IT SHOULD COME AS NO SURPRISE THAT IM-ages of doughnuts and coffee have escalatedrecently, with doughnut stores erupting onmany street corners and online stores avail-
able for any coffee item imaginable. In keepingwith this latest trend, we created an activity for mid-dle-grades students to apply mathematical formulaswhile exploring the geometry of a doughnut. Therationale for this activity was for the students to de-velop more sophisticated thinking of surface areaand volume. We conducted this activity in twosixth-grade classes at an all-girls school. An entireforty-five-minute class period was used to completethe activity and answer the majority of the discov-ery questions. The remaining questions were com-pleted as a homework assignment.
PAULA MAIDA, [email protected], teachesmathematics courses to preservice teachers at
Western Connecticut State University inDanbury, Connecticut. She is interested
in activities involving mathematicalconnections and encourages her stu-dents to find the mathematics ineverything around them. MICHAELMAIDA, [email protected], teaches middle school mathe-matics at Convent of the SacredHeart in Greenwich, Connecticut.He is interested in incorporatinghands-on activities into his classes to
strengthen understanding and pro-mote enjoyment of mathematics. PH
• Double-sided, two-page, stapled handout (figs. 1–3)• One doughnut per student (use a typical dough-
nut that has a hole in the center)• Empty doughnut boxes that hold a dozen doughnuts
(ask a doughnut store manager for these and explainthat they are for a school experiment)
• Centimeter cubes• Flexible tape measures
• Calculators• Napkins
Each student received a napkin, doughnut, a hand-ful of centimeter cubes, and the stapled handout.Students brought calculators to class. The stu-dents shared the tape measures and the doughnutboxes. Because of taste preferences, the dough-nuts used in this activity were chocolate andvanilla frosted, however, any variety would be ap-propriate (e.g., plain, glazed) as long as they haveholes in the center.
How Does Your Doughnut Measure Up?
A. INITIAL THOUGHTS
Estimate the volume of your doughnut in cubiccentimeters. ___________
Estimate the surface area of your doughnut insquare centimeters. ___________
Estimate the volume of the doughnut box incubic centimeters. ___________
Estimate the surface area of the doughnut boxin square centimeters. ___________
B. MEASUREMENTS
Measure the following dimensions of yourdoughnut and box. Record them in the chart.
Fig. 1 Parts A and B started students rolling on the activity.
Circumference of outer circle (in cm)
Circumference of inner circle (in cm)
Diameter of outer circle (in cm)
Diameter of inner circle (in cm)
Height of doughnut (in cm)
Height of box (in cm)
Length of box (in cm)
Width of box (in cm)
How Does Your Doughnut Measure Up?—Continued
C. CALCULATIONS
Calculate the following areas, showing your work in the work-space that is provided. Note that your calculations will be ap-proximations because of the irregularities of your doughnut.Record your answer in the answer column. Do not forget tolabel the units.
Fig. 2 In part C, students calculate measurements.
Area of outer circle
Area of inner circle
Area of top of doughnut
Area of outer lateral surface of doughnut
Area of inner lateral surface of doughnut
Area of front face of box
Area of side face of box
Area of top face of box
Workspace Answer
214 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
Initial ThoughtsWE BEGAN BY ASKING THE STUDENTS TO CON-centrate on their doughnut and list its various attrib-utes. Replies included “has a hole in the middle”and “has frosting on the top.” We continued by ask-ing them to reflect on its mathematical characteris-tics. Responses included direct comparison mea-sures (“Mine is wider than hers”), nonstandardmeasures (“About one large marshmallow can fit inthe space inside my doughnut”), and standard mea-sures (“My doughnut is about three and a half cen-timeters high”). We questioned students further:“What types of attributes might a doughnut man-ager be concerned about and why?” Students com-mented that he or she would want a dozen dough-nuts to fit inside a box “without getting smushed”and that the box should have enough room to leavespace for the frosting. We also asked them to distin-guish among one-, two-, and three-dimensional mea-sures. Sample responses included the “height of thedoughnut” as one-dimensional; the “volume of thedoughnut box” as three-dimensional; the “area ofthe top of the doughnut box or of the top of thedoughnut” as two-dimensional; and the “diameter ofthe doughnut” as one-dimensional. The studentshad knowledge of geometric terminology and for-mulas from previous class lessons, so this activityreinforced these concepts. Consequently, this activ-ity was more self-guided for students than teacher-guided. After hearing many more answers, we con-sidered this set of questions to be a smooth segueinto the planned activity.
