Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Geometry
Document related concepts
Transcript
Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Geometry - Unit Outline First Semester Page Unit Abbreviated Name TEKS 3-7 Logical Arguments and Constructions LOGICON G.1A, G.1B, G.1C, G.2A, G.2B, G.3A, G.3B, G.3C, G.3D, G.3E, G.4, G.9A, G.10A 25 8-10 Coordinate Geometry COGEO G.1C, G.5A, G.7A, G.7B, G.7C 15 11-12 Triangles and Polygons TRIPOLY G.1A, G.2A, G.2B, G.3A, G.3B, G.3C, G.3D, G.5A, G.5B, G.9B 20 13-14 Proof and Congruence (and Rigid Transf) PROCON G.2B, G.3A, G.3B, G.3C, G.3D, G.3E, G.5C, G.7A, G.9B, G.10A, G.10B 18 Time # of Days 78 Second Semester 15-16 Similarity and Proof (& Dilations) SIMPRO G.2B, G.3A, G.3C, G.5B, G.5C, G.7A, G.11A, G.11B, G.11C 17 17-18 Right Triangles RTTRI G.5D, G.8C, G.8F, G.11C 15 19-20 Quadrilaterals QUAD G.1A, G.2A, G.2B, G.5D, G.7A, G.7B, G.7C, G.8C, G.9B, G.10A 15 2DFIG G.2B, G.4, G.5B, G.5D, G.8A, G.8C, G.11B, G.11D 10 3DFIG G.2B, G.4, G.5B, G.5D, G.6A, G.6B, G.6C, G.8D, G.9D, G.11B, G.11D 12 21-23 24-25 Two-Dimensional Figures Three-Dimensional Figures 26 Circles CIRC G.2A, G.2B, G.4, G.5A, G.5B, G.8B, G.9C 15 27 Probability PROB G.8E, New TEKS: G.13A, G.13B, G.13C, G.13D, G.13E 15 99 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry 2015 Process Standards (1) Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to: (A) apply mathematics to problems arising in everyday life, society, and the workplace; (B) 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; (C) select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems; (D) communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate; (E) create and use representations to organize, record, and communicate mathematical ideas; (F) analyze mathematical relationships to connect and communicate mathematical ideas; and (G) display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication. 2 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Logical Arguments and Constructions (25 days) Enduring Understandings The student understands that logical reasoning can be used to justify and prove mathematical statements. The student understands that geometric relationships can be represented in a variety of ways. The student understands that geometric relationships can be analyzed and interpreted through conjectures and proofs. The student understands the structure of, and relationships within, an axiomatic system. Vocabulary conditional, hypothesis, conclusion, theorem, postulate, proof, validity, contrapositive, converse, inverse, counter example, deductive reasoning, inductive reasoning, point, line, plane, space, intersection, betweenness, collinear, non-collinear, coplanar, non-coplanar, ray, segment, midpoint, angle, interior and exterior of an angle, acute, obtuse, right angle, congruent, angle bisector, linear pair, parallel, perpendicular, skew, adjacent angles, vertical angles, complementary, supplementary, transversal, alternate interior (exterior) angles, corresponding angles, same side interior (exterior) angles, circle, radius, diameter, center (of a circle), arc Logical reasoning G.3A(S) Geometric Structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to determine the validity of a conditional statement, its converse, inverse, and contrapositive. G.3C(R) Geometric Structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to use logical reasoning to prove statements are true and find counter examples to disprove statements that are false. G.3D(S) Geometric Structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to use inductive reasoning to formulate a conjecture. G.3E(S) Geometric Structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to use deductive reasoning to prove a statement. The student will know… The student will be able to… The structure of a conditional Write definitions, postulates, and theorems as conditional (if-then) statement (hypothesis and statements and determine their validity. conclusion) and how to recognize Identify the hypothesis and conclusion of a conditional. each part. Write the converse of a conditional and determine its validity. Introduce the concept of inverse and contrapositive of a conditional Use a counter example to disprove a false statement. Apply inductive and deductive reasoning strategies to informal proofs of conditional statements which are true. 3 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Logical Arguments and Constructions, continued Foundation and structure G.4(S) Geometric Structure. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. G.1A Geometric Structure. The student understands the structure of, and relationships within, an axiomatic system. The student is expected to develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. G.1B(S) Geometric Structure The student understands the structure of, and relationships within, an axiomatic system. The student is expected to recognize the historical development of geometric systems and know mathematics is developed for a variety of purposes. G.1C(S) Geometric Structure. The student understands the structure of, and relationships within, an axiomatic system. The student is expected to compare and contrast the structures and implications of Euclidean and nonEuclidean geometries. The student will know… The student will be able to… Identify a basic figure (point, ray, angle, etc.) or concept (skew, That mathematical concepts are congruent, etc.) given a definition, verbal description, or diagram. interconnected. Use concrete models to make and verify conjectures regarding Standard notation for point, line, undefined terms such as point, line, or plane (Ex. Two points ray, etc. determine a line). (Examples: point A, line m or line Give examples and non-examples of each vocabulary term. AB , ray AB , etc.) Describe the attributes of each vocabulary term (Ex. A line has no How postulates and theorems thickness and extends forever in two directions). were developed from the basic Describe the intersection of points, lines, planes, etc. definitions of Euclidean Geometry. Make connections between definitions and related postulates and Plane (Euclidean) geometry is the theorems. geometry of our natural Introduce concept of Euclidean surroundings but there are other Compare and contrast Euclidean and non-Euclidean geometries types of geometries evident in the (spherical, taxicab) in order to emphasize the importance of precise world. definitions and application of postulates. 4 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Logical Arguments and Constructions, continued Constructions and verifications G.2A(S) Geometric Structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. G.2B(R) Geometric Structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to make conjectures about angles, lines, [polygons, circles, and threedimensional figures] and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. G.3B(S) Geometric Structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to construct and justify statements about geometric figures and their properties. The student will know… The student will be able to… That the following can be assumed Represent a point, line, plane, ray, segment in a diagram and label from a drawing: points on a line, appropriately. ray, segment, angle or plane, Represent relationships between/among points, lines, planes, collinear, coplanar, betweenness. segments, rays, etc. such as perpendicularity, parallelism, Constructions can be performed collinearity, etc. using a variety of tools. Use construction methods (i.e. patty paper, safety compass, The difference between a Geometer’s Sketchpad). geometric construction and a Copy a segment sketch/drawing. Add segments Circles are the basis for Find the midpoint of a segment constructions using a compass Use logical reasoning to verify conjectures about basic geometric and straightedge. (Note to teacher: principles. The arc is the “part of a circle” as Betweenness used in constructions.) Segment addition Symmetry is the basis for Midpoint constructions using patty paper. Identify parts of a circle (diameter, center, radius, arc) The vocabulary associated with Relate concept of congruent radii within a circle to using arcs in circles. constructions. Radii of the same circle are Arcs are part of a circle congruent. Radii of a circle are congruent Two circles with congruent radii Arcs drawn with the same radius represent equal distances from the are congruent. center of the circle. 