From Fibonacci Numbers to Geometric Sequences and the Binet Formula by Way of the Golden Ratio!

1997 ◽  
Vol 90 (5) ◽  
pp. 386-389
Author(s):  
Angelo S. DiDomenico
Keyword(s):  
Golden Ratio ◽  
Fibonacci Numbers ◽  
School Mathematics ◽  
Evaluation Standards ◽  
Binet Formula

Involving students in open exploration and a search for patterns and relations is a very exciting and rewarding experience. These activities promote stimulating discussions, the application of known skills and relations to unfamiliar settings, and the development of what the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) has called “mathematical power”—the ability to probe and connect and to reason both inductively and deductively. Involvement of this kind can also lead to findings that capture the imagination and that foster a lasting interest in mathematics and an appreciation for its beauty.

scholarly journals On Generalization of Fibonacci, Lucas and Mulatu Numbers

2020 ◽  
Vol 1 (3) ◽  
pp. 112-122
Author(s):  
Agung Prabowo
Keyword(s):  
Continued Fraction ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Binet Formula

Fibonacci numbers, Lucas numbers and Mulatu numbers are built in the same method. The three numbers differ in the first term, while the second term is entirely the same. The next terms are the sum of two successive terms. In this article, generalizations of Fibonacci, Lucas and Mulatu (GFLM) numbers are built which are generalizations of the three types of numbers. The Binet formula is then built for the GFLM numbers, and determines the golden ratio, silver ratio and Bronze ratio of the GFLM numbers. This article also presents generalizations of these three types of ratios, called Metallic ratios. In the last part we state the Metallic ratio in the form of continued fraction and nested radicals.

Beyond the Golden Ratio: A Calculator-Based Investigation

2001 ◽  
Vol 94 (2) ◽  
pp. 138-144
Author(s):  
Peter L. Glidden
Keyword(s):  
Problem Solving ◽  
Mathematical Reasoning ◽  
Golden Ratio ◽  
School Mathematics ◽  
Opportunities To Learn ◽  
Mathematical Communication ◽  
Evaluation Standards ◽  
Mathematical Connections ◽  
Principles And Standards ◽  
Classroom Use

NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) calls for increased emphasis on problem solving, mathematical reasoning, mathematical communication, and mathematical connections. This call is reaffirmed in Principles and Standards for School Mathematics (NCTM 2000). A preferred way of achieving these goals is by having students perform mathematical investigations in which they explore mathematics, search for patterns, and use technology when appropriate. In short, students should be given opportunities to learn mathematics by doing mathematics. Of course, if students are to learn mathematics through investigations, teachers must have a ready supply of such investigations available for classroom use.

The Grand Periodic Function

2018 ◽  
Author(s):  
Jan C. A. Boeyens
Keyword(s):  
Number Theory ◽  
Periodic Function ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Magic Number ◽  
Self Similarity ◽  
Covalent Interaction ◽  
Great Pyramid ◽  
Relationship Of ◽  
The Relationship

The discovery of material periodicity must rank as one of the major achievements of mankind. It reveals an ordered reality despite the gloomy pronouncements of quantum philosophers. Periodicity only appears in closed systems with well-defined boundary conditions. This condition excludes an infinite Euclidean universe and all forms of a chaotic multiverse. Manifestations of cosmic order were observed and misinterpreted by the ancients as divine regulation of terrestrial events, dictated by celestial intervention. Analysis of observed patterns developed into the ancient sciences of astrology, alchemy and numerology, which appeared to magically predict the effects of the macrocosm on the microcosm. The sciences of astronomy and chemistry have by now managed to outgrow the magic connotation, but number theory remains suspect as a scientific pursuit. The relationship between Fibonacci numbers and cosmic self-similarity is constantly being confused with spurious claims of religious and mystic codes, imagined to be revealed through the golden ratio in the architecture of the Great Pyramid and other structures such as the Temple of Luxor. The terminology which is shared by number theory and numerology, such as perfect number, magic number, tetrahedral number and many more, contributes to the confusion. It is not immediately obvious that number theory does not treat 3 as a sacred number, 13 as unlucky and 666 as an apocalyptic threat. The relationship of physical systems to numbers is no more mysterious nor less potent than to differential calculus. Like a differential equation, number theory does not dictate, but only describes physical behavior. The way in which number theory describes the periodicity of matter, atomic structure, superconductivity, electronegativity, bond order, and covalent interaction was summarized in a recent volume. The following brief summary of these results is augmented here by a discussion of atomic and molecular polarizabilities, as derived by number theory, and in all cases specified in relation to the grand periodic function that embodies self-similarity over all space-time.

scholarly journals Fibonacci, Golden Ratio, and Vector Bundles

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 426
Author(s):  
Noah Giansiracusa
Keyword(s):  
Lie Algebra ◽  
Moduli Space ◽  
Vector Bundles ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Theoretical Physics ◽  
Stable Curves ◽  
Exceptional Lie Algebra ◽  
Summation Formulas

There is a family of vector bundles over the moduli space of stable curves that, while first appearing in theoretical physics, has been an active topic of study for algebraic geometers since the 1990s. By computing the rank of the exceptional Lie algebra g2 case of these bundles in three different ways, a family of summation formulas for Fibonacci numbers in terms of the golden ratio is derived.