In Battista’s research regardingstudents’ thinking about area and vol-
ume measurement, he claims, “Instruc-tional tasks must also encourage and support
students’ construction of personally meaningfulenumeration strategies (i.e., those that are based onproperly structured mental models). Students’ con-struction of such strategies is facilitated, not by ‘giv-ing’ them formulas, but by encouraging students toinvent, reflect on, test, and discuss enumerationstrategies in a spirit of inquiry and problem solving”(Battista 2003, p. 135). After the students exploredand discussed possible attributes to be measured,we prompted them to decide on sensible units fortheir measurements. They unanimously agreed oncentimeters (which they confessed was becausethey saw the centimeter cubes on the table andthought this manipulative could be useful in theirestimations). As a class, we explored other units ofmeasurement, such as millimeters (metric) andinches (customary), and agreed that although manyof the choices given would be suitable, the centime-ter was the unit with the greatest vote.
Students began estimating the volume and surfacearea of their doughnut and the volume and surface areaof the doughnut box in part A of this activity (see fig. 1).All students chose to use the centimeter cubes for theirestimation. Some students had difficulty doing this partquickly. As opposed to an educated guess, they wantedto measure and use their calculators to “get it right.”Others attempted to completely build a replica of thedoughnut using their centimeter cubes. “An importantpoint from a mathematical perspective is that such(layer) structuring is more general and powerful thanusing standard area and volume formulas. For example,layer structuring is extremely useful for thinking aboutthe volumes of cylinders and many problems in calcu-lus” (Battista 2003, p. 129).
The teachers briefly reminded the studentsabout sensible ranges versus exact answers and theability to choose an estimate rapidly. This part ofthe activity illustrated that although these studentshave had frequent opportunities to estimate, multi-ple estimation experiences and frequent exposureare necessary for students to build comfort and skillin their estimation abilities.
Before leaving part A of this activity, we spentmore time focusing on the volume and surface areaof the doughnut only. The students had just finishedmaking their estimates and now we wanted them tothink of ways to refine those measurements withbetter accuracy. It took awhile for them to recognizean approach. A student commented that she knewthe volume of a right circular cylinder and that thedoughnuts looked like short right circular cylinders.Another student debated that this was not true be-
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cause of the doughnut’s hole. After another mo-ment, another student reasoned that the hole itselflooked like it was a right circular cylinder that hadbeen removed. A brief discussion continued as stu-dents reasoned that they could calculate the differ-ence between the volumes of two right circularcylinders, the “larger” one minus the “smaller” one,to calculate the volume of the doughnut. Althoughthe teachers supported this discussion, we could notlet the activity branch off without citing some mathe-matical inaccuracies. First, we showed students amanipulative of a torus, which is a doughnut-shapedsolid. Second, we used a miniature Slinky toy and acircle to illustrate how a circle can be revolved togenerate (or trace out) that torus shape. We brieflyexplained that determining the volume of this solidof revolution requires calculus, which is discussedin later grades, and that since integrals are not read-ily accessible to them, we would be content withclose approximations of a short right circular cylin-der with an inner cylinder removed. Actually, onestudent commented that these particular frosteddoughnuts did actually look more “cylinder-like”than “torus-like,” possibly because of the specificcutter used and that doughnut’s consistency. Weadmit that her observation was accurate butnonetheless agreed that the students’ real-lifedoughnuts with miscellaneous bumps could not beidentified as any perfect shape. Therefore, weagreed to be comfortable with their suggestions ofusing cylinder formulas, acknowledging their finalcalculations as very clever approximations.
Similar discussions occurred relating to the sur-face area of the doughnut, as discussed in detail inthe discovery section addressed in figure 3. Inci-dentally, this initial discussion already raised thestudents to a level of thinking that was higher thantheir previous work with geometric formulas. Thisbrainstorming session produced greater resultsthan the teachers had anticipated.
Measurements
IN PART B OF THE ACTIVITY IN FIGURE 1, TITLED“Measurements,” students measured and recordedvarious dimensions of their doughnut. Although theworksheet had been predesigned by the teachers,the students had already suggested measuring themajority of these dimensions during the initial class-room discussion. We encouraged students to mea-sure as accurately as possible, to the nearest tenth ofa centimeter. They shared flexible tape measures forthis part of the experiment. The students did an excellent job with labeling units. Comments madeby the students such as “This is cool” and “This isawesome” suggested that they were enjoying the
Fig. 3 Part D, titled “Discoveries,” asked students to explain their doughnutexploration.
How Does Your Doughnut Measure Up?—Continued
D. DISCOVERIES
1. (a) Calculate the volume of your doughnut. Show the step(s) youused. Label the units. Note that your calculations will be ap-proximations because of the irregularities of your doughnut.
(b) Do you think the volume you calculated for your doughnut isan overestimate or an underestimate of the actual volume ofthe doughnut? Why?
2. (a) Calculate the surface area of your doughnut. Show thestep(s) you used. Label the units. Note that your calculationswill be approximations because of the irregularities of yourdoughnut.
(b) Your younger sister asks you, “What do you mean by surfacearea of that doughnut?” What types of doughnuts might behelpful in your explanation? How would you word your an-swer to your sister?
3. (a) Calculate the volume of the air inside the doughnut box. Showthe step(s) you used. Label the units.
(b) If you put a dozen of your doughnuts into the box, explain inwords how you would determine the volume of the air aroundthe doughnuts inside the box.
(c) Now use the method you described to find the volume ofthis air.
4. (a) Calculate the surface area of the doughnut box. Show thestep(s) you used. Label the units.
(b) Compare this calculation with your original estimate of thebox’s surface area in part A. What did you notice?
5. (a) According to a Dunkin’ Donuts Web site, “Dunkin’ Donutssells more than 6 million donuts a day, a whopping 2.3 billiona year.” If the 2.3 billion doughnuts sold each year were onlypowdered doughnuts, estimate the amount of powdered sugarneeded to cover them. Source: Dunkin’ Donuts. “Dunkin’ Donuts 5 Points.”www.dd5points.com/donuts.htm
(b) Which did you rely on for your answer: the surface area of thedoughnut or the volume of the doughnut? Why?
216 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
activity. They were truly immersed at thispoint and were also interested in compar-ing their own measurement with their
peers’ as they progressed. We also empha-sized our expectations of clear mathematical
communication. For example, if they were in-terested in the diameter measure of a classmate’s
doughnut, they needed to use the term diameter in-stead of asking, “How far across is yours?” The aver-age of the measurements found by the twenty-fourstudents is as follows:
Circumference of outer circle (in cm): 30.3 cmCircumference of inner circle (in cm): 7.5 cm Diameter of outer circle (in cm): 9.4 cmDiameter of inner circle (in cm): 2.7 cmHeight of doughnut (in cm): 3.3 cmHeight of box (in cm): 5.0 cmLength of box (in cm): 38.0 cmWidth of box (in cm): 29.3 cm
This activity helped students realize that whenreal-world objects do not perfectly “fit the mold,”then adjusting measurements might be considered.For instance, when some doughnuts were more ovalthan circular, students made reasonable adjustmentsfor their doughnut’s diameter. They learned thattweaking measurements can be an acceptable andsensible way of working with real-world information.
