Texarkana Independent School District
|Scope and Sequence |
|2009-2010 |
|Texarkana Independent School District |
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|I = Introduced P = Practiced M= Mastered |
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| | 111.34 Geometry Pre-AP (One Credit), | |
| |Grade 10. High School | |
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| |Grading Period | |
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| |(1) The student understands the structure of, and relationships within, an axiomatic system. The student is expected to: | |
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| |develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems | |
| |Including the use of direct proofs, manipulatives and technology to draw conclusions and discover relationships about geometric shapes and their properties | |
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| |recognize the historical development of geometric systems and know mathematics is developed for a variety of purposes | |
| |Including: | |
| |•The discovery of Pi and it’s applications | |
| |•A historical discussion of Euclid’s elements and how they are used in the development of modern geometry | |
| |•A time line of geometry’s developments | |
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| |compare and contrast the structures and implications of Euclidean and non-Euclidean geometries | |
| |Including parallelism as exhibited in Euclid’s 5th postulate. | |
| |Non-Euclidian geometries include: | |
| |•Spherical to show parallel lines do not exist as defined in Euclidean geometry | |
| |•Cylindrical to show parallel lines do exist as defined in Euclidean geometry | |
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| |(2) The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to: | |
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| |use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships | |
| |Including: | |
| |•The use of manipulatives and technology | |
| |•The construction of angle bisectors, perpendicular bisectors, parallel lines, congruent angles, congruent segments, perpendicular lines at a point on a | |
| |line, perpendicular lines from a point to a line and segment bisectors | |
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| |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 | |
| |Including: | |
| |•Reflections | |
| |•Translations | |
| |•Rotations | |
| |•The use of direct proofs, manipulatives and technology to draw conclusions and discover relationships about geometric shapes and their properties | |
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| |(3) The student applies logical reasoning to justify and prove mathematical statements. The student is expected to: | |
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| |(A) determine the validity of a conditional statement, its converse, inverse, and contrapositive | |
| |Including consistent usage as it applies to geometric figures and relationships | |
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| |construct and justify statements about geometric figures and their properties | |
| |Including: | |
| |•The formulation of conclusions in the form of a conditional statement | |
| |•The use of manipulatives and technology to draw conclusions about geometric figures | |
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| |(C) use logical reasoning to prove statements are true and find counter examples to disprove statements that are false | |
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| |(D) use inductive reasoning to formulate a conjecture Including: | |
| |•The student discovery of the sum of the interior angles of a polygon | |
| |•Finding the volume of cones and pyramids | |
| |•The student discovery of relationships among similar polygons and solids | |
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| |(E) use deductive reasoning to prove a statement | |
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| |(4) The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to: | |
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| |(A) select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems Including: | |
| |•Interpreting real-world geometric situations in terms of graphs, tables, and literal equations | |
| |•Describing real-world geometric situations that fit appropriate representations | |
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| |(5) The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to: | |
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| |(A) use numeric and geometric patterns to develop algebraic expressions representing geometric properties Including describing functional | |
| |relationships in writing equations or inequalities as they pertain to: | |
| |•Areas of circles and polygons | |
