M.1 
Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:


M.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.

M.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.

M.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.

M.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.

M.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.

M.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.

M.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.

M.5 
Mathematical modeling in science and engineering. The student applies mathematical processes with algebraic techniques to study patterns and analyze data as it applies to science. The student is expected to:


M.5A 
Use proportionality and inverse variation to describe physical laws such as Hook's Law, Newton's Second Law of Motion, and Boyle's Law.

Use
PROPORTIONALITY AND INVERSE VARIATION
Including, but not limited to:
 Direct variation – a linear relationship between two variables, x (independent) and y (dependent), that always has a constant, unchanged ratio, k, and can be represented by y = kx
 Characteristics of direct variation
 Linear proportional relationship
 Linear
 Passes through the origin (0, 0)
 Represented by y = kx
 Constant of proportionality represented as
 When b = 0 in y = mx + b, then k = the slope, m
 Inverse variation – a relationship between two variables, x (independent) and y (dependent), that always has a constant, unchanged ratio, k, and can be represented by
 Characteristics of inverse variation
 Proportional relationship
 Nonlinear
 Does not pass through the origin (0, 0)
 Represented by
 Constant of proportionality represented as k = xy
To Describe
PHYSICAL LAWS SUCH AS HOOKE'S LAW, NEWTON'S SECOND LAW OF MOTION, AND BOYLE'S LAW
Including, but not limited to:
 Modeling physical laws through data collection and analysis
 Hooke’s Law – the force required to stretch or compress a spring a given distance is directly proportional to the distance. Hooke’s Law is represented by F = kx, where F is the amount of force, k is the constant factor associated with the spring (stiffness), and x is the amount of displacement or change.
 Newton’s Second Law of Motion – the force required to move an object is directly proportional to the mass of the object and the acceleration desired. Newton’s Second Law of Motion is represented by F_{net} = ma, where F_{net} is the net force, m is the mass of the object, and a is the acceleration.
 Boyle’s Law – if temperature is constant, then pressure of an ideal gas is inversely related to the volume of the gas. Boyle’s Law is represented by P_{a}V_{a} = P_{b}V_{b}, where P is the pressure on a volume of gas and V is the volume of the gas.
Note(s):
 Grade Level(s)
 Grade 8 studied direct variation and proportionality.
 Algebra I studied the linear parent function f(x) = x.
 Mathematical Models with Applications introduces application of inverse variation.
 Algebra II will study inverse variation and the rational function, f(x) = .
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 VI.C. Functions – Model realworld situations with functions
 VI.C.1. Apply known functions to model realworld situations.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 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.
 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.
 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.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.
 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.

M.5B 
Use exponential models available through technology to model growth and decay in areas, including radioactive decay.

Use
EXPONENTIAL MODELS AVAILABLE THROUGH TECHNOLOGY
Including, but not limited to:
 Exponential function: f(x) = a • b^{x}
 Exponential growth: b > 1
 Exponential decay: 0 < b < 1
To Model
GROWTH AND DECAY IN AREAS, INCLUDING RADIOACTIVE DECAY
Including, but not limited to:
 Population growth
 Halflife
 Radioactive decay
 Absorption of substances by the body
 Growth of bacteria
Note(s):
 Grade Level(s)
 Algebra I studied the exponent parent function f(x) = ab^{x}, with both growth and decay.
 Algebra II will study the exponential function f(x) = ab^{x}, as well as the logarithm function, f(x) = log_{b}(x), where b is 2, 10, or e.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 VI.C. Functions – Model realworld situations with functions
 VI.C.1. Apply known functions to model realworld situations.
 VI.C.2. Develop a function to model a situation.
 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.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.1. Interpret results of the mathematical problem in terms of the original realworld situation.
 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.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.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.
 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.

M.5C 
Use quadratic functions to model motion.

