INSTRUCTIONAL FOCUS DOCUMENT Geometry

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1 UNIT OVERVIEW This unit bundles student expectations that address perimeter and area of two-dimensional figures and composite figures including regular polygons. Effects of dimensional changes and arc length and sector area of circles are also addressed. Concepts are incorporated into both non-contextual and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace. Prior to this unit, in Grade 2, student used concrete models to determine area. In Grade 3, students determined perimeter and area of rectangles and composite figures. In Grade 4, students used formulas to determine perimeter and area of rectangles and squares. In Grade 5, students solve problems related to perimeter. In Grade 6, students determine solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles. In Grade 7, students determine the circumference and area of circles and the area of composite figures. In Grade 8, students model the effect on linear and area measurements of dilated two-dimensional shapes. In Unit 07, students studied the relationships of two- and three-dimensional figures. During this unit, students apply processes for finding area, perimeter, and circumference of two-dimensional figures and investigate dimensional change of two-dimensional figures. Students explore relationships in regular polygons and derive the formula for area of a regular polygon. Students investigate various methods for finding the area of regular polygons in mathematical problems. Students find perimeter, circumference and area of two-dimensional figures and area of regular polygons in problem situations, including proportional and non-proportional dimensional change. Students explore perimeter and area of composite figures, including compositions with regular polygons in problem situations. Students define the arc length of a sector of a circle and the area of a sector of a circle. Students explore the proportional relationships between circumference of circle and arc length and area of circle and area of sector. Students find perimeter and area of composite figures, including compositions with regular polygons in problem situations. Students address changes in scale or measurement units. Students use proportional relationships to find the length of arcs and area of sectors of circles in problem situations. After this unit, in Unit 09, students will find the surface area and volume of three-dimensional figures and composite figures, including dimensional change. In subsequent courses in mathematics, these concepts will continue to be applied to problem situations involving two-dimensional figures. This unit is supporting the development of Texas College Career Readiness Standards (TxCCRS): III. Geometric Reasoning A1, C1, C3; IV. Measurement Reasoning C1, C2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections. According to the National Council of Teachers of Mathematics (2009) in Focus in High School Mathematics: Reasoning and Sense Making, the key elements of reasoning and sense making with geometry must include multiple representations of functions. In this unit, students gather data from geometric figures, and organize this information into tables, graphs or diagrams. This leads to the development of symbolic expressions and verbal descriptions. A variety of representations helps make relationships more understandable to more students than working with symbolic representations alone. These approaches serve as the basis for this unit on polygons and circle. At the conclusion of this unit, students are asked to create graphic organizers. TxCCRS cites many skills related to the communication and representation of mathematical ideas. The National Council of Teachers of Mathematics (2000) said that all students in grades 9 12 should explore relationships in two-dimensional geometric figures, make and Last Updated 10/17/2015 Page 1 of 24

2 test conjectures about two-dimensional geometric figures, and solve problems involving two-dimensional geometric figures. According to the National Council of Teachers of Mathematics, using diagrams and constructions to interpret and communicate geometric relationships is essential in geometry. Using definitions of figures to characterize figures in terms of their properties is another essential in geometry. In geometry, the proving process involves working with diagrams, variation and invariance, conjectures, and definitions. (2012, p. 92) National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc. National Council of Teachers of Mathematics. (2009). Focus in High School Mathematics: Reasoning and Sense Making. Reston, VA: National Council of Teachers of Mathematics, Inc. National Council of Teachers of Mathematics. (2012). Developing essential understanding of for Teaching Mathematics in Grades Reston, VA: National Council of Teachers of Mathematics, Inc. Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from OVERARCHING UNDERSTANDINGS AND QUESTIONS Geometric and spatial reasoning are necessary to describe and analyze geometric relationships in mathematics and the real-world. Why are geometric and spatial reasoning necessary in the development of an understanding of geometric relationships? Why is it important to visualize and use diagrams to effectively communicate/illustrate geometric relationships? Attributes and properties of two-dimensional geometric shapes are foundational to developing geometric and measurement reasoning. Why is it important to compare and contrast attributes and properties of two-dimensional geometric shapes? How does analyzing the attributes and properties of two-dimensional geometric shapes develop geometric and measurement reasoning? Application of attributes and measures of figures can be generalized to describe geometric relationships which can be used to solve problem situations. Why are attributes and measures of figures used to generalize geometric relationships? Last Updated 10/17/2015 Page 2 of 24

3 How can numeric patterns be used to formulate geometric relationships? Why is it important to distinguish measureable attributes? How do geometric relationships relate to other geometric relationships? Why is it essential to develop generalizations for geometric relationships? How are geometric relationships applied to solve problem situations? How do different systems of measure relate to one another? PERFORMANCE ASSESSMENT(S) Unit 08 PA A circular, advertising layout has a radius of 8 inches. a. Find the circumference and area of the logo. b. In order to customize the logo for display at a convention, the company would like to enlarge the circular logo by a scale factor of 4. Find the circumference and area of the enlarged logo. c. Buttons were created to hand out at the convention using the logo compressed by a scale factor of. nd the circumference and area of the compressed logo. d. What geometric relationship can be used to justify the effects on circumference and area OVERARCHING CONCEPTS UNIT CONCEPTS Geometric Reasoning Geometric Attributes/Properties Geometric Relationships Scale Factors Similarity Two-Dimensional Figures Measurement Reasoning Length Dimensional Change Formulas Perimeter/Circumference Systems of Measurement Associated Mathematical Processes UNIT UNDERSTANDINGS Diagrams can be used to visualize and illustrate geometric relationships and aid in solving problems. Why are diagrams necessary for visualizing the geometric relationships found in the problem situation? How are diagrams used to organize information from the problem situation? How do diagrams aid in calculations when solving problems? If all dimensions of a two-dimensional figure are changed by the same scale factor, the result is a proportional change in perimeter, circumference, and area; whereas, if only one dimension of a twodimensional figure is changed by a scale factor, the result is a nonproportional change in perimeter, circumference, and area. How are similar figures generated? What geometric relationship exists between linear dimensions, perimeter and area of a two-dimensional object that has Last Updated 10/17/2015 Page 3 of 24

4 PERFORMANCE ASSESSMENT(S) when enlarging and compressing the circular logo? 2. The children s rectangular doorway to the mouse house at a local amusement park is similar to the adult doorway, but the area of the children s doorway is as large as the adult doorway. a. Determine the ratio of the perimeters of the children s doorway to the adult s doorway. b. If the adult doorway is 200 cm tall and 110 cm wide, what are the dimensions of the children s doorway? c. What are the perimeter and area of both the children s doorway and the adult doorway? d. What geometric relationship exists between the similarity ratio of the linear dimensions, perimeter, and area of similar polygons? e. Suppose only the width of the adult doorway needed to be widened by 30 cm. How would this affect the perimeter and area of the adult doorway? 3. To renovate the kitchen, a couple decides to replace the painted area behind the stove top with OVERARCHING CONCEPTS UNIT CONCEPTS Application Tools and Techniques Problem Solving Model Communication Representations Relationships Justification UNIT UNDERSTANDINGS undergone a proportional dimension change? What geometric relationship exists between linear dimensions, perimeter and area of a two-dimensional object that has undergone a non-proportional dimension change? What are some of the possible effects on the perimeter and area of a two-dimensional object when a scale factor is applied to just one of the dimensions? Scale factor on two dimensions? How can the resulting effects on the perimeter and area of the scaled two-dimensional object be predicted? A geometric relationship exists between the apothem, perimeter, and area of a regular polygon. How can the area of a regular polygon be found if the side length is given? How can the area of a regular polygon be found if the apothem is known? How can the area of a regular polygon be found if the perimeter is known? How can the perimeter of a regular polygon be found if the apothem is known? How can the area of a regular polygon be found using nonoverlapping right isosceles triangles? How can the radius of a regular polygon be used to find its area? Last Updated 10/17/2015 Page 4 of 24

