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 Instructional Focus DocumentMathematical Models with Applications
 TITLE : Unit 01: Algebraic Models in Science, Engineering, and Social Sciences SUGGESTED DURATION : 19 days

#### Unit Overview

Introduction
This unit bundles student expectations that address proportional relationships, quadratic functions, and exponential growth and decay functions to model and make predictions in real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Grades 6 – 8, students solved problems involving proportional relationships using tables, graphs, and equations. In Algebra I, students represented linear, quadratic, and exponential functions in problem situations using tables, graphs, and model functions to make predictions and draw conclusions in terms of the problem situation. Students investigated and compared the domain and range and other key attributes for each of the function models. Students used technology to determine a best fit model for data that could be represented by linear, quadratic, or exponential functions and applied the models to make predictions for real-world problems.

During this Unit
Students collect real data, study patterns, and use algebraic techniques to describe physical laws of science that are modeled by proportionality and inverse variation such as Hook’s Law, Newton’s Second Law of Motion, and Boyle’s Law. Exponential growth and decay functions are explored using technology to model problems in science and social studies, including radioactive decay. Students use quadratic functions to model motion by collecting data and determining quadratic functions to fit the data with and without technology. Students also use regression methods available through technology to generate linear and exponential functions and determine the strength of the model for making predictions using the correlation coefficient.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool HS MMA

After this Unit
Students will continue to revisit the applications of proportionality and the functions addressed in this unit, including linear functions, direct variation, inverse variation, exponential growth and decay functions, and quadratic functions and their attributes as they apply to given problem situations. Proper use of the regression feature using technology is discussed and practiced throughout the course. Throughout the fields of Mathematical Models with Applications, students will be required to take given information or collected data and determine tools and methods needed to solve the problem situation. The concepts in this unit will also be applied in subsequent mathematics courses.

This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning D1, D2; V. Statistical Reasoning A1, B4, C2, C3; VI. Functions B2, C1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to the Principles and Standards for School Mathematics (1989) from the National Council of Teachers of Mathematics (NCTM), students will understand relations and functions and select, convert flexibly among, and use various representations for them. According to Principles and Standards for School Mathematics: Algebra Standard for Grades 9 – 12 (2000), “High school students' algebra experience should enable them to create and use tabular, symbolic, graphical, and verbal representations and to analyze and understand patterns, relations, and functions with more sophistication than in the middle grades. In helping high school students learn about the characteristics of particular classes of functions, teachers may find it helpful to compare and contrast situations that are modeled by functions from various classes” (NCTM, p. 297). In this unit, students must think at a level higher than simply converting between representations. Instead, the activities in this unit require decisions on which representation is the most appropriate. Principles and Standards for School Mathematics (1989) describes this as the need to identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationship (NCTM). Additionally, “When involved in these activities, students may encounter ‘messy data,’ for which a line or a curve might not be an exact fit. They will need experience with such situations and assistance from the teacher to develop their ability to find a function that fits the data well enough to be useful as a prediction tool (NCTM, 2000, p. 228)” (as cited by Van de Walle, 2004, p. 448).

National Council of Teachers of Mathematics. (1989). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics: Algebra standard for grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.
Van de Walle, J. (2004). Elementary and middle school mathematics: Teaching developmentally. Boston: Pearson Education, Inc.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding how two quantities vary together (covariation) builds flexible functional reasoning in order to make predictions and critical judgements about the relationship.
• How can it be determined if a relationship between two variables can be represented by a function?
• How can functions be used to model problem situations efficiently?
• How can functions be used to model relationships in science, engineering, and social sciences?
• What are the strengths and limitations of a particular function model for a problem situation?
• How is function notation used to define and describe a function rule?
• How does the numeric pattern in the data help determine the function model for a problem situation?
• Proportional relationships, including direct and inverse variation, can model and describe scientific laws and be applied to make predictions and critical judgments in real-world problem situations.
• How do direct variation and inverse variation compare?
• How can proportional relationships, including direct variation and inverse variation, be used to describe physical laws in science?
• What proportional relationships are described by …
• Hook’s Law?
• Newton’s Second Law of Motion?
• Boyle’s Law?
• Using technology, how can experimental data be collected, analyzed to determine an appropriate model, and interpreted to make predictions in science?
• Mathematical Modeling
• Science and Engineering
• Proportional and inverse variation
• Social Sciences
• Data
• Conclusions and predictions
• Regression methods
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding how two quantities vary together (covariation) builds flexible functional reasoning in order to make predictions and critical judgements about the relationship.
• How can it be determined if a relationship between two variables can be represented by a function?
• How can functions be used to model problem situations efficiently?
• How can functions be used to model relationships in science, engineering, and social sciences?
• What are the strengths and limitations of a particular function model for a problem situation?
• How is function notation used to define and describe a function rule?
• How does the numeric pattern in the data help determine the function model for a problem situation?
• Exponential functions can model growth and decay in real-world problem situations and can be applied to make predictions and critical judgments in terms of problem situations.
• What representations can be used to model exponential functions?
• What are the characteristics and attributes of an exponential function, and how do they relate to the problem situation?
• How is exponential growth distinguished from exponential decay?
• How are exponential growth and exponential decay used to represent problems in …
• science?
• social science?
• Using technology, how can experimental data be collected, analyzed to determine an appropriate model, and interpreted to make predictions?
• Mathematical Modeling
• Science and Engineering
• Exponential models
• Social Sciences
• Data
• Conclusions and predictions
• Regression methods
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding how two quantities vary together (covariation) builds flexible functional reasoning in order to make predictions and critical judgements about the relationship.
• How can it be determined if a relationship between two variables can be represented by a function?
• How can functions be used to model problem situations efficiently?
• How can functions be used to model relationships in science, engineering, and social sciences?
• What are the strengths and limitations of a particular function model for a problem situation?
• How is function notation used to define and describe a function rule?
• How does the numeric pattern in the data help determine if the function model in a problem situation is linear or quadratic?
• Quadratic and linear functions can model motion of falling objects in real-world problem situations and can be applied to make predictions and critical judgments in terms of problem situations.
• What representations can be used to model quadratic functions?
• What are the characteristics and attributes of quadratic functions and how do they relate to the problem situation?
• Why is motion modeled by quadratic functions?
• Using technology, how can experimental data be collected, analyzed to determine an appropriate model, and interpreted to make predictions in science?
• How can forces in nature, including those which cause parabolic motion, be represented mathematically?
• Mathematical Modeling
• Science and Engineering
• Social Sciences
• Data
• Conclusions and predictions
• Regression methods
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that a graph of time versus distance on which the y-values representing distance increase means that the object in motion goes up rather than understanding that it is a representation of distance over time or velocity.

