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 TITLE : Unit 08: Volume and Surface Area SUGGESTED DURATION : 12 days

#### Unit Overview

Introduction
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.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool Grade 7

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 a 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 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 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 A2, B1; II. Algebraic Reasoning D1, D2; III. Geometric and Spatial Reasoning A2, C1, D2, D3; V. Statistical Reasoning A1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
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. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.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.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Illustrating and analyzing geometric relationships in models and diagrams aid in representing attributes of geometric figures with quantifiable measures and equations in order to generalize geometric relationships and solve problems.
• How can the relationship between the volume of a rectangular prism and the volume of a rectangular pyramid having both congruent bases and heights be …
• modeled?
• described verbally?
• generalized symbolically?
• How can the relationship between the volume of a triangular prism and the volume of a triangular pyramid having both congruent bases and heights be …
• modeled?
• described verbally?
• generalized symbolically?
• Expressions, Equations, & Relationships
• Geometric Representations
• Two-dimensional figures
• Three-dimensional figures
• Geometric Relationships
• Formulas
• Volume
• Measure relationships
• Geometric properties
• Representations
• Associated Mathematical Processes
• Application
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Illustrating and analyzing geometric relationships in models and diagrams aid in representing attributes of geometric figures with quantifiable measures and equations in order to generalize geometric relationships and solve problems.
• What is the process for determining the volume of a …
• rectangular prism?
• rectangular pyramid?
• triangular prism?
• triangular pyramid?
• How can the height of a …
• rectangular prism
• rectangular pyramid
• triangular prism
• triangular pyramid
… be determined when given the area of the base and its volume?
• How can the area of the base of a …
• rectangular prism
• rectangular pyramid
• triangular prism
• triangular pyramid
… be determined when given the height and its volume?
• What relationship exists between the volume of a rectangular prism and the volume of a rectangular pyramid having both congruent bases and heights?
• What relationship exists between the volume of a triangular prism and the volume of a triangular pyramid having both congruent bases and heights?
• What is the difference between the lateral surface area and total surface area of a figure?
• How can the area of the shape’s net be used to find the …
• lateral surface area
• total surface area
… of a …
• rectangular prism?
• rectangular pyramid?
• triangular prism?
• triangular pyramid?
• Expressions, Equations, & Relationships
• Geometric Representations
• Two-dimensional figures
• Three-dimensional figures
• Geometric Relationships
• Formulas
• Area, lateral surface area, and total surface area
• Volume
• Measure relationships
• Geometric properties
• Representations
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

#### 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.
• 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 pyramid – a rectangular face opposite the common vertex (apex) where the 4 triangular faces meet
• Base of a triangular pyramid – a triangular face opposite the common vertex (apex) where the 3 triangular faces meet
• Bases of a rectangular prism – any two congruent, opposite, and parallel faces shaped like rectangles; exactly 3 possible sets
• Bases of a triangular prism – the two congruent, opposite and parallel faces shaped like triangles
• Congruent – of equal measure, having exactly the same size and same shape
• Edge – where the sides of two faces meet on the three-dimensional figure
• 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 and b are counting (natural) numbers
• 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)
• Vertex (vertices) in the three-dimensional figure – the point (corner) where three or more edges of the three-dimensional figure meet
• Volume – the measurement attribute of the amount of space occupied by matter

Related Vocabulary:

 Base Height Length Parallel Perpendicular Rectangle Rectangular prism Rectangular pyramid Triangle Triangular pyramid Triangular prism Width
Unit Assessment Items System Resources Other Resources

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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 Center if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Grade 7 Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Student Expectations (TEKS) are labeled Readiness as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Supporting as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Process standards as identified by TEA of the assessed curriculum.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
TEKS# SE# TEKS SPECIFICITY
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.
Process Standard

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
• 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:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
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.
Process Standard

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
• 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:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.A. Statistical Reasoning – Design a study
• V.A.1. Formulate a statistical question, plan an investigation, and collect data.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.2. Formulate a plan or strategy.
• VII.A.3. Determine a solution.
• VII.A.4. Justify the solution.
• VII.A.5. Evaluate the problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving process.
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.
Process Standard

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
• 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:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
7.1D

Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Process Standard

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
• 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:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
7.1E Create and use representations to organize, record, and communicate mathematical ideas.
Process Standard

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

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.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
7.1F Analyze mathematical relationships to connect and communicate mathematical ideas.
Process Standard

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.
• Current knowledge to new learning

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:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
7.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Process Standard

