Course Description: This course is designed for students who have successfully completed the standards for TJ Math 1. Students continue the study of geometric topics, with a focus on transitioning to algebraic methods for solving problems. Topics of study include transformations, right triangle trigonometry, area of polygons and circles, surface area and volume, vectors in 2D (including component form and unit vector topics), solving equations and inequalities, linear systems up to 3D, and basic matrix algebra. Use of technology is integrated throughout the course. The process standard focus will be representation. The process standard focus will be reasoning. Upon completion of this course, students take the SOL end-of-course test in Geometry. This set of standards includes emphasis on two- and three-dimensional reasoning skills, coordinate and transformational geometry, and the use of geometric models to solve problems. A variety of applications and some general problem-solving techniques, including algebraic skills, shall be used to implement these standards. Appropriate technology tools will be used to assist in teaching and learning. Any technology that will enhance student learning shall be used. Note: Because topics are grouped in a way to emphasize algebraic methods utilizing a variety of geometric topics, standards appear throughout various units. The same essential understandings apply. Therefore, once a standard has been defined and its appropriate essential understandings stated, the standard will implicitly appear throughout this document. page 1 of 17
Unit of Study / Textbook Correlation Unit 1: Lines, Distances and Coordinate Proofs Prentice Hall Geometry 1.7, 3.7, 3.8, 6.7, 6.8, 6.9 Objectives Standard 2: USE ANGLE RELATIONSHIPS TO DETERMINE IF TWO LINES ARE PARALLEL The student will use the relationships between angles formed by two lines cut by a transversal to a) determine whether two lines are parallel; b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs; and c) solve real-world problems involving angles formed when parallel lines are cut by a transversal. Parallel lines intersected by a transversal form angles with specific relationships. Some angle relationships may be used when proving two lines intersected by a transversal are parallel. The Parallel Postulate differentiates Euclidean from non-euclidean geometries such as spherical geometry and hyperbolic geometry. Resources and Activities Benchmark 2.b Verify Parallelism Using Algebraic, Coordinate, and Deductive Methods The student will use the relationships between angles formed by two lines cut by a transversal to verify the parallelism, using algebraic and coordinate methods as well as deductive proofs. Indicator 2.b.1 The student will verify whether two lines are parallel using algebraic and coordinate methods. Benchmark 2.c Solve real-world problems involving angle relationships The student will use the relationships between angles formed by two lines cut by a transversal to solve real-world problems involving angles formed when parallel lines are cut by a transversal. Indicator 2.c.1 The student will solve real world problems with intersecting and parallel lines in a plane. page 2 of 17
Standard 3: SOLVE PROBLEMS INVOLVING SYMMETRY AND TRANSFORMATION The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include a) investigating and using formulas for finding distance, midpoint, and slope; b) applying slope to verify and determine whether lines are parallel or perpendicular; c) investigating symmetry and determining whether a figure is symmetric with respect to a line or a point; and d) determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Transformations and combinations of transformations can be used to describe movement of objects in a plane. The distance formula is an application of the Pythagorean Theorem. Geometric figures can be represented in the coordinate plane. Techniques for investigating symmetry may include paper folding, coordinate methods, and dynamic geometry software. Parallel lines have the same slope. The product of the slopes of perpendicular lines is -1. The image of an object or function graph after an isomorphic transformation is congruent to the preimage of the object. Benchmark 3.a Investigate & Use Formulas for Finding Distance, Midpoint, & Slope The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include investigating and using formulas for finding distance, midpoint, and slope. Indicator 3.a.1 The student will use the midpoint formula to find the coordinates of the midpoint. Indicator 3.a.2 The student will use a formula to find the slope of a line. Indicator 3.a.3 The student will apply the distance formula to find the length of a line segment. Indicator 3.a.4 The student will derive the distance formula. Benchmark 3.b Apply Slope to Verify and Determine Parallelism and Perpendicularity The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include applying slope to verify and determine whether lines are parallel or perpendicular. Indicator 3.b.1 The student will compare slopes to determine parallelism or perpendicularity. Indicator 3.b.2 The student will write equations of parallel and perpendicular lines. page 3 of 17
