 Hello, Guest!
 TITLE : Unit 05: Proportional and Non-Proportional Functions SUGGESTED DURATION : 17 days

#### Unit Overview

Introduction
This unit bundles student expectations that address problems involving proportional and non-proportional situations, direct variation, identifying functions, saving for college, and the effect of long-term investments. 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 Unit 04, students used similar right triangles to develop an understanding of slope. Students determined that the ratio of the change in y-values and x-values is the same for any two points on the same line. Students used data from a table or graph to determine the rate of change, or slope, and the y-intercept.

During this Unit
Students extend their previous understandings of slope and y-intercept to represent proportional and non-proportional linear situations with tables, graphs, and equations. These representations are used as students distinguish between proportional and non-proportional linear situations. Students specifically examine the relationship between the unit rate and slope of a line that represents a proportional linear situation. Problem situations involving direct variation are included within this unit as they are also proportional linear situations. Graphical representations of linear equations are examined closely as students begin to develop the understandings of systems of equations. Students are expected to identify the values of x and y that simultaneously satisfy two linear equations in the form y = mx + b from the intersections of the graphed equations. Students must also verify these values algebraically with the equations that represent the two graphed linear equations. The study of proportional and non-proportional linear situations allows students to enrich their understanding of financial situations by explaining how small amounts of money, without interest, invested regularly grow over time. Students also examine how periodic savings plans can be used to contribute to the cost of attending a two-year or four-year college after estimating the financial costs associated with obtaining a college education. Exploring money invested over time helps students begin to consider the benefits of savings for retirement. Students are formally introduced to functions as a relation in which each element of the input (x) is paired with exactly one element of the output (y). Students must identify functions using sets of ordered pairs, tables, mappings, and graphs. Examining proportional and non-proportional linear relationships is extended to include identifying proportional and non-proportional linear functions in mathematical and real-world problems. A deep understanding of the characteristics of functions is essential to future mathematics coursework beyond Grade 8.

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

After this Unit
In Unit 06, students will continue to examine proportional and non-proportional linear relationships as they contrast bivariate sets of data that suggest a linear relationship with bivariate sets of data that do not suggest a linear relationship from a graphical representation. Students construct scatterplots to address questions of association and make predictions from trend lines that approximate the linear relationship between bivariate sets of data. In Algebra I, students will write linear equations in two variables in various forms and when given a table of values, a graph, and a verbal description. Students will write linear equations in two variables in various forms, including y = mx + b, Ax + By = C, and yy1 = m(xx1), given one point and the slope and given two points. In addition, students will write and solve equations involving direct variation and calculate the rate of change of a linear function represented tabularly, graphically, or algebraically in context of mathematical and real-world problems. Students will also graph systems of two linear equations in two variables on the coordinate plane and determine the solutions if they exist, solve systems of linear equations using concrete models, graphs, tables, and algebraic methods, and estimate graphically the solutions to systems of two linear equations with two variables in real-world problems. In Unit 11, students will further study financial planning.

In Grade 8, graphing proportional relationships while interpreting the unit rate as the slope of the line that models the relationship and identifying functions using sets of ordered pairs, tables, mappings, and graphs is identified as STAAR Readiness Standards 8.4B and 8.5G. Representing and distinguishing between linear proportional and non-proportional situations with tables, graphs, and equations are identified as STAAR Supporting Standards 8.5A, 8.5B, and 8.5F. Solving problems involving direct variation and identifying examples of proportional and non-proportional functions are addressed as STAAR Supporting Standards 8.5E and 8.5H. All of these standards are subsumed under the Grade 8 Reporting Category 2: Computations and Algebraic Relationships as well as the Grade 8 Texas Response to Curriculum Focal Points (TxRCFP): Representing, Applying and Analyzing Proportional Relationships. STAAR Supporting Standard 8.9A is described as identifying and verifying the values of x and y that simultaneously satisfy two linear equations in the form y = mx + b from the intersections of the graphed equations and is part of the Grade 8 Reporting Category 2: Computations and Algebraic Relationships as well as Grade 8 Focal Point: Using Expressions and Equations to Describe Relationships, Including the Pythagorean Theorem (TxRCFP). Explaining how small amounts of money invested regularly, including money saved for college and retirement, grow over time as well as estimating the cost of a two-year and four-year college education and devising a periodic savings plan for accumulating the money needed to contribute to the total cost of attendance for at least the first year of college are identified as STAAR Supporting Standards 8.12C and 8.12G. 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 B1; II. Algebraic Reasoning A1, C1, D1, D2; V. Statistical Reasoning A1, C2; VI. Functions A1, A2, B2, 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 Van de Walle, Karp, and Bay-Williams (2010), “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). According to research published by the Institute of Education Sciences (2009), “A recent survey of algebra teachers associated with the report identified key deficiencies of students entering algebra, including aspects of whole number arithmetic, fractions, ratios, and proportions” (p. 4). The National Council of Teachers of Mathematics (2010) states that “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. 5).