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
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII. A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII. C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
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
• Edge – where the sides of two faces meet on a three-dimensional figure
• Vertex (vertices) in a three-dimensional figure – the point (corner) where three or more edges of a three-dimensional figure meet
• Face – a flat surface of a three-dimensional figure
• Bases of a rectangular prism – any two congruent, opposite, and parallel faces shaped like rectangles; exactly 3 possible sets
• Height of a rectangular prism – the length of a side that is perpendicular to both bases
• Base of a rectangular pyramid – a rectangular face opposite the common vertex (apex) where the 4 triangular faces meet
• Height of a rectangular pyramid – the length of a perpendicular line segment from the vertex of the pyramid to the base
• Prism – a three-dimensional figure containing two congruent and parallel faces that are polygons
• Rectangular prism, including a cube or square prism
• 6 rectangular faces (2 parallel rectangular faces [bases], 4 rectangular faces)
• 12 edges
• 8 vertices
• Pyramid – a three-dimensional figure containing a base that is a polygon and triangular faces that share a common vertex, also known as an apex
• Rectangular pyramid, including a square pyramid
• 5 faces (1 rectangular face [base], 4 triangular faces)
• 8 edges
• 5 vertices (1 apex, 4 vertices)
• 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 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 found using V = Bh or V = (bh)h or V = (lw)h.

Note(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:
• III.A. Geometric and Spatial Reasoning – Figures and their properties
• III.A.2. Form and validate conjectures about one-, two-, and three-dimensional figures and their properties.
• III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
• III.C.1. Make connections between geometry and algebraic equations.
• III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
• III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
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:

• Three-dimensional figure – a figure that has measurements including length, width (depth), and height
• Edge – where the sides of two faces meet on a three-dimensional figure
• Vertex (vertices) in a three-dimensional figure – the point (corner) where three or more edges of a three-dimensional figure meet
• Face – a flat surface of a three-dimensional figure
• Bases 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
• Base of a triangular pyramid – a triangular face opposite the common vertex (apex) where the 3 triangular faces meet
• Height of a triangular pyramid – the length of a perpendicular line segment from the vertex of the pyramid to the base
• 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
• 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 (1 apex, 4 vertices)
• 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.
• 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 = or V = .

Note(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:
• III.A. Geometric and Spatial Reasoning – Figures and their properties
• III.A.2. Form and validate about one-, two-, and three-dimensional figures and their properties.
• III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
• III.C.1. Make connections between geometry and algebraic equations.
• III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
• III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
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.

Solve

PROBLEMS INVOLVING THE VOLUME OF RECTANGULAR PRISMS, TRIANGULAR PRISMS, RECTANGULAR PYRAMIDS, AND TRIANGULAR PYRAMIDS

Including, but not limited to:

• Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
• Various forms of positive rational numbers
• Counting (natural) numbers
• Decimals
• Fractions
• 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, including a cube or square prism
• Triangular prism
• 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, including a square pyramid
• Triangular pyramid
• 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 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 may be found using V = Bh or V = or V = .

Note(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.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
• III.D.2. Determine the surface area and volume of three-dimensional figures.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.
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

Including, but not limited to:

• Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
• Various forms of positive rational numbers
• Counting (natural) numbers
• Decimals
• Fractions
• Three-dimensional figure – a figure that has measurements including length, width (depth), and height
• Edge – where the sides of two faces meet on a three-dimensional figure
• Vertex (vertices) in a three-dimensional figure – the point (corner) where three or more edges of a three-dimensional figure meet
• Face – a flat surface of a three-dimensional figure
• Bases of a rectangular prism – any two congruent, opposite, and parallel faces shaped like rectangles; exactly 3 possible sets
• Height of a rectangular prism – the length of a side that is perpendicular to both bases
• Bases 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
• Base of a rectangular pyramid – a rectangular face opposite the common vertex (apex) where the 4 triangular faces meet
• Height of a rectangular pyramid – the length of a perpendicular line segment from the vertex of the pyramid to the base
• Base of a triangular pyramid – a triangular face opposite the common vertex (apex) where the 3 triangular faces meet
• Height of a triangular pyramid – the length of a perpendicular line segment from the vertex of the pyramid to the base
• Prism – a three-dimensional figure containing two congruent and parallel faces that are polygons
• Rectangular prism, including a cube or square prism
• 6 rectangular faces (2 parallel rectangular faces [bases], 4 rectangular faces)
• 12 edges
• 8 vertices
• Triangular prism
• 5 faces (2 triangular faces [bases], 3 rectangular faces)
• 9 edges
• 6 vertices
• 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, including a square pyramid
• 5 faces (1 rectangular face [base], 4 triangular faces)
• 8 edges
• 5 vertices
• Triangular pyramid
• 4 faces (1 triangular face [base], 3 triangular faces)
• 6 edges
• 4 vertices
• 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 three-dimensional figure)
• 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)
• Net – a two-dimensional model or drawing that can be folded into a three-dimensional solid

Note(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.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
• III.D.2. Determine the surface area and volume of three-dimensional figures.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution. 