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

#### Unit Overview

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.

After this unit, in Geometry 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): III. Geometric Reasoning C1, C3; IV. Measurement Reasoning C1, C2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

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/collegereadiness/crs.pdf

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

Geometric and spatial reasoning are necessary to describe and analyze geometric relationships in mathematics and the real-world.

• Why are geometric and spatial reasoning necessary in the development of an understanding of geometric relationships?
• Why is it important to visualize and use diagrams to effectively communicate/illustrate geometric relationships?

Attributes and properties of two-dimensional geometric shapes are foundational to developing geometric and measurement reasoning.

• Why is it important to compare and contrast attributes and properties of two-dimensional geometric shapes?
• How does analyzing the attributes and properties of two-dimensional geometric shapes develop geometric and measurement reasoning?

Application of attributes and measures of figures can be generalized to describe geometric relationships which can be used to solve problem situations.

• Why are attributes and measures of figures used to generalize geometric relationships?
• How can numeric patterns be used to formulate geometric relationships?
• Why is it important to distinguish measureable attributes?
• How do geometric relationships relate to other geometric relationships?
• Why is it essential to develop generalizations for geometric relationships?
• How are geometric relationships applied to solve problem situations?
• How do different systems of measure relate to one another?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 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.

Geometric Reasoning

• Geometric Attributes/Properties
• Geometric Relationships
• Scale Factors
• Similarity
• Two-Dimensional Figures
• Three-Dimensional Figures

Measurement Reasoning

• Length
• Dimensional Change
• Formulas
• Perimeter/Circumference
• Systems of Measurement
• Volume

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Diagrams can be used to visualize and illustrate geometric relationships and aid in solving problems.

• Why are diagrams necessary for visualizing the geometric relationships found in the problem situation?
• How are diagrams used to organize information from the problem situation?
• How do diagrams aid in calculations when solving problems?

The area of composite two-dimensional figures can be determined by calculating and combining the areas of the shapes that comprise the figure.

• How can a two-dimensional figure be comprised of a combination of shapes?
• What shapes could be used to create a composite two-dimensional figure?
• Why might it be necessary to break down an irregular figure into its component shapes?
• How can the perimeter and area of a composite two-dimensional figure be determined?

If all dimensions of a three-dimensional figure are changed by the same scale factor, the result is a proportional change in surface area and volume; whereas, if only one or two dimensions of a three-dimensional figure are changed by a scale factor, the result is a non-proportional change in surface area and volume.

• How are similar figures generated?
• What geometric relationship exists between linear dimensions, surface area, and volume of a three-dimensional object that has undergone a proportional dimension change?
• What geometric relationship exists between linear dimensions, surface area, and volume of a three-dimensional object that has undergone a non-proportional dimension change?
• What are some of the possible effects on the surface area and volume of a three-dimensional object when a scale factor is applied to just one of the dimensions?  Scale factor on two dimensions?
• How can the resulting effects on the surface area an volume of the scaled two-dimensional object be predicted?

Lateral area and total surface area of three-dimensional figures can be found by using various concrete tools and formulas.

• How can a net be used to find the surface area of a three-dimensional figure?
• How can the perimeter and area of the base and the height be used to find the surface area of a three-dimensional figure?
• How are the formulas for lateral and total surface area of three-dimensional figures connected to the nets of the figures?
• How can the total and lateral surface area of three-dimensional figures and/or three-dimensional composite figures be used to solve problems?
• How can the total and lateral surface area of a three-dimensional figure comprised of a cylinder and two hemispheres be found?
• How can a missing measurement, such as the radius, of a three-dimensional figure comprised of a cylinder and two hemispheres be found?

The volume of a three-dimensional figure can be determined using a formula and applied to solve real-world problems.

• What are the formulas for volume of three-dimensional figures?
• What relationship exists between the area of the base and the height of a cone?
• How can a missing measurement of a cone be found when given the volume and the radius?
• What relationship exists between the height of a cylinder and the base area?
• What relationship exists between a cylinder and a cone that have the same base and height?
• How can the volume of a pyramid be found given the height and the area of the base?
• How can a missing measurement of a pyramid be found when given the volume and the height?
• How can the volume of a prism be found when the length, width and height are given?
• How can a missing measurement of a prism be found when given the volume and two other dimensions, such as the length and width?
• How is the volume of a pyramid related to the volume of a prism with the same base and height?

#### 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.
• Arc length of a circle – a fractional distance of the circumference of a circle defined by the arc
• 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)
• Sector of a circle – a region of the circle bounded by a central angle and its intercepted arc
• 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 Creator 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 (select CCRS from Standard Set dropdown menu)

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

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)
• Blue text: Supporting information / Clarifications from TCMPC (Specificity)
• Blue text in italics: Unit-specific clarification
• Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
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:
• X. Connections
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:
• VIII. Problem Solving and Reasoning
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:
• VIII. Problem Solving and Reasoning
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:
• IX. Communication and Representation
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:
• IX. Communication and Representation
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:
• X. Connections
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:
• IX. Communication and Representation
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
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• C3 – Make connections between geometry and measurement.
• IV. Measurement Reasoning
• C1 – Find the perimeter and area of two-dimensional figures.
• C2 – Determine the surface area and volume of three-dimensional figure.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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
• III. Geometric Reasoning
• C3 – Make connections between geometry and measurement.
• IV. Measurement Reasoning
• C1 – Find the perimeter and area of two-dimensional figures.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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
• III. Geometric Reasoning
• C3 – Make connections between geometry and measurement.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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:

• Formulas for total surface area
• Formulas for lateral surface area
• Connections between formulas and models, including nets
• Three-dimensional figures
• Prisms
• Pyramids
• Cones
• Cylinders
• Spheres
• Use of appropriate units of measure
• Composite figures
• Applications to real-world situations

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
• III. Geometric Reasoning
• C3 – Make connections between geometry and measurement.
• IV. Measurement Reasoning
• C1 – Find the perimeter and area of two-dimensional figures.
• C2 – Determine the surface area and volume of three-dimensional figure.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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:

• Formulas for volume
• Connections between formulas and models
• Three-dimensional figures
• Prisms
• Pyramids
• Cylinders
• Cones
• Spheres
• Use of appropriate units of measure
• Composite figures
• Applications to real-world situations

Note(s):

• Grades 7 and 8 introduced determining and applying formulas to solve problems involving lateral and total surface area and volume of three-dimensional figures.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III. Geometric Reasoning
• C3 – Make connections between geometry and measurement.
• IV. Measurement Reasoning
• C1 – Find the perimeter and area of two-dimensional figures.
• C2 – Determine the surface area and volume of three-dimensional figure.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections