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Mathematical Models with Applications
TITLE : Unit 01: Algebraic Models in Science and Social Studies SUGGESTED DURATION : 19 days

Unit Overview

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.

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): II. Algebraic Reasoning D1, D2; VI. Statistical Reasoning B1, B2, B4, C3; VII Functions A1, A2, B1, B2, C1, C2; VIII Problem Solving and Reasoning B2, C1; IX Communication and Representation A1, A2, A3, B1, C1; X. Connections A1, A2, B1, B2.

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.
Texas Education Agency. (2009). Texas College and Career Readiness Standards. Retrieved from http://www.thecb.state.tx.us/collegereadiness/crs.pdf
Van de Walle, J. (2004). Elementary and middle school mathematics: Teaching developmentally. Boston: Pearson Education, Inc.

OVERARCHING UNDERSTANDINGS and QUESTIONS

Algebraic relationships can be used to describe mathematical and real-world patterns.

  • Why is it important to describe the algebraic relationships found in numeric patterns?
  • What algebraic relationships can be found in patterns?

Proportional reasoning can be used to describe and solve problems in everyday life.

  • Why can proportional reasoning be used to make predictions and comparisons in problem situations?
  • How are direct proportion (direct variation) and inverse proportion (inverse variation) related?
  • How is proportional change distinguished from non-proportional change?
  • How are ratios used in a proportional relationship?

Function models for problem situations can be determined by collecting and analyzing data using a variety of representations and applied to make predictions and critical judgments in terms of the problem situation.

  • Why is it important to determine and apply function models for problem situations?
  • What representations can be used to analyze collected data and how are the representations interrelated?
  • Why is it important to analyze various representations of data when determining appropriate function models for problem situations?  
  • How do the key attributes and characteristics of the function differ from the key attributes and characteristics of the function model for the problem situation?
  • How does technology aid in the analysis and application of modeling and solving problem situations?

Statistical data is collected, analyzed graphically and numerically, and interpreted to make predictions and draw conclusions.

  • Why is it important to understand the analysis of statistical and interpretation of statistical data?
  • Why is it important to use appropriate data collection methods?
  • How does the type of data determine the type of graphical analysis?
  • How does the type of data determine the type of numerical analysis?
  • What is the purpose of analyzing statistical data?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
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.

Algebraic Reasoning

  • Equations
  • Expressions
  • Patterns/Rules
  • Ratios/Rates

Functions

  • Attributes of Functions
  • Constant of Proportionality
  • Direct/Inverse Variation
  • Independent/Dependent Variables
  • Linear Functions
  • Non-Linear Functions
  • Non-Proportional Relationships
  • Proportional Relationships

Statistical Reasoning

  • Conclusions/Predictions
  • Data
  • Summary Statistics
  • Statistical Representations

Associated Mathematical Processes

  • Application
  • Tools and Techniques
  • Problem Solving Model
  • Communication
  • Representations
  • Relationships 
  • Justification

Proportional relationships, including direct and inverse variation, can be used to model scientific and social relationships.

  • 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 do Hook’s Law, Newton’s Second Law of Motion, and Boyle’s Law represent?
  • How does the numeric pattern in the data help determine the function model for a problem situation?
  • Using technology, how can experimental data be collected, analyzed to determine an appropriate model, and interpreted to make predictions?
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.

Algebraic Reasoning

  • Equations
  • Expressions
  • Patterns/Rules
  • Ratios

Functions

  • Attributes of Functions
  • Independent/Dependent Variables
  • Non-Linear Functions
  • Non-Proportional Relationships

Statistical Reasoning

  • Conclusions/Predictions
  • Data
  • Summary Statistics
  • Statistical Representations

Associated Mathematical Processes

  • Application
  • Tools and Techniques
  • Problem Solving Model
  • Communication
  • Representations
  • Relationships 
  • Justification

Exponential functions model growth and decay in science and social studies applications.

  • What representations can be used to model exponential functions?
  • What are the characteristics and attributes of exponential functions and how do they relate to the problem situation?
  • How is exponential growth distinguished from exponential decay?
  • What are real-world examples of exponential growth and exponential decay?
  • How does the numeric pattern in the data help determine the function model for a problem situation?
  • Using technology, how can experimental data be collected, analyzed to determine an appropriate model, and interpreted to make predictions?
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.

Quadratic functions model motion of objects rising and falling in the earth's atmosphere.

