Contents COORDINATE METHODS REGRESSION AND CORRELATION

Transcription

1 Contents UNIT 3 UNIT 4 COORDINATE METHODS Lesson 1 A Coordinate Model of a Plane Investigations 1 Representing Geometric Ideas with Coordinates Reasoning with Slopes and Lengths Representing and Reasoning with Circles On Your Own Lesson 2 Coordinate Models of Transformations Investigations 1 Modeling Rigid Transformations Modeling Size Transformations Combining Transformations On Your Own Lesson 3 Transformations, Matrices, and Animation Investigations 1 Building and Using Rotation Matrices Building and Using Size Transformation Matrices On Your Own Lesson 4 Looking Back REGRESSION AND CORRELATION Lesson 1 Bivariate Relationships Investigations 1 Rank Correlation Shapes of Clouds of Points On Your Own Lesson 2 Least Squares Regression and Correlation Investigations 1 How Good Is the Fit? Behavior of the Regression Line How Strong Is the Association? Association and Causation On Your Own Lesson 3 Looking Back ix

2 UNIT 3 The use of coordinates to specify locations in two dimensions, as on a map, is familiar to you. The general idea of representing points in terms of coordinates has important mathematical applications in computer modeling. It enables designers and engineers to describe geometric ideas in algebraic language. Objects such as lines, circles, and other curves in two dimensions and surfaces in three dimensions can be expressed in terms of functions and equations. Transformations such as rotation or enlargement of these shapes and surfaces can be accomplished by operations on coordinates. In this unit, you will study coordinate methods for representing, analyzing, and transforming two-dimensional shapes. Key ideas will be developed through your work on problems in three lessons. Coordinate Methods Lessons 1 A Coordinate Model of a Plane Use coordinates to represent points, lines, and geometric figures in a plane and on a computer or calculator screen and to analyze properties of shapes. 2 Coordinate Models of Transformations Use coordinates to describe transformations of the plane and to investigate properties of shapes that are preserved under various types of transformations. 3 Transformations, Matrices, and Animation Develop and use matrix representations of polygonal shapes and transformations to create computer animations.

3 Check Your Understanding Consider quadrilateral PQRS with vertex matrix PQRS = a. Draw quadrilateral PQRS on a coordinate grid. b. What special kind of quadrilateral is PQRS? Use coordinates to justify your answer. Investigation 3 Representing and Reasoning with Circles In Investigations 1 and 2, you learned how to represent and analyze polygons in a coordinate plane. You can describe their sides using linear equations and study their properties using ideas of distance and slope. Polygons, particularly triangles and quadrilaterals, are the building blocks for architectural designs. Industrial, automotive, and aerospace designs often require that shapes have circular components. Your work on the problems in this investigation will help you answer these questions: What information is needed to create a circle in a coordinate plane? How can you represent circles in a coordinate plane with equations? How can you use general coordinates of points to reason about special properties of circles? LESSON 1 A Coordinate Model of a Plane 175

4 1 As a class, explore how interactive geometry software could be used to create the design shown at the right. a. What information was needed by the software to draw each circle? Why do you think that information is sufficient? b. Clear the window and redraw the square, centered at the origin, with side length 10 units. c. Draw a circle inscribed in the square, that is, a circle that touches each side of the square at one point. Describe the points of contact of the circle and square. d. Draw a circle circumscribed about the square, that is, a circle that passes through each vertex of the square. e. What is the radius of each circle in Parts c and d? 2 Here are two circles with center at the origin O and radius 10 drawn in a coordinate plane. Diagram I Diagram II y D(?,?) C(a, a) B(8,?) y P(x, y) O A(?,?) x O x F(?, -5) E(-2,?) a. What must be true about the distance from point O to any other point on the circle? b. Without the help of software, find the missing coordinate(s) of points A through F on the circle in Diagram I. c. Suppose P(x, y) is any point on the circle in Diagram II. i. What must be true about the distance OP? ii. Write an equation showing the relationship between x, y, and the radius of the circle. d. Write an equation for a circle with its center at the origin and with radius 7. With radius 3. With radius r. 176 UNIT 3 Coordinate Methods

