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 Instructional Focus DocumentGeometry
 TITLE : Unit 09: Measurement of Three-Dimensional Figures SUGGESTED DURATION : 10 days

#### Unit Overview

Introduction
This unit bundles student expectations that address perimeter, area, and volume of two- and three-dimensional figures and composite figures, including effects of proportional and non-proportional dimensional changes. Concepts are incorporated into both non-contextual and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Grades 2 – 5, students solved problems related to perimeter or regular and irregular figures, including finding missing side length(s). In Grade 6, students determine solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles. In Grade 7, students determine the circumference and area of circles and the area of composite figures. In Grade 8, students model the effect on linear and area measurements of dilated two-dimensional shapes. In Geometry Unit 07, students studied the relationships of two- and three-dimensional figures. In Geometry Unit 08, students were introduced to regular polygons and determining the perimeter, circumference, and area of two-dimensional figures, including composite figures.

During this Unit
Students explore surface area using concrete objects and nets of prisms, pyramids, cylinders and cones and make connections to formulas for lateral and total surface area for prisms, pyramids, cylinders and cones. Students are introduced to the formula for surface area of a sphere. Students apply formulas to determine lateral and total surface area of prisms, pyramids, cylinders, cones, and spheres from diagrams and attribute information, including composite figures. Students apply formulas to determine lateral and total surface area of prisms, pyramids, cylinders, cones, and spheres in real-world problem situations with appropriate measures, including effects of proportional and non-proportional linear dimension changes. Students explore volume using concrete models and investigate the differences between cylinders/cones and prisms/pyramids with congruent bases and heights and make connections to formulas for volume. Students are introduced to the formula for volume of a sphere. Students apply formulas to determine volume of prisms, pyramids, cylinders, cones, and spheres from diagrams and attribute information, including composite figures. Students apply formulas to determine lateral and total surface area and volume of prisms, pyramids, cylinders, cones, spheres, and composite figures in real-world problem situations with appropriate measures, including effects of proportional and non-proportional linear dimension changes.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool HS Geometry

After this Unit
In Unit 11, students will apply concepts of two- and three-dimensional measurement to complete an engineering design project. The concepts in this unit will also be applied in subsequent mathematics courses.

This unit is supporting the development of Texas College Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1, C1; II. Algebraic Reasoning A1, D1, D2; III. Geometric and Spatial Reasoning A2, B1, B2, C1, D1, 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 (2009) in Focus in High School Mathematics: Reasoning and Sense Making, the key elements of reasoning and sense making with geometry must include multiple representations of functions. In this unit, students gather data from geometric figures, and organize this information into tables, graphs or diagrams. This leads to the development of symbolic expressions and verbal descriptions. A variety of representations helps make relationships more understandable to more students than working with symbolic representations alone. These approaches serve as the basis for this unit on polygons and circle. At the conclusion of this unit, students are asked to create graphic organizers. TxCCRS cites many skills related to the communication and representation of mathematical ideas. The National Council of Teachers of Mathematics (2000) said that all students in grades 9 – 12 should explore relationships in two-dimensional geometric figures, make and test conjectures about two-dimensional geometric figures, and solve problems involving two-dimensional geometric figures. According to the National Council of Teachers of Mathematics (2012), using diagrams and constructions to interpret and communicate geometric relationships is essential in geometry. Using definitions of figures to characterize figures in terms of their properties is another essential in geometry. In geometry, the “proving process involves working with diagrams, variation and invariance, conjectures, and definitions.” (p. 92)

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2009). Focus in High School Mathematics: Reasoning and Sense Making. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2012). Developing essential understanding of Geometry for Teaching Mathematics in Grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9

