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:
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.1. Interpret results of the mathematical problem in terms of the original realworld situation.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
 IX.A.2. Connect mathematics to the study of other disciplines.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.
 IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
 IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.

G.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.
 TxCCRS:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.
 V.A. Statistical Reasoning – Design a study
 V.A.1. Formulate a statistical question, plan an investigation, and collect data.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VII.A.2. Formulate a plan or strategy.
 VII.A.3. Determine a solution.
 VII.A.4. Justify the solution.
 VII.A.5. Evaluate the problemsolving process.
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.2. Evaluate the problemsolving process.

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:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.
 V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
 V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.

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:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.

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:
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.

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 nonexamples, patterns, etc.
 Current knowledge to new learning
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxCCRS:
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
 IX.A.2. Connect mathematics to the study of other disciplines.

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:
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.4. Justify the solution.
 VII.B. Problem Solving and Reasoning – Proportional reasoning
 VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
 VII.C. Problem Solving and Reasoning – Logical reasoning
 VII.C.1. Develop and evaluate convincing arguments.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII. C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.

G.10 
Twodimensional and threedimensional figures. The student uses the process skills to recognize characteristics and dimensional changes of two and threedimensional 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 nonproportional dimensional change.

Determine, Describe
HOW CHANGES IN THE LINEAR DIMENSIONS OF A SHAPE AFFECT ITS PERIMETER OR AREA INCLUDING PROPORTIONAL AND NONPROPORTIONAL DIMENSIONAL CHANGE
Including, but not limited to:
 Verbal and written description
 Dimensional change
 Perimeter and circumference
 Area
 Proportional change
 Twodimensional proportional change – two dimensions multiplied by the same scale factor
 Nonproportional change
 Twodimensional nonproportional 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 nonproportional dimensional change
 Emphasis on connections to units
 Dimension changes in realworld problem situations
Note(s):
 Grade Level(s)
 Grade 7 and 8 modeled the effect on linear and area measurements of dilated twodimensional shapes.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 I.C. Numeric Reasoning – Systems of measurement
 I.C.1. Select or use the appropriate type of method, unit, and tool for the attribute being measured.
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.2. Form and validate conjectures about one, two, and threedimensional figures and their properties.
 III.B. Geometric and Spatial Reasoning – Transformations and symmetry
 III.B.1. Identify transformations and symmetries of figures.
 III.B.2. Use transformations to investigate congruence, similarity, and symmetries of figures.
 III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
 III.C.1. Make connections between geometry and algebraic equations.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.1. Find the perimeter and area of twodimensional figures.
 III.D.2. Determine the surface area and volume of threedimensional figures.
 III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.

G.11 
Twodimensional and threedimensional figures. The student uses the process skills in the application of formulas to determine measures of two and threedimensional 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
 Realworld problem situations involving area
 Emphasis on appropriate units of measure
Note(s):
 Grade Level(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 nsided polygon.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 II.A. Algebraic Reasoning – Identifying expressions equations
 II.A.1. Explain the difference between expressions and equations.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.1. Find the perimeter and area of twodimensional figures.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.

G.11B 
Determine the area of composite twodimensional 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 TWODIMENSIONAL 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 twodimensional figures
 Triangles
 Parallelograms
 Trapezoids
 Kites
 Regular polygons
 Sectors of circles
 Applications to realworld situations
 Appropriate use of units of measure
Note(s):
 Grade Level(s)
 Previous grade levels used units, tools, and formulas to find the area of figures in problem situations.
 Previous grade levels introduced composites of twodimensional 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
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.1. Find the perimeter and area of twodimensional figures.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.

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
 Measure of arc length of is denoted as
 Connecting the proportional relationship or
 Applications to realworld problem situations
 Use of appropriate units of measure
 Use of various tools
 Protractor and straightedge
 Dynamic geometric software
 Patty paper
Note(s):
 Grade Level(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
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.2. Form and validate conjectures about one, two, and threedimensional figures and their properties.
 III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
 III.C.1. Make connections between geometry and algebraic equations.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.1. Find the perimeter and area of twodimensional figures.
 III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VII.A.3. Determine a solution.
 VII.B. Problem Solving and Reasoning – Proportional reasoning
 VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.

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 realworld problem situations
 Use of appropriate units of measure
 Use of various tools
 Protractor and straightedge
 Dynamic geometric software
 Patty paper
Note(s):
 Grade Level(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
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.2. Form and validate conjectures about one, two, and threedimensional figures and their properties.
 III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
 III.C.1. Make connections between geometry and algebraic equations.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.1. Find the perimeter and area of twodimensional figures.
 III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VII.A.3. Determine a solution.
 VII.B. Problem Solving and Reasoning – Proportional reasoning
 VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.