5 3 A calculator-produced circle is shown below. The Zsquare window has a scale on both axes of 1 unit. a. What is the radius of the circle? b. Write an equation for this circle. c. What expressions could be placed in the Y= menu to produce the circle? Do your expressions produce a circle with the same radius? d. Use your calculator to produce a copy of the circle shown in the computer display on page Some of the circles you created in Problem 1 did not have their centers at the origin. However, you can use reasoning similar to that in Problem 2 to find equations for these circles. a. What is the center and radius of the circle whose center is on the positive x-axis? i. Suppose P(x, y) is any point on that circle. Explain why it must be the case that (x - 5) 2 + y 2 = 5. ii. Use that information to write an equation for the circle that does not involve a radical symbol. b. Write similar equations for: i. the circle whose center is on the positive y-axis. ii. the circle whose center is on the negative x-axis. iii. the circle whose center is on the negative y-axis. c. Verify that the coordinates of the vertices of the square satisfy your equations of the four circles that contain those vertices. Share the workload with your classmates. 5 Now try to generalize your work in Problems 2 4 to a circle whose center is not on an axis. a. Use reasoning similar to that in y P(x, y) Problem 4 to find the equation of a circle with center C(h, k) and radius r. r b. Compare your equation with those of C(h, k) your classmates. Resolve any differences. c. Rewrite your equation in Part b for the case when C(h, k) is the origin. What do you notice? x LESSON 1 A Coordinate Model of a Plane 177

6 6 Without using technology, determine which of the following equations describe a circle in a coordinate plane. For each equation that represents a circle, determine the center, the radius, and one point on the circle. For each equation that does not represent a circle, explain why not. a. x 2 + y 2 = 25 b. x 2 + y = 16 c. 3x 2 + 3y 2 = 108 d. (x - 5) 2 + (y - 1) 2 = 81 e. 3x 2 + y 2 = 9 f. x 2 + (y + 5) 2 = 1 7 Coordinates as employed by interactive geometry software open new windows to geometry by allowing you to easily create figures and search for patterns in them. Complete Parts a c using your software. You can create the figures yourself or use the Explore Angles in Circles custom tool. a. Draw a circle with center A and diameter with endpoint B. Label the other endpoint C. b. Construct a new point D on the circle. Then draw BD and CD. c. Click and drag point D along the circumference of the circle. i. What appears to be true about CDB in all cases? ii. How is your conjecture supported by calculations from the Measurements menu? d. State your conjecture in the form: An angle inscribed in a semicircle.... Compare your conjecture with your classmates and resolve any differences. 8 As you saw in Investigation 2, coordinates can provide a powerful way to justify conjectures you make about geometric figures. The key is to position the figure in a coordinate plane so that general coordinates are easy to work with. A circle with center at the origin and radius r is shown below. Point A(a, b) is a general point on the circle, different from points P and Q which are endpoints of a diameter on the x-axis. y A(a, b) P(-r, 0) O r Q(r, 0) x 178 UNIT 3 Coordinate Methods

7 Use these general coordinates and the following questions to help justify the conjecture you made in Problem 7: An angle inscribed in a semicircle is a right angle. a. What are some possible methods you could use to justify that PAQ is a right angle? b. What are the coordinates of points P and Q? c. Since point A(a, b) is on the circle, what must be true about the distance OA? How is that distance related to the coordinates a and b? d. Study Jack s argument below. He shows that PAQ is a right triangle, and so PAQ is a right angle. Check the correctness of Jack s reasoning and give reason(s) justifying each step. If there are any errors in Jack s reasoning, correct them. Jack s argument The length of PA = (a + r) 2 + b 2, so (PA) 2 = (a + r) 2 + b 2. (1) The length of AQ = (r a) 2 + b 2, so (AQ) 2 = (r a) 2 + b 2. (2) The length of PQ = 2r, so (PQ) 2 = 4r 2. (3) (PA) 2 + (AQ) 2 = (a + r) 2 + b 2 + (r a) 2 + b 2 (4) = (a 2 + 2ar + r 2 + b 2 ) + (r 2 2ar + a 2 + b 2 ) (5) = 2a 2 + 2r 2 + 2b 2 (6) = 2(a 2 + b 2 ) + 2r 2 (7) = 2r 2 + 2r 2 (8) = 4r 2 (9) = (PQ) 2 (10) Therefore, PAQ is a right triangle with PAQ a right angle. (11) e. Now examine Malaya s argument justifying the conjecture that PAQ is a right angle. Check the correctness of Malaya s reasoning and give reason(s) justifying each step. Correct any errors in Malaya s reasoning. Malaya s argument The slope of PA is b_ a + r. (1) The slope of QA is b_ a r. (2) The product of the slopes is (_ a b )( _ + r b b a r ) = _ 2 a 2 r. (3) 2 Since a 2 + b 2 = r 2, it follows that a 2 r 2 = b 2. (4) This means that the product of the slopes is _ b2 = 1. (5) b2 So, PA AQ and PAQ is a right angle. (6) LESSON 1 A Coordinate Model of a Plane 179

8 Summarize the Mathematics In this investigation, you discovered how to write equations for circles in a coordinate plane and used coordinates to make general arguments. a What is the equation of a circle with center at the origin and radius r? b What is the equation of a circle with center at (h, k) and radius r? c What formula was the key to deriving the equation of a circle? d How can you tell by looking at an equation whether its graph is a circle? e Why are general coordinates such as (a, b) used in reasoning about geometric properties? Be prepared to share your equations and thinking with the class. Check Your Understanding A circle with center at (3, -4) is drawn so that it is tangent to the x-axis. That is, the circle touches the x-axis at only one point called the point of tangency. a. Draw the circle on a coordinate plane. b. What are the coordinates of the point of tangency? c. Write an equation for the circle. d. Write an equation for a congruent circle with center at the origin. e. Graph the circle in Part d on your graphing calculator. 180 UNIT 3 Coordinate Methods

9 COORDINATE METHODS UNIT 3 Unit Overview This unit focuses on coordinate methods in geometry. Just as Rene Descartes ( ) work in coordinate geometry permitted the development of both algebra and geometry to move forward, the introduction of coordinate methods in Course 2 of the Core-Plus Mathematics curriculum advances and connects the algebra and geometry concepts begun in Course 1. The representations of shapes and transformations in a plane now are explored systematically using coordinates. Coordinate methods also lead to a simple, yet powerful, base for computer animations. The use of matrices as a representation tool is extended to describing shapes and transformations, thereby underscoring the connectedness of strands within the curriculum and the utility of matrices as a connector. Technology as a Context for the Unit Using interactive geometry software in this unit allows students to raise questions about how the software is able to create, measure, and reposition shapes. Additionally, working in an interactive technology environment allows students to investigate text-provided questions as well as their own questions. The unit provides a gentle introduction to the use of interactive geometry software as a tool for exploring and applying important geometric ideas. In Unit 6, Trigonometric Methods, students will continue to use the geometry software. In Course 3 and Course 4 units, students will become more proficient with geometry software as a tool for discovering and applying geometric concepts. The time invested in becoming familiar with this tool will pay dividends in later units. Therefore, it is valuable for all students to have individual computer time to explore the commands and functionality of interactive geometry software so they can flexibly interact with it in a variety of instructional settings. We realize that access to technology varies across schools and among classrooms. Some schools have chosen to support student learning with a classroom set of laptop computers. Other schools have more limited access to computers; consequently, teachers sign up to take their classes to the computer lab, use a single classroom computer for demonstration and shared student work, or provide a few computer stations within the classroom for groups to choose to use as they work on particular problems. Since CPMP-Tools is available on the StudentWorks CD-ROM and from the Internet ( CPMP-Tools/), you may ask students to complete an investigation problem or two outside of class and be prepared to share results with classmates. This allows individual computer time and will be particularly helpful for students needing more time to investigate and process the material. Overview T161

10 This unit can be effectively taught using one computer and a projection system. When a single classroom computer is used, we envision an interactive classroom setting in which all students are engaged in the exploration of mathematical situations seeking patterns in displays, making and testing conjectures, and posing related questions. The computer operator can be a classroom teacher, a student, or a group of students. Using interactive geometry software, such as what resides on CPMP-Tools, on a single computer can provide the stage on which a problem situation is posed for the entire class, and classroom discourse can be the vehicle through which the mathematics emerges. If a teacher is facilitating an investigation and acting as the computer operator, the exploration should still belong to the students. Hence, it will be important for the teacher to pose questions to both prompt and push student thinking. Alternatively, the teacher could facilitate an investigation allowing a student to act as the computer operator. In either case, when a problem indicates student choice of figures and/or transformations, the choice(s) should be solicited from students. In this unit, students should consistently have access to grid or dot paper. Student Master 4 can be copied for this purpose. Using square dot paper and colored pencils will allow students to more easily identify shapes and their transformed images, particularly when horizontal and vertical segments are displayed. Unit Objectives Unit 3 Use coordinates to represent points, lines, and geometric figures in a plane and on a computer or calculator screen Use coordinate representations of figures to analyze and reason about their properties Use coordinate methods and programming techniques as a tool to implement computational algorithms, to model rigid transformations and similarity transformations, and to investigate properties of shapes that are preserved under various transformations Build and use matrix representations of polygons and transformations and use these representations to create computer animations CPMP-Tools Although all investigations in this unit are written intending use of CPMP-Tools Interactive Geometry software, the investigations are easily adapted for use with commercial software such as The Geometer s Sketchpad, Cabri Geometre, or Cinderella. Recommendations and options for instruction when one or a few computers are available are at the beginning of investigations and at point of use. When accessing CPMP-Tools software for use in this unit, be sure to select Course 2 from the Course menu on the left; then under the Geometry menu, select Coordinate Geometry. T161A UNIT 3 Coordinate Methods

11 In addition to the general purpose interactive geometry tool, CPMP-Tools geometry software includes several custom tools: Design by Robot, Explore Angles in Circles, Animate Shuttle, and Roll Over. Lesson 1 Investigation 3 Problem 7 (page 178) uses the Explore Angles in Circles custom tool. This investigation does not call for other technology use, and thus lab access is not needed for this task. It can be facilitated, as suggested on page T161A, using a single class computer, having multiple computers available for student use as they reach Problem 7, or by student investigation individually outside of class time. Ideally, for Lesson 1 Investigation 1, students will have access, in pairs, to computers and CPMP-Tools or other interactive geometry software. Doing so will enable students to become experienced using the software and better engage later in other whole-class discussions facilitated using one computer and projector. Selected investigation problems and homework tasks that require computer access can be done outside of class time. See page T164 for specific guidelines. Lesson 2 Investigation 3, Combining Transformations, can be completed using grid paper, but using the interactive geometry software in CPMP-Tools will allow students to quickly perform compositions (in both orders) and request lengths and areas from the Measurement menu to compare sizes of image and preimage polygons. In Lesson 3, students are asked to build transformation matrices and study planning algorithms and programs using a simplified programming language embedded in CPMP-Tools. Computer access during class time is not necessary for Lesson 3 unless you wish to provide students programming time during class. Students should complete Lesson 1 Extensions Task 27 (page 190) and Lesson 2 Extensions Task 31 (page 227) in preparation for developing their own animations in Lesson 3. See the introductory section on page T161 for other technology-related information. Unit 3 Overview T161B

12 Unit 3 Teaching Resources STUDENT MASTERS The key geometric ideas listed here are on a student activity master with additional space to record key ideas as they are developed in this unit. Student Masters 1 2. UNIT 3 Coordinate Methods Name Date Key Geometric Ideas Definitions Isosceles triangle A triangle with at least two sides of equal length Parallelogram A quadrilateral with opposite sides of equal length Rectangle A quadrilateral with four right angles Kite A convex quadrilateral with two distinct pairs of consecutive sides the same length Rhombus A quadrilateral with all four sides the same length Congruent figures Figures that have the same shape and size, regardless of position or orientation Perpendicular bisector of a segment A line that is perpendicular to a segment and contains its midpoint Relationships Pythagorean Theorem If the lengths of the sides of a right triangle are a, b, c, with the side of length c opposite the right angle, then a 2 + b 2 = c 2. Converse of the Pythagorean Theorem If the sum of the squares of the lengths of two sides of a triangle equals the square of the length of the third side, then the triangle is a right triangle. Triangle Inequality The sum of the lengths of any two sides of a triangle is always greater than the length of the third side. Triangle Angle Sum Property The sum of the measures of the angles in a triangle is 180. Quadrilateral Angle Sum Property The sum of the measures of the angles in a quadrilateral is 360. Angle Sum for Regular Polygons The sum of the measures of the interior angles of a regular polygon with n sides is (n - 2)180. Base Angles of Isosceles Triangle Angles opposite congruent sides of an isosceles triangle are congruent. Student Master UNIT 3 Coordinate Methods 1 T161C UNIT 3 Coordinate Methods Key Geometric Ideas from Course 1 This unit builds on important geometric concepts and relationships developed in the Course 1 unit, Patterns in Shape. Specifically: Definitions Isosceles triangle a triangle with at least two sides of equal length Parallelogram a quadrilateral with opposite sides of equal length Rectangle a parallelogram with one right angle Kite a convex quadrilateral with two distinct pairs of consecutive sides the same length Rhombus a quadrilateral with all four sides the same length Congruent figures figures that have the same shape and size, regardless of position or orientation Perpendicular bisector of a segment a line that is perpendicular to a segment and contains its midpoint Square a rhombus with one right angle Relationships Pythagorean Theorem If the lengths of the sides of a right triangle are a, b, c, with the side of length c opposite the right angle, then a 2 + b 2 = c 2. Converse of the Pythagorean Theorem If the sum of the squares of the lengths of two sides of a triangle equals the square of the length of the third side, then the triangle is a right triangle. Triangle Inequality The sum of the lengths of any two sides of a triangle is always greater than the length of the third side. Triangle Angle Sum Property The sum of the measures of the angles in a triangle is 180. Quadrilateral Angle Sum Property The sum of the measures of the angles in a quadrilateral is 360. Polygon Angle Sum Property The sum of the measures of the interior angles of a polygon with n sides is (n - 2)180. Base Angles of Isosceles Triangle Angles opposite congruent sides of an isosceles triangle are congruent. Side-Side-Side (SSS) congruence condition If three sides of a triangle are congruent to the corresponding sides of another triangle, then the two triangles are congruent. Side-Angle-Side (SAS) congruence condition If two sides and the angle between the sides of one triangle are congruent to the corresponding parts of another triangle, then the two triangles are congruent. Angle-Side-Angle (ASA) congruence condition If two angles and the side between the angles of one triangle are congruent to the corresponding parts of another triangle, then the two triangles are congruent. Opposite Angles Property of Parallelograms Opposite angles in a parallelogram are congruent. Condition ensuring a parallelogram If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Condition ensuring a rectangle If the diagonals of a parallelogram are the same length, then the parallelogram is a rectangle. Condition ensuring a square If a rectangle has one pair of consecutive sides the same length, then the rectangle is a square right triangle relationship For a right triangle with acute angles of measures 30 and 60, the length of the side opposite the 30 angle is half the length of the hypotenuse. The length of the side opposite the 60 angle is 3 times the length of the side opposite the 30 angle right triangle relationship For a right triangle with acute angles of measures 45, the length of the hypotenuse is 2 times the length of a leg of the right triangle.

