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 Instructional Focus DocumentMathematical Models with Applications
 TITLE : Unit 08: Amortization of Loans SUGGESTED DURATION : 10 days

#### Unit Overview

Introduction
This unit bundles student expectations that address loan amortizations for home mortgages, automobile, and other financed purchases. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Grade 6, students described the information within and value of credit reports. In Grade 8, students investigated loans and the total cost of repaying loans. In Algebra I, students were introduced to mathematical formulas and solving for missing variables in the formulas.

During this Unit
Students use formulas and technology to create amortization tables for calculating principal, interest, and balances over time for financed purchases. Students use technology (graphing calculator, Excel, etc.) to investigate and predict number of payments, interest rate, principal value, payment, and final value in connection to home loans, automobile loans and other financed purchases. Students compare buying a home to renting a home and buying a vehicle to leasing a vehicle. Students demonstrate financial literacy and reasoning by presenting their preferences of financing or renting a home and financing or leasing an automobile. Students support their preferences with mathematical understandings from amortization models and other influential factors such as ownership and equity.

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

After this Unit
In Unit 12, students will connect the concepts and extend their analysis in a research project.

Additional Notes
This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning D1, D2; V. Statistical Reasoning A1, C2; VI. Functions C1; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, C2, 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 Connections Standard for Grades 9-12 from the National Council of Teachers of Mathematics (NCTM), “Instructional programs from pre-kindergarten through grade 12 should enable students to:

• recognize and use connections among mathematical ideas;
• understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
• recognize and apply mathematics in contexts outside of mathematics.

When students can see the connections across different mathematical content areas, they develop a view of mathematics as an integrated whole. As they build on their previous mathematical understandings while learning new concepts, students become increasingly aware of the connections among various mathematical topics. As students' knowledge of mathematics, their ability to use a wide range of mathematical representations, and their access to sophisticated technology and software increase, the connections they make with other academic disciplines, especially the sciences and social sciences, give them greater mathematical power” (NCTM, 2000, p. 354).

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics: Connections standard for grades 9-12. 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

 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)
• In order to maintain financial stability, consumers must be knowledgeable in the analysis of loan amortization for financing a purchase to make predictions and budgetary judgments.
• How are formulas used to create an amortization table?
• How can technology be used to generate amortization models?
• How does the use of technology aid in investigating and comparing financed purchases?
• How does understanding loan amortizations when making a financial purchase promote a more secured financial future?
• How does being financially literate in terms of financed purchases protect an individual’s financial stability?
• Mathematical Modeling
• Personal Finance
• Finance and budgeting
• Loan amortization
• Credit options
• 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)
• In order to maintain financial stability, consumers must be knowledgeable in financing, renting, and leasing homes and automobiles to make predictions and budgetary judgments.
• How does the use of technology aid in investigating and comparing financed purchases such as a(n) …
• home?
• automobile?
• How is technology used to generate and analyze amortization models?
• How can amortization models be used to compare …
• the effects of different components of the loan?
• different loan options?
• What are the advantages and disadvantages of …
• buying or renting a home?
• buying or leasing an automobile?
• How does understanding buying, renting, and leasing homes and automobiles promote a more secured financial future?
• Mathematical Modeling
• Personal Finance
• Finance and budgeting
• Loan amortization
• Credit options
• 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 the variable i in the monthly payment formula, A = P, represents the interest instead of the interest divided by 12 months.
• Some students, when using the TVM solver in the graphing calculator, may want to enter the interest in the decimal form instead of entering it in the percent form.

Underdeveloped Concepts:

• Some students may need explicit explanation of iterations in a table using algebraic expressions.
• Some students may forget to convert the interest rate from percent to decimal form when using it in the finance formulas.

#### Unit Vocabulary

• Amortization – a scheduled, fixed installment repayment that includes both principal and interest that is paid to the lender over time until the loan is paid in full
• Home equity – investment value built up in a home by a homeowner determined by the difference in the fair market value of the home and the amount still owed on the mortgage on a home
• Interest on a loan – a percentage of the principal charged over time by the loan holder
• Principal of a loan – the original amount borrowed or the amount still owed on a loan, not including the interest

Related Vocabulary:

 Amortization generators (online) Amortization tables Balance on a loan Equity Financed purchases Lease Maintenance costs Minimum payment Monthly payment formula Mortgage payment Payment on a loan Rent Residual value TVM Solver
Unit Assessment Items System Resources Other Resources

