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TOWNSHIP OF UNION PUBLIC SCHOOLS MA 202 Geometry Curriculum Guide 2014 Curriculum Readopted August 2015 Table of Contents Board Members………………………………………………………………………………………………………………….....Page 2 Administration……………………………………………………………………………………………………….…………..... Page 3 Department Supervisors………………………………………………………………………………………………….……..... Page 4 Curriculum Committee……………………………………………………………………………………………………...….....Page 5 District Mission Statement………………………………………………………………………………………….…………..... Page 6 District Philosophy Statement………………………………………………………………..…………………….…………..... Page 6 District Goals………………………………………………………………………………………….…………........................... Page 7 Course Description………………………………………………………………………………………….…………...................Page 8 Recommended Textbook………………………………………………………………………………………............................. Page 9 Pacing Guide………………………………………………………………………………………….………….......................... Page 10 Curriculum Units……………………………………………...…………………………………….…………........................... Page 11 Appendix: Common Core State Standards, High School: Geometry………………...………….…………........................... Page 20 Page | 1 TOWNSHIP OF UNION PUBLIC SCHOOLS Board Members Mr. Francis “Ray” Perkins, President Mr. Richard Galante, Vice President Ms. Versie McNeil Mr. David Arminio Ms. Susana Cooley Mr. Guy Francis Ms. Linda Gaglione Mr. Thomas Layden Mr. Vito Nufrio Page | 2 TOWNSHIP OF UNION PUBLIC SCHOOLS Administration Chief School Administrator…………………………………………………………….………………………….. Dr. Patrick Martin Assistant Superintendent……………………………………………..…………………………..…………… Mr. Gregory A. Tatum Assistant Superintendent………………………………………………………………………………………...…. Dr. Noreen Lishak Director of Technology & Student Information ……………………………………………………….………………. Ms. Ann Hart Director of Grades PreK-5 Curriculum/Testing………………………………………..……………………..… Ms. Tiffany Moutis Director of Special Services…………………………………………………………………………….………………. Ms. Kim Conti Director of Athletics, Physical Education, and Nurses………………………………………………………..…….. Ms. Linda Ionta Page | 3 TOWNSHIP OF UNION PUBLIC SCHOOLS Department Supervisors Mathematics: K-6, Science: K-5……………………………………………………………………………..…….. Ms. Deborah Ford Social Studies: 7-12, Business………………………………………………………………………………...……. Ms. Libby Galante English: K-6, Social Studies: K-6………………………………………………………………………………… Mr. Robert Ghiretti Science: 6-12…………………………………………………………………………………………….……… Ms. Maureen Guilfoyle Career Education, World Languages, ESL, Computers, G&T………………………………….……………. Ms. Yvonne Lorenzo English: 7-12, Library/Media………………………………………………………………………………..…….. Ms. Mary Malyska Mathematics: 7-12……………………………………………………………………………………..…………. Mr. Jason Mauriello Art, Music………………………………………………………………………………………………………………… Mr. Ron Rago Page | 4 TOWNSHIP OF UNION PUBLIC SCHOOLS Curriculum Committee Academic Area Mr. Jason Mauriello – Supervisor Ms. Ana Lytle Ms. Annie Polinger Page | 5 District 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 conducive 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. District 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 the formulation of a learning climate conducive to the needs of all students in general, providing therein for individual differences. The school operates as a partner with the home and community. Page | 6 Statement of District Goals 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. Page | 7 TOWNSHIP OF UNION PUBLIC SCHOOLS Course Description Geometry Geometry students will focus on reasoning about two and three dimensional figures and their properties. They will learn about relationships between geometric figures, including polygons and circles. Students will be making conjectures, proving theorems, and finding counterexamples to refute false conjectures. This course expands on geometry concepts encountered in middle school and applies some concepts addressed in Algebra I. Students will gain an appreciation for the importance of geometry in their everyday lives. Unit 1: Unit 2: Unit 3: Unit 4: Unit 5: Unit 6: Geometric Representations and Reasoning Congruence, Relationships Within Triangles, Quadrilaterals Similarity, Right Triangles, Trigonometry Circles Area, Geometric Probability, Surface Area, Volume Transformations Page | 8 Recommended Textbook Charles, R. I. (2011). Geometry. Boston, Mass: Pearson Prentice Hall. Page | 9 Pacing Guide Content Number of Days Unit 1: Geometric Representations and Reasoning 40 Unit 2: Congruence, Relationships Within Triangles, Quadrilaterals 40 Unit 3: Similarity, Right Triangles, Trigonometry 40 Unit 4: Circles 25 Unit 5: Area, Geometric Probability, Surface Area, Volume 20 Unit 6: Transformations 10 Page | 10 Curriculum Units Unit 1: Geometric Representations and Reasoning Unit 2: Congruence, Relationships Within Triangles, Quadrilaterals