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 Instructional Focus DocumentGrade 8 Mathematics
 TITLE : Unit 09: Measurement of Three-Dimensional Figures SUGGESTED DURATION : 12 days

Unit Overview

Introduction
This unit bundles student expectations that address the volume of cylinders, cones, and spheres, and the surface area of rectangular prisms, triangular prisms, and cylinders. 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.” Additionally, the availability of graphing technology is required during STAAR testing.

Prior to this Unit
In Grade 7, students modeled the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights and connected that relationship to the formulas. Students explained verbally and symbolically the relationship between the volume of a triangular prism and a triangular pyramid having both congruent bases and heights and connected that relationship to the formulas. They also solved problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids as well as problems involving the lateral and total surface area of rectangular prisms, rectangular pyramids, triangular prisms, and triangular pyramids by determining the area of the shape's net. Students described π as the ratio of the circumference of a circle to its diameter and determined the circumference and area of circles.

During this Unit
Students blend previous understandings of the volume of a prism with calculating the area of a circle to determining the volume of a cylinder in terms of its base area and height. As with previous grade level investigations of the volume of three-dimensional figures, students are expected to model the relationship between the volume of a cylinder and a cone having both congruent bases and heights. Students connect these models to the actual formulas for determining the volume of a cylinder and cone, which directly coincides with formulas used for determining the volume of prisms and pyramids on the STAAR Grade 8 Mathematics Reference Materials. Students solve problems involving the volume of cylinders, cones, and spheres. The concept of surface area is extended from finding the sum of the areas of the faces from the net to abstract formulas for lateral and total surface area. Students are expected to 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.

After this Unit
In high school mathematics coursework, students will apply the formulas for the total and lateral surface area of three-dimensional figures, including prisms, pyramids, cones, cylinders, spheres, and composite figures, to solve problems using appropriate units of measure. They will also determine and describe how changes in the linear dimensions of a shape affect its perimeter, area, surface area, or volume, including proportional and non-proportional dimensional change.

In Grade 8, describing the volume formula V = Bh of a cylinder in terms of its base area and its height is STAAR Supporting Standard 8.6A. Solving problems involving the volume of cylinders, cones, and spheres and using previous knowledge of surface area to make connections to the formulas for lateral and total surface area and determining solutions for problems involving rectangular prisms, triangular prisms, and cylinders are STAAR Readiness Standards 8.7A and 8.7B. These standards are part of Grade 8 STAAR Reporting Category 3: Geometry and Measurement. Modeling the relationship between the volume of a cylinder and a cone having both congruent bases and heights and connecting that relationship to the formulas is standard 8.6B and is neither Supporting nor Readiness, but is foundational to the conceptual understanding of geometry and measurement. All of the standards in this unit are part of the Grade 8 Texas Response to Curriculum Focal Points (TxRCFP):Using expressions and equations to describe relationships, including the Pythagorean Theorem. 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.

Research
According to the National Council of Teachers of Mathematics (2010), “In grade 8, problem-solving experiences should include rich problems that require students to bring together different concepts, such as measurement, similarity, ratio and proportion, area and volume, angle relationships, and the Pythagorean theorem” (p. 75). The National Mathematics Advisory Panel (2008) reports that “…students should be able to analyze the properties of two- and three-dimensional shapes using formulas to determine perimeter, area, volume, and surface area” (p. 18). They also conclude that “By learning about how length, area, and volume are measured, students mentally structure and revise their construction of space, both large-scale and small-scale” (p. 281).

National Council of Teachers of Mathematics. (2010). Focus in grade 8: Teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the national mathematics advisory panel. Washington, DC: U.S. Department of Education.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9
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

