Geometry

one credit course

Introduction

Unit 01: Introduction to Logic and Euclidean Geometry

Unit 02: Coordinate Geometry and Transformations

Unit 03: Relationships of Lines and Transversals

Unit 04: Relationships of Triangles, including Congruence and Similarity

Unit 05: Relationships of Right Triangles, including Trigonometry

Unit 06: Relationships of Circles, including Radian Measure and Equations of Circles

Unit 07: Relationships of Two- and Three-Dimensional Figures

Unit 08: Measurement of Two-Dimensional Figures

Unit 09: Measurement of Three-Dimensional Figures

Unit 10: Probability

Unit 11: Engineering Design

Texas Essential Knowledge & Skills (TEKS)

Introduction

(1) The desire to achieve educational excellence is the driving force behind the Texas essential knowledge and skills for mathematics, guided by the college and career readiness standards. By embedding statistics, probability, and finance, while focusing on fluency and solid understanding, Texas will lead the way in mathematics education and prepare all Texas students for the challenges they will face in the 21st century.

(2) The process standards describe ways in which students are expected to engage in the content. The placement of the process standards at the beginning of the knowledge and skills listed for each grade and course is intentional. The process standards weave the other knowledge and skills together so that students may be successful problem solvers and use mathematics efficiently and effectively in daily life. The process standards are integrated at every grade level and course. When possible, students will apply mathematics to problems arising in everyday life, society, and the workplace. Students will use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution. Students will select appropriate tools such as real objects, manipulatives, paper and pencil, and technology and techniques such as mental math, estimation, and number sense to solve problems. Students will effectively communicate mathematical ideas, reasoning, and their implications using multiple representations such as symbols, diagrams, graphs, and language. Students will use mathematical relationships to generate solutions and make connections and predictions. Students will analyze mathematical relationships to connect and communicate mathematical ideas. Students will display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

(3) In Geometry, students will build on the knowledge and skills for mathematics in Kindergarten-Grade 8 and Algebra I to strengthen their mathematical reasoning skills in geometric contexts. Within the course, students will begin to focus on more precise terminology, symbolic representations, and the development of proofs. Students will explore concepts covering coordinate and transformational geometry; logical argument and constructions; proof and congruence; similarity, proof, and trigonometry; two- and three-dimensional figures; circles; and probability. Students will connect previous knowledge from Algebra I to Geometry through the coordinate and transformational geometry strand. In the logical arguments and constructions strand, students are expected to create formal constructions using a straight edge and compass. Though this course is primarily Euclidean geometry, students should complete the course with an understanding that non-Euclidean geometries exist. In proof and congruence, students will use deductive reasoning to justify, prove and apply theorems about geometric figures. Throughout the standards, the term "prove" means a formal proof to be shown in a paragraph, a flow chart, or two-column formats. Proportionality is the unifying component of the similarity, proof, and trigonometry strand. Students will use their proportional reasoning skills to prove and apply theorems and solve problems in this strand. The two- and three-dimensional figure strand focuses on the application of formulas in multi-step situations since students have developed background knowledge in two- and three-dimensional figures. Using patterns to identify geometric properties, students will apply theorems about circles to determine relationships between special segments and angles in circles. Due to the emphasis of probability and statistics in the college and career readiness standards, standards dealing with probability have been added to the geometry curriculum to ensure students have proper exposure to these topics before pursuing their post-secondary education.

(4) These standards are meant to provide clarity and specificity in regards to the content covered in the high school geometry course. These standards are not meant to limit the methodologies used to convey this knowledge to students. Though the standards are written in a particular order, they are not necessarily meant to be taught in the given order. In the standards, the phrase "to solve problems" includes both contextual and non-contextual problems unless specifically stated.

Unit 01: Introduction to Logic and Euclidean Geometry

(10 classes for the entire unit)

Students lay the foundation for geometry by developing an understanding of the structure of a geometric system through examination of the relationship between undefined terms (point, line, and plane), definitions, conjectures, and postulates. Students examine one-dimensional distance relationships in line segments, including fractional distances and midpoints, and make connections to the number line and segment addition. They also examine relationships in rays and angles making connections to the angle measure and angle addition. Constructions are used to explore and make conjectures about congruent geometric relationships in line segments and angles. They connect their understanding of definitions and postulates of lines, angles, and other geometric vocabulary to the context of the real world. Students also investigate logic statements and the conditions that make them true or false. Students explore conditional statements and their related statements (converse, inverse, and contrapositive) in both a real world and mathematical setting to develop an understanding of logic and the role it plays in geometry and the real world. Students are expected to recognize the connection between a biconditional statement and a true conditional statement with a true converse. Deductive reasoning and inductive reasoning are introduced and applied to make conjectures. Students verify that a conjecture is false using counterexamples.

