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 TITLE : Unit 03: One-Variable Equations, Inequalities, and their Applications SUGGESTED DURATION : 13 days

#### Unit Overview

Introduction
This unit bundles student expectations that address modeling, writing, and solving one variable equations with variables on both sides of the equality sign; writing inequalities with variables on both sides of the equality sign; writing a corresponding real-world problem when given a one-variable equation or inequality with variables on both sides of the equality sign; and solving problems comparing interest rates on loans and savings accounts, loan lengths, and total cost of repaying a loan. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, 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. The introduction to the grade level standards state, “While the use of all types of technology is important, the emphasis on algebra readiness skills necessitates the implementation of graphing technology.” Additionally, the availability of graphing technology is required during STAAR testing.

Prior to this Unit
In Grade 6, students distinguished between debit cards and credit cards and explained why it is important to establish a positive credit history. In Grade 7, students modeled and solved one-variable, two-step equations and inequalities; wrote one-variable, two-step equations and inequalities to represent constraints or conditions within problems; wrote corresponding real-world problems given a one-variable, two-step equation or inequality; and calculated and compared simple interest and compound interest earnings.

During this Unit
Students extend their understanding of modeling and solving one-variable equations that represent mathematical and real-world problems from variables on one-side of the equality sign to variables on both sides of the equality sign using rational number coefficients and constants. When solving one-variable equations with variables on both sides of the equality sign, students distinguish between types of solutions as one solution, no solution, and infinite solutions (all real numbers). Students also extend their knowledge of writing one-variable equations or inequalities from variables on one-side of the equality sign to variables on both sides of the equality sign to represent problems using rational number coefficients and constants. Financial literacy contexts, such as calculating and comparing simple and compound interest rates and how those rates affect earnings in a savings account or the total cost of repaying a loan or credit card, are embedded in this unit.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool Grade 8

After this Unit
In Unit 11, students will explain how small amounts of money invested regularly, including money saved for college and retirement, grow over time. In Algebra I, students will solve linear inequalities with one variable, including those for which the application of the distributive property is necessary and in which variables are included on both sides.

In Grade 8, writing one-variable equations or inequalities with variables on both sides that represent problems using rational number coefficients and constants as well as writing a corresponding real-world problem when given a one-variable equation or inequality with variables on both sides of the equal sign using rational number coefficients and constants are STAAR Supporting Standards 8.8A and 8.8B. Modeling and solving one-variable equations with variables on both sides of the equal sign that represent mathematical and real-world problems using rational number coefficients and constants is identified as STAAR Readiness Standard 8.8C. These standards are subsumed under the Grade 8 STAAR Reporting Category 2: Computations and Algebraic Relationships and the Grade 8 Texas Response to Curriculum Focal Points (TxRCFP): Using expressions and equations to describe relationships, including the Pythagorean Theorem. Solving real-world problems comparing how interest rate and loan length affect the cost of credit is STAAR Supporting Standard 8.12A, while calculating and comparing simple interest and compound interest earnings is identified as STAAR Readiness Standard 8.12D. Calculating the total cost of repaying a loan, including credit cards and easy access loans, under various rates of interest and over different periods using an online calculator is standard 8.12B and is neither Supporting nor Readiness, but is foundational to the conceptual understanding of financial literacy. These standards are subsumed under the Grade 8 STAAR Reporting Category 4: Data Analysis and Personal Financial Literacy as well as the Grade 8 Focal Point: Financial Literacy (TxRCFP). This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning A1, B1; II. Algebraic Reasoning A1, C3, D1, D2; V. Statistical Reasoning A1, 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 National Council of Teachers of Mathematics (2010), students “can benefit from productive time spent in grade 8 further developing their use of tables, graphs, and equations, along with properties of arithmetic, to represent and find solutions to problems. A strong general mathematics course in grade 8, focused on building students’ skills in using symbols to represent their mathematical thinking, is essential for increasing these students’ readiness for algebra in high school” (p. 8). Research published by the Institute of Education Sciences (2009) notes that, “A major problem for students who struggle with mathematics is weak understanding of the relationships between the abstract symbols of mathematics and the various visual representations. Student understanding of these relationships can be strengthened through the use of visual representations of mathematical concepts such as solving equations” (p. 30). Van de Walle, Karp & Bay-Williams (2010) note that, “Algebraic thinking or algebraic reasoning involves forming generalizations from experiences with number and computation, formalizing these ideas with the use of a meaningful symbol system, and exploring the concepts of pattern and functions. Far from a topic with little real-world use, algebraic thinking pervades all of mathematics and is essential for making mathematics useful in daily life” (p. 254).

Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. (2009). Assisting students struggling with mathematics: Response to intervention (RtI) for elementary and middle schools (NCEE 2009-4060). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/publications/practiceguides/
National Council of Teachers of Mathematics. (2010). Focus in grade 8 teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013
Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. Boston, MA: Pearson Education, Inc.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgments 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)
• Equations and inequalities can be modeled and written and equations can be solved using various methods to gain insight into the context of the situation and make critical judgments about algebraic relationships and efficient strategies.
• Why are expressions considered foundational to equations and inequalities?
• How are constraints or conditions within a problem situation represented in an …
• equation?
• inequality?
• How does the context of a problem situation, relationships within and between operations, and properties of operations aid in writing an equation and/or inequality to represent the problem situation?
• What is the process for writing a real-world problem to represent constraints or conditions within an equation or inequality?
• How can a(n) …
• concrete model
• pictorial model
• algebraic representation
… be used to represent and solve an equation?
• How can a(n)
• concrete model
• pictorial model
• algebraic representation
… be used to represent an inequality?
• What models effectively and efficiently represent how to solve equations?
• What is the process for solving an equation with variables on both sides, and how can the process be …
• described verbally?
• represented algebraically?
• When considering equations, …
• why is the variable isolated in order to solve?
• how are negative values represented in concrete and pictorial models?
• why must the solution be justified in terms of the problem situation?
• why does equivalence play an important role in the solving process?
• Why is it important to understand when and how to use standard algorithms?
• How does knowing more than one solution strategy build mathematical flexibility?
• Expressions, Equations, and Relationships
• Numeric and Algebraic Representations
• Expressions
• Equations
• Inequalities
• Equivalence
• Operations
• Properties of operations
• Order of operations
• Representations
• 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.

 Financial and economic knowledge leads to informed and rational decisions allowing for effective management of financial resources when planning for a lifetime of financial security.  Why is financial stability important in everyday life? What economic and financial knowledge is critical for planning for a lifetime of financial security? How can mapping one’s financial future lead to significant short and long-term benefits? How can current financial and economic factors in everyday life impact daily decisions and future opportunities?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding loans, repayment of loans, interest rates, and savings helps one make informed financial management decisions, which promotes a more secured financial future.
• How does interest rate on a …
• loan affect the total cost of the loan?
• savings account or investment affect the balance?
• How does understanding interest rate and loan length affect the cost of credit promote a more secured financial future?
• How does the interest rate and the length of time it takes to pay off a loan or money borrowed on credit affect the …
• monthly payment?
• total cost of the loan?
• What is the purpose of different types of loans and how do they differ in repayment expectations?
• How can an online calculator be used to …
• compare various interest rates?
• compare various lengths of time for loans?
• How does understanding loans and the repayment of loans promote a more secured financial future?
• What is the process for determining …
• simple interest?
• compound interest?
• How does the equation for simple interest differ from the equation of compound interest?
• How does understanding interest rates promote a more secured financial future?
• Personal Financial Literacy
• Cost
• Credit
• Credit Cards
• Financial Responsibility
• Interest
• Simple
• Compound
• Loans and Loan Rates
• Payment Methods
• Savings
• 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 constant term can be combined with a variable term (e.g., 2x + 5 = 7x).
• Some students may think that an inequality can only have one solution value that makes the inequality true.
• Some students may think that whenever all terms with the variable cancel on both sides of the equal sign, then the equation has no solutions instead of possibly having an infinite number of solutions.

Underdeveloped Concepts:

• Some students may think that answers to both equations and inequalities are exact answers instead of correctly identifying the solutions to equations as exact answers and the solutions to inequalities as range of answers.
• Some students may think variables are letters representing an object as opposed to representing a number or quantity of objects.
• Some students may think the equal sign means, “solve this” or “the answer is” rather than understanding that equal sign represents a quantitative and balanced relationship.

#### Unit Vocabulary

• Annual percentage rate (APR) – annual percentage rate applied to the balance on a loan compounded monthly
• Car title loan – a high-interest, short term loan of cash for which an automobile title is required as collateral
• Coefficient – a number that is multiplied by a variable(s)
• Collateral – something which is pledged to secure repayment of a loan; in the event of default on the loan, the collateral is forfeited
• Compound interest for a loan – interest that is calculated on the latest balance, including all compounded interest that has been added to the original loan principal
• Compound interest for an investment – interest that is calculated on the latest balance, including alll compounded earned interest that has been added to the original principal investment
• Constant – a fixed value that does not appear with a variable(s)
• Credit – buying or obtaining goods or services now with an agreement to pay in the future
• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• Inequality – a mathematical statement composed of algebraic and/or numeric expressions set apart by an inequality symbol
• Order of operations – the rules of which calculations are performed first when simplifying an expression
• Payday loan – a high-interest, short term loan that is repaid when the borrower receives their next paycheck
• Principal of a loan – the original amount borrowed
• Principal of an investment – the original amount invested
• Simple interest for a loan – interest that is calculated only on the principal amount of the loan
• Simple interest for an investment – interest paid on the original principal in an account, disregarding any previously earned interest
• Solution set – a set of all values of the variable(s) that satisfy the equation or inequality
• Variable – a letter or symbol that represents a number

Related Vocabulary:

 Annual Constraint Condition Balance Default Equality Equal to Expression Exponent Evaluate Forfeit Greater than Greater than or equal to Infinite solutions Interest rate Integer Less than Less than or equal to No solution Not equal to One solution Parentheses/brackets Rational number Repayment period Simplify Solve Solution Title Whole number
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 – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

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 Grade 8 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.
• Student Expectations (TEKS) are labeled Readiness as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Supporting as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Process standards as identified by TEA of the assessed curriculum.
• 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.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
TEKS# SE# TEKS SPECIFICITY
8.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
8.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
Process Standard

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.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• 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.
8.1B 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.
Process Standard

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.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• 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.
8.1C 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.
Process Standard

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.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• 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.
8.1D

Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Process Standard

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• 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.
8.1E Create and use representations to organize, record, and communicate mathematical ideas.
Process Standard

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.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• 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.
8.1F Analyze mathematical relationships to connect and communicate mathematical ideas.
Process Standard

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.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• 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.
8.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Process Standard

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.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• 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.
8.8 Expressions, equations, and relationships. The student applies mathematical process standards to use one-variable equations or inequalities in problem situations. The student is expected to:
8.8A Write one-variable equations or inequalities with variables on both sides that represent problems using rational number coefficients and constants.
Supporting Standard

Write

ONE-VARIABLE EQUATIONS OR INEQUALITIES WITH VARIABLES ON BOTH SIDES THAT REPRESENT PROBLEMS USING RATIONAL NUMBER COEFFICIENTS AND CONSTANTS

Including, but not limited to:

• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• Inequality – a mathematical statement composed of algebraic and/or numeric expressions set apart by an inequality symbol
• Variable – a letter or symbol that represents a number
• One variable on both sides of the equation or inequality
• Coefficient – a number that is multiplied by a variable(s)
• Integers
• Decimals (positive or negative)
• Fractions (positive or negative)
• Constant – a fixed value that does not appear with a variable(s)
• Integers
• Decimals (positive or negative)
• Fractions (positive or negative)
• Solution set – a set of all values of the variable(s) that satisfy the equation or inequality
• Constraints or conditions
• Distinguishing between equations and inequalities
• Characteristics of equations
• Equates two expressions
• Equality of variable
• One solution
• Characteristics of inequalities
• Shows the relationship between two expressions in terms of >, <, ≥, ≤, or ≠
• Inequality of the variable
• One or more solutions
• Equality and inequality words and symbols
• Equal to, =
• Greater than, >
• Greater than or equal to, ≥
• Less than, <
• Less than or equal to, ≤
• Not equal to, ≠
• Relationship of order of operations within an equation or inequality
•  Order of operations – the rules of which calculations are performed first when simplifying an expression
• Parentheses/brackets: simplify expressions inside parentheses or brackets in order from left to right
• Exponents: rewrite in standard numerical form and simplify from left to right
• Multiplication/division: simplify expressions involving multiplication and/or division in order from left to right
• Addition/subtraction: simplify expressions involving addition and/or subtraction in order from left to right
• One-variable equations with variables on both sides from a problem situation
• One-variable inequalities with variables on both sides from a problem situation

Note(s):

• Grade 7 wrote one-variable, two-step equations and inequalities to represent constraints or conditions within problems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• TxCCRS:
• II.D. Algebraic Reasoning – Representing 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.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.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.
8.8B Write a corresponding real-world problem when given a one-variable equation or inequality with variables on both sides of the equal sign using rational number coefficients and constants.
Supporting Standard

Write

A CORRESPONDING REAL-WORLD PROBLEM WHEN GIVEN A ONE-VARIABLE EQUATION OR INEQUALITY WITH VARIABLES ON BOTH SIDES OF THE EQUAL SIGN USING RATIONAL NUMBER COEFFICIENTS AND CONSTANTS

Including, but not limited to:

• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• Inequality – a mathematical statement composed of algebraic and/or numeric expressions set apart by an inequality symbol
• Variable – a letter or symbol that represents a number
• One variable on both sides of the equation or inequality
• Coefficient – a number that is multiplied by a variable(s)
• Integers
• Decimals (positive or negative)
• Fractions (positive or negative)
• Constant – a fixed value that does not appear with a variable(s)
• Integers
• Decimals (positive or negative)
• Fractions (positive or negative)
• Solution set – a set of all values of the variable(s) that satisfy the equation or inequality
• Constraints or conditions
• Distinguishing between equations and inequalities
• Characteristics of equations
• Equates two expressions
• Equality of the variable
• One solution
• Characteristics of inequalities
• Shows the relationship between to expressions in terms of >, <, ≥, ≤, or ≠
• Inequality of the variable
• One or more solutions
• Equality and inequality words and symbols
• Equal to, =
• Greater than, >
• Greater than or equal to, ≥
• Less than, <
• Less than or equal to, ≤
• Not equal to, ≠
• Relationship of order of operations within an equation or inequality
•  Order of operations – the rules of which calculations are performed first when simplifying an expression
• Parentheses/brackets: simplify expressions inside parentheses or brackets in order from left to right
• Exponents: rewrite in standard numerical form and simplify from left to right
• Multiplication/division: simplify expressions involving multiplication and/or division in order from left to right
• Addition/subtraction: simplify expressions involving addition and/or subtraction in order from left to right
• Corresponding real-world problem situation from a one-variable equation with variables on both sides of the equal sign
• Corresponding real-world problem situation from a one-variable inequality with variables on both sides of the inequality symbol

Note(s):

• Grade 7 wrote corresponding real-world problems given a one-variable, two-step equation or inequality.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• 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.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.
• 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.
8.8C Model and solve one-variable equations with variables on both sides of the equal sign that represent mathematical and real-world problems using rational number coefficients and constants.