Standard 6: PROVE TWO TRIANGLES ARE CONGRUENT USING A VARIETY OF METHODS The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs. Congruence has real-world applications in a variety of areas, including art, architecture, and the sciences. Congruence does not depend on the position of the triangle. Concepts of logic can demonstrate congruence or similarity. Congruent figures are also similar, but similar figures are not necessarily congruent. Benchmark 6.a Prove Triangle Congruence Using Algebraic/Coordinate/Deductive Methods The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs. Indicator 6.a.2 The student will use coordinate methods to prove two triangles are congruent. Indicator 6.a.3 The student will use algebraic methods to prove two triangles are congruent. Standard 7: PROVE TWO TRIANGLES ARE SIMILAR USING A VARIETY OF METHODS The student, given information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate methods as well as deductive proofs. Similarity has real-world applications in a variety of areas, including art, architecture, and the sciences. Similarity does not depend on the position of the triangle. Congruent figures are also similar, but similar figures are not necessarily congruent. Benchmark 7.a Prove Triangle Similarity Using Algebraic/Coordinate/Deductive Methods The student, given information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate methods as well as deductive proofs. Indicator 7.a.2 The student will use algebraic methods to prove that triangles are similar. Indicator 7.a.3 The student will use coordinate methods to prove that triangles are similar. Standard 11: INVESTIGATE AND SOLVE PROBLEMS INVOLVING CIRCLES The student will use angles, arcs, chords, tangents, and secants to a) investigate, verify, and apply properties of circles; b) solve real-world problems involving properties of circles; and page 4 of 17
Many relationships exist between and among angles, arcs, secants, chords, and tangents of a circle. All circles are similar. A chord is part of a secant. Real-world applications may be drawn from architecture, art, and construction Benchmark 11.a Investigate, Verify, and Apply Properties of Circles The student will use angles, arcs, chords, tangents, and secants to investigate, verify, and apply properties of circles. Indicator 11.a.4 The student will verify properties of circles using algebraic and coordinate methods. Vocabulary: midpoint formula, distance formula, parallel lines, perpendicular lines, coordinate proof. page 5 of 17
Unit of Study / Textbook Correlation Unit 2: Solving Systems of Equations and Matrices Objectives Standard 3: THE STUDENT WILL REVIEW SOLVING SYSTEMS OF EQUATIONS GRAPHICALLY, BY SUBSTITUTION AND LINEAR COMBINATION FROM ALGEBRA 1. Benchmark 3.e Use Matrices to Organize and Manipulate Data The student will use matrices to organize and manipulate data, including matrix addition, subtraction, and scalar multiplication. Indicator 3.e.1 The student will represent data from practical problems in matrix form. Indicator 3.e.2 The student will calculate the sum or difference of two given matrices. Indicator 3.e.3 The student will calculate the product of a scalar and a matrix. Standard 5: APPLY THE TRIANGLE INEQUALITY PROPERTIES TO SOLVE PRACTICAL PROBLEMS The student, given information concerning the lengths of sides and/or measures of angles in triangles, will a) order the sides by length, given the angle measures; b) order the angles by degree measure, given the side lengths; c) determine whether a triangle exists; and d) determine the range in which the length of the third side must lie. These concepts will be considered in the context of real-world situations. The longest side of a triangle is opposite the largest angle of the triangle and the shortest side is opposite the smallest angle. In a triangle, the length of two sides and the included angle determine the length of the side opposite the angle. In order for a triangle to exist, the length of each side must be within a range that is determined by the lengths of the other two sides. Resources and Activities page 6 of 17