  • 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?
  • What real-world examples can be modeled by quadratic functions?
  • How does the numeric pattern in the data help determine the function model for a problem situation?
  • Using technology, how can experimental data be collected, analyzed to determine an appropriate model, and interpreted to make predictions?
  • Which function is used to model parabolic motion in science?
  • What force causes parabolic motion in nature? 

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 Creator 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 (select CCRS from Standard Set dropdown menu)

Texas Instruments – Graphing Calculator Tutorials

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – Mathematics TEKS: Supporting Information

Texas Education Agency – Interactive Mathematics Glossary

TEKS# SE# TEKS Unit Level Specificity
 
  • Bold black text in italics: Knowledge and Skills Statement (TEKS)
  • Bold black text: Student Expectation (TEKS)
  • Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)
  • Blue text: Supporting information / Clarifications from TCMPC (Specificity)
  • Blue text in italics: Unit-specific clarification
  • Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
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:
    • X. Connections
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:
    • VIII. Problem Solving and Reasoning
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:
    • VIII. Problem Solving and Reasoning
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:
    • IX. Communication and Representation
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:
    • IX. Communication and Representation
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:
    • X. Connections
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:
    • IX. Communication and Representation
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 , 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
    • I. Numeric Reasoning
      • B1 – Perform computations with real and complex numbers.
    • II. Algebraic reasoning
      • B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions.
    • VII. Functions
      • A2 – Recognize and distinguish between different types of functions.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
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 • bx
    • 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
    • Radioactive decay
    • Absorption of substances by the body
  • Growth of bacteria
    • Spread of disease

Note(s):

  • Grade Level(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
    • I. Numeric Reasoning
      • B1 – Perform computations with real and complex numbers.
    • II. Algebraic reasoning
      • B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions.
    • VII. Functions
      • A2 – Recognize and distinguish between different types of functions.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
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) = ax2bx + 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),/span>
    • 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):

  • Grade Level(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
    • I. Numeric Reasoning
      • B1 – Perform computations with real and complex numbers.
    • II. Algebraic reasoning
      • B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions.
    • VII. Functions
      • A2 – Recognize and distinguish between different types of functions.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
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 the 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):

  • 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
    • III. Geometrical Reasoning
      • C2 – Make connections between geometry, statistics, and probability.
    • VI. Statistical Reasoning
      • C3 – Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
    • VII. Functions
      • C2 – Develop a function to model the situation.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
The English Language Proficiency Standards (ELPS), as required by 19 Texas Administrative Code, Chapter 74, Subchapter A, §74.4, outline English language proficiency level descriptors and student expectations for English language learners (ELLs). School districts are required to implement ELPS as an integral part of each subject in the required curriculum.

School districts shall provide instruction in the knowledge and skills of the foundation and enrichment curriculum in a manner that is linguistically accommodated commensurate with the student’s levels of English language proficiency to ensure that the student learns the knowledge and skills in the required curriculum.


School districts shall provide content-based instruction including the cross-curricular second language acquisition essential knowledge and skills in subsection (c) of the ELPS in a manner that is linguistically accommodated to help the student acquire English language proficiency.

http://ritter.tea.state.tx.us/rules/tac/chapter074/ch074a.html#74.4 


Choose appropriate ELPS to support instruction.