 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 cylinder and volume of a cone 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.
• How can the volume of a cylinder be described in terms of its base area and height?
• What is the process for determining the volume of a …
• cylinder?
• cone?
• sphere?
• How can the height of a …
• cylinder
• cone
… be determined when given the radius and its volume?
• How can the radius of a …
• cylinder
• cone
… be determined when given the height and its volume?
• How can the radius of a sphere be determined when given the volume?
• Why would it be beneficial to state the volume of a cylinder, cone, or sphere in terms of pi instead of a decimal approximation?
• What relationship exists between the volume of a cylinder and the volume of a cone having both congruent bases and heights?
• How does previous knowledge of surface area, including nets, aid in generalizing symbolically the formulas for lateral and total surface area of a …
• rectangular prism?
• triangular prism?
• cylinder?
• What is the difference between lateral surface area and total surface area of a figure?
• What is the process for determining the lateral surface area of a …
• rectangular prism?
• triangular prism?
• cylinder?
• What is the process for determining the total surface area of a …
• rectangular prism?
• triangular prism?
• cylinder?
• How can the height of a prism be determined when given the …
• perimeter of the base and lateral surface area?
• perimeter of the base, area of the base, and total surface area?
• How can the height of a cylinder be determined when given the …
• radius and its lateral surface area?
• radius and its total surface area?
• Why would it be beneficial to state the lateral or total surface area of a cylinder in terms of pi instead of a decimal approximation?
• Expressions, Equations, & Relationships
• Geometric Representations
• Two-dimensional figures
• Three-dimensional figures
• Geometric Relationships
• Formulas
• Perimeter and circumference
• 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.

Underdeveloped Concepts:

• Some students may confuse the formulas for the circumference and area of a circle.
• Some students may multiply the radius by 2 rather than squaring it when determining the area of a circle or multiply the radius by 3 rather than cubing it when determining the volume of a sphere.

Unit Vocabulary

• Area – the measurement attribute that describes the number of square units a figure or region covers
• Base of a cone – the circular base opposite the vertex (apex)
• 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
• Bases of a cylinder – the two congruent, opposite circular bases
• Congruent – of equal measure, having exactly the same size and same shape
• Edge – where the two faces meet on a three-dimensional figure
• Face – a flat surface of a three-dimensional figure
• Height of a cone – the length of a perpendicular line segment from the vertex of the cone to the base
• Height of a cylinder – the length of a line segment that is perpendicular to both bases
• Height of a rectangular prism – the length of a side that is perpendicular to both bases
• Height of a triangular prism – the length of a side that is perpendicular to both bases
• 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)
• Pi ( ) – the ratio of the circumference to the diameter of a circle
• Prism – a three-dimensional figure containing two congruent and parallel faces that are polygons
• 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 a three dimensional figure – the point (corner) where three or more edges of a three-dimensional figure meet
• Volume – the measurement attribute of the amount of space occupied by matter

Related Vocabulary:

 Base Circle Cone Cylinder Diameter Height Parallel Perimeter Perpendicular Radius Rectangle Rectangular prism Side Sphere Triangle Triangular prism
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 8 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.
TEKS# SE# TEKS SPECIFICITY
8.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
8.1A 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:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• X. Connections
8.1B 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:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• VIII. Problem Solving and Reasoning
8.1C 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:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• VIII. Problem Solving and Reasoning
8.1D 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, GRAPHS, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• IX. Communication and Representation
8.1E 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:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• IX. Communication and Representation
8.1F 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:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• X. Connections
8.1G 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:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• IX. Communication and Representation
8.6 Expressions, equations, and relationships. The student applies mathematical process standards to develop mathematical relationships and make connections to geometric formulas. The student is expected to:
8.6A Describe the volume formula V = Bh of a cylinder in terms of its base area and its height.
Supporting Standard

Describe

THE VOLUME FORMULA V = Bh OF A CYLINDER IN TERMS OF ITS BASE AREA AND ITS HEIGHT

Including, but not limited to:

