Unit 8 Volume & Surface Area
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
UNIT OVERVIEW This unit bundles student expectations that address investigating the relationship between the volume of a triangular prism and triangular pyramid as well as the volume of a rectangular prism and rectangular pyramid, and solving problems involving the volume and surface area of these figures. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, 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. The introduction to the grade level standards state, “While the use of all types of technology is important, the emphasis on algebra readiness skills necessitates the implementation of graphing technology." Prior to this unit, in Grade 6, students modeled area formulas for parallelograms, trapezoids, and triangles by decomposing and rearranging parts of these shapes, and determined solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions were positive rational numbers. In Grade 7 Unit 07, students used models to determine the approximate formulas for the circumference and area of a circle and connected the models to the actual formulas. During this unit, students model the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights as well as connect that relationship to their respective formulas (e.g. the volume of a rectangular prism is three times the volume of a rectangular pyramid; the volume of a rectangular pyramid is the volume of a rectangular prism). Students are expected to explain verbally and symbolically the relationship between the volume of a triangular prism and a triangular pyramid having both congruent bases and heights as well as connect that relationship to their respective formulas (e.g., the volume of a triangular prism is three times the volume of a triangular pyramid; the volume of a triangular pyramid is the volume of a triangular prism). Students solve problems involving volume, including the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids. Students also solve problems involving the lateral and total surface area of a rectangular prism, rectangular pyramid, triangular prism, and triangular pyramid by determining the area of the shape’s net. After this unit, in Unit 12, students will again solve problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids. In Grade 8, students will describe the volume formula V = Bh of a cylinder in terms of its base area and its height and will model the relationship between the volume of a cylinder and a cone having both congruent bases and heights as well as connect that relationship to their respective formulas. Students will solve problems involving the volume of cylinders, cones, and spheres, and will use previous knowledge of surface area to make connections to the formulas for lateral and total surface area to determine solutions for problems involving rectangular prisms, triangular prisms, and cylinders. In Grade 7, solving problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids is identified as STAAR Readiness Standard 7.9A. Solving problems involving the lateral and total surface area of a rectangular prism, rectangular pyramid, triangular prism, and triangular pyramid by determining the area of the shape's net is identified as STAAR Supporting Standard 7.9D. These two standards are listed under the Grade 7 STAAR Reporting Category: Geometry and Measurement. Modeling the relationship between the volume of rectangular prism and a rectangular pyramid having both congruent bases and heights as well as connect that relationship to their respective formulas is identified as standard 7.8A while explaining verbally and symbolically the relationship between the volume of a triangular prism and Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
the triangular pyramid having both congruent bases and heights as well as connect that relationship to their respective formulas is identified as standard 7.8B. These two standards are neither Supporting nor Readiness, but are foundational to the conceptual understanding of geometry and measurement. All of these standards are a subsumed in the Grade 7 Texas Response to Curriculum Focal Points (TxRCFP): Using expressions and equations to describe relationships in a variety of contexts, including geometric problems. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning, III.C. Geometric Reasoning – Connections between geometry and other mathematical content strands, IV. Measurement Reasoning, VIII. Problem Solving and Reasoning, IX. Communication and Representation, and X Connections. According to the National Council of Teachers of Mathematics (NCTM), Curriculum and Evaluation Standards for School Mathematics (1989), “Students discover relationships and develop spatial sense by constructing, drawing, measuring, visualizing, comparing, transforming, and classifying geometric figures. Discussing ideas, conjecturing, and testing hypotheses precede the development of more formal summary statements. In the process, definitions become meaningful, relationships among figures are understood, and students are prepared to use these ideas to develop informal arguments. At the middle school level, geometry should focus on investigating and using geometric ideas and relationships rather than on memorizing definitions and formulas” (p. 112). Additionally, “In their work with three-dimensional objects, students can make use of what they know about two-dimensional shapes. For example, they can relate the surface area of a three-dimensional object to the area of its two-dimensional net” (NCTM, 2000, p. 245). In regards to the algorithm for calculating volume, “students who understand where formulas come from, do not seem as mysterious, tend to remember them, and reinforce the idea that mathematics makes sense. Rote use of formulas from a book offers none of these advantages” (Van de Walle, Karp, & Bay-Williams, 2010, p. 391). National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. 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 http://www.thecb.state.tx.us/collegereadiness/crs.pdf Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from http://projectsharetexas.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013 Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. Boston, MA: Pearson Education, Inc.
OVERARCHING UNDERSTANDINGS AND QUESTIONS Geometric and spatial reasoning are necessary to describe and analyze art, objects, structures, and the environment. How does visualization of various figures help in understanding the world around us?
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
Geometric attributes and their measures can be used to describe figures and their relationships. Why should measurements be considered when describing geometric figures? How are mathematical attributes of figures used to determine measures? Algebraic reasoning facilitates representing, generalizing, and formalizing patterns and relationships in everyday life. How can situations be identified and described algebraically? A problem-solving model can be applied to critically reason through various problem situations in order to solve problems and analyze solutions. How can the information in a problem be analyzed to determine the question being asked and the relevant information provided and/or needed? What types of plans and/or strategies can be used to solve problems? How can solutions to problems be determined? How can solutions to problems be justified? How can the reasonableness of solutions and the problem solving process be evaluated?
PERFORMANCE ASSESSMENT(S)
OVERARCHING CONCEPTS UNIT CONCEPTS Geometric Reasoning
Mathematics Grade 7 Unit 08 PA 01 Click on the PA title to view related rubric. Provide various pairs of rectangular prisms and rectangular pyramids having both congruent bases and heights for students to select along with measuring containers and rice or sand. Analyze the problem situation(s) described below.
Congruence Geometric Attributes/Properties Geometric Relationships Three-Dimensional Figures Two-Dimensional Figures Measurement Reasoning
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UNIT UNDERSTANDINGS The volume of a triangular prism is three times the volume of a triangular pyramid when the prism and pyramid have both a congruent base and height. How does the relationship between the volumes of a rectangular prism and rectangular pyramid having both congruent bases and heights relate to the formulas for volume? How does the relationship between the volumes of a triangular prism and triangular pyramid having both congruent bases and
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area PERFORMANCE ASSESSMENT(S) Organize and record your work for each of the following tasks. Using precise mathematical language, justify and explain each solution process. 1. Select a rectangular prism and a rectangular pyramid having both congruent bases and heights. Fill the rectangular prism with rice or sand and empty the contents into the measuring container. Record this volume. Pour the contents from the measuring container back into the rectangular prism. Empty the contents of the rectangular prism into the rectangular pyramid, until the rectangular pyramid in completely full. Empty the contents of the rectangular pyramid into the measuring container. Record this volume.
SUGGESTED DURATION : 12 days OVERARCHING CONCEPTS UNIT CONCEPTS Area Formulas Volume Associated Mathematical Processes Application Tools and Techniques Communication Representations Relationships Justification
UNIT UNDERSTANDINGS heights relate to the formulas for volume of prisms and pyramids? The volume of a rectangular prism is three times the volume of a rectangular pyramid when the prism and pyramid have both a congruent base and height. How can the relationship between the volumes of a rectangular prism and rectangular pyramid having both congruent bases and heights be described verbally and symbolically? How does the relationship between the volumes of a rectangular prism and rectangular pyramid having both congruent bases and heights relate to the formulas for volume of prisms and pyramids?
a. Generalize, verbally and symbolically, the relationship between the volume of the rectangular prism and rectangular pyramid that have congruent bases and heights. b. Describe how this relationship relates to the formulas for volume of a rectangular prism (V = Bh) and rectangular pyramid (V = Bh).
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area PERFORMANCE ASSESSMENT(S)
SUGGESTED DURATION : 12 days OVERARCHING CONCEPTS UNIT CONCEPTS
UNIT UNDERSTANDINGS
2. Consider a triangular prism and a triangular pyramid having both congruent bases and heights. a. Generalize, verbally and symbolically, the relationship between the volume of a triangular prism and a triangular pyramid having both congruent bases and heights. b. Describe how this relationship relates to the formulas for volume of a triangular prism (V = Bh) and triangular pyramid (V = Bh). Standard(s): 7.1A , 7.1C , 7.1D , 7.1E , 7.1F , 7.1G , 7.8A , 7.8B ELPS.c.1A , ELPS.c.2C , ELPS.c.2D , ELPS.c.2E , ELPS.c.3C , ELPS.c.3D , ELPS.c.3H , ELPS.c.4C , ELPS.c.4D , ELPS.c.4H Algebraic Reasoning Mathematics Grade 7 Unit 08 PA 02 Click on the PA title to view related rubric. Analyze the problem situation(s) described below. Organize and record your work for each of the following tasks. Using precise mathematical language, justify and explain each solution process. A popcorn company is experimenting with various packaging options for their popcorn. The company is
Solve Geometric Reasoning Geometric Attributes/Properties Geometric Relationships Three-Dimensional Figures Measurement Reasoning
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The volume of prisms and pyramids can be determined with formulas. What is the process for solving problems involving the volume of a rectangular or triangular prism? How can the height of prism be determined when given the area of the base of the prism and its volume? What is the process for solving problems involving the volume of a rectangular or triangular pyramid? How can the height of pyramid be determined when given the area of the base of the prism and its volume? Page 5 of 38
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area PERFORMANCE ASSESSMENT(S) considering a rectangular prism, a rectangular pyramid, a triangular prism, and a triangular pyramid cut from cardboard. Option 1 Base Length: 24 cm; Base Height (or Base Width): 15.5 cm; Height of Triangle A's Face: 31.4; Height of Triangle B's Face: 30 cm; Height of Rectangular Pyramid: 29 cm
SUGGESTED DURATION : 12 days OVERARCHING CONCEPTS UNIT CONCEPTS Formulas Surface Area Volume Associated Mathematical Processes Application Problem Solving Model Tools and Techniques Communication Representations Relationships Justification
UNIT UNDERSTANDINGS
The lateral and total surface area of prisms and pyramids can be determined from the shape’s net. What is the difference between the lateral and total surface area of a figure? How is a net used to solve problems involving the lateral surface area of a prism? How is a net used to solve problems involving the total surface area of a prism? How is a net used to solve problems involving the lateral surface area of a pyramid? How is a net used to solve problems involving the total surface area of a pyramid?
