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Change Bell Ringing and Mathematics: a High School Student Excursion Into Graph Theory and Group Theory

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Change Bell Ringing and Mathematics: a High School Student Excursion Into Graph Theory and Group Theory Western Michigan University ScholarWorks at WMU Honors Theses Lee Honors College 12-9-2005 Change Bell Ringing and Mathematics: A High School Student Excursion into Graph Theory and Group Theory Kristi Carlson Western Michigan University, [email protected] Follow this and additional works at: https://scholarworks.wmich.edu/honors_theses Part of the Curriculum and Instruction Commons, Mathematics Commons, and the Science and Mathematics Education Commons Recommended Citation Carlson, Kristi, "Change Bell Ringing and Mathematics: A High School Student Excursion into Graph Theory and Group Theory" (2005). Honors Theses. 660. https://scholarworks.wmich.edu/honors_theses/660 This Honors Thesis-Open Access is brought to you for free and open access by the Lee Honors College at ScholarWorks at WMU. It has been accepted for inclusion in Honors Theses by an authorized administrator of ScholarWorks at WMU. For more information, please contact [email protected]. l w lZivIn 1VI1 vyil 1vj/\lN KJ INI I V lIiyo! I Y The Carl and Winifred Lee Honors College THE CARL AND WINIFRED LEE HONORS COLLEGE CERTIFICATE OF ORAL EXAMINATION Kristi Carlson, having been admitted to the Carl and Winifred Lee Honors College in Fall 2002 successfully presented the Lee Honors College Thesis on December 9, 2005. The title of the paper is: "Change Bell Ringing and Mathematics: A High School Student Excursion into Graph Theory and Group Theory" (^LuO^y "UsCbk. Dr. Arthur White, Mathematics Dr. Tabitha Mingus, Mathematics Dr. Allison Kelaher-Young, Teachin 1903W. Michigan Ave. PHONE: (269) 387-3230 FAX: (269) 387-3903 Change Bell Ringing And Mathematics: A High School Student Excursion into Graph Theory and Group Theory Kristi Carlson Honors Thesis Monday Tuesday Wednesday Thursday Friday 1. Intro to Change 2. Methods in Change 3. Principles in Change 4. Calls in Change 5. The Nine Tailors Ringing Ringing Ringing Ringing 6. Intro to Graphs 7. Graphs 8. Hamiltonian Graphs 9. Graphs from Change 10. Extents from Change and Terminology Ringing Ringing 11. Intro to Groups 12. Permutation Groups 13. Permutation Groups 14. Even and Odd 15. Extents from Groups in Change Ringing Permutations 16. Change Ringing in 17. How Many Extents? 18.Extents ofMinor 19. Extents ofMinor ABEL Introduction This unit was created as a way to introduce higher level mathematics concepts to advancedhigh school students. If the concepts ofgraph theory and group theory were presented traditionally to students in high school, most would be overwhelmed, but by presentingthem in the context of changebell ringing, studentswill be able to work with the concepts and hopefully understand them. All five ofthe National Council ofTeachers ofMathematics Process Standards are found in this unit, though none ofthe Content Standards are explicitly addressed, because the content ofthis unit is beyond what would be studied in the traditional high school curriculum. Thefirst process standard found is problem solving. Throughout the unit, students will be introduced to a problem and then by solving it, will learn new mathematical content. For instance, in their introduction to graph theory, students are given the Konigsberg Bridge Problem to solve. While students work within the context ofthe problem, they explore the problem that ledto the creation of graph theory. As students try to find a solution, they find thatthere is no solution possible, and they must explain their reasoning that there is no solution possible. This leads right into thenext process standard, which is reasoning & proof. Throughout the entire unit, students willbe expected to explain theirreasoning fortheir answers. It maybe something as simple as citing a definition to saythat a particular example is nota graph in theGraphs lesson or may be more complex, like explaining whythere are only4 extents of Plain Bob Doubles and what restricts the number ofdifferent extents that are possible. Communicationwill be an essential process standard in this unit. For most of the unit, students will be working within small groups. Withinthe small groups students will needto explain their reasoning for their answers to oneanother so that each canunderstand what the others are thinking. Also, once students learnthe correctmathematical terminology, they will be expected to use it in their discussions. Representations will be another process standard that is used extensively in this unit. Students will work out an extentof Plain Bob Doubles by hand in one ofthe early lessons and quickly find that this is bulky and not very fun to do. Once they are introduced to the concepts in graph theory and group theory, students