Benchmark 5.f Investigate Special Lines/Segments/Points Related to Triangles The student will investigate special lines, segments, and points related to triangles. Indicator 5.f.4 The student will find the coordinates of the points of concurrency in a triangle. TJHSST Expanded POS The student will find the product of matrices by hand and with a calculator. The student will represent and solve a system of linear equations in two variables in matrix form The student will find the inverse of a 2 x 2 matrix by hand and with a graphing calculator. The student will find the determinant of a 2 x 2 matrix by hand and with a graphing calculator The student will identify the identity matrix and its properties. The student will solve real-world problems using a variety of techniques. Vocabulary: matrix, scalar, determinant, linear combination, substitution, system of equations page 7 of 17
Unit of Study / Textbook Correlation Objectives Resources and Activities Unit 3: Right Triangles and Trigonometry Prentice Hall Geometry 8.1 8.4 Standard 8: SOLVE REAL-WORLD PROBLEMS INVOLVING RIGHT TRIANGLES The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of special right triangles, and right triangle trigonometry. The Pythagorean Theorem is essential for solving problems involving right triangles. Many historical and algebraic proofs of the Pythagorean Theorem exist. The relationships between the sides and angles of right triangles are useful in many applied fields. Some practical problems can be solved by choosing an efficient representation of the problem. Another formula for the area of a triangle is Area = ½ ABsinC The ratios of side lengths in similar right triangles (adjacent/hypotenuse or opposite/hypotenuse) are independent of the scale factor and depend only on the angle the hypotenuse makes with the adjacent side, thus justifying the definition and calculation of trigonometric functions using the ratios of side lengths for similar right triangles. Benchmark 8.a Solve Real-World Problems Involving Right Triangles The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of special right triangles, and right triangle trigonometry. Indicator 8.a.1 The student will, given two side lengths, use the Pythagorean Theorem to find the third. Indicator 8.a.2 The student will, given three side lengths, determine if the triangle formed is right, obtuse or acute. Indicator 8.a.3 The student will solve for missing lengths using properties of 45-45-90 triangle. page 8 of 17
Indicator 8.a.4 The student will solve for missing lengths using properties of 30-60-90 triangles. Indicator 8.a.6 The student will use properties and right triangle trigonometry to solve problems. Indicator 8.a.7 The student will explain/use the relationship between sine and cosine. Indicator 8.a.8 The student will prove the Pythagorean Theorem and its converse. Indicator 8.a.9 The student will find exact values for trigonometric ratios in a right triangle. Indicator 8.a.10 The student will solve problems using angle of elevation and angle of depression. Vocabulary: Pythagorean Theorem, Pythagorean triple, special right triangles, sine, cosine, tangent, angle of elevation, angle of depression page 9 of 17
Unit of Study / Textbook Correlation Objectives Resources and Activities Unit 4: Additional Trigonometric Topics Vectors Prentice Hall Geometry 8.5, Vector Packet Benchmark 3.f Investigate Vectors and Their Applications Investigate vectors and their applications Indicator 3.f.1 The student will identify, draw, and label vectors using appropriate notation. Indicator 3.f.2 The student will express the addition and scalar multiplication of vectors. Indicator 3.f.3 The student will represent vectors as matrices and determine resultants. Indicator 3.f.4 The student will apply vectors to practical situations. TJHSST Expanded POS The student will solve problems using the laws of sines and cosines. The student will apply the ambiguous case. The student will resolve vectors into their components given the magnitude and direction. The student will learn and use vector notation The student will graph two-dimensional vectors and resultant vectors The student will define a two-dimensional unit vector and find unit vectors The student will perform vector addition and scalar multiplication with twodimensional vectors. The student will find vector components. The student will use vectors in application problems. Vocabulary: initial side, terminal side, Pythagorean identities, standard position, unit circle, component, scalar multiplication, vector, magnitude, direction, initial point, terminal point, vector addition, bearing, gravity, parallel force, perpendicular force, resolve vectors, resultant, direction, resolution, true bearing, law of sines, law of cosines, ambiguous case page 10 of 17
