Hello, Guest!
 Instructional Focus DocumentAlgebra II
 TITLE : Unit 11: Linear, Quadratic, and Exponential Data Models SUGGESTED DURATION : 5 days

#### Unit Overview

This unit bundles student expectations that address analyzing sets of data using technology to determine if the data is best represented using linear, quadratic, or exponential models and applying the selected models to make predictions and critical judgments in terms of the data. Concepts are incorporated into both mathematical and 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 Algebra I Units 03, 08, and 09, students were introduced to linear, quadratic, and exponential functions. In Algebra II Units 05 and 09, students analyzed quadratic and exponential functions. Linear functions were interspersed throughout Algebra II.

During this unit, students analyze and compare linear, quadratic, and exponential data sets using graphs, tables, verbal descriptions, and technology to determine which function can be selected to best model the data. Students use regression methods available through technology to write the appropriate regression function (linear, quadratic, or exponential) to model the data. Students apply the regression model to predict and make decisions and critical judgments in terms of the data.

After this unit, in Algebra II Units 12 and 13, students will review and continue to apply linear, quadratic, and exponential functions. In subsequent mathematics courses, students will also continue to apply these concepts when linear, quadratic, and exponential functions and equations arise in problem situations.

In Algebra II, analysis of sets of data to select an appropriate regression equation and application of that regression equation to make predictions and critical judgments are identified as STAAR Readiness Standard 2A.8C and STAAR Supporting Standards 2A.8A and 2A.8B. These standards are subsumed under STAAR Reporting Category 2: Describing and Graphing Functions and Their Inverses. This unit is supporting the development of Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning A1, B1, C1, D1, D2; III. Geometric Reasoning B2, C1; VII. Functions A2, B1, B2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

According to the National Council of Teachers of Mathematics (2011), Developing Essential Understanding of Functions, Grades 9-12, understanding of the function concept is essential to describing and analyzing quantities which vary with respect to one another. According to research from the National Council of Teachers of Mathematics (2000), Principles And Standards For School Mathematics, high school algebra should provide students with insights into mathematical abstraction and structure. 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 a higher degree of sophistication. Students should develop an understanding of the algebraic properties that govern manipulation of symbols in expressions, equations, and inequalities.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2011). Developing essential understanding of expressions, equations, and functions, grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

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 can function models be used to evaluate one or more elements in their domains?
• 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 are 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

• Multiple Representations

Functions

• Attributes of Functions
• Linear Functions
• Non-Linear Functions

Statistical Reasoning

• Data
• Regression
• Statistical Representations
• Summary Statistics

Associated Mathematical Processes

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

Transformations of the parent functions can be used to determine graphs and equations of representative functions to model problem situations.

• What are the effects of changes on the graph of the parent function, f(x), when f(x) is replaced by af(x), for specific positive and negative values of a?
• What are the effects of changes on the graph of the parent function, f(x), when f(x) is replaced by f(x) + d, for specific positive and negative values of d?
• What are the effects of changes on the graph of the parent function, f(x), when f(x) is replaced by f(bx), for specific positive and negative values of b?
• What are the effects of changes on the graph of the parent function, f(x), when f(x) is replaced by f(x - c) for specific positive and negative values of c?

Unique characteristics of data, scatterplots of data, calculation of regression equations, and comparison of correlation coefficients and coefficients of determination can be used to select an appropriate model from among linear, quadratic, or exponential functions.

• What characteristics of sequential data can be used to determine if the data is linear, quadratic, or exponential?
• How does the rate of change compare for linear, quadratic, and exponential data?
• How are scatterplots used to determine if data is best represented by a linear, quadratic, or exponential function?
• How can a regression equation be determined using a graphing calculator?
• How does the correlation coefficient help determine whether to represent the data by a linear or exponential function?
• How does the coefficient of determination help determine whether to represent the data by a linear, quadratic, or exponential function?

Predictions and critical judgments can be made from models selected to represent linear, quadratic, or exponential data.

• How can representations of the data be used to make predictions and critical judgments?
• How can the coefficient of determination help determine the validity of predictions using the model regression equation?

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think a scatterplot is linear because the points appear to be almost a line, rather than doing a regression analysis and comparing correlation coefficients. Parts of a quadratic relationship and parts of an exponential relationship may appear linear without in-depth analysis.
• Some students may fail to put data in sequential order before checking first and second differences and common ratios.
• Some students may check on first and second differences and fail to check common ratios when analyzing sequential data.
• Some students may not put values in the correct order when calculating first and second differences (not y1y2, but y2y1).
• Some students may not put values in the correct order when calculating the common ratio (not , but ).

