Course Description This course involves the integration of logical reasoning and spatial visualization skills. It includes a study of deductive proofs and applications from Algebra, an intense study of polygons, and an introduction to Trigonometry. Students will be required to "think visually" while transferring information to real life problems. Scope And Sequence Timeframe Unit Instructional Topics 8 Week(s) 3 Week(s) 6 Week(s) Course Rationale In alignment with Common Core State Standards, the Park Hill School District's Mathematics courses provide students with a solid foundation in number sense while building to the application of more demanding math concepts and procedures. The courses focus on procedural skills and conceptual understandings to ensure coherence and depth in mathematical practices and application to real world issues and challenges. Mathematics is a language of patterns and relationships. Physical situations can be modeled mathematically. People use a variety of strategies and operations to solve problems in everyday life. Mathematical understanding is built through problem solving and reasoning. Relationships are represented through the attributes of shapes and objects and influence how we measure. Key Resources Larson Geometry Book by Holt McDougal 2011 www.classzone.com Board Approval Date January 10, 2013 Unit: Congruence Congruence Similarity and Right Triangles Geometric Measurement and Dimension 1. Building blocks 2. Logic 3. Relationships when lines are cut by a transversal 4. Triangles 5. Segments in a triangle 6. Transformations 7. Constructions 1. Similarity 2. Right triangles and trigonometric ratios 1. Quadrilaterals 2. Two-dimensional figures 3. Three-dimensional figures 4. Circles 5. Probability Course Details Duration: 8 Week(s) Page 1
Unit Overview Logical reasoning will be used to formulate proofs. Points, lines and planes are the undefined terms that are used to define other figures in Geometry. Using properties, postulates and theorems together will allow logical proofs to be formed. Use postulates and properties of parallel lines and transversals to solve problems. Transformations and constructions can be used to show the relationship of congruent figures. Essential Questions How do you define/describe and draw both defined and undefined geometric figures? How do you put statements in a logical order to draw a valid conclusion? How do you use the special angles formed by parallel lines cut by a transversal to find missing angle measures? How do you perform and describe given transformations in a coordinate plane? Example Assessment Items Draw and label examples of the undefined and defined terms. Given statements and reasons, students can place them in a logical order. Given two lines cut by a transversal and two angles (with unknowns), students can apply theorems and postulates to find the variable that prove the lines are parallel. Given a transformation mapping and the preimage points in the coordinate plane, the student can construct the image points. Academic Vocabulary Adjacent Angles Altitude Bisect Complementary Angles Supplementary Angles Congruence Coplanar Midsegment Postulate Skew Lines Theorem Transversal Vertex Line Symmetry Rotational Symmetry Topic: Building blocks Duration: 8 Day(s) 1a The student will know the precise definitions of congruence, line segment, and angle, based on the undefined notions of point, line and plane. 1b The student will find the length and midpoint of a line segment on a number line and in a coordinate plane. 1c The student will be able to measure and classify angles. 1d The student will be able to describe angle pair relationships. 1e The student will know the precise definition of polygons and be able to name them. 1f The student will find the point on a directed line segment between two given points that partitions the segment in a given ratio. Topic: Duration: 8 Day(s) Logic 2a The student will understand all aspects of conditional statements and how to write definitions as conditional statements. 2b The student will know and apply the definition of perpendicular lines and planes. 2c The student will know and apply the laws of logic to draw appropriate conclusions. 2d The student will know and apply the algebraic properties to form proofs. 2e The student will be able to prove that vertical angles are congruent. Topic: Duration: 7 Day(s) Relationships when lines are cut by a transversal 3a The student will be able to identify the angles formed by two lines cut by a transversal and the relationship of those angles when the two lines are parallel. Page 2
3b The student will be able to prove angle relationships given parallel lines cut by a transversal and prove that lines are parallel given angle relationships. 3c The student will 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). Topic: Duration: 9 Day(s) Triangles 4a The student will prove that the sum of the interior angles of a triangle is 180 degrees. 4b The student will 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. 4c The student will explain how the criteria for triangle congruence (ASA,SAS,and SSS) follow from the definition of congruence in terms of rigid motions. 4d The student will be able to prove that the base angles of isosceles triangles are congruent. Topic: Duration: 3 Day(s) Segments in a triangle 5a The student will prove that the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length. 5b The student will prove points on a perpendicular bisector of a line segment are equidistant from the segment's endpoints. 5c The student will prove the medians of a triangle meet at a point called the centroid. Topic: Duration: 4 Day(s) Transformations 4e The student will be able to represent transformations in the plane and describe transformations as functions that take points in the plane as inputs and give other points as outputs. T(4.8) The student will develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Ta The student will be able to describe the rotations and reflections that carry a rectangle, parallelogram, trapezoid, or regular polygon onto itself (reflectional and rotational symmetry). Tb The student will be able to rotate, reflect, or translate a geometric figure and 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. Topic: Duration: 3 Day(s) Constructions Ca The student will make formal geometric constructions including copying a segment, copying an angle, bisecting a segment, bisecting an angle, constructing perpendicular lines, constructing the perpendicular bisector of a line segment and constructing a line parallel to a given line through a point not on the line, a tangent line from a point outside a given circle to the circle with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.) Cb The student will construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Cc The student will construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Unit: Similarity and Right Triangles Duration: 3 Week(s) Page 3