5 PERFORMANCE ASSESSMENT(S) a backsplash composed of regular, hexagonal tiles. The wall area that they need to cover is 6 feet by 2 feet. The hexagonal tiles they selected have side length of 4 inches. a. Find the area of one hexagonal tile using the formula for the area of a regular polygon. b. Justify your answer in part a) by using congruent isosceles triangles. Explain your reasoning. c. How many tiles will be needed to cover the backsplash area of their kitchen? Explain your reasoning. Standard(s): G.1A, G.1B, G.1C, G.1D, G.1E, G.1F, G.1G, G.10B, G.11A OVERARCHING CONCEPTS UNIT CONCEPTS UNIT UNDERSTANDINGS Unit 08 PA A polygonal region is formed by plotting the following points on a coordinate plane: (0, 0), (0, 2), (1, 3), (4, 3), (5, 2), (6, 1), and (5, 0). a. Plot the points on a coordinate plane. b. Find the perimeter and area of the polygonal region determined by the ordered pairs. Geometric Reasoning Composition/Decomposition of Figures Composite Figures Geometric Attributes/Properties Geometric Relationships Scale Factors Similarity Two-Dimensional Figures Diagrams can be used to visualize and illustrate geometric relationships and aid in solving problems. Why are diagrams necessary for visualizing the geometric relationships found in the problem situation? How are diagrams used to organize information from the problem situation? How do diagrams aid in calculations when solving problems? The area of composite two-dimensional figures can be determined by Last Updated 10/17/2015 Page 5 of 24

6 PERFORMANCE ASSESSMENT(S) 2. A blueprint for a football field and track is shown below. The track and field is composed of rectangles with semicircles on each end to complete the curves of the track. On the blueprint the outer running track has a width of 5 inches. The inner rectangle measures 55 inches by 34 inches. OVERARCHING CONCEPTS UNIT CONCEPTS Measurement Reasoning Conversions Length Dimensional Change Formulas Perimeter/Circumference Systems of Measurement Associated Mathematical Processes UNIT UNDERSTANDINGS calculating and combining the areas of the shapes that comprise the figure. How can a two-dimensional figure be comprised of a combination of shapes? What shapes could be used to create a composite twodimensional figure? Why might it be necessary to break down an irregular figure into its component shapes? How can the perimeter and area of a composite twodimensional figure be determined? a. Using the blueprint measurements, find the actual dimensions of the running track and football field if the blueprint follows a scale of 1 inch : 2 meters. b. Steeple High School athletic department must present the actual perimeter and area that will be encompassed by the new track and field to the school board. Find the perimeter and area that is taken up by the actual track and field. Round to the nearest hundredth as necessary. Application Tools and Techniques Problem Solving Model Communication Representations Relationships Justification A geometric relationship exists between the measure of an arc length of a circle and the circumference of the circle. What relationship exists between the arc length of a circle and its circumference? How can a specific arc length of a circle be used to find the circumference of the circle? How can the circumference of the circle be used to find a specific arc length of the circle? A geometric relationship exists between the measure of the area of a sector of a circle and its area. Last Updated 10/17/2015 Page 6 of 24

7 PERFORMANCE ASSESSMENT(S) 3. CAP is a central angle of circle A. The m CAP is 130 o, and the radius of circle A = 10 inches. a. Find the length of to the nearest inch. Justify your answer. b. Find the area of sector CAP to the nearest tenth of a square inch. Justify your answer. 4. An automatic sprinkler is set so that it turns through an angle of 100 o then returns back to its original setting. The sprinkler shoots water up to 15 feet away from the sprinkler head. a. Find the area of grass watered by the sprinkler. b. The flower garden to be watered is in the shape of an isosceles triangle with a base of 18 feet and an altitude of 6 feet. Where should the sprinkler head be located to sufficiently water the flower garden? Explain your reasoning. Standard(s): G.1A, G.1B, G.1C, G.1D, G.1E, G.1F, G.1G, G.11B, G.12B, G.12C OVERARCHING CONCEPTS UNIT CONCEPTS UNIT UNDERSTANDINGS What relationship is found between the measure of the area of a sector of a circle and its area? How can the area of a sector of a circle be used to find the area of the circle? How can the area of a circle be used to find the measure of the area of a sector? MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS Misconceptions: Last Updated 10/17/2015 Page 7 of 24

