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The Illinois Mathematics Teacher

EditorsDaniel JordanColumbia College Chicago623 S Wabash Ave, Suite 500Chicago, IL 60605

Christopher ShawColumbia College Chicago623 S Wabash Ave, Suite 500Chicago, IL 60605

Reviewers

Edna BazikSusan Beal

Carol BensonPatty Bruzek

Dane CampBill Carroll

Mike CartonChip Day

Lesley EbelTodd Edwards

Linda FosnaughDianna GalanteLinda Gilmore

Linda HankeyPat HerringtonAlan Holverson

John JohnsonRobin Levine-Wissing

Cheryl LubinskiGeorge Marino

Karen MeyerJackie MurawskaNicolette Norris

Jacqueline PalmquistJames Pelech

Randy Pippen

Sue PippenAdam Poetzel

Anne Marie SherryAurelia Skiba

Joe SticklesMary ThomasRobert Urbain

Nicole Wessman-EnzingerDarlene Whitkanack

Peter WilesLara Willox

ICTM Governing Board

OfficersPresident: George Reese Secretary: Lannette Jennings

Past President: Bob Mann Treasurer: Rich WyllieBoard Chair: Jackie Murawska Conference Coordinator: Marshall Lassak

Directors

Early Childhood

Denise BrownCarly Morales

Grades 5–8

Eric BrightAnita Reid

Grades 9–12

Jeremy BabelMartin Funk

Community College/University

Craig CullenPeter Wiles

At-Large

Zachary HerrmannJackie MurawskaSendhil Revuluri

The Illinois Mathematics Teacher is the official journal of the Illinois Council of Teachers of Mathematics and is de-voted to providing ideas and information to support the professional development of teachers at all levels of the curriculum(K-16). Current issues and select back-issues are available online at http://www.ictm.org/illinois-math-teacher.

This work is licensed under the Creative Commons Attribution 3.0 United States License. To view acopy of this license, visit http://creativecommons.org/licenses/by/3.0/us/.

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Table of Contents

Editors’ Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2015 ICTM Awards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Tina NocellaRichard KaplanDanyel Larsen

Articles

Dynamic Discourse with Dynamic Statistics Software . . . . . . 6Randall Groth

Is Edgar Allan Poe Really a Mathematician? . . . . . . . . . . . . . . . . 13Judy Brown, Jennifer Helton, Tamra C. Ragland, M. Todd Edwards

Developing Meaning in Trigononometry . . . . . . . . . . . . . . . . . . . . . . 25Valerie May & Scott Courtney

Aiming a Basketball for a Rebound:Student Solutions Using Dynamic Geometry Software . . . . . . 34Diana Cheng, Tetyana Berezovski, Asli Sezen-Barrie

Taking Calculus to New Heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Ken Gasser & Darl Rassi

Puzzle

“Do the Math,” a crossword puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Damon Gulczynski

Editors’ Note

Welcome to volume 63 of Illinois MathematicsTeacher, the journal of the Illinois Council ofTeachers of Mathematics.

For this issue, we would like to extend a specialthanks to our hard-working reviewers, many ofwhom refereed multiple articles just for this issue.We greatly appreciate the experience they havebrought to bear their reviews, and all of thearticles in this issue have been improved by theircareful eyes.

In this issue

Each year, at the annual meeting of the ICTM,winners of the ICTM awards are honored in a cer-emony. This year, we begin a tradition of invit-ing these accomplished educators to share theirthoughts with this journals readers. Among theawardees this year were Tina Nocella (Illinois Pro-mising New Teacher of Mathematics), RichardKaplan (T. E. Rine Secondary Mathematics Teach-ing Award), and Danyel Larsen (Lee Yunker Math-ematics Leadership Award), each of whom sub-mitted remarks to the journal for publication.

Every article in this issue presents aninnovative way of using interesting examplesand mathematical software to teach classicalmathematics topics to students. In fact, amongthe five articles in this issue, the authors usefour different programs: GeoGebra, TinkerPlots,Desmos, and Geometer’s Sketchpad.

“Dynamic Discourse with Dynamic StatisticsSoftware,” by Randall Groth, describes waysto use visualization software and a structureddiscourse model to help prospective mathematicsteachers learn to grasp foundational statisticsconcepts. Judy Brown, Jennifer Helton, TamraC. Ragland, and M. Todd Edwards findconnections to trigonometry, geometry, physics,and more, in a novel multi-week project basedon the predicament of the prisoner from EdgarAllan Poe’s short story, “The Pit and the

Pendulum,” in their article, “Is Edgar Allan PoeReally a Mathematician?” From Valerie May andScott Courtney, we have “Developing Meaningin Trigononometry,” which details a sequence ofactivities designed to help students grasp theconnections between right-triangle trigonometryand the idea of trigonometric functions asreal-valued functions. Diana Cheng, TetyanaBerezovski, and Asli Sezen-Barrie describe aproject in which their students use dynamicgeometry to analyze the trajectory of a basketballbeing shot at a basket, in “Aiming a Basketballfor a Rebound: Student Solutions Using DynamicGeometry Software.” And Ken Gasser and DarlRassi describe a project in which students usecalculus to design, analyze, and fly real hot airballoons in class, in “Taking Calculus to NewHeights.”

The final contribution to this issue is agiant-sized crossword puzzle with a mathematicaltheme, from constructor Damon Gulczynski.

We remind our readers that the IMT journalis published exclusively online through the ICTMwebsite. New articles are published as they areaccepted on the journal website and periodicallycollected into complete issues.

If you would like to join ICTM, we encourageyou to visit the ICTM website, http://ictm.org.To submit an article or volunteer to be a reviewerfor the IMT, please go to the journal website athttp://www.ictm.org/illinois-math-teacher,where you will find a detailed set of guidelines forsubmission to the journal. You may also send anemail to the editors at [email protected].

Thank you for reading, and we look forwardto working with you.

Daniel Jordan & Christopher Shaw

Illinois Mathematics Teacher 1

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mailto:[email protected]

2015 ICTM Awards

From the editors: The editors congratulate thewinners of the 2015 ICTM awards, which werepresented at the 65th Annual Meeting of theIllinois Council of Teachers of Mathematics. Theawardees are listed below and were invited tosubmit remarks. Those that did are indicated byan * and their remarks follow. Information aboutthe awards and nomination instructions can befound at http://www.ictm.org/ictm-awards.

Illinois Promising New Teacher of Mathematics

Tina Nocella*Niles North High School, Niles

Middle School Mathematics Teaching Award

Martha ReillyByron Middle School, Byron

T.E. Rine Secondary Mathematics Teaching Award

Richard Kaplan*Evanston Township High School, Evanston

Max Beberman Mathematics Educator Award

Martha EggersMcKendree University, Lebanon

Fred Flener Award: Beyond the Classroom

Leona MirzaNorth Park University, Chicago

Lee Yunker Mathematics Leadership Award

Danyel Larsen*Oregon High School, Oregon

Distinguished Life Achievement Award

Peter BraunfeldUniversity of Illinois, Urbana-Champaign

Five Years of CollaborationRemarks by Tina Nocella

Throughout my first five years as an educator,I have learned that the education of our studentsis not something that can be easily done byone individual teacher. Thankfully, the teachingprofession encourages and even requires thecollaboration and communication of everyoneinvolved.

Let’s begin with our students. Our studentsprovide meaningful feedback to us daily in avariety of forms. Whether it be through abi-weekly meeting with each class’s steeringcommittee or brutally honest feedback after alesson that didn’t go as planned, students are anamazing source of rich and insightful ideas to helpimprove our classrooms. They know what helpsthem learn, what motivates them, what engagesthem. It is up to us to listen to this feedback andincorporate it into our lessons and classrooms tocreate an environment that meets the needs of ourstudents.

In addition to our students, we have ourcolleagues. These are the professionals we workwith every day, each week, once a month, oreven twice a year. Whether it is a mentor, aveteran teacher who knows the ins and outs ofthe curriculum, a team member who wants to trynew ideas, a director, or a best friend, the morewe learn from each other and grow together, thestronger the impact we can have as professionals.

I was recently at a conference on VisibleLearning where John Hattie informed us thatthe number one influence on student achievementis teacher collaboration. Collectively, we cancontinue to grow by pushing each other to beour best for our students and for each other, andby actively choosing to rely on each other forinspiration and motivation.

Outside of our schools, we have a varietyof professional development opportunities for

2 Illinois Mathematics Teacher

http://www.ictm.org/ictm-awards

2015 ICTM Awards

educators. Some possibilities include theIllinois Council for Teachers of Mathematics, theMetropolitan Mathematics Club of Chicago, andthe Complex Instruction Consortium. Theseorganizations provide us spaces where we canmeet many outstanding educators to share stories,collaborate on lessons, develop plans to overcomeobstacles, celebrate accomplishments, and somuch more. This vast collection of insights andreflections provide all of us with new ideas andquestions and spark our curiosity, which we canthen share with others to further develop, analyze,answer, and improve.

In addition to these organizations, I havelearned that professional development does notexist only in the form of formal conferences andorganizations. It can also take place via socialmedia through a variety of Twitter chats orhashtags. Someone once told me that Twitterwas one of the best professional developmentopportunities out there for educators. At first Ithought that was an exaggeration, but now I amthankful that I was encouraged to get connected.

These organizations and professionaldevelopment opportunities continue to show methat individually we do not know everythingabout education, but we all know something.Collectively, we can continue to improve and makea difference for our current and future students.All we need to do is collaborate with the amazingeducators that we are privileged to work with inour field.

As a fifth-year teacher, I continue to beinspired by the unbelievable work so manyeducators are doing in their classrooms each year.

I am looking forward to a long career oflearning from and with an amazing group ofprofessionals! I hope we can all work together,in any way that seems fit, to continue supportingeach other on our journeys to improve ourteaching for all students.

Tina Nocella

NILES NORTH HIGH SCHOOL

9800 LAWLER AVE

SKOKIE, IL 60077

E-mail : [email protected]

Advice from a Veteran TeacherRemarks by Richard Kaplan

My experiences as a math teacher for thirtyyears—at Lake View High School, a ChicagoPublic Schools neighborhood school, and atEvanston Township High School—tell a storyof the vast potential and real desire to growand learn of urban minority students; and ofbuilding successful programs in two very differentChicago area high schools where lots of studentsof color successfully took AP Calculus andgrew tremendously from this experience. AtLake View, our work resulted in 40% of theseniors taking AP Calculus. At Evanston, ourwork resulted in an increase in the number ofAfrican American and Latino students taking APCalculus from under ten to over one hundred.At both schools, we started summer math classesto accelerate our targeted populations. We builtpeer support for academic success so that studentshelped us recruit and motivate. And in our mathclasses, we built relationships with our studentsand worked really hard at teaching with the aimof getting students to believe that with hard workthey could achieve amazing results. What followsis a summary of my ideas about teaching:

1. It is fundamental that teachers believethat students can do challenging andintellectually rigorous work.

2. Many students have hard, often chaoticlives. Our job as teachers is, whateverstudent or school roadblocks appear, tomotivate and inspire our students toachieve.

3. Human beings in many situationssolve really difficult problems by workingcollectively, creatively, and resourcefully.Teachers need these attributes to inspirestudents to achieve academically. I cannotimagine a more intellectually difficult taskthan figuring out how to successfullymotivate and teach urban minority studentsa rigorous curriculum.

4. Teachers need to develop close relationshipswith their students. Students need to knowthat we care about them as people. As


Harvard professor Ronald Ferguson states,referring to Black and Hispanic students,“a teacher’s demand (for hard work andachievement) may be understood as eitherfriendly or hostile, depending upon whetherthe teacher is perceived as caring or not.Once these kids know you care they’ll walkthrough walls for you” (Sparks, 2003).

5. Teachers need to make their classroomssafe and supportive learning environmentswhere students are empowered to askquestions, work collectively, and celebratehard work and success. Teachers need tomanage their classrooms so that behavioralissues do not interfere with the learningprocess. This needs to be done in a strong,effective, but loving way where the personaldignity of students is maintained.

6. Teachers need to teach in a compelling waythat inspires students to fully engage inclass and with an excitement for the subjectmatter and for the students.

7. Teachers need to be there for studentsevery day. The everydayness of teaching,which Hollywood movies totally miss, isvital for students developing respect fortheir teachers and moving forward.

8. Teachers need to work really hard atall aspects of teaching—developing lessons,explaining concepts, organizing and runningthe classroom, building relationships withstudents, grading, collaborating with otherteachers, and brainstorming new ways ofgetting through to students.

9. Teachers need to be really smart in theircontent areas. They need to be able toexplain concepts clearly in a variety of ways,to develop lessons that take students fromwhere they are academically to a higherlevel, to understand students’ questionsand respond helpfully, and to adjust theirlessons and catch the teachable moment. Inany class, students have a wide range ofabilities. It takes extraordinary content andteaching knowledge to be able to challengeand not frustrate students and to supportthem without watering-down the content.

10. Teachers need to collaborate with eachother. This takes a high level of mutualtrust. Teachers need to be able to openlydiscuss the ways in which poverty and raceaffect students. They need to be able todiscuss curriculum, instruction, pedagogy,and teacher-student relationships, andshould visit each others classes and discusstheir successes and failures with the goal ofcollectively improving their teaching.

11. Teachers need to experience success. Manyteachers genuinely want to change students’lives. This work is extremely difficult.Students have lots of complex issues. Cynicsand naysayers are everywhere. Teachers’efforts need to be supported and celebrated.

