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Loci

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The document outlines a 40-minute lesson plan on finding algebraic expressions for loci that are circles. It reviews how a circle is defined as the locus of points equidistant from a fixed point. Students then learn to find the equation of a circle given the center and radius through an online video example. The lesson also covers finding the equation of a circle when a moving point forms perpendicular line segments to two fixed points, using the concept of angles in semi-circles. Students practice these skills through worksheet questions, with the teacher demonstrating an example before each practice session.

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Lesson Plan: S5 (core) Locus Lesson Plan Date N/A Gender N/A Duration 40 min Textbook Mathematics in Focus (Book D) Class S5 St ability N/A No. of st. N/A TOPIC Algebraic Expressions of Loci - circles Previous Knowledge: Students should be able to 1) sketch and describe the locus of a moving point which maintains a fixed distance from a fixed point. [The locus is a circle.] 2) apply the distance formula. 3) find the slope of a line segment when two points on the line are given. 4) find the algebraic equation of the locus of a moving point that maintains a fixed distance from a straight line. [The locus is a straight line] 5) apply “angle in a semi-circle” and “the converse of angle in a semi-circle.” Teaching Objective: Student should be able to 1) find the algebraic equation of the locus of a moving point that maintains a fixed distance from a fixed point. 2) find the algebraic equation of the locus of a moving point which forms two perpendicular line segments to two different fixed points. 3) verify whether a locus passes through a particular point or not. Teaching Tool(s): 1) Worksheet - S5 (core) Locus Reference: 1) Mathematics in Focus (2009), Hung Chun Wah et al., Educational Publishing House Ltd. 2) New Centuries Mathematics (Oxford) P.1
Lesson Plan: S5 (core) Locus Time Allocation and Teaching contents: Time (min) Teaching content or activity Objective 1 Introduction “Ladies and gentlemen, we’ve changed our focus on locus problems in the last lesson. We drew loci on a coordinate plane and found the equation that the points on the loci must satisfy. We’ve done several examples of loci of straight lines. In the following, we’ll discuss some loci that are circles.” 10 Students watch the following video: 1 http://www.youtube.com/watch?v=Qy1I2ehReVA The video (1) reviewed previous knowledge (4), (2) discussed teaching objective (1), and (3) discussed the characteristic of an equation of a circle. The problem solved in the video for teaching objective (1): P moves on a rectangular coordinate plane. It maintains a distance of 2 units from A(1,2). Find the equation of the locus of P. 3 Ask students questions: 1  What is the locus formed by a moving point that keeps a fixed distance from a fixed point?  For an equation of a circle, what is the relationship between the coefficient of x2 and the coefficient of y2?  For an equation of a circle, is there any xy cross term? 6 1st Class work session: page 3 of worksheet Q1 and Q2 1,3 (1) Q moves on a rectangular coordinate plane. It maintains a distance of 3 units from H(–2,1). Find the equation of the locus of Q. (2) S moves on a rectangular coordinate plane. It maintains a distance of 1 unit from R(1,–4). (a) Find the equation of the locus of S. (b) Show that the locus of S passes through (1,–3). 9 Teacher demonstrates: 2, 3 e.g.2: There are two points on a coordinates plane, A(4,3) and B(1,–2). P moves on the coordinate plane such that AP is perpendicular to BP. (a) Find the equation of the locus of P. P.2
Lesson Plan: S5 (core) Locus (b) Does the locus of P pass through A(4,3)? Discuss: the reason that the locus of P is a circle (converse of angle in a semi-circle nd 7 2 Class work session: page 3 of worksheet Q3 and Q4 2,3 (3) M (–2,0) and N(2,0) are on a rectangle coordinate plane. T is a moving point such that MT is perpendicular to NT. Find the equation of the locus of T. (4) P moves on a rectangular coordinate plane. A(–1,3) and B(1,–1) are two fixed points. Find the equation of the locus of P if AP⊥BP. 2 Summary and homework “Ok. Let’s summarize what we’ve learned in this lesson. First, we find an algebraic expression for the locus of a moving point P, which maintains a fixed distance from a fixed point. Then, we went over one more condition that gives a locus of a circle: A moving point P forms two line segments with two fixed points and those two line segments are always perpendicular to each other. This condition implicitly uses the converse of angle in a semi-circle. Homework: Please finish Textbook Exercise 14.2, Q9 and Q12, on homework book B P.3