One Point of View: Reclaiming School Mathematics

1990 ◽  
Vol 37 (8) ◽  
pp. 4-5
Author(s):  
Portia Elliott
Keyword(s):  
Mathematics Education ◽  
Mathematics Curriculum ◽  
Point Of View ◽  
School Mathematics ◽  
Evaluation Standards ◽  
Design Change

The framers of the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) call for a radical “design change” in all aspects of mathematics education. They believe that “evaluation is a tool for implementing the Standards and effecting change systematically” (p. 189). They warn, however, that “without changes in how mathematics is assessed, the vision of the mathematics curriculum described in the standards will not be implemented in classrooms, regardless of how texts or local curricula change” (p. 252).

Transition Boards: A Good Idea Made Better

1991 ◽  
Vol 38 (5) ◽  
pp. 4-8
Author(s):  
John T. Sutton ◽  
Tonya D. Urbatsch
Keyword(s):  
Conceptual Understanding ◽  
Mathematics Curriculum ◽  
School Mathematics ◽  
Evaluation Standards ◽  
Addition And Subtraction ◽  
School Mathematics Curriculum

The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) recognizes that addition and subtraction computations remain an important part of the school mathematics curriculum and recommends that the emphasis be shifted to the understanding of concepts. Transition boards are simple devices to aid students' conceptual understanding.

Implementing the Standards: the Measurement Standards

1989 ◽  
Vol 37 (2) ◽  
pp. 22-26
Author(s):  
Mary Montgomery Lindquist
Keyword(s):  
School Mathematics ◽  
Evaluation Standards ◽  
Measurement Standards

Clear expectations for the measurement curricula of grades K–8 are expressed in the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989). The statements in figure 1 are discussed in the Standards. Central to both the K–4 and 5–8 standards is the process of measuring, which can help students build understanding about measuring and make connections among various measurement concepts and skills.

ONE POINT OF VIEW: We Must Have “Designated Math Leaders” in the Elementary School!

1991 ◽  
Vol 39 (1) ◽  
pp. 7-9
Author(s):  
James V. Bruni
Keyword(s):  
Elementary School ◽  
Elementary School Teachers ◽  
Mathematics Learning ◽  
Point Of View ◽  
School Mathematics ◽  
School Level ◽  
Leadership Roles ◽  
Evaluation Standards ◽  
Agents Of Change

NCTM's development of the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) and the recent companion document, Professional Standards for Teaching Mathematics (NCTM 1991), is an extraordinary achievement. At a time when many agree that we urgently need change in mathematics education, these sets of standards project an exciting vision of what mathematics learning can be and how all students can develop “mathematical power.” They establish a broad framework to guide reform efforts and challenge everyone interested in the quality of school mathematics programs to work collaboratively to use them as a basis for change. How will we meet this challenge? The Editorial Panel believes that translating that vision into reality at the elementary school level will be possible only if elementary school teachers are involved in taking leadership roles as agents of change.

Informing Learning Through the Clinical Interview

1991 ◽  
Vol 38 (6) ◽  
pp. 44-46
Author(s):  
Madeleine J. Long ◽  
Meir Ben-Hur
Keyword(s):  
Student Outcomes ◽  
Clinical Interview ◽  
School Mathematics ◽  
Assessment Tools ◽  
Teaching Mathematics ◽  
National Council ◽  
Evaluation Standards ◽  
Paper And Pencil ◽  
Primary Assessment

The National Council of Teachers of Mathematics's Curriculum and Evaluation Standards for School Mathematics (1989) and Professional Srandards for Teaching Mathematics (1989) endorse the view that assessment should be made an integral part of teaching. Although many of the student outcomes described in the Srandards cannot properly be assessed using paper-and-pencil tests, such tests remain the primary assessment tools in today's classroom.

TEACHING MATHEMATICS WITH TECHNOLOGY: Computation and Estimation

1992 ◽  
Vol 40 (1) ◽  
pp. 48-51
Author(s):  
Janet Parker ◽  
Connie Carroll Widmer
Keyword(s):  
School Mathematics ◽  
Teaching Mathematics ◽  
Evaluation Standards ◽  
Mental Mathematics

As we prepare for the day envisioned by the Curriculum ond Evaluation Standards for School Mathematics (NCTM 1989), when every student will have a calculator and every class will have at least one computer available at all times, we need to reexamine the roles of computation, estimation, and mental mathematics in the teaching and practice of mathematics. It is true that calculators and computers can perform virtually all computations, relieving us and our students of much drudgery; however, this is not their only role. Calculators and computers also make it easy for us to solve problems in a new mode, T-E-M-T-T: trial, error, and modified trial through technology.

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