Density of Primitive Pythagorean Triples
Duncan A. Killen
|Duncan Killen received BA and MD degrees from Vanderbilt University in the 1950’s. Since retirement he has been a part time student at the Johnson County Community College, Overland Park, Kansas. He has a particular interest in mathematics. E-mail: firstname.lastname@example.org|
|Based on the properties of a Primitive Pythagorean Triple (PPT), a computer program was written to generate, print, and count all PPTs is an arbitrarily chosen integer. The Density of Primitive Pythagorean Triples may be defined as the ratio of the number of PPTs whose hypotenuse is less than or equal to . The PPT Density for all PPTs with a primitive hypotenuse less than or equal to , remains rather stable, even as is increased from 5 to 1,000,000.|
Using a TI-83 calculator, a linear regression correlation between the number of PPTs and the value of Ix, using 36 data points distributed between = 1,000 and = 1,000,000 was determined and the results are as shown: (Back to Top)
|Robert Levine lived in New York City for 30 years before moving to Tucson, Arizona in 1976. He’s always loved math and science, even though he never passed calculus in his youth. He went back to college at age 51 and took the four semesters of calculus. He first became acquainted with the cross-product in Calculus III which led to his discovery.|
The cross-product is a mathematical operation that is performed between two 3-dimentional vectors. The result is a vector that is orthogonal or perpendicular to both of them. Learning about this for the first time while taking Calculus-III, the class was taught that if A×B = A×C, it does not necessarily follow that B = C. This seemed baffling. The author reasoned that if this were true, there should be a way to alter the B vector in such a way that the result of the cross-product is still unchanged, but was told that this was impossible.
When the course ended and there was time to think about it again, the author went to work trying to solve the impossible, and quickly succeeded. At the same time, an interesting fact was discovered about the cross-product that allowed for success. The proof was not so quick and easy though, but eventually it was accomplished as well. The proof involves an interesting twist where I present the finale, although still unproven, along with several related equations. The flow of proven equations then skips over that unproven group and eventually proves one of the equations in the group, which in turn proves the entire group. (Back to Top)
Student Engagement in a Quantitative Literacy Course
William L. Briggs, Nora Sullivan, Mitchell M. Handelsman
|William Briggs has been on the mathematics faculty at the University of Colorado at Denver for 20 years. He received his MS and PhD in applied mathematics from Harvard University. His research is in mathematical problems that arise in biology and medicine. He is a University of Colorado President’s Teaching Scholar and the recipient of a Fulbright Fellowship to Ireland. E-mail: email@example.com|
Nora Sullivan received degrees from Amherst College (BA 1996) and the University of Colorado at Denver (MA 2001). She was an All- American rugby player in college. She currently works as a therapist at an Adolescent Day Treatment in Denver and is working toward her licensure. E-mail: nora firstname.lastname@example.org
Mitchell M. Handelsman holds degrees from Haverford College and the University of Kansas. He is currently professor of psychology and a CU President’s Teaching Scholar at the University of Colorado at Denver. In 1992 he was the Colorado Professor of the Year, named by the Council for Advancement and Support of Education. E-mail: email@example.com
|The purpose of this paper is to describe the rationale, design, objectives, and methods underlying a liberal arts mathematics course that has been taught at the University of Colorado at Denver since 1992. The course is well aligned with recent recommendations for introducing quantitative literacy into the undergraduate curriculum. Surveys administered at the beginning and end of the course revealed that student engagement takes many different forms and is related to student performance. This study provides practical insights about effective strategies for teaching such a course. (Back to Top)|
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Mathematics: Assessment & Integration of Success Skills
Roxane Barrows & Bernita Crawford
|Roxane Barrows has 15 years of experience in the field of education. She earned a Bachelor’s Degree in information systems from The Ohio State University and a Master’s Degree in mathematics from Ohio University. She is currently working on a PhD in higher education at Ohio University. She has been employed at Hocking College for 12 years, first as a professor of mathematics and then as the mathematics coordinator. She currently is an Associate Dean of the School of Arts and Sciences and an adjunct mathematics professor.|
Bernita Crawford has 33 years of experience in the field of education. She earned a Bachelor’s Degree in education from The Ohio State University and a Master’s Degree in higher education from Ohio University. She taught high school sciences for twenty-four years including physics, chemistry, biology, and general science. She has been employed at Hocking College in Nelsonville, Ohio, since 1991, first as an adjunct and then as an assistant professor in the School of Health and Nursing. Three years ago, she became the Coordinator for the Assessment of Student Academic Achievement. Among her other responsibilities are membership in the Success Skills Learning Community and Curriculum Council. She is also an active member in the Ohio Two-Year College Assessment Network.
