Content Area: Math Course Title: Geometry Elements Grade Level: 9-12 Geometric Structure 23 blocks Congruence 22 blocks Similarity 19 blocks Measurement 25 blocks Date Created by: Daphne Rodriguez and Sharon Kilmer, June 2012 Board Approved on: 8/16/2012
Content Area: Math Unit Title: Geometric Structure Geometry Elements Unit 1 Target Course/Grade Level: Geometry Elements/ 9th 12th Duration: 25 blocks Unit Summary: The unit begins with laying down the foundation for a formal study of Euclidean geometry. Students will be engaged in the study of the three undefined terms in geometry. Concepts first introduced in middle school and Algebra I will be revisited and elaborated upon. Students will also be introduced to basic constructions using compass and straight edge and patty paper. Students will further extend their knowledge of parallel lines, perpendicular lines and angle pair relationships. Formative assessments will conducted through daily quizzes and class discussions. A summative assessment will be administered once a concept is learned. Primary interdisciplinary connections: Science, Business, Technology, Art 21 st century themes: Life and career skills (Initiative and Self-Direction, Social and Cross Cultural Skills, Productivity and Accountability); Learning and Innovation skills (Creativity and Innovation, Critical Thinking and Problem Solving, Communication and Collaboration) Learning Targets Standards: G-CO 1, 9, 12; G-GPE 4, 5, 6; A-CED 1, 4 Unit Essential Questions What are the building blocks of geometry? What are the tools of geometry and how they used to sketch constructions? How are special angle pairs related? How can two lines be proved to be parallel? How is slope used to describe a line, and how does slope help you classify lines as parallel or perpendicular? Unit Enduring Understandings Geometry has a specific structure that includes definitions, postulates, and theorems. Each geometric figure has a specific naming convention. Geometric constructions are important in visualizing relationships between geometric figures and are a critical aid in solving problems in geometry. Expressing an argument in a logical manner is a mathematical skill that can be extended to everyday life. There are specific relationships between lines and angles in a coordinate plane and in space. Concept: Basics of Geometry Standards: G-CO 1, G-CO 12 A-CED 4 Describe the three undefined terms in geometry: point, line and plane. Identify and model points, lines and planes. Identify intersecting lines and planes. Find the distance between two points on a number and in a coordinate plane using the Distance Formula. Find the midpoint of a segment by using the Midpoint Formula. Construct a segment and an angle using a compass, straight edge and patty paper. Construct a segment bisector and angle bisector using compass, straight edge and patty paper. Construct Perpendicular lines using compass, straight edge and patty paper. Construct perpendicular bisectors of a segment using compass, straight edge, and patty paper. Construct a line Parallel to a given line through a point not on the line using compass, straight edge and patty paper.
Measure and classify angles according to their measures. Identify and use congruent angles and the bisector of an angle. Identify and use special angle pairs. Identify perpendicular lines. Identify and name polygons. Find perimeter, circumference and area of two-dimensional figures. Identify and name three dimensional figures. Find surface area and volume. Vocabulary: undefined term, point line, plane, collinear, coplanar, intersection, space, distance, midpoint, segment bisector, ray, opposite rays, angle side, vertex, interior, exterior, acute angle, obtuse angle, right angle, angle bisector, adjacent angles, linear pair, vertical angles, complementary angles, supplementary angles, perpendicular, polygon, vertex of a polygon, concave, convex, n-gon, equilateral, equiangular, regular, perimeter, circumference, area, polyhedron, face, edge, prism, base, pyramid, cylinder, cone, sphere, regular polyhedron, Platonic Solid, surface area, volume. Concept: Reasoning and Proof Standards: G-CO 9 Use inductive reasoning to formulate a conjecture. Use logical reasoning to prove statements are true and find counterexamples that are false. Determine the validity of a conditional statement, its converse, inverse and contrapositive. Use deductive reasoning to make inferences about statements Vocabulary: inductive reasoning, conjecture, counterexample, compound statement, conditional statement, if-then statement, hypothesis, conclusion, converse, inverse, contrapositive, deductive reasoning, postulate, theorem, congruent