{"title":"A Novel Method for Completing the Square","authors":"Ellen Lehet, Kuo-Liang Chang","doi":"10.1080/07468342.2023.2273195","DOIUrl":"https://doi.org/10.1080/07468342.2023.2273195","url":null,"abstract":"SummaryIn this paper, the authors present a new approach for teaching completing the square that does not rely on geometrical or algebraic models. Instead the authors’ novel approach relies on purely numerical reasoning. Additional informationNotes on contributorsEllen LehetEllen Lehet (elehet@alumni.nd.edu) works in mathematics curriculum and is interested in the intersection of mathematical practice, philosophy, and education. She earned her Ph.D. in philosophy from the University of Notre Dame in 2020 with a focus in philosophy of mathematics. Her research interests are focused in mathematical explanation and understanding from both a philosophical and a pedagogical perspective.Kuo-Liang Chang Kuo-Liang Chang (kchang@uvu.edu) is currently a professor in the department of Mathematical and Quantitative Reasoning at Utah Valley University. He earned his Ph.D. degree in mathematics education (2010) and master’s degree in applied mathematics (2003) from Michigan State University. His research interests center on the originality and flexibility of mathematical reasoning, and mathematical problem solving. He believes offering students alternative perspectives in problem solving and helping students complete their own reasoning attempts (for obtaining their perspectives) could be a way of enhancing students’ learning experience.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136283985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Hermite-Hadamard Inequalities and Applications","authors":"Roger Nelsen","doi":"10.1080/07468342.2023.2273181","DOIUrl":"https://doi.org/10.1080/07468342.2023.2273181","url":null,"abstract":"SummaryWe derive the inequalities in the title and illustrate their use in establishing a variety of inequalities encountered in undergraduate mathematics.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136283971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Problems and Solutions","authors":"Greg Oman, Charles N. Curtis","doi":"10.1080/07468342.2023.2271821","DOIUrl":"https://doi.org/10.1080/07468342.2023.2271821","url":null,"abstract":"","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136283974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"How to Compute the Volumes of Hyperbolic Solids of Revolution","authors":"Robert L. Lamphere","doi":"10.1080/07468342.2023.2276651","DOIUrl":"https://doi.org/10.1080/07468342.2023.2276651","url":null,"abstract":"SummaryWe give two formulas for finding the volumes of solids of revolution in hyperbolic geometry. We also prove each formula. These formulas and their proofs are analogous to the ones in Euclidean geometry. We also provide several examples of their use. These formulas may be useful in college geometry courses that include a section on hyperbolic geometry. Additional informationNotes on contributorsRobert L. Lamphere Robert L. Lamphere (robert.Lamphere@kctcs.edu) received his Masters in mathematics from University of Illinois and his Masters in computer science from Northern Illinois University. He is an emeritus professor at the Elizabethtown Community and Technical College. His research interests are Non-Euclidean geometry and Newton’s Principia.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135093110","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Proof Without Words: Bisecting a Quadrilateral’s Perimeter","authors":"Thomas E. Cooper","doi":"10.1080/07468342.2023.2274250","DOIUrl":"https://doi.org/10.1080/07468342.2023.2274250","url":null,"abstract":"SummaryWe extend a result from triangles to quadrilaterals and prove without words that circles externally tangent to a side and tangent to the extended adjacent sides can be used to bisect the perimeter of the quadrilateral.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135093239","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Weight of a Ten-Pound Bag of Potatoes","authors":"Rus May","doi":"10.1080/07468342.2023.2270891","DOIUrl":"https://doi.org/10.1080/07468342.2023.2270891","url":null,"abstract":"SummaryThe weight of a ten-pound bag of potatoes is almost certainly not exactly ten pounds. Rather, it is a random variable with a minimum of ten pounds. We investigate the distribution of these weights in the context of a renewal theorem from the theory of stochastic processes and provide a straightforward demonstration of this theorem using basic tools from calculus. Additional informationNotes on contributorsRus MayRus May (r.may@moreheadstate.edu) teaches math at Morehead State University. He is interested in probability and combinatorics, especially the burgeoning field of analytic combinatorics in several variables. Outside of the classroom he enjoys cooking and eats a lot of potatoes.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135432425","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Timo Tossavainen, Pentti Haukkanen, Jorma K. Merikoski, Mika Mattila