8 Some students may think that the length of the arc of a sector is the same as the degree measure of the arc instead of it being a measure in dimensions of length. Some students may think that the same scale factor is used for area of proportional dimensional change as is used for perimeter (circumference) instead of using the scale factor squared. Some students may think the apothem is the same as the radius from the center of a regular polygon to the vertex instead of the perpendicular distance from the center to the side of a regular polygon. Underdeveloped Concepts: Some students may not be able to distinguish the components of composite figures. Some students may not know which formulas to use when determining the perimeter, circumference, and area of the components of a composite figure. UNIT VOCABULARY Apothem a segment that extends from the center of a regular polygon perpendicular to a side of the regular polygon. The apothem bisects the side of the regular polygon to which it is drawn. Arc length of a circle a fractional distance of the circumference of a circle defined by the arc Radius of a regular polygon a segment that extends from the center of a regular polygon to a vertex. The radius of a regular polygon bisects the vertex angle to which it is drawn. Regular polygon a convex polygon in which all sides are congruent (equilateral) and all angles are congruent (equiangular) Sector of a circle a region of the circle bounded by a central angle and its intercepted arc Two-dimensional non-proportional change either only one dimension multiplied by a scale factor or the two dimensions are multiplied by different scale factors Two-dimensional proportional change two dimensions multiplied by the same scale factor Related Vocabulary: Area Bisect Composite figure Linear dimension Perpendicular Radius Last Updated 10/17/2015 Page 8 of 24

9 Circumference Perimeter Scale factor SYSTEM RESOURCES Mathematics Implementation TAG Tool Mathematics Concepts Tree OTHER RESOURCES Texas Education Agency Revised Mathematics TEKS: Side-by-Side TEKS Comparison Texas Education Agency Texas College and Career Readiness Standards Texas Education Agency - Revised Mathematics TEKS: Vertical Alignment Charts Texas Instruments - Graphing Calculator Tutorials Texas Education Agency Mathematics Curriculum Texas Education Agency Assessment TEKS# SE# TEKS UNIT LEVEL SPECIFICITY Bold black text in italics: Knowledge and Skills Statement (TEKS) Bold black text: Student Expectation (TEKS) Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s) Blue text: Supporting information / Clarifications from TCMPC (Specificity) Blue text in italics: Unit-specific clarification Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS) G.1 Mathematical process standards. The student uses Last Updated 10/17/2015 Page 9 of 24

10 TEKS# SE# mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to: TEKS UNIT LEVEL SPECIFICITY G.1A Apply mathematics to problems arising in everyday life, society, and the workplace. Apply MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE Note(s): The mathematical process standards may be applied to all content standards as appropriate. TxCCRS: X. Connections G.1B 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. 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 Note(s): The mathematical process standards may be applied to all content standards as appropriate. TxCCRS: VIII. Problem Solving and Reasoning G.1C Select tools, including real objects, manipulatives, paper and pencil, and technology Select Last Updated 10/17/2015 Page 10 of 24

11 TEKS# SE# TEKS UNIT LEVEL SPECIFICITY as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, TO SOLVE PROBLEMS Select TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS Note(s): The mathematical process standards may be applied to all content standards as appropriate. TxCCRS: VIII. Problem Solving and Reasoning G.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate. Communicate MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE Note(s): The mathematical process standards may be applied to all content standards as appropriate. TxCCRS: IX. Communication and Representation G.1E Create and use representations to organize, record, and communicate mathematical ideas. Create, Use Last Updated 10/17/2015 Page 11 of 24

12 TEKS# SE# TEKS UNIT LEVEL SPECIFICITY REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS Note(s): The mathematical process standards may be applied to all content standards as appropriate. TxCCRS: IX. Communication and Representation G.1F Analyze mathematical relationships to connect and communicate mathematical ideas. Analyze MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS Note(s): The mathematical process standards may be applied to all content standards as appropriate. TxCCRS: X. Connections G.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication. Display, Explain, Justify MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION Note(s): The mathematical process standards may be applied to all content standards as appropriate. TxCCRS: IX. Communication and Representation Last Updated 10/17/2015 Page 12 of 24

13 TEKS# SE# TEKS UNIT LEVEL SPECIFICITY G.10 Two-dimensional and three-dimensional figures. The student uses the process skills to recognize characteristics and dimensional changes of twoand three-dimensional figures. The student is expected to: G.10B Determine and describe how changes in the linear dimensions of a shape affect its perimeter, area, surface area, or volume, including proportional and non-proportional dimensional change. Determine, Describe HOW CHANGES IN THE LINEAR DIMENSIONS OF A SHAPE AFFECT ITS PERIMETER, OR AREA INCLUDING PROPORTIONAL AND NON-PROPORTIONAL DIMENSIONAL CHANGE Including, but not limited to: Note(s): Verbal and written description Dimensional change Perimeter and circumference Area Proportional change Two-dimensional proportional change two dimensions multiplied by the same scale factor Non-proportional change Two-dimensional non-proportional change either only one dimension multiplied by a scale factor or the two dimensions are multiplied by different scale factors Comparison of the effect of proportional and non-proportional dimensional change Emphasis on connections to units Dimension changes in real-world problem situations Grade Level(s) Last Updated 10/17/2015 Page 13 of 24