ELPS# Subsection C: Cross-curricular second language acquisition essential knowledge and skills.
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ELPS.c.1 The ELL uses language learning strategies to develop an awareness of his or her own learning processes in all content areas. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.1A use prior knowledge and experiences to understand meanings in English
ELPS.c.1B monitor oral and written language production and employ self-corrective techniques or other resources
ELPS.c.1C use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and grade-level vocabulary
ELPS.c.1D speak using learning strategies such as requesting assistance, employing non-verbal cues, and using synonyms and circumlocution (conveying ideas by defining or describing when exact English words are not known)
ELPS.c.1E internalize new basic and academic language by using and reusing it in meaningful ways in speaking and writing activities that build concept and language attainment
ELPS.c.1F use accessible language and learn new and essential language in the process
ELPS.c.1G demonstrate an increasing ability to distinguish between formal and informal English and an increasing knowledge of when to use each one commensurate with grade-level learning expectations
ELPS.c.1H develop and expand repertoire of learning strategies such as reasoning inductively or deductively, looking for patterns in language, and analyzing sayings and expressions commensurate with grade-level learning expectations.
ELPS.c.2 The ELL listens to a variety of speakers including teachers, peers, and electronic media to gain an increasing level of comprehension of newly acquired language in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in listening. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.2A distinguish sounds and intonation patterns of English with increasing ease
ELPS.c.2B recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters
ELPS.c.2C learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions
ELPS.c.2D monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed
ELPS.c.2E use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language
ELPS.c.2F listen to and derive meaning from a variety of media such as audio tape, video, DVD, and CD ROM to build and reinforce concept and language attainment
ELPS.c.2G understand the general meaning, main points, and important details of spoken language ranging from situations in which topics, language, and contexts are familiar to unfamiliar
ELPS.c.2H understand implicit ideas and information in increasingly complex spoken language commensurate with grade-level learning expectations
ELPS.c.2I demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs.
ELPS.c.3 The ELL speaks in a variety of modes for a variety of purposes with an awareness of different language registers (formal/informal) using vocabulary with increasing fluency and accuracy in language arts and all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in speaking. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.3A practice producing sounds of newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters to pronounce English words in a manner that is increasingly comprehensible
ELPS.c.3B expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication
ELPS.c.3C speak using a variety of grammatical structures, sentence lengths, sentence types, and connecting words with increasing accuracy and ease as more English is acquired
ELPS.c.3D speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency
ELPS.c.3E share information in cooperative learning interactions
ELPS.c.3F ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended speaking assignments
ELPS.c.3G express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics
ELPS.c.3H narrate, describe, and explain with increasing specificity and detail as more English is acquired
ELPS.c.3I adapt spoken language appropriately for formal and informal purposes
ELPS.c.3J respond orally to information presented in a wide variety of print, electronic, audio, and visual media to build and reinforce concept and language attainment.
ELPS.c.4 The ELL reads a variety of texts for a variety of purposes with an increasing level of comprehension in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in reading. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations apply to text read aloud for students not yet at the stage of decoding written text. The student is expected to:
ELPS.c.4A learn relationships between sounds and letters of the English language and decode (sound out) words using a combination of skills such as recognizing sound-letter relationships and identifying cognates, affixes, roots, and base words
ELPS.c.4B recognize directionality of English reading such as left to right and top to bottom
ELPS.c.4C develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials
ELPS.c.4D use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary and other prereading activities to enhance comprehension of written text
ELPS.c.4E read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned
ELPS.c.4F use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language
ELPS.c.4G demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding to questions, and taking notes commensurate with content area and grade level needs
ELPS.c.4H read silently with increasing ease and comprehension for longer periods
ELPS.c.4I demonstrate English comprehension and expand reading skills by employing basic reading skills such as demonstrating understanding of supporting ideas and details in text and graphic sources, summarizing text, and distinguishing main ideas from details commensurate with content area needs
ELPS.c.4J demonstrate English comprehension and expand reading skills by employing inferential skills such as predicting, making connections between ideas, drawing inferences and conclusions from text and graphic sources, and finding supporting text evidence commensurate with content area needs
ELPS.c.4K demonstrate English comprehension and expand reading skills by employing analytical skills such as evaluating written information and performing critical analyses commensurate with content area and grade-level needs.
ELPS.c.5 The ELL writes in a variety of forms with increasing accuracy to effectively address a specific purpose and audience in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in writing. In order for the ELL to meet grade-level learning expectations across foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations do not apply until the student has reached the stage of generating original written text using a standard writing system. The student is expected to:
ELPS.c.5A learn relationships between sounds and letters of the English language to represent sounds when writing in English
ELPS.c.5B write using newly acquired basic vocabulary and content-based grade-level vocabulary
ELPS.c.5C spell familiar English words with increasing accuracy, and employ English spelling patterns and rules with increasing accuracy as more English is acquired
ELPS.c.5D edit writing for standard grammar and usage, including subject-verb agreement, pronoun agreement, and appropriate verb tenses commensurate with grade-level expectations as more English is acquired
ELPS.c.5E employ increasingly complex grammatical structures in content area writing commensurate with grade-level expectations, such as:
ELPS.c.5F write using a variety of grade-appropriate sentence lengths, patterns, and connecting words to combine phrases, clauses, and sentences in increasingly accurate ways as more English is acquired
ELPS.c.5G narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired.
Last Updated 09/01/2016
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