• Three-dimensional figure – a figure that has measurements including length, width (depth), and height
• Bases of a cylinder – the two congruent, opposite circular faces
• Height of a cylinder – the length of a line segment that is perpendicular to both bases
• Cylinder
• 2 congruent, parallel circular bases
• 1 curved surface
• Pi (π) – the ratio of the circumference to the diameter of a circle
• Volume – the measurement attribute of the amount of space occupied by matter
• One way to measure volume is a three-dimensional cubic measure
• Formulas for volume from STAAR Grade 8 Mathematics Reference Materials
• Cylinder
• V = Bh, where B represents the base area and represents the height of the cylinder, which is the number of times the base area is repeated or layered
• The base of a cylinder is a circle whose area may be found with the formula, A = πr2 meaning the base area, B, may be found with the formula B = πr2h; therefore, the volume of a cylinder may be found using V = Bh or V = πr2h.
• Relationship between volume of a prism and volume of a cylinder
• The formula used to determine volume of a prism is V = Bh, and the formula to determine the volume of a cylinder is V = Bh.
• The base area depends on the shape of the base (e.g., the shape of the base of a triangular prism is a triangle; the shape of the base of a cylinder is a circle, etc.).
• Relationship between volume of a cylinder, its base area, and height
• The volume of a cylinder is the product of its base area and its height. (V = Bh)
• The base area of a cylinder is the quotient of its volume and its height. ( )
• The height of a cylinder is the quotient of its volume and its base area. ( )

Note(s):

• Grade 5 introduced the volume of a rectangular prism using V = Bh.
• Grade 6 applied the volume of a rectangular prism using V = Bh.
• Grade 7 introduced and applied the volume of a triangular prism using V = Bh.
• Grade 7 modeled the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights and connected 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, including the Pythagorean Theorem
• TxCCRS:
• 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
8.6B 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.

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

Including, but not limited to:

• Three-dimensional figure – a figure that has measurements including length, width (depth), and height
• Bases of a cylinder – the two congruent, opposite circular faces
• Height of a cylinder – the length of a line segment that is perpendicular to both bases
• Base of a cone – the circular face opposite the vertex (apex)
• Height of a cone – the length of a perpendicular line segment from the vertex of the cone to the base
• Cylinder
• 2 congruent, parallel circular bases
• 1 curved surface
• Cone
• 1 curved surface
• 1 vertex (apex)
• Pi (π) – the ratio of the circumference to the diameter of a circle
• 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 cylinder and a cone having both congruent bases and heights
• Filling the cone with a measurable unit (e.g., rice, sand, water, etc.) and emptying the contents into the cylinder until the cylinder is completely full.
• The contents of the cone will need to be emptied three times in order to fill the cylinder completely.
• Creating a replica of the cone and cylinder with clay and comparing their masses.
• The mass of the cylinder will be three times the mass of the cone, whereas the mass of the cone is the mass of the cylinder.
• Generalizations from models used to represent the relationship between the volume of a cylinder and a cone having congruent bases and heights.
• The volume of a cylinder is three times the volume of a cone.
• The volume of a cone is the volume of a cylinder.
• Connections between models to represent volume of a cylinder and cone having both congruent bases and heights to the formulas for volume
• Formulas for volume from STAAR Grade 8 Mathematics Reference Materials
• Cylinder
• V = Bh, where B represents the base area and h represents the height of the cylinder, which is the number of times the base area is repeated or layered
• The base of a cylinder is a circle whose area may be found with the formula, A = πr2 meaning the base area, B, may be found with the formula B = πr2; therefore, the volume of a cylinder may be found using V = Bh or V = πr2h.
• Cone
• V = Bh, where B represents the base area and h represents the height of the cone
• The base of a cone is a circle whose area may be found with the formula, A = πr2 meaning the base area, B, may be found with the formula B = πr2; therefore, the volume of a cone may be found using V = Bh or V = πr2h.
• Relationship between the volume of prisms and cylinders as compared to the volume of pyramids and cones
• The formula used to determine volume of a prism is V = Bh, and the formula to determine the volume of a cylinder is V = Bh.
• The formula used to determine volume of a pyramid is V = Bh, and the formula to determine the volume of a cone is V = Bh.

Note(s):

• Grade 7 modeled 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.
• Grade 7 explained verbally and symbolically the relationship between the volume of a triangular prism and a triangular pyramid having both congruent bases and heights and connected 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, including the Pythagorean Theorem
• TxCCRS:
• 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
8.7 Expressions, equations, and relationships. The student applies mathematical process standards to use geometry to solve problems. The student is expected to:
8.7A Solve problems involving the volume of cylinders, cones, and spheres.