I feel fortunate to have chosen this professionof teaching high school math. I feel fortunateto have a job where I get to help changethe trajectory of young people, in particularurban and minority youth. I feel fortunate tobe teaching young people that most importantlesson—that through hard work they can besuccessful—and see so many of them light upwhen they internalize that lesson.

Richard Kaplan

EVANSTON TOWNSHIP HIGH SCHOOL

1600 DODGE AVE

EVANSTON, IL 60201


Listening and LeadingRemarks by Danyel Larsen

As Stephen Covey has told us, “Most peopledo not listen with the intent to understand; theylisten with the intent to reply” (Covey, 1989).In my experience, a good leader is a person whotakes time to listen to what other people have tosay. If you, as a leader, have a vision of where youwant a classroom, department, committee, school,or district to go, and you want others to get onboard with that vision, the key is to listen to theirvarious perspectives, demonstrate understandingof those perspectives, and use your understanding


2015 ICTM Awards

to develop a roadmap of how to get from whereyou are to where you want to be.

As teachers, we all know that education isa business of people. If we want to improveeducation in every aspect, from counting andnumeracy in Kindergarten to AP Calculus andStatistics in high school, we must take the timeto listen with the intent to understand andcultivate meaningful relationships with everyoneand anyone involved. This means that you reachout to your students, your faculty and staff, youradministration, your community, and possiblybeyond to governing bodies. I believe we won’timprove with more standards or fewer standards,or equal funding, or whatever system it is thatwe have today. I know we can’t get there withsmart-boards, tablets, white boards, or slate andchalk. Up until two years ago, I still had theoriginal 1930s slate at the front of my room! WhatI do know is that we must focus on creating ashared vision for what it means to be educatedand to contribute meaningfully to our society. Itbegins with listening and cultivating relationshipswith everyone involved. Simply stated, this meansworking to include and understand others. Andas you exhibit and hone these skills, others willrecognize these qualities in you and the traits willgrow into leadership qualities. Suddenly you willfind that you are a leader, even if that was notyour original intent.

Danyel Larsen

OREGON HIGH SCHOOL

210 S 10TH ST

OREGON, IL 61061


References

Covey, S. R. (1989). The 7 Habits of Highly EffectivePeople. New York: Free Press.

Sparks, D. (2003). We care, therefore they learn. Harvardlecturer Ronald F. Ferguson points to key componentsthat affect achievement: content, pedagogy, and,importantly, students’ relationships with their teachers.Journal of Staff Development , 24 , 42–47.


Dynamic Discourse with Dynamic Statistics Software

Randall E. Groth

Abstract

This article describes how dynamic statistics software was used to catalyze discourse among a groupof prospective elementary and middle school teachers. The prospective teachers formed their ownstatistical questions and then investigated them with the software. Their course instructor thenpurposefully selected and sequenced work they had produced to become the basis for classroomdiscourse. As the prospective teachers presented their work, the course instructor helped the classmake connections among the different strategies and statistical questions that were shared. The overallexperience illustrates a model for the facilitation of technology-enhanced statistical discourse.

Keywords: statistics, discourse, teacher education, dynamic statistics software, TinkerPlots, FivePractices Model

Published online by Illinois Mathematics Teacher on October 21, 2015.

Classroom discourse about statistics canbe exceptionally rich and engaging. Everyday, people use statistics to make arguments.Statistics are used to explore questions suchas, “Who is the best pitcher in baseball?,”“How much allowance does a typical sixth-grader receive?,” and “Is there a relationshipbetween grades and hours spent playing videogames?” These types of questions are opento multiple solution strategies and multiplesolutions. Individuals must use statisticscreatively and convincingly to persuade others oftheir positions.

Although statistical questions have thepotential to be engaging catalysts for classroomdiscourse, posing and arguing statistical questionsdoes not always come naturally to students.Statistical classroom discourse can be hamperedby trouble understanding content (Shaughnessy,2007). Students also may not understand thatsome statistical questions can have multiplesolutions, since mathematics questions often havea single solution, even if that solution canbe obtained using multiple strategies (Rossmanet al., 2006). The act of posing questionsitself can be unnatural for students, as student-posed questions are relatively rare in mathematicsclassrooms (Knuth, 2002). Such hurdles must

be overcome if students are to attain CommonCore State Standards for Mathematical Practicesuch as “Make sense of problems and persevere insolving them,” and “Construct viable argumentsand critique the reasoning of others” (CommonCore State Standards Initiative and others, 2010).

In this article, I discuss ways in which I havefostered the activities of posing statistical ques-tions and discussing solutions to them. Two maintools I have used are the dynamic statistics soft-ware TinkerPlots (Konold & Miller, 2005) and themodel of classroom discourse described in 5 Prac-tices for Orchestrating Productive MathematicsDiscussions (Stein & Smith, 2011). By using thesetogether, I have supported prospective teachers’attempts to engage in the types of thinking theyultimately would have to help their students at-tain, including: constructing viable arguments,critiquing the reasoning of others, describing thetypical value of a distribution, comparing distri-butions, and looking for relationships between vari-ables (Common Core State Standards Initiativeand others, 2010).

1. Posing and Investigating Questions withTinkerPlots

TinkerPlots can be used as a tool to helpstudents form statistical questions and explore



Figure 1: TinkerPlots sample file with data for 100 cats

them. For example, a dataset about a groupof 100 cats is shown in figure 1. The data andaccompanying questions are included with thesoftware. The upper left corner contains a stackof data cards with information about each cat.The card at the top of the stack, for example,is for a cat named “Bits,” who is a male thatweighs 13 pounds and has yellowish eyes (amongother attributes). The 100 cats are initially shownas a haphazard scatter of points in the top rightcorner of figure 1. The software can be used toorganize the data into a useful form for answeringthe questions.

Figures 2 and 3 show how a student could useTinkerPlots to approach the question, “How longdoes it generally take for a kitten to grow intoa cat?” Figure 2 represents the idea that weightmight be used to define the difference betweenkittens and cats. It was produced by clickingon the “age” attribute in the data cards andthen using the “order horizontally” command toput the data in order by age. The next stepwas to click on the “weight” attribute. Thisproduced a color gradient, with the heaviestweights represented as dark red and the lightestweights represented in white. The graph in figure2 shows that many of the white and pale red datapoints belong to cats that are one-half year old orless (the data card for each specific cat appears at

Figure 2: Using TinkerPlots and the “weight” attribute todefine the difference between kittens and cats

Figure 3: Using TinkerPlots and the “body length”attribute to define the difference between kittens and cats

the top of the stack when one clicks on any givenpoint on the graph). This observation can help seta boundary between “kitten” and “cat.” Somemight argue that the “length” attribute couldalso be used to set the boundary. They couldclick on “length” in the data cards to produce acolor gradient graph for that attribute (figure 3).This might lead to a slightly different boundarythan that obtained by looking only at weight.Classroom discussions in which the two differentapproaches are shared help bring out the idea thatstatistical questions are often open to multiplesolution strategies and multiple solutions.

After my students had worked with and dis-cussed several sample data sets included with Tin-kerPlots, I challenged them to explore sample datasets included with the program to formulate andinvestigate their own statistical questions (assign-ment shown in figure 4).

2. Using the 5 Practices Model to SupportClassroom Discourse

I believed that my students’ responses to thehomework assignment shown in figure 4 wouldprovide the basis for rich classroom discourse.


Randall E. Groth

Figure 4: TinkerPlots file exploration assignment

However, since this was the first time they hadbeen responsible for formulating and investigatingtheir own statistical questions, I also believed thatcareful guidance of the classroom conversationwould be necessary.

I employed the “5 practices” model to helpstructure the conversation. The 5 practicesare: (i) anticipating students’ responses, (ii)monitoring responses, (iii) purposefully selectingresponses, (iv) purposefully sequencing responses,and (v) connecting responses (Stein & Smith,2011). After assigning the homework task, Ianticipated how students might interpret it andthe array of strategies (correct and incorrect) theywould use. I then monitored their responses. Tomake monitoring easier, I required students tosubmit their responses to me online at least twohours before class. As I read their responses, Iconcentrated on deciding which strategies wouldbe productive to include in whole-class discussion.I then purposefully selected strategies to beshared so that a diverse array of reasoningwas represented in the discussion. Next, Ipurposefully sequenced the selected responses.My overall objective in sequencing was toconstruct a coherent classroom narrative aroundthe responses. Finally, as I called on students toshare, I helped in connecting their responses. This

involved comparing and contrasting the presentedstrategies, asking students to do the same,emphasizing important concepts in the responses,and discussing the efficiency and generalizabilityof the different strategies. Below, I explain inmore detail how I used each practice to facilitateclass discussion.

2.1. Practice 1: Anticipating Responses

In giving the homework assignment (figure4), I anticipated that my students would havesome difficulties posing statistical questions. Asnoted in the report, “Guidelines for assessmentand instruction in statistics education (GAISE),”

The formulation of a statisticsquestion requires an understanding ofthe difference between a question thatanticipates a deterministic answer anda question that anticipates an answerbased on data that vary (Franklinet al., 2007).

Many of the questions in the “statistics”portions of textbooks do not fit this definition.For example, the question, “What was the meantest score for our last mathematics test?” canbe answered simply by using the algorithm forthe mean and reporting the result. In contrast,the question, “What was the typical score forour last mathematics test?” is more open.Students can examine the distribution of data anddetermine the “typical” score in a number of ways.They might create a plot and identify a centerdata cluster. If a one-number summary of thetypical score is needed, they then might plausiblydescribe the location of the center cluster usingmean, median, mode, or some other method.There is room for personal judgment and reasonedargument in determining a typical value, but notin determining the mean value. I anticipated thatsubtle, but important, differences of this naturemight be challenging for students to grasp.

I also anticipated that students may needassistance using the color gradient feature ofTinkerPlots to its fullest extent. Conventionally,relationships between two variables such as “age”



and “weight” are represented in scatterplotsrather than with color gradients (as in figure2). Because of this, I anticipated thatthose who posed statistical questions aboutrelationships between two variables would tend touse TinkerPlots to produce a scatterplot ratherthan a color gradient diagram. Although doingso is not incorrect, it also does not take fulladvantage of the color gradient representation.I believed it was important for the prospectiveteachers in my class to understand the colorgradient representation because it has provento be an intuitive way for some children toorganize data (Konold, 2002). Hence, I thought itwould be necessary to connect their knowledge ofscatterplots to the color gradient representationat some point.

2.2. Practice 2: Monitoring Responses

As I monitored student responses, I noticeddifficulties they had in posing statistical questions.Many of them posed questions that involved sim-ply finding the largest or smallest values in a dataset. For example, one student chose to examinea dataset about cereals in a grocery store, andasked which cereal had the most sugar per serv-ing. Another student chose a dataset with infor-mation about U. S. Presidents and looked for theone who served the most years. Some of the ques-tions students posed required using TinkerPlotsto compute measures of center without relatingthe choice of measure to the appearance of thedistribution. In one such case, a student askedhow far the mean height for girls in a datasetwas from the mean height for boys in the samedataset. Although these types of questions maybe useful to address for some purposes, they fallshort of being rich statistical questions because oftheir deterministic, closed nature.

Some students did pose questions thatinvolved looking for relationships between twovariables. For example, one student asked if therewas a relationship between a basketball player’sheight and number of points scored. Anotherstudent asked about the relationship betweenheight and the number of rebounds. Thesestudents used scatterplots to examine the data.

Figure 5: Describing the relationship between calories andnutritional values of a cereal

In one case, a student used both a color gradientdiagram and a scatterplot-like representation toexplore a relationship between variables. She wasinterested in looking at how the fastest times onrecord for the men’s 100-meter dash had changedover time. Her representations and accompanyingexplanations are shown in figure 6.

2.3. Practice 3: Purposefully Selecting Responses

The response shown in figure 6 immediatelystood out as an important strategy to includein class discussion, because it linked the colorgradient representation from TinkerPlots withthe more familiar scatterplot representation. Allof the others who had chosen to investigate arelationship between two variables used either thecolor gradient or a scatterplot, but not both.Nonetheless, some who used scatterplots analyzedthem in ways I had not anticipated. A studentexploring the relationship between calories andnutritional value rating for cereals used a verticaldivider to separate the graph and count howmany cereals had relatively high and relativelylow ratings (figure 5). Another student wenta step further and sliced the scatterplot bothhorizontally and vertically by computing themean values for the variables on the horizontaland vertical axes (figure 7). By slicing thegraph twice in this manner, she gained a betterunderstanding of the relationship between abasketball player’s height and number of pointsscored. These unusual uses of scatterplots stoodout as examples of strategies with the potentialto spark discussion and thought during class.

Along with encouraging discussion about


Randall E. Groth

Figure 6: Describing the sharp decrease in Olympic 100m times

questions involving the relationship between twovariables, I wanted to bring out the idea thatTinkerPlots can be used to make comparisonsbetween two groups. Therefore, I selected theresponses shown in figures 8 and 9 for inclusionin class discussion. Figure 8 compares the heightsof females to males. The triangle under each plotrepresents the mean height for each group. Figure9 shows a comparison of calories in different typesof cereal. It was made by plotting the variable“calories” along the horizontal axis and thendragging the categorical attribute “manufacturer”to the vertical. The ⊥-symbol beneath each plotrepresents the median number of calories for eachmanufacturer.

