MINIMAL EDGE DARI GRAF 2-CONNECTED DENGAN CIRCUMFERENCE TERTENTU (On Edge Minimal 2-Connected Graphs with Prescribed Circumference)

Misal G adalah graf 2-connected dengan n vertex dan m edge yang mempunyai circumference c ≥.4. Dalam makalah ini ditentukan banyaknya minimum edge dari graf 2-connected G dengan n vertex dan m edge yang mempunyai circumference c ≥.4. Kata kunci : circumference, graf 2-connecte

EXTREMAL PROBLEMS CONCERNING CYCLES IN GRAPHS AND THEIR COMPLEMENTS

Let Gt(n) be the class of connected graphs on n vertices having the longest cycle of length t and let G ∈ Gt(n). Woodall (1976) determined the maximum number of edges of G, ε(G) ≤ w(n,t), where w(n, t) = (n - 1) t/2 - r(t – r - 1)/2 and r = (n - 1 ) - (t - 1) ⎣(n - 1)/(t - 1)⎦. An alternative proof and characterization of the extremal (edge-maximal) graphs given by Caccetta and Vijayan (1991). The edge- maximal graphs have the property that their complements are either disconnected or have a cycle going through each vertex (i.e. they are hamiltonian). This motivates us to investigate connected graphs with prescribed circumference (length of the longest cycle) having connected complements with cycles . More specifically, we focus our investigations on : Let G(n, c, c ) denote the class of connected graphs on n vertices having circumference c and whose connected complements have circumference c . The problem of interest is that of determining the bounds of the number of edges of a graph G ∈ G(n, c, c ) and characterize the extremal graphs of G(n, c, c ). We discuss the class G(n, c, c ) and present some results for small c. In particular for c = 4 and c = n - 2, we provide a complete solution. Key words : extremal graph, circumferenc

THE ECCENTRIC DIGRAPH OF FRIENDSHIP GRAPH, AND FIRECRACKER GRAPH

Let G be a graph with a set of vertices V(G) and a set of edges E(G). The distance from vertex u to vertex v in G, denoted by d(u,v), is a length of the shortest path from vertex u to v. The eccentricity of vertex u in graph G is the maximum distance from vertex u to any other vertices in G, denoted by e(u). Vertex v is an eccentric vertex from u if d(u,v) = e(u). The eccentric digraph ED(G) of a graph G is a graph that has the same set of vertices as G, and there is an arc (directed edge) joining vertex u to v if v is an eccentric vertex from u. The purposes of this reserarch are to determine the eccentric digraphs of some\ud classes of graphs, in particular the friendship graphs, and the firecracker graphs

THE ECCENTRIC DIGRAPH OF FRIENDSHIP GRAPH AND FIRECRACKER GRAPH

Let G be a graph with a set of vertices and a set of edges Then the distance from vertex u to vertex v in G, denoted by is the length of the shortest path from vertex u to v. The eccentricity of vertex u in a graph G is the maximum distance from vertex u to any other vertices in G, denoted by Vertex v is an eccentric vertex from u if The eccentric digraph of a graph G is a graph that has the same set of vertices as G, and there is an arc (directed edge) joining vertex u to v if v is an eccentric vertex from u. In this paper, we determine the eccentric digraph of a class of graphs called the friendship graph and firecracker grap

ASSESSMENT OF EXAM QUESTIONS QUALITY ACCORDING TO COGNITIVE DOMAIN OF BLOOM’S TAXONOMY

2013 curriculum specially prepared students to compete in a global education. Therefore, it provided students an exam with Higher Order Thinking questions. In the learning process, the questions would play a central role. The aim of this research is to describe cognitive domain of midle school semester 1 exam questions composed by the teacher. This research is qualitative-descriptive research. The subjects were two teachers in Senior High School 1 Cawas. One teacher was a senior subject and the other was a beginner subject. The techniques of collecting data was documentation. The data were validated using triangulation method. All data were analyzed using the model of Miles and Huberman. The result on the study showed that the most common middle school semester 1 exam questions used by all subjects were application aspect whereas analysis aspect was least used. Even, in beginner subjects, analysis aspect was not used at all. The quality of both questions arranged by the two subject was different. Question arranged by senior subject could not be resolved directly. Students had to interpret the questions to other form. Even, the question arranged by beginner subject could be resolved directly without interpreting the intent question first. Questions of comprehension aspect arranged by senior subject demanded skills of students to change the question to other form. Even in beginner subject, questions could be solved simply by applying the formula. Question of analysis aspect was only used by senior subject. In this aspect, question was around of conclusion of logarithm.Keywords: Bloom’s Taxonomy, Cognitive Domain, Mathematics Teache