5 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Logical Arguments and Constructions, continued Foundation and structure G.1A Geometric Structure. The student understands the structure of, and relationships within, an axiomatic system. The student is expected to develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. G.2B(R) Geometric Structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to make conjectures about angles, lines, [polygons, circles, and threedimensional figures] and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. The student will know… The student will be able to… The standard diagram notations Make and verify basic postulates regarding point, line, and plane (Ex. for line, ray, segment, point, plane, Two points determine a line). congruence, and distance and Make and verify conjectures regarding terms such as collinear, how to label the parts of a coplanar, midpoint, bisectors, etc. diagram. Draw and label points on a line (including number line). That no assumption can be made Classify angles based on their attributes. from a diagram regarding size or Acute, obtuse, right congruence unless explicitly Complementary, supplementary indicated in the diagram. Linear pair (i.e. angles forming a “straight angle”) Vertical angles Adjacent angles Explore and verify basic postulates about angles and triangles. Angle Addition Postulate Identify special pairs of angles formed by two lines and a transversal. Identify pairs of congruent angles when two parallel lines are cut by a transversal. Identify pairs of supplementary angles when two parallel lines are cut by a transversal. Constructions G.2A(S) Geometric Structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. G.3B(S) Geometric Structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to construct and justify statements about geometric figures and their properties. The student will know… The student will be able to… That perpendicular lines form 4 Perform basic constructions involving angles and triangles. congruent (right) angles. - Measure an angle - Copy an angle Circles are the basis for - Bisect an angle constructions using a compass - Add angles and straightedge. (Note to teacher: - Construct a right angle The arc is the “part of a circle” as used in constructions.) Use construction methods to verify conjectures about parallel and perpendicular lines. Symmetry is the basis for Relate concept of congruent radii within a circle to using arcs in constructions using patty paper. constructions. - Arcs are part of a circle - Radii of a circle are congruent - Arcs drawn with the same radius represent equal distances from the center of the circle. 6 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Logical Arguments and Constructions, continued Applications G.4(S) Geometric Structure. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. G.9A(S) Congruence and the Geometry of Size. The student analyzes properties and describes relationships in geometric figures. The student is expected to formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations and concrete models. G.10A(S) Congruence and the Geometry of Size. The student applies the concept of congruence to justify properties of figures and solve problems. The student is expected to use congruence transformations to make conjectures and justify properties of geometric figures including figures represented on a coordinate plane. The student will know… The student will be able to… That when 2 lines are intersected Find missing measures of angles formed by parallel lines cut by a by a transversal, several special transversal. angle pairs are formed: Solve problems using properties and attributes of angles. corresponding, alternate interior, Solve problems using addition and segment bisector. alternate exterior, same-side Write and verify conditional statements about parallel lines interior, and same-side exterior. intersected by a transversal (Ex. If 2 lines intersected by a That when parallel lines are transversal are parallel, then alternate interior angles are congruent). intersected by a transversal, some Write conjectures about parallel and perpendicular lines (Ex. If two pairs of angles are congruent and lines are perpendicular to the same line, then they are parallel to some pairs of angles are each other). supplementary. Write and verify the converse of conditionals regarding parallel and perpendicular lines. 7 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Coordinate Geometry (15 days) Enduring Understandings The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures. The student understands that the concept of congruence can be used to justify properties of figures and solve problems. Vocabulary distance, midpoint, endpoints, bisector, transversal, parallel, perpendicular, slope, y-intercept, rate of change Distance and midpoint G.1C(S) Geometric Structure. The student understands the structure of, and relationships within, an axiomatic system. The student is expected to compare and contrast the structures and implications of Euclidean and nonEuclidean geometries. G.5A(R) Geometric Patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to use numeric and geometric patterns to develop algebraic expressions representing geometric patterns. G.7A(S) Dimensionality and the Geometry of Location. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. The student is expected to use [one-and] two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures. G.7B(R) Dimensionality and the Geometry of Location. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. The student is expected to use slopes and equations of lines to investigate geometric relationships, including [parallel lines, perpendicular lines, and] special segments of triangles and other polygons. G.7C(R) Dimensionality and the Geometry of Location. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. The student is expected to derive and use formulas involving length[, slope,] and midpoint. The student will know… The student will be able to… 2 2 That x stands for “the change in x” Conceptually develop the distance formula, x y , using the and y stands for “the change in y”. Pythagorean Theorem and the coordinate plane. (Do NOT simplify Context clues for recognizing when radicals here – use calculator approximations as needed.) a problem situation calls for the use Develop the midpoint formula using the coordinate plane and tables. of the distance or midpoint Apply distance and midpoint formulas to verify properties of figures formulas. represented on a coordinate plane (Ex. For triangle ABC, given That the concepts of distance, coordinates of vertices, verify if it is an equilateral triangle). midpoint, and perpendicularity for Apply the distance and midpoint formulas to real-life problem Euclidean geometry may look situations, including problems involving length. different in non-Euclidean Solve problems involving distance and midpoint. geometries. Optional: Compare and contrast Euclidean and non-Euclidean geometries (taxicab) in order to emphasize the importance of precise definitions and application of postulates. 8 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Coordinate Geometry, continued Identifying and representing patterns – algebra G.5A(R) Geometric Patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to use numeric and geometric patterns to develop algebraic expressions representing geometric patterns. The student will know… The student will be able to… A sequence can be represented as Determine the next element in a sequence of geometric figures a function where the independent based on the pattern for the sequence. (x) values represent the number of Determine the next element in a sequence of numbers based on the each term in the sequence and the pattern for the sequence. dependent (y) values represent the - Linear sequence value of each term. - Quadratic sequence, y = ax2 + c only - y = ax2 + bx + c (K level) Write an algebraic expression to represent the value of the nth term in a linear or quadratic sequence. Represent the relationship between term numbers and term values in a linear or quadratic sequence in a variety of forms (tabular, algebraic, graphical). Use finite differences to find a function rule for a table of data (linear and quadratic). Identifying and representing patterns – algebra G.7C(R) Dimensionality and the Geometry of Location. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. The student is expected to derive and use formulas involving [length,] slope, [and midpoint]. The student will know… The student will be able to… Formulas involved in writing Determine the slope of a line from given information. equations of lines: slope-intercept rise y , or ) - Two points (on a graph, form, slope formula, point-slope run x formula and standard form. - y and x from table That the slope of a line can be - Make connections between graphical, tabular, and numerical found in a variety of ways representations depending on the information Write the equation for a line in slope-intercept and point-slope forms: given. - Given two points - Given a point and slope - Given the y-intercept and slope - Given the x- and y-intercepts Manipulate equations to convert to slope-intercept form. Manipulate equations to convert to standard form (Optional for K) 9 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Coordinate Geometry, continued Parallel or perpendicular lines G.7A(S) Dimensionality and the Geometry of Location. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. The student is expected to use [one- and] two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures. G.7B(R) Dimensionality and the Geometry of Location. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. The student is expected to use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons. The student will know… The student will be able to… Parallel lines (except vertical lines) Investigate relationships among parallel and perpendicular lines, rays, have slopes which are equal to or segments and form conjectures (Ex. Parallel lines have equal each other. slopes). All vertical lines are parallel. Use the coordinate plane to explore slopes of lines which are parallel, perpendicular, or neither. The product of the slopes of lines which are perpendicular is -1(i.e. Determine if 2 lines are parallel, perpendicular, or neither given their the slopes are opposite-signed graphs, equations or verbal descriptions. reciprocals of each other). Write the equation of a line that is parallel or perpendicular to a given line. Use properties of parallel and perpendicular to verify conjectures about angles, sides, special segments, etc. of polygons. 10 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Triangles and Polygons (20 days) Enduring Understandings The student understands the structure of, and relationships within, an axiomatic system. The student understands that logical reasoning can be used to justify and prove mathematical statements. The student understands that geometric relationships can be represented in a variety of ways. The student understands that geometric figures can be described and classified by their attributes. Vocabulary corollary (optional), triangle, acute, obtuse, right, scalene, isosceles, legs, base, base angles, vertex angle, equilateral, equiangular, polygon, concave, convex, regular, pentagon, hexagon, heptagon, octagon, decagon, n-gon, altitude, median, midsegment, perpendicular bisector, angle bisector, quadrilateral, centroid, circumcenter, incenter, orthocenter, Euler line (optional) Foundation and Structure G.1A(R) Geometric Structure. The student is expected to develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. G.3A(S) Geometric Structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to determine the validity of a conditional statement, its converse, [inverse, and contrapositive]. G.3C(R) Geometric Structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to use logical reasoning to prove statements are true and find counter examples to disprove statements that are false. G.3D(S) Geometric Structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to use inductive reasoning to formulate a conjecture. The student will know… The student will be able to… Triangles and other polygons can Represent the interrelationship of different types of angles, triangles, be classified by their attributes and other polygons. related to angles, sides, and Classify triangles based on their attributes as described verbally, number of sides. symbolically or from a given diagram (isosceles, acute, etc.). Some attributes of triangles and Explore and verify basic postulates about angles and triangles. other polygons are generalized Triangle Sum Theorem through theorems and postulates. Exterior Angle Theorem Sum of exterior angles Equilateral triangles Isosceles triangles Classify polygons based on their attributes as described verbally, symbolically or from a given diagram. N-sided polygons (pentagon, hexagon, etc.) Regular and non-regular Concave and convex 11 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Triangles and Polygons, continued Applications G.5A(R) Geometric Patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to use numeric and geometric patterns to develop algebraic expressions representing geometric patterns. G.5B(S) Geometric Patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, [ratios in similar figures and solids, and] angle relationships in polygons and circles. The student will know… The student will be able to… An interior angle of a polygon and Develop and apply the Triangle Inequality Postulate. its adjacent exterior angle form a Compare the angle measures of a triangle with the lengths of its linear pair and are supplementary. sides. The sum of the measures of the Extend Triangle Sum Theorem to the sum of interior angles for other interior angles of a triangle is 180. polygons. Constructing diagonals from a Represent the pattern for the measure of one interior angle of a single vertex in a polygon partitions regular polygon of n sides in a variety of ways (algebraically, the polygons into triangular regions. graphically, etc.). Find missing measurements (angles, sides, etc.) in triangles and other polygons based on given information and the attributes and properties of the polygons. Solve problems using properties and attributes of triangles and other polygons. Constructions and explorations G.2A(S) Geometric Structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to use constructions to explore attributes of geometric figures [and to make conjectures about geometric relationships]. G.3B(S) Geometric Structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to construct and justify statements about geometric figures and their properties. The student will know… The student will be able to… Constructions can be performed Perform basic constructions involving angles and triangles, including using a variety of tools. constructing an equilateral triangle. The difference between a geometric Use constructions to verify postulates about triangles. construction and a sketch/drawing. Geometer’s Sketchpad Constructions depend on a Concrete models such as patty paper knowledge of fundamental Compass and straightedge definitions, theorems and Define and construct special segments (median, altitude, bisectors, postulates. midsegment) of angles and triangles. Identify and construct points of concurrency (incenter, circumcenter, orthocenter, centroid). Connect points of concurrency to Euler line. (optional) Verifying conjectures G.2B(R) Geometric Structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to make conjectures about angles, [lines,] polygons,[circles, and threedimensional figures] and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. G.9B(S) Congruence and the Geometry of Size. The student analyzes properties and describes relationships in geometric figures. The student is expected to formulate and test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models. The student will know… The student will be able to… The definitions of median, altitude, Write and verify conjectures about isosceles triangles involving and angle bisector and how it median, altitude, base angles, and angle bisectors. relates to the sides and angles of a triangle. 12 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Proof and Congruence (18 days) Enduring Understandings The student understands that logical reasoning can be used to justify and prove mathematical statements. The student understands the importance of congruence relationships to justify properties of figures and solve problems. The student understands that congruence can be verified in a variety of ways. The student understands that mathematical statements can be proved and justified in a variety of ways. The student understands that geometric relationships can be used to make and verify conjectures. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures. The student understands that the concept of congruence can be used to justify properties of figures and solve problems. Vocabulary proof, congruence, right triangle, hypotenuse, leg, included angle/side, opposite angle/side, corresponding parts/sides/angles, isometry, transformation, image, pre-image, symmetry, mirror, reflection, translation, rotation, similar, tessellation, rigid Logical reasoning - proofs G.3A(S) Geometric Structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to determine the validity of a conditional statement, its converse,[ inverse, and contrapositive]. G.3B(S) Geometric Structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to construct and justify statements about geometric figures and their properties. G.3C(R) Geometric Structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to use logical reasoning to prove statements are true and find counter examples to disprove statements that are false. G.3D(S) Geometric Structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to use inductive reasoning to formulate a conjecture. G.3E(S) Geometric Structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to use deductive reasoning to prove a statement. G.10A(S) Congruence and the Geometry of Size. The student applies the concept of congruence to justify properties of figures and solve problems. The student is expected to use congruence transformations to make conjectures and justify properties of geometric figures including figures represented on a coordinate plane. G.10B(R) Congruence and the Geometry of Size. The student applies the concept of congruence to justify properties of figures and solve problems. The student is expected to justify and apply triangle congruence relationships. The student will know… The student will be able to… The meaning of the acronym Prove congruence relationships using a variety of methods such as CPCTC – “corresponding parts of flowchart proof, coordinate proof, and proof by transformation (patty congruent triangles are congruent”. paper or Geometer’s Sketchpad). That known measures of Prove triangle congruence using Side-side side-(SSS), Side-anglecorresponding parts of congruent side (SAS), Angle-side-angle (ASA), Angle-angle-side (AAS), and triangles can be used to find Hypotenuse leg (HL). unknown measures of the triangles. Use deductive reasoning to prove corresponding parts of congruent triangles congruent (CPCTC). Context clues for recognizing when triangle congruence can be used in Find missing sides of triangles and other figures based on real-life problem situations. congruence relationships. Use triangle congruence to solve problems in real-life situations (Ex. Surveying, p.199). Apply the concept of congruence to other polygons. 13 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Proof and Congruence continued Congruence (Rigid) Transformations G.2B(R) Geometric Structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to make conjectures about angles, lines, polygons, circles, [and threedimensional figures] and determine the validity of the conjectures choosing from a variety of approaches such as coordinate, transformational, or axiomatic. G.7A(S) Dimensionality and the Geometry of Location. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. The student is expected to use [one- and] two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures. G.9B(S) Congruence and the Geometry of Size. The student analyzes properties and describes relationships in geometric figures. The student is expected to formulate and test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models. G.10A(S) Congruence and the Geometry of Size. The student applies the concept of congruence to justify properties of figures and solve problems. The student is expected to use congruence transformations to make conjectures and justify properties of geometric figures including figures represented on a coordinate plane. The student will know… The student will be able to… Transformations such as Identify lines and/or points of symmetry in figures. translations, reflections, and Draw the reflection, translation, or rotation of a figure using a variety rotations do not change the size or of methods (coordinate plane, patty paper, Geometer’s Sketchpad). shape of a figure. - Translations Transformations can be Concept of Rule: (x,y)(x+h,y+k) represented graphically, verbally, Verbal description: vertical / horizontal shifts and symbolically. - Reflections The line of reflection is the Draw reflection over any line perpendicular bisector of the Recognize/draw the line of reflection between a figure and its segment joining a point and its image given the graph. image. Concept of Rules Across the x-axis: (x,y)(x,-y) Across the y-axis: (x,y)(-x,y) Across y = x: (x,y)(y,x) - Rotations Clockwise and counterclockwise 90, 180, 270 Concept of Rules Ex. 90: (x,y) (-y,x) Perform a composition of two or more types of rigid transformations on the same figure. Include transformations that are not centered on an axis or origin. Patterns in the real world G.5C(S) Geometric Patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to use properties of transformations and their compositions to make connections between mathematics and the real world, such as tessellations. The student will know… The student will be able to… Transformations are used in the Describe at least one example of how geometric transformations are real world to create patterns. applied in real-life situations, such as tessellations and graphic design. Nature contains patterns that are related to geometric Recognize the existence of patterns in nature and describe the types transformations (Ex. petals on a of transformations represented. flower). Determine whether or not a plane figure tessellates. End of Semester 14 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Similarity and Proof (17 days) Enduring Understandings The student understands that logical reasoning can be used to justify and prove mathematical statements. The student understands that the concept of similarity can be used to justify properties of figures and solve problems. The student understands that geometric relationships can be represented in a variety of ways. The student understands that geometric relationships can be used to make and verify conjectures. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures. Vocabulary corresponding parts, ratio, proportion, scale factor, similarity, triangle midsegment, geometric mean, transformation, image, pre-image, dilation, compression/reduction, expansion/enlargement, scale factor, non-rigid Logical reasoning & proofs G.3A(S) Geometric Structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to determine the validity of a conditional statement, its converse, [inverse, and contrapositive]. G.3C(R) Geometric Structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to use logical reasoning to prove statements are true and find counter examples to disprove statements that are false. G.11A(S) Similarity and the Geometry of Shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to use and extend similarity properties and transformations to explore and justify conjectures about geometric figures. The student will know… The student will be able to… Corresponding angles of similar Make connections between the dilation of a geometric figure and the figures are congruent. concept of similarity. Corresponding sides of similar Use ratios and proportions to determine if two figures are similar. figures are proportional. Prove (informally) two triangles are similar using AA, SAS, and SSS Apply definition of triangle midsegment to solve problems. Patterns in the real world G.5C(S) Geometric Patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to use properties of transformations and their compositions to make connections between mathematics and the real world, such as tessellations. The student will know… The student will be able to… Transformations are used in the Describe at least one example of how geometric transformations are real world to create patterns. applied in real-life situations, such as graphic design, and fractals. Recognize the existence of patterns in nature and describe the types Nature contains patterns that are of transformations represented. related to geometric transformations (Ex. perspective drawings, object moves closer to viewer, concentric circles, ripples in a pond). 