| |Perimeters of polygons and circumference of circles Including: | |
| |•Finding the sum of the interior angles of polygons | |
| |•Deriving volume formulas | |
| |•Discovering the area formulas for a regular polygon | |
| |Apply on Exit Level TAKS formulas for volume and area in problem solving situations. | |
| |•Discovering the relationship among the sides of 45-45-90 and 30-60-90 triangles | |
| |Tested on Exit Level TAKS | |
| |Apply the relationship of 45-45-90 and 30-60-90 triangles to problem solve. | |
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| |(B) 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 Including properties of polygons, ratios in | |
| |similar figures and solids, and angle relationships in polygons and circles | |
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| |(C) use properties of transformations and their compositions to make connections between mathematics and the real world, such as tessalations | |
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| |(D) 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 Including trigonometric ratios sine, | |
| |cosine, tangent | |
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| |(6) The student analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these | |
| |representations to solve problems. The student is expected to: | |
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| |(A) describe and draw the intersection of a given plane with various three-dimensional geometric figures Including conics | |
| |and other cross-sectional views of geometric solids | |
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| |(B) use nets to represent and construct three-dimensional geometric figures | |
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| |(C) use orthographic and isometric views of three-dimensional geometric figures to represent and construct three-dimensional geometric figures and solve | |
| |problems Including the use of unit blocks to explore concrete models. | |
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| |(7) The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. The| |
| |student is expected to: | |
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| |(A) use one- and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures Including triangles and | |
| |quadrilaterals. | |
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| |(B) use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of | |
| |triangles and other polygons | |
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| |(C) derive and use formulas involving length, slope, and midpoint Including: | |
| |•The relationship between Pythagorean theorem and the distance formula | |
| |•The application of the formulas to prove properties of figures such as rhombi, squares, rectangles, etc… | |
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| |(8) The student uses tools to determine measurement of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem | |
| |situations. The student is expected to: | |
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| |(A) find areas of regular polygons, circles, and composite figures | |
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| |(B) find areas of sectors and arc lengths of circles using proportional reasoning Including: | |
| |• [pic] | |
| |• Area of sector ═ Central Angle | |
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| |Area of circle 3600 | |
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| |(C) derive, extend, and use the Pythagorean Theorem Including: | |
| |•Distance formula | |
| |•Unknown lengths in polygons and circles | |
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| |(D) find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites of these figures in problem situations | |
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| |(9) The student analyzes properties and describes relationships in geometric figures. The student is expected to: | |
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| |(A) formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations and concrete models | |
| |Including: | |
| |•Finding the slopes of lines to determine their relationship (parallel, perpendicular or intersecting) | |
| |•Student discovery of Mid-segment theorem, Dual Parallels theorem, Dual Perpendiculars theorem and Triangle Proportionality theorem | |
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| |(B) formulate and test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models | |
| |Including: | |
| |•Recognizing polygons (through decagons) | |
| |•Properties of regular polygons | |
| |•Properties of quadrilaterals, triangles, and special polygons (e.g. hexagons) | |
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| |(C) formulate and test conjectures about the properties and attributes of circles and the lines that intersect them based on explorations and concrete | |