#### Unit Vocabulary

• Regression equation – line of best fit representing a set of bivariate data
• Correlation coefficient (r-value) – numeric value that assesses the strength of the linear relationship between two quantitative variables in a set of bivariate data
• Coefficient of determination (r2-value) – representation of the percent of data closest to the regression line and used to measure how well the regression line can be used as a prediction model

Related Vocabulary:

 Common ratio Exponential First differences Linear Quadratic Rate of change Regression Regression equation Second differences Sequential data
Unit Assessment Items System Resources Other Resources

Show this message:

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 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 Algebra II Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Bold red text in italics:  Student Expectation identified by TEA as a Readiness Standard for STAAR
• Bold green text in italics: Student Expectation identified by TEA as a Supporting Standard for STAAR
• 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)
2A.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
2A.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
2A.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
2A.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
2A.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
2A.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
2A.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
2A.1G Display, explain, or 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
2A.8 Data. The student applies mathematical processes to analyze data, select appropriate models, write corresponding functions, and make predictions. The student is expected to:
2A.8A Analyze data to select the appropriate model from among linear, quadratic, and exponential models.
Supporting Standard

Analyze

DATA

Including, but not limited to:

• Data collected from data collection devices
• Data given in mathematical problem situations
• Types of data
• Linear
• Constant rate of change (slope)
• If independent values change sequentially, the dependent values have a common first difference.
• Non-constant rate of change
• If independent values change sequentially the dependent values have a common second difference.
• Exponential
• Non-constant rate of change
• If independent values change sequentially the dependent values have a common ratio, .
• Comparisons between sequential data as linear, quadratic, and exponential

To Select

THE APPROPRIATE MODEL FROM AMONG LINEAR, QUADRATIC, AND EXPONENTIAL MODELS

Including, but not limited to:

• Data collected from data collection devices
• Data given in real-world problem situations
• Data relationships
• Linear
• Exponential
• Data representations
• Data tables
• Graphs/scatterplots
• Verbal descriptions
• Algebraic generalizations

Note(s):

• Algebra I introduced the linear, quadratic, and exponential functions.
• Algebra II expands on transformations and applications of exponential functions.
• 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
• A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• III. Geometric Reasoning
• B2 – Identify the symmetries of a plane figure.
• C1 – Make connections between geometry and algebra.
• VII. Functions
• A2 – Recognize and distinguish between different types of functions.
• B1 – Understand and analyze features of a function.
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.8B Use regression methods available through technology to write a linear function, a quadratic function, and an exponential function from a given set of data.
Supporting Standard

Use

REGRESSION METHODS AVAILABLE THROUGH TECHNOLOGY TO WRITE A LINEAR FUNCTION, A QUADRATIC FUNCTION, AND AN EXPONENTIAL FUNCTION FROM A GIVEN SET OF DATA

Including, but not limited to:

• Data collected from data collection devices
• Data given in mathematical and real-world problem situations
• Data relationships
• Linear
• Exponential
• Regression equation – line of best fit representing a set of bivariate data
• 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
• Correlation coefficients closest to ±1 indicate the best model for mathematical and real-world problem situations.
• Value of the correlation coefficient, –1 ≤ r ≤ 1
• Perfect correlation, r = 1 or –1
• Strong correlation, 0.68 < |r| < 1.00
• Moderate correlation, 0.34 ≤ |r| ≤ 0.68
• Weak, 0 < |r| < 0.34
• No correlation, r = 0
• Coefficient of determination (r2-value) – representation of the percent of data closest to the regression line and used to measure how well the regression line can be used as a prediction model
• When the coefficient of determination, r2, is given in the regression calculations, it can be used to determine the strength of the regression model to represent and make predictions in mathematical and real-world problem situations.
• The coefficient of determination, r2, can be used to analyze and compare all types of relationships such as linear, quadratic, and exponential

Note(s):

• Algebra I introduced the linear, quadratic, and exponential functions.
• Algebra I calculated, using technology, the correlation coefficient between two quantitative variables and interpreted this quantity as a measure of the strength of association.
• 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
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• A2 – Recognize and distinguish between different types of functions.
• B1 – Understand and analyze features of a function.
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.8C Predict and make decisions and critical judgments from a given set of data using linear, quadratic, and exponential models.

Predict, Make

DECISIONS AND CRITICAL JUDGMENTS FROM A GIVEN SET OF DATA USING LINEAR, QUADRATIC, AND EXPONENTIAL MODELS

Including, but not limited to:

• Mathematical and real-world problem situations modeled by linear, quadratic, and exponential functions and equations
• Predictions, decisions, and critical judgments from function models
• Justification of reasonableness of solutions in terms of mathematical and real-world problem situations
• Mathematical justification
• Substitution in original problem
• Justification for predictions using the coefficient of determination, r2

Note(s):

• Algebra I introduced the linear, quadratic, and exponential functions.
• Algebra I introduced the correlation coefficient as a measure of the strength of linear association.
• Algebra I applied linear, quadratic, and exponential functions to model and make predictions in real-world problem situations.
• 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
• A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g. polynomials, radicals, rational expressions).
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• A2 – Recognize and distinguish between different types of functions.
• B1 – Understand and analyze features of a function.
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections