490 MATHEMATICS TEACHING IN THE MIDDLE SCHOOLMATHEMATICS TEACHING IN THE MIDDLE SCHOOL

Teacher to Teacher

UnexpectedRiches froma GeoboardQuadrilateralActivity

UnexpectedRiches froma GeoboardQuadrilateralActivityB A R B A R A J. B R I T T O N A N D S H E R Y L L. S T U M P

Copyright © 2001 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.

VOL. 6, NO. 8 . APRIL 2001 491

EXPLORING THE SIMPLE QUADRILATERALS IN Athree-peg-by-three-peg section of a geoboardhas proved to be a very stimulating activity inelementary mathematics content and methods

classes. In the April 1998 issue of MathematicsTeacher, Kennedy and McDowell described a won-derful activity that focused on counting quadrilater-als on geoboards of various sizes. Our focus in thisarticle is on the properties of quadrilaterals. The ac-tivity that we describe is often used as an introduc-tion to a unit on geometry. Tom Lewis, who is withthe Moline Public Schools in Illinois, also usedthese activities in his fifth-grade classroom and re-ports similar results.

Beginning the Activity

THE CLASSROOM ACTIVITY BEGINS WITH STU-dents working in groups to find all the quadrilat-erals that they can on the three-peg-by-three-peggeoboard section. They keep track of their fig-ures on dot paper. The groups discuss the mean-ing of different quadrilaterals. Most groups un-derstand that different means noncongruent andlaunch into the activity. Other groups are puz-zled. Typically, these students ask the instructorwhether two squares of different sizes are consid-ered different, or whether they are the same be-cause they are the same shape. The instructorasks whether the students would consider thesquares to be the same. The group usually saysno. Through discussion, the students decide thatthe shapes are different if they do not fit exactlyon top of each other. The students begin to iden-tify figures related by reflections, rotations, andtranslations as congruent, although they may notuse that terminology.

When all groups think that they have found allthe quadrilaterals, the class comes together toshare results. The geoboards themselves can bepropped up in the chalk tray for this discussion, orstudents can be asked to transfer their figures tolarge sheets of paper with three-by-three dot gridsto allow the figures to be displayed and viewed eas-ily. One by one, each group selects a figure and pre-sents it to the class. The selected figure must be dif-ferent from any other shapes that have already

been presented. Occasionally, a congruent figure ispresented, leading to a whole-class discussion of rotations, reflections, and translations. The namesand properties of the various shapes are discussedas the shapes are presented.

When all the groups have presented theirquadrilaterals, the instructor notes whether all six-teen have been found, usually stating how many areleft, and the groups attack the problem of findingthe missing shapes. One semester, none of thegroups considered the possibility of outlining con-cave quadrilaterals. When one of the students dis-covered this possibility, the “Aha!” could be heardthroughout the room. All the remaining concavequadrilaterals were quickly found.

One of the most interesting discussions stemmingfrom this activity arose when a student claimed that theshape shown in figure 1 was a quadrilateral. The in-structor asked the classto decide whether thefigure was, in fact, aquadrilateral. The classdecided that it was not.The students realizedthat although the pegshad diameter, concep-tually they were meantto be points and therubber bands weremeant to be lines.Hence, being on oneside or the other of apeg did not mean thatthe rubber band was ac-tually passing through adifferent point.

BARBARA BRITTON, barbara.britton@emich.edu, is on thefaculty at Eastern Michigan University, Ypsilanti, MI48197. Her professional interests include the history ofmathematics and the beliefs of preservice teachers. SHERYLSTUMP, sstump@gw.bsu.edu, teaches at Ball StateUniversity, Munice, IN 47306. Her interests include teach-ers’ knowledge and performance assessment.

Fig. 1 The students debated whether thisshape qualified as a quadrilateral.

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492 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL

Sorting the QuadrilateralsAFTER THE STUDENTS HAVE FOUND AND DIS-cussed all the quadrilaterals, the instructor leads theclass in a sorting activity. By moving the geoboardsor large sheets of dot paper, the instructor sorts thesixteen quadrilaterals into two groups and tells theclass that all the quadrilaterals in one group have aproperty that is shared by none of the quadrilateralsin the other group. The students then try to identifythe property. Some examples of properties that havebeen used include having one pair of parallel sides,two pairs of parallel sides, a right angle, four rightangles, an obtuse angle, a convex or nonconvexshape, and line symmetry. The sorting activity givesstudents an opportunity to communicate their math-ematical thinking out loud. Some students struggleto find the language to describe the geometric prop-erties that they see. The instructor may help the stu-dents connect their ideas with formal mathematicalterminology. If time permits, individual studentsmay come to the chalkboard to demonstrate differ-ent methods of sorting. The notion of sorting itself isa challenge for some students.

For homework, the students receive a sheet con-Fig. 2 Homework assignment for sorting quadrilaterals

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taining the sixteen quadrilaterals drawn on three-by-three dot grids (see fig. 2). Students cut apartthe set of sixteen quadrilaterals and sort the shapesby pasting them in two groups onto a sheet ofpaper. The students also write explanations of theproperties used in their methods of sorting. Stu-dents must use a property that has not previouslybeen discussed in the whole-class sorting activity.

Determining the Areas of theQuadrilateralsAS A FURTHER EXTENSION OF THE ACTIVITY, STU-dents are asked to find the areas of the quadrilat-erals. This activity can be done in groups in classor as a homework assignment. After the studentshave found the areas, they share their answersand methods with the class. The variety of meth-ods is astonishing. Students discuss numerousgeometry concepts as they make convincing argu-ments that their methods aresound. Many students use acut-and-paste approach (seefig. 3). They see that one partof the figure fits into an un-covered par t o f anothersquare and must convince therest of the class that thepieces are congruent. Theirarguments lead to a review ofmany geometric principlesabout congruent triangles be-cause the cut pieces are usu-ally triangular. Occasionally,a student will say that he orshe actually cut the figuresinto pieces and thus the stu-dent knows that the pieces fit.

Another approach to findingthe area of these quadrilateralsis illustrated in the statement“That part looked like one-fourth and that part looked likeone-half” (see fig. 4). Suchstatements are usually correct,but the reasoning behind themis not intuitively obvious.Teachers can introduce a prob-lem-solving opportunity byhaving students prove that thesection of the shape believedto be one-fourth or one-half orone, in some instances, does infact have that area measure.

Of course, some studentsuse area formulas to determine

areas. Other approaches that wehave seen include determiningthe area outside the figures andsubtracting that value from 4,thinking of triangles as half-rectangles on the geoboard, andusing previously found areas toaid in finding new ones. Nodoubt, students can come upwith other strategies, as well.Every now and then, studentssurprise us with new strategies.

Conclusion

WHAT STARTED OUT AS AN AC-tivity to introduce students tothe geoboard turned out to besurprisingly rich. A recognitionof shapes and their attributes;concepts of point and line, re-flection, rotation, and transla-tion; congruent figures; geomet-ric proof; and area concepts areall represented in one problem.This activity also clearly showsthe students that problems canbe solved in a variety of ways.

Reference

Kennedy, Joe, and Eric McDowell.“Geoboard Quadrilaterals.”Mathematics Teacher 91 (April1998): 288–90. C

Fig. 3 Cut-and-paste method for finding thearea of a quadrilateral

Fig. 4 Another student method for finding thearea of a quadrilateral

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