Download - The Archaeological Dig Site: Using Geometry to Reconstruct the Past

The Archaeological Dig Site: Using Geometry to Reconstruct the PastAuthor(s): Patricia S. Moyer and Wei Shen HsiaSource: The Mathematics Teacher, Vol. 94, No. 3 (March 2001), pp. 193-199, 206-207Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/20870633 .

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/Activities Patricia S. Moyer and Wei Shen Hsia

roc

The Archaeologicpl^Dij Using Geometry In secondary mathematics, students often see]

connection between geometry and the realjfe mathematical situations around them. When.

asked to describe geometric figures, their desc tions are sometimes no more than an identifi of sides and angles. They have not had e? in using more than one property in a mafl situation or in describing how two geomet erties are related. The van Hiele model off dents learn geometry proposes that studen

standings of geometry move from recognition! description to analysis (Fuys, Geddes, and

'

1988). For students to make this transition to I lytic thinking, teachers need to create problem Sit uations that enhance development of students' intuitive understandings. These investigations allow students to explore relationships among geoA metric shapes and to make conjectures about prop-] erties. The conjectures can then be stated formally as theorems.

Since many secondary school students enter foj mal geometry classes without adequate inform^p geometry experiences (Senk 1989), teachers ]

engage students in investigations that bridgej gap between their naive notions of geometry i the more formal relationships that we want Aem to

comprehend. Although students may lack the

vocabulary and conceptual understanding to

express geometric relationships in formal terms, teachers can develop their repertoire by furnishing

meaningful investigations that begin at the very basic level and move to more complex analytical activity.

The investigation described in this article is one in which students apply very basic understandings of geometric properties to investigate polygons and

iti ttruct the Past

ami

luces a method for

^e measurements of ration of the circular

use several kinds of math

!ssed in Standard 3: Geometry ense in Principi? $ Standards for

?cs(NCTM2000J.r on the following ex

Ses 9-12: "students] os (including cong

sesoftwo-j explc similarit dimensional about them, .establish the

faction^ maAJ^tfcersjTWuse guoinuuii luuau lo solve

problems in, and gain insight into, other disciplines and other areas ofinterest such as art and architec

E>. 308). the investigation, the students

japping a dig site for examina sfer scale drawings of a

gized dig site by using string, jvices.

In the second part , stiments are given broken facts and asked to use mea

twn.Tk

given pIF stakes, of the id pieces of circular

33 meast)

the broken pieces to determine the

Patricia S. Moyer, [email protected], teaches mathematics education courses at George Mason University, Fairfax, VA 22030-4444. Her research interests include teachers' and students' uses of representation in mathematics and teacher development Wei Shen Hsia, [email protected], is the chair of the department of mathematics at the Uni versity of Alabama, Tuscaloosa, AL 35487. His research interests include operations research and problem-solving techniques for secondary school mathematics teachers.

'6

Students often see little connection

between

geometry and

real-life situations

Edited by A Darien Lauten, [email protected], Rivier College, Nashua, NH 03061 This section is designed to provide in reproducible formats mathematics activities appropriate for students in grades 7-12. This material may be reproduced by cfassroom teachers for use in their own classes. Readers who have developed success ful classroom activities are encouraged to submit manuscripts, in a format similar to the "Activities" already published, to the senior journal editor for review. Of particular interest are activities focusing on the Council's curriculum standards, its expanded concept of bask skills, problem solving and applications, and the uses of calculators and computers. Write to NCTM, attention: infocentral, or send e-mail to [email protected], for the catalog of educational materials, which lists compilations of "Activities" in bound form.?Ed.

Vol. 94, No. 3 ? March 2001 193



This activity serves as an

entry point for

discussions

about

geometric

relationships

original circumferences of the artifacts.