Unit 3: Similarity, Right Triangles, Trigonometry Unit 4: Circles Unit 5: Area, Geometric Probability, Surface Area, Volume Unit 6: Transformations Page | 11 Unit 1: Geometric Representations and Reasoning Essential Questions *How can coordinates be used to describe and analyze geometric objects? Instructional Objectives/ Skills and Benchmarks (CPIs) G-CO-1 Use coordinates and algebraic techniques to describe and define geometric figures and to interpret, represent, and verify geometric relationships. Activities Assessments 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). Given the coordinates of the vertices of quadrilateral, determine whether it is a parallelogram. Study coordinates in space as an extension of 3D geometry. Interdisciplinary Connections (History): Interpret the role of the parallel postulate as the key postulate in the formulation of Euclidean geometry. Page | 12 Unit 1: Geometric Representations and Reasoning, continued Essential Questions *How can you decide if a conjecture is false? Instructional Objectives/ Skills and Benchmarks (CPIs) G-CO-9 Use reasoning and some form of proof to verify or refute conjectures and theorems through the following: -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. Activities Assessments Use online textbook or SmartBoard to create and test conjectures about relationships. For example, use the SmartBoard to test the impact on one vertical angle if the other angle is changed. Determine if the conjecture is true or false; if false, give a counterexample: Supplementary angles form a linear pair. Page | 13 Unit 1: Geometric Representations and Reasoning, continued Essential Questions *How can you determine if two angles have a special relationship? Instructional Objectives/ Skills and Benchmarks (CPIs) G-CO-9 Develop and test conjectures about angles and lines, including 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 *How can you use a compass to perform constructions? G-CO-12,13 Analyze, execute, explain, and apply simple geometric constructions; using a variety of methods (e.g., straightedge and compass, patty/tracing paper, or technology), students should be able to execute the following constructions: Activities Assessments Use online textbook or SmartBoard to create and test conjectures about relationships between angles. For example, use the SmartBoard to explore the relationships among angles on two parallel lines intersected by a transversal. Draw a pof parallel line and a transversal; label the eight angles formed; identify a pair of angles with each of the following relationships: Students will use a compass and a straight-edge to perform constructions. Given a line and a point not on the line, construct a line through the point that is perpendicular to the original line. -Vertical Angles -Supplementary Angles -Linear Pair -Alternate Interior Angles -Corresponding Angles -Angle bisector -Midpoint and perpendicular bisector of a segment Page | 14 Unit 2: Congruence, Relationships Within Triangles, Quadrilaterals Essential Questions Instructional Objectives/ Skills and Benchmarks (CPIs) G-CO-8 Determine and apply conditions that guarantee congruence of triangles. Activities Assessments Use online textbook or SmartBoard to explore conditions under which two triangles may or may not be congruent. Write a formal two-column proof proving two triangles congruent using one of the following postulates: *How can you find the measures of angles inside polygons? G-CO-10 Develop and use the triangle sum and angle measure theorems for polygons and the triangle inequality theorem. Have students find the pattern for the sum of interior angles in a convex polygon by extending the triangle sum theorem. *How can you use the properties of special segments inside a triangle to identify them? G-CO-9 Identify the special parts of triangles and use their properties to solve problems, including the following: Have students find the circumcenter of a If the measure of the midsegment of a right triangle by graphing. triangle is 2x + 3 and the measure of the side parallel to the midsegment is 6x – 4, find x. *How can you determine if two triangles are congruent? -SSS -SAS -ASA -AAS -HL Find the sum of the measures of the interior angles of a convex hexagon. -Median -Altitude -Perpendicular Bisector -Angle Bisector -Midsegment *How can you determine if a quadrilateral is a parallelogram? G-CO-11 Develop and test conjectures about quadrilaterals, using minimal conditions for a shape to be a special quadrilateral. Solve algebraic problems using properties of a parallelogram. Write a paragraph explaining the differences between the four types of parallelograms (parallelogram, rectangle, rhombus, square). Page | 15 Unit 3: Similarity, Right Triangles, Trigonometry Essential Questions *How can you tell if