Concept: Perpendicular and Parallel Lines Standards: G-CO 1, 9; G-GPE 4, 5, 6; A-CED 1, 4 Identify the relationship between two lines and two planes. Identify angle relationships that occur with parallel lines and a transversal. Show lines are parallel from given angle relationships. Calculate the slope of a line to analyze the line and write its equation. Use theorems to determine the relationships between specific pairs of angles. Use the properties of parallel lines and special angle pairs to solve problems algebraically. Write the equation of a line from the given information in a graph. Vocabulary: parallel lines, skew lines, parallel planes, transversal, alternate interior angles, alternate exterior angles, consecutive interior angles, corresponding angles, slope, slope-intercept form, point-slope form, equidistant Unit Learning Targets/Objectives Students will... Complete notes during the class lesson. Use teacher created worksheets and the textbook to complete practice problems for skills learned during class. Evidence of Learning
Formative Assessment (Daily): Daily quizzes will be given to check for understanding. Summative Assessment (3 blocks): Three major tests will be given to evaluate learning from this unit. The three tests correspond to the three concepts in this unit. Students will be graded in part on their understanding of the concepts and in part on their ability to execute the operations used in solving problems. Partial credit will be given only if the student exhibits understanding of the problem but made an error in the execution. Equipment needed: pencil, notes, textbook, calculator, compass, metric ruler, patty paper. Teacher resources: Geometry by Larson, Boswell, Stiff; Geometry by Carter, Cuevas, Day, Malloy and Cummins; Interwrite board
Content Area: Math Unit Title: Congruence Geometry Elements Curriculum Unit 2 Target Course/Grade Level: Geometry Elements/ 9 th 12 th Duration: 24 blocks Unit Summary: This unit further extends students knowledge about the congruence and the properties of triangles. Students will deepen their knowledge and understanding of congruent figures by studying the effects that rigid transformations have on geometric figures. This understanding will be reinforced through the process of showing two triangles are congruent. Students will also see the connections between special lines and points of concurrency in triangles and how these are used to solve problems in a real world setting. Students will be able to construct congruent triangles according to specific given conditions by using compass and straight edge or patty paper. Through the study of quadrilaterals, students will understand that geometric figures can be classified according to their attributes and that these attributes can be used to show that a quadrilateral is of a particular type. Formative assessments will conducted through daily quizzes and class discussions. A summative assessment will be administered once a concept is learned. Primary interdisciplinary connections: Science, Business, Technology, Art 21 st century themes: Life and career skills (Initiative and Self-Direction, Social and Cross Cultural Skills, Productivity and Accountability); Learning and Innovation skills (Creativity and Innovation, Critical Thinking and Problem Solving, Communication and Collaboration) Learning Targets Standards: : G-CO 7, 8, 10, 11, G-GPE 4; A-CED 1, 4 Unit Essential Questions What are the minimum conditions that need to be met in order to show that two triangles are congruent? How do rigid transformations help us to understand congruence in figures? How can it be determined if three sides form a triangle? How can we use the special properties of a triangle to solve problems? What are the conditions that are required to determine if a quadrilateral is a parallelogram? How can we use the characteristics of special quadrilaterals to sort parallelograms? Unit Enduring Understandings Triangles are fundamental aesthetic, structural elements that are useful in many disciplines such as art, architecture, and engineering. Rigid Transformations are closely related to the concept of congruence. Relationships exist between the angles, sides and diagonals of quadrilaterals. Properties of triangles and quadrilaterals can be used to solve problems in real world settings such as construction and design. Concept: Congruent Triangles Standards: G-CO 7, 8, 10; A-CED 1 Apply special relationships about the interior and exterior angles of triangles. Identify corresponding parts of congruent triangles. Show triangles are congruent by using the definition of congruence. Identify and classify triangles by angle measures and side measures.