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 judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life? 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 how two quantities vary together (covariation) in situations involving invariant (constant) relationships builds flexible proportional reasoning in order to make predictions and critical judgements about the relationship.
• The unit rate can be determined from the graph of a proportional relationship and used to describe the constant rate of change, the slope of the line.
• How can a graph of a proportional relationship be used to interpret the slope of a line and the unit rate?
• Proportional and non-proportional relationships can be presented using multiple representations, and those representations can be examined to distinguish between linear and non-linear proportional situations and identify attributes of linear relationships.
• What are the characteristics of a linear proportional situation in a(n) …
• table?
• graph?
• equation in the form of y = kx?
• What is the relationship between the slope of a line, the constant of proportionality, and the unit rate of a situation that represents a linear proportional relationship?
• How can the equation of a linear proportional situation be manipulated to prove that the constant of proportionality exists within the relationship?
• What is the relationship between the constant of proportionality and problems involving direct variation or linear proportional situations?
• How is the process of identifying the slope or y-intercept from a problem involving direct variation similar to the process of identifying the slope or y-intercept from a problem involving a linear proportional situation?
• What are the key characteristics of a linear proportional and non-proportional situations?
• What are the similarities and differences between the …
• graphs
• tables
• equations
… of a linear proportional and linear non-proportional situations?
• Understanding how investments grow over time and saving for college and retirement helps one make informed financial management decisions, which promotes a more secured financial future.
• What is the long-term effect of small amounts of money invested regularly over time?
• How does the length of time money stays in a savings account or investment affect the balance?
• How does understanding investing money over time promote a more secured financial future?
• What are some various savings options for college and what are their benefits?
• Why would investing a small amount of money over time be more favorable than investing one lump sum for college and/or retirement?
• What factors must be considered when estimating the school related costs of a two-year or four-year college education?
• What is the effect on the total cost of a college education if an individual selects a(n) …
• in-state college versus an out-of-state college?
• public versus private college?
• What are the benefits of a periodic savings plan to pay for college?
• What is the process to devise a periodic savings plan, with and without family contributions, to accumulate money needed to pay for the cost of attending the first year of college?
• How does understanding planning for college promote a more secured financial future?
• Proportionality
• Ratios and Rates
• Unit rates
• Slope
• Relationships and Generalizations
• Equivalence
• Direct variation
• Constant of proportionality
• Linear proportional
• Linear non-proportional
• Representations
• Personal Financial Literacy
• College
• Cost
• Payment options
• Savings plans
• Interest
• Investments
• Financial Responsibility
• 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.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding how two quantities vary together (covariation) in situations involving invariant (constant) relationships builds flexible proportional reasoning in order to make predictions and critical judgements about the relationship.
• Proportional and non-proportional relationships can be presented using multiple representations, and those representations can be examined to distinguish between linear and non-linear proportional situations and identify attributes of linear relationships, all of which develops foundational concepts of functions.
• What are the key characteristics of a linear proportional and non-proportional situations?
• What are the similarities and differences between the …
• graphs
• tables
• equations
… of a linear proportional and linear non-proportional situations?
• What are the key characteristics of linear proportional and linear non-proportional …
• situations?
• functions?
• What are the similarities and differences between the …
• graphs
• tables
• equations
… of linear proportional and linear non-proportional …
• situations?
• functions?
• What are some examples of …
• proportional functions
• non-proportional functions
… that arise from …
• mathematical problems?
• real-world problems?
• What are the characteristics of a function?
• How can sets of …
• ordered pairs
• tables
• mappings
• graphs
… be used to determine if a relationship is a function?
• The ability to visualize the point of intersection of two simultaneously graphed linear equations aids in understanding how the point satisfies both equations, which introduces solutions to systems of equations.
• When two linear equations in the form of y = mx + b are graphed and the graphs intersect at one point, what does the point of intersection represent?
• What does the ordered pair at the point of intersection of two graphed linear equations in the form of y = mx + b represent in terms of the input and output values?
• How can the ordered pair at the point of intersection of two graphed linear equations in the form of y = mx + b be verified algebraically?
• What generalization can be made about the values of x and y that simultaneously satisfy two linear equations?
• Proportionality
• Relationships and Generalizations
• Proportional
• Linear proportional
• Linear non-proportional
• Relations
• Functions
• Proportional functions
• Non-proportional functions
• Representations
• Expressions, Equations, and Relationships
• Algebraic Relationships
• Linear
• Numeric and Algebraic Representations
• Equations
• 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.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may not relate the constant rate of change or unit rate to m in the equation y = mx + b.
• Some students may not relate the constant of proportionality or unit rate as k in the equation y = kx or m in the equation y = mx + b, when b = 0.
• Some students may think that a constant rate of change always means the situation is proportional.
• Some students may not associate slope represented as whole number as a rational number that can be represented as .
• Some students may think that a function can have multiple outputs (y) for the same input (x).
• Some students may think that a function cannot have multiple inputs (x) that correspond to the same output (y).

Underdeveloped Concepts:

• Some students may think that the slope in a linear relationship is , since the x coordinate (horizontal) always comes before the y coordinate (vertical) in an ordered pair. Instead, the correct representation of slope in a linear relationship is .
• Some students may think that the intercept coordinate is the zero term instead of the non-zero term, since intercepts are associated with zeros. In other words, students may think (0, 4) would be the x-intercept because the 0 is in the x coordinate.
• Students may not graph lines correctly on the coordinate plane.
• Students may use (y, x) as the ordered pair instead of (x, y).

#### Unit Vocabulary

• 401(k) – a set amount of money, or percentage of pay, that is set aside from an employee’s pay check by their employer, before the employee’s wages are taxed. The employer may or may not contribute as well to the employee’s 401(k) fund depending on employer’s policy. The money is taxed when it is withdrawn at retirement age. In addition, if withdrawn prior to retirement age, an additional penalty tax is assessed.
• 403(b) – a set amount of money, or percentage of pay, that is set aside from an employee’s pay check by their employer, before the employee’s wages are taxed. The money is taxed when it is withdrawn at retirement age. In addition, if withdrawn prior to retirement age, an additional penalty tax is assessed.
• 529 account – educational savings account managed by the state, and is usually tax-deferred
• Annuity – deductible and non-deductible contributions may be made, taxes may be waived if used for higher education
• 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
• Function – relation in which each element of the input (x) is paired with exactly one element of the output (y)
• Individual retirement account (IRA) – a set amount of money, or percentage of pay, that is invested by an individual with a bank, mutual fund, or brokerage
• Inflation – the general increase in prices and decrease in the purchasing value of money
• Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
• Mappings – the process of pairing input and output in a function. Mappings are usually demonstrated by a diagram consisting of two lists, usually in ovals, with arrows associating items from the first list to items in the second list.
• Principal of an investment – the original amount invested
• Relation – a set of ordered pairs (x, y) where the x is associated with a y
• Retirement savings – optional savings plans or accounts to which the employer can make direct deposits of an amount deducted from the employee's pay at the request of the employee
• Savings account – a bank or credit union account in which the money deposited earns interest so there will be more money in the future than originally deposited
• Slope – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or or , denoted as m in y = mx + b
• Social Security – a percentage of an employee's pay required by law that the employer withholds from the employee's pay for social security savings which is deposited into the federal retirement system; payment toward that employee's eventual retirement; the employer also is required to pay a matching amount for the employee into the federal retirement system
• Taxable investment account – many companies will create an investment portfolio with the specific purpose of saving and building a strong portfolio to be used to pay for college
• Traditional savings accounts – money put into a savings account much like paying a monthly expense such as a light bill or phone bill
• U.S. savings bond – money saved for a specific length of time and guaranteed by the federal government
• Unit rate – a ratio between two different units where one of the terms is 1
• y-intercepty coordinate of a point at which the relationship crosses the y-axis meaning the x coordinate is equal to zero, denoted as b in y = mx + b and the ordered pair (0, b)

Related Vocabulary:

 Constant of proportionality Interest Intersection Investment Linear Mapping Non-proportional Origin Ordered pair Portfolio Proportional Rate of change Retirement Rise Room and board Run Tuition x-axis x coordinate x-value y-axis y coordinate y-value
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 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 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
• 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, 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.
• 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.
8.4 Proportionality. The student applies mathematical process standards to explain proportional and non-proportional relationships involving slope. The student is expected to:
8.4B Graph proportional relationships, interpreting the unit rate as the slope of the line that models the relationship.

Graph

PROPORTIONAL RELATIONSHIPS, INTERPRETING THE UNIT RATE AS THE SLOPE OF THE LINE THAT MODELS THE RELATIONSHIP

Including, but not limited to:

• Unit rate – a ratio between two different units where one of the terms is 1
• Slope – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or or , denoted as m in y = mx + b
• 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
• Graphing unit rate from various representations
• Verbal
• Numeric
• Tabular(horizontal/vertical)
• Symbolic/algebraic
• Connections between unit rate in proportional relationships to the slope of a line

Note(s):

• Algebra I will calculate the rate of change of a linear function represented tabularly, graphically, or algebraically in context of mathematical and real-world problems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• TxCCRS:
• VI.C. Functions – Model real-world situations with functions
• VI.C.2. Develop a function to model a situation.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• 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.
8.5 Proportionality. The student applies mathematical process standards to use proportional and non-proportional relationships to develop foundational concepts of functions. The student is expected to:
8.5A Represent linear proportional situations with tables, graphs, and equations in the form of y = kx.
Supporting Standard

Represent

LINEAR PROPORTIONAL SITUATIONS WITH TABLES, GRAPHS, AND EQUATIONS IN THE FORM OF y = kx

Including, but not limited to:

• Slope – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or or , denoted as m in y = mx + b
• y-intercept – y coordinate of a point at which the relationship crosses the y-axis meaning the x coordinate is equal to zero, denoted as b in y = mx + b and the ordered pair (0, b)
• Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
• One quantity is dependent on the other
• Two quantities may be directly proportional to each other
• Can be classified as a positive or negative relationship
• Can be expressed as a pair of values that can be graphed as ordered pairs
• Graph of the ordered pairs matching the relationship will form a line
• Linear proportional problem situations
• 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.
• Multiple representations of linear proportional problem situations
• Verbal
• Table (horizontal/vertical)
• Graph
• Algebraic
• Both y = kx and kx = y forms
• Association of k as multiplication by a given constant factor (including unit rate)
• Rational number coefficients and constants
• Manipulation of equations

Note(s):

• Grade 7 represented constant rates of change in mathematical and real-world problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt.
• Grade 7 converted between measurement systems, including the use of proportions and the use of unit rates.
• Algebra I will write and solve equations involving direct variation.
• Algebra I will use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions.
• Algebra I will write linear equations with two variables given a table of values, a graph, and a verbal description.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• 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.
• 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.
8.5B Represent linear non-proportional situations with tables, graphs, and equations in the form of y = mx + b, where b  0.
Supporting Standard

Represent

LINEAR NON-PROPORTIONAL SITUATIONS WITH TABLES, GRAPHS, AND EQUATIONS IN THE FORM OF y = mx + b, WHERE b ≠ 0

Including, but not limited to:

• Slope – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or or , denoted as m in y = mx + b
• y-intercept – y coordinate of a point at which the relationship crosses the y-axis meaning the x coordinate is equal to zero, denoted as b in y = mx + b and the ordered pair (0, b)
• Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
• One quantity is dependent on the other
• Two quantities may be directly proportional to each other
• Can be classified as a positive or negative relationship
• Can be expressed as a pair of values that can be graphed as ordered pairs
• Graph of the ordered pairs matching the relationship will form a line
• Linear non-proportional problem situations
• Linear
• Does not pass through the origin (0, 0)
• Represented by y = mx + b, where b ≠ 0
• Constant slope represented as m = or m = or m = • Multiple representations of linear non-proportional problem situations
• Verbal
• Table (horizontal/vertical)
• Graph
• Algebraic
• Both y = mx + b and mx + b = y forms
• Rational number coefficients and constants
• Manipulation of equations