TEKS in this unit: G.1A, G.1B, G.1C, G.1D, G.1E, G.1F, G.1G, G.2A, G.4A, G.4B, G.4C, G.5B, G.5C

Unit 02: Coordinate Geometry and Transformations

(15 classes for the entire unit)

Students investigate the undefined terms point, line, and plane in a two-dimensional coordinate system in Euclidean and spherical geometries. Using coordinate points, students derive the distance formula and apply the distance formula to determine lengths and congruence of line segments and fractional distances less than one from one end of a line segment to the other. Coordinate points are also used to derive and apply the midpoint formula and slope formula. Slope is applied to define and investigate parallel and perpendicular lines, including comparison of parallel lines in Euclidean and spherical geometry. Students algebraically determine the equation of a line when given a point on the line and a line parallel or perpendicular to the line. In addition, students build upon their knowledge of coordinate geometry to analyze the critical attributes of transformations, including translations, reflections, rotations with points of rotation other than the origin, and dilations where the center of dilation can be any point on the coordinate plane. Students examine patterns to generalize rigid transformations (translations, reflections, and rotations) in the coordinate plane. Students also explore non-rigid transformations or dilations in the coordinate plane using scale factors. They compare and contrast dilations to other geometric transformations and examine relationships in terms of similarity. Students perform rigid transformations, non-rigid transformations, and composite transformations using coordinate notation. Students identify the sequence of transformations performed for a given pre-image or image on or off a coordinate plane. Reflection symmetry and rotational symmetry in plane figures are identified and differentiated.

TEKS in this unit: G.1A, G.1B, G.1C, G.1D, G.1E, G.1F, G.1G, G.2A, G.2B, G.2C, G.3A, G.3B, G.3C, G.3D, G.4D

Unit 03: Relationships of Lines and Transversals

(6 classes for the entire unit)

Students explore angle relationships formed by one line and one transversal including vertical angles, linear pairs, and adjacent angles. Students construct congruent angles and a line parallel to a given line through a point not on a line using a compass and a straightedge. Students investigate patterns to make conjectures and define angles formed by parallel lines cut by a transversal. Students explore angle relationships formed by two parallel lines and one or more transversal(s) including corresponding angles, same side interior angles, alternate interior angles, and alternate exterior angles. Students use a variety of tools such as patty paper, folding techniques, etc. to investigate these relationships between angle pairs formed when parallel lines are cut by a transversal(s). Students formulate deductive proofs for conjectures about angles formed by parallel lines and transversals and apply these relationships to solve mathematical and real-world problems. Students explore and apply the converse of theorems and postulates for parallel lines cut by a transversal to solve mathematical and real-world problems.

TEKS in this unit: G.1A, G.1B, G.1C, G.1D, G.1E, G.1F, G.1G, G.4A, G.5A, G.5B, G.5C, G.6A

Unit 04: Relationships of Triangles, including Congruence and Similarity

(19 classes for the entire unit)

Students explore patterns and properties of triangles according to sides and angles (interior and exterior angles) using a variety of tools. Students verify theorems involving the sum of the interior angles of a triangle and theorems involving the base angles of isosceles triangles and apply these geometric relationships to solve mathematical and real-world problems. Students compare geometric relationships between Euclidean and spherical geometries, including the sum of the angles in a triangle. Students use constructions to verify the Triangle Inequality theorem and apply the theorem to solve problems. Students construct angle bisectors, segment bisectors, perpendicular lines, and perpendicular bisectors using a compass and a straightedge in order to investigate patterns and make conjectures about geometric relationships of special segments in triangles (altitudes, angle bisectors, medians, perpendicular bisectors, midsegments). Students verify and formalize properties and theorems of special segments and apply the geometric relationships to solve problems. Students analyze patterns of congruent triangles using a variety of methods to identify congruent figures and their corresponding congruent sides and angles. Students use rigid transformations of triangles and constructions to explore triangle congruency. Students formalize a definition of triangle congruency establishing necessary criterion for congruency, as well as formalize postulates and theorems for triangle congruency (Side-Angle-Side, Angle-Side-Angle, Side-Side-Side, Angle-Angle-Side, and Hypotenuse-Leg). Students apply triangle congruency and corresponding parts of congruent triangles are congruent (CPCTC) to prove two triangles are congruent using a variety of proofs. Students apply triangle congruency theorems and CPCTC to solve problems. Students use dilations of triangles and constructions to investigate and explore similarity. Students formalize a definition of triangle similarity establishing corresponding sides of triangles are proportional and corresponding angles of triangles are congruent. Students formalize postulates and theorems to prove triangles are similar using Apply Angle-Angle similarity and the Triangle Proportionality theorem. Students apply triangle similarity to prove two triangles are similar using a variety of proofs. Students apply triangle similarity theorems and proportional understanding to solve problems.

