Let G be a connected graph. Dominating set is a set of vertices which each vertex D has at least one neighbor in G. The minimum cardinality of D is called the domination number G(γ(G)). The metric dimension of G is the minimum cardinality of a series of vertices so that each vertex G is uniquely. It is determined by the distance of vector to the selected vertices. A dominating metric dimension set is a set of vertices has a dominating set D which has condition of metric dimension. The minimum cardinality is called the resolving domination number of G, (DomDim (G)). We analyze the resolving domination number of helm graph and it’s operation. We study combine the existence concept of dominating set and metric dimension. We have obtained the minimum cardinality of dominating number.

Abstrak. Inquiry based learning has been promoted as a student-centered approach that can strengthen the relationship between teaching and research. Inquiry can be defined as seeking for truth, information or knowledge/understanding and is used in all facets and phases of life. In this study, researches tried to apply inquiry-based learning with mathematical proving skills. The classes used are the control class and the experimental class. This study uses a mixed method namely quantitative and qualitative methods. This study aims to develop inquiry-based learning tools and produce learning products in the form of student worksheets (LKM), learning outcomes tests (THB), and monographs find out significant differences between control class and experimental class. This research uses 4D development methods (define, design, development, and disseminate). To find out the effect of inquiry based learning. Rshiny program is used. The overall result of product validation is 50 % with a valid category. The effectiveness of learning outcomes is 0.304 with the medium category. The practicality of the teacher’s response results was 75.55 with a very practical category, and the student response result was 83.46 with a very practical category. With the testing criteria accept H0 if the significance value or probability value > 0.05 then H0 is accepted and H1 if the significance value or probability value < 0.05 then H0 is rejected. The results showed that there was an impact of the development of inquiry-based learning tools on mathematical proving skills of students to prove the problem of dominating metric dimention numbers.

Abstrak. Conjecturing plays a role in mathematics learning, namely: (1) conjecturing as a way of solving problems (2) conjecturing as a process that helps students in understanding material, and (3) conjecturing as a process that trains students in reasoning. One learning model that can be applied in lectures in order to improve students’ conjecturing skills in solving expected mathematical problems is a discovery learning model, this paper will describe the development of learning material using discovery learning models and the implementation to know the effect of student conjecturing skills on combinatorics problems. The problem will be focused on discrete modeling with dominating metric dimension number studies. Further we analyze their conjecturing skills by using a triangulation methods, namely a combination of qualitative and quantitative methods. The subjects of this study are students majoring in Mathematics Education at Jember University. Research subjects using experimental class and control class. The result are 22% of students in the control class have fulfilled the completeness criteria study. While in the experimental class there are 29% has met the criteria for mastery learning. It means after the application of discovery learning, the students conjecturing skills are improved signicantly. Overall the results of the study indicate that the stages in conjecturing skill were done sequentially although not all steps were done.