Option 2 Base Length: 13.5 cm; Base Height (or Base Width): 16 cm; Height of figure: 14 cm
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days OVERARCHING CONCEPTS UNIT CONCEPTS
PERFORMANCE ASSESSMENT(S)
UNIT UNDERSTANDINGS
Option 3 Base Length: 48 cm; Base Height (or Base Width): 21 cm; Height of Triangles A, B Faces: 17.8 cm each; Height of Triangular C's Face: 16.55 cm; Height of Triangular Pyramid: 15 cm
Option 4 Base Length: 17 cm; Base Height (or Base
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area PERFORMANCE ASSESSMENT(S)
SUGGESTED DURATION : 12 days OVERARCHING CONCEPTS UNIT CONCEPTS
UNIT UNDERSTANDINGS
Width): 19 cm; Height of figure: 22 cm
1. The logo of the popcorn company will be printed on the sides of the prisms and pyramids for $0.005 per square centimeter. The printing company charges for the total surface that will be printed, even if the entire surface of a side will not be printed on. a. Describe which packaging option would be most cost effective if the company logo was printed on only the lateral faces. b. Describe which packaging option would be most cost effective if the company logo was printed on all the faces and bases. 2. The popcorn company has decided that it would like to have one prism and one pyramid packaging Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area PERFORMANCE ASSESSMENT(S)
SUGGESTED DURATION : 12 days OVERARCHING CONCEPTS UNIT CONCEPTS
UNIT UNDERSTANDINGS
option and would like both options to hold the same amount of popcorn. a. Calculate the volumes of the four packaging options to determine which prism and pyramid options will hold the least amount of popcorn. b. The company has decided that they would like to adjust the height of the rectangular prism with the least volume so that it has the same volume as the triangular pyramid with the least volume. Determine the adjusted height of the rectangular prism, to the nearest hundredth, so that its volume is the same as the triangular pyramid.. Standard(s): 7.1A , 7.1B , 7.1C , 7.1D , 7.1E , 7.1F , 7.1G , 7.9A , 7.9D ELPS.c.1A , ELPS.c.2C , ELPS.c.2D , ELPS.c.2E , ELPS.c.3C , ELPS.c.3D , ELPS.c.3H , ELPS.c.4C , ELPS.c.4D , ELPS.c.4F , ELPS.c.4H , ELPS.c.5B , ELPS.c.5F , ELPS.c.5G
MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS Misconceptions: Some students may think that “B” is synonymous with “b”, the length of the base, instead of “B”, which represents the area of the base of a three-dimensional figure. Some students may confuse the height of the two-dimensional face as the height of the three-dimensional prism or pyramid. Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
Some students may not associate the faces and bases of a prism or pyramid to the correct parts of the corresponding net. Some students may think that the face that rests on the bottom of the prism is the base.
UNIT VOCABULARY Area – the measurement attribute that describes the number of square units a figure or region covers Base of a rectangular prism – any two congruent, opposite and parallel faces shaped like rectangles; possibly more than one set Base of a rectangular pyramid – a rectangle attached to 4 triangular faces that meet at one common point, a vertex Base of a triangular prism – the two congruent, opposite and parallel faces shaped like triangles Base of a triangular pyramid – a triangle attached to 3 triangular faces that meet at one common point, a vertex Congruent – of equal measure, having exactly the same size and same shape Face – a flat surface of a three-dimensional figure Height of a rectangular prism – the length of a side that is perpendicular to both bases Height of a rectangular pyramid – the length of a perpendicular line segment from the vertex of the pyramid to the base Height of a triangular prism – the length of a side that is perpendicular to both bases Height of a triangular pyramid – the length of a perpendicular line segment from the vertex of the pyramid to the base Lateral surface area – the sum of all the lateral surface areas of a figure; the number of square units needed to cover the lateral view (area excluding the base(s) of a three-dimensional figure) Net – a two-dimensional model or drawing that can be folded into a three-dimensional solid Positive rational numbers – the set of numbers that can be expressed as a fraction where a > 0 and b > 0 Prism – a three-dimensional figure containing two congruent and parallel faces that are polygons Pyramid – a three-dimensional figure containing a base that is a polygon and the faces are triangles that share a common vertex, also known as an apex Three-dimensional figure – a figure that has measurements including length, width (depth), and height Total surface area – the sum of all the surface areas of a figure; the number of square units needed to cover all of the surfaces (bases and lateral area) Volume – the measurement attribute of the amount of space occupied by matter Related Vocabulary: Base Edge
Perpendicular Rectangle
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Triangular pyramid Triangular prism Page 10 of 38
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area Height Length Parallel
SUGGESTED DURATION : 12 days Rectangular prism Rectangular pyramid Triangle
UNIT ASSESSMENT ITEMS Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Creator if your district has granted access to that tool.
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Mathematics Concepts Tree Mathematics Grade 7 STAAR Analysis_Unit 08_TEKS Resource System Mathematics Grade 7 STAAR Enhanced Blueprint Mathematics Grade 7 TEKS Introduction and Focal Points with Aligned Standards Mathematics Grade 7 TEKS Supporting Information (with TEKS Resource System Comments)
Texas Education Agency – Texas College and Career Readiness Standards Texas Education Agency – Grade 7 Reference Materials Texas Education Agency - Revised Mathematics TEKS: Vertical Alignment Charts Texas Instruments - Graphing Calculator Tutorials Texas Education Agency – Mathematics Curriculum Texas Education Agency – Assessment Texas Education Agency – Mathematics TEKS: Supporting Information Texas Education Agency – Interactive Mathematics Glossary
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
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Bold black text in italics: Knowledge and Skills Statement (TEKS) Bold black text: Student Expectation (TEKS) Bold red text in italics: Student Expectation identified by TEA as a Readiness Standard for STAAR Bold green text in italics: Student Expectation identified by TEA as a Supporting Standard for STAAR Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit 7.1
Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
7.1A
Apply mathematics to problems arising in everyday life, society, and the workplace.
Blue text: Supporting Information / Clarifications from TCMPC (Specificity) Blue Italic text: Unit-specific clarification Black text: TEA Texas Response to Curriculum Focal Points (TxRCFP); Texas College and Career Readiness Standards (TxCCRS); TEA STAAR
Apply MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE Including, but not limited to: Mathematical problem situations within and between disciplines Everyday life Society
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
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Workplace Note(s): The mathematical process standards may be applied to all content standards as appropriate. TxRCFP: Developing fluency with rational numbers and operations to solve problems in a variety of contexts Representing and applying proportional relationships Using expressions and equations to describe relationships in a variety of contexts, including geometric problems Comparing sets of data TxCCRS: X. Connections 7.1B
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 Including, but not limited to: Problem-solving model Analyze given information Formulate a plan or strategy Determine a solution Justify the solution
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
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Evaluate 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. TxRCFP: Developing fluency with rational numbers and operations to solve problems in a variety of contexts Representing and applying proportional relationships Using expressions and equations to describe relationships in a variety of contexts, including geometric problems Comparing sets of data TxCCRS: VIII. Problem Solving and Reasoning 7.1C
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.
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 Including, but not limited to: Appropriate selection of tool(s) and techniques to apply in order to solve problems Tools Real objects Manipulatives Paper and pencil Technology
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
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Techniques Mental math Estimation Number sense Note(s): The mathematical process standards may be applied to all content standards as appropriate. TxRCFP: Developing fluency with rational numbers and operations to solve problems in a variety of contexts Representing and applying proportional relationships Using expressions and equations to describe relationships in a variety of contexts, including geometric problems Comparing sets of data TxCCRS: VIII. Problem Solving and Reasoning 7.1D
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, AND LANGUAGE AS APPROPRIATE Including, but not limited to: Mathematical ideas, reasoning, and their implications Multiple representations, as appropriate Symbols Diagrams
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
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Language Note(s): The mathematical process standards may be applied to all content standards as appropriate. TxRCFP: Developing fluency with rational numbers and operations to solve problems in a variety of contexts Representing and applying proportional relationships Using expressions and equations to describe relationships in a variety of contexts, including geometric problems Comparing sets of data TxCCRS: IX. Communication and Representation 7.1E
Create and use representations to organize, record, and communicate mathematical ideas.
Create, Use REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS Including, but not limited to: Representations of mathematical ideas Organize Record Communicate Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
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Note(s): The mathematical process standards may be applied to all content standards as appropriate. TxRCFP: Developing fluency with rational numbers and operations to solve problems in a variety of contexts Representing and applying proportional relationships Using expressions and equations to describe relationships in a variety of contexts, including geometric problems Comparing sets of data TxCCRS: IX. Communication and Representation 7.1F
Analyze mathematical relationships to connect and communicate mathematical ideas.