Hocking College, like many institutions of higher learning, has struggled to define, document, and assess those general skills deemed necessary for success in the workplace and life. The mathematics faculty have spent many years developing appropriate tools for assessment of student academic achievement. Although the process has taken several years, it has evolved into an ongoing method utilized by faculty to improve instruction and learning.
Not only are math faculty assessing student academic achievement, but they are also integrating "Success Skills” into their mathematics classes. Two Success Skills, "Communicates Effectively” and "Maintains Professional Skills and Attitudes,” have been integrated into the mathematics curriculum and strategies for assessment of them have been started. The mathematics faculty also developed a test to address the Success Skill "Demonstrates Mathematics.” The information from this test is shared with technology/program coordinators. Future steps involve integrating all eight Success Skills into the mathematics classes. (Back to Top)
Insights into the Area Model When Connecting Multiplication
with Whole Numbers to Decimal Numbers
Connie Yarema & Carol Williams
Connie Yarema is associate professor of mathematics at Abilene Christian University. She works with pre-service mathematics teachers as well as classroom teachers involved in Texas Teacher Quality grants. Her research interests include lesson studies designed in cooperation with classroom teachers.
Carol Williams is professor of mathematics at Abilene Christian University and Dean of the Graduate School. Her interests include encouraging high school girls and college women to persist in mathematics. She has been the recipient of three grants from the Mathematical Association of America in this area. E-mail: firstname.lastname@example.org
This article describes a valuable lesson that university mathematics faculty members learned from fifth grade and middle school teachers in a professional development workshop. The goal of the workshop was to show how models for whole number operations could be linked to models for rational numbers and to connect the traditional algorithms to the models. While working problems, most of the teachers modeled multiplication of whole numbers, fractions, and decimals using the newly taught area model.
Several presented an area model for decimals in an unanticipated way that led to an incorrect answer. The lesson learned was that to attain correct products from models, consistency in setting up the factors in an area model for multiplication is needed as the factors change from whole numbers to fractions and decimals. In the case of using base-10 blocks to represent the factors, the name of the block must be the same as the upper surface area of the block so that correct answers can be interpreted when modeling multiplication of whole numbers and decimals. (Back to Top)
A Mathematics Teacher’s Transition toward Inquiry-Based Discourse
in a Course for Prospective Elementary Teachers
|Lisa Clement is an assistant professor of mathematics education at San Diego State University. She co-directs a Master of Arts program in Education with a concentration in K–8 Mathematics Education, and trains mathematics tutors of seventh grade students in the Sweetwater Union High School District.|
|Using Kazemi’s and Stipek’s (2001) framework of classroom practice, the discourse between students enrolled in a mathematics-for-teachers course and their instructor is examined. The teacher’s practice is in transition from a focus on having students share multiple strategies toward a practice that additionally includes the mathematical justifications for those strategies, and pressing students to explore their errors. Field observations, classroom transcripts, teacher interviews, and student interviews were analyzed and triangulated for this study. (Back to Top)|
Edited by Sandra DeLozier Coleman
THE GOLDEN RATIO: THE STORY OF PHI, the World’s Most Astonishing Number, Mario Livio, Broadway Books, New York, 2002, ISBN 0-7679- 0816-3. (Back to Top)
Mathematics For Learning With Inflammatory Notes For The Education Of Educologists
Chapter 1: Counting With Number-Phrases
|In the Spring 2004 issue of The AMATYC Review, Schremmer introduced his idea for an open-source serialized text: Mathematics For Learning. The Preface to the text appeared in the Spring issue. This issue contains the beginning of Part 1, "Arithmetic: Numbers specified directly,” and contains Chapter 1: Counting With Number-Phrases. There are two sections to this chapter: "Accounting for Money” and "Addition Leads to Large Collections.”(Back to Top)|
The Problems Section The Problems Section
Edited by Stephen Plett & Robert Stong
New Problems: The AS Problem Set consists of four new problems.
Set AQ Solutions: Solutions are given to the four problems from the AQ Problem Set that were in the Fall 2003 issue of The AMATYC Review. (Back to Top)