Moyer, a coauthor of this article, originally designed this lesson with colleagues as an integrat ed unit involving mathematics, science, history, and

language arts. Students mapped the archaeological

dig site, collected artifacts placed in the site by the

teacher, conducted scientific and mathematical

analyses on the artifacts to determine information

about them, and used language-arts skills to

describe and classify the contents of the dig site.

This article begins by describing the first part of the investigation, in which students construct the

dig site; it can be adapted for students in the mid

dle grades. The second part of the investigation provides a bridge between basic geometric concepts learned in the middle grades and the more

advanced topics of high school geometry. For all

grades, this activity serves as an entry point for

discussions about geometric relationships and fur

ther investigations of properties of circles.

TEACHERS' GUIDE The purpose of this activity is to engage students in an investigation that models a real-life mathemati cal situation and that explores the properties of cir

cles in a meaningful follow-up experience. It can be

completed in two or more class periods. The first class period should be scheduled for constructing the polygon, or "dig site," in an outdoor area. Dur

ing the second class period, students can investi

gate the artifact pieces. If a grassy area is not read

ily available or if the weather does not cooperate on

the date of the dig-site construction, the activity can be completed indoors using string and masking tape to construct the polygon on the floor or on

large pieces of chart paper.

Parti

Measuring angles is an important skill for this

activity. Because students need to use stakes and

string to measure angles at the dig site, a practice session with these materials and a review of angle

measurement is helpful. Divide the students into

groups of three; each group needs a protractor and a piece of string that is approximately eight feet

long. One student in each group is the vertex of the

angle, and the other two students are points along the rays of the angle, as shown in figure 1. The

students at the vertex can use pencils or similar

straight objects to hold the strings horizontally in front of themselves. When the groups are ready, call out angle measures and ask students to create

the given angle with the string and protractor. The

students holding the strings along the rays of the

angle should move from side to side until the group member at the vertex of the angle indicates that

they have obtained the correct angle measure. The

Fig. 1 One student is the vertex of the angle, and

the other two are points along the rays of the angle.

students holding the rays of the angles must esti mate the size of each angle, and the student at the vertex must use the protractor to measure the

angle that the string forms and give directions to

group members to move until the measure of the

angle is accurate. This method helps students become familiar

with the estimated measures of angles and gives them practice using a protractor. Students should take turns being the vertex and the points on the

rays of the angle. As students work, you should walk around to be sure that they are correctly read

ing their protractors. You should include such angle pairs as 70 degrees and 110 degrees so that stu

dents can discuss how they know which is which.

Having one group do 70 degrees and another do 110

degrees might also be useful. The class could then discuss common protractor errors.

After students have reviewed measuring and

constructing angles, the real lesson can begin. Dis cuss careers that involve measurement and geome

try skills, such as surveyor, archaeologist, architect, and engineer. Tell the students that they will work in teams of three to five people to construct a life sized dig site in an outdoor grassy area; then give each team a drawing of its dig site?a scale draw

ing of a four- or five-sided polygon that indicates the lengths of the sides in meters. See figure 2. Before students begin constructing the dig site,

they should measure the angles of their dig-site drawings on their papers. Students should review the definition of similarity and observe that pre

serving angles is part of the definition of enlarging. Therefore, the measure of each angle on the scale

drawing remains the same even when the figure is

enlarged to create the dig site. Students may also use large pieces of chart paper to practice enlarging the scale drawings. Several trials indoors may increase the accuracy and precision of their con

194 MATHEMATICS TEACHER



Fig. 2 A Student measures the angles of the dig site

on the scale drawing.

structions of the outdoor dig sites. The following materials should be ready for con

structing the dig site: balls of string; wooden or metal stakes, or large ten-inch nails if the area is

Fig. 3 ~

Students gather their materials for the dig site.

grassy; hammers for pounding the stakes into the

ground; protractors, trundle wheels, or tape mea

sures; and metersticks. See figure 3. After these items have been obtained and scale

drawing measurements have been made, students can go to a designated grassy area where each

group has enough room to construct a dig site. Students use the scale drawing and its measure ments to create the dig site with stakes, string, and measuring devices. See figure 4. Students

might discuss a variety of strategies to begin the

dig site. After putting the first stake in the ground, some groups might begin by measuring the angle and others might begin by measuring the string's length to create the sides.