TEKS in this unit: G.1A, G.1B, G.1C, G.1D, G.1E, G.1F, G.1G, G.4D, G.5A, G.5B, G.5C, G.5D, G.6A, G.6B, G.6C, G.6D, G.7A, G.7B, G.8A

Unit 05: Relationships of Right Triangles, including Trigonometry

(22 classes for the entire unit)

Students concretely explore and prove the Pythagorean Theorem and its converse. Students examine patterns of given Pythagorean triples to discover other Pythagorean triples. Students use constructions of right triangles and their altitudes to investigate and analyze geometric relationships, including geometric mean. Students explore geometric relationships in special right triangles (30°-60°-90° and 45°-45°-90°) and similar triangles. Students develop right triangle trigonometry and trigonometric ratios to determine side lengths and angle measures in right triangles. Students apply all right triangle geometric relationships to solve both mathematical and real-world problem situations.

TEKS in this unit: G.1A, G.1B, G.1C, G.1D, G.1E, G.1F, G.1G, G.5B, G.5C, G.6D, G.7A, G.8B, G.9A, G.9B

Unit 06: Relationships of Circles, including Radian Measure and Equations of Circles

(12 classes for the entire unit)

Students define new circle vocabulary, including special segments and angles of circles, using diagrams and definitions. Students use patterns, diagrams, and a variety of tools to investigate circles, chords, secants, tangents, and their angle relationships, including central and inscribed angles. Students use patterns, diagrams, and a variety of tools to investigate circles, chords, secants, tangents, and their segment length relationships. Students apply theorems about combined circle angle/segment length relationships, including central and inscribed angles, in non-contextual problems. Students describe and develop the concept of radian measure of an angle as the ratio of the length of an arc intercepted by a central angle and the radius of the circle. Students convert between degree and radian measures. Students develop the equation of a circle, x² + y² = r², using the coordinate grid and the Pythagorean Theorem, given the radius, r, and center at the origin. Students determine the equation of a circle, (x – h)² + (y – k)² = r², given the radius and a center of (h, k). Students represent real-world situations using equations of circles.

TEKS in this unit: G.1A, G.1B, G.1C, G.1D, G.1E, G.1F, G.1G, G.2B, G.5A, G.12A, G.12D, G.12E

Unit 07: Relationships of Two- and Three-Dimensional Figures

(13 classes for the entire unit)

Students define types of quadrilaterals with a focus on identifying the characteristics (including sides, angles, and diagonal relationships) of parallelograms, rectangles, rhombi, and squares. Students use the characteristics of the quadrilateral to prove quadrilaterals are parallelograms, rectangles, rhombi, or squares using coordinate geometry, two column proofs, paragraph proofs, and flow charts. Students apply quadrilateral relationships to solve real-world problems involving lengths of sides, angles, and midpoints. Students define and identify polygons, including regular polygons by number of sides. Students use tabular, graphical, and symbolic generalization to develop formulas for interior and exterior angles in terms of number of sides. Students extend and apply the properties of quadrilaterals and other polygons and interior and exterior angle theorems to determine lengths of sides, diagonals, midpoints, and all angle measures. Students identify shapes of two-dimensional cross sections of prisms, pyramids, cylinders, cones, and spheres. Students explore and identify three-dimensional objects generated by rotations of two-dimensional shapes.