Unit of Study / Textbook Correlation Objectives Resources and Activities Unit 5: Tranformations Prentice Hall Geometry 9.1-9.7 Benchmark 3.c Determine Whether a Figure is Symmetric The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include investigating symmetry and determining whether a figure is symmetric with respect to a line or a point. Indicator 3.c.1 The student will determine if a figure has point symmetry/line symmetry/both/neither. Indicator 3.c.2 The student will identify and use appropriate vocabulary for transformations. Benchmark 3.d Determine the Geometric Transformation Applied to a Figure The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Indicator 3.d.1 The student will identify the transformation that has taken place including reflection, rotation, dilation or translation. Indicator 3.d.2 The student will represent transformations using coordinate notation. Standard 10: SOLVE REAL-WORLD PROBLEMS INVOLVING ANGLES OF POLYGONS The student will solve real-world problems involving angles of polygons. A regular polygon will tessellate the plane if the measure of an interior angle is a factor of 360. Both regular and nonregular polygons can tessellate the plane. Two intersecting lines form angles with specific relationships. An exterior angle is formed by extending a side of a polygon. The exterior angle and the corresponding interior angle form a linear pair. The sum of the measures of the interior angles of a convex polygon may be found by dividing the interior of the polygon into nonoverlapping triangles page 11 of 17
Benchmark 10.a Use Polygon Angle Measures to Solve Real-World Problems The student will solve real-world problems involving angles of polygons Indicator 10.a.1 The student will identify tessellations in art, construction and nature. Vocabulary: image, preimage, transformation, rigid transformations, isometry, dilation, reflection, line of reflection, line of symmetry, rotation, center of rotation, angle of rotation, rotational symmetry, point symmetry, line symmetry, translation, glide reflection, composition of transformations, frieze pattern, tesslation, ratio, proportion, extremes, means, geometric mean, similar polygons, scale factor, dilation, reduction, enlargement page 12 of 17
Unit of Study / Textbook Correlation Objectives Resources and Activities Unit 6: Areas of Plane Figures Prentice Hall Geometry 1.8, 10.1-10.3, 10.5-10.8 Benchmark 10.a Use Polygon Angle Measures to Solve Real-World Problems The student will solve real-world problems involving angles of polygons. Indicator 10.a.7 The student will find the perimeter of a regular polygon. Indicator 10.a.8 The student will find the area of a regular polygon. Benchmark 11.c Find Arc Length and Areas of Sectors in Circles The student will use angles, arcs, chords, tangents, and secants to find arc lengths and areas of sectors in circles. Indicator 11.c.1 The student will calculate the area of a sector and the length of an arc of a circle. Indicator 11.c.2 The student will calculate the area of bounded regions. Indicator 11.c.3 The student will calculate geometric probability. Vocabulary: perimeter, area, sector, geometric probability page 13 of 17
Unit of Study / Textbook Correlation Objectives Resources and Activities Unit 7: Surface Areas and Volumes of Solid Figures Prentice Hall Geometry 11.1-11.7 Standard 13: USE FORMULAS FOR SURFACE AREA & VOLUME TO SOLVE PROBLEMS The student will use formulas for surface area and volume of three-dimensional objects to solve real-world problems. The surface area of a three-dimensional object is the sum of the areas of all its faces. The volume of a three-dimensional object is the number of unit cubes that would fill the object. Benchmark 13.a Use Formulas for Surface Area and Volume to Solve Practical Problems The student will use formulas for surface area and volume of three-dimensional objects to solve real-world problems. Indicator 13.a.1 The student will find the total surface area of 3D shapes using appropriate formulas. Indicator 13.a.2 The student will calculate the volume of 3D shapes using appropriate formulas. Indicator 13.a.3 The student will solve problems involving total surface area and volume of 3D figures. Indicator 13.a.4 The student will use calculators to find decimal approximations for the results. Indicator 13.a.5 The student will find total surface area of 3D figures using nets. Indicator 13.a.6 The student will find the lateral surface area of 3D shapes using appropriate formulas. page 14 of 17