{"title":"We Can Differentiate Numbers, Too","authors":"Timo Tossavainen, Pentti Haukkanen, Jorma K. Merikoski, Mika Mattila","doi":"10.1080/07468342.2023.2268494","DOIUrl":"https://doi.org/10.1080/07468342.2023.2268494","url":null,"abstract":"SummaryWe survey the history of the arithmetic derivative and more recent advances in research on this topic. Among other things, we discuss a few generalizations of the original arithmetic derivative and some arithmetic differential equations that are related to Goldbach’s conjecture and the twin prime conjecture. Our primary purpose is to give an overview of this field, but we also aim at providing supplementary material for an introductory course on discrete mathematics or number theory. Therefore, our survey contains ten exercises. Additional informationNotes on contributorsTimo TossavainenTimo Tossavainen (timo.tossavainen@ltu.se) is professor of mathematics education at Lulea University of Technology in Sweden. He received his Ph.D. in mathematics from Jyväskylä University, Finland, under the supervision of Pekka Koskela. He is interested in recreational mathematics, nonfiction literature, progressive rock, and cross-country skiing.Pentti HaukkanenPentti Haukkanen (pentti.haukkanen@tuni.fi) received his Ph.D. in mathematics from Tampere University, Finland, under the supervision of Seppo Hyyrö. Currently, he is university lecturer of mathematics at his alma mater. In his free time, he enjoys various sports and culture.Jorma K. MerikoskiJorma K. Merikoski (jorma.merikoski@tuni.fi) is emeritus professor of mathematics at Tampere University. He received his Ph.D. in mathematics from this university under the supervision of Seppo Hyyrö. Besides mathematics, he enjoys running, cross-country skiing, and literature.Mika MattilaMika Mattila (mika.mattila@tuni.fi) is university teacher of mathematics at Tampere University, where he also received his Ph.D. in mathematics under the supervision of Pentti Haukkanen. In addition to mathematics and some physical activities, he is interested in literature, movies, and retro console games.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135818872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Proof Without Words: The American Flag Inspires a Proof that the Sum of the First n Positive Odd Integers is <i>n</i> <sup>2</sup>","authors":"James Schultz","doi":"10.1080/07468342.2023.2273197","DOIUrl":"https://doi.org/10.1080/07468342.2023.2273197","url":null,"abstract":"\"Proof Without Words: The American Flag Inspires a Proof that the Sum of the First n Positive Odd Integers is n2.\" The College Mathematics Journal, ahead-of-print(ahead-of-print), p. 1","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135818308","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Why the Golden Spiral is Golden","authors":"Otis F. Graf","doi":"10.1080/07468342.2023.2273194","DOIUrl":"https://doi.org/10.1080/07468342.2023.2273194","url":null,"abstract":"SummaryIt is shown that there is a logarithmic spiral that segments a straight line through its pole in the golden ratio ϕ. This is the same spiral that is often referred to in the literature as the “Golden Spiral” Using the tools of analytic geometry and parametric vector equations, it is shown that the golden spiral is actually a member of a family of spirals defined by a generalized equation. The spirals exist as a continuum defined by a real number q > 1. When q=ϕ, the spiral segments a line through its pole in the golden ratio. That is the characteristic feature that sets it apart from all the other spirals in the continuum. Additional informationNotes on contributorsOtis F. GrafOtis F. Graf Jr. (otis.graf@hccs.edu) received a BS in physics and math and a Ph.D. in aerospace engineering from the University of Texas at Austin. He began his career doing trajectory planning, mission analysis and software development for NASA at the Johnson Space Center in Houston, TX. Later he joined IBM and designed large data storage systems for US and international research organizations. After retirement from IBM he joined the adjunct faculty at Houston Community College where he tutors math and physics students who are on an academic path to university engineering degrees.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135818908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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