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TOWNSHIP OF UNION PUBLIC SCHOOLS MA 280 Honors Geometry Curriculum Guide 2012 1 Board Members See web site: twpunionschools.org 2 TOWNSHIP OF UNION PUBLIC SCHOOLS Administration See web site: twpunionschools.org 3 DEPARTMENT SUPERVISORS See web site: twpunionschools.org 4 Honors Geometry Curriculum Committee Steven Chaneski – Teacher of Mathematics 5 Table of Contents Title Page 1. Board Members/Administration/Department Supervisors/Curriculum Committee 2. Table of Contents 3. District Mission/Philosophy Statement/District Goals 4 Course Description/Recommended Texts 5 Course Proficiencies 6 Pacing Guide 7 Curriculum Units 8 Appendix: New Jersey Core Curriculum Content Standards 13 6 Mission Statement The Township of Union board of Education believes that every child is entitled to an education, designed to meet his or her individual needs, in an environment that is conductive to learning. State standards, federal and state mandates, and local goals and objectives, along with community input, must be reviewed and evaluated on a regular basis to ensure that an atmosphere of learning is both encouraged and implemented. Furthermore, any disruption to or interference with a healthy and safe educational environment must be addressed, corrected, or when necessary removed in order for the district to maintain the appropriate educational setting. Philosophy Statement The Township of Union Public School District, as a societal agency, reflects democratic ideals and concepts through its educational practices. It is the belief of the Board of Education that a primary function of the Township of Union Public School System is formulation of a learning climate conductive to the needs of all students in general, providing therein for individual differences. The school operates as a partner with the home and community. Statement of District Goals 7 Develop reading, writing, speaking, listening, and mathematical skills. Develop a pride in work and a feeling of self-worth, self-reliance, and self discipline. Acquire and use the skills and habits involved in critical and constructive thinking. Develop a code of behavior based on moral and ethical principals. To be able to work with others cooperatively. Acquire a knowledge and appreciation of the historical record of human achievement and failures and current societal issues. Acquire a knowledge and understanding of the physical and biological sciences. Efficient and effective participation in economic life and the development of skills to enter a specific field of work. Appreciate and understand literature, art, music, and other cultural activities. Develop an understanding of the historical and cultural heritage. Develop a concern for the proper use and/or preservation of natural resources. Develop basic skills in sports and other forms of recreation. Course Description Geometry Geometry students will focus on reasoning about two and three dimensional figures and their properties. They will develop facility with a broad range of ways of representing geometric ideas, including coordinates, transformations and vectors. Students will be making conjectures, proving theorems, and finding counterexamples to refute false claims. This course builds on geometry students encountered in middle school and extends some concepts addressed in Algebra I. Students will be able to understand how geometry relates to the real world and how often math is used in their lives today and in the future. Curriculum Units Unit 1: Unit 2: Unit 3: Unit 4: Unit 5: Geometric Representations Reasoning in Geometric Situations Congruence, Similarity, and Right Triangle Trigonometry Circles Three – Dimensional Geometry Recommended Texts Geometry, Ron Larson, McDougal Littell, 2007 Flatland, Edwin A. Abbott, Dover Thrift Edition, 1992 8 Course Proficiencies Unit 1: Geometric Representations: SWBAT: - Understand and solve problems using geometric representations. - Show position and represent motion in the coordinate plane using vectors Unit 2: Reasoning in Geometric Situations: SWBAT: - Use the laws of inductive and deductive reasoning to draw conclusions. - Prove geometric theorems and statements using formal proofs. - Apply properties of parallel and perpendicular lines. - Understand and apply properties of triangles - Make geometric constructions. Unit 3: Congruence, Similarity, and Right Triangle trigonometry SWBAT: - Understand congruence of triangles and apply properties for problem solving. - Understand similarity, right triangles and trigonometry. - Understand and apply the properties of quadrilaterals to solve problems. - Understand and perform transformations. Unit 4: Circles SWBAT: - Understand and apply properties of circles to solve problems. Unit 5: Three-Dimensional Geometry SWBAT: - Understand and calculate surface area and volume of figures. 9 Pacing Guide- Course Content Number of Days Unit 1: Geometric Representations 20 Unit 2: Reasoning in Geometric Situations 20 Unit 3: Congruence, Similarity and Right Triangle Trigonometry 90 Unit 4: Circles 30 Unit 5: Three-Dimensional Geometry 20 10 Curriculum Units Unit 1: Geometry Representations: Essential Questions Instructional Objectives/ Skills and Benchmarks Activities Assessments Identify objects in the classroom that can be represented by geometric shapes. Calculate distances on the coordinate plane. Find distances and midpoints. Students, in their study of algebra, learned ways to find the point of intersection of two lines. They should connect this to problems associated with specific geometric representations (e.g., finding the point at which a perpendicular bisector intersects a line segment on the coordinate plane). Enrichment: Study coordinates in space as an extension of 3D geometry.. How can geometric shapes be used to represent the real world. How can coordinates be used to describe and analyze geometric objects? Model points, lines and planes. (G-CO.1) Use segments, midpoints, distance formula and angles to solve problems (G-CO.1) Use coordinates and algebraic techniques to describe and define geometric figures and to interpret, represent, and verify geometric relationships. (G-GPE.4 – 7) How are vectors used? 11 Show position and represent motion in the coordinate plane using vectors (G-CO.2) Students will perform transformations using vectors. Extended Constructed Response (ECR): Given the coordinates of the vertices of quadrilateral, determine whether it is a parallelogram. SCR: The rectangular coordinates of three points in a plane are Q(-3, -1), R(-2, 3), and S(1, -3). A fourth point T is chosen so that the magnitude of vector ST is equal to twice the magnitude of vector QR. What is the ycoordinate of T? Calculate positions using vector notation. Unit 2: Reasoning in Geometric Situations: Essential Questions Instructional Objectives/ Skills and Benchmarks Activities Assessments *How can mathematical reasoning help you make generalizations? Complete a proof for an algebraic solution. Prove the same statement (e.g., the diagonals of a rectangle are congruent) using a proof. ECR: Write the inverse, converse, and contrapositive of an advertising slogan and identify which of these statements are logically equivalent. Technology Integration: Use Classzone.com to analyze angle relationships. ECR: Prove that the diagonals of a rhombus are perpendicular. (This might be done using Euclidean methods or coordinate geometry). ECR: Given a line and a point, construct a line through the point that is perpendicular to the original line. Justify the steps of the construction. (Note that students might use tools of their choice to answer this question.) MC: The vertices of a triangle PQR are the points P(1, 2), Q(4, 6), and R(-4, 12). Which one of the following statements about triangle PQR must be true? *How do you know when you have proven something? *How can geometric/algebraic relationships best be represented and verified? 12 Use reasoning and some form of proof to verify or refute conjectures and theorems about lines, angles and polygons. (G-CO.9) Students should be able to: Make, test, and confirm or refute geometric conjectures using a variety of methods, including technology. Demonstrate through example or explanation how indirect reasoning can be used to establish a claim. Recognize flaws or gaps in the reasoning supporting an argument. Develop a counterexample to refute an invalid statement. Determine the correct conclusions based on interpreting a theorem in which necessary or sufficient conditions in the theorem or hypothesis are satisfied. (G-GPE.4 – 7) Basic concepts include the following: Vertical Angles Supplementary and complementary angles Linear pairs of angles Angles formed by parallel lines (alternate interior and corresponding angles) Perpendicular bisector Angle bisector Midpoint and perpendicular bisector of a segment Perpendicular or parallel lines (G-CO.9-11) A. PQR is a right triangle with the right angle at P. B. PQR is a right triangle with the right angle at Q. C. PQR is a right triangle with the right angle at R. D. PQR is not a right triangle. Unit 3: Congruence, Similarity, and Right Triangle Trigonometry Essential Questions Instructional Objectives/ Skills and Benchmarks Activities Assessments What makes shapes alike and different? Determine and apply conditions that guarantee congruence of triangles. How are similarity, congruence, and symmetry related? Analyze special segments within triangles. ECR: Use congruent triangles to prove that the bisector of the angle opposite the base of an isosceles triangle is the perpendicular bisector of the base. Identify and apply conditions that are sufficient to guarantee similarity of triangles. Find the scale factor for two congruent polygons. What situations can be analyzed using transformations and symmetries? Extend the concepts of similarity and congruence to other polygons in the plane. How can transformations be described mathematically? Show how similarity of right triangles allows the trigonometric functions sine, cosine, and tangent to be properly defined as ratios of sides. How can we describe and analyze iterative geometric patterns? 13 Analyze properties of parallelograms and other quadrilaterals. (G-C0.6-8, G-SRT.1-11) Interdisciplinary Connections: Use similarity to calculate the measures of corresponding parts of similar figures, and apply similarity in a variety of problem solving contexts within mathematics and other disciplines. Classroom Task: Have each student draw their own triangle and then construct a line parallel to one side. Ask them how the smaller triangle is related to the larger one. Have them explain and justify their conjectures. Design the perfect bedroom and make a scale drawing. Enrichment: Derive, interpret, and use the identity sin2θ + cos2θ = 1 for angles θ between 0° and 90° as a special representation of the Pythagorean Theorem. Performance Assessment Task: Measure the heights of a number of objects and the lengths of their shadows on a sunny day. Create a scatterplot of the data, and write an equation that relates shadow length to object height. Write a brief report of your work and your findings, describing how you used similar triangles. (NCTM Navigating Through Geometry) SCR: The maximum slope for handicapped ramps in new construction is 1/12. What angle will such a ramp make with the ground? Unit 4: Circles Essential Questions What geometric relationships can be found in circles Instructional Objectives/ Skills and Benchmarks Activities