Underdeveloped Concepts:

• Some students may not have a deep understanding of functions and the connection of the different representations and the relationship with real-life data.
• Some students still struggle with algebraic manipulations, solving equations, and connecting the equations and their solutions to problem situations.
• Although some students may be very proficient with the graphing calculator, some do not connect the representations from one screen of the calculator to another.

#### Unit Vocabulary

• 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 PaVa = PbVb, where P is the pressure on a volume of gas and V is the volume of the gas.
• Conclusions and predictions – use of the regression model that fits the data to predict data points beyond those collected
• Correlation coefficient (r-value) – numeric value that assesses the strength of the linear relationship between two quantitative variables in a set of bivariate data
• Data table – table of values of collected data
• 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
• Domain – set of input values for the independent variable over which the function is defined
• 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.
• Graphical analysis – line graph or scatterplot
• Hook'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.
• 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 y = • 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.
• Maximum or minimum value – largest or smallest y-coordinate or value a function takes over a given interval of the curve
• 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 Fnet = ma, where Fnet is the net force, m is the mass of the object, and a is the acceleration.
• Numerical analysis – correlation coefficient and inferences
• Range – set of output values for the dependent variable over which the function is defined
• 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.
• x-intercept(s)x coordinate of a point at which the relation crosses the x-axis, meaning the y coordinate equals zero, (x, 0)
• y-intercept(s)y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)

Related Vocabulary:

 Asymptote Axis of symmetry Coordinate plane Dependent variable Domain Exponent Exponential Function Increase Independent variable Intercept Inverse Linear Maximum Minimum Parabolic motion Proportional Quadratic Range Rate of change Scatterplot Slope Symmetry Variable Variation Vertex x-axis x-intercept y-axis y-intercept
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Center if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

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

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Mathematical Models with Applications Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE 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.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
TEKS# SE# TEKS SPECIFICITY
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 – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world 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, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world 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 problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• 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 problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving 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, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world 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 non-examples, 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
• Non-linear
• 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 Fnet = ma, where Fnet 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 PaVa = PbVb, where P is the pressure on a volume of gas and V is the volume of the gas.

Note(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 real-world situations with functions
• VI.C.1. Apply known functions to model real-world 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 – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world 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, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world 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) = abx
• 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
• Half-life
• Absorption of substances by the body
• Growth of bacteria

Note(s):

• Algebra I studied the exponent parent function f(x) = abx, with both growth and decay.
• Algebra II will study the exponential function f(x) = abx, as well as the logarithm function, f(x) = logb(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 real-world situations with functions
• VI.C.1. Apply known functions to model real-world 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 – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world 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, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world 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

Including, but not limited to:

• Representations
• Algebraic generalization
• Graph
• Table
• Verbal description
• Quadratic function, f(x) = ax2 + 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
• x-intercept(s) – x coordinate of a point at which the relation crosses the x-axis, meaning the y coordinate equals zero, (x, 0)
• y-intercept(s) – y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)
• Maximum or minimum value – largest or smallest y-coordinate 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
• Technology-based regression when given at least three data points

Note(s):

• Algebra I studied quadratic parent function f(x) = x2, and its attributes; graphing and solving.
• Algebra II continues the study of the quadratic function f(x) = x2, 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 real-world situations with functions
• VI.C.1. Apply known functions to model real-world situations.
• VI.C.2. Develop a function to model a situation.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world 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, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world 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 (r-value) – 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 real-world 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 r-value 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 non-linear 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):

• 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 real-world 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, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world 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. 