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:

8.1B 
Use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution.
Process Standard

Use
A PROBLEMSOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEMSOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:
 Problemsolving model
 Analyze given information
 Formulate a plan or strategy
 Determine a solution
 Justify the solution
 Evaluate the problemsolving 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 nonexamples, 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:

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:
 Threedimensional 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 threedimensional cubic measure
 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 = πr^{2} meaning the base area, B, may be found with the formula B = πr^{2}h; therefore, the volume of a cylinder may be found using V = Bh or V = πr^{2}h.
 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 Level(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:
 Threedimensional 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 threedimensional 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 = πr^{2} meaning the base area, B, may be found with the formula B = πr^{2}; therefore, the volume of a cylinder may be found using V = Bh or V = πr^{2}h.
 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 = πr^{2} meaning the base area, B, may be found with the formula B = πr^{2}; therefore, the volume of a cone may be found using V = Bh or V = πr^{2}h.
 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 Level(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.
Readiness Standard

Solve
PROBLEMS INVOLVING THE VOLUME OF CYLINDERS, CONES, AND SPHERES
Including, but not limited to:
 Threedimensional figure – a figure that has measurements including length, width (depth), and height
 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 threedimensional cubic measure
 Positive rational number side lengths
 Recognition of volume embedded in mathematical and realworld problem situations
 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 = πr^{2} meaning the base area, B, may be found with the formula B = πr^{2}; therefore, the volume of a cylinder may be found using V = Bh or V = πr^{2}h.
 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 = πr^{2} meaning the base area, B, may be found with the formula B = πr^{2}; therefore, the volume of a cone may be found using V = Bh or V = πr^{2}h.
 Sphere
 V = πr^{3}, where r represents the radius of the sphere
 Composite figures
Note(s):
 Grade Level(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.
Readiness Standard

Use
PREVIOUS KNOWLEDGE OF SURFACE AREA TO MAKE CONNECTIONS TO THE FORMULAS FOR LATERAL AND TOTAL SURFACE AREA
Including, but not limited to:
 Threedimensional figure – a figure that has measurements including length, width (depth), and height
 Edge – where the sides of two faces meet on a threedimensional figure
 Vertex (vertices) in a threedimensional figure – the point (corner) where three or more edges of a threedimensional figure meet
 Face – a flat surface of a threedimensional 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 threedimensional 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 twodimensional 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 threedimensional 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 + 2πr^{2}, 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:
 Threedimensional figure – a figure that has measurements including length, width (depth), and height
 Prism – a threedimensional figure containing two congruent and parallel faces that are polygons
 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 twodimensional 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 h 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 + 2πr^{2}, 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 Level(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