Analyze MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS Including, but not limited to: Mathematical relationships Connect and communicate mathematical ideas Conjectures and generalizations from sets of examples and non-examples, patterns, etc. Note(s): The mathematical process standards may be applied to all content standards as appropriate. TxRCFP: Developing fluency with rational numbers and operations to solve problems in a variety of
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
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contexts Representing and applying proportional relationships Using expressions and equations to describe relationships in a variety of contexts, including geometric problems Comparing sets of data TxCCRS: X. Connections 7.1G
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 Including, but not limited to: Mathematical ideas and arguments Validation of conclusions Displays to make work visible to others Diagrams, visual aids, written work, etc. Explanations and justifications Precise mathematical language in written or oral communication Note(s): The mathematical process standards may be applied to all content standards as appropriate. TxRCFP: Developing fluency with rational numbers and operations to solve problems in a variety of contexts Representing and applying proportional relationships
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
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Using expressions and equations to describe relationships in a variety of contexts, including geometric problems Comparing sets of data TxCCRS: IX. Communication and Representation 7.8
Expressions, equations, and relationships. The student applies mathematical process standards to develop geometric relationships with volume. The student is expected to:
7.8A
Model the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights and connect that relationship to the formulas.
Model THE RELATIONSHIP BETWEEN THE VOLUME OF A RECTANGULAR PRISM AND A RECTANGULAR PYRAMID HAVING BOTH CONGRUENT BASES AND HEIGHTS AND CONNECT THAT RELATIONSHIP TO THE FORMULAS Including, but not limited to: Three-dimensional figure – a figure that has measurements including length, width (depth), and height Prism – a three-dimensional figure containing two congruent and parallel faces that are polygons Rectangular prism 6 rectangular faces (2 parallel rectangular faces [bases], 4 rectangular faces) 12 edges 8 vertices Face – a flat surface of a three-dimensional figure
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
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Base of a rectangular prism – any two congruent, opposite and parallel faces shaped like rectangles; possibly more than one set Height of a rectangular prism – the length of a side that is perpendicular to both bases Pyramid – a three-dimensional figure containing a base that is a polygon and the faces are triangles that share a common vertex, also known as an apex Rectangular pyramid 5 faces (1 rectangular face [base], 4 triangular faces) 8 edges 5 vertices Base of a rectangular pyramid – a rectangle attached to 4 triangular faces that meet at one common point, a vertex Height of a rectangular pyramid – the length of a perpendicular line segment from the vertex of the pyramid to the base Volume – the measurement attribute of the amount of space occupied by matter One way to measure volume is a three-dimensional cubic measure Congruent – of equal measure, having exactly the same size and same shape Various models to represent the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights Filling the rectangular pyramid with a measurable unit (e.g., rice, sand, water, etc.) and emptying the contents into the rectangular prism until the rectangular prism is completely full The contents of the rectangular pyramid will need to be emptied three times in order to fill the rectangular prism completely. Creating a replica of the rectangular pyramid and rectangular prisms with clay and comparing their masses The mass of the rectangular prism will be three times the mass of the rectangular pyramid, whereas the mass of the rectangular pyramid is the mass of the
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
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UNIT LEVEL SPECIFICITY
SE#
rectangular prism. Generalizations from models used to represent the relationship between the volume of a rectangular prism and a rectangular pyramid having congruent bases and heights The volume of a rectangular prism is three times the volume of a rectangular pyramid. The volume of a rectangular pyramid is the volume of a rectangular prism. Connections between models to represent volume of a rectangular prism and rectangular pyramid having both congruent bases and heights to the formulas for volume Formulas for volume from STAAR Grade 7 Mathematics Reference Materials Prism V = Bh, where B represents the base area and h represents the height of the prism which is the number of times the base area is repeated or layered Rectangular prism The base of a rectangular prism is a rectangle whose area may be found with the formula, A = bh or A = lw, meaning the base area, B, may be found with the formula B = bh or B = lw; therefore, the volume of a rectangular prism may be found using V = Bh or V =(bh)h or V = (lw)h. Pyramid V = Bh, where B represents the base area and h represents the height of the pyramid Rectangular pyramid The base of a rectangular pyramid is a rectangle whose area may be found with the formula, A = bh or A = lw, meaning the base area, B, may be found with the formula B = bh or B = lw; therefore, the volume of a rectangular pyramid may be
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
found using V = Bh or V = (bh)h or V = (lw)h. Note(s): Grade Level(s): Grade 6 modeled area formulas for parallelograms, trapezoids, and triangles by decomposing and rearranging parts of these shapes. Grade 8 will describe the volume formula V = Bh of a cylinder in terms of its base area and its height. Grade 8 will model the relationship between the volume of a cylinder and a cone having both congruent bases and heights and connect that relationship to the formulas. Various mathematical process standards will be applied to this student expectation as appropriate. TxRCFP: Using expressions and equations to describe relationships in a variety of contexts, including geometric problems TxCCRS: IV. Measurement Reasoning VIII. Problem Solving and Reasoning IX. Communication and Representation 7.8B
Explain verbally and symbolically the relationship between the volume of a triangular prism and a triangular pyramid having both congruent bases and heights and connect that relationship to the formulas.
Explain VERBALLY AND SYMBOLICALLY THE RELATIONSHIP BETWEEN THE VOLUME OF A TRIANGULAR PRISM AND A TRIANGULAR PYRAMID HAVING BOTH CONGRUENT BASES AND HEIGHTS AND CONNECT THAT RELATIONSHIP TO THE FORMULAS Including, but not limited to:
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
Page 22 of 38
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Three-dimensional figure – a figure that has measurements including length, width (depth), and height Prism – a three-dimensional figure containing two congruent and parallel faces that are polygons Triangular prism 5 faces (2 triangular faces [bases], 3 rectangular faces) 9 edges 6 vertices Face – a flat surface of a three-dimensional figure Base of a triangular prism – the two congruent, opposite and parallel faces shaped like triangles Height of a triangular prism – the length of a side that is perpendicular to both bases Pyramid – a three-dimensional figure containing a base that is a polygon and the faces are triangles that share a common vertex, also known as an apex Triangular pyramid 4 faces (1 triangular face [base], 3 triangular faces) 6 edges 4 vertices Base of a triangular pyramid – a triangle attached to 3 triangular faces that meet at one common point, a vertex Height of a triangular pyramid – the length of a perpendicular line segment from the vertex of the pyramid to the base Volume – the measurement attribute of the amount of space occupied by matter One way to measure volume is a three-dimensional cubic measure Congruent – of equal measure, having exactly the same size and same shape Generalizations of the relationship between the volume of a triangular prism and a triangular pyramid having congruent bases and heights The volume of a triangular prism is three times the volume of a triangular pyramid. Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
The volume of a triangular pyramid is the volume of a triangular prism. Connections between models to represent volume of a triangular prism and triangular pyramid having both congruent bases and heights to the formulas for volume Formulas for volume from STAAR Grade 7 Mathematics Reference Materials Prism V = Bh, where B represents the base area and h represents the height of the prism which is the number of times the base area is repeated or layered) Triangular prism The base of a triangular prism is a triangle whose area may be found with the formula, A = bh, meaning the base area, B, may be found using B = bh; therefore, the volume of a triangular prism may be found using V = Bh or V = . Pyramid V = Bh, where B represents the base area and h represents the height of the pyramid Triangular pyramid The base of a triangular pyramid is a triangle whose area may be found with the formula, A = bh, meaning the base area, B, may be found using B = bh; therefore, the volume of a triangular pyramid may be found using V = Bh or V =
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
or V =
Page 24 of 38
.
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Note(s): Grade Level(s): Grade 7 introduces explaining verbally and symbolically the relationship between the volume of a triangular prism and a triangular pyramid having both congruent bases and heights and connecting that relationship to the formulas. Grade 8 will model the relationship between the volume of a cylinder and a cone having both congruent bases and heights and connect that relationship to the formulas. Various mathematical process standards will be applied to this student expectation as appropriate. TxRCFP: Using expressions and equations to describe relationships in a variety of contexts, including geometric problems TxCCRS: I. Numeric Reasoning IV. Measurement Reasoning VIII. Problem Solving and Reasoning IX. Communication and Representation 7.9
Expressions, equations, and relationships. The student applies mathematical process standards to solve geometric problems. The student is expected to:
7.9A
Solve problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids. Readiness Standard
Solve PROBLEMS INVOLVING THE VOLUME OF RECTANGULAR PRISMS, TRIANGULAR PRISMS, RECTANGULAR PYRAMIDS, AND TRIANGULAR PYRAMIDS
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
Page 25 of 38
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Including, but not limited to: Positive rational numbers – the set of numbers that can be expressed as a fraction , where a > 0 and b > 0 Various forms of positive rational numbers Counting (natural) numbers Decimals (greater than zero) Fractions (proper, improper, and mixed numbers greater than zero) Three-dimensional figure – a figure that has measurements including length, width (depth), and height Prism – a three-dimensional figure containing two congruent and parallel faces that are polygons Rectangular prism 6 rectangular faces (2 parallel rectangular faces [bases], 4 rectangular faces) 12 edges 8 vertices Face – a flat surface of a three-dimensional figure Base of a rectangular prism – any two congruent, opposite and parallel faces shaped like rectangles; possibly more than one set Height of a rectangular prism – the length of a side that is perpendicular to both bases Triangular prism 5 faces (2 triangular faces [bases], 3 rectangular faces) 9 edges 6 vertices Base of a triangular prism – the two congruent, opposite and parallel faces shaped like triangles Height of a triangular prism – the length of a side that is perpendicular to Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
Page 26 of 38
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
both bases Pyramid – a three-dimensional figure containing a base that is a polygon and the faces are triangles that share a common vertex, also known as an apex Rectangular pyramid 5 faces (1 rectangular face [base], 4 triangular faces) 8 edges 5 vertices Base of a rectangular pyramid – a rectangle attached to 4 triangular faces that meet a one common point, a vertex Height of a rectangular pyramid – the length of a perpendicular line segment from the vertex of the pyramid to the base Triangular pyramid 4 faces (1 triangular face [base], 3 triangular faces) 6 edges 4 vertices Base of a triangular pyramid – a triangle attached to 3 triangular faces that meet a one common point, a vertex Height of a triangular pyramid – the length of a perpendicular line segment from the vertex of the pyramid to the base Volume – the measurement attribute of the amount of space occupied by matter One way to measure volume is a three-dimensional cubic measure Positive rational number side lengths Recognition of volume embedded in mathematical and real-world problem situations Formulas for volume from STAAR Grade 7 Mathematics Reference Materials Prism V = Bh, where B represents the base area and h represents the height of the prism which is the number of times the base area is repeated or layered) Rectangular prism The base of a rectangular prism is a rectangle whose area may be
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
Page 27 of 38
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
found with the formula, A = bh or A = lw, meaning the base area, B, may be found with the formula B = bh or B = lw; therefore, the volume of a rectangular prism may be found using V = Bh or V = (bh)h or V = (lw)h. Triangular prism The base of a triangular prism is a triangle whose area may be found with the formula, A = bh, meaning the base area, B, may be found using B = bh; therefore, the volume of a triangular prism may be found using V = Bh or V =
.