15 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Similarity and Proof Continued Dilations (Non-rigid Transformations) G.2B(R) Geometric Structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to make conjectures about angles, lines, polygons, circles, [and threedimensional figures] and determine the validity of the conjectures choosing from a variety of approaches such as coordinate, transformational, or axiomatic. G.7A(S) Dimensionality and the Geometry of Location. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. The student is expected to use [one- and] two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures. G.11A(S) Similarity and the Geometry of Shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to use and extend similarity properties and transformations to explore and justify conjectures about geometric figures. The student will know… The student will be able to… A dilation of a figure does not Draw the dilation of a figure given the scale factor of the dilation using change the shape of a figure but it the coordinate plane, ruler, or Geometer’s Sketchpad. will change the size according to Identify coordinates of a dilated figure given the pre-image and scale the scale factor used. factor or a pair of corresponding coordinates. A scale factor of 1 does not change Describe the relationship between a pre-image and its dilated image the size of a figure. using the scale factor. Include dilations not centered on the origin. Applications G.5B(S) Geometric Patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to use numeric and geometric patterns to make generalizations about geometric properties, including [properties of polygons,] ratios in similar figures and solids [, and angle relationships in polygons and circles]. G.11B(S) Similarity and the Geometry of Shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to use ratios to solve problems involving similar figures. G.11C(R) Similarity and the Geometry of Shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to develop, apply, and justify triangle similarity relationships, such as right triangle ratios, [trigonometric ratios, and Pythagorean triples] using a variety of methods. The student will know… The student will be able to… Many problem situations can be Use ratios to find missing measures in similar figures. represented by using similar Use similarity and ratios to solve real-life applications. figures and therefore solved using Determine scale factor between 2 similar figures. properties of similar figures. - Scale factor between 0 and 1 when image is reduction of pre Ratio is the comparison between image two measures while scale factor is - Scale factor >1 when image is enlargement of pre-image specifically the multiplier used to Find measurements of similar triangles in a right triangle with an determine a measurement on the altitude drawn to the hypotenuse, including geometric mean in a right: image of a dilated figure. triangle. A scale factor between 0 and 1 reduces a number. A scale factor > 1 enlarges a number. A scale factor = 1 results in a congruence. 16 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Right Triangles (15) Enduring Understandings The student understands the structure of, and relationships within, an axiomatic system. The student understands that the concept of similarity can be used to justify properties of figures and solve problems. The student understands that geometric relationships can be represented in a variety of ways. The student understands that a variety of tools such as the Pythagorean Theorem can be used to determine measurements of geometric figures. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures. Vocabulary hypotenuse, short leg, long leg, ratio, sine, cosine, tangent, angle of elevation/depression Pythagorean Theorem G.5D(R) Geometric Patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to identify and apply patterns from right triangles to solve meaningful problems, including [special right triangles (45-45-90 and 30-60-90) and] triangles whose sides are Pythagorean triples. G.8C(R) Congruence and the Geometry of Size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to derive, extend, and use the Pythagorean Theorem. G.8F(S) Congruence and the Geometry of Size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to use conversions between measurement systems to solve problems in real-world situations. G.11C(R) Similarity and the Geometry of Shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to develop, apply, and justify [triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and] Pythagorean triples using a variety of methods. The student will know… The student will be able to… Estimate the value of radicals (mentally and with a calculator). Right triangles have certain properties that do not exist in non- Simplify basic radical expressions (rationalize denominators – right triangles. optional). For a right triangle with sides a, b, Derive/verify the Pythagorean Theorem using a variety of methods. and c where c is the longest side, Use the Pythagorean Theorem to find the missing sides of a right then a2+ b2 = c2. triangle. For a triangle with sides a, b, and Use the Pythagorean Theorem to classify a triangle as acute, right, or c where c is the longest side, if obtuse. a2+ b2 c2 , then the triangle is Determine whether or not a triangle is a right triangle when given the not a right triangle. side lengths. A radical expression can be Use the Pythagorean Theorem to solve real-life problems involving estimated as having a value right triangles. between two consecutive Convert measurements to a consistent unit of measure both within the integers. same measurement system and between measurement systems. Measurements used in a formula must be of the same unit (Ex. Using the Pythagorean Theorem with one side length in inches and the others in centimeters will not give a correct answer). 17 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Right Triangles continued Special Right Triangles G.5D(R) Geometric Patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to identify and apply patterns from right triangles to solve meaningful problems, including special right triangles (45-45-90 and 30-60-90) [and triangles whose sides are Pythagorean triples]. G.11C(R) Similarity and the Geometry of Shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to develop, apply, and justify triangle similarity relationships, such as right triangle ratios,[ trigonometric ratios, and Pythagorean triples] using a variety of methods. The student will know… The student will be able to… Triangles whose corresponding Identify the patterns found in the side lengths of special right triangles. angles are congruent, have Use special right triangle relationships to find missing sides of 30-60corresponding sides which are 90 and 45-45-90 triangles, approximating values when necessary. proportional and ratios of certain sides are consistent. All 45-45-90 triangles are right isosceles triangles. A 45-45-90 triangle is half of a square. A 30-60-90 triangle is half of an equilateral triangle. Trig ratios and similar right triangles G.11C(R) Similarity and the Geometry of Shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to develop, apply, and justify [triangle similarity relationships, such as right triangle ratios,] trigonometric ratios, [and Pythagorean triples] using a variety of methods. The student will know… The student will be able to… Triangles whose corresponding Develop/verify the sine, cosine, and tangent ratios for right triangles in angles are congruent, have a variety of ways. corresponding sides which are - Measure sides of similar triangles and angles proportional and ratios of certain - Use coordinate plane sides are consistent. - Geometer’s Sketchpad Use sine, cosine, and tangent ratios to find lengths of missing sides and/or measures of missing angles in right triangles, estimating when necessary. Use sine, cosine, and tangent ratios to solve real-life application problems. - Angles of elevation and depression - Engineering/surveying - Construction Develop and apply the Law of Sines (K only – optional – NOTE: must allow time to discuss ambiguous case). 