Apply the Triangle Sum Theorem. Apply the Exterior Angle Theorem. Apply the SSS, SAS, ASA Triangle Congruence Postulates to test for and show two triangles are congruent. Apply the AAS Triangle Congruence Theorem to test for and show two triangles are congruent. Make congruent triangle constructions by using: three sides, two sides and an included angle, two angles and an included side. Apply the properties of isosceles and equilateral triangles to solve problems. Identify reflections, translations and rotations. Verify congruence after congruence transformation. Vocabulary: acute triangle, obtuse triangle, right triangle, equilateral triangle, isosceles triangle, scalene triangle, exterior angle, remote interior angles, corollary, corresponding parts, included angle, included side, legs of an isosceles triangle, vertex angle, base angles, transformation, congruence transformation, preimage, image, reflection, translation, rotation. Concept: Properties of Triangles Standards: G-CO 10; A-CED 1,4 Construct the perpendicular and angle bisectors of a triangle. Identify and use perpendicular bisectors and angle bisectors in triangles. Construct the medians and altitudes of triangles. Identify and use the medians and altitudes of triangles. Recognize and apply properties of inequalities to the measures of the angles of a triangle. Recognize and apply properties of inequalities to the relationships between the angles and sides of a triangle. Construct the inscribed and circumscribed circles of a triangle. Vocabulary: perpendicular bisector, median, centroid, altitude, indirect reasoning Concept: Quadrilaterals Standards: G-CO 11, G-GPE 4; A-CED 1 Find and use the sum of the measures of the interior and exterior angles of a polygon. Recognize and apply properties of the sides and angles of parallelograms. Recognize and apply properties of the diagonals of parallelograms. Recognize the conditions that ensure a quadrilateral is a parallelogram. Show that a set of points in the coordinate plane form a parallelogram. Recognize and apply the properties of rectangles. Determine whether a parallelogram is a rectangle. Recognize and apply the properties of rhombi and squares. Determine whether quadrilaterals are rhombi or squares. Recognize and apply the properties of trapezoids and kites. Show properties of angles for a quadrilateral inscribed in a circle. Vocabulary: diagonal, parallelogram, rectangle, rhombus, square, trapezoid, bases, legs of a trapezoid, base angles, isosceles trapezoid, midsegment of a trapezoid, kite Unit Learning Targets/Objectives Students will...
Complete notes during the class lesson. Use the textbook to complete practice problems for skills learned during class. Evidence of Learning Formative Assessment (Daily): Daily quizzes will be given to check for understanding. Summative Assessment (3 blocks): Three major tests will be given to evaluate learning from this unit. The three tests correspond to the three concepts in this unit. Students will be graded in part on their understanding of the concepts and in part on their ability to execute the operations used in solving problems. Partial credit will be given only if the student exhibits understanding of the problem but made an error in the execution. Equipment needed: pencil, notes, textbook, calculator, compass, metric ruler, patty paper. Teacher resources: Geometry by Larson, Boswell, Stiff; Geometry by Carter, Cuevas, Day, Malloy and Cummins; Interwrite board
Geometry Elements Curriculum Unit 3 Content Area: Math Duration: blocks 21 Unit Title: Similarity Target Course/Grade Level: Geometry Elements/ 9th 12th Unit Summary: During the course of this unit, students will explore the concepts of proportionality and similarity. Students will understand how proportionality is used to relate two similar figures. This concept of proportionality will be extended to scale models and students will examine the practical use of this concept in real world settings. Students will deepen their knowledge of the Pythagorean Theorem within the context of solving right triangles. Trigonometric ratios will be used to solve right triangles. Students will be engaged in a more formal study of transformation and understand the connection between transformations and similar figures. Formative assessments will conducted through daily quizzes and class discussions. A summative assessment will be administered once a concept is learned. Primary interdisciplinary connections: Science, Business, Technology, Art 21 st century themes: Life and career skills (Initiative and Self-Direction, Social and Cross Cultural Skills, Productivity and Accountability); Learning and Innovation skills (Creativity and Innovation, Critical Thinking and Problem Solving, Communication and Collaboration) Learning Targets Standards: G-CO 2, 3, 4, 5, 6; C-SRT 1a, 1b, 2, 3, 4, 5, 6, 7, 8, 10, 11; N-VM 1, 2, 4; A-CED 1, 4 Unit Essential Questions What does it mean for two figures to be similar? What are the minimum requirements to determine similarity between two triangles? What is the special relationship between parallel lines and proportionality? How is a figure affected after a similarity transformation? How is proportionality used in creating scale models? How does trigonometry help in finding the missing measures of triangles? What are the special attributes of right triangles? How are transformations and symmetry related? Does the order of composition of transformations matter? Unit Enduring Understandings Proportions are useful in relating two quantities and are useful in solving a variety of problems. Similar geometric figures have proportional attributes. Concepts of similarity are used to relate twodimensional and three-dimensional figures. The Pythagorean Theorem is used to find the missing sides of a right triangle. Trigonometry has many useful applications including solving triangles. Transformations are used in creating tessellations and can be found in architecture and art. Concept: Proportions and Similarity Standards: G-SRT 2, 3, 4, 5; A-CED 1 Identify similar polygons and use ratios and proportions to solve problems. Use proportions to identify similar polygons. Solve problems using the properties of similar polygons.