2.4. Practice 4: Purposefully SequencingResponses

In structuring our class discussion of the se-lected responses, I made a distinction betweenthose who posed questions involving the compar-ison of two or more distributions (figures 8 and 9)and those who looked for relationships betweentwo variables (figures 5, 6, and 7). During thefirst half of class discussion, I asked those who hadcompared two or more distributions to present. I

Figure 7: Describing the relationship between basketballplayer height and points scored



Figure 8: Comparing height distributions for females andmales

then asked those who looked for relationships be-tween variables to present. In regard to the lat-ter group, I decided to ask a student who hadused a simple, familiar scatterplot representationto present first. Then, I asked the students whohad produced figures 8 and 9 to present, in orderto illustrate how slicing a conventional scatterplotmay help facilitate its analysis. Finally, I had thestudent who produced figure 6 present, becauseher TinkerPlots file showed how both a scatter-plot and color gradient diagram can be used toexplore relationships.

2.5. Practice 5: Connecting Responses

As students presented their work, I contributedquestions and observations to help connect re-sponses and deepen class discussion. In regardto the two students whose work involved com-parison of distributions (figures 8 and 9), I askedwhy the mean and median should be used as one-number summaries to compare the distributions.I reminded the class that the presence of outliersand skewed distributions can sometimes distortconventional measures of center. This led to adiscussion of how the “divider” feature in Tinker-Plots can be used to highlight the central clus-ter of data. Conventional measures of center thatfall within the highlighted central cluster can betrusted as reasonable one-number summaries ofdata. This discussion helped emphasize the im-portance of visually inspecting data and identify-ing central clusters before simply computing sum-

Figure 9: Comparing calories per serving for several cereals

mary statistics to compare groups.As the responses involving relationships

between variables were shared, I asked theclass to notice how all of the representationsshown provided options for exploring suchrelationships. The conventional scatterplot wasthe most familiar to my group of prospectiveteachers. I pointed out, though, that childrenwho have not yet used scatterplots may findrepresentations such as sliced scatterplots andcolor gradients to be initially more natural andaccessible. This led to a discussion of how one’sbackground knowledge may influence their choiceof representation. This point about children’scognition was a valuable take-away from thediscussion.

3. Conclusion

The 5 practices model proved to be an im-portant tool for managing and guiding discourseabout questions posed with dynamic statistics soft-ware. In some classroom conversations, teacherscall on students at random or ask for volunteers.Although such practices are sometimes useful, it


Randall E. Groth

is possible for important responses that may helpmove the discussion forward to be lost or insuf-ficiently considered. The 5 practices model pro-vided a structure to help me make decisions aboutwhich strategies were most important to share,how the strategies could be sequenced, and howthey could be connected. Without such a struc-ture, I may have missed opportunities to draw at-tention toward creative patterns of thinking andresponses that linked important ideas to one an-other (e.g., responses containing various represen-tations for exploring relationships between vari-ables). Using the 5 practices model in conjunctionwith dynamic statistics software allowed me toconstruct a coherent classroom narrative aroundstudent responses while addressing important sta-tistical ideas from multiple perspectives.

References

Common Core State Standards Initiative and others(2010). Common Core State Standards for Mathemat-ics. Washington, DC: National Governors AssociationCenter for Best Practices and the Council of Chief StateSchool Officers.

Franklin, C., Kader, G., Mewborn, D., Moreno, J., Peck,R., Perry, M., & Scheaffer, R. (2007). Guidelinesfor assessment and instruction in statistics education(GAISE) report: A Pre-K-12 Curriculum Framework .Alexandria, VA: American Statistical Association.

Knuth, E. J. (2002). Fostering mathematical curiosity. TheMathematics Teacher , 95 (2), 126–130.

Konold, C. (2002). Teaching concepts rather than con-ventions. New England Journal of Mathematics, 34 (2),69–81.

Konold, C., & Miller, C. D. (2005). Tinkerplots: Dynamicdata exploration, version 1.1. (Software). http://www.keycurriculum.com/products/tinkerplots. Ac-cessed Feb. 2015.

Rossman, A., Chance, B., & Medina, E. (2006). Someimportant comparisons between statistics and mathe-matics, and why teachers should care. In G. F. Burril,& P. C. Elliot (Eds.), Thinking and reasoning with dataand chance (pp. 323–333). Reston, VA: National Coun-cil of Teachers of Mathematics.

Shaughnessy, J. M. (2007). Research on statistics learning.In F. K. Lester (Ed.), Second handbook of research onmathematics teaching and learning (pp. 957–1009). Re-ston, VA: National Council of Teachers of Mathematics.

Stein, M. K., & Smith, M. (2011). 5 Practices for Orches-trating Productive Mathematics Discussions. Reston,VA: National Council of Teachers of Mathematics.

Randall E. Groth

DEPARTMENT OF EDUCATION SPECIALTIES

SALISBURY UNIVERSITY

1101 CAMDEN AVENUE

SALISBURY, MD 21801



http://www.keycurriculum.com/products/tinkerplots

http://www.keycurriculum.com/products/tinkerplots

Is Edgar Allan Poe Really a Mathematician?

Judy Brown∗, Jennifer Helton, Tamra C. Ragland, M. Todd Edwards

Abstract

In the following article, the authors discuss work associated with The Pit and the Pendulum, athree-week project-based learning unit. Inspired by Edgar Allan Poe’s story of the same name, theproject encourages students to connect their knowledge of trigonometry, physics, and language arts asthey analyze Poe’s classic work. In particular, students use clues from Poe’s written descriptions todetermine whether the prisoner in “The Pit and the Pendulum” has time to escape.

Keywords: dynamic geometry software, project-based learning, inquiry, secondary-level mathematics

Published online by Illinois Mathematics Teacher on December 4, 2015.

The adoption of the Common Core StateStandards for Mathematics (Common CoreState Standards Initiative and others, 2010)has precipitated a shift in the teaching andlearning of mathematics in classrooms throughoutthe country (National Council of Teachers ofMathematics, 2010). In particular, the CommonCore’s emphasis on deep learning and applicationhas brought into question the “one day, onetopic” lesson plan model that has long been afixture in school classrooms. Recognizing this newreality, the authors of this paper have shifted theirinstructional methods from a standard “lecture-homework-lecture” model to one that more fullyengages small teams of students in project-based learning activities. Using the project-basedlearning (PBL) framework (Hallerman & Larmer,2013), our instructional team has constructeda number of projects that require students toconnect their knowledge of mathematics to othercontent areas as they solve meaningful questionsin context.

In the paragraphs that follow, we highlightour work on one such project, a three-weekproject-based unit we refer to as The Pitand the Pendulum. Based on Edgar AllanPoe’s story of the same name (Poe, 1843),the project encourages students to apply theirknowledge of trigonometry, physics, and language

∗Corresponding author

arts as they analyze Poe’s classic work. Inparticular, students use clues from Poe’s writtendescriptions to determine whether the prisonerin “The Pit and the Pendulum” has time toescape. We describe the project in threesections. First, we share instructional detailsof the unit itself including sequencing, pacing,the use of technological tools, and samples ofstudent work. Next, we describe steps thatwe took to address the needs of all learners,including those with individual education plans(IEP), as we highlight specific differentiationstrategies employed at various stages within theproject. Lastly, we assess the effectiveness ofthe project, providing revision ideas and furtherareas of possible investigation. For this project,our mathematics instructional team consisted offour educators: Judy, a mathematics teacherwho taught an Algebra III/Trigonometry class;Jennifer, an Intervention Specialist who workedwith several students in Judy’s class; and Tamraand Todd, both mathematics education professorsat local universities at the time of writing.

1. Instructional details

1.1. Phase 1: Project launch

We introduce students to the project witha YouTube video of Poe’s The Pit and thePendulum (Corman, 1961). Our students siton the edge of their seats as narrator Vincent


Judy Brown, Jennifer Helton, Tamra C. Ragland, M. Todd Edwards

Figure 1: Bocce ball pendulum suspended from classroomceiling

Price describes the lowering of a sharp pendulum,swinging and slowly descending toward a prisonertied down to a platform. As the torture devicebegins to slice the flesh of the victim, we stop themovie abruptly. Not surprisingly, this frustratesstudents. Aw! Mrs. Brown, why did you stop themovie? Does the prisoner die? Does he escape?Reviewing the clip is a natural transition into thedriving questions, which are “Does the prisonerreally have time to escape?” and “Is EdgarAllan Poe really a mathematician?” based on theinformation given in the short story. Poe’s writingprovides us with clues regarding both questions.Using mathematics, physics, and careful analysisof Poe’s text, students have all the tools they needto answer the questions posed above.

1.2. Phase 2: Exploring a model

Following a careful viewing of the film excerpt,students explore the plausibility of Poe’s textusing a pendulum we construct with fishing wireand a bocce ball suspended from the classroomceiling. Students begin their analysis by observinga pendulum released from the tip of a volunteer’snose, as shown in figure 1.

Measuring the period

As the pendulum swings back and forth, somestudents are surprised (and relieved) to observethat the bocce ball never swings beyond theball’s original position. In other words, no nosesare harmed in the demonstration. This initialexploration provides students with opportunitiesto discuss period, the time it takes for the

ball to complete one full cycle away from theparticipant’s nose and back again. Classmatesrecord one period of the pendulum with astopwatch as the pendulum completes one cycle.

Discussion points

The pendulum demonstration motivates stu-dent discussion on a variety of topics suitable forstudents with a wide range of ability levels, suchas:

• Potential and kinetic energy. At what point(or points) does the pendulum possess po-tential energy? At what point (or points)does the pendulum exhibit kinetic energy?(Suitable for Algebra I students)

• Period. How is the period of the pendu-lum affected by the height at which it is re-leased? (Suitable for Algebra II students)

• Speed. Is the speed of the pendulum con-stant? If not, when are minimum and max-imum speeds achieved? Will the pendulumeventually stop? If so, why? If not, whynot? (Suitable for Algebra III/Trigonometrystudents)

• Acceleration. Is the acceleration of the pen-dulum constant? If not, when are minimumand maximum acceleration achieved? (Suit-able for Calculus students)

1.3. Phase 3: Student-constructed models

Position and velocity graphs

On the following day, students anchor weightsto a string to make their own pendulums. Insmall groups, they release the pendulums fromring stands and use motion detectors and graph-ing calculators to graph the following with respectto time: (a) the position of the pendulums; and(b) the velocity of the pendulums. These graphsare used to explore conjectures from the previ-ous day’s activities. Figure 2 illustrates two suchgraphs generated by students.

Analysis of position and velocity graphs gen-erates lively discussion among our students. Someindicate that the graphs look quite similar (e.g.,



Figure 2: Top: position with respect to time, Bottom:velocity with respect to time (generated by a TI-NspireCAS graphing calculator connected to a Vernier MotionDetector)

both “go up and down like a roller coaster”). Oth-ers notice some damping in the graphs (e.g., “thehighest and lowest positions get closer to zero witheach cycle”). Yet others focus on differences be-tween the graphs (e.g., the times at which max-ima and minima occur are not the same in thetwo graphs).

Geometric models

Using Geometer’s Sketchpad (Jackiw, 2014) tomodel the motion of the pendulum allows the stu-dents to make further mathematical connections.In the construction illustrated in figure 3, fromthe activity A Sine Wave Tracer (Bennett, 2002)the student can drag point D right or left usingGeometer’s Sketchpad and see that point F moveshorizontally across the sketch as well. As point Emoves around the circle, point F moves up anddown like a sewing machine needle. Starting withpoint D to the right of the circle and combiningthese two motions creates a sine curve, a periodicfunction that models the motion of the pendulum.The right semicircle represents the pendulum onits side, and as F moves up and down, it repre-sents the period of the pendulum, which tracesthe diameter of the circle.

Adding the coordinate grid behind the sketchwith the origin at the center of the circle allowsstudents to estimate the circumference of the cir-

Figure 3: Model of pendulum movement with Geometer’sSketchpad

cle in grid units. Counting the horizontal distanceof the curve and dividing by the radius, studentsestimate that the circumference of the unit circleis about 6.28, a close approximation for 2π.

1.4. Phase 4: Building arguments

Story Synopsis

The next few days in mathematics class werespent using the story to organize the informationpertinent to answer the driving question, “Doesthe prisoner really have time to escape?” Theprisoner tells us, “Looking upward, I surveyed theceiling of my prison. It was some 30 or 40 feetoverhead” (Poe, 1843, p. 12). He said the blade ofthe pendulum swung above him for what seemedlike hours until it was 3 inches above his chest.“I saw that some ten or twelve vibrations wouldbring the steel in actual contact with my robe”(Poe, 1843, p. 15). He rubbed food from a nearbybowl onto the ropes that tied his hands down.He thought that the swarming rats could chewthrough the ropes to free him. “Yet one minuteand I felt the struggle would be over” (Poe, 1843,p. 17). Perhaps it was one and a half minutesbecause he had been swooning in and out of con-sciousness.

Twice again it swung, and a sharp senseof pain shot through every nerve. Butthe moment of escape had arrived. Ata wave of my hand my deliverers hur-ried tumultuously away. With a steadymovement—cautious, sidelong, shrink-ing, and slow—I slid from the embraceof the bandage and beyond the reach



Figure 4: Students testing one of the three variables

of the scimitar. For the moment atleast, I was free. (Poe, 1843, p. 17)

The plan worked! This is how the story goes,but is it actually possible? The students realizedthat they needed to find a formula to be able tocalculate the period of the pendulum and deter-mine if 10-12 swings of 30-40 feet of rope wouldtake between 1 and 11

2minutes, which is the time

the prisoner estimated for the rats to free him.The students realized they needed to investi-

gate the parts of the pendulum that may affectthe period in order to model a relationship be-tween that factor and the period. Students weredivided into six groups, each of which constructedpendulums using ring stands, strings, and weightsborrowed from the physics classroom (see figure4). They tested three variables: amplitude (an-gle of release), the length of the string, and theweight of the bob. Each variable was exploredby two of the six groups. Upon completion, thegroups reported that, in their experiments, of thethree factors that were tested, only the length ofthe string affected the period of the pendulum.