Model, Solve

ONE-VARIABLE EQUATIONS WITH VARIABLES ON BOTH SIDES OF THE EQUAL SIGN THAT REPRESENT MATHEMATICAL AND REAL-WORLD PROBLEMS USING RATIONAL NUMBER COEFFICIENTS AND CONSTANTS

Including, but not limited to:

• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• Variable – a letter or symbol that represents a number
• One variable on both sides of the equation
• Coefficient – a number that is multiplied by a variable(s)
• Integers
• Decimals (positive or negative)
• Fractions (positive or negative)
• Constant – a fixed value that does not appear with a variable(s)
• Integers
• Decimals (positive or negative)
• Fractions (positive or negative)
• Characteristics of equations
• Equates two expressions
• Equality of the variable
• One solution
• Equality words and symbol
• Equal to, =
• Relationship of order of operations within an equation
•  Order of operations – the rules of which calculations are performed first when simplifying an expression
• Parentheses/brackets: simplify expressions inside parentheses or brackets in order from left to right
• Exponents: rewrite in standard numerical form and simplify from left to right
• Multiplication/division: simplify expressions involving multiplication and/or division in order from left to right
• Addition/subtraction: simplify expressions involving addition and/or subtraction in order from left to right
• Models to solve one-variable equations with variables on both sides of the equal sign (concrete, pictorial, algebraic)
• Solutions to one-variable equations with variables on both sides of the equal sign from mathematical and real-world problem situations
• Possible solutions
• One real solution
• No solution
• Infinite solutions (all real solutions)

Note(s):

• Grade 7 modeled and solved one-variable, two-step equations and inequalities.
• Algebra I will solve linear equations in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• TxCCRS:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
• II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
• II.D. Algebraic Reasoning – Representing relationships
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.
• 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.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.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.
8.12 Personal financial literacy. The student applies mathematical process standards to develop an economic way of thinking and problem solving useful in one's life as a knowledgeable consumer and investor. The student is expected to:
8.12A Solve real-world problems comparing how interest rate and loan length affect the cost of credit.
Supporting Standard

Solve

REAL-WORLD PROBLEMS COMPARING HOW INTEREST RATE AND LOAN LENGTH AFFECT THE COST OF CREDIT

Including, but not limited to:

• Credit – buying or obtaining goods or services now with an agreement to pay in the future
• Annual percentage rate (APR) – annual percentage rate applied to the balance on a loan compounded monthly
• Principal of a loan – the original amount borrowed
• Interest
• Simple interest for a loan – interest that is calculated only on the principal amount of the loan
• Formula for simple interest from STAAR Grade 8 Mathematics Reference Materials
• I = Prt, where I represents the interest, P represents the principal amount borrowed, r represents the interest rate in decimal form, and t represents the number of years the principal amount is borrowed
• Compound interest for a loan – interest that is computed on the latest balance, including all compounded interest that has been added to the original loan principal
• Formula for compound interest from STAAR Grade 8 Mathematics Reference Material
• A = P(1+ r)t, where A represents the total amount of money borrowed including the principal and accumulated compounded interest, P represents the principal amount borrowed, r represents the interest rate in decimal form, and t represents the number of years the pricipal amount is borrowed
• Longer the repayment period, usually the higher the interest rate
• Longer the repayment period, the lower the monthly payment
• Longer the repayment period, the greater the amount of money repaid over the life of the loan
• Real-world problem situations comparing interest rates, loan length, and/or cost of credit

Note(s):

• Grade 6 distinguished between debit cards and credit cards.
• Grade 7 calculated and compared simple interest and compound interest earnings.
• Grade 8 solves real-world problems comparing how interest rate and loan length affect the cost of credit.
• Algebra I will refer to 1 + r in the compound interest formula, A = P(1 + r)t, as the factor and will be given the variable b.
• Mathematical Models with Applications will introduce analyzing compound interest for multiple compounding periods within a year.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Financial Literacy
• TxCCRS:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• 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.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.
8.12B Calculate the total cost of repaying a loan, including credit cards and easy access loans, under various rates of interest and over different periods using an online calculator.