14 TEKS# SE# TEKS UNIT LEVEL SPECIFICITY Grade 7 and 8 modeled the effect on linear and area measurements of dilated twodimensional shapes. Various mathematical process standards will be applied to this student expectation as appropriate. TxCCRS III. Geometric Reasoning C1 Make connections between geometry and algebra. C3 Make connections between geometry and measurement. IV. Measurement Reasoning C1 Find the perimeter and area of two-dimensional figures. C2 Determine the surface area and volume of three-dimensional figure. VIII. Problem Solving and Reasoning IX. Communication and Representation X. Connections G.11 Two-dimensional and three-dimensional figures. The student uses the process skills in the application of formulas to determine measures of two- and three-dimensional figures. The student is expected to: G.11A Apply the formula for the area of regular polygons to solve problems using appropriate units of measure. Apply THE FORMULA FOR THE AREA OF REGULAR POLYGONS TO SOLVE PROBLEMS USING APPROPRIATE UNITS OF MEASURE Including, but not limited to: Regular polygon a convex polygon in which all sides are congruent (equilateral) and all angles Last Updated 10/17/2015 Page 14 of 24

15 TEKS# SE# TEKS UNIT LEVEL SPECIFICITY are congruent (equiangular) Radius of a regular polygon a segment that extends from the center of a regular polygon to a vertex. The radius of a regular polygon bisects the vertex angle to which it is drawn. Apothem a segment that extends from the center of a regular polygon perpendicular to a side of the regular polygon. The apothem bisects the side of the regular polygon to which it is drawn. Formula for the area of regular polygons A = ap where P represents the perimeter and a represents the apothem. Note(s): Connection to area of a triangle: A = bh Real-world problem situations involving area Emphasis on appropriate units of measure Grade Level(s) Previous grade levels used units, tools, and formulas to find the area of figures in problem situations. Previous grade levels introduced the language of regular polygons. Various mathematical process standards will be applied to this student expectation as appropriate. TxCCRS III. Geometric Reasoning C3 Make connections between geometry and measurement. IV. Measurement Reasoning C1 Find the perimeter and area of two-dimensional figures. VIII. Problem Solving and Reasoning IX. Communication and Representation Last Updated 10/17/2015 Page 15 of 24

16 TEKS# SE# TEKS UNIT LEVEL SPECIFICITY X. Connections G.11B Determine the area of composite two-dimensional figures comprised of a combination of triangles, parallelograms, trapezoids, kites, regular polygons, or sectors of circles to solve problems using appropriate units of measure. Determine THE AREA OF COMPOSITE TWO-DIMENSIONAL FIGURES COMPRISED OF A COMBINATION OF TRIANGLES, PARALLELOGRAMS, TRAPEZOIDS, KITES, REGULAR POLYGONS, OR SECTORS OF CIRCLES TO SOLVE PROBLEMS USING APPROPRIATE UNITS OF MEASURE Including, but not limited to: Note(s): Composites of two-dimensional figures Triangles Parallelograms Trapezoids Kites Regular polygons Sectors of circles Applications to real-world situations Appropriate use of units of measure Grade Level(s) Previous grade levels used units, tools, and formulas to find the area of figures in problem situations. Previous grade levels introduced composites of two-dimensional figures. Various mathematical process standards will be applied to this student expectation as appropriate. TxCCRS Last Updated 10/17/2015 Page 16 of 24