18 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Quadrilaterals (15 days) Enduring Understandings The student understands the structure of, and relationships within, an axiomatic system. The student understands that logical reasoning can be used to justify and prove mathematical statements. The student understands that the concept of congruence can be used to justify properties of figures and solve problems. The student understands that geometric relationships can be represented in a variety of ways. The student understands that a variety of tools such as the Pythagorean Theorem can be used to determine measurements of geometric figures. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures. Vocabulary polygon, quadrilateral, parallelogram, rectangle, square, rhombus, trapezoid, midsegment, isosceles trapezoid, right trapezoid, kite, diagonals, base angles, bisect, consecutive, opposite, supplementary Attributes and properties G.1A(R) Geometric Structure. The student understands the structure of, and relationships within, an axiomatic system. The student is expected to develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. G.2A(S) Geometric Structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. G.2B(R) Geometric Structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to make conjectures about angles, lines, polygons, [circles, and threedimensional figures] and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. G.9B(S) Congruence and the Geometry of Size. The student analyzes properties and describes relationships in geometric figures. The student is expected to formulate and test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models. G.10A(S) Congruence and the Geometry of Size. The student applies the concept of congruence to justify properties of figures and solve problems. The student is expected to use congruence transformations to make conjectures and justify properties of geometric figures including figures represented on a coordinate plane. The student will know… The student will be able to… The defining attributes of a Define: parallelogram, rectangle, square, rhombus, trapezoid parallelogram, rectangle, square, (including right and isosceles), kite according to the attributes rhombus, trapezoid, and kite. of each. The different kinds of quadrilaterals that Investigate/verify properties of parallelograms regarding are inter-related (i.e. all quads are either opposite sides, opposite angles, consecutive angles, and parallelograms or non-parallelograms; a diagonals. square is also a rhombus, rectangle, and Compare properties of a rectangle, square, rhombus, trapezoid parallelogram). (including right and isosceles), and kite to those of a The family of quadrilaterals can be parallelogram. organized by specificity of attributes (Ex. Make and verify conjectures about the angles, sides, diagonals quadrilateral, parallelogram, rectangle, of various quadrilaterals. square). - Example: A diagonal drawn in a rectangle forms two The diagram notations for congruence congruent triangles. (side and angle), right angle, and - Example: AC BD because Quad ABCD is a rhombus. parallel. Use various methods to verify conjectures about quadrilaterals, The meaning of opposite and including patty paper, coordinate graphing, transformations, consecutive as it applies to parts of a Geometer’s Sketchpad, compass and straightedge. polygon. A kite cannot be a rhombus. A trapezoid cannot be a parallelogram. 19 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Quadrilaterals, continued Problems and applications G.5D(R) Geometric Patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to identify and apply patterns from right triangles to solve meaningful problems, including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples. G.8C(R) Congruence and the Geometry of Size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to [derive], extend, and use the Pythagorean Theorem. G.7A(S) Dimensionality and the Geometry of Location. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. The student is expected to use one- and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures. G.7B(R) Dimensionality and the Geometry of Location. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. The student is expected to use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons. G.7C(R) Dimensionality and the Geometry of Location. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. The student is expected to derive and use formulas involving length, slope, and midpoint. The student will know… The student will be able to… The slope of a line can be used to Identify a quadrilateral based on a description or diagram. determine if segments represented Identify the type of quadrilateral represented on a coordinate plane on a coordinate plane are parallel or or from given points using slope to determine parallel or perpendicular. perpendicular segments. Parallel lines have equal slopes. Write and solve equations to find missing sides, angles, or other Perpendicular lines have slopes measures of quadrilaterals based on the properties and attributes which are opposite signed of the quadrilateral and justify the use of the equation. reciprocals of each other. Identify parts of a quadrilateral which are congruent, The segment connecting the supplementary, parallel, perpendicular, etc. based on the midpoints of the non-parallel sides properties and attributes of the given quadrilateral. of a trapezoid is called the Solve real-life application problems involving properties of midsegment. quadrilaterals. Define the midsegment of a trapezoid and use it to solve problems. – Connect to midsegment of triangle. Determine the length of a trapezoid’s midsegment given the lengths of the bases or determine the length of a base given the length of the midsegment and other base. 20 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Two Dimensional Figures (10 days) Enduring Understandings The student understands how perimeter and area are related to a geometric figure. The student understands that two-dimensional figures have unique properties which can be used to solve problems. The student understands that the concept of similarity can be used to justify properties of figures and solve problems. Vocabulary area, base, height, altitude, perimeter, proportional, scale factor, ratio, vertices Finding Area, Perimeter, Circumference G.4(S) Geometric Structure. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. G.5B(S) Geometric Patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons,[ratios in similar figures and solids,] and angle relationships in polygons and circles. G.5D(R) Geometric Patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to identify and apply patterns from right triangles to solve meaningful problems, including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples. G.8A(R) Congruence and the Geometry of Size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to find areas of regular polygons, circles, and composite figures. G.8C(R) Congruence and the Geometry of Size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to [derive], extend, and use the Pythagorean Theorem. The student will know… The student will be able to… Make algebraic connections to area and perimeter. Some geometric formulas can be - Rectangles written as functions and therefore - Squares represented in a variety of ways - Triangles (including equilateral) (graph, table, equation). - Parallelograms The basic area and perimeter - Trapezoids formulas included on the formula - Rhombi chart. - Regular polygons (L – triangle, square, and hexagon or others Missing information in a figure can when apothem or side is given; K – include regular polygons be obtained by applying a where trig may be required to find apothem) definition, property, or theorem Circle (area and circumference) previously learned (Ex. Use the Choose the appropriate formula(s) or strategy for finding the area Pythagorean Theorem to obtain and/or perimeter (circumference). the length of the base so that the Find the area and perimeter (circumference) of geometric figures, area of the triangle can be found). both simple and composite. Context clues for finding area or perimeter (circumference) in real Use the Pythagorean Theorem, patterns of special right triangles, life problem situation. and/or properties and attributes of polygons when necessary to find missing measurements needed to solve an area or perimeter problem. Use Heron’s Formula to find area of triangle (K optional). Find area and perimeter (circumference) in real-life problem situations. 