TEKS in this unit: G.1A, G.1B, G.1C, G.1D, G.1E, G.1F, G.1G, G.2B, G.5A, G.6E, G.10A

Unit 08: Measurement of Two-Dimensional Figures

(15 classes for the entire unit)

Students apply processes for finding area, perimeter, and circumference of two-dimensional figures and investigate dimensional change of two-dimensional figures. Students explore relationships in regular polygons and derive the formula for area of a regular polygon. Students investigate various methods for finding the area of regular polygons in mathematical problems. Students find perimeter, circumference and area of two-dimensional figures and area of regular polygons in problem situations, including proportional and non-proportional dimensional change. Students explore perimeter and area of composite figures, including compositions with regular polygons in problem situations. Students define the arc length of a sector of a circle and the area of a sector of a circle. Students explore the proportional relationships between circumference of circle and arc length and area of circle and area of sector. Students find perimeter and area of composite figures, including compositions with regular polygons in problem situations. Students address changes in scale or measurement units. Students use proportional relationships to find the length of arcs and area of sectors of circles in problem situations.

TEKS in this unit: G.1A, G.1B, G.1C, G.1D, G.1E, G.1F, G.1G, G.10B, G.11A, G.11B, G.12B, G.12C

Unit 09: Measurement of Three-Dimensional Figures

(10 classes for the entire unit)

Students explore surface area using concrete objects and nets of prisms, pyramids, cylinders and cones and make connections to formulas for lateral and total surface area for prisms, pyramids, cylinders and cones. Students are introduced to the formula for surface area of a sphere. Students apply formulas to determine lateral and total surface area of prisms, pyramids, cylinders, cones, and spheres from diagrams and attribute information, including composite figures. Students apply formulas to determine lateral and total surface area of prisms, pyramids, cylinders, cones, and spheres in real-world problem situations with appropriate measures, including effects of proportional and non-proportional linear dimension changes. Students explore volume using concrete models and investigate the differences between cylinders/cones and prisms/pyramids with congruent bases and heights and make connections to formulas for volume. Students are introduced to the formula for volume of a sphere. Students apply formulas to determine volume of prisms, pyramids, cylinders, cones, and spheres from diagrams and attribute information, including composite figures. Students apply formulas to determine lateral and total surface area and volume of prisms, pyramids, cylinders, cones, spheres, and composite figures in real-world problem situations with appropriate measures, including effects of proportional and non-proportional linear dimension changes.

TEKS in this unit: G.1A, G.1B, G.1C, G.1D, G.1E, G.1F, G.1G, G.10B, G.11A, G.11B, G.11C, G.11D

Unit 10: Probability

(12 classes for the entire unit)

Students define and develop formulas for permutations and combinations and apply permutations and combinations to solve contextual problems. Students determine number of possible outcomes of an event, including combinations, permutations, and the Fundamental Counting Principle. Students investigate and define probability in terms of possible outcomes and desired event. Students determine the theoretical and experimental probability based on area models in problem situations. Students identify events as being independent or dependent, and connect independence and dependence to with and without replacement. Students apply the concept of probabilities of independent and dependent events to solve contextual problems. Students are introduced to conditional probability, including notation, and apply conditional probability in contextual problems.

TEKS in this unit: G.1A, G.1B, G.1C, G.1D, G.1E, G.1F, G.1G, G.13A, G.13B, G.13C, G.13D, G.13E

Unit 11: Engineering Design

(10 classes for the entire unit)

Students apply prior knowledge to construct a three-dimensional structure and analyze the structure using two-dimensional views. Student groups research and select a structure composed of polygons and circles that meets at least five specified geometric elements. Students construct a three-dimensional scale model of the structure with lengths labeled to the nearest centimeter. Students draw two-dimensional diagrams of the top, front, and side views of the scale model, labeling all lengths to the nearest centimeter and all angles to the nearest degree. Students determine equations of the lines that represent each edge of the diagrams of the top, front, and side views of the scale model that could be used to create blueprints in a computer graphics software program. Students calculate the exposed surface area and volume of the scale model and summarize how the surface area and volume of the scale model could be used to find the surface area and volume of the original structure in square feet and cubic feet, respectively. Students create a list of at least three conjectures about the geometric relationships in the structure and verify these conjectures using measurement, deductive reasoning, coordinate geometry, or other techniques.

TEKS in this unit: G.1A, G.1B, G.1C, G.1D, G.1E, G.1F, G.1G, G.2C, G.3A, G.5C, G.6D, G.6E, G.7A, G.9A, G.9B, G.10B, G.11B, G.11C, G.11D, G.12B, G.12C

Texas Essential Knowledge & Skills (TEKS)

TEKS - Math - Geometry.pdf

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