| |models Including: | |
| |•Identifying tangents, secants, chords, diameters, radii, inscribed angles, central angles | |
| |•Student exploration of the properties of intersecting chords, secants and tangents | |
| |•Exploration of the relationships among angles in circles | |
| |•Application of central angles to the reading of circle graphs | |
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| |(D) analyze the characteristics of polyhedra and other three-dimensional figures and their component parts based on explorations and concrete models | |
| |Including: | |
| |•Prisms (with regular polygon bases to 10 sides) | |
| |•Pyramids | |
| |•Cones | |
| |•Cylinders | |
| |•Spheres | |
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| |(10) The student applies the concept of congruence to justify properties of figures and solve problems. The student is expected to: | |
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| |(A) use congruence transformations to make conjectures and justify properties of geometric figures including figures represented on a coordinate plane | |
| |Including rotations, reflections, translations, and combinations of these | |
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| |(B) justify and apply triangle congruence relationships Including | |
| |•SAS, SSS, ASA, AAS, HL | |
| |•The use of triangle congruence to prove corresponding parts of triangles are congruent | |
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| |(11) The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to: | |
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| |(A) use and extend similarity properties and transformations to explore and justify conjectures about geometric figures Including: | |
| |•Dilations | |
| |•Rotations | |
| |•Reflections | |
| |•Translations | |
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| |(B) use ratios to solve problems involving similar figures Including: | |
| |•Comparing the areas, perimeters and volumes of similar polygons and solids | |
| |•Dilations | |
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| |(C) develop, apply, and justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples using a | |
| |variety of methods | |
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| |(D) 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 | |
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|Additional TEKS |
|111.24 Mathematics, |Grading Period |
|Grade 8. Middle School | |
| |1 |2 |3 |4 |5 |6 |
|(8.1) The student understands that different forms of numbers are appropriate for different | | | | | | |
|situations. The student is expected to: | | | | | | |
|(8.3) Patterns, relationships, and algebraic thinking. The student identifies proportional or | | | | | | |
|non-proportional linear relationships in problem situations and solves problems. | | | | | | |
|The student is expected to: | | | | | | |
|(8.6) The student uses transformational geometry to develop spatial sense. The student is expected| | | | | | |
|to: | | | | | | |
|(B) graph dilations, reflections, and translations on a coordinate plane | | | |IPM | | |
|(8.7) The student uses geometry to model and describe the physical world. The student is expected | | | | | | |
|to: | | | | | | |
|(B) use geometric concepts and properties to solve problems in fields such as art and | | | | | | |
|architecture | | | | | | |
|Include: | | | | | | |
|•Using the given data to solve for perimeter, circumference, area, volume, or dimension |I |P |P |P |P |PM |
|•Various representations of limits of measures | | | | | | |
|(C) use pictures or models to demonstrate the Pythagorean Theorem | | | | | | |
|Including: | | | | | | |
|•When inscribed in a circle or polygon and/or real life pictorial examples | | |I |P |PM | |
|•Vocabulary: (i.e. hypotenuse, leg, radius, diameter) | | | | | | |
|(D) locate and name points on a coordinate plane using ordered pairs of rational numbers | | | | | | |
|Including: | | | | | | |
|•Using all four quadrants | | | | | | |
|•Vocabulary (i.e. x-axis, y-axis, x-coordinate, y-coordinate, quadrants, origin) |I |P |P |P |P |PM |
|(8.8) The student uses procedures to determine measures of three-dimensional figures. The student | | | | | | |
|is expected to: | | | | | | |
|(B) connect models of prisms, cylinders, pyramids, spheres, and cones to formulas for volume of | | | | | | |
|these objects | | | | | | |
|Including: | | | | |IPM | |
|•Matching nets and models to appropriate formulas to problem solve | | | | | | |
|•Real-life models (i.e. sphere-basketball) | | | | | | |
|(C) estimate measurements and use formulas to solve application problems involving lateral and | | | | | | |
|total surface area and volume | | | | | | |
|Including: | | | | | | |