Unit Overview The examination of ratios and proportions of similar figures and how they relate to geometric figures. Prove figures are similar and understand the relationships of the angles and sides of similar figures. Know and apply the Pythagorean Theorem and special right triangles. Know the basic trigonometric ratios and apply them to real world problems. Essential Questions How can you identify similar triangles by using the AA Triangle Similarity Postulate, the SSS or SAS Triangle Similarity Theorems? How can you use the Pythagorean Theorem to solve problems? How can you use the trigonometric ratios to solve real-world problems? Example Assessment Items Given two triangles with angle and/or side measurements, students can decide whether the triangles are similar or not. Given a right triangle and two side lengths of the triangle, students can use the Pythagorean Theorem to find the missing side length. Given a real - world example, students can apply the trigonometric ratios to solve for the appropriate unknown. Academic Vocabulary Dilation Hypotenuse Similarity Cosine Sine Tangent Topic: Similarity Duration: 7 Day(s) 6a The student will use the definition of similarity to identify similar polygons and solve problems involving similar polygons. 6b The student will use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 6c The student will prove a line parallel to one side of a triangle divides the other two sides proportionally. 6d The student will use the properties of dilations given by a center and a scale factor. 6e The student will compare transformations that preserve distance and angle measure to those that do not (e.g., translation versus horizontal stretch). Topic: Duration: 9 Day(s) Right triangles and trigonometric ratios 7a The student will 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. 7b The student will explain and use the relationship between the sine and cosine of complementary angles. 7c The student will be able to apply the Pythagorean Theorem and trigonometric ratios to solve right triangles in applied problems. 7d The student will be able to prove the Pythagorean Theorem by using triangle similarity. 7e The student will be able to use special right triangles to solve problems. 7e+ The student will prove the Laws of Sines and Cosines and use them to solve problems. 7e+ The student will be able to explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. 7e+ The student will be able to choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. 7e+ The student will be able to prove the Pythagorean identity sin²(theta) + cos²(theta) = 1 and use it to find sin(theta), cos(theta), or tan (theta) given sin(theta), cos(theta), or tan(theta) and the quadrant of the angle. Unit: Geometric Measurement and Dimension Duration: 6 Week(s) Page 4
Unit Overview Explore the ratio of surface area and volume in three-dimensional figures. Know the definitions and properties of quadrilaterals. Identify and describe relationships of segments and angles in circles. Find and apply area of two-dimensional figures, including arc length and area of sectors in a circle. Find and apply surface area and volume of three-dimensional figures. Essential Questions How can you use the properties of quadrilaterals to find missing lengths and angle measures? How can you apply concepts of angle relationships in circles to find missing measurements? How can you find the geometric probability that a randomly chosen point will be in the shaded region of a figure? How can you apply the results of surface area and volume in real world situations? Example Assessment Items Given a picture of a special quadrilateral with segment lengths and/or angle measurements, students can find all the missing lengths and /or measurements. Given a picture with chords, arcs, radii, tangents, secants, etc drawn and given measurements, students can find the measurements of missing angle measures and/or arc measures. Given a shaded region, students can find the geometric probability of randomly choosing a point in the shaded region. Given the dimensions of a right prism, pyramid, cylinder, cone or sphere, students can find the surface area and volume of the figure. Academic Vocabulary Chord Secant Tangent Radian Topic: Quadrilaterals Duration: 8 Day(s) 8a The student will prove opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. 8b The student will use coordinates to prove simple geometric theorems algebraically. Topic: Duration: 8 Day(s) Two-dimensional figures 11 The student will know how to find area of two-dimensional figures. 11a The student will understand that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of an angle as the constant of proportionality; derive the formula for the area of a sector. 11b The student will be able to find the area of regular polygons. 11c The student will be able to find geometric probability. Topic: Duration: 7 Day(s) Three-dimensional figures 12a The student will find surface area and volume of three dimensional shapes including prisms, cylinders, pyramids, cones and spheres in real world applications. Understand Cavalieri's principle for the formulas for the volume of a sphere and other solid figures. 12b The student will identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. 12c The student will apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). 12d The student will 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). Topic: Duration: 4 Day(s) Circles 10a The student will understand that all circles are similar. Page 5
10b The student will identify and describe relationships among inscribed angles, radii, and chords including 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. 10c The student will derive the equation of a circle of given center and radius using the Pythagorean Theorem and be able to complete the square to find the center and radius of a circle given by an equation. Topic: Duration: 3 Day(s) Probability Description Use and understand probability to make informed decisions. The student will understand independence and conditional probability and use them to interpret data. The student will use the rules of probability to compute probabilities of compound events in a uniform probability model. The student will use probability to evaluate outcomes of decisions. Page 6