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

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

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

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

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

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

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

Texas Instruments – Graphing Calculator Tutorials

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
TEKS# SE# TEKS SPECIFICITY
M.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
M.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
M.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

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

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.A. Statistical Reasoning – Design a study
• V.A.1. Formulate a statistical question, plan an investigation, and collect data.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.2. Formulate a plan or strategy.
• VII.A.3. Determine a solution.
• VII.A.4. Justify the solution.
• VII.A.5. Evaluate the problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving process.
M.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS
Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
M.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE
Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert [among multiple representations of equations, inequalities, and relationships.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
M.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
M.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
M.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION
Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
M.3 Mathematical modeling in personal finance. The student uses mathematical processes with algebraic formulas, graphs, and amortization modeling to solve problems involving credit. The student is expected to:
M.3A Use formulas to generate tables to display series of payments for loan amortizations resulting from financed purchases.

Use

FORMULAS FOR LOAN AMORTIZATIONS

Including, but not limited to:

• Principal of a loan – the original amount borrowed or the amount still owed on a loan, not including the interest
• Interest on a loan – a percentage of the principal charged over time by the loan holder
• Amortization – a scheduled, fixed installment repayment that includes both principal and interest that is paid to the lender over time until the loan is paid in full
• Amortization formula calculates the monthly payment amount due
• A = P, where A is the monthly payment amount, P is the initial principal, i is the monthly interest rate; and n is the number of monthly payments

To Generate

TABLES TO DISPLAY SERIES OF PAYMENTS FOR LOAN AMORTIZATIONS RESULTING FROM FINANCED PURCHASES

Including, but not limited to:

• Amortization tables display payments and remaining principal on a loan and are generated by using a monthly payment formula and iterative calculations.
• Manually create a table and generate cell values with manual calculations.
• Spreadsheet using calculated monthly payment

Note(s):

• Grade Level Note
• Algebra I studied the exponential parent function f(x) = abx and its characteristics.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• VI.C. Functions – Model real-world situations with functions
• VI.C.1. Apply known functions to model real-world situations.
• VII.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.2. Understand attributes and relationships with inductive and deductive reasoning.
• 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.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.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.3. Know and understand the use of mathematics in a variety of careers and professions.
M.3C Use technology to create amortization models to investigate home financing and compare buying a home to renting a home.

Use

TECHNOLOGY TO CREATE AMORTIZATION MODELS TO INVESTIGATE HOME FINANCING

Including, but not limited to:

• Principal of a loan –  the original amount borrowed or the amount still owed on a loan, not including the interest
• Interest on a loan –  a percentage of the principal charged over time by the loan holder
• Amortization – a scheduled, fixed installment repayment that includes both principal and interest that is paid to the lender of time until the loan is paid in full
• Monthly payment formula: A = P, where A is the monthly payment amount; P is the initial principal; i is the monthly interest rate, and n is the number of monthly payments
• Graphing calculator
• Finance capabilities of the graphing calculator to generate information on interest rates and monthly payments
• Microsoft Excel (or other spreadsheet)
• Amortization generators online

Compare

BUYING A HOME TO RENTING A HOME

Including, but not limited to:

• Costs for renting a home
• Lease agreements, including security and cleaning deposits, length of time, responsibilities of property owner and of renter
• Lease amenities, including use of office machines, concierge service, pool and other recreational facilities, common areas such as meeting rooms or party rooms,  office staff to receive mail or other deliveries in tenant's absence
• Maintenance costs not covered by property owner
• Costs for buying a home
• Costs of acquiring a mortgage and closing costs of home purchase, including down payment, inspection, appraisal, loan closing costs, possible repairs to property
• Mortgage payments, including fixed loan payments, are set for the life of the loan
• Homeowner association fees for neighborhood maintenance and amenities, when applicable
• Maintenance costs (e.g., lawn maintenance equipment, replacement of filters for the HVAC system, repair or replacement of heating/cooling equipment, exterior maintenance such as painting, roof repair or replacement, maintenance or replacement of appliances, etc.)
• Investment advantages, equity in the property that accumulates over time; safety against rising rent over time
• Home equity – investment value built up in a home by a homeowner determined by the difference in the fair market value of the home and the amount still owed on the mortgage on a home
• Comparison of the options of buying and leasing a home will yield various conclusions depending on many personal factors and preferences of each individual, such as length of time one plans to live in a location, ability to secure a mortgage, desire and ability to maintain a property as an owner, etc.