Calculations
THE STUDENTS HAD BEEN DROOLING FROM THEsmell of their doughnuts while taking measurements,so we said that they could go ahead and eat themafter they performed their calculations, since all mea-surements were complete. Students used their own
calculators to complete the table in figure 2. The pictorial representations displayed on the
worksheet provided additional clarification of the ter-minology. Those diagrams were created in TheGeometer’s Sketchpad computer program using cor-responding measurements. This means, for instance,that the rectangle illustrating the “area of the outerlateral surface of the doughnut” was purposely cre-ated with its length being equal to the circumferenceof the circle depicted for “area of the outer circle.”
The only recurring question in part C of figure 2was the meaning of the outer and inner lateral sur-face of the doughnut. Foreseeing the need for thisexplanation, we prepared a manipulative in advance.It was created from a Styrofoam right circular cylin-der purchased from a craft store. The cylinder wasapproximately the size of a doughnut, so we simplycut and discarded a small cylinder from the center.A rectangle was cut from bright colored paper toperfectly fit the outer lateral surface of the manipu-lative and another rectangle was cut to fit the innerlateral surface. When a student asked for clarifica-tion of the outer lateral surface, we asked her towrap the colored rectangle around the correspond-ing surface of the doughnut manipulative. Each stu-dent who completed this task recognized that thelength of the rectangle is the circumference of theouter circle and that the height of the rectangle isthe height of the doughnut. Since they knew how tocalculate the area of a rectangle, they consequentlycould obtain the area of the outer lateral surface.They recognized that the inner lateral surface usedsimilar reasoning.
Discoveries
“IN THE SPATIAL STRUCTURING PROCESS, INDIV-iduals abstract an object’s composition and form byidentifying, interrelating, and organizing its compo-nents” (Battista 2003, p. 123). The teachers wit-nessed this structuring process and observed thislevel of thinking during conversations that studentshad with one another as they worked on part D ofthis activity, shown in figure 3. In part D, studentswere asked to reflect on their work by answeringdiscovery questions designed by the teachers. Thestudents chose their own approaches to answer thequestions by recognizing the fundamental compo-nent pieces from their earlier work and relatingthem to the perceived object composition. Again,the students were engaged and active, applying for-mulas and clarifying terminology with one anotherwhile the teachers provided very little assistance.
Two approaches were used to determine the vol-ume of their doughnut. In one approach, studentscalculated the product of “area of top of doughnut”
1. (a) Calculate the volume of your doughnut. Show thestep(s) you used. Label the units. Note that your calcu-lations will be approximations because of the irregulari-ties of your doughnut.
Fig. 4 These students showed how they calculated the volume of their doughnuts.
VOL. 11, NO. 5 . DECEMBER 2005/JANUARY 2006 217
and “height of doughnut.” In another approach, stu-dents calculated the volume of two right circularcylinders and found their difference (see fig. 4).
The students were asked to consider if their calcu-lated volume was an overestimate or an underesti-mate of the actual volume of the doughnut. Some stu-dents reasoned that their calculated volume was anoverestimate because their measurements werebased on a right circular cylinder with a right circularcylinder removed. That is, in their calculations, thelateral surfaces meet the bases at “sharp” rims,whereas the curvature of real doughnuts creates a“shaved off” appearance and consequently a smallertrue volume (see fig. 5). A student who answered“underestimate” reasoned that she did not accountfor the bumps that were on her doughnut, giving itextra actual volume. Although there was no correctanswer to this question, it was interesting to note thatthe majority of students (sixteen versus eight) an-swered “overestimate.” Many students who answered“overestimate” used their perception of size to answerthis question, claiming that their doughnut simply didnot look as big as the number they calculated.
To determine the surface area of their doughnut,most students calculated it this way: twice the areaof the doughnut’s base + outer lateral surface +inner lateral surface (see fig. 6), clarifying againthat their surface area calculation was an approxi-mation to the doughnut’s true surface area. Stu-dents who answered this question incorrectly ne-glected the doughnut’s hole. They calculated it thisway: twice the area of the outer circle + outer lateralsurface. This latter response prompted the ques-tion, “Are there doughnut choices matching thatsurface area?” Students replied, “Yes, the filled onessuch as Boston cream or jelly-filled powdereddoughnuts.” While answering question 2a, “Calcu-late the surface area of your doughnut,” some stu-dents made extra work for themselves by referenc-
ing their initial measurements and recalculating itin the space provided, instead of using the calcula-tions they had already recorded in part C’s table.
Next, students were asked to describe the sur-face area, relevant to the doughnut, to a youngerchild. They were encouraged to consider doughnutflavors to aid in their explanation. The responses tothis question were right on target. Their clear, cor-rect explanations suggested that they had a goodconceptual grasp of surface area (see fig. 7). Onestudent reasoned that the surface area relates to theamount of doughnut you can see without taking abite; another described it as the area of everythingon the outside of the doughnut that you can touch.
1. (b) Do you think the volume you calculatedfor your doughnut is an overestimate oran underestimate of the actual volume ofthe doughnut? Why?
Overestimate because if it is squared offthere is extra space.
I think I overestimated the volume because thedonut is not a perfect cylinder. Since I measuredthe widest part of the donut, my calculation overestimated the volume of the donut.
Fig. 5 Students discussed their estimates.
2. (a) Calculate the surface area of your doughnut. Show thestep(s) you used. Label the units. Note that your calcu-lations will be approximations because of the irregular-ities of your doughnut.
Fig. 6 A doughnut’s surface area calculations are illustrated.
2. (b) Your younger sister asks you, “What doyou mean by surface area of that doughnut?”What types of doughnuts might be helpful inyour explanation? How would you word youranswer to your sister?
The flavor that would be most helpful in my explanation would probably be a chocolateglazed. This is because unlike the frosted donutsthe glazed donuts cover not only the top of thedonut in glazed, but the entire donut. I wouldword my answer by saying something along thelines of “The surface area of a donut is the partof the donut that is covered in glaze. Not the inside, but outside surface of the donut.”
It’s like the area that covers the surface. In otherwords, what is wrapped around it. For example aglazed doughnut. The glaze is the surface areabecause it covers all of the outside. Same thing withpowdered doughnuts; they are covered in powder.
Fig. 7 These students explained surface area.
The focus then turned from their doughnuts to thedoughnut box. The students calculated the volume ofthe air inside the empty box and then imagined placinga dozen doughnuts inside the box. Using descriptivesentences first (asked in question 3b), then mathemati-cal calculations (requested in question 3c), studentsdetermined the volume of the air around the dozendoughnuts inside the box (see fig. 8). (It should benoted that the students’ calculations were slightlyflawed since they measured the exterior of the box,and the volume of the air inside the empty box doesnot include the cardboard material. We discussed thisimportant detail with the students, and they adjustedtheir measurements accordingly.)
The students did a good job calculating the sur-face area of the doughnut box. However, as was thesituation for question 2a, “Calculate the surface areaof your doughnut,” we recognized that many stu-dents duplicated their work. They showed thoroughcalculations using their initial measurements and re-calculated the areas of the faces of the box. After
questioning the students, we learned thattheir extra work was a result of the di-
rections asking them to “Show thestep(s) you used.” The intention
was for them to quickly refer-ence the last three rows oftheir table in part C, which al-ready contained the neces-sary work shown in theworkspace, and only showthe summation step calcula-tion. In future renditions ofthis activity, the teacherswill place greater emphasison using results from part C.
According to a Dunkin’
Donuts Web site, “Dunkin’ Donuts sells more than6 million donuts a day, a whopping 2.3 billion ayear” (Dunkin’ Donuts 2004). The students wereasked to estimate the amount of powdered sugarneeded to cover those 2.3 billion doughnuts. Theyused various units for their answers (including cm2,in.2, cups, grams). Most students multiplied the sur-face area of their doughnut (found in question 2a)by 2.3 billion, using cm2 as units. We intended forthe students to think in terms of units different thancm2, such as cups, tablespoons, or teaspoons. How-ever, the question was not clear on this point. Whenthe activity is implemented again, we will considerrevising the last sentence in question 5a so that itreads “estimate the number of cups of powderedsugar needed to cover them.” All students an-swered “surface area” for question 5b (see fig. 9).