Pyramid V = Bh, where B represents the base area and h represents the height of the pyramid Rectangular pyramid The base of a rectangular pyramid is a rectangle whose area may be found with the formula, A = bh or A = lw, meaning the base area, B, may be found with the formula B = bh or B = lw; therefore, the volume of a rectangular pyramid may be found using V = Bh or V = (bh)h or V = (lw)h. Triangular pyramid The base of a triangular pyramid is a triangle whose area may be found with the formula, A = bh, meaning the base area, B, may be found using B = bh; therefore, the volume of a triangular pyramid
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
Page 28 of 38
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
may be found using V = Bh or V =
or V =
.
Note(s): Grade Level(s): Grade 6 determined solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers. Grade 8 will solve problems involving the volume of cylinders, cones, and spheres. Various mathematical process standards will be applied to this student expectation as appropriate. TxRCFP: Using expressions and equations to describe relationships in a variety of contexts, including geometric problems TxCCRS: I. Numeric Reasoning III.C. Geometric Reasoning – Connections between geometry and other mathematical content strands IV. Measurement Reasoning VIII. Problem Solving and Reasoning IX. Communication and Representation X. Connections 7.9D
Solve problems involving the lateral and total surface area of a rectangular prism, rectangular pyramid, triangular prism, and triangular pyramid by determining the area of the shape's net. Supporting Standard
Solve PROBLEMS INVOLVING THE LATERAL AND TOTAL SURFACE AREA OF A RECTANGULAR PRISM, RECTANGULAR PYRAMID, TRIANGULAR PRISM, AND TRIANGULAR PYRAMID BY DETERMINING THE AREA OF THE SHAPE'S NET
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
Page 29 of 38
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Including, but not limited to: Positive rational numbers – the set of numbers that can be expressed as a fraction , where a > 0 and b > 0 Various forms of positive rational numbers Counting (natural) numbers Decimals (greater than zero) Fractions (proper, improper, and mixed numbers greater than zero) Three-dimensional figure – a figure that has measurements including length, width (depth), and height Prism – a three-dimensional figure containing two congruent and parallel faces that are polygons Rectangular prism 6 rectangular faces (2 parallel rectangular faces [bases], 4 rectangular faces) 12 edges 8 vertices Face – a flat surface of a three-dimensional figure Base of a rectangular prism – any two congruent, opposite and parallel faces shaped like rectangles; possibly more than one set Height of a rectangular prism – the length of a side that is perpendicular to both bases Triangular prism 5 faces (2 triangular faces [bases], 3 rectangular faces) 9 edges 6 vertices Base of a triangular prism – the two congruent, opposite and parallel faces shaped like triangles Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
Page 30 of 38
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Height of a triangular prism – the length of a side that is perpendicular to both bases Pyramid – a three-dimensional figure containing a base that is a polygon and the faces are triangles that share a common vertex, also known as an apex Rectangular pyramid 5 faces (1 rectangular face [base], 4 triangular faces) 8 edges 5 vertices Base of a rectangular pyramid – a rectangle attached to 4 triangular faces that meet at one common point, a vertex Height of a rectangular pyramid – the length of a perpendicular line segment from the vertex of the pyramid to the base Triangular pyramid 4 faces (1 triangular face [base], 3 triangular faces) 6 edges 4 vertices Base of a triangular pyramid – a triangle attached to 3 triangular faces that meet at one common point, a vertex Height of a triangular pyramid – the length of a perpendicular line segment from the vertex of the pyramid to the base Area – the measurement attribute that describes the number of square units a figure or region covers Area is a two-dimensional square unit measure. Positive rational number side lengths Surface Area Lateral surface area – the sum of all the lateral surface areas of a figure; the number of square units needed to cover the lateral view (area excluding the base(s) of a threedimensional figure) Total surface area – the sum of all the surface areas of a figure; the number of square
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
Page 31 of 38
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
units needed to cover all of the surfaces (bases and lateral area) Net – a two-dimensional model or drawing that can be folded into a three-dimensional solid Note(s): Grade Level(s): Grade 6 determined solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers. Grade 8 will use previous knowledge of surface area to make connections to the formulas for lateral and total surface area and determine solutions for problems involving rectangular prisms, triangular prisms, and cylinders. Various mathematical process standards will be applied to this student expectation as appropriate. TxRCFP: Using expressions and equations to describe relationships in a variety of contexts, including geometric problems TxCCRS: I. Numeric Reasoning III.C. Geometric Reasoning – Connections between geometry and other mathematical content strands IV. Measurement Reasoning VIII. Problem Solving and Reasoning IX. Communication and Representation X. Connections
ELPS#
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS.
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area ELPS#
SUGGESTED DURATION : 12 days
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. http://ritter.tea.state.tx.us/rules/tac/chapter074/ch074a.html#74.4 Choose appropriate ELPS to support instruction. ELPS.c.1
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:
ELPS.c.1A
use prior knowledge and experiences to understand meanings in English
ELPS.c.1B
monitor oral and written language production and employ self-corrective techniques or other resources
ELPS.c.1C
use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and grade-level vocabulary
ELPS.c.1D
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)
ELPS.c.1E
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
ELPS.c.1F
use accessible language and learn new and essential language in the process
ELPS.c.1G
demonstrate an increasing ability to distinguish between formal and informal English and an increasing knowledge of when to use each one
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area ELPS#
SUGGESTED DURATION : 12 days
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS. commensurate with grade-level learning expectations
ELPS.c.1H
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.
ELPS.c.2
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:
ELPS.c.2A
distinguish sounds and intonation patterns of English with increasing ease
ELPS.c.2B
recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters
ELPS.c.2C
learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions
ELPS.c.2D
monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed
ELPS.c.2E
use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language
ELPS.c.2F
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
ELPS.c.2G
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
ELPS.c.2H
understand implicit ideas and information in increasingly complex spoken language commensurate with grade-level learning expectations
ELPS.c.2I
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.
ELPS.c.3
The ELL speaks in a variety of modes for a variety of purposes with an awareness of different language registers (formal/informal) using vocabulary
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area ELPS#
SUGGESTED DURATION : 12 days
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS. 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:
ELPS.c.3A
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
ELPS.c.3B
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
ELPS.c.3C
speak using a variety of grammatical structures, sentence lengths, sentence types, and connecting words with increasing accuracy and ease as more English is acquired
ELPS.c.3D
speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency
ELPS.c.3E
share information in cooperative learning interactions
ELPS.c.3F
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
ELPS.c.3G
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
ELPS.c.3H
narrate, describe, and explain with increasing specificity and detail as more English is acquired
ELPS.c.3I
adapt spoken language appropriately for formal and informal purposes
ELPS.c.3J
respond orally to information presented in a wide variety of print, electronic, audio, and visual media to build and reinforce concept and language attainment.
ELPS.c.4
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
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area ELPS#
SUGGESTED DURATION : 12 days
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS. 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:
ELPS.c.4A
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
ELPS.c.4B
recognize directionality of English reading such as left to right and top to bottom
ELPS.c.4C
develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials
ELPS.c.4D
use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary and other prereading activities to enhance comprehension of written text
ELPS.c.4E
read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned
ELPS.c.4F
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
ELPS.c.4G
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
ELPS.c.4H
read silently with increasing ease and comprehension for longer periods
ELPS.c.4I
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
ELPS.c.4J
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
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area ELPS#
SUGGESTED DURATION : 12 days
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS.
ELPS.c.4K
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.
ELPS.c.5
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:
ELPS.c.5A
learn relationships between sounds and letters of the English language to represent sounds when writing in English
ELPS.c.5B
write using newly acquired basic vocabulary and content-based grade-level vocabulary
ELPS.c.5C
spell familiar English words with increasing accuracy, and employ English spelling patterns and rules with increasing accuracy as more English is acquired
ELPS.c.5D
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
ELPS.c.5E
employ increasingly complex grammatical structures in content area writing commensurate with grade-level expectations, such as:
ELPS.c.5F
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
ELPS.c.5G
narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired.