two triangles are similar? *How can you find the lengths of the sides of similar figures? *What is the best method to use for determining side lengths and/or angle measures in a given right triangle? Instructional Objectives/ Skills and Benchmarks (CPIs) G-CO-7;G-SRT-2,3 Identify and apply conditions that are sufficient to guarantee similarity of triangles (SAS~,SSS~,AA~) Activities Assessments Write a paragraph proof showing the postulate used to prove two triangles similar. The ratio of the perimeter of ∆U to the perimeter of ∆V is 1:2. G-SRT-4,5,6 Set up and solve proportions based on similar figures and right triangles with altitudes. Students will design a scale drawing of a bedroom, which must include a minimum of 4 geometric shapes, labeled with actual dimensions. G-SRT-7,8,9,10,11 Use Pythagorean theorem, special right triangles, and trigonometry to solve for side lengths and/or angle measures in right triangles. To solve application problems involving The maximum slope for handicapped angles of elevation an angles of ramps in new construction is 1/12. What depression. angle will such a ramp make with the ground? 2 4 U 5 X y V Z 1. 2. Determine if the triangles are similar. What is the value of x + y? Page | 16 Unit 4: Circles Essential Questions *How do mathematical representations assist you in solving problems? *What geometric relationships can be found in circles? *How can you find the measures of angles and arcs using circles? Instructional Objectives/ Skills and Benchmarks (CPIs) G-C-1,2 To recognize and apply the definitions and properties of a circle Activities Assessments To use online textbook or SmartBoard to investigate properties of circles Prove that if a radius of a circle is perpendicular to a chord of the circle, then it bisects the chord G-C-4 To verify and apply relationships associated with tagents to a circle To investigate the intersecting chords theorem and other segment relationships using the online textbook or SmartBoard If the distance from a point outside a circle to the point of tangency is 8in and the radius is 6in, find the distance from the point to the center of the circle. G-C-2,3,4,5 To determine the relationships between angles and arcs in circles: Use the smartboard to show the relationships between arc and angle measures as you shift the vertex from the center to the circle to inside and to outside. Show that a triangle inscribed on the diameter of a circle is a right triangle Use a graphing calculator to graph the equations of circles. Find the equation of a circle with diameter 20in and center at (3,-1). -Central Angles -Inscribed Angles -Vertex Inside -Vertex Outside *How can you determine the equation of a circle given a graph? G-GPE-1 To relate the equation of a circle to its characteristics and its graph Page | 17 Unit 5: Area, Geometric Probability, Surface Area, Volume Essential Questions *How are 1-, 2-, and 3dimensional shapes related? Instructional Objectives/ Skills and Benchmarks (CPIs) G-GMD-3 Calculate the areas of different types of polygons and the probability that a randomly selected point lies in the shaded region of the figure. Activities Assessments Review basic probability, using examples of selecting marbles, cards, flipping coins, etc. Given a square inscribed in a circle, find the probability that a randomly selected point in the circle is not in the square. *How can 3-dimensional objects be represented in 2 dimensions? G-GMD-4;G-MG-1, -To create, interpret, and use 2- Use manipulatives to allow students to dimensional representations of explore the nets of 3-dimensional 3-dimensional shapes figures. *How can 3-dimensional objects be measured? G-GMD-1,3 -To determine the surface area and volume of solid figures. Measure 3-dimensional figures and find the surface area and volume. Find the surface area of a square pyramid whose sides have length 2cm. Find the volume of a cone with radius 2in and height 4in. Page | 18 Unit 6: Transformations Essential Questions *How can you transform a figure? Instructional Objectives/ Skills and Benchmarks (CPIs) G-CO-2,4,5,G-SRT-1 Identify and find translations, reflections, rotations, and dilations of figures *What are the different types of symmetry? G-CO-2,3 Determine if a figure has line symmetry, rotational symmetry, or both Have student draw figures that have the following symmetries: G-CO-5,6 Explore tessellations though hands-on activities Give students dot paper and allow them to create a tessellation. *Can a given figure tesselate? Activities Assessments Use the smartbord to visually show all transformations. Determine the scale factor of a given dilation. How many capital letters of the alphabet have rotational symmetry? -No symmetry -Line symmetry -Rotational symmetry -Line & Rotational symmetry Do octagons tessellate? Explain how you can mathematically determine if a figure can tessellate? Page | 19 Common Core State Standards High School: Geometry 1. Congruence o Experiment with transformations in the plane CCSS.Math.Content.HSG-CO.A.