Note(s):

• Grade Level(s)
• Grade 8 calculated 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.
• Grade 8 identified and explained the advantages and disadvantages of different payment methods.
• Algebra I studied the exponential parent function f(x) = abx.
• Mathematical Models with Applications introduces a monthly payment formula and amortization of loans.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• VI.C. Functions – Model real-world situations with functions
• VI.C.1. Apply known functions to model real-world situations.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• 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.2. Understand attributes and relationships with inductive and deductive reasoning.
• 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.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.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.3. Know and understand the use of mathematics in a variety of careers and professions.
M.3D Use technology to create amortization models to investigate automobile financing and compare buying a vehicle to leasing a vehicle.

Use

TECHNOLOGY TO CREATE AMORTIZATION MODELS TO INVESTIGATE AUTOMOBILE FINANCING

Including, but not limited to:

• Principal of a loan – the original amount borrowed or the amount still owed on a loan, not including the interest
• Interest on a loan – a percentage of the principal charged over time by the loan holder
• Amortization – a scheduled, fixed installment repayment that includes both principal and interest that is paid to the lender over time until the loan is paid in full
• Monthly payment formula: A = P, where A is the monthly payment amount, P is the initial principal, i is the monthly interest rate, and n is the number of monthly payments
• Graphing calculator
• Finance capabilities of the graphing calculator to generate information on interest rates and monthly payments.
• Microsoft Excel (or other spreadsheet)
• Amortization generators online

Compare

BUYING A VEHICLE TO LEASING A VEHICLE

Including, but not limited to:

• Total cost for buying a vehicle
• No mileage limits
• Down payment may be required
• Insurance
• Maintenance cost
• Ownership/equity
• Total cost of leasing a vehicle
• Lease mileage limits
• Down payment required
• Insurance
• Maintenance costs
• No ownership/equity
• Comparison of the options of buying and leasing a vehicle yield various conclusions depending on many personal factors and preferences of each individual (e.g., average miles driven, driving habits, frequency of purchasing a vehicle, etc.).

Note(s):