Calculate

THE TOTAL COST OF REPAYING A LOAN, INCLUDING CREDIT CARDS AND EASY ACCESS LOANS, UNDER VARIOUS RATES OF INTEREST AND OVER DIFFERENT PERIODS USING AN ONLINE CALCULATOR

Including, but not limited to:

• Credit – buying or obtaining goods or services now with an agreement to pay in the future
• Annual percentage rate (APR) – annual percentage rate applied to the balance on a loan compounded monthly
• Principal of a loan – the original amount borrowed
• Collateral – something which is pledged to secure repayment of a loan; in the event of default on the loan, the collateral is forfeited
• Interest
• Simple interest for a loan – interest that is calculated only on the principal amount of the loan
• Formula for simple interest from STAAR Grade 8 Mathematics Reference Materials
• I = Prt, where I represents the interest, P represents the principal amount borrowed, r represents the interest rate in decimal form, and t represents the number of years the principal amount is borrowed
• Compound interest for a loan – interest that is calculated on the latest balance, including all compounded interest that has been added to the original loan principal
• Formula for compound interest from STAAR Grade 8 Mathematics Reference Materials
• A = P(1+ r)t, where A represents the total amount of money borrowed including the pricipal and accumulated compounded interest, P represents the principal amount borrowed, r represents the interest rate in decimal form, and t represents the number of years the principal amount is borrowed
• Longer the repayment period, the higher the interest rate
• Longer the repayment period, the lower the monthly payment
• Longer the repayment period, the higher the effective interest rate
• Various types of loans
• Easy access loan
• Payday loan – a high-interest, short term loan that is repaid when the borrower receives their next paycheck
• Car title loan – a high-interest, short term loan of cash for which an automobile title is required as collateral
• Credit card
• Tend to have higher interest rates than other types of loans
• Various fees may be associated
• Longer the repayment period, the higher the effective interest rate
• Calculates compound interest
• Online calculator to determine the costs of loans

Note(s):

• Grade 5 identified the advantages and disadvantages of different methods of payment, including check, credit card, and electronic payments.
• Grade 6 explained why it is important to establish a positive credit history.
• Grade 7 calculated and compared simple interest and compound interest earnings.
• Grade 8 introduces calculating and comparing interest on simple and compound on loans.
• Algebra I will refer to 1 + r in the compound interest formula, A = P(1 + r)t, as the factor and will be given the variable b.
• Mathematical Models with Applications will introduce analyzing compound interest for multiple compounding periods within a year.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Financial Literacy
• TxCCRS:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• 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.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.
8.12D Calculate and compare simple interest and compound interest earnings.

Calculate, Compare

SIMPLE INTEREST AND COMPOUND INTEREST EARNINGS

Including, but not limited to:

• Principal of an investment – the original amount invested
• Simple interest for an investment – interest paid on the original principal in an account, disregarding any previously earned interest
• Compound interest for an investment – interest that is calculated on the latest balance, including all compounded interest that has been added to the original principal investment
• Formulas for interest from STAAR Grade 8 Mathematics Reference Materials
• Simple interest
• I = Prt, where I represents the interest, P represents the principal amount deposited, r represents the interest rate in decimal form, and t represents the number of years the principal amount is deposited
• Compound interest
• A = P(1+ r)t, where A represents the total accumulated amount including the principal and earned compounded interest, P represents the principal amount deposited, r represents the interest rate in decimal form, and t represents the number of years the principal amount is deposited
• Comparing simple and compound interest earnings

Note(s):

• Grade 7 calculated and compared simple interest and compound interest earnings.
• Grade 8 solves real-world problems comparing how interest rate and loan length affect the cost of credit.
• Algebra I will refer to 1 + r in the compound interest formula, A = P(1 + r)t, as the factor and will be given the variable b.
• Mathematical Models with Applications will introduce analyzing compound interest for multiple compounding periods within a year.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Financial Literacy
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• 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.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. 