
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
strikethrough.

Legend:  Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
 Unitspecific 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 twodimensional and threedimensional geometric figures, including measurable attributes
 TxCCRS:

3.1B 
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.
Process Standard

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:
 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 twodimensional and threedimensional 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 twodimensional and threedimensional 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 twodimensional and threedimensional 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 twodimensional and threedimensional 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 nonexamples, 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 twodimensional and threedimensional geometric figures, including measurable attributes
 TxCCRS:

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 twodimensional and threedimensional 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 twodimensional geometric figures to develop generalizations about their properties. The student is expected to:


3.6A 
Classify and sort two and threedimensional figures, including cones, cylinders, spheres, triangular and rectangular prisms, and cubes, based on attributes using formal geometric language.
Readiness Standard

Classify, Sort
TWODIMENSIONAL FIGURES BASED ON ATTRIBUTES USING FORMAL GEOMETRIC LANGUAGE
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 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 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 (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 twodimensional 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
 Quadrilateral
 4 sides
 4 vertices
 Types of quadrilaterals
 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
 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”)
Classify, Sort
THREEDIMENSIONAL FIGURES, INCLUDING CONES, CYLINDERS, SPHERES, TRIANGULAR AND RECTANGULAR PRISMS, AND CUBES, 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., edges, vertices, faces [bases], 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 [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 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
 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 threedimensional figure
 Vertex (vertices) in a threedimensional figure – the point (corner) where three or more edges of a threedimensional figure meet
 Congruent – of equal measure
 Threedimensional figures
 Curved surface threedimensional 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.), 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):
 Kindergarten classified and sorted a variety of regular and irregular two and threedimensional figures regardless of orientation or size.
 Grade 1 classified and sorted threedimensional 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 twodimensional 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 twodimensional and threedimensional 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 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 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 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 (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
 Subcategories of quadrilaterals
 Types of quadrilaterals
 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 Level(s):
 Grade 2 created twodimensional 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 twodimensional 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 twodimensional and threedimensional geometric figures, including measurable attributes
 TxCCRS:
 III.A. Geometric Reasoning – Figures and their properties
 IX. Communication and Representation

3.6E 
Decompose two congruent twodimensional 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 TWODIMENSIONAL FIGURES INTO PARTS WITH EQUAL AREAS
Including, but not limited to:
 Twodimensional 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 TWODIMENSIONAL 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
 Twodimensional 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 nonzero 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):
 Grade Level(s):
 Grade 1 partitioned twodimensional figures into two and four fair shares or equal parts and described the parts using words.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Understanding fractions as numbers and representing equivalent fractions
 TxCCRS:
 IV.C. Measurement Reasoning – Measurement involving geometry and algebra
 IX. Communication and Representation
 X. Connections