Figure 5: Scatterplot of period with respect to the pendu-lum length using the TI-Nspire CAS graphing calculator

The following day each group took a differentstring length: 15 cm, 30 cm, 60 cm, 100 cm, 125cm, and 150 cm, and used the procedure from theprevious days to find the period of the pendulum.The class created a scatterplot to illustrate theirdata, with the period on the x-axis, and the stringlength on the y-axis.

The graph suggested a quadratic relationship

L = g

(T

2π

)2

, where L is the length of the string,

T is the time of the period in seconds, and g is ac-celeration due to gravity (which in our problem weapproximated as 32.2 ft/s2). Figure 5 illustratesthe graph of the inverse function, with length onthe x-axis and period on the y-axis, yielding asquare root function

T = 2π

√L

g.

Techniques from calculus are required to de-rive the formula for the period of the pendulum,but from previous activities the students under-stood where the parts of the period of the pendu-lum equation came from.

Students used the formula to generate a pre-diction for the period of the pendulum for the ropelengths, as shown in the leftmost column of thetable in figure 6. They multiplied by 10 and 12to get the second and third columns of the table.The students spent a day working in their groupsto analyze the table and answer the driving ques-



Figure 6: Poe worksheet with student work

tions. This analysis was not as straightforwardand readily apparent as we expected, as shown inthe student work.

An Aha! Moment

Table 1 uses the time for one period of the pen-dulum to calculate the time for 10 and 12 swingsat 30 and 40 feet only. Initially, we saw the storylike this:

40 ft 30 ftMin: 10(7.0) = 70 sec Min: 10(6.06) = 60.6 secMax: 12(7.0) = 84 sec Max: 12(6.06) = 72.72 sec

Table 1: Initial period calculations

This answer seemed very clear cut. The pris-oner has time to escape if the rats chew thoughin 60 seconds, but does not have time to escape ifthe rats chew through in 90 seconds.

Tamra brought a mathematics teacher candi-date with her. The candidate was present dur-ing the pendulum investigation and in his solutiontable, he investigated the lengths of the rope in5 feet increments between 30 and 40 feet ratherthan just the minimum and maximum ends of thespectrum. Each of the student groups used timeframes of 60 seconds (1 min) and 90 seconds (1.5min), excluding the times in between. The an-swer actually depended upon all of these factors:

Figure 7: Student response to the driving question

length of the rope, number of swing cycles remain-ing, and time for the rats to chew through therope. Since all of these factors vary within thestory, the students’ solutions did as well. As thecalculations in table 1 suggest, when the rope is 40feet long and has 10 swings left, the prisoner willbe cut at 70 seconds. So, if the rats chew throughin 60 seconds the prisoner will escape with 10 sec-onds to spare; but if the rats do not chew throughuntil 90 seconds, then he would have been slicedthrough 20 seconds earlier and would not haveescaped. The work suggests that there was nounique correct answer to this question.

Even with this variation, the general consen-sus was that at some point in the given frame-work the prisoner would have had time to escape:Edgar Allan Poe was a mathematician after all!

Student Work Samples

Student work in figure 7 suggests that thestudents have it backwards. Can your studentsidentify why this is not a correct solution andwhat the students were thinking?

In fact, there isn’t a single correct solution,but the entire project is full of mathematics andlends itself to further instruction. In AdvancedPre-Calculus, we continued to study distance,velocity, and acceleration problems, finding thegraphs and the equations, which led into a studyof derivatives and integrals and how they relateto these problems. Some of these studentsalso took Physics, which the lead teacher andstudent candidate visited, and they were studyingthe pendulum and simple harmonic motion aswell and making connections between the topicsbeing studied in the two classes. One studentcommented, “This is magical!” after another



student elaborated on the connected concepts.

1.5. Phase 5: Foucault pendulum

The culminating piece of the project wasa field trip to a local university that housesa Foucault pendulum. We timed the periodand determined the length of the rope withoutdirect measurement and discussed our findingswith a physics professor who demonstrated simpleharmonic motion to small groups of our students.Our students were surprised when the professornoted that he often forgets if gravity is dividedby the length or length is divided by gravity inthe period equation. He reminded us that muchof what we study is logical and related to priorknowledge so that memorization isn’t necessary.If students recall that the length of the periodincreases as the length of the rope increases,a result that was previously observed throughexperimentation, they can reason that the lengthmust be in the numerator of the equation withgravity in the denominator. He also asked if wetook into account the rate at which the pendulumwas descending in the story. He thought it couldbe calculated from the information given. Basedon this comment, the students realized that theirproject could be further explored. In science,initial findings lead to new questions in a never-ending cycle of inquiry.

2. Differentiation/Intervention

The main strategy for differentiatinginstruction is to address it in the planning process.We planned the project together over a seriesof about four meetings, each approximately twohours in length. We started with the overalllearning targets, broke down the strategies neededto attain the targets, and planned activitiesto lead the students to answer the essentialquestion. As the math teachers discussed ideasgained from previous pendulum explorations, theintervention specialist asked clarifying questions,identified areas of possible struggle, and suggestedideas for scaffolding instruction. For example,the cumbersome computations and mathematicalprocesses on the worksheets were broken into

simpler steps using the row headings, such as“divide row C by 10,” to find the averageof the period. An example of cross-curricularmodification happened in the English classes.The students on individualized education planswere given copies of the story with importantinformation underlined, and had to explain why itwas important. The other students had to extractthe important information from the unmarkedstory.

Additionally, we practiced the activities asa group, placing ourselves in the roles of thestudents, following the directions to perform theexperiments, use the technology, and completethe investigations. This helped alleviate confusionfor the students, and made their work time moreseamless and productive. Involving teachers withdifferent perspectives enhanced the quality of theunit.

Finally, it was beneficial for the teachers tosupport one another by team-teaching the lesson.Judy, the classroom teacher, lead the instructionand Jennifer, the intervention specialist, waspresent in the classroom throughout the project.On the days of experimentation, Tamra and Toddattended as well and worked with small groups ofstudents, asked necessary questions, and alertedthe lead teacher when students were confused sonecessary alterations could be made.

3. Assessment of Lesson

Continuous informal assessment occurredthroughout the lesson. The teacher askedquestions, making note of student responses. Asthe students showed understanding, the lessoncontinued. When the students were not ableto respond correctly, adjustments were made inthe lesson to clarify concepts. The presence ofmultiple teachers made it easier for the neededchanges to occur. For instance, when the scientificconcepts of the pendulum were introduced, wedecided the students needed to time the periodseveral times in order to understand the conceptof an unchanging period. This allowed thestudents to experience the idea of measuring aperiod, therefore clearly defining it. We also



noticed during the lesson that the students wouldneed to understand the concept of the unchangingperiod to have a better understanding in futureactivities.

More formal assessment occurred using thegraphs the students created and the worksheetsthey completed. Again, the teacher graded theirwork immediately in order to give feedback tothe students before moving forward to ensure theconcepts were understood and prior knowledgewas properly utilized.

4. Acknowledgments

The authors would like to acknowledge theteachers, Jackie Harris (English), Jenn Reid(English) and Hope Strickland (Physics) of theDayton Regional STEM School, Dr. Todd Smith,Associate Professor of Physics at the Universityof Dayton, and the secondary mathematicsteacher candidate Cody Brannum of Central StateUniversity for their contributions to this project.

Author Note: All worksheets for this project-lesson are found at the end of the article.

References

Bennett, D. (2002). Exploring Geometry with the Geome-ter’s Sketchpad . Key Curriculum Press.


Corman, R. (1961). The pit and the pendulum (Film). viahttp://www.youtube.com/watch?v=uPG9YqKx5A. Ac-cessed Oct. 2015.

Hallerman, S., & Larmer, J. (2013). The role ofPBL in making the shift to Common Core.http://bie.org/blog/the_role_of_pbl_in_

making_the_shift_to_common_core. AccessedOct. 2015.

Jackiw, N. (2014). Geometer’s sketchpad, version 5. (Soft-ware). http://www.dynamicgeometry.com. AccessedOct. 2015.

National Council of Teachers of Mathematics (2010). Mak-ing it Happen: A Guide to Interpreting and Implement-ing Common Core State Standards for Mathematics.Reston, VA. PDF E-book.

Poe, E. A. (1843). The Pit and the Pendu-lum. http://www.ibiblio.org/ebooks/Poe/Pit_

Pendulum.htm. Accessed Oct. 2015.

Judy Brown

NATIONAL TRAIL HIGH SCHOOL

6940 OXFORD-GETTYSBURG ROAD

NEW PARIS, OHIO 45347


Jennifer Helton

MONTGOMERY COUNTY LEARNING CENTER WEST

3500 KETTERING BOULEVARD

KETTERING, OH 45439


Tamra C. Ragland

WINTON WOODS CITY SCHOOLS DISTRICT

1215 W. KEMPER ROAD

CINCINNATI, OH 45240


M. Todd Edwards

DEPARTMENT OF TEACHER EDUCATION

MIAMI UNIVERSITY

OXFORD, OH 45056



http://www.youtube.com/watch?v=uPG9YqKx5A

http://bie.org/blog/the_role_of_pbl_in_making_the_shift_to_common_core

http://bie.org/blog/the_role_of_pbl_in_making_the_shift_to_common_core

http://www.dynamicgeometry.com

http://www.ibiblio.org/ebooks/Poe/Pit_Pendulum.htm

http://www.ibiblio.org/ebooks/Poe/Pit_Pendulum.htm


Appendix APendulum Unit: Amplitude

1. The variable to test is the amplitude or angle of release. You want to determine ifchanging the angle of release of the pendulum changes the time for the period of thependulum. That means the length of the string and the weight of the bob or bobtype must be controlled but the amplitude or angle of release will vary.

2. Choose the length of string and keep it the same for all three trials. This does notneed to be measured, just keep it the same each time.

3. Choose the bob. This does not need to be weighed, just keep using the same one foreach trial.

4. Tie the string to a T-stand. Tie the bob to the end of the string. This will be thesame for each trial. Release the bob from an angle of chosen degrees and record it inthe first column of the table below. This will change for each trial. Release the boband time the periods of the pendulum for 10 full swings. Record that in the tablebelow. Then divide that number by 10 to get the average time for 1 period of thependulum at that angle. Do three trials and divide the total time by 3 to get theaverage in the final row of the table.

5. Change the angle of release and record the angle used in the second column of thetable and repeat step 4 above. Complete the second column of the table.

6. Choose a third and final angle of release that is different from the two already triedand complete the third column of the table below.

Note: You don’t necessarily need to measure the angle of release. You could just markthree separate angles on a protractor as a, b, and c, and release the bob from those pointseach time to keep this variable controlled. Whether or not it affects the period is what isimportant. If the times change as the angle of release changes then you will know amplitudehas an effect on the period of the pendulum.

Angle of release

A Trial 1: 10 periods in seconds

B Trial 1: 1 period in seconds (divide row A by 10)

C Trial 2: 10 periods in seconds

D Trial 2: 1 period in seconds (divide row C by 10)

E Trial 3: 10 periods in seconds

F Trial 3: 1 period in seconds (divide row E by 10)

G Total time for all three trials of the pendulum (addrows B, D, and F)

Average time for three trials of the period of thependulum (divide row G by 3)



Pendulum Unit: Length of the string

1. The variable to test is the length of the string. You want to determine if changing thelength of the string changes the time for the period of the pendulum. That meansthe weight of the bob or bob type and the angle of release must be controlled but thelength of the string will vary.

2. Choose the bob and keep it the same for all trials. This does not need to be weighed,just keep it the same each time.

3. Choose the angle of release. It doesn’t matter what the measure of the angle is. Whatmatters is that the angle of release stays the same for each trial and does not change.So once you choose an angle, keep using the same one for each trial. You could justuse halfway between 0◦ and 90◦.

4. Tie one unit of string to a T stand. This can be one foot or 20 centimeters or whateveryou choose. Record this length in the first column of the table below.

5. Release the bob from the chosen angle, and time the periods of the pendulum for10 full swings. Record that in the table below. Then divide that number by 10 toget the average time for 1 period of the pendulum at that angle. Do three trials anddivide the total time by 3 to get the average in the final row of the table.

6. Change the length of the string by making it longer or shorter and record that in thesecond column and repeat step 5 above. Complete the second column of the table.

7. Choose a third and final string length that is different from the two already tried andcomplete the third column of the table below.

Note: You don’t necessarily need to measure the length of the string. You could just markthree separate lengths as a, b, and c or double and triple the length of a to get b and c.Whether or not it affects the period is what is important. If the times change as the lengthof the string changes then you will know that string length has an effect on the period ofthe pendulum.

Length of string











Pendulum Unit: Weight of the bob (bob type)

1. The variable to test is the bob. You want to determine if changing the weight or typeof bob changes the time for the period of the pendulum. That means the length ofthe string and the angle of release must be controlled but the bob weight and typewill vary.

2. Choose the length of the string and keep it the same for all trials. This does not needto be measured, just keep it the same each time.

3. Choose the amplitude (or angle of release). It doesn’t matter what the measure ofthe angle is. What matters is that the angle of release stays the same for each trialand does not change. So once you choose an angle, keep using the same one for eachtrial. You could just use halfway between 0◦ and 90◦.