Last Updated 04/26/2016
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
SUGGESTED DURATION : 12 days
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SUGGESTED DURATION : 12 days
UNIT OVERVIEW This unit bundles student expectations that address investigating the relationship between the volume of a triangular prism and triangular pyramid as well as the volume of a rectangular prism and rectangular pyramid, and solving problems involving the volume and surface area of these figures. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, 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. The introduction to the grade level standards state, “While the use of all types of technology is important, the emphasis on algebra readiness skills necessitates the implementation of graphing technology." Prior to this unit, in Grade 6, students modeled area formulas for parallelograms, trapezoids, and triangles by decomposing and rearranging parts of these shapes, and determined solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions were positive rational numbers. In Grade 7 Unit 07, students used models to determine the approximate formulas for the circumference and area of a circle and connected the models to the actual formulas. During this unit, students model the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights as well as connect that relationship to their respective formulas (e.g. the volume of a rectangular prism is three times the volume of a rectangular pyramid; the volume of a rectangular pyramid is the volume of a rectangular prism). Students are expected to explain verbally and symbolically the relationship between the volume of a triangular prism and a triangular pyramid having both congruent bases and heights as well as connect that relationship to their respective formulas (e.g., the volume of a triangular prism is three times the volume of a triangular pyramid; the volume of a triangular pyramid is the volume of a triangular prism). Students solve problems involving volume, including the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids. Students also solve problems involving the lateral and total surface area of a rectangular prism, rectangular pyramid, triangular prism, and triangular pyramid by determining the area of the shape’s net. After this unit, in Unit 12, students will again solve problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids. In Grade 8, students will describe the volume formula V = Bh of a cylinder in terms of its base area and its height and will model the relationship between the volume of a cylinder and a cone having both congruent bases and heights as well as connect that relationship to their respective formulas. Students will solve problems involving the volume of cylinders, cones, and spheres, and will use previous knowledge of surface area to make connections to the formulas for lateral and total surface area to determine solutions for problems involving rectangular prisms, triangular prisms, and cylinders. In Grade 7, solving problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids is identified as STAAR Readiness Standard 7.9A. Solving problems involving the lateral and total surface area of a rectangular prism, rectangular pyramid, triangular prism, and triangular pyramid by determining the area of the shape's net is identified as STAAR Supporting Standard 7.9D. These two standards are listed under the Grade 7 STAAR Reporting Category: Geometry and Measurement. Modeling the relationship between the volume of rectangular prism and a rectangular pyramid having both congruent bases and heights as well as connect that relationship to their respective formulas is identified as standard 7.8A while explaining verbally and symbolically the relationship between the volume of a triangular prism and Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
the triangular pyramid having both congruent bases and heights as well as connect that relationship to their respective formulas is identified as standard 7.8B. These two standards are neither Supporting nor Readiness, but are foundational to the conceptual understanding of geometry and measurement. All of these standards are a subsumed in the Grade 7 Texas Response to Curriculum Focal Points (TxRCFP): Using expressions and equations to describe relationships in a variety of contexts, including geometric problems. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning, III.C. Geometric Reasoning – Connections between geometry and other mathematical content strands, IV. Measurement Reasoning, VIII. Problem Solving and Reasoning, IX. Communication and Representation, and X Connections. According to the National Council of Teachers of Mathematics (NCTM), Curriculum and Evaluation Standards for School Mathematics (1989), “Students discover relationships and develop spatial sense by constructing, drawing, measuring, visualizing, comparing, transforming, and classifying geometric figures. Discussing ideas, conjecturing, and testing hypotheses precede the development of more formal summary statements. In the process, definitions become meaningful, relationships among figures are understood, and students are prepared to use these ideas to develop informal arguments. At the middle school level, geometry should focus on investigating and using geometric ideas and relationships rather than on memorizing definitions and formulas” (p. 112). Additionally, “In their work with three-dimensional objects, students can make use of what they know about two-dimensional shapes. For example, they can relate the surface area of a three-dimensional object to the area of its two-dimensional net” (NCTM, 2000, p. 245). In regards to the algorithm for calculating volume, “students who understand where formulas come from, do not seem as mysterious, tend to remember them, and reinforce the idea that mathematics makes sense. Rote use of formulas from a book offers none of these advantages” (Van de Walle, Karp, & Bay-Williams, 2010, p. 391). National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. 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 http://www.thecb.state.tx.us/collegereadiness/crs.pdf Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from http://projectsharetexas.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013 Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. Boston, MA: Pearson Education, Inc.
OVERARCHING UNDERSTANDINGS AND QUESTIONS Geometric and spatial reasoning are necessary to describe and analyze art, objects, structures, and the environment. How does visualization of various figures help in understanding the world around us?
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
Geometric attributes and their measures can be used to describe figures and their relationships. Why should measurements be considered when describing geometric figures? How are mathematical attributes of figures used to determine measures? Algebraic reasoning facilitates representing, generalizing, and formalizing patterns and relationships in everyday life. How can situations be identified and described algebraically? A problem-solving model can be applied to critically reason through various problem situations in order to solve problems and analyze solutions. How can the information in a problem be analyzed to determine the question being asked and the relevant information provided and/or needed? What types of plans and/or strategies can be used to solve problems? How can solutions to problems be determined? How can solutions to problems be justified? How can the reasonableness of solutions and the problem solving process be evaluated?
PERFORMANCE ASSESSMENT(S)
OVERARCHING CONCEPTS UNIT CONCEPTS Geometric Reasoning
Mathematics Grade 7 Unit 08 PA 01 Click on the PA title to view related rubric. Provide various pairs of rectangular prisms and rectangular pyramids having both congruent bases and heights for students to select along with measuring containers and rice or sand. Analyze the problem situation(s) described below.
Congruence Geometric Attributes/Properties Geometric Relationships Three-Dimensional Figures Two-Dimensional Figures Measurement Reasoning
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UNIT UNDERSTANDINGS The volume of a triangular prism is three times the volume of a triangular pyramid when the prism and pyramid have both a congruent base and height. How does the relationship between the volumes of a rectangular prism and rectangular pyramid having both congruent bases and heights relate to the formulas for volume? How does the relationship between the volumes of a triangular prism and triangular pyramid having both congruent bases and
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area PERFORMANCE ASSESSMENT(S) Organize and record your work for each of the following tasks. Using precise mathematical language, justify and explain each solution process. 1. Select a rectangular prism and a rectangular pyramid having both congruent bases and heights. Fill the rectangular prism with rice or sand and empty the contents into the measuring container. Record this volume. Pour the contents from the measuring container back into the rectangular prism. Empty the contents of the rectangular prism into the rectangular pyramid, until the rectangular pyramid in completely full. Empty the contents of the rectangular pyramid into the measuring container. Record this volume.
SUGGESTED DURATION : 12 days OVERARCHING CONCEPTS UNIT CONCEPTS Area Formulas Volume Associated Mathematical Processes Application Tools and Techniques Communication Representations Relationships Justification
UNIT UNDERSTANDINGS heights relate to the formulas for volume of prisms and pyramids? The volume of a rectangular prism is three times the volume of a rectangular pyramid when the prism and pyramid have both a congruent base and height. How can the relationship between the volumes of a rectangular prism and rectangular pyramid having both congruent bases and heights be described verbally and symbolically? How does the relationship between the volumes of a rectangular prism and rectangular pyramid having both congruent bases and heights relate to the formulas for volume of prisms and pyramids?
a. Generalize, verbally and symbolically, the relationship between the volume of the rectangular prism and rectangular pyramid that have congruent bases and heights. b. Describe how this relationship relates to the formulas for volume of a rectangular prism (V = Bh) and rectangular pyramid (V = Bh).
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area PERFORMANCE ASSESSMENT(S)
SUGGESTED DURATION : 12 days OVERARCHING CONCEPTS UNIT CONCEPTS
UNIT UNDERSTANDINGS
2. Consider a triangular prism and a triangular pyramid having both congruent bases and heights. a. Generalize, verbally and symbolically, the relationship between the volume of a triangular prism and a triangular pyramid having both congruent bases and heights. b. Describe how this relationship relates to the formulas for volume of a triangular prism (V = Bh) and triangular pyramid (V = Bh). Standard(s): 7.1A , 7.1C , 7.1D , 7.1E , 7.1F , 7.1G , 7.8A , 7.8B ELPS.c.1A , ELPS.c.2C , ELPS.c.2D , ELPS.c.2E , ELPS.c.3C , ELPS.c.3D , ELPS.c.3H , ELPS.c.4C , ELPS.c.4D , ELPS.c.4H Algebraic Reasoning Mathematics Grade 7 Unit 08 PA 02 Click on the PA title to view related rubric. Analyze the problem situation(s) described below. Organize and record your work for each of the following tasks. Using precise mathematical language, justify and explain each solution process. A popcorn company is experimenting with various packaging options for their popcorn. The company is
Solve Geometric Reasoning Geometric Attributes/Properties Geometric Relationships Three-Dimensional Figures Measurement Reasoning
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The volume of prisms and pyramids can be determined with formulas. What is the process for solving problems involving the volume of a rectangular or triangular prism? How can the height of prism be determined when given the area of the base of the prism and its volume? What is the process for solving problems involving the volume of a rectangular or triangular pyramid? How can the height of pyramid be determined when given the area of the base of the prism and its volume? Page 5 of 38
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area PERFORMANCE ASSESSMENT(S) considering a rectangular prism, a rectangular pyramid, a triangular prism, and a triangular pyramid cut from cardboard. Option 1 Base Length: 24 cm; Base Height (or Base Width): 15.5 cm; Height of Triangle A's Face: 31.4; Height of Triangle B's Face: 30 cm; Height of Rectangular Pyramid: 29 cm
SUGGESTED DURATION : 12 days OVERARCHING CONCEPTS UNIT CONCEPTS Formulas Surface Area Volume Associated Mathematical Processes Application Problem Solving Model Tools and Techniques Communication Representations Relationships Justification
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The lateral and total surface area of prisms and pyramids can be determined from the shape’s net. What is the difference between the lateral and total surface area of a figure? How is a net used to solve problems involving the lateral surface area of a prism? How is a net used to solve problems involving the total surface area of a prism? How is a net used to solve problems involving the lateral surface area of a pyramid? How is a net used to solve problems involving the total surface area of a pyramid?