Solve

PROBLEMS INVOLVING THE VOLUME OF CYLINDERS, CONES, AND SPHERES

Including, but not limited to:

• Three-dimensional figure – a figure that has measurements including length, width (depth), and height
• Cylinder
• Cone
• Sphere
• Pi (π) – the ratio of the circumference to the diameter of a circle
• 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 8 Mathematics Reference Materials
• Cylinder
• V = Bh, where B represents the base area and represents the height of the cylinder, which is the number of times the base area is repeated or layered
• The base of a cylinder is a circle whose area may be found with the formula, A = πr2 meaning the base area, B, may be found with the formula B = πr2; therefore, the volume of a cylinder may be found using V = Bh or V = πr2h.
• Cone
• V = Bh, where B represents the base area and h represents the height of the cone
• The base of a cone is a circle whose area may be found with the formula, A = πr2 meaning the base area, B, may be found with the formula B = πr2; therefore, the volume of a cone may be found using V = Bh or V = πr2h.
• Sphere
• V = πr3, where r represents the radius of the sphere
• Composite figures

Note(s):

• Grade 7 solved problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• 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
8.7B 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.

Use

PREVIOUS KNOWLEDGE OF SURFACE AREA TO MAKE CONNECTIONS TO THE FORMULAS FOR LATERAL AND TOTAL SURFACE AREA

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
• 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
• Bases of a cylinder – the two congruent, opposite circular faces
• Height of a cylinder – the length of a line segment that is perpendicular to both bases
• 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
• Triangular prism
• 5 faces (2 triangular faces [bases], 3 rectangular faces)
• 9 edges
• 6 vertices
• Cylinder
• 2 congruent, parallel circular bases
• 1 curved surface
• Pi (π) – the ratio of the circumference to the diameter of a circle
• Area – the measurement attribute that describes the number of square units a figure or region covers
• Area is a two-dimensional square unit measure.
• Surface area
• Lateral surface area – 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 number of square units needed to cover all of the surfaces (bases and lateral area)
• Connections between nets used to find lateral and total surface area and the formulas
• Formulas for surface area from STAAR Grade 8 Mathematics Reference Materials
• Lateral surface area
• Prism
• S = Ph, where P represents the perimeter of the base and h represents the height of the prism
• Cylinder
• S = 2πrh, where r represents the radius of the circular base and h represents the height of the cylinder
• Total surface area
• Prism
• S = Ph + 2B, where P represents the perimeter of the base, h represents the height of the prism, and B represents the base area of the prism
• Cylinder
• S = 2πrh + 2πr2, where r represents the radius of the circular base and h represents the height of the cylinder

Determine

SOLUTIONS FOR PROBLEMS INVOLVING LATERAL AND TOTAL SURFACE AREA FOR RECTANGULAR PRISMS, TRIANGULAR PRISMS, AND CYLINDERS

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
• Triangular prism
• Cylinder
• Pi (π) – the ratio of the circumference to the diameter of a circle
• 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
• Formulas for surface area from STAAR Grade 8 Mathematics Reference Materials
• Lateral surface area
• Prism
• S = Ph, where P represents the perimeter of the base and represents the height of the prism
• Rectangular prism
• Triangular prism
• Cylinder
• S = 2πrh, where r represents the radius of the circular base and h represents the height of the cylinder
• Total surface area
• Prism
• S = Ph + 2B, where P represents the perimeter of the base, h represents the height of the prism, and B represents the base area of the prism
• Rectangular prism
• Triangular prism
• Cylinder
• S = 2πrh + 2πr2, where r represents the radius of the circular base and h represents the height of the cylinder
• Lateral and total surface area involving composite figures including rectangular prisms, triangular prisms, and cylinders

Note(s):

• Grade 7 determined the area of composite figures containing combinations of rectangles, squares, parallelograms, trapezoids, triangles, semicircles, and quarter circles.
• Grade 7 solved problems involving the lateral and total surface area of a rectangular prisms, rectangular pyramids, triangular prisms, and triangular pyramids by determining the area of the shape's net.
• Grade 8 introduces determining lateral and total surface area using a formula.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• 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 