• Grade Level(s)
• Grade 8 calculated 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.
• Grade 8 identified and explained the advantages and disadvantages of different payment methods.
• Algebra I studied parent functions f(x) = x, f(x) = x2, and f(x) = abx.
• Mathematical Models with Applications introduces monthly payment formula and amortization of loans.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• VI.C. Functions – Model real-world situations with functions
• VI.C.1. Apply known functions to model real-world situations.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• 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.2. Understand attributes and relationships with inductive and deductive reasoning.
• 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.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.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.3. Know and understand the use of mathematics in a variety of careers and professions.
ELPS# Subsection C: Cross-curricular second language acquisition essential knowledge and skills. The English Language Proficiency Standards (ELPS), as required by 19 Texas Administrative Code, Chapter 74, Subchapter A, §74.4, outline English language proficiency level descriptors and student expectations for English language learners (ELLs). School districts are required to implement ELPS as an integral part of each subject in the required curriculum. School districts shall provide instruction in the knowledge and skills of the foundation and enrichment curriculum in a manner that is linguistically accommodated commensurate with the student’s levels of English language proficiency to ensure that the student learns the knowledge and skills in the required curriculum.School districts shall provide content-based instruction including the cross-curricular second language acquisition essential knowledge and skills in subsection (c) of the ELPS in a manner that is linguistically accommodated to help the student acquire English language proficiency. Choose appropriate ELPS to support instruction. Click here to collapse or expand this section. ELPS.c.1 The ELL uses language learning strategies to develop an awareness of his or her own learning processes in all content areas. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to: ELPS.c.1A use prior knowledge and experiences to understand meanings in English ELPS.c.1B monitor oral and written language production and employ self-corrective techniques or other resources ELPS.c.1C use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and grade-level vocabulary ELPS.c.1D speak using learning strategies such as requesting assistance, employing non-verbal cues, and using synonyms and circumlocution (conveying ideas by defining or describing when exact English words are not known) ELPS.c.1E internalize new basic and academic language by using and reusing it in meaningful ways in speaking and writing activities that build concept and language attainment ELPS.c.1F use accessible language and learn new and essential language in the process ELPS.c.1G demonstrate an increasing ability to distinguish between formal and informal English and an increasing knowledge of when to use each one commensurate with grade-level learning expectations ELPS.c.1H develop and expand repertoire of learning strategies such as reasoning inductively or deductively, looking for patterns in language, and analyzing sayings and expressions commensurate with grade-level learning expectations. ELPS.c.2 The ELL listens to a variety of speakers including teachers, peers, and electronic media to gain an increasing level of comprehension of newly acquired language in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in listening. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to: ELPS.c.2A distinguish sounds and intonation patterns of English with increasing ease ELPS.c.2B recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters ELPS.c.2C learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions ELPS.c.2D monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed ELPS.c.2E use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language ELPS.c.2F listen to and derive meaning from a variety of media such as audio tape, video, DVD, and CD ROM to build and reinforce concept and language attainment ELPS.c.2G understand the general meaning, main points, and important details of spoken language ranging from situations in which topics, language, and contexts are familiar to unfamiliar ELPS.c.2H understand implicit ideas and information in increasingly complex spoken language commensurate with grade-level learning expectations ELPS.c.2I demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs. ELPS.c.3 The ELL speaks in a variety of modes for a variety of purposes with an awareness of different language registers (formal/informal) using vocabulary with increasing fluency and accuracy in language arts and all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in speaking. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to: ELPS.c.3A practice producing sounds of newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters to pronounce English words in a manner that is increasingly comprehensible ELPS.c.3B expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication ELPS.c.3C speak using a variety of grammatical structures, sentence lengths, sentence types, and connecting words with increasing accuracy and ease as more English is acquired ELPS.c.3D speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency ELPS.c.3E share information in cooperative learning interactions ELPS.c.3F ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended speaking assignments ELPS.c.3G express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics ELPS.c.3H narrate, describe, and explain with increasing specificity and detail as more English is acquired ELPS.c.3I adapt spoken language appropriately for formal and informal purposes ELPS.c.3J respond orally to information presented in a wide variety of print, electronic, audio, and visual media to build and reinforce concept and language attainment. ELPS.c.4 The ELL reads a variety of texts for a variety of purposes with an increasing level of comprehension in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in reading. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations apply to text read aloud for students not yet at the stage of decoding written text. The student is expected to: ELPS.c.4A learn relationships between sounds and letters of the English language and decode (sound out) words using a combination of skills such as recognizing sound-letter relationships and identifying cognates, affixes, roots, and base words ELPS.c.4B recognize directionality of English reading such as left to right and top to bottom ELPS.c.4C develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials ELPS.c.4D use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary and other prereading activities to enhance comprehension of written text ELPS.c.4E read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned ELPS.c.4F use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language ELPS.c.4G demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding to questions, and taking notes commensurate with content area and grade level needs ELPS.c.4H read silently with increasing ease and comprehension for longer periods ELPS.c.4I demonstrate English comprehension and expand reading skills by employing basic reading skills such as demonstrating understanding of supporting ideas and details in text and graphic sources, summarizing text, and distinguishing main ideas from details commensurate with content area needs ELPS.c.4J demonstrate English comprehension and expand reading skills by employing inferential skills such as predicting, making connections between ideas, drawing inferences and conclusions from text and graphic sources, and finding supporting text evidence commensurate with content area needs ELPS.c.4K demonstrate English comprehension and expand reading skills by employing analytical skills such as evaluating written information and performing critical analyses commensurate with content area and grade-level needs. ELPS.c.5 The ELL writes in a variety of forms with increasing accuracy to effectively address a specific purpose and audience in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in writing. In order for the ELL to meet grade-level learning expectations across foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations do not apply until the student has reached the stage of generating original written text using a standard writing system. The student is expected to: ELPS.c.5A learn relationships between sounds and letters of the English language to represent sounds when writing in English ELPS.c.5B write using newly acquired basic vocabulary and content-based grade-level vocabulary ELPS.c.5C spell familiar English words with increasing accuracy, and employ English spelling patterns and rules with increasing accuracy as more English is acquired ELPS.c.5D edit writing for standard grammar and usage, including subject-verb agreement, pronoun agreement, and appropriate verb tenses commensurate with grade-level expectations as more English is acquired ELPS.c.5E employ increasingly complex grammatical structures in content area writing commensurate with grade-level expectations, such as: ELPS.c.5F write using a variety of grade-appropriate sentence lengths, patterns, and connecting words to combine phrases, clauses, and sentences in increasingly accurate ways as more English is acquired ELPS.c.5G narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired.
Last Updated 08/01/2018

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