17 TEKS# SE# TEKS UNIT LEVEL SPECIFICITY III. Geometric Reasoning C3 Make connections between geometry and measurement. VIII. Problem Solving and Reasoning IX. Communication and Representation X. Connections G.12 Circles. The student uses the process skills to understand geometric relationships and apply theorems and equations about circles. The student is expected to: G.12B Apply the proportional relationship between the measure of an arc length of a circle and the circumference of the circle to solve problems. Apply THE PROPORTIONAL RELATIONSHIP BETWEEN THE MEASURE OF AN ARC LENGTH OF A CIRCLE AND THE CIRCUMFERENCE OF THE CIRCLE TO SOLVE PROBLEMS Including, but not limited to: Arc length of a circle a fractional distance of the circumference of a circle defined by the arc Connecting the proportional relationship Applications to real-world problem situations Use of appropriate units of measure Use of various tools Protractor and straightedge Dynamic geometric software Patty paper Last Updated 10/17/2015 Page 17 of 24

18 TEKS# SE# TEKS UNIT LEVEL SPECIFICITY Note(s): Grade Level(s) Previous grade levels explored characteristics of circles and proportional relationships. Various mathematical process standards will be applied to this student expectation as appropriate. TxCCRS III. Geometric Reasoning A1 Identify and represent the features of plane and space figures. C3 Make connections between geometry and measurement. VIII. Problem Solving and Reasoning IX. Communication and Representation X. Connections G.12C Apply the proportional relationship between the measure of the area of a sector of a circle and the area of the circle to solve problems. Apply THE PROPORTIONAL RELATIONSHIP BETWEEN THE MEASURE OF THE AREA OF A SECTOR OF A CIRCLE AND THE AREA OF THE CIRCLE TO SOLVE PROBLEMS Including, but not limited to: Sector of a circle a region of the circle bounded by a central angle and its intercepted arc Connecting the proportional relationship Applications to real-world problem situations Use of appropriate units of measure Use of various tools Protractor and straightedge Last Updated 10/17/2015 Page 18 of 24

19 TEKS# SE# TEKS UNIT LEVEL SPECIFICITY Dynamic geometric software Patty paper Note(s): Grade Level(s) Previous grade levels explored characteristics of circles and proportional relationships. Various mathematical process standards will be applied to this student expectation as appropriate. TxCCRS III. Geometric Reasoning A1 Identify and represent the features of plane and space figures. C3 Make connections between geometry and measurement. VIII. Problem Solving and Reasoning IX. Communication and Representation X. Connections ELPS# SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS. The English Language Proficiency Standards (ELPS), as required by 19 Texas Administrative Code, Chapter 74, Subchapter A, 74.4, outline English language proficiency level descriptors and student expectations for English language learners (ELLs). School districts are required to implement ELPS as an integral part of each subject in the required curriculum. School districts shall provide instruction in the knowledge and skills of the foundation and enrichment curriculum in a manner that is linguistically accommodated commensurate with the student s levels of English language proficiency to ensure that the student learns the knowledge and skills in the required curriculum. School districts shall provide content-based instruction including the cross-curricular second language acquisition essential knowledge and skills in subsection (c) of the ELPS in a manner that is linguistically accommodated to help the student acquire English language proficiency. Last Updated 10/17/2015 Page 19 of 24

20 ELPS# Choose appropriate ELPS to support instruction. SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS. ELPS.c.1 ELPS.c.1A ELPS.c.1B ELPS.c.1C ELPS.c.1D ELPS.c.1E ELPS.c.1F ELPS.c.1G ELPS.c.1H ELPS.c.2 The ELL uses language learning strategies to develop an awareness of his or her own learning processes in all content areas. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to: use prior knowledge and experiences to understand meanings in English monitor oral and written language production and employ self-corrective techniques or other resources use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and grade-level vocabulary speak using learning strategies such as requesting assistance, employing non-verbal cues, and using synonyms and circumlocution (conveying ideas by defining or describing when exact English words are not known) internalize new basic and academic language by using and reusing it in meaningful ways in speaking and writing activities that build concept and language attainment use accessible language and learn new and essential language in the process demonstrate an increasing ability to distinguish between formal and informal English and an increasing knowledge of when to use each one commensurate with grade-level learning expectations develop and expand repertoire of learning strategies such as reasoning inductively or deductively, looking for patterns in language, and analyzing sayings and expressions commensurate with grade-level learning expectations. The ELL listens to a variety of speakers including teachers, peers, and electronic media to gain an increasing level of comprehension of newly acquired language in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in listening. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to: Last Updated 10/17/2015 Page 20 of 24

