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 TITLE : Unit 10: Two- and Three-Dimensional Figures SUGGESTED DURATION : 10 days

#### Unit Overview

Introduction
This unit bundles student expectations that address sorting and classifying two- and three-dimensional figures, identifying examples and non-examples of quadrilaterals, and decomposing two congruent two-dimensional figures into parts with equal areas and expressing the area of each part as a unit fraction of the whole. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, tools and techniques, 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, students analyzed attributes of two-dimensional shapes and three-dimensional solids in order to develop generalizations about their properties. Using formal geometric language, students classified and sorted polygons with 12 or fewer sides by identifying the number of sides and number of vertices.

During this Unit
Students continue to develop their understanding of geometric figures by sorting and classifying two- and three-dimensional figures that may vary in size, shape, and orientation based on attributes using formal geometric language. Students focus their exploration of two-dimensional figures as they explore subcategories of quadrilaterals, including rhombuses, parallelograms, trapezoids, rectangles, and squares. Students use formal language to describe the attributes and properties of each subcategory of quadrilaterals as well as recognizing and drawing quadrilaterals that do not fit into any of the subcategories. Students also apply previous understanding of area and fractions to their exploration of two-dimensional figures. Students decompose two congruent two-dimensional figures into parts with equal areas and express the area of each part as a unit fraction of the whole. Students discover that equal shares of the same whole do not always have the same shape but are equal if the areas of each part are equal. A solid understanding of the properties and attributes of geometric figures is critical to students’ future success in the study of geometry.

After this Unit
In Unit 12, students will continue to explore the concept of area in rectangles and composite figures. In Grade 4, students will identify and explain points, lines, line segments, rays, angles, and perpendicular and parallel lines in geometric figures.

In Grade 3, classifying and sorting two- and three-dimensional solids based on attributes and using attributes to identify examples and non-examples of quadrilaterals are identified as STAAR Readiness Standard 3.6A and STAAR Supporting Standard 3.6B. These standards are included in the Grade 3 Texas Response to Curriculum Focal Points (TxRCFP): Describing characteristics of two-dimensional and three-dimensional geometric figures, including measurable attributes. Decomposing congruent two-dimensional figures into parts with equal areas and expressing the area of each part as a unit fraction is identified as STAAR Supporting Standard 3.6E and included in the Grade 3 Texas Response to Curriculum Focal Points (TxRCFP): Understanding fractions as numbers and representing equivalent fractions. All three standards are included in Grade 3 STAAR Reporting Category: Geometry and Measurement. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): III.A. Geometric Reasoning – Figures and their properties, IV.C. Measurement Reasoning – Measurement involving geometry and algebra, IX. Communication and Representation, and X. Connections.

Research
According to the National Research Council (2001), “There is much pedagogical value in returning geometry to its roots in spatial measure” (p. 281). The National Council of Teachers of Mathematics (2009) explains the importance of Grade 3 focusing on analyzing, describing, comparing, and classifying properties of shapes to lay a foundation that will enable students to extend this understanding to the study of perimeter and area (p. 65-66).

National Council of Teachers of Mathematics. (2009). Focus in grade 3: Teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Research Council. (2001). Adding it up: Helping children learn mathematics. Kilpatrick, J., Swafford, J., and Findell, B. (Eds.) Mathematics Learning Study Committee, Center for Education Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013

 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)
• Illustrating and analyzing geometric relationships in models and diagrams aid in representing and describing the attributes of geometric figures in order to generalize geometric relationships and solve problem situations.
• What attributes and properties exist in …
• two-dimensional figures?
• three-dimensional figures?
• How are attributes and properties used to classify …
• geometric figures?
• What relationships exist between …
• two-dimensional figures and three-dimensional figures?
• the sides and vertices in a quadrilateral?
• How are …
• two-dimensional figures and three-dimensional figures
• the attributes of circles and other two-dimensional figures
• figures with curved surfaces and figures with only flat surfaces
… alike and different?
• How can a collection of …
• two-dimensional
• three-dimensional
… figures be sorted and classified in more than one way?
• Geometry
• Geometric Attributes and Properties
• Classification
• Geometric Representations
• Two-dimensional figures
• Three-dimensional figures
• Associated Mathematical Processes
• 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.