13 U nit 3 Planning Guide Lesson Objectives On Your Own Assignments* Suggested Pacing Materials Lesson 1 A Coordinate Model of a Plane Use coordinates to represent points, lines, and geometric figures in a plane Develop and use coordinate representations of geometric ideas such as distance, slope, and midpoint to analyze properties of lines and shapes Design algorithms for programming calculators or computers to perform routine geometry-related computations Develop and use equations for circles in a coordinate plane Reason with general coordinates to establish properties of triangles, quadrilaterals, and circles After Investigation 1: A1 or A2, A3, A4 or A5, A6, choose two of C12 C15, choose two of R20 R22, E27 or E28, Rv33 Rv35 After Investigation 2: choose two of A7 A10, C16, choose two of R23 R25, choose one of E29 E31, Rv36 Rv38 After Investigation 3: A11, C17 C19, R26, E32, Rv39 Rv42 10 days (including assessment) Internet access to site referred to on page T162A CPMP-Tools interactive geometry software or similar software Computer access for pairs of students Computer and projection system Grid or graph paper Rulers Unit resources Lesson 2 Coordinate Models of Transformations Use coordinates to develop function rules modeling translations, line reflections, and rotations and size transformations centered at the origin Use coordinates to investigate properties of figures under one or more rigid transformations or under similarity transformations Explore the concept of function composition using successive application of two transformations After Investigation 1: A1, A3 A5, A6 or A7, R23, E29, Rv36 Rv38 After Investigation 2: A8 A10, C15, R24, R25, R26 or R27, E27, choose one of E30 E34, Rv39, Rv40 After Investigation 3: A11, A12 or A13, A14, choose one of C16 C19, C20, C21, C22, R28, E35, Rv41 Rv43 11 days (including assessment) Grid or graph paper Rulers CPMP-Tools interactive geometry software or similar software Computer and projection system Unit resources Lesson 3 Transformations, Matrices, and Animation Use coordinate rules for rotations about the origin to develop corresponding matrix representations Use coordinate rules for size transformations centered at the origin to develop corresponding matrix representations Use matrix representations of shapes and transformations to create simple animations involving rotations and size transformations After Investigation 1: A1, A2, A3 or A4, C7, C8, C9 or C10, R12 R14, Rv22, Rv23 After Investigation 2: A5, A6, C11, R15, E16, choose one of E17 E21, Rv24, Rv25 6 days (including assessment) Grid or graph paper Rulers CPMP-Tools interactive geometry software or similar software Computer and projection system Unit resources Unit 3 Lesson 4 Looking Back Review and synthesize the major objectives of the unit 3 days (including assessment) Grid or graph paper Rulers CPMP-Tools interactive geometry software Computer access for pairs of students Unit resources * When choice is indicated, it is important to leave the choice to the student. Note: It is best if Connections tasks are discussed as a whole class after they have been assigned as homework. Overview T161D