Option 2 Base Length: 13.5 cm; Base Height (or Base Width): 16 cm; Height of figure: 14 cm
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days OVERARCHING CONCEPTS UNIT CONCEPTS
PERFORMANCE ASSESSMENT(S)
UNIT UNDERSTANDINGS
Option 3 Base Length: 48 cm; Base Height (or Base Width): 21 cm; Height of Triangles A, B Faces: 17.8 cm each; Height of Triangular C's Face: 16.55 cm; Height of Triangular Pyramid: 15 cm
Option 4 Base Length: 17 cm; Base Height (or Base
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area PERFORMANCE ASSESSMENT(S)
SUGGESTED DURATION : 12 days OVERARCHING CONCEPTS UNIT CONCEPTS
UNIT UNDERSTANDINGS
Width): 19 cm; Height of figure: 22 cm
1. The logo of the popcorn company will be printed on the sides of the prisms and pyramids for $0.005 per square centimeter. The printing company charges for the total surface that will be printed, even if the entire surface of a side will not be printed on. a. Describe which packaging option would be most cost effective if the company logo was printed on only the lateral faces. b. Describe which packaging option would be most cost effective if the company logo was printed on all the faces and bases. 2. The popcorn company has decided that it would like to have one prism and one pyramid packaging Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area PERFORMANCE ASSESSMENT(S)
SUGGESTED DURATION : 12 days OVERARCHING CONCEPTS UNIT CONCEPTS
UNIT UNDERSTANDINGS
option and would like both options to hold the same amount of popcorn. a. Calculate the volumes of the four packaging options to determine which prism and pyramid options will hold the least amount of popcorn. b. The company has decided that they would like to adjust the height of the rectangular prism with the least volume so that it has the same volume as the triangular pyramid with the least volume. Determine the adjusted height of the rectangular prism, to the nearest hundredth, so that its volume is the same as the triangular pyramid.. Standard(s): 7.1A , 7.1B , 7.1C , 7.1D , 7.1E , 7.1F , 7.1G , 7.9A , 7.9D ELPS.c.1A , ELPS.c.2C , ELPS.c.2D , ELPS.c.2E , ELPS.c.3C , ELPS.c.3D , ELPS.c.3H , ELPS.c.4C , ELPS.c.4D , ELPS.c.4F , ELPS.c.4H , ELPS.c.5B , ELPS.c.5F , ELPS.c.5G
MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS Misconceptions: Some students may think that “B” is synonymous with “b”, the length of the base, instead of “B”, which represents the area of the base of a three-dimensional figure. Some students may confuse the height of the two-dimensional face as the height of the three-dimensional prism or pyramid. Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
Some students may not associate the faces and bases of a prism or pyramid to the correct parts of the corresponding net. Some students may think that the face that rests on the bottom of the prism is the base.
UNIT VOCABULARY Area – the measurement attribute that describes the number of square units a figure or region covers Base of a rectangular prism – any two congruent, opposite and parallel faces shaped like rectangles; possibly more than one set Base of a rectangular pyramid – a rectangle attached to 4 triangular faces that meet at one common point, a vertex Base of a triangular prism – the two congruent, opposite and parallel faces shaped like triangles Base of a triangular pyramid – a triangle attached to 3 triangular faces that meet at one common point, a vertex Congruent – of equal measure, having exactly the same size and same shape Face – a flat surface of a three-dimensional figure Height of a rectangular prism – the length of a side that is perpendicular to both bases Height of a rectangular pyramid – the length of a perpendicular line segment from the vertex of the pyramid to the base Height of a triangular prism – the length of a side that is perpendicular to both bases Height of a triangular pyramid – the length of a perpendicular line segment from the vertex of the pyramid to the base Lateral surface area – the sum of all the lateral surface areas of a figure; the number of square units needed to cover the lateral view (area excluding the base(s) of a three-dimensional figure) Net – a two-dimensional model or drawing that can be folded into a three-dimensional solid Positive rational numbers – the set of numbers that can be expressed as a fraction where a > 0 and b > 0 Prism – a three-dimensional figure containing two congruent and parallel faces that are polygons Pyramid – a three-dimensional figure containing a base that is a polygon and the faces are triangles that share a common vertex, also known as an apex Three-dimensional figure – a figure that has measurements including length, width (depth), and height Total surface area – the sum of all the surface areas of a figure; the number of square units needed to cover all of the surfaces (bases and lateral area) Volume – the measurement attribute of the amount of space occupied by matter Related Vocabulary: Base Edge
Perpendicular Rectangle
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Triangular pyramid Triangular prism Page 10 of 38
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area Height Length Parallel
SUGGESTED DURATION : 12 days Rectangular prism Rectangular pyramid Triangle
UNIT ASSESSMENT ITEMS Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Creator if your district has granted access to that tool.
Vertex Width
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Mathematics Concepts Tree Mathematics Grade 7 STAAR Analysis_Unit 08_TEKS Resource System Mathematics Grade 7 STAAR Enhanced Blueprint Mathematics Grade 7 TEKS Introduction and Focal Points with Aligned Standards Mathematics Grade 7 TEKS Supporting Information (with TEKS Resource System Comments)
Texas Education Agency – Texas College and Career Readiness Standards Texas Education Agency – Grade 7 Reference Materials Texas Education Agency - Revised Mathematics TEKS: Vertical Alignment Charts Texas Instruments - Graphing Calculator Tutorials Texas Education Agency – Mathematics Curriculum Texas Education Agency – Assessment Texas Education Agency – Mathematics TEKS: Supporting Information Texas Education Agency – Interactive Mathematics Glossary
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
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Bold black text in italics: Knowledge and Skills Statement (TEKS) Bold black text: Student Expectation (TEKS) Bold red text in italics: Student Expectation identified by TEA as a Readiness Standard for STAAR Bold green text in italics: Student Expectation identified by TEA as a Supporting Standard for STAAR Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit 7.1
Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
7.1A
Apply mathematics to problems arising in everyday life, society, and the workplace.
Blue text: Supporting Information / Clarifications from TCMPC (Specificity) Blue Italic text: Unit-specific clarification Black text: TEA Texas Response to Curriculum Focal Points (TxRCFP); Texas College and Career Readiness Standards (TxCCRS); TEA STAAR
Apply MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE Including, but not limited to: Mathematical problem situations within and between disciplines Everyday life Society
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
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Workplace Note(s): The mathematical process standards may be applied to all content standards as appropriate. TxRCFP: Developing fluency with rational numbers and operations to solve problems in a variety of contexts Representing and applying proportional relationships Using expressions and equations to describe relationships in a variety of contexts, including geometric problems Comparing sets of data TxCCRS: X. Connections 7.1B
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 Including, but not limited to: Problem-solving model Analyze given information Formulate a plan or strategy Determine a solution Justify the solution
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
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Evaluate 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. TxRCFP: Developing fluency with rational numbers and operations to solve problems in a variety of contexts Representing and applying proportional relationships Using expressions and equations to describe relationships in a variety of contexts, including geometric problems Comparing sets of data TxCCRS: VIII. Problem Solving and Reasoning 7.1C
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.
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 Including, but not limited to: Appropriate selection of tool(s) and techniques to apply in order to solve problems Tools Real objects Manipulatives Paper and pencil Technology
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
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Techniques Mental math Estimation Number sense Note(s): The mathematical process standards may be applied to all content standards as appropriate. TxRCFP: Developing fluency with rational numbers and operations to solve problems in a variety of contexts Representing and applying proportional relationships Using expressions and equations to describe relationships in a variety of contexts, including geometric problems Comparing sets of data TxCCRS: VIII. Problem Solving and Reasoning 7.1D
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, AND LANGUAGE AS APPROPRIATE Including, but not limited to: Mathematical ideas, reasoning, and their implications Multiple representations, as appropriate Symbols Diagrams
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
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Language Note(s): The mathematical process standards may be applied to all content standards as appropriate. TxRCFP: Developing fluency with rational numbers and operations to solve problems in a variety of contexts Representing and applying proportional relationships Using expressions and equations to describe relationships in a variety of contexts, including geometric problems Comparing sets of data TxCCRS: IX. Communication and Representation 7.1E
Create and use representations to organize, record, and communicate mathematical ideas.
Create, Use REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS Including, but not limited to: Representations of mathematical ideas Organize Record Communicate Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
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Note(s): The mathematical process standards may be applied to all content standards as appropriate. TxRCFP: Developing fluency with rational numbers and operations to solve problems in a variety of contexts Representing and applying proportional relationships Using expressions and equations to describe relationships in a variety of contexts, including geometric problems Comparing sets of data TxCCRS: IX. Communication and Representation 7.1F
Analyze mathematical relationships to connect and communicate mathematical ideas.