Students' Epistemological Obstacles on Geometric-algebraic Relations of Transformation Geometry

Learning obstacles are naturally experienced by students, including the topic of transformation geometry. This study aims to describe the epistemological obstacles experienced by students on geometric-algebraic relations of transformation geometry. A qualitative approach is used with a case study as a research model. The subjects were 9 students of 11th grade obtained through snowball sampling. The data were collected through paper tests and think aloud method then analyzed through data reduction, data presentation, and concluding. The results showed that there are epistemological obstacles found in students' understanding of the geometric-algebraic relations of transformation geometry. Students know the geometric meaning of a transformation process they have done, indicated by the absence of errors that occur in the tests they take or in their articulation of thinking through think aloud. However, there are some limitations of knowledge experienced by students including the limited basic knowledge about matrix algebra operations, limitations in realizing the geometric-algebraic connections of the cases they encounter, and do not know the matrix of transformation that should be used. These limitations then become the epistemological obstacles for students in understanding and problem solving the transformation problem in the form of algebraic (analytically). These epistemological obstacles cause students to be unable to understand the algebraic meaning of transformation geometry. Thus, geometric-algebraic relationships are not formed completely and result in hindering students in understanding transformation geometry

THE EFECTIVENESS OF NUMBERED HEADS TOGETHER WITH GUIDED DISCOVERY LEARNING AND JIGSAW II WITH GUIDED DISCOVERY LEARNING VIEWED FROM ADVERSITY QUOTIENT

This research was a quasi-experimental research with 2×3 factorial design aimed to find out the influence of learning model NHT with guided discovery learning and Jigsaw II with guided discovery learning for students’ mathematics achievement. The population of this study were all of the eleventh grade students of Junior High School in Karanganyar regency and sampling was done by stratified cluster random sampling. The data was collected by test, questionnaire, and documentation.The test of hypothesis used two-way analysis of variance with unequal cell, past analysis of variance with Scheffe’ method and significance level was 0.05. Based on hypothesis test, it could be concluded that (1) the learning model of Jigsaw II with guided discovery learning approach results students’ mathematics achivement better than NHT with guided discovery learning., (2) students’ mathematics achievement with the climbers type was as good as students’ mathematics achievement with the campers type, and students’ mathematics achievement with the campers type result better than students’ mathematics achievement the quitters type, (3) for each learning model, students’ mathematics achievement with the climbers type was as good as with students’mathematics achievement with the campers type, and students’ mathematics achievement the campers type result better than students’ mathematics achievement the quitters type, (4) for each category AQ, the learning model of Jigsaw II with guided discovery learning approach results better than students’ mathematics achivement learning model NHT with guided discovery learning.Keywords: Jigsaw II, Numbered Heads Together, Guided Discovery Learning, Adversity Quotient, Mathematics Achievemen

Profile of Visual-Spatial Intelligence In Solving Geometric of 11th Grades Viewed From Gender Differences

Visual-spatial intelligence is one of the multiple bits of intelligence that important to solve a mathematics problem, especially in geometry. This present research investigates the profile of students’ visual-spatial intelligence. This research focuses on analysis and description of students’ visual-spatial intelligence level generally and its aspect when solving the geometric problem. Visual-spatial intelligence aspect, there is imagination, pattern seeking, problem-solving, and conceptualization. Qualitative research with case study strategy was used in this research. The subject in this research involved 12 students of 11th grades chosen with purposive sampling. Data in this research were students’ visual-spatial intelligence test result and task based interviews. They were asked to complete visual-spatial intelligence test before interview. The data was analyzed based on visual-spatial intelligence aspect of female and male students. The results of this research show that female students have better pattern seeking and conceptualization. Meanwhile, male students have better in imagination and problem-solving

Mathematical Abstraction of Junior High School Students on Function Based on Gender Perspectives

The purpose of this study is to describe the mathematical abstraction of junior high school students in learning the concept of function between male and female students. This is a qualitative descriptive study with total participants were 28 students in grade 8th. There were 14 female students and 14 male students in two different classes. The data were collected through written tests and interviews. In this study, the researcher gave abstraction test questions. Furthermore, from the test results, the interview was conducted. The researcher chose 4 communicative students from the high ability. Based on the data analysis result, mathematical abstraction of the students can be classified into 4 levels: 1) Recognition, 2) Representation, 3) Structural Abstraction and 4) Structural Awareness. The result of research showed that female students achieved mathematical abstraction levels, namely: recognition, representation, structural abstraction and structural awareness. Meanwhile, male students only achieved the level of recognition, it means that commonly the male students do not yet have the mathematical abstraction ability in function concepts. Therefore, it is necessary to conduct further research on the differences in the abstraction process of male and female students in learning context using the SCL approach