructed in activity sheet 1 is indeed a regular hexagon.