21 ELPS# ELPS.c.2A ELPS.c.2B ELPS.c.2C ELPS.c.2D ELPS.c.2E ELPS.c.2F ELPS.c.2G ELPS.c.2H ELPS.c.2I ELPS.c.3 ELPS.c.3A ELPS.c.3B SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS. distinguish sounds and intonation patterns of English with increasing ease recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language listen to and derive meaning from a variety of media such as audio tape, video, DVD, and CD ROM to build and reinforce concept and language attainment understand the general meaning, main points, and important details of spoken language ranging from situations in which topics, language, and contexts are familiar to unfamiliar understand implicit ideas and information in increasingly complex spoken language commensurate with grade-level learning expectations demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs. The ELL speaks in a variety of modes for a variety of purposes with an awareness of different language registers (formal/informal) using vocabulary with increasing fluency and accuracy in language arts and all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in speaking. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to: practice producing sounds of newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters to pronounce English words in a manner that is increasingly comprehensible expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication Last Updated 10/17/2015 Page 21 of 24

22 ELPS# ELPS.c.3C ELPS.c.3D ELPS.c.3E ELPS.c.3F ELPS.c.3G ELPS.c.3H ELPS.c.3I ELPS.c.3J ELPS.c.4 ELPS.c.4A ELPS.c.4B ELPS.c.4C SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS. speak using a variety of grammatical structures, sentence lengths, sentence types, and connecting words with increasing accuracy and ease as more English is acquired speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency share information in cooperative learning interactions ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended speaking assignments express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics narrate, describe, and explain with increasing specificity and detail as more English is acquired adapt spoken language appropriately for formal and informal purposes respond orally to information presented in a wide variety of print, electronic, audio, and visual media to build and reinforce concept and language attainment. The ELL reads a variety of texts for a variety of purposes with an increasing level of comprehension in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in reading. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations apply to text read aloud for students not yet at the stage of decoding written text. The student is expected to: learn relationships between sounds and letters of the English language and decode (sound out) words using a combination of skills such as recognizing sound-letter relationships and identifying cognates, affixes, roots, and base words recognize directionality of English reading such as left to right and top to bottom develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials Last Updated 10/17/2015 Page 22 of 24

23 ELPS# ELPS.c.4D ELPS.c.4E ELPS.c.4F ELPS.c.4G ELPS.c.4H ELPS.c.4I ELPS.c.4J ELPS.c.4K ELPS.c.5 ELPS.c.5A SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS. use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary and other prereading activities to enhance comprehension of written text read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding to questions, and taking notes commensurate with content area and grade level needs read silently with increasing ease and comprehension for longer periods demonstrate English comprehension and expand reading skills by employing basic reading skills such as demonstrating understanding of supporting ideas and details in text and graphic sources, summarizing text, and distinguishing main ideas from details commensurate with content area needs demonstrate English comprehension and expand reading skills by employing inferential skills such as predicting, making connections between ideas, drawing inferences and conclusions from text and graphic sources, and finding supporting text evidence commensurate with content area needs demonstrate English comprehension and expand reading skills by employing analytical skills such as evaluating written information and performing critical analyses commensurate with content area and grade-level needs. The ELL writes in a variety of forms with increasing accuracy to effectively address a specific purpose and audience in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in writing. In order for the ELL to meet gradelevel learning expectations across foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations do not apply until the student has reached the stage of generating original written text using a standard writing system. The student is expected to: learn relationships between sounds and letters of the English language to represent sounds when writing in English Last Updated 10/17/2015 Page 23 of 24

24 ELPS# ELPS.c.5B ELPS.c.5C ELPS.c.5D ELPS.c.5E ELPS.c.5F ELPS.c.5G SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS. write using newly acquired basic vocabulary and content-based grade-level vocabulary spell familiar English words with increasing accuracy, and employ English spelling patterns and rules with increasing accuracy as more English is acquired edit writing for standard grammar and usage, including subject-verb agreement, pronoun agreement, and appropriate verb tenses commensurate with grade-level expectations as more English is acquired employ increasingly complex grammatical structures in content area writing commensurate with grade-level expectations, such as: write using a variety of grade-appropriate sentence lengths, patterns, and connecting words to combine phrases, clauses, and sentences in increasingly accurate ways as more English is acquired narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired. Last Updated 10/17/2015 Last Updated 10/17/2015 Page 24 of 24