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)

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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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Nội dung Text: 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 create situations in which students must understand the problems, do the work needed to solve problems, adjust his thinking, and attempt to obtain new information.” [2; 93]. In each teaching situation, we believe that the teacher needs to design a structure that contains three basic situations: situation of action, situations of comunication and situations of validation [4]. From the point of view that “doing mathematics properly implies that one is dealing with problems" [5; 22], we need to design a teaching situation where students are given face-to-face situations, and they work on their own and together to solve the problem. At that point students must adjust their thinking to the new information obtained and establish or 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 "will be 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 of teaching analytic geometry, we "need to pay attention on both the axiomatic method and the method of coordinates." The two methods complement each other, contributing to the improvement of the quality of teaching geometry and the ability to learn it. From theories presented in research, textbooks and teacher’s books, we believe that it is feasible to design geometric teaching situations that are based upon opinions of activities and ideas of the theory of situations. 2. Content In this paper, we present the results of our study: designing the teaching situation to create 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 the following problem: “In space, there is a plane (α) : Ax + By + Cz + D = 0 and there is a point M (x0 ; y0 ; z0 ). Let’s determine the formula of calculating the distance from a point M 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 ′ where M ′ is the perpendicular projection of the point M in plane (α). Then, students can propose the process (basic process) like this: Step 1: Determine the point M ′ which is the perpendicular projection of point Min plane (α); Step 2: Determine the length of segment MM ′ . So, at the moment, all students have the tools: They have gone through the process of calculating this distance previously. However, this process gives them only the segment length (the length of segment MM ′ ) and not a formula to determine the length. They have the qualitative methods but not the quantitative method (formula) to arrive at what they are looking for, with hope and faith in the results. Thus, the problem of calculating the distance becomes a problem of calculating the length of a segment or the distance between two points. This kind of problem, for students, is a known problem (in analytic geometry, they have got the formula to solve this problem). So, every student believes that they can do this problem. After a discussion time, each group or individual student can be asked to perform the tasks in the following ways: * Action 2 (situation of action) Option 1: (Use the base process combined with what one knows of vectors in space). Let M ′ (x1 ; y1 ; z1 ) be the perpendicular projection of point M on plane (α), we 48
Designing a teaching situation: developing formula to calculate the distance... −−−→ −−−→ have MM ′ ⊥ (α) ⇔ MM ′ //~n(α) , we have −−−→′ MM = t~n(α) , (t ∈ R) (2.1) −−−→ Moreover, we have MM ′ = (x1 − x0 ; y1 − y0 ; z1 − z0 ) , ~n(α) = (A; B; C) so the equation (2.1) become this system of equations    x1 − x0 = tA  x1 = x0 + tA y1 − y0 = tB ⇔ y1 = y0 + tB (2.2) z1 − z0 = tC z1 = z0 + tC   Applying the coordinates of point M ′ (in the equation (2.2)) in the equation of plane (α), we have A (x0 + tA) + B (y0 + tB) + C (z0 + tC) + D = 0 Ax0 + By0 + Cz0 + D ⇔t=− A2 + B 2 + C 2 Then, we have
−−−→
d(M; (α)) =
MM ′
q = (At)2 + (Bt)2 + (Ct)2 p = (A2 + B 2 + C 2 ) t2 √ = |t| A2 + B 2 + C 2 Applying the value of t in the above formula, we have
Ax0 + By0 + Cz0 + D
√
d (M; (α)) =
−
. A2 + B 2 + C 2 A2 + B 2 + C 2
|Ax0 + By0 + Cz0 + D| = √ A2 + B 2 + C 2 Option 2: (Use the base process combined with knowledge of the vector in space which is presented in the textbook). LetM ′ (x1 ; y1 ; z1 ) be the perpendicular projection −−−→ of point M on plane (α). We have two vectors MM ′ = (x1 − x0 ; y1 − y0 ; z1 − z0 ) and ~n(α) = (A, B, C) which are parallel to each other because they are perpendicular to plane (α). So, we have 49
Bui Van Nghi and Nguyen Tien Trung
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