4. Tie string to a T stand. Tie one bob to other end of the string. You can choose fromwashers, ping-pong balls, quarters, tennis balls, etc. These will change for each trialsince this is the test variable. Record the bob type in the first column of the table.

5. Release the bob from an angle of chosen degrees. Release the bob and time theperiods of the pendulum for 10 full swings. Record that in the table below. Thendivide that number by 10 to get the average time for 1 period of the pendulum atthat angle. Do three trials and divide the total time by 3 to get the average in thefinal row of the table.

6. Change the bob at the end of the string and record the type used in the secondcolumn below and repeat step 6 above. Complete the second column of the table.

7. Choose a third and final bob that is different from the two already tried and completethe third column of the table.

Note: You don’t necessarily need to measure the weight of the bob. You could just usethree different materials that seem to be lighter or heavier than each other. Whether or notit affects the period is what is important. If the times change as the weight or type of bobchanges then you will know that the bob has an effect on the period of the pendulum.

Bob type or weight











Pendulum Unit: The period of the pendulum

Now that we know that the length of the string is the variable that affects the period of thependulum, we can make a graph to see the pattern in these two relationships. Set up yourpendulum using the string lengths assigned to your group. Measure to the bottom of thebob. Use the same procedure you used when testing the pendulum variables and completeyour portion of the chart below to share with the class.

Length of string (in cm)15 30 60 100 125 150









Graph the results using the x-axis for the length of the string (in cm) and the y-axis forthe period of the pendulum (in sec). What type of graph below does this model?

• A linear function: y = mx+ b

• A quadratic function: y = x2

• A cubic function: y = x3

• An exponential function: y = a · bx

• An inverse variation function: y = 1x

• A square root function: y =√x

The formula for the period of the pendulum is

p = 2π

√L

g

where p is the period of the pendulum, L is the length of the string, and g is the accelerationdue to gravity (9.8 m/s2, or 32.2 ft/s2).

Graph the inverse of the results above. Let the x-axis be the period of the pendulum andthe y-axis be the length of the string. What function above does this model? The formulafor the period of the pendulum is also:

L = g

(p

2π

)2



Appendix BPoe problem summary

In the actual story, “The Pit and the Pendulum,” Edgar Allan Poe writes that the pendulumis 30-40 feet long. He says that he thinks there are 10-12 swings left and that he thinks hehas about 1 to 11

2 minutes to escape. Use the formula for the period of the pendulum tocomplete the table below. (Hint use 32.2 ft/s2 instead of 9.8 m/s2 for gravity)

Length (ft) Time (1 period) Time (10 periods) Time (12 periods)

40

35

30

25

20

15

10

5

Does the prisoner in “The Pit and the Pendulum” have time to escape? Given his assump-tions, are there times when he will be able to escape and other times when he will not?Is Edgar Allan Poe really a mathematician? Based on the information extracted from thestory (look for details), if he does escape, when do you think that happens? In other words,which assumptions are true? Brainstorm here. Use charts, tables, graphs, etc.

At 25 ft, is there time for a prisoner to escape?

At what length do you think a person would NOT have time to escape? Explain yourreasoning.

Will he always be able to escape if the rats chew through the rope in 1 minute? Will heever be able to escape if it takes the rats 11

2 minutes to chew through the rope? Explainyour reasoning.


Developing Meaning in Trigonometry

Valerie May, Scott Courtney∗

Abstract

Trigonometric concepts and ideas continue to be an important component of the high schoolmathematics curriculum. In spite of its importance to both high school and advanced mathematics andscience, research has shown that trigonometry remains a difficult topic for both students and teachers.In this article, we describe a sequence of activities designed for students to identify relationshipsbetween the sine, cosine, and tangent functions; we derive trigonometric identities related to the sumof angles; and we connect trigonometry to other areas of mathematics. The sequence of activities isdesigned to support students’ development of coherent trigonometric meanings by building on students’prior knowledge in an active learning environment.

Keywords: trigonometry, trigonometric functions, trigonometric identities

Published online by Illinois Mathematics Teacher on February 29, 2016.

In alignment with the Common Core StateStandards for Mathematics (Common CoreState Standards Initiative and others, 2010),trigonometric functions and other trigonometricideas and concepts (e.g., trigonometric ratios,the laws of sines and cosines) continue tobe important components of the high schoolmathematics curriculum. Further, the capacityto work with trigonometric functions is requisitefor study in both advanced mathematics (e.g.,Fourier series and the Fourier transform) andthe sciences (e.g., modeling the behavior of lightand sound waves). Yet despite its importance toboth high school and advanced mathematics andscience, research has shown that trigonometryremains a difficult topic for both students andteachers (Brown, 2006; Thompson et al., 2007;Weber, 2005).

Thompson describes the difficulties that mid-dle and secondary school students in the UnitedStates have with trigonometry, resulting from an“incoherence of foundational meanings developedin grades 5 through 10” (Thompson, 2008, p. 48).Rather than developing a meaning of angle mea-sure that supports a single trigonometry, encom-passing both triangle similarity and periodic be-


havior, typical curricula develop them separatelyand unrelatedly. Specifically, middle school andsecondary mathematics textbooks develop two un-related approaches to trigonometry: the trigonom-etry of triangles and the trigonometry of periodicfunctions.

In this article, we describe a sequence of activ-ities (a multiple-day lesson) designed for studentsto: conjecture relationships between the sine, co-sine, and tangent functions; derive trigonometricidentities related to the sum of angles; and con-nect trigonometry to other areas of mathemat-ics. In an active learning environment, and bybuilding on students’ prior and developing un-derstandings and ways of thinking, the sequenceof activities is designed to support the develop-ment of coherent meanings for several trigono-metric concepts and ideas, including trigonomet-ric functions.

The activities are designed with several im-portant goals. Firstly, the activities help studentsconnect concepts in geometry and algebra. Sec-ondly, the activities help students develop an un-derstanding of how mathematics ideas develop.Thirdly, students come to see that mathematicsprinciples and properties have their foundationsin logic and are not just arbitrary rules madeby mathematicians—contrary to what many stu-


Valerie May, Scott Courtney

ConceptualCategory

Functions (F) Functions (F) Functions (F) Geometry (G)

Domain TrigonometricFunctions (TF)

TrigonometricFunctions (TF)

TrigonometricFunctions (TF)

Similarity, RightTriangles, andTrigonometry (SRT)

Cluster A. Extend the domainof trigonometricfunctions using theunit circle

C. Prove and applytrigonometricidentities

C. Prove and applytrigonometricidentities

C. Definetrigonometric ratiosand solve problemsinvolving righttriangles

Standard 3. Use specialtriangles to determinegeometrically thevalues of sine, cosine,tangent for π

3 , π4 , and

π6 , and use the unitcircle to express thevalues of sine, cosine,and tangent for π − x,π + x, and 2π − x interms of their valuesfor x, where x is anyreal number.

8. Prove thePythagorean identitysin2(θ) + cos2(θ) = 1and use it to findsin(θ), cos(θ), ortan(θ) given sin(θ),cos(θ), or tan(θ) andthe quadrant of theangle.

9. Prove the additionand subtractionformulas for sine,cosine, and tangentand use them to solveproblems.

7. Explain and use therelationship betweenthe sine and cosine ofcomplementary angles.

Table 1: Common Core content standards addressed by the lesson

dents believe. Lastly, and possibly most impor-tantly, the activities require students to engage inmathematical discourse designed to promote co-operation and high-level cognitive thinking.

1. The Trigonometry Lesson (Sequence ofActivities)

As designed, the lesson could be implementedat the beginning of a trigonometry unit in highschool Algebra II, Integrated Mathematics II orIII, or a fourth-year mathematics course (e.g.,Precalculus). The specific Common Core contentstandards addressed by the lesson are presentedin table 1.

In order to participate productively in thelesson, students should be able to use a functionto create a table of values, recall and utilizethe Pythagorean theorem and its consequences,and think about angle measure in terms of themeasure of a subtended arc. With such anunderstanding of angle measure, when given anangle, students can imagine an arbitrary circlecentered at the angle’s vertex. Then, “degree”is viewed as an arc on the circle whose length

Lesson Component Component Description Timeline

Activity 1 Filling the table with data Day 1

Activities 2A, 2B Trigonometric identities Day 2

Activity 3 Making connections Day 3

Activity 4 Determine the function equations Day 4

Table 2: Lesson Components and Timeline

is 1/360 of the circle’s circumference and “anglemeasure in degrees” is viewed as the length ofthe arc subtended by the angle, measured inarcs of length 1/360 of the circle’s circumference(Dubinsky & Harel, 1992; Thompson et al., 2007).As indicated by Thompson, this way of thinkingabout angle measure describes the “principle bywhich a protractor works” (Thompson, 2008,p. 50).

In the following sections, we describe students’engagement with the activities over multiple days,and specify how each activity supports the de-velopment of significant trigonometric meanings.Although original, the sequence of activities wasinspired by Avila’s article in which Algebra stu-dents used functions to draw engaging artwork(Avila, 2013). Table 2 displays the timeline forthe different components of the lesson.



2. Activity 1: Filling the Table with Data

Depending on students’ prior knowledge andskills, the first component of the lesson, complet-ing the trigonometric function table (or “Trig Ta-ble”), could be done for homework as prepara-tion for subsequent in-school activities. However,for the purposes of this article, we will treat thetrigonometric function table component as an in-class activity. Depending on students’ prior un-derstanding and experience, we sometimes needto demonstrate how to use a compass and pro-tractor. Whereas such demonstration will involvea whole-class discussion, the remaining activitiesin the lesson are done in small groups of two tothree, and managed in ways that promote reflec-tive mathematical discourse (Cobb et al., 1997).

The initial activity requires the use of a pencil,graph paper, a compass, and a protractor. Theinstructions provided to students are displayed inappendix A.

Throughout Activity 1, we ask students toimagine a right triangle embedded in the circleso that the triangle’s hypotenuse is the circle’sradius and one angle is formed by the angle inquestion (10◦, 20◦, etc.). Furthermore, we definesine and cosine as follows:

By “sine of an angle” we mean thepercent of the radius’ length made bythe length of the side “opposite” theorigin in the embedded right triangle.By “cosine of an angle” we mean thepercent of the radius’ length made bythe length of the side “adjacent” theorigin.(Thompson et al., 2007, p. 417)

Such meanings allow the cosine and sine functionsto be explored as functions, where the output ofthe cosine function is the abscissa (x-coordinate)of the terminus of the arc subtended by theangle and the output of the sine function is theordinate (y-coordinate) of the terminus of the arcsubtended by the same angle, with both measuredas a fraction of one radius (Moore, 2010, p. 5).

3. Activity 2: Trigonometric Identities

The second activity requires the use of thecompleted table of values (Trig Table) from Ac-tivity 1. The instructions provided to students forActivity 2A are shown in appendix B.

At this stage of the lesson, we bring the groupstogether and engage the class in a discussion re-garding the group results for Activity 2A. Groupsare asked to explain their mathematical thinkingas they articulate their results. Next, we againpartition the class into groups of two to three stu-dents (the same groups as for Activity 2A) to be-gin Activity 2B (see appendix C).

At this stage of the lesson, we bring the groupstogether again and engage the class in a discus-sion regarding the group results for Activity 2B.Groups are asked to explain their mathematicalthinking as they articulate their results.

4. Activity 3: Making Connections

The third activity requires the use of the com-pleted table of values (Trig Table) from Activity 1and the group results from Activities 2A and 2B.We inform students that they will explore cos(α)and sin(α) in connection with algebra. The in-structions provided to students for Activity 3 areshown in appendix D.

At this stage of the lesson, we bring the groupstogether and engage the class in a discussion re-garding the group results for Activity 3. Groupsare asked to explain their mathematical thinkingas they articulate their results. Throughout thisdiscussion, we support students in making con-nections between the Pythagorean trigonometricidentity, the equation of a circle with center (h, k),and meanings for sine and cosine in terms of theembedded right triangle with “adjacent” and “op-posite” sides relating to the origin.

Next, we inform the class that we will continueto think in terms of algebra. Students have usedCartesian coordinates to graph a circle and canrecognize certain functions (and function families)by looking at their graphical representation. Weinform students that they are to construct a graphof the sine function using graph paper and thegiven table of values (see figure 1).



Value of angle x(radians)

0 π6

π4

π3

π2

3π4 π 5π

43π2

7π4 2π 9π

45π2

11π4 3π 13π

47π2

15π4 4π

Value of sinx

Figure 1: Table for graphing the sine function. This figure displays the table of values that students are requested tocomplete, and then use to graph the sine function.

Once groups have completed the graph ofthe sine function, they are asked to explainany patterns that they notice (“What does thefunction look like?”), and to explain the patternof the sine function in term of angles around thecircle. Students will need to explain how pointson the unit circle correspond to points on the sinegraph, and describe how the pattern of the sinefunction emerges through continuously evaluatingthe ordinate (y-coordinate) of the terminus of thearc subtended by the angle as one moves aroundthe unit circle with increasing angle values.

5. Activity 4: Determine the FunctionEquations

The culminating activity is designed tobe hands-on and challenging, yet enjoyable,especially for students who like to use technologyto manipulate graphical representations offunctions. For this activity, students are requiredto employ their prior knowledge and skills,including recalling the graphical representationsof several families of functions and determiningthe domain and range of a function (particularly,functions with restricted domains and ranges).In addition, students will learn that distinctfunction equations can generate similar graphicalrepresentations. Finally, the activity is designedto motivate students to utilize online graphingcalculator technology to explore functions andquickly ascertain how they might need to modifytheir functions to create their diagrams. Forexample, students will need to construct twocircular functions in order to draw the moon. Thedirections and diagram (or “picture”) given to thestudents are shown in figure 2.