Use
QUADRATIC FUNCTIONS
Including, but not limited to:
 Representations
 Algebraic generalization
 Graph
 Table
 Verbal description
 Quadratic function, f(x) = ax^{2} + bx + c
 If the value of a is positive, then the vertex is a minimum and the parabola opens upward.
 If the value of a is negative, then the vertex is a maximum and the parabola opens downward.
 Key attributes of quadratic functions
 xintercept(s) – x coordinate of a point at which the relation crosses the xaxis, meaning the y coordinate equals zero, (x, 0)
 yintercept(s) – y coordinate of a point at which the relation crosses the yaxis, meaning the x coordinate equals zero, (0, y)
 Maximum or minimum value – largest or smallest ycoordinate or value a function takes over a given interval of the curve
 Domain – set of input values for the independent variable over which the function is defined
 Domain of the quadratic function is all real numbers.
 Range – set of output values for the dependent variable over which the function is defined
 Range of the quadratic function depends on the maximum or minimum.
 Function has a maximum, the range will be all numbers less than or equal to the maximum.
 Function has a minimum, the range will be all numbers greater than or equal to the minimum.
To Model
MOTION
Including, but not limited to:
 Analysis of quadratic functions as related to the context of motion problem situations
 Analysis of key attributes of quadratic functions as related to the context of motion problem situations
 Limiting domain and/or range based on context of a motion problem situation
 Validity and reasonableness of solution(s) in terms of a motion problem situation
 Interpreting meaning of an ordered pair in context of a motion problem situation
 Technologybased regression when given at least three data points
Note(s):
 Grade Level(s)
 Algebra I studied quadratic parent function f(x) = x^{2}, and its attributes; graphing and solving.
 Algebra II continues the study of the quadratic function f(x) = x^{2}, writing, solving, graphing, and applying.
 Precalculus will study quadratic relations as conic sections.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 VI.C. Functions – Model realworld situations with functions
 VI.C.1. Apply known functions to model realworld situations.
 VI.C.2. Develop a function to model a situation.
 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.
 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. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 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.
 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.

M.9 
Mathematical modeling in social sciences. The student applies mathematical processes and mathematical models to analyze data as it applies to social sciences. The student is expected to:


M.9F 
Use regression methods available through technology to model linear and exponential functions, interpret correlations, and make predictions.

Use
REGRESSION METHODS AVAILABLE THROUGH TECHNOLOGY
Including, but not limited to:
 Regression model – equation of best fit representing a set of bivariate data. Calculators find the equation of regression by calculating the lowest sum of the squares of differences in the predicted values and the observed values.
 Correlation coefficient (rvalue) – numeric value that assesses the strength of the linear relationship between two quantitative variables in a set of bivariate data
 When the correlation coefficient, r, is given in regression calculations, it can be used to determine the strength of the regression model as a representation of mathematical and realworld problem situations.
 The correlation coefficient, r, can only be used to analyze linear relationships or relationships that can be linearized such as exponential.
 The closer the rvalue is to 1 or –1, the stronger the relationship.
 Linear regression – regression used to determine the linear function that is the best fit for the data. The linear regression function on the graphing calculator can be used to calculate the best linear function to model the data.
 Exponential regression – regression used to determine the exponential function that is the best fit for the data. The exponential regression function on the graphing calculator can be used to calculate the best exponential function to model the data.
To Model
LINEAR AND EXPONENTIAL FUNCTIONS, INTERPRET CORRELATIONS, AND MAKE PREDICTIONS
Including, but not limited to:
 Graphic organizers
 Data table – table of values of collected data
 Graphical analysis – line graph or scatterplot
 Numerical analysis – correlation coefficient and inferences
 Positive or negative correlation
 No correlation
 Linear and nonlinear correlation
 Conclusions and predictions – use of the regression model that fits the data to predict data points beyond those collected
 Correlations may or may not be used to justify predictions.
 Linear Model
 Exponential Model
Note(s):
 Grade Level(s)
 Algebra I introduced correlation coefficients with linear functions.
 Algebra I studied linear functions and introduced exponential functions.
 Algebra II will introduce and use regression methods with graphing technology.
 Mathematical Models with Applications uses regression available through graphing technology to study data.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 V.B. Statistical Reasoning – Describe data
 V.B.4. Describe patterns and departure from patterns in the study of data.
 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.
 V.C.3. Make predictions using summary statistics.
 VI.B. Functions – Analysis of functions
 VI.B.2. Algebraically construct and analyze new functions.
 VI.C. Functions – Model realworld situations with functions
 VI.C.2. Develop a function to model a situation.
 IX.A. Connections – Connections among the strands of mathematics
 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.