Analyze MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS Including, but not limited to: Mathematical relationships Connect and communicate mathematical ideas Conjectures and generalizations from sets of examples and non-examples, patterns, etc. Note(s): The mathematical process standards may be applied to all content standards as appropriate. TxRCFP: Developing fluency with rational numbers and operations to solve problems in a variety of
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
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contexts Representing and applying proportional relationships Using expressions and equations to describe relationships in a variety of contexts, including geometric problems Comparing sets of data TxCCRS: X. Connections 7.1G
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 Including, but not limited to: Mathematical ideas and arguments Validation of conclusions Displays to make work visible to others Diagrams, visual aids, written work, etc. Explanations and justifications Precise mathematical language in written or oral communication Note(s): The mathematical process standards may be applied to all content standards as appropriate. TxRCFP: Developing fluency with rational numbers and operations to solve problems in a variety of contexts Representing and applying proportional relationships
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
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Using expressions and equations to describe relationships in a variety of contexts, including geometric problems Comparing sets of data TxCCRS: IX. Communication and Representation 7.8
Expressions, equations, and relationships. The student applies mathematical process standards to develop geometric relationships with volume. The student is expected to:
7.8A
Model the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights and connect that relationship to the formulas.
Model THE RELATIONSHIP BETWEEN THE VOLUME OF A RECTANGULAR PRISM AND A RECTANGULAR PYRAMID HAVING BOTH CONGRUENT BASES AND HEIGHTS AND CONNECT THAT RELATIONSHIP TO THE FORMULAS Including, but not limited to: Three-dimensional figure – a figure that has measurements including length, width (depth), and height Prism – a three-dimensional figure containing two congruent and parallel faces that are polygons Rectangular prism 6 rectangular faces (2 parallel rectangular faces [bases], 4 rectangular faces) 12 edges 8 vertices Face – a flat surface of a three-dimensional figure
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
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Base of a rectangular prism – any two congruent, opposite and parallel faces shaped like rectangles; possibly more than one set Height of a rectangular prism – the length of a side that is perpendicular to both bases Pyramid – a three-dimensional figure containing a base that is a polygon and the faces are triangles that share a common vertex, also known as an apex Rectangular pyramid 5 faces (1 rectangular face [base], 4 triangular faces) 8 edges 5 vertices Base of a rectangular pyramid – a rectangle attached to 4 triangular faces that meet at one common point, a vertex Height of a rectangular pyramid – the length of a perpendicular line segment from the vertex of the pyramid to the base Volume – the measurement attribute of the amount of space occupied by matter One way to measure volume is a three-dimensional cubic measure Congruent – of equal measure, having exactly the same size and same shape Various models to represent the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights Filling the rectangular pyramid with a measurable unit (e.g., rice, sand, water, etc.) and emptying the contents into the rectangular prism until the rectangular prism is completely full The contents of the rectangular pyramid will need to be emptied three times in order to fill the rectangular prism completely. Creating a replica of the rectangular pyramid and rectangular prisms with clay and comparing their masses The mass of the rectangular prism will be three times the mass of the rectangular pyramid, whereas the mass of the rectangular pyramid is the mass of the
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
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UNIT LEVEL SPECIFICITY
SE#
rectangular prism. Generalizations from models used to represent the relationship between the volume of a rectangular prism and a rectangular pyramid having congruent bases and heights The volume of a rectangular prism is three times the volume of a rectangular pyramid. The volume of a rectangular pyramid is the volume of a rectangular prism. Connections between models to represent volume of a rectangular prism and rectangular pyramid having both congruent bases and heights to the formulas for volume Formulas for volume from STAAR Grade 7 Mathematics Reference Materials Prism V = Bh, where B represents the base area and h represents the height of the prism which is the number of times the base area is repeated or layered Rectangular prism The base of a rectangular prism is a rectangle whose area may be found with the formula, A = bh or A = lw, meaning the base area, B, may be found with the formula B = bh or B = lw; therefore, the volume of a rectangular prism may be found using V = Bh or V =(bh)h or V = (lw)h. Pyramid V = Bh, where B represents the base area and h represents the height of the pyramid Rectangular pyramid The base of a rectangular pyramid is a rectangle whose area may be found with the formula, A = bh or A = lw, meaning the base area, B, may be found with the formula B = bh or B = lw; therefore, the volume of a rectangular pyramid may be
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
found using V = Bh or V = (bh)h or V = (lw)h. Note(s): Grade Level(s): Grade 6 modeled area formulas for parallelograms, trapezoids, and triangles by decomposing and rearranging parts of these shapes. Grade 8 will describe the volume formula V = Bh of a cylinder in terms of its base area and its height. Grade 8 will model the relationship between the volume of a cylinder and a cone having both congruent bases and heights and connect that relationship to the formulas. Various mathematical process standards will be applied to this student expectation as appropriate. TxRCFP: Using expressions and equations to describe relationships in a variety of contexts, including geometric problems TxCCRS: IV. Measurement Reasoning VIII. Problem Solving and Reasoning IX. Communication and Representation 7.8B
Explain verbally and symbolically the relationship between the volume of a triangular prism and a triangular pyramid having both congruent bases and heights and connect that relationship to the formulas.
Explain VERBALLY AND SYMBOLICALLY THE RELATIONSHIP BETWEEN THE VOLUME OF A TRIANGULAR PRISM AND A TRIANGULAR PYRAMID HAVING BOTH CONGRUENT BASES AND HEIGHTS AND CONNECT THAT RELATIONSHIP TO THE FORMULAS Including, but not limited to:
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Three-dimensional figure – a figure that has measurements including length, width (depth), and height Prism – a three-dimensional figure containing two congruent and parallel faces that are polygons Triangular prism 5 faces (2 triangular faces [bases], 3 rectangular faces) 9 edges 6 vertices Face – a flat surface of a three-dimensional figure Base of a triangular prism – the two congruent, opposite and parallel faces shaped like triangles Height of a triangular prism – the length of a side that is perpendicular to both bases Pyramid – a three-dimensional figure containing a base that is a polygon and the faces are triangles that share a common vertex, also known as an apex Triangular pyramid 4 faces (1 triangular face [base], 3 triangular faces) 6 edges 4 vertices Base of a triangular pyramid – a triangle attached to 3 triangular faces that meet at one common point, a vertex Height of a triangular pyramid – the length of a perpendicular line segment from the vertex of the pyramid to the base Volume – the measurement attribute of the amount of space occupied by matter One way to measure volume is a three-dimensional cubic measure Congruent – of equal measure, having exactly the same size and same shape Generalizations of the relationship between the volume of a triangular prism and a triangular pyramid having congruent bases and heights The volume of a triangular prism is three times the volume of a triangular pyramid. Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
The volume of a triangular pyramid is the volume of a triangular prism. Connections between models to represent volume of a triangular prism and triangular pyramid having both congruent bases and heights to the formulas for volume Formulas for volume from STAAR Grade 7 Mathematics Reference Materials Prism V = Bh, where B represents the base area and h represents the height of the prism which is the number of times the base area is repeated or layered) Triangular prism The base of a triangular prism is a triangle whose area may be found with the formula, A = bh, meaning the base area, B, may be found using B = bh; therefore, the volume of a triangular prism may be found using V = Bh or V = . Pyramid V = Bh, where B represents the base area and h represents the height of the pyramid Triangular pyramid The base of a triangular pyramid is a triangle whose area may be found with the formula, A = bh, meaning the base area, B, may be found using B = bh; therefore, the volume of a triangular pyramid may be found using V = Bh or V =
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
or V =
Page 24 of 38
.
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Note(s): Grade Level(s): Grade 7 introduces explaining verbally and symbolically the relationship between the volume of a triangular prism and a triangular pyramid having both congruent bases and heights and connecting that relationship to the formulas. Grade 8 will model the relationship between the volume of a cylinder and a cone having both congruent bases and heights and connect that relationship to the formulas. Various mathematical process standards will be applied to this student expectation as appropriate. TxRCFP: Using expressions and equations to describe relationships in a variety of contexts, including geometric problems TxCCRS: I. Numeric Reasoning IV. Measurement Reasoning VIII. Problem Solving and Reasoning IX. Communication and Representation 7.9
Expressions, equations, and relationships. The student applies mathematical process standards to solve geometric problems. The student is expected to:
7.9A
Solve problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids. Readiness Standard
Solve PROBLEMS INVOLVING THE VOLUME OF RECTANGULAR PRISMS, TRIANGULAR PRISMS, RECTANGULAR PYRAMIDS, AND TRIANGULAR PYRAMIDS
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Page 25 of 38
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Including, but not limited to: Positive rational numbers – the set of numbers that can be expressed as a fraction , where a > 0 and b > 0 Various forms of positive rational numbers Counting (natural) numbers Decimals (greater than zero) Fractions (proper, improper, and mixed numbers greater than zero) Three-dimensional figure – a figure that has measurements including length, width (depth), and height Prism – a three-dimensional figure containing two congruent and parallel faces that are polygons Rectangular prism 6 rectangular faces (2 parallel rectangular faces [bases], 4 rectangular faces) 12 edges 8 vertices Face – a flat surface of a three-dimensional figure Base of a rectangular prism – any two congruent, opposite and parallel faces shaped like rectangles; possibly more than one set Height of a rectangular prism – the length of a side that is perpendicular to both bases Triangular prism 5 faces (2 triangular faces [bases], 3 rectangular faces) 9 edges 6 vertices Base of a triangular prism – the two congruent, opposite and parallel faces shaped like triangles Height of a triangular prism – the length of a side that is perpendicular to Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
Page 26 of 38
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
both bases Pyramid – a three-dimensional figure containing a base that is a polygon and the faces are triangles that share a common vertex, also known as an apex Rectangular pyramid 5 faces (1 rectangular face [base], 4 triangular faces) 8 edges 5 vertices Base of a rectangular pyramid – a rectangle attached to 4 triangular faces that meet a one common point, a vertex Height of a rectangular pyramid – the length of a perpendicular line segment from the vertex of the pyramid to the base Triangular pyramid 4 faces (1 triangular face [base], 3 triangular faces) 6 edges 4 vertices Base of a triangular pyramid – a triangle attached to 3 triangular faces that meet a one common point, a vertex Height of a triangular pyramid – the length of a perpendicular line segment from the vertex of the pyramid to the base Volume – the measurement attribute of the amount of space occupied by matter One way to measure volume is a three-dimensional cubic measure Positive rational number side lengths Recognition of volume embedded in mathematical and real-world problem situations Formulas for volume from STAAR Grade 7 Mathematics Reference Materials Prism V = Bh, where B represents the base area and h represents the height of the prism which is the number of times the base area is repeated or layered) Rectangular prism The base of a rectangular prism is a rectangle whose area may be
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
Page 27 of 38
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
found with the formula, A = bh or A = lw, meaning the base area, B, may be found with the formula B = bh or B = lw; therefore, the volume of a rectangular prism may be found using V = Bh or V = (bh)h or V = (lw)h. Triangular prism The base of a triangular prism is a triangle whose area may be found with the formula, A = bh, meaning the base area, B, may be found using B = bh; therefore, the volume of a triangular prism may be found using V = Bh or V =
.