Assessments To investigate the intersecting chords theorem and other segment relationships using the computer Prove that if a radius if a circle is perpendicular to a chord of the circle, then it bisects the chord Find approximate area of a running track consisting of a rectangle and two semi-circles Show that a triangle inscribed on the diameter of a circle is a right triangle Technology Integration: Use a graphing calculator to graph equations of circles An apple pie is cut into six equal slices. The diameter of the pie is 10 inches. Find the approximate arc length of one slice of pie and the approximate area of the top of one slice of pie Find the equation of a circle with diameter 20in and center at the (3,-1) 14 To recognize and apply the definitions and properties of a circle To verify and apply relationships associated with a circle To determine the length of line segments and arcs, the measure of angles, and the areas of shapes that they define in complex geometric drawings To relate the equation of a circle to its characteristics and its graph (G-C.1-4) Unit 5: Three-Dimensional Geometry Essential Questions Instructional Objectives/ Skills and Benchmarks Activities Assessments How are 1-, 2-, and 3dimensional shapes related? How can 3-dimensional objects be represented in 2 dimensions? How can 3D objects be measured? 15 To determine the surface area and volume of solid figures, recognizing and using relationships among volumes of common solids To create, interpret, and use twodimensional representations of 3dimensional shapes To analyze cross-sections of basic three-dimensional objects and identify the resulting shapes To describe the characteristics of 3-dimensional objects traced out when a one- or two- dimensional figure is rotated about an axis To analyze all possible relationships among two or three planes in space and identify their intersections (enrichment topics not included in CCCS) Read Flatland and discuss how 1D, 2D, 3D and 4D world are related. Use appropriate formula to calculate 3D measures. Use a clear transparency to construct a net of a 3-D figure and then make the object Use different shaped boxes to show how they are different or the same from the mathematical planes that they represent. Complete a project based on Flatland to evaluate the themes of the book. Find the surface area of a square pyramid whose sides all have length 2 cm. Find the volume of a cone with radius 2in, and height 4in Sketch the figure formed of blocks from perspective views using isometric dot paper. Describe all possible results of the intersection of a plane with a cube A triangle has vertices at (0,0), (2,0), and (0,10). What is the volume of the shape generated when this triangle is rotated about the x-axis? New Jersey Common Core State Standards Geometry Congruence Experiment with transformations in the plane G.CO.1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. G-CO.2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G-CO.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G-CO.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G-CO.5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions G-CO.6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. G-CO.7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. G-CO.8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Prove geometric theorems 16 G-CO.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. G-CO.10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. G-CO.11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Make geometric constructions G-CO.12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. G-CO.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Similarity, right triangles and Trigonometry Understand similarity in terms of similarity transformations G-SRT.1. Verify experimentally the properties of dilations given by a center and a scale factor: A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G-SRT.2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G-SRT.3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity G-SRT.4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G-SRT.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 17 Define trigonometric ratios and solve problems involving right triangles G-SRT.6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G-SRT.7. Explain and use the relationship between the sine and cosine of complementary angles. G-SRT.8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ★ Apply trigonometry to general triangles G-SRT.9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. G-SRT.10. (+) Prove the Laws of Sines and Cosines and use them to solve problems. G-SRT.11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). Circles Understand and apply theorems about circles G-C.1. Prove that all circles are similar. G-C.2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. G-C.3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. G-C.4. (+) Construct a tangent line from a point outside a given circle to the circle. Find arc lengths and areas of sectors of circles 18 G-C.5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Expressing Geometric Properties with equations Translate between the geometric description and the equation for a conic section G-GPE.1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. G-GPE.2. Derive the equation of a parabola given a focus and directrix. G-GPE.3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. Use coordinates to prove simple geometric theorems algebraically G-GPE.4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). G-GPE.5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). G-GPE.6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. G-GPE.7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. 19