2.1 
Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:


2.1A 
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.
 TxRCFP:
 Developing proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:

2.1B 
Use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution.

Use
A PROBLEMSOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEMSOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:
 Problemsolving model
 Analyze given information
 Formulate a plan or strategy
 Determine a solution
 Justify the solution
 Evaluate the problemsolving process and the reasonableness of the solution
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:
 VIII. Problem Solving and Reasoning

2.1C 
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, 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:
 Developing proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:
 VIII. Problem Solving and Reasoning

2.1D 
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.
 TxRCFP:
 Developing proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:
 IX. Communication and Representation

2.1E 
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.
 TxRCFP:
 Developing proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:
 IX. Communication and Representation

2.1F 
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 nonexamples, patterns, etc.
 Current knowledge to new learning
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:

2.1G 
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.
 TxRCFP:
 Developing proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:
 IX. Communication and Representation

2.8 
Geometry and measurement. The student applies mathematical process standards to analyze attributes of twodimensional shapes and threedimensional solids to develop generalizations about their properties. The student is expected to:


2.8A 
Create twodimensional shapes based on given attributes, including number of sides and vertices.

Create
TWODIMENSIONAL SHAPES BASED ON GIVEN ATTRIBUTES, INCLUDING NUMBER OF SIDES AND VERTICES
Including, but not limited to:
 Variety of materials and drawings
 Computer programs
 Art materials
 Twodimensional figure – a figure with two basic units of measure, usually length and width
 Polygon – a closed figure with at least 3 sides, where all sides are straight (no curves)
 Spatial visualization – creation and manipulation of mental representations of shapes
 Attributes of twodimensional figures – characteristics that define a geometric figure (e.g., sides, vertices, etc.)
 Properties of twodimensional figures – relationship of attributes within a geometric figure (e.g., a square has 4 sides equal in length 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 sides equal in length but a rectangle has only opposite sides equal in length; etc.)
 Attributes of twodimensional figures
 Side – a straight outer boundary between two vertices (line segment) of a twodimensional figure
 Number of sides
 Length of sides
 Vertex (vertices) in a twodimensional figure – the point (corner) where two sides of a twodimensional figure meet
 Types of vertices
 Square corners
 Square corners 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
 Not square corners
 Opposite corners
 Attributes that do not identify twodimensional figures
 Orientation
 Size
 Color
 Texture
 Regular figure – a polygon with all sides and corners that appear to be the same or equal
 Irregular figure – a polygon with sides and/or corners that appear to be different or unequal
 Create regular and irregular twodimensional figures based on attributes and properties.
 Circle
 A figure formed by a closed curve with all points equal distance from the center
 No straight sides
 No vertices
 Triangle
 Rectangle
 4 sides
 4 vertices
 Opposite sides equal in length
 4 square corners
 Rhombus
 4 sides
 4 vertices
 All sides equal in length
 Opposite corners equal
 Square (a special type of rectangle and a special type of rhombus)
 4 sides
 4 vertices
 All sides equal in length
 Opposite sides equal in length
 4 square corners
 Opposite corners equal
 Pentagon
 Hexagon
 7gon (heptagon)
 Octagon
 9gon (nonagon)
 Decagon
 11gon (hendecagon)
 12gon (dodecagon)
Note(s):
 Grade Level(s):
 Grade 1 created twodimensional figures, including circles, triangles, rectangles, and squares, as special rectangles, rhombuses, and hexagons.
 Grade 1 identified twodimensional shapes, including circles, triangles, rectangles, and squares, as special rectangles, rhombuses, and hexagons and describe their attributes using formal geometric language.
 Grade 3 will 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.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:
 III.A. Geometric Reasoning – Figures and their properties
 IX. Communication and Representation

2.8B 
Classify and sort threedimensional solids, including spheres, cones, cylinders, rectangular prisms (including cubes as special rectangular prisms), and triangular prisms, based on attributes using formal geometric language.

Classify, Sort
THREEDIMENSIONAL SOLIDS, INCLUDING SPHERES, CONES, CYLINDERS, RECTANGULAR PRISMS (INCLUDING CUBES AS SPECIAL RECTANGULAR PRISMS), AND TRIANGULAR PRISMS, BASED ON ATTRIBUTES USING FORMAL GEOMETRIC LANGUAGE
Including, but not limited to:
 Threedimensional 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 threedimensional figures – characteristics that define a geometric figure (e.g., faces, curved surfaces, edges, vertices, etc.)
 Properties of threedimensional figures – relationship of attributes within a geometric figure (e.g., a rectangular prism has 6 faces and each pair of opposite faces are the same size and shape, etc.) and between a group of geometric figures (e.g., a cube and a rectangular prism both have 6 faces with opposite faces equal in size and shape; however, a cube has only square faces but a rectangular prism can have square or rectangular faces; etc.)
 Attributes of threedimensional 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
 Edge – where the sides of two faces meet on a threedimensional figure
 Vertex (vertices) in a threedimensional figure – the point (corner) where three or more edges of a threedimensional figure meet
 Curved surface threedimensional figures
 Cone
 1 flat surface shaped like a circle
 1 curved surface
 1 vertex
 Cylinder
 2 equal, opposite, flat surfaces shaped like circles
 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, 3 rectangular faces)
 9 edges
 6 vertices
 Rectangular prism
 6 rectangular faces
 12 edges
 8 vertices
 Cube (special rectangular prism or square prism)
 6 square faces
 12 edges
 8 vertices
 Pyramids
 Triangular pyramid
 4 triangular faces
 6 edges
 4 vertices
 Rectangular pyramid (including square pyramid)
 5 faces (1 rectangular/square face, 4 triangular faces)
 8 edges
 5 vertices
 Concrete models (e.g., wood or plastic figures, etc.), realworld objects (e.g., a cereal box, can of beans, etc.), and pictorial models (e.g., drawings, images, etc.)
 Collection of threedimensional 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):
 Grade Level(s):
 Grade 1 classified and sorted regular and irregular twodimensional shapes based on attributes using informal geometric language.
 Grade 3 will classify and sort two and threedimensional figures, including cones, cylinders, spheres, triangular and rectangular prisms, and cubes, based on attributes using formal geometric language.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:
 III.A. Geometric Reasoning – Figures and their properties
 IX. Communication and Representation

2.8C 
Classify and sort polygons with 12 or fewer sides according to attributes, including identifying the number of sides and number of vertices.

Classify, Sort
POLYGONS WITH 12 OR FEWER SIDES ACCORDING TO ATTRIBUTES, INCLUDING IDENTIFYING THE NUMBER OF SIDES AND NUMBER OF VERTICES
Including, but not limited to:
 Twodimensional 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 twodimensional figures – characteristics that define a geometric figure (e.g., sides, vertices, etc.)
 Properties of twodimensional figures – relationship of attributes within a geometric figure (e.g., a square has 4 sides equal in length 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 sides equal in length but a rectangle has only opposite sides equal in length; etc.)
 Regular figure – a polygon with all sides and corners that appear to be the same or equal
 Irregular figure – a polygon with sides and/or corners that appear to be different or unequal
 Attributes of twodimensional figures
 Side – a straight outer boundary between two vertices (line segment) of a twodimensional figure
 Number of sides
 Length of sides
 Vertex (vertices) in a twodimensional figure – the point (corner) where two sides of a twodimensional figure meet
 Types of vertices
 Square corners
 Square corners 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
 Not square corners
 Opposite corners
 Polygon – a closed figure with at least 3 sides, where all sides are straight (no curves)
 Types of polygons
 Triangle
 Rectangle
 4 sides
 4 vertices
 Opposite sides equal in length
 4 square corners
 Rhombus
 4 sides
 4 vertices
 All sides equal in length
 Opposite corners equal
 Square (a special type of rectangle and a special type of rhombus)
 4 sides
 4 vertices
 All sides equal in length
 Opposite sides equal in length
 4 square corners
 Opposite corners equal
 Pentagon
 Hexagon
 7gon (heptagon)
 Octagon
 9gon (nonagon)
 Decagon
 11gon (hendecagon)
 12gon (dodecagon)
 Concrete models (e.g., wood or plastic figures, etc.) and pictorial models (e.g., drawings, images, etc.)
 Collection of twodimensional 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):
 Grade Level(s):
 Grade 1 classified and sorted regular and irregular twodimensional shapes based on attributes using informal geometric language.
 Grade 3 will classify and sort two and threedimensional figures, including cones, cylinders, spheres, triangular and rectangular prisms, and cubes, based on attributes using formal geometric language.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:
 III.A. Geometric Reasoning – Figures and their properties
 IX. Communication and Representation

2.8D 
Compose twodimensional shapes and threedimensional solids with given properties or attributes.