 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)
• Illustrating and analyzing geometric relationships in models and diagrams aid in representing and describing the attributes of geometric figures in order to generalize geometric relationships and solve problem situations.
• How can a two-dimensional shape be decomposed in more than one way?
• What strategies can be used to …
• decompose a two-dimensional figure in to parts with equal areas?
• determine if the resulting parts of two identical decomposed two-dimensional shapes are equal in area?
• How can each part of a two-dimensional figure, decomposed into equivalent parts, be described as fractions?
• Why can equal shares of two identical wholes …
• be named using the same unit fraction?
• appear different in shape?
• Geometry
• Decomposition of Figures
• Fractions
• Congruence
• Geometric Representations
• Two-dimensional figures
• 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 a quadrilateral must fall into one of the subcategories of trapezoids, rectangles, rhombuses, or squares rather that recognizing any four-sided figure as a quadrilateral.
• Some students may think figures with equal area must look the same rather than recognizing various combinations of length and width that equal the same area.

Underdeveloped Concepts:

• Although some students may be able to identify regular figures, they may not be able to identify irregular figures due to limited exposure to a variety of images and lack of understanding regarding the attributes of a given figure (e.g., a student may be able to identify a regular hexagon from exposure to pattern blocks, but fail to recognize any six-sided figure as a hexagon).
• Some students may have difficulty recognizing geometric figures if the figures have been transformed by orientation or size.
• Some students may list attributes of a figure separately but not see the interrelationships between figures (e.g., a square and rectangle as the only examples of quadrilaterals).
• Some students may categorize two-dimensional figures incorrectly based on only a few attributes of the figure rather than considering all of the figure’s defining attributes (e.g., a student may say, “If the shape has four sides, it is a square,” although this may not be true because a four-sided figure could also be a rectangle or rhombus).
• Some students may call a three-dimensional figure by the name of one of its two-dimensional faces (e.g., a student may refer to a cube as a square, etc.).

#### Unit Vocabulary

• Area – the measurement attribute that describes the number of unit squares (or square units) a figure or region covers
• Attributes of three-dimensional figures – characteristics that define a geometric figure (e.g., edges, vertices, faces [bases], etc.)
• Attributes of two-dimensional figures – characteristics that define a geometric figure (e.g., sides, vertices, etc. )
• Base of a cone – the flat surface shaped like a circle
• Base of a pyramid – a face opposite the common vertex (apex) where the triangular faces meet
• Bases of a cylinder – the two congruent, opposite flat surfaces shaped like circles
• Bases of a prism – the two congruent, opposite faces that are connected by rectangular faces
• Classify – applying an attribute to categorize a sorted group
• Congruent – of equal measure
• Congruent figures – figures that are the same size and same shape
• Decompose figures – to break a geometric figure into two or more smaller geometric figures
• Edge – where the sides of two faces meet on a three-dimensional figure
• Face of a prism – a polygon that forms a surface of a prism
• Irregular figure – a polygon with sides and/or corners (angles) that are not all congruent
• Polygon – a closed figure with at least 3 sides, where all sides are straight (no curves)
• Properties of three-dimensional figures – relationship of attributes within a geometric figure (e.g., a rectangular prism has 6 faces and each pair of opposite faces [bases] are congruent, etc.) and between a group of geometric figures (e.g., a cube and a rectangular prism both have 6 faces with opposite faces [bases] congruent; however, a cube has only square faces but a rectangular prism can have square or rectangular faces; etc.)
• Properties of two-dimensional figures – relationship of attributes within a geometric figure (e.g., a square has 4 congruent sides and 4 square corners, etc.) and between a group of geometric figures (e.g., a square and a rectangle both have 4 sides and 4 square corners; however, a square has 4 congruent sides but a rectangle has only opposite sides congruent; etc.)
• Quadrilateral – a polygon with 4 sides and 4 vertices
• Regular figure – a polygon with all sides and corners (angles) congruent
• Side – a straight outer boundary between two vertices (line segment) of a two-dimensional figure
• Sort – grouping objects or figures by a shared characteristic or attribute
• Three-dimensional figure – a figure that has measurements including length, width (depth), and height
• Two-dimensional figure – a figure with two basic units of measure, usually length and width
• Unit fraction – a fraction in the form representing the quantity formed by one part of a whole that has been partitioned into b equal parts where b is a non-zero whole number
• Vertex (vertices) in a three-dimensional figure – the point (corner) where three or more edges of a three-dimensional figure meet
• Vertex (vertices) in a two-dimensional figure – the point (corner) where two sides of a two-dimensional figure meet

Related Vocabulary:

 7-gon (heptagon) 9-gon (nonagon) 11-gon (hendecagon) 12-gon (dodecagon) Circle Closed figure Cone Cube Cylinder Decagon Denominator Equilateral triangle Hexagon Isosceles trapezoid Isosceles triangle Length Numerator Octagon Open figure Parallelogram Partition Pentagon Rectangle Rectangular prism Rectangular pyramid Rhombus Scalene triangle Sphere Square Square corner Square unit Surface of a three-dimensional figure Trapezoid Triangle Triangular prism Trianglular pyramid Width
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

Texas Education Agency – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Grade 3 Mathematics TEKS

TEKS# SE# Unit Level Taught Directly TEKS Unit Level Specificity

Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Student Expectations (TEKS) are labeled Readiness as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Supporting as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Process standards as identified by TEA of the assessed curriculum.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

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.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
3.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
3.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
Process Standard

Apply

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

Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Understanding and applying place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Solving problems with multiplication and division within 100
• Understanding fractions as numbers and representing equivalent fractions
• Describing characteristics of two-dimensional and three-dimensional geometric figures, including measurable attributes
• TxCCRS:
• X. Connections
3.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.
Process Standard

Use

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

Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Understanding and applying place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Solving problems with multiplication and division within 100
• Understanding fractions as numbers and representing equivalent fractions
• Describing characteristics of two-dimensional and three-dimensional geometric figures, including measurable attributes
• TxCCRS:
• VIII. Problem Solving and Reasoning
3.1C

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

Process Standard

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE 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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Understanding and applying place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Solving problems with multiplication and division within 100
• Understanding fractions as numbers and representing equivalent fractions
• Describing characteristics of two-dimensional and three-dimensional geometric figures, including measurable attributes
• TxCCRS:
• VIII. Problem Solving and Reasoning
3.1D

Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Process Standard

Communicate

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

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Understanding and applying place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Solving problems with multiplication and division within 100
• Understanding fractions as numbers and representing equivalent fractions
• Describing characteristics of two-dimensional and three-dimensional geometric figures, including measurable attributes
• TxCCRS:
• IX. Communication and Representation
3.1E Create and use representations to organize, record, and communicate mathematical ideas.
Process Standard

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Understanding and applying place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Solving problems with multiplication and division within 100
• Understanding fractions as numbers and representing equivalent fractions
• Describing characteristics of two-dimensional and three-dimensional geometric figures, including measurable attributes
• TxCCRS:
• IX. Communication and Representation
3.1F Analyze mathematical relationships to connect and communicate mathematical ideas.
Process Standard

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Understanding and applying place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Solving problems with multiplication and division within 100
• Understanding fractions as numbers and representing equivalent fractions
• Describing characteristics of two-dimensional and three-dimensional geometric figures, including measurable attributes
• TxCCRS:
• X. Connections
3.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Process Standard

Display, Explain, Justify

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

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Understanding and applying place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Solving problems with multiplication and division within 100
• Understanding fractions as numbers and representing equivalent fractions
• Describing characteristics of two-dimensional and three-dimensional geometric figures, including measurable attributes
• TxCCRS:
• IX. Communication and Representation
3.6 Geometry and measurement. The student applies mathematical process standards to analyze attributes of two-dimensional geometric figures to develop generalizations about their properties. The student is expected to:
3.6A Classify and sort two- and three-dimensional figures, including cones, cylinders, spheres, triangular and rectangular prisms, and cubes, based on attributes using formal geometric language.

Classify, Sort

TWO-DIMENSIONAL FIGURES BASED ON ATTRIBUTES USING FORMAL GEOMETRIC LANGUAGE

Including, but not limited to:

• Two-dimensional figure – a figure with two basic units of measure, usually length and width
• Sort – grouping objects or figures by a shared characteristic or attribute
• Classify – applying an attribute to categorize a sorted group
• Attributes of two-dimensional figures – characteristics that define a geometric figure (e.g., sides, vertices, etc.)
• Properties of two-dimensional figures – relationship of attributes within a geometric figure (e.g., a square has 4 congruent sides and 4 square corners, etc.) and between a group of geometric figures (e.g., a square and a rectangle both have 4 sides and 4 square corners; however, a square has 4 congruent sides but a rectangle has only opposite sides congruent; etc.)
• Regular figure – a polygon with all sides and corners (angles) congruent
• Irregular figure – a polygon with sides and/or corners (angles) that are not all congruent
• Attributes of two-dimensional figures
• Side – a straight outer boundary between two vertices (line segment) of a two-dimensional figure
• Number of sides
• Length of sides
• Vertex (vertices) in a two-dimensional figure – the point (corner) where two sides of a two-dimensional figure meet
• Number of vertices
• Types of vertices
• Square corners (right angles)
• Square corners (right angles) can be determined using the corner of a known square or rectangle (e.g., sticky note, sheet of paper, etc.).
• May have a box in corner to represent square corner (right angle)
• Not square corners (not right angles)
• Opposite corners (angles)
• Congruent – of equal measure
• Types of two-dimensional figures
• Circle
• A figure formed by a closed curve with all points equal distance from the center
• No straight sides
• No vertices
• Polygon – a closed figure with at least 3 sides, where all sides are straight (no curves)
• Types of polygons
• Triangle
• 3 sides
• 3 vertices
• Types of triangles
• Scalene triangle
• 3 sides
• 3 vertices
• No congruent sides
• No congruent corners (angles)
• Isosceles triangle
• 3 sides
• 3 vertices
• At least 2 congruent sides
• At least 2 congruent corners (angles)
• Equilateral triangle (a special type of isosceles triangle)
• 3 sides
• 3 vertices
• All sides congruent
• All corners (angles) congruent
• 4 sides
• 4 vertices
• Trapezoid
• 4 sides
• 4 vertices
• Exactly one pair of sides equal distance apart
• Types of trapezoids
• Isosceles trapezoid
• 4 sides
• 4 vertices
• Exactly one pair of sides equal distance apart
• At least 2 congruent sides, where 2 of the sides are opposite each other
• Parallelogram
• 4 sides
• 4 vertices
• Opposite sides congruent
• Opposite sides equal distance apart
• Opposite corners (angles) congruent
• Types of parallelograms
• Rectangle
• 4 sides
• 4 vertices
• Opposite sides congruent
• Opposite sides equal distance apart
• 4 square corners (right angles)
• Rhombus
• 4 sides
• 4 vertices
• All sides congruent
• Opposite sides equal distance apart
• Opposite corners (angles) congruent
• Square (a special type of rectangle and a special type of rhombus)
• 4 sides
• 4 vertices
• All sides congruent
• Opposite sides congruent
• Opposite sides equal distance apart
• 4 square corners (right angles)
• Opposite corners (angles) congruent
• Pentagon
• 5 sides
• 5 vertices
• Hexagon
• 6 sides
• 6 vertices
• 7-gon (heptagon)
• 7 sides
• 7 vertices
• Octagon
• 8 sides
• 8 vertices
• 9-gon (nonagon)
• 9 sides
• 9 vertices
• Decagon
• 10 sides
• 10 vertices
• 11-gon (hendecagon)
• 11 sides
• 11 vertices
• 12-gon (dodecagon)
• 12 sides
• 12 vertices
• Concrete models (e.g., wood or plastic figures, etc.) and pictorial models (e.g., drawings, images, etc.)
• Collection of two-dimensional figures
• Sort and justify
• Rule used for sorting expressed
• Attributes and properties of geometric figures expressed
• Existence (have) and absence (do not have) of attributes and properties expressed (e.g., figures that have “a common attribute” and figures that do not have “a common attribute”)

Classify, Sort

THREE-DIMENSIONAL FIGURES, INCLUDING CONES, CYLINDERS, SPHERES, TRIANGULAR AND RECTANGULAR PRISMS, AND CUBES, BASED ON ATTRIBUTES USING FORMAL GEOMETRIC LANGUAGE

Including, but not limited to:

• Three-dimensional figure – a figure that has measurements including length, width (depth), and height
• Sort – grouping objects or figures by a shared characteristic or attribute
• Classify – applying an attribute to categorize a sorted group
• Attributes of three-dimensional figures – characteristics that define a geometric figure (e.g., edges, vertices, faces [bases], etc.)
• Properties of three-dimensional figures – relationship of attributes within a geometric figure (e.g., a rectangular prism has 6 faces and each pair of opposite faces [bases] are congruent, etc.) and between a group of geometric figures (e.g., a cube and a rectangular prism both have 6 faces with opposite faces [bases] congruent; however, a cube has only square faces but a rectangular prism can have square or rectangular faces; etc.)
• Attributes of three-dimensional figures
• Surfaces
• Curved surface
• Flat surface
• Face of a prism – a polygon that forms a surface of a prism
• Number of faces
• Shape of faces
• Bases of a prism – the two congruent, opposite faces that are connected by rectangular faces
• Bases of a cylinder – the two congruent, opposite flat surfaces shaped like circles
• Base of a cone – the flat surface shaped like a circle
• Base of a pyramid – a face opposite the common vertex (apex) where the triangular faces meet
• Edge – where the sides of two faces meet on a three-dimensional figure
• Number of edges
• Vertex (vertices) in a three-dimensional figure – the point (corner) where three or more edges of a three-dimensional figure meet
• Number of vertices
• Congruent – of equal measure
• Three-dimensional figures
• Curved surface three-dimensional figures
• Cone
• 1 flat surface shaped like a circle (base)
• 1 curved surface
• 1 vertex (apex)
• Cylinder
• 2 congruent, opposite, flat surfaces shaped like circles (bases)
• 1 curved surface
• Sphere
• 1 curved surface with all points on the surface equal distance from the center
• Prisms
• Triangular prism
• 5 faces (2 triangular faces [bases], 3 rectangular faces)
• 9 edges
• 6 vertices
• Rectangular prism
• 6 faces (2 rectangular faces [bases], 4 rectangular faces)
• 12 edges
• 8 vertices
• Cube (special rectangular prism or square prism)
• 6 faces (2 square faces [bases], 4 square faces)
• 12 edges
• 8 vertices
• Pyramids
• Triangular pyramid
• 4 faces (1 triangular face [base], 3 triangular faces)
• 6 edges
• 4 vertices (1 apex, 3 vertices)
• Rectangular pyramid (including square pyramid)
• 5 faces (1 rectangular/square face [base], 4 triangular faces)
• 8 edges
• 5 vertices (1 apex, 4 vertices)
• Concrete models (e.g., wood or plastic figures, etc.), real-world objects (e.g., a cereal box, can of beans, etc.), and pictorial models (e.g., drawings, images, etc.)
• Collection of three-dimensional figures
• Sort and justify
• Rule used for sorting expressed
• Attributes and properties of geometric figures expressed
• Existence (have) and absence (do not have) of attributes and properties expressed (e.g., figures that have “a common attribute” and figures that do not have “a common attribute”)

Note(s):

• Kindergarten classified and sorted a variety of regular and irregular two- and three-dimensional figures regardless of orientation or size.
• Grade 1 classified and sorted three-dimensional solids, including spheres, cones, cylinders, rectangular prisms (including cubes as special rectangular prisms), triangular prisms, rectangular pyramids (including square pyramids), and triangular pyramids, based on attributes using formal geometric language.
• Grade 2 classified and sorted polygons with 12 or fewer sides according to attributes, including identifying the number of sides and number of vertices.
• Grade 4 will classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines or the presence or absence of angles of a specified size.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Describing characteristics of two-dimensional and three-dimensional geometric figures, including measurable attributes
• TxCCRS:
• III.A. Geometric Reasoning – Figures and their properties
• IX. Communication and Representation
3.6B Use attributes to recognize rhombuses, parallelograms, trapezoids, rectangles, and squares as examples of quadrilaterals and draw examples of quadrilaterals that do not belong to any of these subcategories.
Supporting Standard

Use

ATTRIBUTES TO RECOGNIZE RHOMBUSES, PARALLELOGRAMS, TRAPEZOIDS, RECTANGLES, AND SQUARES AS EXAMPLES OF QUADRILATERALS

Including, but not limited to:

• Attributes of two-dimensional figures – characteristics that define a geometric figure (e.g., sides, vertices, etc.)
• Properties of two-dimensional figures – relationship of attributes within a geometric figure (e.g., a square has 4 congruent sides and 4 square corners, etc.) and between a group of geometric figures (e.g., a square and a rectangle both have 4 sides and 4 square corners; however, a square has 4 congruent sides but a rectangle has only opposite sides congruent; etc.)
• Attributes of two-dimensional figures
• Side – a straight outer boundary between two vertices (line segment) of a two-dimensional figure
• Number of sides
• Length of sides
• Vertex (vertices) in a two-dimensional figure – the point (corner) where two sides of a two-dimensional figure meet
• Number of vertices
• Types of vertices
• Square corners (right angles)
• Square corners (right angles) can be determined using the corner of a known square or rectangle (e.g., sticky note, sheet of paper, etc.).
• May have a box in corner to represent square corner (right angle)
• Not square corners (not right angles)
• Opposite corners (angles)
• Congruent – of equal measure
• Polygon – a closed figure with at least 3 sides, where all sides are straight (no curves)
• Quadrilateral – a polygon with 4 sides and 4 vertices
• Trapezoid
• 4 sides
• 4 vertices
• Exactly one pair of sides equal distance apart
• Types of trapezoids
• Isosceles trapezoid
• 4 sides
• 4 vertices
• Exactly one pair of sides equal distance apart
• At least 2 congruent sides, where 2 of the sides are opposite each other
• Parallelogram
• 4 sides
• 4 vertices
• Opposite sides congruent
• Opposite sides equal distance apart
• Opposite corners (angles) congruent
• Types of parallelograms
• Rectangle
• 4 sides
• 4 vertices
• Opposite sides congruent
• Opposite sides equal distance apart
• 4 square corners (right angles)
• Rhombus
• 4 sides
• 4 vertices
• All sides congruent
• Opposite sides equal distance apart
• Opposite corners (angles) congruent
• Square (a special type of rectangle and a special type of rhombus)
• 4 sides
• 4 vertices
• All sides congruent
• Opposite sides congruent
• Opposite sides equal distance apart
• 4 square corners (right angles)
• Opposite corners (angles) congruent

Draw

EXAMPLES OF QUADRILATERALS THAT DO NOT BELONG TO ANY OF THE SUBCATEGORIES OF QUADRILATERALS

Including, but not limited to:

• Quadrilateral – a polygon with 4 sides and 4 vertices
• Attributes and properties of quadrilaterals that do not belong to any of the subcategories of quadrilaterals
• 4 sides
• 4 vertices
• All or opposite sides not congruent
• All or opposite sides not equal distance apart
• All or opposite corners (angles) not congruent

Note(s):

• Grade 2 created two-dimensional shapes based on given attributes, including number of sides and vertices.
• Grade 2 classified and sorted polygons with 12 or fewer sides according to attributes, including identifying the number of sides and number of vertices.
• Grade 4 will identify points, lines, line segments, rays, angles, and perpendicular and parallel lines.
• Grade 4 will identify and draw one or more lines of symmetry, if they exist, for a two-dimensional figure.
• Grade 4 will apply knowledge of right angles to identify acute, right, and obtuse triangles.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Describing characteristics of two-dimensional and three-dimensional geometric figures, including measurable attributes
• TxCCRS:
• III.A. Geometric Reasoning – Figures and their properties
• IX. Communication and Representation
3.6E Decompose two congruent two-dimensional figures into parts with equal areas and express the area of each part as a unit fraction of the whole and recognize that equal shares of identical wholes need not have the same shape.
Supporting Standard

Decompose

TWO CONGRUENT TWO-DIMENSIONAL FIGURES INTO PARTS WITH EQUAL AREAS

Including, but not limited to:

• Two-dimensional figure – a figure with two basic units of measure, usually length and width
• Congruent figures – figures that are the same size and same shape
• Decompose figures into equal parts.
• Decompose figures – to break a geometric figure into two or more smaller geometric figures
• Equal sized parts of congruent wholes have equal areas.
• Area – the measurement attribute that describes the number of unit squares (or square units) a figure or region covers

Express

THE AREA OF EACH PART OF A TWO-DIMENSIONAL FIGURE DECOMPOSED INTO EQUAL PARTS AS A UNIT FRACTION OF THE WHOLE

Including, but not limited to:

• Area – the measurement attribute that describes the number of unit squares (or square units) a figure or region covers
• Two-dimensional figure – a figure with two basic units of measure, usually length and width
• Express equal sized parts as unit fractions of the whole.
• Unit fraction – a fraction in the form representing the quantity formed by one part of a whole that has been partitioned into b equal parts where bis a non-zero whole number
• Numerator of 1 written above the fraction bar represents 1 equal part being specified or considered.
• Denominator (b) written below the fraction bar tells the total number of equal parts in the whole or set.
• Whole number denominators of 2, 3, 4, 6, and 8

Recognize

THAT EQUAL SHARES OF IDENTICAL WHOLES NEED NOT HAVE THE SAME SHAPE

Including, but not limited to:

• Equal sized parts of congruent wholes have equal area.
• Area – the measurement attribute that describes the number of unit squares (or square units) a figure or region covers
• Congruent figures – figures that are the same size and same shape
• Equal sized parts of congruent wholes need not have the same shape.

Note(s):