|•Measurements in metric and standard units for cubes, cylinders, cone, spheres, and prisms | | | | | | |
|•Rounding all dimensions to whole numbers | | | | | | |
|•Using “3” for (pi symbol) | | | | | | |
|•The capital B on the formula chart is the area of the base | | | | |IPM | |
|•Vocabulary: (i.e. surface area, prism, rectangular prism, triangular prism, cylinder, pyramid, | | | | | | |
|lateral surface area, edge, face, vertex, height, base, total surface area, net, volume) | | | | | | |
|•Real-life models (i.e. rectangular prism = a present or a shoe box) | | | | | | |
|(9) The student uses indirect measurement to solve problems. The student is expected to: | | | | | | |
|(B) use proportional relationships in similar two-dimensional figures or similar | | | | | | |
|three-dimensional figures to find missing measurements | | | | | | |
|Including: | | | | | | |
|•Setting up proportions or using a scale factor | | | | | | |
|•Identifying the corresponding sides of similar figures when the figure is rotated and/or not | | |I |P |PM | |
|rotated | | | | | | |
|•Vocabulary: (i.e. similar, corresponding, scale factor, dimensions, rotation, proportional and | | | | | | |
|two- and three-dimensional figures) | | | | | | |
|10) The student describes how changes in dimensions affect linear, area, and volume measures. The | | | | | | |
|student is expected to: | | | | | | |
|(B) describe the resulting effect on volume when dimensions of a solid are changed proportionally| | | | | | |
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|(11) The student applies concepts of theoretical and experimental probability to make predictions.| | | | | | |
|The student is expected to: | | | | | | |
|(B) use theoretical probabilities and experimental results to make predictions and decisions | | | | | | |
|Including: | | | | | | |
|•Displaying the results as a fraction or a decimal or percent | | | | | | |
|•Working the problem from a verbal description | | | | | | |
|•Analyzing data from a table or graph | | | | |IP |PM |
|•Using experimental results and comparing those results with the theoretical results. | | | | | | |
|(12) The student uses statistical procedures to describe data. The student is expected to: | | | | | | |
|C) select and use an appropriate representation for presenting and displaying relationships | | | | | | |
|among collected data, including line plots, line graphs, stem and leaf plots, circle graphs, bar | | | | | | |
|graphs, box and whisker plots, histograms, and Venn diagrams, with and without the use of | | | | | | |
|technology | | | | |IP |PM |
|Including: | | | | | | |
|•Frequency tables | | | | | | |
|•Vocabulary (i.e. intervals, scale) | | | | | | |
|(13) The student evaluates predictions and conclusions based on statistical data. The student is | | | | | | |
|expected to: | | | | | | |
|14) The student applies Grade 8 mathematics to solve problems connected to everyday experiences, | | | | | | |
|investigations in other disciplines, and activities in and outside of school. The student is | | | | | | |
|expected to: | | | | | | |
|B) use a problem-solving model that incorporates understanding the problem, making a plan, | | | | | | |
|carrying out the plan, and evaluating the solution for reasonableness | | | | | | |
|This student expectation can be tested in every strand including one or more than one TEKS at a |I |P |P |P |PM | |
|time. | | | | | | |
|C) select or develop an appropriate problem-solving strategy from a variety of different types, | | | | | | |
|including drawing a picture, looking for a pattern, systematic guessing and checking, acting it | | | | | | |
|out, making a table, working a simpler problem, or working backwards to solve a problem |I |P |P |P |PM | |
|This student expectation can be tested in every strand including one or more than one TEKS at a | | | | | | |
|time. | | | | | | |
|15) The student communicates about Grade 8 mathematics through informal and mathematical | | | | | | |
|language, representations, and models. The student is expected to: | | | | | | |
|16) The student uses logical reasoning to make conjectures and verify conclusions. The student is | | | | | | |
|expected to: | | | | | | |
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|B) validate his/her conclusions using mathematical properties and relationships | | | | | | |
|This student expectation can be tested in every strand including one or more than one TEKS at a |I |P |P |P |PM | |
|time. | | | | | | |
|111.32 Algebra I (One Credit). | | | | | | |
|Grade 9. High School | | | | | | |
|(A.1)The student understands that a function represents a dependence of one quantity on another | | | | | | |
|and can be described in a variety of ways. The student is expected to: | | | | | | |
|(B) gather and record data and use data sets, to determine functional relationships between | | | | | | |
|quantities | | | | | | |
|Including: | | | | | | |
|•Students collecting data that models linear and quadratic functions |I |P |P |P |P |PM |
|•Writing equations from a table of data | | | | | | |
|•Generating a list of data from a functional relationship | | | | | | |
|•Using a graphing calculator (specifically using the table function in the calculator). An option| | | | | | |
|would be to teach linear regression using the calculator | | | | | | |
|(C) describe functional relationships for given problem situations and write equations or | | | | | | |
|inequalities to answer questions arising from the situations | | | | | | |
|Including: | | | | | | |
|•Areas of circles and squares |I |P |P |P |P |PM |
|•Perimeters of squares, equilateral triangles, and circumference | | | | | | |
|•Constant rate of change (i.e. slope) | | | | | | |
|•Literal equations (a = lw solve for l) | | | | | | |
| (D) represent relationships among quantities using concrete models, tables, graphs, diagrams, | | | | | | |
|verbal descriptions, equations, and inequalities | | | | | | |
| | | | | | | |
| |I |P |P |P |P |PM |
|(E) interpret and make decisions, predictions, and critical judgments from functional | | | | | | |
|relationships | | | | | | |
|Including linear relationships (constant rate of change), | | | | | | |
|quadratic relationships communicated with concrete models, tables, graphs, diagrams, verbal |I |P |P |P |P |PM |
|descriptions, equations, and inequalities. | | | | | | |
|(A.2)The student uses the properties and attributes of functions. The student is expected to: | | | | | | |
|(B) identify mathematical domains and ranges and determine reasonable domain and range values for | | | | | | |
|given situations, both continuous and discrete | | | | | | |
|Including: | | | | | | |
|•Values displayed in a table | | | | |IP |PM |
|•Values displayed by an equation | | | | | | |
|•Values displayed in a graph. | | | | | | |
|•Values displayed by an inequality. | | | | | | |
|•Values from a verbal description of everyday experiences such as temperature, money, height, etc.| | | | | | |
|(C) interpret situations in terms of given graphs or create situations that fit given graphs | | | | | | |
|Including interpreting real-world situations in terms of graphs and also describing a real-world | | | | | | |
|situation that fits a graph | | | | |IP |PM |
|(D) collect and organize data, make and interpret scatterplots (including recognizing positive, | | | | | | |
|negative, or no correlation for data approximating linear situations), and model, predict, and | | | | | | |
|make decisions and critical judgments in problem situations | | | | | | |
|Including organizing data that demonstrates a positive linear correlation, a negative linear | | | | |IP |PM |
|correlation, and no correlation with and without a graphing calculator | | | | | | |
|(A.3)The student understands how algebra can be used to express generalizations and recognizes and| | | | | | |
|uses the power of symbols to represent situations. The student is expected to: | | | | | | |
|(B) look for patterns and represent generalizations algebraically | | | | | | |
|Including expressions in the form of, but not limited to: | | | | | | |
|•an, an±b, a/n, n/a, a/n ± b, n/a ± b, a ±n, n – a |I |P |P |P |P |PM |
|•geometric sequence | | | | | | |
|•arithmetic sequence | | | | | | |
|•common ratios and differences | | | | | | |
|(A.4) The student understands the importance of the skills required to manipulate symbols in order| | | | | | |
|to solve problems and uses the necessary algebraic skills required to simplify algebraic | | | | | | |
|expressions and solve equations and inequalities in problem situations. The student is expected | | | | | | |
|to: | | | | | | |
|(B) use the commutative, associative, and distributive properties to simplify algebraic | | | | | | |
|expressions |I |P |P |P |PM | |
|(C) use the commutative, associative, and distributive properties to simplify algebraic | | | | | | |
|expressions |I |P |P |P |PM | |
|(A.5) The student understands that linear functions can be represented in different ways and | | | | | | |
|translates among their various representations. The student is expected to | | | | | | |
|(C) use, translate, and make connections among algebraic, tabular, graphical, or verbal | | | | | | |
|descriptions of linear functions | | | | | | |
|Including: | | | | | | |
|•Real-world verbal descriptions of a constant rate of change such as earning an hourly wage or a | | | | | | |
|constant speed. |I |P |P |P |PM | |
|•Connecting the graph of a line to a description of a real-world experience. | | | | | | |
|•Connecting an algebraic expression to a description of a real-world experience. | | | | | | |
|•Using a graphing calculator. | | | | | | |
|(A.6) The student understands the meaning of the slope and intercepts of the graphs of linear | | | | | | |