Note(s):

• Grade 7 represented linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b.
• Algebra I will write linear equations in two variables in various forms, including y = mx + b, Ax + By = C, and yy1 = m(xx1), given one point and the slope and given two points.
• Algebra I will use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions.
• Algebra I will write linear equations with two variables given a table of values, a graph, and a verbal description.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• 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.
8.5E Solve problems involving direct variation.
Supporting Standard

Solve

PROBLEMS INVOLVING DIRECT 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
• 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.
• Recognition that when a problem situation has two variable quantities with a constant ratio, then the variable quantities have a relationship reflecting direct variation.
• Ratio, k, represents constant of variation or constant of proportionality
• Direct variation can also be phrased as direct proportion or directly proportional.
• Problem situations (e.g., circumference, conversions, unit rates, similarity, percents, etc.) involving prediction and comparison with justification
• Various solution methods for solving problems involving direct variation
• Table (horizontal/vertical)
• Graph
• Algebraic

Note(s):

• Grade 7 determined the constant of proportionality ( ) within mathematical and real-world problems.
• Algebra I will write and solve equations involving direct variation.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships.
• TxCCRS:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• VI.B. Functions – Analysis of functions
• VI.B.2. Algebraically construct and analyze new functions.
• 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.
• 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.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
8.5F Distinguish between proportional and non-proportional situations using tables, graphs, and equations in the form y = kx or y = mx + b, where b  0.
Supporting Standard

Distinguish

BETWEEN PROPORTIONAL AND NON-PROPORTIONAL SITUATIONS USING TABLES, GRAPHS, AND EQUATIONS IN THE FORM y = kx OR y = mx + b, WHERE b ≠ 0

Including, but not limited to:

• Slope – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or or , denoted as m in y = mx + b
• y-intercept – coordinate of a point at which the relationship crosses the y-axis meaning the x coordinate is equal to zero, denoted as b in y = mx + b and the ordered pair (0, b)
• Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
• One quantity is dependent on the other
• Two quantities may be directly proportional to each other
• Can be classified as a positive or negative relationship
• Can be expressed as a pair of values that can be graphed as ordered pairs
• Graph of the ordered pairs matching the relationship will form a line
• 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.
• Linear non-proportional relationship
• Linear
• Does not pass through the origin (0, 0)
• Represented by y = mx + b, where b ≠ 0
• Constant slope represented as m = or m = or m = • Various representations
• Table (horizontal/vertical)
• Graph
• Equation

Note(s):

• Grade 7 represented linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = kx.
• Algebra I will write linear equations in two variables given a table of values, a graph, and a verbal description.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships.
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• VI.A. Functions – Recognition and representation of functions
• VI.A.2. Recognize and distinguish between different types of functions.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
8.5G Identify functions using sets of ordered pairs, tables, mappings, and graphs.

Identify

FUNCTIONS USING SETS OF ORDERED PAIRS, TABLES, MAPPINGS, AND GRAPHS

Including, but not limited to:

• Relation – a set of ordered pairs (x, y) where the x is associated with a specific y
• Function – relation in which each element of the input (x) is paired with exactly one element of the output (y)
• Distinguish between relations and functions
• All functions are relations but not all relations are functions.
• Various representations
• Sets of ordered pairs
• Tables (horizontal/vertical)
• Mappings – the process of pairing input and output in a function. Mappings are usually demonstrated by a diagram consisting of two lists, usually in ovals, with arrows associating items from the first list to items in the second list.
• Visual representation of a relation or a pairing of inputs with outputs
• Arrows connect inputs to corresponding outputs
• Can be used to quickly determine if a relation is a function
• Graphs
• Vertical line test can be used to determine if a relation is a function when graphing

Note(s):

• Grade 8 introduces identifying functions using sets of ordered pairs, tables, mappings, and graphs.
• Algebra I will introduce function notation.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• TxCCRS:
• VI.A. Functions – Recognition and representation of functions
• VI.A.1. Recognize if a relation is a function.
8.5H Identify examples of proportional and non-proportional functions that arise from mathematical and real-world problems.
Supporting Standard