21 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Two Dimensional Figures, continued Dimensional change G.2B(R) Geometric Structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to make conjectures about angles, lines, [polygons, circles, and three-dimensional figures] and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. G.5B(S) Geometric Patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to use numeric and geometric patterns to make generalizations about geometric properties, including [properties of polygons,] ratios in similar figures [and solids, and angle relationships in polygons and circles]. G.11B(S) Similarity and the Geometry of Shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to use ratios to solve problems involving similar figures. G.11D(R) Similarity and the Geometry of Shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to describe the effect on perimeter, area, [and volume] when one or more dimensions of a figure are changed and apply this idea in solving problems. The student will know… The student will be able to… Perimeter is a measure of length Estimate/determine the area or perimeter of a new 2-D figure when and is one-dimensional. one or more dimensions of an original figure are changed proportionally or nonproportionally Area is a two-dimensional measure. o Change both dimensions in the same way Ratio is the comparison between o Change both dimensions in different ways two measures while scale factor is o Change only one dimension specifically the multiplier used to o Type of change: determine a measurement on the x->3x, etc image of a dilated figure. x->x+1, etc Scaling up (or down) both o Compare resulting perimeter or area to original. dimensions of a 2-D figure will increase (or decrease) the perimeter Ex: Given a rectangle whose length is doubled but the width stays the same, how is the area affected? by the value of the scale factor. Answer: The area changes from A to 2A. Scaling up (or down) both Ex. Given a rectangle whose length changes from x to x + 1but the dimensions of a 2-D figure will width stays the same, how is the perimeter affected? increase (or decrease) the area by Answer: Perimeter changes from P to P+2. the square of the scale factor. Determine the scale factor of change in perimeter or area between Scaling up (or down) only one two figures when the dimensions have been changed proportionally. dimension of a 2-D figure will change the area by the same value as the scale factor. 22 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Three Dimensional Figures (12 days) Enduring Understandings The student understands the relationship between three-dimensional geometric figures and related two-dimensional representations. The student understands that three-dimensional figures have unique properties which can be used to solve problems. The student understands that the concept of similarity can be used to justify properties of figures and solve problems. Vocabulary prism, pyramid, sphere, cylinder, cone, vertices, base, edge, face, lateral face, volume, surface area, altitude, slant height, height, isometric drawing, net, orthographic view, proportional, scale factor Attributes and representations G.1C(S) Geometric Structure. The student understands the structure of, and relationships within, an axiomatic system. The student is expected to compare and contrast the structures and implications of Euclidean and nonEuclidean geometries. G.6A(S) Dimensionality and the Geometry of Location. The student analyzes the relationship between threedimensional geometric figures and related two-dimensional representations and uses these representations to solve problems. The student is expected to describe and draw the intersection of a given plane with various threedimensional geometric figures. G.6B(S) Dimensionality and the Geometry of Location. The student analyzes the relationship between threedimensional geometric figures and related two-dimensional representations and uses these representations to solve problems. The student is expected to use nets to represent and construct three-dimensional geometric figures. G.6C(S) Dimensionality and the Geometry of Location. The student analyzes the relationship between threedimensional geometric figures and related two-dimensional representations and uses these representations to solve problems. The student is expected to use orthographic and isometric views of three-dimensional geometric figures to represent and construct three-dimensional geometric figures and solve problems. G.9D(S) Congruence and the Geometry of Size. The student analyzes properties and describes relationships in geometric figures. The student is expected to analyze the characteristics of polyhedra and other three-dimensional figures and their component parts based on explorations and concrete models. The student will know… The student will be able to… The names and attributes (number Name, identify, classify, or describe a 3D figure based on its of bases, edges, etc.) of 3D figures. attributes. A 3D figure can be represented with Draw nets for prisms, pyramids, cylinders, and cones. a two-dimensional drawing of its Identify, construct and draw a 3D figure given its net. faces (net). Identify and describe cross sections of a given 3D figure from A 3D figure can be represented with various angles. a series of two-dimensional drawings Identify and describe the attributes of a 3D figure (edges, faces, representing its front, side, and top vertices, etc.). view (orthographic views). Given a 3D figure (visually or by verbal description), sketch and That some concepts defined for describe the orthographic views (front, top, and side) in an Euclidean geometry may look isometric drawing (Ex. Sketch the top, front, and side views of a different in non-Euclidean prism). geometries. Given a 3D figure, sketch an isometric view. Compare and contrast Euclidean and non-Euclidean geometries (spherical, hyperbolic, taxicab) in order to emphasize the importance of precise definitions and application of postulates. 23 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Three Dimensional Figures, continued Finding surface area and volume G.4(S) Geometric Structure. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. G.5B(S) Geometric Patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles. G.5D(R) Geometric Patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to identify and apply patterns from right triangles to solve meaningful problems, including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples. G.8D(R) Congruence and the Geometry of Size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composite of these figures in problems situations. The student will know… The student will be able to… Context clues for surface area and Find the surface area and volume in problem situations involving volume. cubes, prisms (any base), pyramids (any regular base), cylinders, The word “height” can apply to the cones, spheres, and composite figures. base of a figure or the 3D figure Determine missing measures (radius, height, etc.) of a 3D figure itself. given its surface area and/or volume and other dimensions or The area of the base (B) is an measurements. integral part of surface area and Choose the appropriate formula(s) or strategy for finding the surface volume formulas. area and/or volume of a 3D figure. Use the Pythagorean Theorem, patterns of special right triangles, and/or properties and attributes of polygons when necessary to find missing measurements needed to solve a surface area or volume problem. Dimensional Change G.2B(R) Geometric Structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to make conjectures about [angles, lines, polygons, circles, and] threedimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. G.5B(S) Geometric Patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to use numeric and geometric patterns to make generalizations about geometric properties, including [properties of polygons,] ratios in similar figures and solids, and angle relationships in polygons [and circles]. G.11B(S) Similarity and the Geometry of Shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to use ratios to solve problems involving similar figures. G.11D(R) Similarity and the Geometry of Shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to describe the effect on [perimeter,] area, and volume when one or more dimensions of a figure are changed and apply this idea in solving problems. The student will know… The student will be able to… The ratio of surface areas (or Investigate properties of similar solids. volumes) of two figures is - Use proportional relationships and scale factor to identify similar determined by the scale factors solids. between each of the corresponding - Use proportional reasoning to relate surface area and volume to dimensions of the two figures. the scale factor of similar solids. Investigate proportional dimension changes and their impact on the attributes of a solid object. - Investigate and describe the resulting changes in surface area or volume when one or more dimensions of a three-dimensional object are changed. Solve problems relating to dimensional change in a solid object. 