Standard 14: USE PROPORTIONAL REASONING/DETERMINE AFFECT OF CHANGE TO DIMENSIONS The student will use similar geometric objects in two- or three-dimensions to a) compare ratios between side lengths, perimeters, areas, and volumes; b) determine how changes in one or more dimensions of an object affect area and/or volume of the object; c) determine how changes in area and/or volume of an object affect one or more dimensions of the object; and d) solve real-world problems about similar geometric objects. A change in one dimension of an object results in predictable changes in area and/or volume. A constant ratio exists between corresponding lengths of sides of similar figures. Proportional reasoning is integral to comparing attribute measures in similar objects. Benchmark 14.a Compare Ratios Between Side Lengths/Perimeters/Areas/Volumes The student will use similar geometric objects in two- or three-dimensions to compare ratios between side lengths, perimeters, areas, and volumes. Indicator 14.a.1 The student will compare ratios between side lengths/perimeters/area/volumes. Indicator 14.a.2 The student will find the lengths of sides of similar geometric solids. Indicator 14.a.3 The student will find the perimeter, area, or volume of similar geometric objects. Benchmark 14.b Determine Affect of Changes in Dimensions on Area/Volume The student will use similar geometric objects in two- or three-dimensions to determine how changes in one or more dimensions of an object affect area and/or volume of the object. Indicator 14.b.1 The student will describe how changes in dimensions affect other derived measures. page 15 of 17
Benchmark 14.c Determine the Affect of Changes in Area and/or Volume The student will use similar geometric objects in two- or three-dimensions to determine how changes in area and/or volume of an object affect one or more dimensions of the object. Indicator 14.c.1 The student will describe how changes in one or more measures affect other measures. Indicator 14.b.1 The student will solve real-world problems involving similar objects. Vocabulary: surface area, lateral area, pyramid, sphere, cylinder, rectangular prism, cube, cone page 16 of 17
Unit of Study / Textbook Correlation Objectives Resources and Activities Unit 8: Loci, Circles and Ellipses Prentice Hall Geometry 12.5-12.6, Virginia 12 Standard 12: WRITE THE EQUATION OF A CIRCLE The student, given the coordinates of the center of a circle and a point on the circle, will write the equation of the circle. A circle is a locus of points equidistant from a given point, the center. Standard form for the equation of a circle is ( x h ) 2 + ( y k ) 2 = r 2, where the coordinates of the center of the circle are ( h, k ) and r is the length of the radius. The circle is a conic section. Benchmark 12.a Write the Equation of a Circle The student, given the coordinates of the center of a circle and a point on the circle, will write the equation of the circle. Indicator 12.a.1 The student will identify the center, radius and diameter of a circle from a given equation. Indicator 12.a.2 The student will use the distance formula to find the radius of a circle. Indicator 12.a.3 The student will identify on the circle, given the center coordinates and radius. Indicator 12.a.4 The student will find the equation of the circle, identify center coordinates and radius given the diameter endpoints. Indicator 12.a.6 The student will find the equation of the circle given the center and a point on the circle. Indicator 12.a.7 The student will recognize the equation of a circle is derived from the Pythagorean Theorem. Indicator 12.a.8 The student will write the equation of a line tangent to a circle. Indicator 12.a.9 The student will write the equation of a circle given three points on the circle. Indicator 12.a.10 The student will identify an ellipse from its equation. Indicator 12.a.11 The student will sketch the graph of an ellipse/circle given in (h, k) form. Indicator 12.a.12 The student will write an equation in (h, k) form given the properties of an ellipse. Indicator 12.a.13 The student will state whether the graphs are circles or ellipses given the standard form. Vocabulary: circle, ellipse, standard form, (h, k) form page 17 of 17