Pyramid V = Bh, where B represents the base area and h represents the height of the pyramid Rectangular pyramid The base of a rectangular pyramid is a rectangle whose area may be found with the formula, A = bh or A = lw, meaning the base area, B, may be found with the formula B = bh or B = lw; therefore, the volume of a rectangular pyramid may be found using V = Bh or V = (bh)h or V = (lw)h. Triangular pyramid The base of a triangular pyramid is a triangle whose area may be found with the formula, A = bh, meaning the base area, B, may be found using B = bh; therefore, the volume of a triangular pyramid
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
Page 28 of 38
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
may be found using V = Bh or V =
or V =
.
Note(s): Grade Level(s): Grade 6 determined solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers. Grade 8 will solve problems involving the volume of cylinders, cones, and spheres. Various mathematical process standards will be applied to this student expectation as appropriate. TxRCFP: Using expressions and equations to describe relationships in a variety of contexts, including geometric problems TxCCRS: I. Numeric Reasoning III.C. Geometric Reasoning – Connections between geometry and other mathematical content strands IV. Measurement Reasoning VIII. Problem Solving and Reasoning IX. Communication and Representation X. Connections 7.9D
Solve problems involving the lateral and total surface area of a rectangular prism, rectangular pyramid, triangular prism, and triangular pyramid by determining the area of the shape's net. Supporting Standard
Solve PROBLEMS INVOLVING THE LATERAL AND TOTAL SURFACE AREA OF A RECTANGULAR PRISM, RECTANGULAR PYRAMID, TRIANGULAR PRISM, AND TRIANGULAR PYRAMID BY DETERMINING THE AREA OF THE SHAPE'S NET
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
Page 29 of 38
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Including, but not limited to: Positive rational numbers – the set of numbers that can be expressed as a fraction , where a > 0 and b > 0 Various forms of positive rational numbers Counting (natural) numbers Decimals (greater than zero) Fractions (proper, improper, and mixed numbers greater than zero) Three-dimensional figure – a figure that has measurements including length, width (depth), and height Prism – a three-dimensional figure containing two congruent and parallel faces that are polygons Rectangular prism 6 rectangular faces (2 parallel rectangular faces [bases], 4 rectangular faces) 12 edges 8 vertices Face – a flat surface of a three-dimensional figure Base of a rectangular prism – any two congruent, opposite and parallel faces shaped like rectangles; possibly more than one set Height of a rectangular prism – the length of a side that is perpendicular to both bases Triangular prism 5 faces (2 triangular faces [bases], 3 rectangular faces) 9 edges 6 vertices Base of a triangular prism – the two congruent, opposite and parallel faces shaped like triangles Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
Page 30 of 38
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Height of a triangular prism – the length of a side that is perpendicular to both bases Pyramid – a three-dimensional figure containing a base that is a polygon and the faces are triangles that share a common vertex, also known as an apex Rectangular pyramid 5 faces (1 rectangular face [base], 4 triangular faces) 8 edges 5 vertices Base of a rectangular pyramid – a rectangle attached to 4 triangular faces that meet at one common point, a vertex Height of a rectangular pyramid – the length of a perpendicular line segment from the vertex of the pyramid to the base Triangular pyramid 4 faces (1 triangular face [base], 3 triangular faces) 6 edges 4 vertices Base of a triangular pyramid – a triangle attached to 3 triangular faces that meet at one common point, a vertex Height of a triangular pyramid – the length of a perpendicular line segment from the vertex of the pyramid to the base Area – the measurement attribute that describes the number of square units a figure or region covers Area is a two-dimensional square unit measure. Positive rational number side lengths Surface Area Lateral surface area – the sum of all the lateral surface areas of a figure; the number of square units needed to cover the lateral view (area excluding the base(s) of a threedimensional figure) Total surface area – the sum of all the surface areas of a figure; the number of square
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
Page 31 of 38
INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
SUGGESTED DURATION : 12 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
units needed to cover all of the surfaces (bases and lateral area) Net – a two-dimensional model or drawing that can be folded into a three-dimensional solid Note(s): Grade Level(s): Grade 6 determined solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers. Grade 8 will use previous knowledge of surface area to make connections to the formulas for lateral and total surface area and determine solutions for problems involving rectangular prisms, triangular prisms, and cylinders. Various mathematical process standards will be applied to this student expectation as appropriate. TxRCFP: Using expressions and equations to describe relationships in a variety of contexts, including geometric problems TxCCRS: I. Numeric Reasoning III.C. Geometric Reasoning – Connections between geometry and other mathematical content strands IV. Measurement Reasoning VIII. Problem Solving and Reasoning IX. Communication and Representation X. Connections
ELPS#
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS.
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area ELPS#
SUGGESTED DURATION : 12 days
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. http://ritter.tea.state.tx.us/rules/tac/chapter074/ch074a.html#74.4 Choose appropriate ELPS to support instruction. ELPS.c.1
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:
ELPS.c.1A
use prior knowledge and experiences to understand meanings in English
ELPS.c.1B
monitor oral and written language production and employ self-corrective techniques or other resources
ELPS.c.1C
use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and grade-level vocabulary
ELPS.c.1D
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)
ELPS.c.1E
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
ELPS.c.1F
use accessible language and learn new and essential language in the process
ELPS.c.1G
demonstrate an increasing ability to distinguish between formal and informal English and an increasing knowledge of when to use each one
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area ELPS#
SUGGESTED DURATION : 12 days
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS. commensurate with grade-level learning expectations
ELPS.c.1H
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.
ELPS.c.2
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:
ELPS.c.2A
distinguish sounds and intonation patterns of English with increasing ease
ELPS.c.2B
recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters
ELPS.c.2C
learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions
ELPS.c.2D
monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed
ELPS.c.2E
use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language
ELPS.c.2F
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
ELPS.c.2G
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
ELPS.c.2H
understand implicit ideas and information in increasingly complex spoken language commensurate with grade-level learning expectations
ELPS.c.2I
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.
ELPS.c.3
The ELL speaks in a variety of modes for a variety of purposes with an awareness of different language registers (formal/informal) using vocabulary
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area ELPS#
SUGGESTED DURATION : 12 days
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS. 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:
ELPS.c.3A
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
ELPS.c.3B
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
ELPS.c.3C
speak using a variety of grammatical structures, sentence lengths, sentence types, and connecting words with increasing accuracy and ease as more English is acquired
ELPS.c.3D
speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency
ELPS.c.3E
share information in cooperative learning interactions
ELPS.c.3F
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
ELPS.c.3G
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
ELPS.c.3H
narrate, describe, and explain with increasing specificity and detail as more English is acquired
ELPS.c.3I
adapt spoken language appropriately for formal and informal purposes
ELPS.c.3J
respond orally to information presented in a wide variety of print, electronic, audio, and visual media to build and reinforce concept and language attainment.
ELPS.c.4
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
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area ELPS#
SUGGESTED DURATION : 12 days
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS. 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:
ELPS.c.4A
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
ELPS.c.4B
recognize directionality of English reading such as left to right and top to bottom
ELPS.c.4C
develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials
ELPS.c.4D
use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary and other prereading activities to enhance comprehension of written text
ELPS.c.4E
read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned
ELPS.c.4F
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
ELPS.c.4G
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
ELPS.c.4H
read silently with increasing ease and comprehension for longer periods
ELPS.c.4I
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
ELPS.c.4J
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
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area ELPS#
SUGGESTED DURATION : 12 days
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS.
ELPS.c.4K
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.
ELPS.c.5
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:
ELPS.c.5A
learn relationships between sounds and letters of the English language to represent sounds when writing in English
ELPS.c.5B
write using newly acquired basic vocabulary and content-based grade-level vocabulary
ELPS.c.5C
spell familiar English words with increasing accuracy, and employ English spelling patterns and rules with increasing accuracy as more English is acquired
ELPS.c.5D
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
ELPS.c.5E
employ increasingly complex grammatical structures in content area writing commensurate with grade-level expectations, such as:
ELPS.c.5F
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
ELPS.c.5G
narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired.
Last Updated 04/26/2016
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
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INSTRUCTIONAL FOCUS DOCUMENT Grade 7 Mathematics TITLE : Unit 08: Volume and Surface Area
Last Updated 04/26/2016 Print Date 08/19/2016 Printed By Jaicee Sawyer, NOCONA MIDDLE
SUGGESTED DURATION : 12 days
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