Designing a teachiang situation: Developing formula to calculate the distance from a point to a plane in space (geometry for the 12th grade, chapter 3, lesson 2)

In this article, the author designs a teaching situation: developing formula to calculate the distance from a point to a plane in space. In these situations, all learning activities of students will have been planned by the teacher.
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Designing a teachiang situation: Developing formula to calculate the distance from a point to a plane in space (geometry for the 12th grade, chapter 3, lesson 2) JOURNAL OF SCIENCE OF HNUE Interdisciplinary Science, 2013, Vol. 58, No. 5, pp. 47-52 This paper is available online at http://stdb.hnue.edu.vn DESIGNING A TEACHING SITUATION: DEVELOPING FORMULA TO CALCULATE THE DISTANCE FROM A POINT TO A PLANE IN SPACE (Geometry for the 12th grade, Chapter 3, Lesson 2) Bui Van Nghi1 and Nguyen Tien Trung2 1 Faculty of Mathematics, Ha Noi National University of Education 2 University of Education Publishing House Abstract. In this article, the author designs a teaching situation: developing formula to calculate the distance from a point to a plane in space. In these situations, all learning activities of students will have been planned by the teacher. The formula to calculate the distance from a point to a plane will be created through the process of two different student’s activities: the first is the process of determining the distance in synthetic geometry and the second is the similarity in the formula for calculating the distance from a point to a line in the plane (something that students already know). In this teaching situation, students learn through their own activities but in a manner that was part of the teacher’s plan. In the process of implementing the scenario, from time to time teachers will need to orient students at a minimum level in such a manner that students will find the desired formula. Keywords: Teaching situation, the distance from a point to a plane in space.1. Introduction When teaching mathematics, particularly geometry, The teacher needs to createsituations in which students must understand the problems, do the work needed to solveproblems, adjust his thinking, and attempt to obtain new information.” [2; 93]. In each teaching situation, we believe that the teacher needs to design a structurethat contains three basic situations: situation of action, situations of comunication andsituations of validation [4]. From the point of view that “doing mathematics properly implies that one is dealingwith problems [5; 22], we need to design a teaching situation where students are givenface-to-face situations, and they work on their own and together to solve the problem. Atthat point students must adjust their thinking to the new information obtained and establishor develop their skills.Received November 05, 2012. Accepted June 25, 2013.Contact Nguyen Tien Trung, e-mail address: trungnt@hnue.edu.vn 47 Bui Van Nghi and Nguyen Tien Trung Therefore, the teaching situation must be designed in such a way that students willbe responsible for the relationship between them and knowledge.” [6; 159] According to a study presented by Bui Van Nghi (2008) [1; 184], in the process ofteaching analytic geometry, we need to pay attention on both the axiomatic method andthe method of coordinates. The two methods complement each other, contributing to theimprovement of the quality of teaching geometry and the ability to learn it. From theories presented in research, textbooks and teacher’s books, we believethat it is feasible to design geometric teaching situations that are based upon opinionsof activities and ideas of the theory of situations.2. Content In this paper, we present the results of our study: designing the teaching situation tocreate formula for calculating the distance from a point to a plane in space. The scenario of the teaching situation involves the following actions: * Action 1 (situation of action) The teacher divides the class into four groups and asks each group to solve thefollowing problem: “In space, there is a plane (α) : Ax + By + Cz + D = 0 and there is apoint M (x0 ; y0 ; z0 ). Let’s determine the formula of calculating the distance from a pointM to a plane (α). Every student knows how to determine the distance from a point to a plane in space:the distance from a point M to a plane (α) is equal to the length of segment MM ′ whereM ′ is the perpendicular projection of the point M in plane (α). Then, students can proposethe process (basic process) like this: Step 1: Determine the point M ′ which is the perpendicular projection of point Minplane (α); Step 2: Determine the length of segment MM ′ . So, at the moment, all students have the tools: They have gone through the processof calculating this distance previously. However, this process gives them only the segmentlength (the length of segment MM ′ ) and not a formula to determine the length. They havethe qualitative methods but not the quantitative method (formula) to arrive at what theyare looking for, with hope and faith in the results. Thus, the problem of calculating the distance becomes a problem of calculatingthe length of a segment or the distance between two points. This kind of problem, forstudents, is a known problem (in analytic geometry, they have got the formula to solvethis problem). So, every student believes that they can do this problem. After a discussion time, each group or individual student can be asked to performthe tasks in the following ways: * Action 2 (situation of action) Option 1: (Use the base process combined with what one knows of vectors inspace). Let M ′ (x1 ; y1 ; z1 ) be the perpendicular projection of point M on plane (α), we48 Designing a teaching situation: developing formula to calculate the distance... −−−→ ...