24 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Circles (15 days) Enduring Understandings The student understands that relationships in circles and involving circles or related parts of circles can be represented in a variety of ways. The student understands that proportional reasoning can be used to determine measurements of circles and related figures. Vocabulary circle, circumference, chord, diameter, radius, semicircle, arc, proportion, arc length/measure, major arc, minor arc, intercepted arc, central angle, inscribed angle, inscribed/circumscribed polygon, secant, tangent, point of tangency, area, sector, circle segment (region between arc and chord - optional) Make and verify conjectures G.2A(S) Geometric Structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. G.2B(R) Geometric Structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to make conjectures about angles, lines, [polygons,] circles, [and threedimensional figures] and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. G.5A(R) Geometric Patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to use numeric and geometric patterns to develop algebraic expressions representing geometric patterns. G.5B(S) Geometric Patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to use numeric and geometric patterns to make generalizations about geometric properties, including [properties of polygons, ratios in similar figures, and solids, and] angle relationships in [polygons and] circles. G.9C(S) Congruence and the Geometry of Size. The student analyzes properties and describes relationships in geometric figures. The student is expected to formulate and test conjectures about the properties and attributes of circles and the lines that intersect them based on explorations and concrete models. The student will know… The student will be able to… The vocabulary associated with Identify parts of a circle and related lines, line segments, and circles. points. Attributes of polygons can be Make conjectures about lines, angles, and circles. useful in solving problems involving - Measure of central angle = measure of intercepted arc circles. - Measure of inscribed angle = ½ measure of intercepted arc - Radius drawn to point of tangency is to tangent - Any radius drawn perpendicular to a chord in a circle bisects the chord - Congruent chords intercept congruent arcs - Others as needed Verify conjectures about lines, angles, and circles using a variety of methods, including patty paper, Geometer’s Sketchpad, Google Sketchup, coordinate plane, transformations, and algebraic patterns. Make and verify conjectures about triangles and quadrilaterals inscribed in a circle. - Connecting the Triangle Sum Theorem to the angles of an inscribed triangle - Opposite angles of a quadrilateral are supplementary - Sum of angles in a quadrilateral = 360 Find the measures of angles formed by radii, chords, tangents, or secants of a circle. Find length of line segments connected to a circle (e.g. chords, tangent segments, secant segments). 25 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Measurement G.4(S) Geometric Structure. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. G.8B(S) Congruence and the Geometry of Size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to find areas of sectors and arc lengths of circles using proportional reasoning. The student will know… The student will be able to… The measure of a central angle is a Use proportional reasoning to find the arc length of a given arc fractional part of 360. arc measure arc length OR using The length of an arc is a fractional 360 circumference part of the circle’s circumference. arc measure arc length circumference . 360 Include composite figures with sectors and other 2D figures. Define, identify, and apply congruent arcs. Develop and apply the arc addition postulate. Find the measures of inscribed angles and their intercepted arcs (including an angle inscribed in a semicircle). Find the measures of angles formed by two chords using the measures of the intercepted arcs and vice versa. Incorporate measures represented by algebraic expressions. Use proportional reasoning to find measurements in a circle. - Arc length - Area of sector - Area of segment (K, optional for L) - Real-life applications 26 Geometry Scope and Sequence 2014-15 DRAFT Prentice Hall Mathematics: Texas Geometry Probability (15 days) Enduring Understandings The student understands that the number of possible outcomes of an event can be determined mathematically. The student understands that the concept of probability can be used to determine the likelihood of certain events. The student understands that knowing the probability of an event can lead to making informed predictions and critical judgments. Vocabulary combinations, permutations, factorial, probability, event, sample, outcomes, independent events, dependent events, replacement, conditional probability G.8E(S) Congruence and the Geometry of Size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to use area models to connect geometry to probability and statistics. New TEKS G.13A Probability. The student uses the process skills to understand probability in real-world situations and how to apply independence and dependence of events. The student is expected to develop strategies to use permutations and combinations to solve contextual problems. G.13B Probability. The student uses the process skills to understand probability in real-world situations and how to apply independence and dependence of events. The student is expected to determine probabilities based on area to solve contextual problems. G.13C Probability. The student uses the process skills to understand probability in real-world situations and how to apply independence and dependence of events. The student is expected to identify whether two events are independent and compute the probability of the two events occurring together with or without replacement. G.13D Probability. The student uses the process skills to understand probability in real-world situations and how to apply independence and dependence of events. The student is expected to apply conditional probability in contextual problems. G.13E Probability. The student uses the process skills to understand probability in real-world situations and how to apply independence and dependence of events. The student is expected to apply independence in contextual problems. The student will know… The student will be able to… In counting the number of Determine the number of possible combinations of selection of objects. combinations of a selection of objects, o models such as tree diagram order does not matter. o Fundamental Counting Principle In counting the number of n! o n Cr permutations of a selection of objects, r ! n r ! order does matter. The meaning and uses of factorial as it Determine the number of permutations of a selection of objects. o Tree diagram relates to probability formulas. n! The formulas for permutations and n Pr combinations. n r ! o Probabilities can be modeled using Use permutations and combinations to solve problems. geometric figures. Solve simple probability problems using area models. A probability can be expressed as a Find the probability of independent events. number between 0 and 1, inclusive or Find the probability of dependent events (conditional probability) with and as a percentage between 0 and 100, without replacement. inclusive. The difference between independent See Algebra II Holt textbook, Sections 11-1 to 11-3 and dependent events. In independent events, the occurrence of one event does not affect the probability of the other event. In dependent events, the occurrence of one event affects the probability of the other event. 27