Compose
TWODIMENSIONAL SHAPES WITH GIVEN PROPERTIES OR ATTRIBUTES
Including, but not limited to:
 Twodimensional figure – a figure with two basic units of measure, usually length and width
 Spatial visualization – creation and manipulation of mental representations of shapes
 Compose figures – to combine smaller geometric figures to form a larger geometric figure
 Attributes of twodimensional figures – characteristics that define a geometric figure (e.g., sides, vertices, etc.)
 Properties of twodimensional figures – relationship of attributes within a geometric figure (e.g., a square has 4 sides equal in length 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 sides equal in length but a rectangle has only opposite sides equal in length; etc.)
 Regular figure – a polygon with all sides and corners that appear to be the same or equal
 Irregular figure – a polygon with sides and/or corners that appear to be different or unequal
 Attributes of twodimensional figures
 Side – a straight outer boundary between two vertices (line segment) of a twodimensional figure
 Number of sides
 Length of sides
 Vertex (vertices) in a twodimensional figure – the point (corner) where two sides of a twodimensional figure meet
 Types of vertices
 Square corners
 Square corners 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
 Not square corners
 Opposite corners
 Attributes that do not identify a twodimensional figure
 Orientation
 Size
 Color
 Texture
 Compose twodimensional figures using a variety of concrete models.
 Compose regular and irregular figures based on attributes and properties.
Compose
THREEDIMENSIONAL SOLIDS WITH GIVEN PROPERTIES OR ATTRIBUTES
Including, but not limited to:
 Threedimensional figure – a figure that has measurements including length, width (depth), and height
 Spatial visualization – creation and manipulation of mental representations of shapes
 Compose figures – to combine smaller geometric figures to form a larger geometric figure
 Attributes of threedimensional figures – characteristics that define a geometric figure (e.g., faces, curved surfaces, edges, vertices, etc.)
 Properties of threedimensional figures – relationship of attributes within a geometric figure (e.g., a rectangular prism has 6 faces and each pair of opposite faces are the same size and shape, etc.) and between a group of geometric figures (e.g., a cube and a rectangular prism both have 6 faces with opposite faces equal in size and shape; however, a cube has only square faces but a rectangular prism can have square or rectangular faces; etc.)
 Attributes of threedimensional 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
 Edge – where the sides of two faces meet on a threedimensional figure
 Vertex (vertices) in a threedimensional figure – the point (corner) where three or more edges of a threedimensional figure meet
 Attributes that do not identify a threedimensional figure
 Orientation
 Size
 Color
 Texture
 Compose threedimensional figures using a variety of concrete models.
 Compose threedimensional figures based on attributes and properties.
Note(s):
 Grade Level(s):
 Grade 1 composed twodimensional shapes by joining two, three, or four figures to produce a target shape in more than one way if possible.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:
 III.A. Geometric Reasoning – Figures and their properties
 IX. Communication and Representation

2.8E 
Decompose twodimensional shapes such as cutting out a square from a rectangle, dividing a shape in half, or partitioning a rectangle into identical triangles and identify the resulting geometric parts.

Decompose
TWODIMENSIONAL SHAPES
Including, but not limited to:
 Twodimensional figure – a figure with two basic units of measure, usually length and width
 Spatial visualization – creation and manipulation of mental representations of shapes
 Decompose figures – to break a geometric figure into two or more smaller geometric figures
 Decompose twodimensional figures by cutting, dividing, or partitioning.
 Such as cutting a square from a rectangle
 Such as dividing a shape in half
 Such as partitioning a rectangle into identical triangles
 Resulting shapes equal or not equal
 Decompose twodimensional shapes using a variety of concrete models and materials.
Identify
THE RESULTING GEOMETRIC PARTS OF A DECOMPOSED TWODIMENSIONAL SHAPE
Including, but not limited to:
 Twodimensional figure – a figure with two basic units of measure, usually length and width
 Name resulting geometric figures (e.g., a rectangle partitioned into smaller rectangles that may or may not be equal in size or shape; a rectangle partitioned into triangles that may or may not be equal in size or shape; etc.)
Note(s):
 Grade Level(s):
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:
 III.A. Geometric Reasoning – Figures and their properties
 IX. Communication and Representation
