Sharing Teaching Ideas: Cos_x + Sin_x and the Trigonometric Sum and Difference Identities

2002 ◽  
Vol 95 (7) ◽  
pp. 510-514
Keyword(s):  
Formal Proofs ◽  
Trigonometric Sum ◽  
Teaching Ideas

That the cosine or sine of a sum or difference is not the sum or difference of the cosines or sines is easy to establish by counterexample. However, the actual expansions for cos (A ± B) and sin (A ± B) are not intuitively obvious for most students. Standard textbooks do cover the derivations of these expansions, but the results have a greater impact if students conjecture them first. This article describes a lesson that introduces the formal proofs of the trigonometric sum and difference identities. In this activity, students discover the expansions for the cosine and sine of a sum and difference by using technology to generate selected graphs and by analyzing patterns in these graphs. The only prerequisite is the ability to find the equation of a sinusoid when the graph of the function is given.

scholarly journals Extremal Values of the Sackin Tree Balance Index

2021 ◽  
Author(s):  
Mareike Fischer
Keyword(s):  
Phylogenetic Trees ◽  
Theoretical Computer Science ◽  
Search Trees ◽  
Formal Proofs ◽  
Birth Process ◽  
Ordered Trees ◽  
Research Areas ◽  
New Findings ◽  
Mathematical Phylogenetics ◽  
Extremal Values

AbstractTree balance plays an important role in different research areas like theoretical computer science and mathematical phylogenetics. For example, it has long been known that under the Yule model, a pure birth process, imbalanced trees are more likely than balanced ones. Also, concerning ordered search trees, more balanced ones allow for more efficient data structuring than imbalanced ones. Therefore, different methods to measure the balance of trees were introduced. The Sackin index is one of the most frequently used measures for this purpose. In many contexts, statements about the minimal and maximal values of this index have been discussed, but formal proofs have only been provided for some of them, and only in the context of ordered binary (search) trees, not for general rooted trees. Moreover, while the number of trees with maximal Sackin index as well as the number of trees with minimal Sackin index when the number of leaves is a power of 2 are relatively easy to understand, the number of trees with minimal Sackin index for all other numbers of leaves has been completely unknown. In this manuscript, we extend the findings on trees with minimal and maximal Sackin indices from the literature on ordered trees and subsequently use our results to provide formulas to explicitly calculate the numbers of such trees. We also extend previous studies by analyzing the case when the underlying trees need not be binary. Finally, we use our results to contribute both to the phylogenetic as well as the computer scientific literature using the new findings on Sackin minimal and maximal trees to derive formulas to calculate the number of both minimal and maximal phylogenetic trees as well as minimal and maximal ordered trees both in the binary and non-binary settings. All our results have been implemented in the Mathematica package SackinMinimizer, which has been made publicly available.

scholarly journals Frameworks for understanding the nature of interactions, networking, and community in a social networking site for academic practice

2011 ◽  
Vol 12 (3) ◽  
pp. 119 ◽  
Author(s):  
Grainne Conole ◽  
Rebecca Galley ◽  
Juliette Culver
Keyword(s):  
Social Networking ◽  
New Technologies ◽  
Social Networking Site ◽  
Current Debate ◽  
User Interactions ◽  
Academic Practice ◽  
User Behaviour ◽  
Learning And Teaching ◽  
Changing Patterns ◽  
Teaching Ideas

This paper describes a new social networking site, Cloudworks, which has been developed to enable discussion and sharing of learning and teaching ideas/designs and to promote reflective academic practice. The site aims to foster new forms of social and participatory practices (peer critiquing, sharing, user-generated content, aggregation, and personalisation) within an educational context. One of the key challenges in the development of the site has been to understand the user interactions and the changing patterns of user behaviour as it evolves. The paper explores the extent to which four frameworks that have been used in researching networked learning contexts can provide insights into the patterns of user behaviour that we see in Cloudworks. The paper considers this within the current debate about the new types of interactions, networking, and community being observed as users adapt to and appropriate new technologies.

From sequential to distributed formal proofs

1997 ◽  
pp. 499-513
Author(s):  
Víctor M. Gulfas ◽  
J. Santiago Jorge ◽  
Juan J. Quintela ◽  
José L. Freire
Keyword(s):  
Formal Proofs

Formal Proofs and Decidibility

1996 ◽  
pp. 50-55
Author(s):  
David Booth ◽  
Renatus Ziegler
Keyword(s):  
Formal Proofs

The Computational Relevance of Formal Logic Through Formal Proofs

2019 ◽  
pp. 81-96 ◽  
Author(s):  
Ariane A. Almeida ◽  
Ana Cristina Rocha-Oliveira ◽  
Thiago M. Ferreira Ramos ◽  
Flávio L. C. de Moura ◽  
Mauricio Ayala-Rincón
Keyword(s):  
Formal Logic ◽  
Formal Proofs

scholarly journals Construction and Implementation of Teaching Mode for Digital Mapping based on Interactive Micro-course Technology

2018 ◽  
Vol 13 (02) ◽  
pp. 21
Author(s):  
Ning Gao
Keyword(s):  
Learning Styles ◽  
Learning Style ◽  
Classroom Organization ◽  
Teaching Mode ◽  
Self Evaluation ◽  
Internet Education ◽  
Digital Mapping ◽  
Engineering Department ◽  
Evaluation Of Learning ◽  
Teaching Ideas

The era of “Internet + education” has caused reforms in teaching ideas, teaching modes, and learning styles. The emergence of micro-course technology provides new strategies for integrating learning styles. Task-driven digital mapping teaching, known as traditional classroom organization, has poor teaching effect due to single learning style and strategy. A new teaching mode for digital mapping was constructed in this study based on micro-course technology by combining interactive micro-course technology and digital mapping teaching to adapt to the demands of modern teaching. This teaching mode mainly included four modules, namely, micro-courseware, micro-video, micro-exercise, and micro-examination. It realized the hierarchical teaching of knowledge points in digital mapping course, simplification of basic principles, simulation of engineering cases, and self-evaluation of learning outcomes. The teaching mode was applied to 114 students from the Mapping Engineering Department of Henan University of Urban Construction. Results indicate that the proposed teaching mode based on interactive micro-course technology promoting the independent after-class learning of the students, stimulating their learning enthusiasm, enhancing their practical abilities of the students, and improving the effect of teaching. This mode of teaching provides a new concept for the teaching mode reform of other courses in mapping engineering.

scholarly journals Instructors’ ideas on the use of code-switching in EFL in Turkey: A case study

2017 ◽  
Vol 7 (1) ◽  
pp. 9-17
Author(s):  
Engin Evrim Onem
Keyword(s):  
Rank Order ◽  
Relevant Literature ◽  
Code Switching ◽  
State Universities ◽  
Future Research ◽  
Efl Teachers ◽  
Language Classrooms ◽  
Foreign Language Instructors ◽  
Teaching Ideas

Abstract   Code switching is a very common phenomenon in EFL language classrooms. The goal of this case study is to find out the possible reasons why EFL instructors employ code switching in ELT classes in Turkey. To achieve this, a brief questionnaire composed of the most common seven reasons mentioned in the relevant literature on code switching in language classrooms was compiled by the researcher and administered to ten EFL teachers working at different state universities in Turkey. The participants were asked to rank order the reasons from the most ideal to the least ideal purpose of employing code switching in classrooms for themselves and were later asked to write the reasons for their choices. It was found that “leaving no confusion about the topic” was the most common reason for the participants and the teachers who prioritized that reason seemed to have similar ideas about employing code switching in EFL. Discussion of the results and implications for future research are presented. Keywords: Code switching, language teaching, ideas about code switching, foreign language instructors.    

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