Identify similar triangles using the AA Similarity Postulate and the SSS and SAS Similarity Theorems. Use similar triangles to solve problems. Use proportional parts within triangles. Use proportional parts with parallel lines. Recognize and use proportional relationships of corresponding angle bisectors, and medians of similar triangles. Identify and apply similarity transformations. Verify similarity after a similarity transformation. Interpret scale models. Use scale models and drawings to solve problems. Vocabulary: ratio, proportions, extremes, means, cross product, similar polygons, similarity ratio, scale factor, midsegment of a triangle, dilation, similarity transformation, center of dilation, scale factor of a dilation, enlargement, reduction, scale model, scale drawing, scale Concept: Right Triangles and Trigonometry Standards: G-SRT 4, 6, 7, 8, 10, 11; N-VM 1, 2, 4; A-CED 1, 4 Find the geometric mean between two numbers. Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse. Use the Pythagorean Theorem to solve problems. Use the Converse of the Pythagorean Theorem to solve problems. Use the properties of 45-45 - 90 triangles to solve problems. Use the properties of 30-60 - 90 triangles to solve problems. Find trigonometric ratios using right triangles. Use trigonometric ratios to find angle and side measures in right triangles. Vocabulary: geometric mean, Pythagorean triple, trigonometry, trigonometric ratio, sine, cosine, tangent, inverse sine, inverse cosine, inverse tangent. Concept: Transformations and Symmetry Standards: G-CO 2, 3, 4, 5, 6; C-SRT 1a, 1b Draw reflections. Draw and describe reflections in the coordinate plane. Draw translations. Draw and describe translations in the coordinate plane. Draw rotations. Draw and describe rotations in the coordinate plane. Draw compositions of rigid transformations. Draw compositions of reflections in parallel and intersecting lines. Identify line and rotational symmetries in two-dimensional figures. Identify line and rotational symmetry in three-dimensional figures. Draw dilations.
Draw and describe dilations in the coordinate plane. Vocabulary: reflection, line of reflection, translation, rotation, center of rotation, angle of rotation, composition of transformations, symmetry, line of symmetry, rotational symmetry. Unit Learning Targets/Objectives Students will... Complete notes during the class lesson. Use the textbook to complete practice problems for skills learned during class. Evidence of Learning Formative Assessment (Daily): Daily quizzes will be given to check for understanding. Summative Assessment (3 blocks): Three major tests will be given to evaluate learning from this unit. The three tests correspond to the three concepts in this unit. Students will be graded in part on their understanding of the concepts and in part on their ability to execute the operations used in solving problems. Partial credit will be given only if the student exhibits understanding of the problem but made an error in the execution. Equipment needed: pencil, notes, textbook, calculator, compass, metric ruler, patty paper. Teacher resources: Geometry by Larson, Boswell, Stiff; Geometry by Carter, Cuevas, Day, Malloy and Cummins; Interwrite board
Content Area: Math Unit Title: Measurement Geometry Elements Curriculum Unit 4 Target Course/Grade Level: Geometry Elements/ 9 th 11 th Duration: 24 blocks Unit Summary: This unit takes students through a study of measurement in geometry. The unit begins with the study of circles. Students will discover the significance of the number pi. The concept of area first introduced in middle school will be elaborated upon to include the area of kites and rhombi as well as the area of regular polygons. Students will also learn to find the surface area and volumes of solids. Finally, the concept of area will be extended to include probability in the form of geometric probability. Students will learn how to determine a sample space and how this sample space, along with counting principles is used to determine the probability of an event. Students will learn that probability is useful in making informed decisions and predictions about a set of data. Formative assessments will conducted through daily quizzes and class discussions. A summative assessment will be administered once a concept is learned. Primary interdisciplinary connections: Science, Business, Technology, Art 21 st century themes: Life and career skills (Initiative and Self-Direction, Social and Cross Cultural Skills, Productivity and Accountability); Learning and Innovation skills (Creativity and Innovation, Critical Thinking and Problem Solving, Communication and Collaboration) Learning Targets Standards: G-CO 1, 13; G-C 1, 2, 3, 5; G-GPE 1, 4, 7; G-GMD 1, 2, 3, 4; G-MG 2, 3; G-MG 1; S-IC 2, 3; S-CP 1, 2, 3, 5, 6, 7, 8, 9; A-CED 1, 4 Unit Essential Questions What are the properties of circles and the relationships among angles, lines and line segments in and around circles? What is the special relationship between the circumference and the diameter of a circle? How are area formulas derived? How are the areas of similar figures related? Is there are general formula or method to find the area of any regular polygon? How is a sample space determined? How is probability determined? What factors affect sample space and probability? Unit Enduring Understandings Measurement consists of three parts: number, unit, precision. The number pi is the special relationship between the circumference and the diameter of every circle. Changing one linear dimension in a figure with two or three dimensions will have a different effect on the figure s perimeter, circumference, area or volume. Probability is useful in making inferences and predictions. The probability of an event occurring can be predicted with varying degrees of confidence. Concept: Circles Standards: G-CO 1, 13; G-C 1, 2, 3, 5; G-GPE 1, 4; A-CED 1, 4 Identify and use parts of circles. Solve problems involving the circumference of a circle. Show all circles are similar. Vocabulary: circle, center, radius, diameter, congruent circles, concentric circles, circumference, pi, segment.