Extensions
IN ONE OF THE TWO CLASSES WE WORKED WITH,we extended the applications by stopping the stu-dents after they calculated their doughnut’s volume(see question 1a in fig. 3). The students were thenasked to estimate the number of doughnuts thatDunkin’ Donuts sells each year. The teachersshared the Web site estimate described above, andasked, “What is the volume of 2,300,000,000 of yourdoughnuts per year?” Students multiplied the vol-ume of their own doughnut by 2.3 billion. Theteachers continued, “If the volume of Earth is ap-proximately 1.09 × 1027 cm3, then how many yearswould it take to fill up Earth with your doughnuts?”We briefly discussed the idea of breaking thedoughnuts up so that gaps and holes were filled in.This was a loaded problem with connections to sci-ence, calculations with large numbers, the use ofscientific notation, and real-world applications. Withanswers near 2 × 1015 years, students were sur-prised at how large planet Earth really is. The teach-ers asked questions: “Will your great, great, great,great grandchildren still be busy filling up Earth
3. (b) If you put a dozen of your doughnuts intothe box, explain in words how you would deter-mine the volume of the air around the dough-nuts inside the box.
First, I would find the volume of onedonut, then multiply it by 12 (dozen). Thiswould let me know how much room thedonuts take up. Then I would find the volume of the box. This shows me how muchair there is. Then I would subtract the 12donuts’ volume from the air in the box toget the air around the donuts in the box.
Fig. 8 The focus having moved from doughnut to doughnutbox, this student investigated the volume of the air aroundthe doughnuts inside the box.
5. (b) Which did you rely on for your answer:the surface area of the doughnut or the volumeof the doughnut? Why?
I relied on the surface area of a donut. This isbecause the surface area is the outside of thedonut, and the part of the donut which you putthe sugar on. If I used the volume, then I wouldbe filling the inside with sugar.
Fig. 9 This student explained how the earlier question wasanswered.
218 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
VOL. 11, NO. 5 . DECEMBER 2005/JANUARY 2006 219
with your doughnuts?” It was difficult for studentsto fathom this number of years. There was a lot ofchatter in the room and comments of amazement.
Conclusions
IN BONOTTO’S RESEARCH REGARDING STUDENTS’learning and understanding of surface area, shewrites, “Traditional classroom teaching often seemsto favor the separation between classroom and real-life experience. This rift is particularly significantwith respect to the concept of surface” (Bonotto2003, p. 157). In addition to surface area, this sameargument can undoubtedly be made for volume aswell. Applications of measurement and geometricvocabulary (including circumference, diameter,face, lateral surface, volume, surface area) enhancecomprehension and recollection of the terminology.If we are to expect our students to recognize thegeometry and measurements surrounding them,we should offer frequent, engaging activities for re-flection on their physical environment.
In this activity, students used verbal discussionsand clever reasoning to further develop their level ofthinking related to measurement. Their written expla-
nations and documentation, feedback, and subse-quent assessments revealed a greater understandingof the concepts of volume and surface area.
Furthermore, this activity encouraged students toanalyze a three-dimensional shape that did not exactlycorrespond with the definition of a right circular cylin-der. The hole and curvature of this concrete model re-quired more discerning thought during measurementsand calculations. This classroom activity using geome-try and measurement allowed students to witness howpowerful and attainable mathematics truly is.
References
Battista, Michael T. “Understanding Students’ Thinkingabout Area and Volume Measurement.” In Learningand Teaching Measurement, 2003 Yearbook of the Na-tional Council of Teachers of Mathematics (NCTM).Reston, VA: NCTM, 2003.
Bonotto, Cinzia. “About Students’ Understanding andLearning of the Concept of Surface Area.” In Learningand Teaching Measurement, 2003 Yearbook of the Na-tional Council of Teachers of Mathematics (NCTM).Reston, VA: NCTM, 2003.