The final activity is followed by a full classdiscussion in which students compare pictures,the function equations students used to re-create

Activity 4Examine the following picture and find a series of function(equations) that, when graphed, generate the picture. Usean online graphing calculator tool (such as desmos.com)to graph your function equations. You should utilize thosefunctions that we have discussed in class or that you haveexperienced in your prior courses. You may need to restrictthe domains and/or ranges of certain functions. Pleaseprint a screen shot of your function equations and thepicture that they generate.

Figure 2: Sailing under the moon. This figure illustrates asample diagram (created using desmos.com) that studentsare required to re-create by determining the functionequations whose graphs generate the diagram. For aresource to help you use Desmos to create graphics, see,e.g., Ebert’s recent article (Ebert, 2015).


http://www.desmos.com

http://www.desmos.com


the pictures, the required domain and/or rangerestrictions, and the strategies students employedto help them develop their series of functionequations.

6. Conclusion

As indicated in this article, the teacherplays a significant role throughout the sequenceof activities in managing student thinking anddiscourse to focus on meanings and the coherenceof meanings. The main difficulty that wehave encountered when engaging students inthese activities involve those students lacking inexperiences that asked them to consider whatit is they are measuring when they measure anangle. For such students, when pushed to considersuch a query, a common reply is the amount theterminal side is rotated from the initial side in atriangle. As described by Thompson, et al., forsuch students, “Angles are measured in fractionsof a rotation [and] trigonometry is about solvingtriangles” (Thompson et al., 2007, p. 430). Suchunderstandings and ways of thinking can be quiteresistant to attempts at their modification.

One of the more productive outcomes thatwe have noted in using these activities is thatstudents exhibit an increased level of comfortin articulating their meanings, thinking, andreasoning, and critiquing the thinking andreasoning of others—including their teacher.Furthermore, the sequencing of the activitiesallows students to start with the conceptions theyhave developed, even those typically deleteriousto connecting right triangle trigonometry with thetrigonometry of periodic functions. Finally, wehave noticed that students who have developedmeanings for angle measure, degree, sine of anangle, and cosine of an angle, as presentedhere, lack the difficulties their classmatesencounter when trying to construct an imageof cos(sin(15◦)); that is, when attempting toconceive of sin(15◦) as an argument to cosine.

References

Avila, C. L. (2013). Graphing art revisited: The evolutionof a good idea. Ohio Journal of School Mathematics,67 , 1–10.

Brown, S. A. (2006). The trigonometric connection: stu-dents’ understanding of sine and cosine. In Proceed-ings 30th Conference of the International Group for thePsychology of Mathematics Education (p. 228). Prague:International Group for the Psychology of MathematicsEducation.

Cobb, P., Boufi, A., McClain, K., & Whitenack, J. (1997).Reflective discourse and collective reflection. Journalfor research in mathematics education, 28 (3), 258–277.


Dubinsky, E., & Harel, G. (1992). The nature of the pro-cess concept of function. In E. Dubinsky, & G. Harel(Eds.), The concept of function: Aspects of epistemologyand pedagogy (pp. 85–106). Volume 25 of MAA Notes.

Ebert, D. (2015). Graphing projects with Desmos. Math-ematics Teacher , 108 (5), 388–391.

Moore, K. C. (2010). The role of quantitative and covaria-tional reasoning in developing precalculus students’ im-ages of angle measure and central concepts of trigonom-etry. In Proceedings for the Thirteenth SIGMAA on Re-search in Undergraduate Mathematics Education Con-ference.

Thompson, P. W. (2008). Conceptual analysis of math-ematical ideas: Some spadework at the foundations ofmathematics education. In Proceedings of the AnnualMeeting of the International Group for the Psychologyof Mathematics Education (pp. 45–64).

Thompson, P. W., Carlson, M. P., & Silverman, J. (2007).The design of tasks in support of teachers’ developmentof coherent mathematical meanings. Journal of Mathe-matics Teacher Education, 10 (4), 415–432.

Weber, K. (2005). Students’ understanding of trigonomet-ric functions. Mathematics Education Research Jour-nal , 17 (3), 91–112.

Valerie May

WADSWORTH HIGH SCHOOL

625 BROAD STREET

WADSWORTH, OH 44281

E-mail : wadc [email protected]

Scott Courtney

KENT STATE UNIVERSITY

401 WHITE HALL

P.O. BOX 5190

KENT, OH 44242-0001




Appendix AInstructions for Activity 1

Please follow and complete the directions below using a pencil, graph paper (prefer-ably 1 cm grid), a compass, and a protractor:

1. Draw a Cartesian coordinate system. Make certain to label the x- and y-axes.

2. Construct a quarter circle (in the first quadrant) of radius 10 cm (or 1 dm).Consider the radius of the circle to be 1 unit.

3. Use your protractor to construct an angle of 10 degrees, that is an arc of length10/360 the circle’s circumference. Use the origin (O) as the vertex and thex-axis for one side of the angle.

4. Label as P the point that intersects the angle segment and the unit circle.

5. Determine the x- and y-coordinates for P and insert these values in the appro-priate row and column of the table.

6. The x-value for P represents the value for the cosine of 10 degrees, i.e., cos(10◦);the y-value for P represents the value for the sine of 10 degrees, i.e., sin(10◦).

7. Determine the remaining values in the table for sine and cosine by repeatingthe process (parts 3-6 above) for angles of measure 20◦, 30◦, etc.

8. Determine the value for the slope of the segment OP , for every angle in thetable.

9. Construct a tangent line to the (quarter) circle at point P , for every angle inthe table; determine the slope of each tangent line.

Trig Table

Angle measure(arc length, indegrees)

Cosine(x-coordinate)

Sine(y-coordinate)

Slope of“triangle”hypotenuse—slope ofsegment OP

Slope oftangentline—slope ofline tangent tocircle at thepoint (x, y)

10◦

20◦

30◦

40◦

45◦

50◦

60◦

70◦

80◦

90◦



Appendix BInstructions for Activity 2A

In small groups of 2–3, please follow and complete the directions below. You willneed to use your completed Trig Table from Activity 1.

1. Examine the four quantities obtained for the 45◦ angle.

2. Discuss any observations (relationships, patterns) that your group makes re-garding data for the 45◦ angle.

3. Describe any relationships that your group identifies between the slope of thehypotenuse and the slope of the tangent line for the 45◦ angle. Use completesentences.

4. Examine the quantities obtained for sine (ordinate, y-coordinate), cosine (ab-scissa, x-coordinate) and the slope of the tangent line at the point (x, y) for the45◦ angle.

5. Describe any relationships that your group identifies between sine, cosine, andtangent (the slope of the tangent line at the point (x, y)) for the 45◦ angle.

6. Determine whether the relationship identified in part 5 holds true for any otherangle measures. Which angles, if any?

7. Examine the four quantities obtained for the 60◦ angle.

8. Discuss any observations (relationships, patterns) that your group makes re-garding data for the 60◦ angle.

9. Describe any relationships that your group identifies between the slope of thehypotenuse and the slope of the tangent line for the 60◦ angle. Use completesentences.

10. Examine the quantities obtained for sine (ordinate, y-coordinate), cosine (ab-scissa, x-coordinate) and the slope of the tangent line at the point (x, y) for the60◦ angle.

11. Describe any relationships that your group identifies between sine, cosine, andtangent (the slope of the tangent line at the point (x, y)) for the 60◦ angle.

12. Determine whether the relationship identified in part 11 holds true for any otherangle measures. Which angles, if any?

13. Can you generalize any relationships between sine, cosine, and tangent regard-less of the angle measure? Explain.

14. From Activity 1, recall the value for the slope of the segment OP . What typeof triangle do you see? Describe any relationships that your group identifiesbetween sine, cosine, and the hypotenuse OP . Explain.

15. Share your results with another group.



Appendix CInstructions for Activity 2B

In small groups of 2–3, please follow and complete the directions below. You willneed to use your completed Trig Table from Activity 1 and your work from Activity2A.

1. Choose an arbitrary angle θ (measured in degrees) from your Trig Table. Con-sider the values for sine and cosine for angle θ.

2. Consider the values that are obtained for cos(90◦ − θ) and sin(90◦ − θ). Whatcan you conclude?

3. Describe how and why your conclusion from part 2 makes sense. Make certainto explain your mathematical thinking and reasoning. Use complete sentenceswhere appropriate.

4. Think of the angle 90◦ as being formed from the sum 90◦ = 60◦ + 30◦.

5. Examine the value of cos(60◦+30◦) in relation to the values of cos(60◦), sin(60◦),cos(30◦), and sin(30◦).

6. Discuss any relationships that you identify between the values cos(60◦ + 30◦),cos(60◦), sin(60◦), cos(30◦), and sin(30◦). What can you conclude? Explain.

7. Describe how and why your conclusion from part 6 makes sense. Make certain toexplain your mathematical thinking. Use complete sentences where appropriate.

8. Formulate a similar, and more general conjecture, for cos(α + β), where α andβ are each angles measured in degrees. Check your conjecture with other anglevalues from your Trig Table.

9. Repeat the process described in parts 4–7 above for sin(60◦ + 30◦).

10. Formulate a general conjecture for sin(α + β), where α and β are each anglesmeasured in degrees. Check your conjecture with other angle values from yourTrig Table.




Appendix DInstructions for Activity 3

In small groups of 2–3, please follow and complete the directions below. You willneed to use your completed Trig Table from Activity 1 and your work from Activities2A (#14) and 2B.

1. Make a table of values for the relation x2 + y2 = 1; that is, make a table of x-and y-values that make the relation true.

2. On a separate piece of graph paper, graph the relation x2 + y2 = 1. Whatgeometric object does the graph of the relation represent?

3. Make a table of values for the relation x2 + y2 = 4; that is make a table of x-and y-values that make the relation true.

4. On the same piece of graph paper used in part 2 above, graph the relationx2 + y2 = 4. What geometric object does the graph of the relation represent?

5. Imagine a graph of the relation (x− 3)2 + (y− 2)2 = 4. Describe the geometricobject that this relation represents. Make certain to explain your mathematicalthinking. Use complete sentences where appropriate.



Aiming a Basketball for a Rebound: Student Solutions Using Dynamic

Geometry Software

Diana Cheng∗, Tetyana Berezovski, Asli Sezen-Barrie

Abstract

Sports can provide interesting contexts for mathematical and scientific problems. In this article, wepresent a high school level geometry problem situated in a basketball context. We present a variety ofstudent solutions to the problem using dynamic geometry software and offer possible extensions.

Keywords: geometry, sports, dynamic geometry software, problem solving strategies

Published online by Illinois Mathematics Teacher on January 21, 2016.

“How can we get a basketball into a hoop?”This is a simply stated, open-ended, and complexproblem faced by athletes, but it is also a problemthat can be motivating for high school students ina geometry or physics class.

We created a two-dimensional model of abasketball hoop with dynamic geometry software,Geometer’s Sketchpad (Jackiw, 1991). Toincrease accessibility options, we wanted studentsto be able to model the problem using apencil-and-paper printout as well. Thus, wechose the scale factor of the model to be suchthat a National Basketball Association (NationalBasketball Association, 2006) regulation-sizedbasketball with a 29 inch circumference (9.23 inchdiameter) could be represented by a US pennycoin with 0.75 inch diameter. Using this scalefactor, we represented an 18 inch diameter NBAhoop as the circle with its center at point B andradius BD = 1.85 cm (see figure 1). The 24inch backboard was represented by line segmentIL = 4.95 cm.

1. Basketball Aiming Problem

We posed this problem to students: “You area basketball player standing at point Q. Yourcoach tells you to always aim for the center of thehoop, point B. Place point F on the backboardIL such that you will be using the backboard to


Figure 1: Basketball hoop model created using Geometer’sSketchpad

help you make your shot. Explain how you foundpoint F . Find two different ways to determinepoint F ’s location.”

This is a cognitively challenging problem(Stein et al., 1996) as there is no explicit solutionpathway given. The problem involves a modelingsituation, as it is realistic that basketball playerswould aim the basketball towards the centerof the hoop, and many shots use a reboundfrom the backboard. Learning goals for thisactivity include having students determine whichquantities (such as coordinate points, distances,angles) are relevant and how these quantitiesshould be used to solve the problem. There aremultiple solutions to this problem, some of which


Aiming a Basketball for a Rebound

are discussed in this article.

2. Method

The participants in this study were sevenin-service middle and high school mathematicsteachers taking a semester-long problem solvinggraduate course in spring 2014. All of theparticipants were enrolled in a master’s degreeprogram in mathematics education offered by themathematics department of a public university.The instructor of the course is the first authorof this paper. The participants were giventhis problem to solve independently as part oftheir final exams taken in class in a computerlaboratory. The participants were provided accessto the Geometer’s Sketchpad file with the diagramshown in figure 1 and a protractor and ruler tosolve the problem. Because participants wereasked to provide two solution methods to theproblem, a total of fourteen solution strategieswere collected, with some used by more thanone participant. Eight solutions provided bythe participants are reported in this article andare categorized into six distinct categories ofstrategies.

3. Solution Strategies

In our problem, we simplified basketball to twodimensions. A mathematically correct solutionto this problem involves recognizing that (a) thetrajectory of the ball is not a single straight linebecause the directions state that the ball mustbounce off the backboard, and (b) the angle atwhich the ball hits the backboard will be the sameangle it reflects off of the backboard. That is,the angle of incidence is congruent to the angle ofreflection. Connections can also be made fromphysical observations of Descartes-Snell’s Law,used in geometrical optics. This law states thatyou will always get a constant number when youdivide the sines of the angles of incidence andof refraction. This constant gives the reflectiveindices of the boundary objects (e.g., water, glass)on which a light wave hits; each object has adifferent reflective index (The Physics Classroom,

Figure 2: Productive strategy 1: Slopes

2014). The way a light wave bounces off asurface is similar to the way a basketball hits thebackboard. While we did not specifically mentionDescartes-Snell’s Law in class, it is possible thatthe teachers heard of it through their high schoolor undergraduate coursework.

4. Productive Strategies

Three strategies used by students wereconsidered to be productive in the sense thatapplying them accurately would yield a correctanswer to the problem. The productive strategieseach took into account that the angle at whichthe basketball hits the backboard is the sameangle at which the basketball bounces off thebackboard. Participants started out by findingpoint F on backboard IL. Auxiliary lines werethen created as described below. If the strategyinvolved a measure of two relevant quantities thatneeded to be equivalent, then the measures ofthese two quantities had to be made explicit inthe participant’s work. Because of the low levelof precision that is possible using Geometer’sSketchpad, if angles were measured, they only hadto be within one degree of each other in order foranswers to be considered sufficiently accurate.

Strategy 1: Slopes

Participants using this strategy created linesegments FQ and FB. They then measured theslopes of FQ and FB. Point F was then draggedalong IL until the magnitudes of these two slopeswere as close as possible (see figure 2).


Diana Cheng, Tetyana Berezovski, Asli Sezen-Barrie

Figure 3: Productive strategy 2: Congruent angles

This strategy is effective because it presumesthat the slopes of FB and FQ need to have thesame magnitude but be opposite of each other.This strategy is equivalent to making ∠IFB and∠LFQ congruent, as these angles are the inversetangents of the slopes.

Strategy 2: Congruent Angles

Participants using this strategy created linesegments FQ and FB and measured ∠IFB and∠LFQ. Point F was dragged until these twoangles were as close as possible. In the workshown in figure 3, the measures of ∠IFB and∠LFQ only differ by 0.1 degree.

This strategy is productive because it createscongruent angles ∠IFB and ∠LFQ such that thebasketball bounces off the backboard at the sameangle at which it bounces onto the backboard.

Strategy 3: Reflection

Participants using this strategy created linesegment FQ. Then they then created a lineperpendicular to IL through point F . This linewas used as a line of reflection for FQ. PointF was dragged until the image of FQ passedthrough point B (see figure 4).

This strategy is effective because it createscongruent angles ∠IFB and ∠LFQ.

Strategy 4: Similar Triangles

Another way to solve the problem, notmentioned by the participants of this study,involves setting up a proportion to comparecorresponding lengths of two similar triangles.The larger triangle is formed in the followingmanner: Draw the line perpendicular to IL

Figure 4: Productive strategy 3: Reflection of FQ

through point Q. Line IL will intersect thisperpendicular line at point A with coordinates(5.03, 1.85). The other two vertices of the largertriangle are Q and F . The smaller triangleis formed in the following manner: Draw theradius of the hoop that is perpendicular tothe backboard, intersecting IL at point E withcoordinates (0, 1.85). The other two vertices ofthe smaller triangle are B and F (see figure 5).

Triangle FEB and triangle FAQ are similarby angle-angle similarity. Angles ∠FEB and∠FAQ are both right angles by construction ofthe perpendicular lines, and ∠EFB and ∠AFQare as close to congruent as possible because theangle of incidence is assumed to be congruent tothe angle of reflection. Since corresponding sidesin similar triangles are proportional, EB/EF =AQ/AF . Since AE is a straight line of length5.03 cm and AE = AF + EF , we can findthat EF = 1.22 cm and the location of F is(1.22, 1.85).

Figure 5: Productive strategy 4: Similar triangles



5. Unproductive Strategies

There were three unproductive strategiesattempted by participants: the use of a fixedangle, the use of a straight line, and the useof irrelevant congruent angles. These strategiesare considered unproductive because they do notinvolve the use of relevant quantities or a relevanttrajectory of the ball. The strategies are furtherdescribed below.

Strategy 5: Fixed Angle

While the use of angles can be a productivestrategy, in the work shown below it is unclearhow participants decided which angle to measureand what the measure of that angle should be.Two different angles were identified by differentparticipants, with different angle measures. Inthe first example (see the top image in figure 6),the participant found point F on IL such thatm∠FBQ = 90◦. In the second example (see thebottom image in figure 6), the participant foundpoint F on IL such that m∠FBD = 45◦. Nojustification of these angle measures was provided.

We speculate that these angle measures werechosen because many diagrams involving trianglesin high school geometry textbooks are righttriangles or isosceles triangles.

Strategy 6: Straight Line

This was the most common strategy usedby participants. Participants using this strategythought that the ball would travel in a straightline through B to the backboard. Thus, theydrew in line segment QB and extended it until itintersected IL at point F . Participants using thisstrategy did not take into consideration ∠LFQ(see figure 7).

We speculate that participants using thisstrategy misunderstood the prompt whereby thecoach tells you to aim for point B. Participantsmay have thought that the only way to aim forpoint B was by going in a straight line from Q toB.

Figure 6: Two examples of unproductive strategy 5: Fixedangle

Figure 7: Unproductive strategy 6: Straight line


Diana Cheng, Tetyana Berezovski, Asli Sezen-Barrie

Figure 8: Two examples of unproductive strategy 7:Irrelevant congruent angles

Strategy 7: Irrelevant Congruent Angles

Participants using this strategy recognizedthat two angles needed to be congruent, but usedirrelevant angles. In the first example (see thetop image in figure 8), the participant found theangle bisector of ∠LQB, and thought that pointF would be located at the intersection of the anglebisector and IL. In the second example (see thebottom image in figure 8), the participant drew inthe midpoint V of IL and created point F alongV L. The participant then drew in V B, FB, FQ,and QL. The participant dragged point F alongV L until the measures of ∠V BF and ∠FQL werealmost the same.

We hypothesize that participants using thisstrategy may have encountered Descartes-Snell’slaw and were trying to apply it, but did notremember which angles needed to be taken intoaccount.

6. Discussion

A rich set of responses to this basketballmodeling problem was obtained and reportedin this article. Strategies that yielded correctsolutions as well as unproductive strategies weredescribed. In order to solve this problemcorrectly, participants needed to synthesizeknowledge across disciplines and recognize that aphysical principle about angles should be appliedin this geometry problem.

Taken as a whole, the solution paths can applya variety of mathematical content standards fromthe Common Core State Standards (CommonCore State Standards Initiative and others, 2010).For example, all of the strategies involved using“geometric shapes, their measures, and theirproperties to describe objects” on the basketballcourt (HSG.MG.A.1) and applying “geometricmethods to solve design problems” of aimingthe basketball to bounce off the backboard(HSG.MG.A.3). The strategies all involvedmaking “formal geometric constructions” withthe dynamic geometry software (HSG.CO.D.12).Strategy 1 involves representing “constraints byequations . . . and interpret[ing] solutions as viableor nonviable options in a modeling context”(HAS.CED.A.3) since the trajectory of thebasketball is being modeled by slopes of lines.Strategy 3 involves representing “transformationsin the plane,” particularly reflections(HSG.CO.A.2). Strategy 4 involves using“congruence and similarity criteria for trianglesto solve problems and prove relationships” insimilar triangles (HSG.SRT.B.4), and establishingthe “angle-angle criterion for two triangles to besimilar” (HSG.SRT.A.3).

Integrating this basketball modeling problemas a science activity is consistent with recommen-dations by the Next Generation Science Standards(NGSS Lead States, 2013). Descartes-Snell’s Lawcould be discussed as a follow-up to this activ-ity, focusing on the core scientific idea that “whenlight shines on an object, it is reflected, absorbed,or transmitted through the object, depending onthe object’s material” (MS-PS4-2) and the relatedscientific and engineering practice of using “math-



ematical representations to describe and/or sup-port scientific conclusions and design solutions”(MS-PS4-1). Understanding the basketball modelas an instance of Descartes-Snell’s Law involvesstudents using analogic reasoning, treating thepath of light against a surface similarly to thepath of the basketball as it bounces off the back-board (English, 2013). Once students have suc-cessfully made the basketball shot, we can pointout that reflecting the hoop across the backboardtransforms the shots into a straight line.

Multiple additional problems can be posedconcerning this two-dimensional model. Twoextension problems are proposed here. They bothinvolve finding a locus of points within the hoopunder different constraints.

• Your coach does not need you to aim forpoint B. Ignoring the backboard IL, shadein a locus of points in the hoop at which theball may be centered while still remainingentirely in the hoop.

• Find the locus of points on the backboardIL for which the ball would rebound intothe hoop (taking into consideration therequirement that the ball must end entirelywithin the hoop).

Readers who want to further extend this modelcan consider a three-dimensional representationof the basketball hoop and/or take into consider-ation the amount of spin that the basketball mayhave.

References


English, L. D. (Ed.) (2013). Mathematical reasoning:Analogies, metaphors, and images. Routledge.

Jackiw, N. (1991). The geometers sketchpad. computersoftware.

National Basketball Association (2006). Rule no. 1—courtdimensions–equipment. URL: http://www.nba.com/

analysis/rules_1.html.

NGSS Lead States (2013). Next Generation ScienceStandards: For States, By States. Washington, DC:National Academies Press.

Stein, M. K., Grover, B. W., & Henningsen, M. (1996).Building student capacity for mathematical thinkingand reasoning: An analysis of mathematical tasks usedin reform classrooms. American Educational ResearchJournal , 33 (2), 455–488.

The Physics Classroom (2014). Snell’s law. URL:http://www.physicsclassroom.com/class/refrn/

Lesson-2/Snell-s-Law.

Diana Cheng

MATHEMATICS DEPARTMENT

TOWSON UNIVERSITY

TOWSON, MD


Tetyana Berezovski

MATHEMATICS DEPARTMENT

ST. JOSEPH’S UNIVERSITY

PHILADELPHIA, PA


Asli Sezen-Barrie

PHYSICS, ASTRONOMY& GEOSCIENCES DEPARTMENT

TOWSON UNIVERSITY

TOWSON, MD



http://www.nba.com/analysis/rules_1.html

http://www.nba.com/analysis/rules_1.html

http://www.physicsclassroom.com/class/refrn/Lesson-2/Snell-s-Law

http://www.physicsclassroom.com/class/refrn/Lesson-2/Snell-s-Law

Taking Calculus to New Heights

Ken Gasser∗, Darl Rassi

Abstract

Students in a high school calculus class attempt to design the ideal tissue-paper hot-air balloon. Thisoutline provides information on project design, construction of balloons, mathematical analysis ofsurface area and volume of each balloon, flying the balloons, and what can be learned regarding solidsof revolution.

Keywords: calculus, project, construction, solid of revolution, volume, surface area, hot-air balloon,21st century skills

Published online by Illinois Mathematics Teacher on March 1, 2016.

1. Introduction

Every year after the Advanced Placement(AP) Calculus exam, I find myself asking thesame question: “How should I best spend mytime with my Calculus students in the fewremaining weeks of school?” As a general rule,AP students have been rigorously working tolearn, prepare, and take their AP exam(s). Bythe end of the second week of May, even themost die-hard math enthusiasts in my classesare ready for a break from the strict regimenof homework and studying. After trying severaldifferent ideas to best utilize our time, I stumbled


on a proposal from a friend and fellow mathteacher. His suggestion was to apply my students’understanding of calculus by having them utilizesolids of revolution to design, analyze, build, andfly tissue-paper hot-air balloons.

2. The Design

Students were given the following prompt toset the stage for the last three weeks of the year:

Mr. Gasser is looking to start aside business of creating model hot-airballoons. Wanting to create the bestballoon on the market, he has hired anumber of firms to inform him whatshape of balloon he should create toprovide a product that stays in theair for the longest time possible andthat flies higher than his competitors’products. He has given each firm aset amount of material with which towork. The firm that submits the mostconvincing argument as to why theirballoon is the best will be paid fortheir diligent efforts (to the tune offive bonus points each).

Students were broken into “firms” of two to fourto tackle the problem and were given a list of thematerials available to them. Each firm had accessto the following materials:



• sixteen 20 × 30 inch sheets of art tissuepaper (which when glued together end-to-end form eight 20 × 59 inch sheets of paper)

• a 36 × 70 inch sheet of bulletin board paperto use as a panel template

• a bottle of white glue and a glue stick

• approximately 30 inches of thin wire

• a few clothespins

• scrap cardboard to protect the desks andfloors from glue

• a set of general directions for designing andbuilding their balloons

The challenge for each firm was to worktogether as a team. This open-ended problemrequired them to think as mathematicians andutilize all eight of the Common Core Standardsfor Mathematical Practice (Common Core StateStandards Initiative and others, 2010). Inaddition to learning how to construct a hot-air balloon from tissue paper, students wereforced to take risks, collaborate, think critically,and communicate their results without directinstruction from me. These are all 21st centuryskills that each of our students should develop(Partnership for 21st Century Skills, 2009).