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. CCSS.Math.Content.HSG-CO.A.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). CCSS.Math.Content.HSG-CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. CCSS.Math.Content.HSG-CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. CCSS.Math.Content.HSG-CO.A.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. o Understand congruence in terms of rigid motions CCSS.Math.Content.HSG-CO.B.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. CCSS.Math.Content.HSG-CO.B.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. CCSS.Math.Content.HSG-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Page | 20 o Prove geometric theorems CCSS.Math.Content.HSG-CO.C.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. CCSS.Math.Content.HSG-CO.C.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. CCSS.Math.Content.HSG-CO.C.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. o Make geometric constructions CCSS.Math.Content.HSG-CO.D.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. CCSS.Math.Content.HSG-CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. 2. Similarity, Right Triangles, & Trigonometry o Understand similarity in terms of similarity transformations CCSS.Math.Content.HSG-SRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor: CCSS.Math.Content.HSG-SRT.A.1a 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. CCSS.Math.Content.HSG-SRT.A.1b The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Page | 21 CCSS.Math.Content.HSG-SRT.A.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. CCSS.Math.Content.HSG-SRT.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. o Prove theorems involving similarity CCSS.Math.Content.HSG-SRT.B.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. CCSS.Math.Content.HSG-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. o Define trigonometric ratios and solve problems involving right triangles CCSS.Math.Content.HSG-SRT.C.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. CCSS.Math.Content.HSG-SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles. CCSS.Math.Content.HSG-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. o Apply trigonometry to general triangles CCSS.Math.Content.HSG-SRT.D.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. CCSS.Math.Content.HSG-SRT.D.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems. CCSS.Math.Content.HSG-SRT.D.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). Page | 22 3. Circles o Understand and apply theorems about circles CCSS.Math.Content.HSG-C.A.1 Prove that all circles are similar. CCSS.Math.Content.HSG-C.A.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. CCSS.Math.Content.HSG-C.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. CCSS.Math.Content.HSG-C.A.4 (+) Construct a tangent line from a point outside a given circle to the circle. o Find arc lengths and areas of sectors of circles CCSS.Math.Content.HSG-C.B.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. 4. Expressing Geometric Properties with Equations o Translate between the geometric description and the equation for a conic section CCSS.Math.Content.HSG-GPE.A.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. CCSS.Math.Content.HSG-GPE.A.2 Derive the equation of a parabola given a focus and directrix. CCSS.Math.Content.HSG-GPE.A.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. Page | 23 o Use coordinates to prove simple geometric theorems algebraically CCSS.Math.Content.HSG-GPE.B.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). CCSS.Math.Content.HSG-GPE.B.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). CCSS.Math.Content.HSG-GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. CCSS.Math.Content.HSG-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. 5. Geometric Measurement & Dimension o Explain volume formulas and use them to solve problems CCSS.Math.Content.HSG-GMD.A.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. CCSS.Math.Content.HSG-GMD.A.2 (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. CCSS.Math.Content.HSG-GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. o Visualize relationships between two-dimensional and three-dimensional objects CCSS.Math.Content.HSG-GMD.B.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Page | 24 6. Modeling with Geometry o Apply geometric concepts in modeling situations CCSS.Math.Content.HSG-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). CCSS.Math.Content.HSG-MG.A.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). CCSS.Math.Content.HSG-MG.A.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Page | 25