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Loci

  • 1. Lesson Plan: S5 (core) Locus Lesson Plan Date N/A Gender N/A Duration 40 min Textbook Mathematics in Focus (Book D) Class S5 St ability N/A No. of st. N/A TOPIC Algebraic Expressions of Loci - circles Previous Knowledge: Students should be able to 1) sketch and describe the locus of a moving point which maintains a fixed distance from a fixed point. [The locus is a circle.] 2) apply the distance formula. 3) find the slope of a line segment when two points on the line are given. 4) find the algebraic equation of the locus of a moving point that maintains a fixed distance from a straight line. [The locus is a straight line] 5) apply “angle in a semi-circle” and “the converse of angle in a semi-circle.” Teaching Objective: Student should be able to 1) find the algebraic equation of the locus of a moving point that maintains a fixed distance from a fixed point. 2) find the algebraic equation of the locus of a moving point which forms two perpendicular line segments to two different fixed points. 3) verify whether a locus passes through a particular point or not. Teaching Tool(s): 1) Worksheet - S5 (core) Locus Reference: 1) Mathematics in Focus (2009), Hung Chun Wah et al., Educational Publishing House Ltd. 2) New Centuries Mathematics (Oxford) P.1
  • 2. Lesson Plan: S5 (core) Locus Time Allocation and Teaching contents: Time (min) Teaching content or activity Objective 1 Introduction “Ladies and gentlemen, we’ve changed our focus on locus problems in the last lesson. We drew loci on a coordinate plane and found the equation that the points on the loci must satisfy. We’ve done several examples of loci of straight lines. In the following, we’ll discuss some loci that are circles.” 10 Students watch the following video: 1 http://www.youtube.com/watch?v=Qy1I2ehReVA The video (1) reviewed previous knowledge (4), (2) discussed teaching objective (1), and (3) discussed the characteristic of an equation of a circle. The problem solved in the video for teaching objective (1): P moves on a rectangular coordinate plane. It maintains a distance of 2 units from A(1,2). Find the equation of the locus of P. 3 Ask students questions: 1  What is the locus formed by a moving point that keeps a fixed distance from a fixed point?  For an equation of a circle, what is the relationship between the coefficient of x2 and the coefficient of y2?  For an equation of a circle, is there any xy cross term? 6 1st Class work session: page 3 of worksheet Q1 and Q2 1,3 (1) Q moves on a rectangular coordinate plane. It maintains a distance of 3 units from H(–2,1). Find the equation of the locus of Q. (2) S moves on a rectangular coordinate plane. It maintains a distance of 1 unit from R(1,–4). (a) Find the equation of the locus of S. (b) Show that the locus of S passes through (1,–3). 9 Teacher demonstrates: 2, 3 e.g.2: There are two points on a coordinates plane, A(4,3) and B(1,–2). P moves on the coordinate plane such that AP is perpendicular to BP. (a) Find the equation of the locus of P. P.2
  • 3. Lesson Plan: S5 (core) Locus (b) Does the locus of P pass through A(4,3)? Discuss: the reason that the locus of P is a circle (converse of angle in a semi-circle nd 7 2 Class work session: page 3 of worksheet Q3 and Q4 2,3 (3) M (–2,0) and N(2,0) are on a rectangle coordinate plane. T is a moving point such that MT is perpendicular to NT. Find the equation of the locus of T. (4) P moves on a rectangular coordinate plane. A(–1,3) and B(1,–1) are two fixed points. Find the equation of the locus of P if AP⊥BP. 2 Summary and homework “Ok. Let’s summarize what we’ve learned in this lesson. First, we find an algebraic expression for the locus of a moving point P, which maintains a fixed distance from a fixed point. Then, we went over one more condition that gives a locus of a circle: A moving point P forms two line segments with two fixed points and those two line segments are always perpendicular to each other. This condition implicitly uses the converse of angle in a semi-circle. Homework: Please finish Textbook Exercise 14.2, Q9 and Q12, on homework book B P.3