 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? Geometric, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life. Why is developing geometric, spatial, and measurement reasoning essential? How does geometric, spatial, and measurement reasoning affect how one sees and works in the world?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding and working with variance and invariance within geometry builds flexible algebraic and geometric reasoning and deepens understanding of intrinsic properties of geometric relationships.
• What invariant (unchanging) and variant (changing) relationships exist within three-dimensional geometric figures, including composite figures?
• How does scaling measurements of geometric figures demonstrate variance and invariance?
• Accurate representations, models, or diagrams within a geometric system allows for visualizing, illustrating, and analyzing geometric relationships to aid in making and validating conjectures about those geometric relationships and is central to geometric thinking.
• What types of problem situations represent relationships in three-dimensional geometric figures, including composite figures?
• How can representations and appropriate geometric language be used to effectively communicate and illustrate geometric relationships in three-dimensional geometric figures, including composite figures?
• What tools and processes can be used to …
• find area, perimeter, and circumference of two-dimensional figures?
• find surface area (lateral and total) and volume of three-dimensional figures, including composite figures?
• describe the effects of proportional changes in linear dimensions on three-dimensional geometric figures?
• make connections between formulas and three-dimensional geometric figures?
• Attributes and quantifiable measures of geometric figures can be generalized to describe, determine, and represent algebraic and geometric relationships and be applied to solve problem situations.
• How can understanding relationships in three-dimensional geometric figures, including composite figures, be applied when solving problem situations?
• How can measurable attributes related to …
• side length
• scale factor
• perimeter and circumference
• area, lateral surface area, and total surface area
• volume
… be distinguished and described in order to generalize geometric relationships of three-dimensional geometric figures, including composite figures?
• What processes can be used to determine the …
• lateral surface area
• total surface area
• volume
• dimensional change
… of three-dimensional geometric figures, including …
• prisms?
• pyramids?
• cylinders?
• cones?
• spheres?
• composite figures?
• Two-Dimensional and Three-Dimensional Figures; Circles
• Circles
• Measure relationships
• Geometric Relationships
• Perpendicularity
• Proportionality
• Corresponding sides and angles
• Perimeter
• Area, lateral surface area, and total surface area
• Volume
• Measure relationships
• Formulas
• Proportional and non-proportional dimensional change
• Geometric Representations
• Two-dimensional figures
• Three-dimensional figures
• Composite figures
• Patterns, Operations, and Properties
• 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 the same scale factor is used for surface area and volume of proportional dimensional change as is used for the linear dimensional change instead of using the scale factor squared or the scale factor cubed, respectively.
• Some students may confuse the slant height and altitude of the pyramid and cone.
• Some students may think that surface area means only lateral surface area instead of total surface area.

Underdeveloped Concepts:

• Students may not be able to correctly identify the face or base of a three-dimensional figure.
• Students may not know how to find the area of a two-dimensional figure when finding the area of the base for a three-dimensional figure
• Students may not have a clear understanding of scale factor.
• Some students may not know how to determine how many times larger one three-dimensional figure is when compared to another three-dimensional figure when all side lengths have changed proportionality.
• Some students confuse total surface area with volume.
• Some students may think that when calculating total surface area of a three-dimensional, composite figure that all surfaces are included in the total surface area (e.g., When given a three-dimensional figure composed of a cylinder and two hemispheres, the top and bottom of the cylinder and the bottom of each hemisphere are not included in the total surface area.).
• Students may not be able to accurately draw a three-dimensional figure.

#### Unit Vocabulary

• Apothem – a segment that extends from the center of a regular polygon perpendicular to a side of the regular polygon. The apothem bisects the side of the regular polygon to which it is drawn.
• Radius of a regular polygon – a segment that extends from the center of a regular polygon to a vertex. The radius of a regular polygon bisects the vertex angle to which it is drawn.
• Regular polygon – a convex polygon in which all sides are congruent (equilateral) and all angles are congruent (equiangular)
• Three-dimensional non-proportional change – either one and/or two dimensions multiplied by a scale factor or the three dimensions are multiplied by different scale factors
• Three-dimensional proportional change – three dimensions multiplied by the same scale factor
• Two-dimensional non-proportional change – either only one dimension multiplied by a scale factor or the two dimensions are multiplied by different scale factors
• Two-dimensional proportional change – two dimensions multiplied by the same scale factor