|functions and zeros of linear functions and interprets and describes the effects of changes in | | | | | | |
|parameters of linear functions in real-world and mathematical situations. The student is expected | | | | | | |
|to: | | | | | | |
|(B) interpret the meaning of slope and intercepts in situations using data, symbolic | | | | | | |
|representations, or graphs | | | | | | |
|Including algebraic equations in slope-intercept form, point-slope form, and standard form with | | | | | | |
|and without a graphing calculator. | | | | | | |
|Such as: |I |P |P |P |PM | |
|•Symbolic representations including use of tables and real world applications | | | | | | |
|•Representation of slope as integers, fractions, decimals and mixed numbers | | | | | | |
|(C) investigate, describe, and predict the effects of changes in m and b on the graph of y = mx + | | | | | | |
|b | | | | | | |
|Including algebraic equations in which the equation is in slope-intercept form, point-slope form, | | | | | | |
|and standard form with and without a graphing calculator. | | | | | | |
|Such as: |I |P |P |P |PM | |
|•Transformation | | | | | | |
|•Changing slope and/or y intercept | | | | | | |
|•Doubling/halving slope | | | | | | |
|•Parallel and perpendicular slope | | | | | | |
|(D) graph and write equations of lines given characteristics such as two points, a point and a | | | | | | |
|slope, or a slope and y-intercept | | | | | | |
|Including algebraic equations in slope-intercept form, point-slope form, and standard form with | | | | | | |
|and without a graphing calculator |I |P |P |P |PM | |
|(E) determine the intercepts of the graphs of linear functions and zeros of linear functions from | | | | | | |
|graphs, tables, and algebraic representations | | | | | | |
|Including algebraic equations in slope-intercept form, point-slope form, and standard form with | | | | |IP |PM |
|and without a graphing calculator. | | | | | | |
|(F) interpret and predict the effects of changing slope and y-intercept in applied situations | | | | | | |
|•Including real-world situations that model a constant change such as a salary, commission, a ride| | | | | | |
|in a taxi, renting a car, speed, buying gasoline, etc. | | | | |IP |PM |
|•Algebraic equations in slope-intercept form, point-slope form, and standard form | | | | | | |
|(G) relate direct variation to linear functions and solve problems involving proportional change | | | | | | |
|Including: | | | | | | |
|•Real-world situations that model a constant change such as a salary, commission, a ride in a | | | | | | |
|taxi, renting a car, speed, buying gasoline, etc. | | | | | | |
|•Algebraic equations in slope-intercept form, point-slope form, and stand form | | |I |P |PM | |
|•Using a graphing calculator | | | | | | |
|(A.7) The student formulates equations and inequalities based on linear functions, uses a variety | | | | | | |
|of methods to solve them, and analyzes the solutions in terms of the situation. The student is | | | | | | |
|expected to | | | | | | |
|(B) investigate methods for solving linear equations and inequalities using concrete models, | | | | | | |
|graphs, and the properties of equality, select a method, and solve the equations and inequalities | | | | | | |
|Including: | | | | | | |
|•Using information from concrete models to write linear equations and inequalities, plot graphs, | | | | | | |
|and solve equations and inequalities | | | | | | |
|•Use graphs to solve linear equations and inequalities | | | | | | |
|•Algebraic equations and inequalities in slope-intercept form, point-slope form, and standard form| | | | |IP |PM |
| | | | | | | |
|•Using a graphing calculator | | | | | | |
|(C) interpret and determine the reasonableness of solutions to linear equations and inequalities | | | | | | |
|Including: | | | | | | |
|•Linear relationships in tables, equations, inequalities, and verbal descriptions | | | | | | |
|•Algebraic equations and inequalities in slope-intercept form, point-slope form, and standard form|I |P |P |P |P |PM |
|•Using a graphing calculator | | | | | | |
|(A.8) The student formulates systems of linear equations from | | | | | | |
|problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms | | | | | | |
|of the situation. The student is expected to: | | | | | | |
|(A.9)The student understands that the graphs of quadratic functions are affected by the parameters| | | | | | |
|of the function and can interpret and describe the effects of changes in the parameters of | | | | | | |
|quadratic functions. Following are performance descriptions. | | | | | | |
|(A.11)The student understands there are situations modeled by functions that are neither linear | | | | | | |
|nor quadratic and models the situations. The student is expected to: | | | | | | |
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