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 Instructional Focus DocumentGeometry
 TITLE : Unit 08: Measurement of Two-Dimensional Figures SUGGESTED DURATION : 15 days

#### Unit Overview

This unit bundles student expectations that address perimeter and area of two-dimensional figures and composite figures including regular polygons. Effects of dimensional changes and arc length and sector area of circles are also addressed. 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 Grade 2, student used concrete models to determine area. In Grade 3, students determined perimeter and area of rectangles and composite figures. In Grade 4, students used formulas to determine perimeter and area of rectangles and squares. In Grade 5, students solve problems related to perimeter. 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.

During this unit, students apply processes for finding area, perimeter, and circumference of two-dimensional figures and investigate dimensional change of two-dimensional figures. Students explore relationships in regular polygons and derive the formula for area of a regular polygon. Students investigate various methods for finding the area of regular polygons in mathematical problems. Students find perimeter, circumference and area of two-dimensional figures and area of regular polygons in problem situations, including proportional and non-proportional dimensional change. Students explore perimeter and area of composite figures, including compositions with regular polygons in problem situations. Students define the arc length of a sector of a circle and the area of a sector of a circle. Students explore the proportional relationships between circumference of circle and arc length and area of circle and area of sector. Students find perimeter and area of composite figures, including compositions with regular polygons in problem situations. Students address changes in scale or measurement units. Students use proportional relationships to find the length of arcs and area of sectors of circles in problem situations.

After this unit, in Geometry Unit 09, students will find the surface area and volume of three-dimensional figures and composite figures, including dimensional change. In subsequent courses in mathematics, these concepts will continue to be applied to problem situations involving two-dimensional figures.

This unit is supporting the development of Texas College Career Readiness Standards (TxCCRS): III. Geometric Reasoning A1, 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, 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.” (2012, 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.

#### 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

Measurement Reasoning

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

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?

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

• How are similar figures generated?
• What geometric relationship exists between linear dimensions, perimeter and area of a two-dimensional object that has undergone a proportional dimension change?
• What geometric relationship exists between linear dimensions, perimeter and area of a two-dimensional object that has undergone a non-proportional dimension change?
• What are some of the possible effects on the perimeter and area of a two-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 perimeter and area of the scaled two-dimensional object be predicted?

A geometric relationship exists between the apothem, perimeter, and area of a regular polygon.

• How can the area of a regular polygon be found if the side length is given?
• How can the area of a regular polygon be found if the apothem is known?
• How can the area of a regular polygon be found if the perimeter is known?
• How can the perimeter of a regular polygon be found if the apothem is known?
• How can the area of a regular polygon be found using non-overlapping right isosceles triangles?
• How can the radius of a regular polygon be used to find its area?
 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

• Composition/Decomposition of Figures
• Composite Figures
• Geometric Attributes/Properties
• Geometric Relationships
• Scale Factors
• Similarity
• Two-Dimensional Figures

Measurement Reasoning

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

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?

A geometric relationship exists between the measure of an arc length of a circle and the circumference of the circle.

• What relationship exists between the arc length of a circle and its circumference?
• How can a specific arc length of a circle be used to find the circumference of the circle?
• How can the circumference of the circle be used to find a specific arc length of the circle?

A geometric relationship exists between the measure of the area of a sector of a circle and its area.

• What relationship is found between the measure of the area of a sector of a circle and its area?
• How can the area of a sector of a circle be used to find the area of the circle?
• How can the area of a circle be used to find the measure of the area of a sector?

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that the length of the arc of a sector is the same as the degree measure of the arc instead of it being a measure in dimensions of length.
• Some students may think that the same scale factor is used for area of proportional dimensional change as is used for perimeter (circumference) instead of using the scale factor squared.
• Some students may think the apothem is the same as the radius from the center of a regular polygon to the vertex instead of the perpendicular distance from the center to the side of a regular polygon.

Underdeveloped Concepts:

• Some students may not be able to distinguish the components of composite figures.
• Some students may not know which formulas to use when determining the perimeter, circumference, and area of the components of a composite 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
• 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 Bisect Circumference Composite figure Linear dimension Perimeter Perpendicular Radius Scale factor
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, OR AREA INCLUDING PROPORTIONAL AND NON-PROPORTIONAL DIMENSIONAL CHANGE

Including, but not limited to:

• Verbal and written description
• Dimensional change
• Perimeter and circumference
• Area
• Proportional change
• Two-dimensional proportional change – two 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
• 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.12 Circles. The student uses the process skills to understand geometric relationships and apply theorems and equations about circles. The student is expected to:
G.12B Apply the proportional relationship between the measure of an arc length of a circle and the circumference of the circle to solve problems.

Apply

THE PROPORTIONAL RELATIONSHIP BETWEEN THE MEASURE OF AN ARC LENGTH OF A CIRCLE AND THE CIRCUMFERENCE OF THE CIRCLE TO SOLVE PROBLEMS

Including, but not limited to:

• Arc length of a circle – a fractional distance of the circumference of a circle defined by the arc
• Connecting the proportional relationship
• Applications to real-world problem situations
• Use of appropriate units of measure
• Use of various tools
• Protractor and straightedge
• Dynamic geometric software
• Patty paper

Note(s):

• Previous grade levels explored characteristics of circles and proportional relationships.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III. Geometric Reasoning
• A1 – Identify and represent the features of plane and space figures.
• C3 – Make connections between geometry and measurement.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
G.12C Apply the proportional relationship between the measure of the area of a sector of a circle and the area of the circle to solve problems.

Apply

THE PROPORTIONAL RELATIONSHIP BETWEEN THE MEASURE OF THE AREA OF A SECTOR OF A CIRCLE AND THE AREA OF THE CIRCLE TO SOLVE PROBLEMS

Including, but not limited to:

• Sector of a circle – a region of the circle bounded by a central angle and its intercepted arc
• Connecting the proportional relationship
• Applications to real-world problem situations
• Use of appropriate units of measure
• Use of various tools
• Protractor and straightedge
• Dynamic geometric software
• Patty paper

Note(s):