Related Vocabulary:

 Area of a sector Base Circumference Composite figure Cone Cube Cylinder Diameter Edge Face Geometric solid Height Hemisphere Hexagonal pyramid Lateral surface area Net Polyhedron Prism Pyramid Radius Sphere Surface area Three-dimensional figure Total surface area Two-dimensional figure Vertex Volume
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 – 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 Geometry 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.
• 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
G.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
G.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE

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.
• 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.
G.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION

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.
• 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.
G.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS

Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• 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.
G.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE

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.
• 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.
G.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• 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.
G.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• 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.
G.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

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.
• 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.
G.10 Two-dimensional and three-dimensional figures. The student uses the process skills to recognize characteristics and dimensional changes of two- and three-dimensional figures. The student is expected to:
G.10B Determine and describe how changes in the linear dimensions of a shape affect its perimeter, area, surface area, or volume, including proportional and non-proportional dimensional change.

Determine, Describe

HOW CHANGES IN THE LINEAR DIMENSIONS OF A SHAPE AFFECT ITS PERIMETER, AREA, SURFACE AREA, OR VOLUME, INCLUDING PROPORTIONAL AND NON-PROPORTIONAL DIMENSIONAL CHANGE

Including, but not limited to:

• Verbal and written description
• Dimensional change
• Perimeter and circumference
• Area and surface area
• Volume
• Proportional change
• Two-dimensional proportional change – two dimensions multiplied by the same scale factor
• Three-dimensional proportional change – three dimensions multiplied by the same scale factor
• Non-proportional change
• Two-dimensional non-proportional change – either only one dimension multiplied by a scale factor or the two dimensions are multiplied by different scale factors
• Three-dimensional non-proportional change – either one and/or two dimensions multiplied by a scale factor or the three dimensions are multiplied by different scale factors
• Comparison of the effect of proportional and non-proportional dimensional change
• Emphasis on connections to units
• Dimension changes in real-world problem situations

Note(s):

• Grade 7 and 8 modeled the effect on linear and area measurements of dilated two-dimensional shapes.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• I.C. Numeric Reasoning – Systems of measurement
• I.C.1. Select or use the appropriate type of method, unit, and tool for the attribute being measured.
• 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.B. Geometric and Spatial Reasoning – Transformations and symmetry
• III.B.1. Identify transformations and symmetries of figures.
• III.B.2. Use transformations to investigate congruence, similarity, and symmetries of figures.
• 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.1. Find the perimeter and area of two-dimensional figures.
• III.D.2. Determine the surface area and volume of three-dimensional figures.
• III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
• 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.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
G.11 Two-dimensional and three-dimensional figures. The student uses the process skills in the application of formulas to determine measures of two- and three-dimensional figures. The student is expected to:
G.11A Apply the formula for the area of regular polygons to solve problems using appropriate units of measure.

Apply

THE FORMULA FOR THE AREA OF REGULAR POLYGONS TO SOLVE PROBLEMS USING APPROPRIATE UNITS OF MEASURE

Including, but not limited to:

• Regular polygon – a convex polygon in which all sides are congruent (equilateral) and all angles are congruent (equiangular)
• Radius of a regular polygon – a segment that extends from the center of a regular polygon to a vertex. The radius of a regular polygon bisects the vertex angle to which it is drawn.
• Apothem – a segment that extends from the center of a regular polygon perpendicular to a side of the regular polygon. The apothem bisects the side of the regular polygon to which it is drawn.
• Formula for the area of regular polygons
• A = aP where P represents the perimeter and a represents the apothem.
• Connection to area of a triangle: A = bh
• Real-world problem situations involving area
• Emphasis on appropriate units of measure

Note(s):

• Previous grade levels used units, tools, and formulas to find the area of figures in problem situations.
• Previous grade levels introduced the language of regular polygons.
• Grade 7 determined the composite area of figures composed of rectangles, squares, parallelograms, trapezoids, triangles, semicircles, and quarter circles.
• Geometry introduces a formula for the area of an n-sided polygon.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• II.A. Algebraic Reasoning – Identifying expressions equations
• II.A.1. Explain the difference between expressions and equations.
• III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
• III.D.1. Find the perimeter and area of two-dimensional figures.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.
G.11B Determine the area of composite two-dimensional figures comprised of a combination of triangles, parallelograms, trapezoids, kites, regular polygons, or sectors of circles to solve problems using appropriate units of measure.

Determine

THE AREA OF COMPOSITE TWO-DIMENSIONAL FIGURES COMPRISED OF A COMBINATION OF TRIANGLES, PARALLELOGRAMS, TRAPEZOIDS, KITES, REGULAR POLYGONS, OR SECTORS OF CIRCLES TO SOLVE PROBLEMS USING APPROPRIATE UNITS OF MEASURE

Including, but not limited to:

• Composites of two-dimensional figures
• Triangles
• Parallelograms
• Trapezoids
• Kites
• Regular polygons
• Sectors of circles
• Applications to real-world situations
• Appropriate use of units of measure

Note(s):

• Previous grade levels used units, tools, and formulas to find the area of figures in problem situations.
• Previous grade levels introduced composites of two-dimensional figures.
• Geometry introduces kites, regular polygons, and sectors as shapes that can make up composite figures.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
• III.D.1. Find the perimeter and area of two-dimensional figures.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.
G.11C 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.

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

Including, but not limited to:

• Connections between formulas and models, including nets
• Formulas for lateral and total surface area
• Prisms
• Lateral surface area
• S = Perimeter of the base times the height
• S = Ph
• Total surface area
• S = Perimeter of the base times the height plus 2 times the area of the base
• S = Ph + 2B
• Pyramids
• Lateral surface area
• S = times the perimeter of the base times the slant height
• S = P
• Total surface area
• S = times the perimeter of the base times the slant height plus the area of the base
• S = Pℓ + B
• Cones
• Lateral surface area
• S = π times the radius times the slant height
• S = πrℓ
• Total surface area
• S = π times the radius times the slant height plus the area of the base
• S = πrℓ +πr2
• Cylinders
• Lateral surface area
• S = 2 times π times the radius times the height
• S = 2πrh
• Total surface area
• S = 2 times π times the radius times the height plus 2 times the area of the base
• S = 2πrh + 2πr2
• Spheres
• Total surface area
• S = 4 times π times the radius squared
• S = 4πr2
• Composite figures
• Applications to real-world situations
• Use of appropriate units of measure

Note(s):

• Previous grade levels used units, tools, and formulas to find area of figures in problem situations.
• Previous grade levels introduced the terminology of regular polygons.
• Grade 7 determined the surface area of pyramids and prisms using nets.
• Grade 8 connects previous knowledge of surface area and nets to the surface area formulas.
• Geometry introduces cones, pyramids, spheres, and composite figures to the surface area formulas.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• 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.
G.11D Apply the formulas for the volume of three-dimensional figures, including prisms, pyramids, cones, cylinders, spheres, and composite figures, to solve problems using appropriate units of measure.

Apply

THE FORMULAS FOR THE VOLUME OF THREE-DIMENSIONAL FIGURES, INCLUDING PRISMS, PYRAMIDS, CONES, CYLINDERS, SPHERES, AND COMPOSITE FIGURES TO SOLVE PROBLEMS USING APPROPRIATE UNITS OF MEASURE

Including, but not limited to:

• Connections between formulas and models
• Formulas for volume
• Prisms
• V = Area of the base times the height
• V = Bh
• Pyramids
• V = times the area of the base times the height
• V = Bh
• Cylinders
• V = Area of the base times the height
• V = πr2h
• Cones
• V = times the area of the base times the height
• V = Bh
• Spheres
• V = times π times the radius cubed
• V = πr3
• Composite figures
• Applications to real-world situations
• Use of appropriate units of measure

Note(s): 