For the final assessment, students submitted aportfolio of their work. They included photos todocument their design and construction processas well as their test flight. Documentation wassupposed to be ordered logically and all work wasto be typed. (This provided a great opportunityto introduce students to Microsoft’s EquationEditor). Below is a list of what I looked for in eachportfolio along with the number of possible pointsthat could be earned (for a total of 60 possiblepoints):

• title page including group name along withnames of all group members (2 pts)

• picture of balloon profile (2 pts)

• picture of panel template (2 pts)

• pictures (minimum three) of balloonconstruction (6 pts)

• piecewise defined function used to modelprofile curve (10 pts)

• detailed sketch of curves on a coordinateplane (5 pts)

• volume of balloon, including all supportingwork (10 pts)

• surface area of balloon, including allsupporting work (10 pts)

• explanation of process used to measure themaximum height of the balloon in its flight(5 pts)

• chart illustrating data collected (flight time,flight height, volume, surface area, basicprofile design) (3 pts)

• revised profile designed after the first testflight, along with a convincing argumentwhy the new, improved design should bechosen over all other designs (5 pts)

3. Constructing the Balloon

A majority of the time spent on theproject was in designing and constructing theballoon. These steps required attention todetail and mathematical thinking. The studentsfirst designed a two-dimensional profile of theirballoon. Then students constructed eight two-dimensional panels, or gores. Finally, each groupfastened the panels together to make a three-dimensional balloon. Examples of some of theprocesses are shown below (see figures 2–6).

To better understand the design and construc-tion process, it may be helpful to think of a beachball (see figure 1). The profile of the ball is a cir-cle. The corresponding panels are shaped like nar-row diamonds with the middle vertices roundedout. When the panels are attached to each otheralong their seams they create a three-dimensionalbeach ball.

For each balloon, firms had to first designhalf of the profile of their theoretical balloon (see


Ken Gasser, Darl Rassi

Figure 1: A beach ball illustrating how, in this case, 6panels sewed together approximate a sphere (public do-main image, obtained from openclipart.org)

Figure 2: Sample profile design

figure 2). This process gave rise to good discussionwithin the groups. Students discussed the sizeand shape of their balloon. They also discussedproperties that they thought would make theirballoon go higher than any other firm’s balloon.It was enjoyable to hear students critiquing theideas and reasoning of their peers and to see themfinally come together on one design that the firmcould support. Each firm also had to make sure itsballoon was small enough to be constructed withthe given amount of art tissue paper, which ledstudents to consider the relationship between theprofile, panels, radius, and circumference. Withsome guidance, students found the arc lengthof their final profile curve could be no longerthan the tissue paper (59 inches) and that theirgreatest circumference must be less than eighttimes the width of the paper (160 inches).

To ensure that their tissue paper would belong enough to form the balloon, firms neededto find the arc length of their balloons’ profiles.To do so, students laid a piece of string alongthe profile of the balloon, marked the endpointsof the profile on the string, and then measured

Figure 3: Choosing a representative sample of points

the length of the taut string between the marks.Instructors could at this time teach studentshow to find the arc length of a curve if theywould like their students to use their curve fittingabilities and calculus to calculate the arc lengthof the profile. Either way, students had to usemathematical thinking to answer the real-life,practical question, “Will we have enough tissuepaper?”

After designing the profile, students thencreated a template for the eight panels, or gores.To save time I helped students through thesesteps, but if time permits, the task could beenriched by asking them to determine how tomake the templates for the gores themselves.A brief outline of the concepts in designing atemplate follows. First, students were asked todraw their profile in the coordinate plane (seefigure 2). Once drawn, the students identifieda representative sample of points lying on theprofile curve after first segmenting the curve intointervals of various widths (generally one to threeinches), as shown in figure 3. When the paneltemplate is constructed, the points along theprofile curve will lie on the center axis of eachpanel. By finding the arc length (string methodor calculus) from an end of the profile to a pointon the profile, one can find a point on the centeraxis of the panel that corresponds to the point onthe profile (see figure 4).

This process is repeated for each point on theprofile. (This took some time as some studentsused over 100 points.) After finding each ofthe points on the center axis of the panel thatcorrespond to the profile points, students neededto realize the width of the panel at each point isone-eighth of the circumference, since there willbe eight equally sized panels creating the balloon.


https://openclipart.org/detail/24008/beach-ball


Distance L along the balloon profile

Same distance through the corresponding gore panel

Figure 4: Finding a corresponding point on a gore panel’scenter axis

Figure 5: Half panel/gore

The circumference can be calculated by using theradius from the matching point of the profile. Ihave found it easier to fold the panel in half, so thewidth of half of the panel would be one-sixteenthof the circumference. Repeating this process ateach point gives the students a series of pointsthat they can connect to create a template forthe eight panels (see figure 5). When their edgesare glued together, the eight panels will create thedesired three-dimensional balloon.

Once students better understood themathematics used to create the panels, studentsused their careful measurements of their profileto build a template for their panel. This was oneof the more critical processes in the constructionphase. Sloppy measurements or unfocused teammembers could lead to panels that, once gluedtogether, create a balloon that looks nothing likewhat was expected.

The panel template (see figure 6) was made

Figure 6: Unfolded panel template. Eight panels gluedtogether will create a balloon.

Figure 7: Gluing panels together

out of bulletin board paper. Once made, thestudents overlaid the template onto the pieces ofart tissue paper to cut out the panels that formedtheir hot-air balloon. After the panels were cutout, students carefully glued them together, edge-to-edge, with a sufficient, yet minimal, amount ofglue (either by using a glue stick or white glue)(see figures 7 and 8). The left edge of panel 1 wasglued to the left edge of panel 2, the right edgeof panel 2 was glued to the right edge of panel 3,and so on to form an accordion of panels. Thenthey curled back panels 2–7 and glued the rightedge of panel 8 to the unglued edge of panel 1,completing the structure of the balloon. Once thepanels were all glued together, a tissue paper capwas glued onto the top of the balloon to seal offthe top seams of all of the panels. In addition, athin piece of wire was used to provide rigidity anda small amount of weight to the bottom openingof the balloon (see figure 9).



Figure 8: An “accordion” of panels glued together

Figure 9: A piece of wire added to the bottom of theballoon

4. Analysis of Design and Flight

After completing construction, and prior toflying their balloons, each firm analyzed itsdesign. The design analysis helped studentsunderstand two underlying components of theirballoon design: the volume and the surface area.

To explore these ideas more precisely, studentsused the coordinate points along the curve oftheir profile along with the regression capabilitiesof their graphing calculators or GeoGebra(Hohenwarter, 2014) to create a piecewisefunction that closely approximated the shape oftheir curve. They then used calculus to determinethe volume and surface area of their balloon.

An example of a set of points from one profile(measured in inches), its corresponding piecewisedefined function, graph, and calculations forsurface area and volume are shown in figures 10and 11, respectively.

Another goal of the project was for each firmto determine a method for measuring the totalheight achieved by its balloon as well as theheights of the balloons produced by the otherfirms. Several methods have been employedover the years to measure heights. Favoriteshave included using a clinometer (coupled witha little trigonometry) and range finders (suppliedby the hunters in the class). Digital photographsfrom a fixed location have also been used soballoon flights can be compared side-by-side togauge height. It was, however, very difficult toaccurately measure the height of a balloon whosepath was so easily changed by a slight breeze. Inaddition, an official time keeper was selected tomeasure the time each balloon stayed in flight.

5. Flying the Balloons

Needless to say, I was very cautious when itcame time to introduce students to somethingthat could burn them, their paper balloon, theschool, or even the lush grass of the schoolbaseball field. I was very aware of the fact thata little flame, some tissue paper, and a bit ofdried glue could turn a pretty balloon into afrightful ball of fire, which would present more



f(x) =

5 0 ≤ x ≤ 50.0848x2 − 0.6509x + 6.0339 5 < x ≤ 10−0.00006x4 + 0.00548x3 − 0.2119x2 + 4.25843x− 18.20301 10 < x ≤ 43

x y0 51 5...

...38 12.56339 11.12540 9.562541 7.62542 4.375

Figure 10: The scatterplot and corresponding piecewise defined function used to model the profile in figure 2 (createdusing GeoGebra)

V = π

∫ b

ar2 dx

= π

[∫ 5

052 dx

+

∫ 10

5

(0.0848x2 − 0.6509x+ 6.0339

)2dx

+

∫ 43

10

(−0.00006x4 + 0.00548x3 − 0.2119x2 + 4.25843x− 18.20301

)2dx

]≈ 24,733.779 cubic inches

SA = 2π

∫ b

af(x)

√1 + (f ′(x))2 dx

= 2π

[∫ 5

05√

1 + (0)2 dx

+

∫ 10

5

(0.0848x2 − 0.6509x+ 6.0339

)√1 + (0.1696x− 0.6509)2 dx

+

∫ 43

10

(−0.00006x4 + 0.00548x3 − 0.2119x2 + 4.25843x− 18.20301

)√1 + (−0.000024x3 + 0.01644x2 − 0.4238x+ 4.25843)2 dx

]≈ 4,164.708 square inches

Figure 11: Volume and surface area calculations



problems than I wanted. Having one or two fireextinguishers at hand along with the principal’sapproval were prerequisites for flight day. Theultimate goal was to have students launch theirballoons outside on a warm, still, spring dayin May. This meant that I had to pay carefulattention to the weather forecast, be patient, andtake advantage of the nice weather when it came.After several experiments over the years, I havefound that a propane turkey fryer provides themost consistent heat. To concentrate the heat,I placed some circular reducing funnels, suppliedby a local heating and air conditioning company,on top of the turkey fryer to force the heat toexit a five inch diameter opening. Students wereto have an opening no smaller than nine inchesin diameter to leave space for their fingers sothey could hold their balloon over the heat source(wearing gloves of course). See the cover image ofthis article for an illustration of the process.

Once the heat source was set up and producinga steady stream of heat, the students held theirballoons over the heat source for up to fiveminutes each. Then they let go and anxiouslyhoped to see their balloon soar to the heavens.Some balloons have flown marvelously. Othersdid not fly so high but stayed in flight longersince they were designed with a large opening thatwould allow the balloon to act as a parachute onits descent. Some balloons leaked so much hot airthat they only lifted a few inches. Team “LeadBalloon” illegally reinforced their seams with tapeto prevent any air leakage. As a result of theirconstruction, the balloon was so heavy that whenreleased above the heat source it immediatelysank down instead of floating up! Balloons heatedwith the turkey fryer have flown in excess of onehundred feet. Some have stayed in the air for overthree minutes, while others ended their flight in atree (at which time the clock officially stops).

6. Tying it all Together

Once all of the firms had an opportunity to flytheir balloons, some even a second time, studentsreturned to the classroom to share with the othergroups. After sharing their flight times, heights

achieved, profile design, volume, and surfacearea, firms then regrouped and created a newtheoretical design. Each firm then prepared awritten analysis about why the members felt theirnew design would produce a superior balloon.

If by this time in the project students have notyet fully discussed what keeps a hot-air balloonafloat, I like to facilitate a discussion on whatexactly makes a hot-air balloon float. Studentsstart to focus on how to maximize the amountof hot air in a balloon and how to keep it ashot as possible for as long as possible. Studentssoon realize that to keep the air hot in a balloonthey need to minimize the surface area of theballoon to reduce the speed at which the hot aircools. I love seeing the light bulbs flick on whenstudents realize that they already knew of a shapethat maximizes volume while minimizing surfacearea—a sphere! Though we did not have time, itwould have been interesting as a class to researchwhy piloted hot-air balloons used in practice arenot spherical.

7. Conclusion

Teaching seniors at the end of the year cansometimes be difficult, especially during thosedays when the sun actually shines in May. APstudents know that the course is essentially overfor them after they complete the AP exam.But in recent years, when I have utilized thisballoon project, I have found my students tobe thoroughly engaged. The remainder of theschool year passed very quickly for all. Whilesome balloons hardly flew at all and would havemade a better wall decoration than an aircraft,some soared higher than I imagined possible. Inthe end, all groups learned firsthand throughthe invention process of success and failure.Calculus served as the tool to explain someof the “whys” behind both the successful andunsuccessful designs.

Throughout the project, students workedcollaboratively with each other, took risks, hadfun, and presented original solutions. These areall skills that will likely benefit them in theglobal marketplace that they will be entering in



a few short years. In addition, students had toutilize the Standards for Mathematical Practice(Common Core State Standards Initiative, 2014).These included making sense of a problem andpersevering in solving it, constructing viablearguments and critiquing the reasoning of others,modeling with mathematics, using appropriatetools strategically, and attending to precision.Without question, students looked forward toclass. Some even begged for more time tomake adjustments after their first flights. Otherstudents were even seen at lunch and in study halldiscussing their designs. This past year multiplestudents commented on how this assignment wastheir favorite school project of all time. In futureyears, I will likely make adjustments to thisactivity, but for now, I have found a new favoriteend-of-the-year calculus project, and I invite youto give it a try in your classroom.

Educators interested in more details on thisparticular project are encouraged to contact theauthors.

References


Hohenwarter, M. (2014). GeoGebra, version 5. (Software).http://geogebra.org. Accessed Feb. 2016.

Partnership for 21st Century Skills (2009). P21 frame-work definitions. http://www.p21.org/storage/

documents/P21_Framework_Definitions.pdf. Ac-cessed Feb. 2016.

Ken Gasser

CHIPPEWA HIGH SCHOOL

100 VALLEY VIEW ROAD

DOYLESTOWN, OH 44230

E-mail : chip [email protected]

Darl Rassi

OLIVET NAZARENE UNIVERSITY

ONE UNIVERSITY AVENUE

BOURBONNAIS, IL 60914-2345



http://geogebra.org

http://www.p21.org/storage/documents/P21_Framework_Definitions.pdf

http://www.p21.org/storage/documents/P21_Framework_Definitions.pdf

Do the Mathcrossword puzzle by Damon Gulczynski


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