will be able to use representations presentin graphtheory and group theory to showthe mathematical ideaspresent in the structure ofchange ringing; students will find these representations much easier to work with than the straight change ringing context. The final process standard that is found is connections. This is found in many areas throughout this unit. The students will have to make connections between how they can use graph theory and group theory to both represent what they see happening in change ringing. They will also have to see the connections between change bell ringing and the mathematics present in both graph theory and group theory. I would like to thank my thesis committee: Dr. Arthur White Dr. Tabitha Mingus Dr. Alison Young And my friends and family for all of their support throughout this process. (And my computer for not deleting everything I had in the middle of the semester, like it threatened to do!) Lesson Title: Introduction to Change Ringing Lesson length: 1 hour Goal: Students will understand the basics ofwhat change ringing is. Materials: Computers, handouts with websites, projector, handbells, signs numbered 1-8 Terms to know: change ringing, change, row, tower, bell, handbell, extent, cross, stand, rounds Lesson: •• Short explanation ofchange ringing o Bells are hung on a full wheel, takes about 2 seconds to ring bell once around before it can ring again o Because ofthis, can't ring tunes, so ring changes instead • Changes are different permutations o Rules ofchanges • The first and last changes are rounds • No changes can be repeated • No bell can change more than one position between changes • No bell stays in the same position for more than 2 changes • The working bells do the same work • Each lead is palindromic o Explain # ofpossible changes on different numbers ofbells (n!) • 3 - 6; 4 - 24; 5 - 120; 6 - 720; 7 - 5,040; 8 - 40,320; 9 - 362,880; 10 - 3,628,800; 11-39,916,800; 12-479,001,600 o Try to ring quarters (>1250 changes) or peals (>5000 changes) • Split students into groups of3-4 and give 1 handout per group • Have students look at websites to find out more information about change ringing, encourage to not just look at the first few sites • Have students ring on bodies o Give 8 students numbered sign and appropriate handbell o Have students cross in pairs (rule #1) - What will happen ifcross again? o Have first/last stand and others cross (rule #2) o Continue alternating rules until back to rounds • Iftime permits, show video ofringing on a projector Websites to look at: http://www.nagcr.org/ - Homepage ofthe North American Guild ofChange Ringers http://www.cccbr.org.uk/ - Homepage ofthe Central Council ofChange Bell Ringers http://web.mit.edu/bellringers/www/ - Homepage for MIT's Guild ofBellringers http://www.kzoo.edu/ringers/index.htm - Homepage ofthe Kalamazoo College Guild ofChange Ringers http://www.ringingworld.co.uk/ - Homepage of The Ringing World http://www.whitechapelbellfoundry.co.uk/ - Homepage ofWhitechapel Bell Foundry http://www.campanophile.co.uk/ - Homepage of Campanophile http://www.cs.ubc.ca/spider/kvdoel/bells/bells.html - The Bells Applet: Change Ringing on the Web http://www.changeringers.karoo.net/ - Homepage ofChange Ringers Web Dictionary http://www.ringing.org/main/pages/intro - Homepage ofringing.org Lesson Title: Methods in Change Ringing Lesson length: 1 hour Goal: Students will understand how methods are constructed and construct their own method. Materials: Overhead with all minimus methods and oftreble hunt/dodge, graph paper to write out methods/plain hunt Terms to know: Method, rounds, cross(x), stand (12) Lesson: • Review rules for methods o The first and last changes are rounds o No changes can be repeated o No bell can change more than one position between changes o No bell stays in thesamepositionfor more than 2 changes o The working bells do the same work o Each lead is palindromic • Methods are treble dominated o Treble can hunt (1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1) o Treble can treble dodge (1, 2, 1, 2, 3, 4, 3, 4, 5, 6, 5, 6, 6, 5, 6, 5, 4, 3, 4, 3, 2, 1, 2, 1) • Examples ofchanges between rows - allowed or not? o 123456-> 124356 o 342165-> 342561 o 542316-> 453216 o 135246-> 142536 • Remember the hunting we did on handbells yesterday? That was plain hunting. o Rule # 1: Every pair crosses, o Rule # 2: First and last bells stand and others cross. o Start with rule # 1 and alternate until back to rounds, o Write out plain hunt on 6. • Split students into groups of3-4 and have groups try to come up with a method (minimus) like plain hunt but that contains all 24 changes (most likely Plain Bob). o Compare different methods found and tell method names. • Ifextra time, compare methods with one another o What is similar between multiple/all methods? What is different? Treble Hunt .....1 .—i Treble Dodge .—i Minimus Methods Reverse Double Plain
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