Identify

EXAMPLES OF PROPORTIONAL AND NON-PROPORTIONAL FUNCTIONS THAT ARISE FROM MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Slope – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or or , denoted as m in y = mx + b
• y-intercept – y coordinate of a point at which the relationship crosses the y-axis meaning the x coordinate is equal to zero, denoted as b in y = mx + b and the ordered pair (0, b)
• Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
• One quantity is dependent on the other
• Two quantities may be directly proportional to each other
• Can be classified as a positive or negative relationship
• Can be expressed as a pair of values that can be graphed as ordered pairs
• Graph of the ordered pairs matching the relationship will form a line
• Function – relation in which each element of the input (x) is paired with exactly one element of the output (y)
• Linear proportional function
• 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.
• Linear non-proportional function
• Linear
• Does not pass through the origin (0, 0)
• Represented by y = mx + b, where b ≠ 0
• Constant slope represented as m = or m = or m = • Various representations
• Verbal
• Table (horizontal/vertical)
• Graph
• Equation
• Generalizations about functions and linear proportional and linear non-proportional relationships in mathematical and real-world problem situations
• All linear proportional and linear non-proportional relationships are functions.
• Not all functions are linear proportional or linear non-proportional functions.
• Not all linear relationships are functions.

Note(s):

• Grade 8 introduces examples of proportional and non-proportional functions that arise from mathematical and real-world problems.
• Algebra I will introduce function notation.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• VI.A. Functions – Recognition and representation of functions
• VI.A.2. Recognize and distinguish between different types of functions.
• 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.
8.9 Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to develop foundational concepts of simultaneous linear equations. The student is expected to:
8.9A Identify and verify the values of x and y that simultaneously satisfy two linear equations in the form y = mx + b from the intersections of the graphed equations.
Supporting Standard

Identify, Verify

THE VALUES OF x AND y THAT SIMULTANEOUSLY SATISFY TWO LINEAR EQUATIONS IN THE FORM y = mx + b FROM THE INTERSECTIONS OF THE GRAPHED EQUATIONS

Including, but not limited to:

• Slope – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or or , denoted as m in y = mx + b
• y-intercept – y coordinate of a point at which the relationship crosses the y-axis meaning the x coordinate is equal to zero, denoted as b in y = mx + b and the ordered pair (0, b)
• Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
• Linear proportional relationship
• Linear
• Represented by y = kx or y = mx + b, where b = 0
• For y = kx and y = mx + b, k = the slope, m
• Passes through the origin (0, 0), meaning the y-intercept, b, is 0
• Constant of proportionality represented as • Constant slope represented as m = or m = or m = • Linear non-proportional relationship
• Linear
• Represented by y = mx + b, where b ≠ 0
• Does not pass through the origin (0, 0), meaning the y-intercept, b, is not 0
• Constant slope represented as m = or m = or m = • Values of x and y that simultaneously satisfy two linear equations from a graph
• Simultaneously satisfy both linear equations means the intersection point or solution will lie on both lines
• Algebraic verification of the intersection of graphed equations as ordered pairs
• Intersection point or solution, when substituted into each equation, will result in true equations. If both equations are true, then the point of intersection simultaneously satisfies both equations.
• Intersection point and any other points on the same line result in equivalent slopes. If two lines contain the same point, then the point simultaneously satisfies both equations.

Note(s):

• Grade 7 determined if the given value(s) make(s) one-variable, two-step equations and inequalities true.
• Algebra I will graph systems of two linear equations in two variables on the coordinate plane and determine the solutions if they exist.
• Algebra I will solve systems of linear equations using concrete models, graphs, tables, and algebraic methods.
• Algebra I will estimate graphically the solutions to systems of two linear equations with two variables in real-world 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.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
• II.C.1. Describe and interpret solution sets of equalities 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.A.4. Justify the solution.
• 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.
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:
8.12C Explain how small amounts of money invested regularly, including money saved for college and retirement, grow over time.
Supporting Standard

Explain

HOW SMALL AMOUNTS OF MONEY INVESTED REGULARLY, INCLUDING MONEY SAVED FOR COLLEGE AND RETIREMENT, GROW OVER TIME

Including, but not limited to:

• Principal of an investment – the original amount invested
• Various types of investments
• Savings account – a bank or credit union account in which the money deposited earns interest so there will be more money in the future than originally deposited
• Traditional savings account – money put into a savings account much like paying a monthly expense such as a light bill or phone bill
• Taxable investment account – many companies will create an investment portfolio with the specific purpose of saving and building a strong portfolio to be used to pay for college
• Annuity – deductible and non-deductible contributions may be made, taxes may be waived if used for higher education; sold by financial institutions
• U.S. savings bond – money saved for a specific length of time and guaranteed by the federal government
• 529 account – educational savings account managed by the state, and is usually tax-deferred
• Retirement savings – optional savings plans or accounts to which the employer can make direct deposits of an amount deducted from the employee's pay at the request of the employee
• 401(k) – a set amount of money, or percentage of pay, that is set aside from an employee’s pay check by their employer, before the employee’s wages are taxed. The employer may or may not contribute as well to the employee’s 401(k) fund depending on employer’s policy. The money is taxed when it is withdrawn at retirement age. In addition, if withdrawn prior to retirement age an additional penalty tax is assessed.
• 403(b) – a set amount of money, or percentage of pay, that is set aside from an employee’s pay check by their employer, before the employee’s wages are taxed. The money is taxed when it is withdrawn at retirement age. In addition, if withdrawn prior to retirement age an additional penalty tax is assessed.
• Similar to a 401(k); however, 403(b) plans are offered by non-profit organizations
• Individual retirement account (IRA) – a set amount of money, or percentage of pay, that is invested by an individual with a bank, mutual fund, or brokerage.
• Social Security – a percentage of an employee's pay required by law that the employer withholds from the employee's pay for social security savings which is deposited into the federal retirement system; payment toward that employee's eventual retirement; the employer also is required to pay a matching amount for the employee into the federal retirement system.
• Generalizations of investing money regularly, including money for college and retirement
• Small amounts of money invested regularly build a larger principal amount to earn more interest
• A small amount of money invested for a longer period of time has the potential to earn as much interest as one large lump sum investment.
• Investing small amounts of money regularly may be more manageable for most people and demonstrates long-term financial planning and responsibility.

Note(s):

• Grade 7 analyzed and compared monetary incentives, including sales, rebates, and coupons.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Financial Literacy
• TxCCRS:
• 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.
8.12G Estimate the cost of a two-year and four-year college education, including family contribution, and devise a periodic savings plan for accumulating the money needed to contribute to the total cost of attendance for at least the first year of college.
Supporting Standard

Estimate

THE COST OF A TWO-YEAR AND FOUR-YEAR COLLEGE EDUCATION, INCLUDING FAMILY CONTRIBUTION

Including, but not limited to:

• Various considerations for each college
• School related costs
• Tuition (in state or out of state)
• Fees
• Room and board
• Books
• Cost of living in location (various costs of living depending on the city and state of college)
• Inflation – the general increase in prices and decrease in the purchasing value of money
• When planning ahead of time for college savings, the increase in all expenses based on inflation must be considered (e.g., tuition, room and board, etc.)
• Family contribution

Devise

A PERIODIC SAVINGS PLAN FOR ACCUMULATING THE MONEY NEEDED TO CONTRIBUTE TO THE TOTAL COST OF ATTENDANCE FOR AT LEAST THE FIRST YEAR OF COLLEGE

Including, but not limited to:

• Periodic savings plan
• Accumulating money to contribute to a savings plan
• Savings account – a bank or credit union account in which the money deposited earns interest so there will be more money in the future than originally deposited
• 529 account – educational savings account managed by the state, and is usually tax-deferred
• Family contribution
• Plan for saving for college
• Estimate the total cost of attendance for each year at the college
• Determine what, if any, family contributions will be received
• Determine if a savings account was established to pay for college
• Calculate the cost of attending college and subtract the amount saved or contributed to determine yearly or monthly payments toward a college savings plan
• Creation of a budget to include a savings plan to cover the cost of college

Note(s):

• Grade 6 explained various methods to pay for college, including through savings, grants, scholarships, student loans, and work-study.
• Grade 7 calculated and compared simple and compound interest earnings.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Financial Literacy
• TxCCRS:
• 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.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions. 