Fig. 4 Students construct their dig site.

When the first stake is in the ground, students can wrap the string around it, and two members of the team can extend the lengths of string that rep resent the lengths of the sides. Students may dis cover that the way that the string is wrapped around the stake determines the location of the ver tex of the angle. See figure 5. For example, if the

string merely wraps around the back of the stake, the angle measure must be taken from the back of

Hard to measure

Easier to measure

Fig. 5 Wrapping the string around the front of

the stake makes the angle easier to measure.

Vol.94,No.3 March2001



the stake. If the string wraps around the stake and crosses in the front of the stake, the vertex of the

angle shifts to the front of the stake. From previous experience, we have found that wrapping the string around the stake so that the string crosses in the front creates fewer difficulties and facilitates accu

rate measurements of the angles. Students should then measure angles and sides

and use the protractors, trundle wheels, stakes, and

string to create the sides of the polygon. This process should continue until all sides and angles have been measured and the dig site is an exact transla tion of the scale drawing. One benefit of this type of

activity is that creating the polygon on the grass

requires that the students work in teams. While one student is checking the measure of the angle at the vertex, other members of the team must hold the lengths of the string to give the correct angle measure for each side. A coordinated effort is need ed for students to accurately obtain the lengths of the sides and the measures of the angles.

If the scale drawings are triangular, then

enlarging the lengths proportionately will guaran tee similarity. However, the example here employs a quadrilateral. When the polygon has more than three sides, teachers should address the issue of

proportionality and how it can affect the final dimensions of the dig site. Minor errors in measure ment can cause major differences when students

attempt to connect the string to the stake on the final side of the dig site and find that the measure ments do not match. When students measure the last side and angle, which should connect the

strings and close the polygon, they often find that the length of the side or the measurement of the

angle are not even close to the measures that they intended. Such a result causes a dilemma for the

group and stimulates discussion, since students must remeasure the lengths of the sides of the fig ure and the angles to see where they went wrong in their construction. A team effort is needed for stu dents to work together to make the precise modifi cations necessary to adjust the measures of the

angles and the sides to match them exactly to their scale drawing.

Teachers may design several dig-site diagrams that have four or more sides so that students are

forced to work with the variations in angles. Teach ers may also want to include an example that is a nonconvex polygon so that students must construct a reflex angle, that is, one that is greater than 180

degrees.

Part 2

After the large figure has been constructed, students can complete the second part of the activity by inves

tigating the mathematical properties of circular

objects placed in the archaeological dig site. The

purpose of this part of the activity is to use the mathematical properties of a broken piece of a plate or pottery to determine its original circumference. Because students are to find relationships and veri

fy their conjectures during this activity, the teacher should not describe the properties of intersecting chords before completing this activity. Students must draw their own conclusions from the guided investi

gations on the activity sheets. With just a small broken piece of the plate, students should be able to reconstruct the original dimensions of the plate.

In this part of the activity, students explore the

properties of circles by using broken pieces of circu lar artifacts from the dig site. To make the activity authentic, obtain a variety of circular dishes or pot tery objects at a thrift or second-hand store. Because circular dishes of different sizes should be

used, plates work particularly well. Broken or

chipped dishes, or any dishes in poor condition, suf

fice, since the teacher should break these artifacts and place them in the dig site for students to inves

tigate. Before constructing the dig site, use a ham mer to break the circular objects. Keep only those

pieces that include a portion of the circumference of the original object. Place pieces from various

objects in each of the dig sites so that students have a variety of objects to measure. These objects are

used for the mathematical investigation of circles that is described on the activity sheets.

The activity sheets guide students through an

investigation in which they reconstruct the circum ference of a circle from a given piece of the circle. On sheet 2, students make an observation about the products of the lengths of the segments of two

intersecting chords in a circle. They find that the

products of these segment lengths are equal. Stu dents next verify their observation by investigating and applying what they have learned to other cir cles. Students can share the patterns that they find

during a brief class discussion. The class can then

agree on a likely conjecture that the group can

apply to do the rest of the work. Such a conjecture provides evidence to support a conclusion that the observation holds true for a circle.

On sheet 3, students make a second observation about the midpoint of a chord and its relationship to the center of the circle. They find that a perpen dicular line through the midpoint of the chord

always passes through the center of the circle. This result allows students to determine the diameter and radius of each circle.

On sheet 4, students apply the information that

they have learned to reconstruct a simulated arti fact. Discussing the first question on this sheet is

important. Elicit all students' ideas, and then advise them of the method that they should employ for the rest of their work. They should use the

length measurements of the segments of a chord

MATHEMATICS TEACHER



divided at its midpoint to calculate the length of a

segment that passes through the center of a circle, thus determining the diameter and radius of the

circle. They can then make a few minor calculations to determine the circumference of the original object.

Students next apply their mathematical investi

gations and conjectures to the broken artifact

pieces found in the dig site. Students trace the arc on the broken piece onto a sheet of paper. Tracing the arc from the artifact onto the paper makes sub

sequent measurements easier to complete. Stu dents then measure the chord from the given arc, find its midpoint, and determine the diameter of the circle. By applying this information, students find the circle's center and radius and can then use a compass to draw the circumference of the circular artifact. In doing so, students have used mathemat ics to essentially reconstruct the artifact.

Although the artifact-reconstruction process shown here is mathematically accurate, today's archaeologists employ tools and a much more effi cient process to determine these measurements.

Archaeologists use a large grid of concentric circles with radii at fixed intervals. The archaeologist visu

ally matches the broken pieces of the artifact on the

grid to determine the original circumference and diameter of the circular artifact.

SOLUTIONS Most answers depend on the size of the figures and artifacts provided.

The answer to question 2 on sheet 2 is that the two products are equal. The answers to question 5 on sheet 2 are as follows: ED = 20, AE = 0.9, EB = 14.76, and CE = 7.5.

The answer to question 1 on sheet 3 is that the

perpendicular to the chord passes through the center. To answer question 1 on sheet 4, you could con

struct a chord of the original circle on the arc of the artifact piece and then construct a perpendicular bisector of that chord. The center of the circle is somewhere along that bisector. Draw a different

chord, and do the same thing. The bisectors inter sect at the center of the circle. You can then draw the original circle. You could also construct a chord on the arc of the artifact and then construct any other chord. Measure the three known lengths, then apply the conjecture that you made in sheets 1 and 2 to find the fourth length.

REFERENCES Fuys, David, Dorothy Geddes, and Rosamond Tischler. The Van Hiele Model of Thinking in Geometry among Adolescents. Journal for Research in Mathematics Education Monograph Series, no. 3. Reston, Va.: National Council of Teachers of Mathematics, 1988.

National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000.

Senk, Sharon L. "Van Hiele Levels and Achievement in Writing Geometry Proofs." Journal for Research in

Mathematics Education 20 (May 1989): 309-21.

The authors extend special thanks to Wesley Gordon and his geometry class at Holt High School in

Tuscaloosa, Alabama, for participating in this

investigation; to Wesley Gordon and Elizabeth Milewicz for their feedback on the manuscript; and to Eugene Futato, senior archaeologist at the

University of Alabama Museums. @

(Worksheets begin on page 198)

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Do You Have Problems? If so, why not share them! Of course, we are talking about

mathematics problems appropriate for the "Calendar that appears each month in the Mathematics Teacher.

Problems can be submitted by individuals or by groups. We especially encourage high school mathematics departments to

work collaboratively to produce a set of "Calendar" problems. Your name will be mentioned when your problems are published.

Please request guidelines for submission from the senior jour nal editor, Mathematics Teacher, National Council of Teachers of

Mathematics, 1906 Association Drive, Reston, VA 20191-9988. You can also request these guidelines by e-mail to [email protected] or by calling NCTM's Fax-on

Demand Service Center,

(800) 220-8483, and

requesting document 803.

Vol. 94, No. 3 ? March 2001 197



DIG-SITE CONSTRUCTION SHEET 1

1. Gather the following materials for construction of the dig site:

String Wooden or metal stakes Hammers Protractors

Long measuring tapes, trundle wheels, or metersticks

2. In groups of three, with one student at the vertex and the other two students at the legs of the angle, use a five-meter-long piece of string and a pencil to create the following angle measurements:

30? 45? 70? 90? 110? 135? 165?

3. Using the scale-model drawing below or the one assigned by your teacher, measure the angles of the drawing. Construct a dig site using these angle measures, the length measures of the sides indicated on the drawing, and the materials that you have gathered. The string repre sents the sides of the figure, and the stakes represent the vertices and hold the figure in place on a grassy area.

7 meters

5 meters

4. After constructing the dig site, answer the following questions on a separate sheet of paper:

Explain the method that your group used to construct the dig site.

Did an approach that your group used work really well? Explain.

What difficulties did your group encounter in constructing the dig site?

How did you solve these problems to arrive at an accurate representation of the scale

drawing?

From the Mathematics Teacher, March 2001



OBSERVATION, INVESTIGATION, AND VERIFICATION SHEET 2

The drawing shows a circle with two chords, AB and CD, intersecting at point E.

1. Measure the lengths of the four segments, and denote them AE, EB, CE, and ED.

Multiply AE and EB. Also multiply CE and ED.

AE = EB =

AExEB =

CE = ED =

CExED =

2. Do you notice anything interesting about the two products?

Draw five circles of any size on a separate sheet of paper. Draw two intersecting chords of any length in each of the circles. Label the chords in each of the circles with the same labels used in the example on sheet 1. That is, one chord should be labeled AB, the other chord should be labeled CD, and the intersection of the two chords should be labeled E.

3. Measure each segment in your drawings, and complete the following chart.

_AE_EB_CE__ED AExEB CE ED_ Circle 1 Circle 2 Circle 3 Circle 4 Circle 5

4. What patterns do you see?

5. Let AB and CD be two chords in a circle with intersection E. Using the conjecture that you made in the previous investigation and the given information, complete the following chart:

AE_EB_CE_ED 8 5 2 _

_ 7 4.5 1.4 2.8 _ 11.48 3.6 3V3 5 _ 2< 3


(Continued on page 206)



(Continued from page 199)

SECOND OBSERVATION SHEET 3

F

1. The three circles each have one chord, FG, and O is the center of each circle. Find the mid

point of each chord, and label it H. Draw a line perpendicular to FG through the midpoint, H.

What do you notice about the line in relation to the center of the circle?

2. Inside each circle, write its radius and diameter.




ARCHAEOLOGICAL APPLICATION AND RECONSTRUCTION OF THE PAST SHEET 4

Suppose that you find a broken artifact that is part of a circular object. It could be something as small as a plate or as large as the Aztec calendar. The artifact piece is not half a circle but only a part of the original circle shown in the diagram.

1. How might you use the mathematical patterns that you have observed to determine the

original dimensions of the circular artifact? ;

Use several pieces of broken artifacts that your teacher gave to you, or use the pieces of the broken x% artifact pictured below to reconstruct the original circular shape of each artifact, or the original circle. Use what you have discovered from the previous examples to help you determine the original dimensions of each artifact.




Download - The Archaeological Dig Site: Using Geometry to Reconstruct the Past

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