Concept: Areas of Polygons and Circles Standards: G-GPE 7; G-MG 2; G-MG 3; A-CED 1, 4 Find perimeters and areas of parallelograms. Find perimeters and areas of triangles. Find areas of trapezoids. Find areas of rhombi and kites. Find areas of circles. Find areas of sectors of circles. Areas of regular polygons. Find areas of composite figures. Find the areas of similar figures by using scale factors. Find scale factors or missing measures given the areas of similar figures. Vocabulary: base of a parallelogram, height of a parallelogram, base of a triangle, height of a triangle, height of a trapezoid, center of a regular polygon, radius of a regular polygon, composite figure Concept: Surface Area and Volume Standards: G-GMD 1, 2, 3, 4; G-MG 1; A-CED 4 Draw isometric views of three dimensional figures. Investigate cross-sections of three dimensional figures. Identify three-dimensional objects generated by rotations of two-dimensional objects. Find lateral areas and surface areas of prisms. Find lateral areas and surface areas of cylinders. Find lateral areas and surface areas of pyramids. Find lateral areas and surface areas of cones. Find volumes of prisms and cylinders. Find volumes of pyramids and cones. Find the surface area and volumes of spheres. Identify congruent or similar solids. Use properties of similar solids. Identify three dimensional objects generated by rotations of two dimensional objects. Vocabulary: isometric view, cross section, lateral face, lateral edge, base edge, altitude, height, lateral area, axis, regular pyramid, slant height, right cone, hemisphere, similar solids, congruent solids Concept: Probability and Measurement Standards: S-IC 2, 3; S-CP 1, 2, 3, 5, 6, 7, 8, 9 Use lists, tables, and tree diagrams to represent sample spaces. Use the Fundamental Counting Principle to count outcomes. Use permutations with probability. Use combinations with probability. Find probabilities by using length. Find probabilities by using area. Find the probabilities of dependent and independent events.
Vocabulary: sample space, tree diagram, Fundamental Counting Principle, permutation, factorial, combination, geometric probability, probability model, compound event, independent events, dependent events, conditional probability, probability tree. Unit Learning Targets/Objectives Students will... Complete notes during the class lesson. Use the textbook to complete practice problems for skills learned during class. Evidence of Learning Formative Assessment (Daily): Daily quizzes will be given to check for understanding. Summative Assessment (4 blocks): Four major tests will be given to evaluate learning from this unit. The three tests correspond to the three concepts in this unit. Students will be graded in part on their understanding of the concepts and in part on their ability to execute the operations used in solving problems. Partial credit will be given only if the student exhibits understanding of the problem but made an error in the execution. Equipment needed: pencil, notes, textbook, calculator, compass, metric ruler, patty paper. Teacher resources: Geometry by Larson, Boswell, Stiff; Geometry by Carter, Cuevas, Day, Malloy and Cummins; Interwrite board
Math Standards Alignment Course Title Standards Met Geometry G-CO 1 13; G-C 1, 2, 3, 5; G-GPE 1, 4, 5, 6, 7; G-GMD 1, 3, 4; G-MG 1, 2, 3; G-SRT 1a, 1b, 2, 3, 4, 5, 6, 7, 8, 10, 11 S-IC 2, 3; S-CP 1, 2, 3, 5, 6, 7, 8, 9 Alignment Status Standard: Number and Quantity met: N-VM 1, 2, 4 not met: N-RN 1 3; N-Q 1 3; N-CN 1 9; N-VM 3, 5 12 Standard: Algebra met: A-CED 1, 4; A-REI 1, 3; not met: A-SSE 1 4; A-APR 1 7; A-CED 2, 3; A-REI 2, 4 12 Standard: Functions Standard: Geometry Standard: Statistics and Probability met: none not met: all met: All except as noted. not met: G-SRT 9; G-C 4; G-GPE 2, 3; G-GMD 2 met: S-IC 2, 3; S-CP 1, 2, 3, 5, 6, 7, 8, 9 not met: S-ID 1 9; S-IC 1, 4, 5, 6; S-CP 4