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

Unit Overview

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, supporting their preferences with mathematical understandings from amortization models and other influential factors such as ownership and equity.

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

This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning B1; VII. Functions A2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

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.

OVERARCHING UNDERSTANDINGS and QUESTIONS

Equations can model problem situations and be solved using various methods.

• Why are equations used to model problem situations?
• How are equations used to model problem situations?
• What methods can be used to solve equations?

Knowledgeable consumers and investors develop an economic way of thinking and problem solving.

• How are models used to investigate and solve financial problems?
• How can technology be used in financial problem solving?
• How does financial literacy affect financial responsibility and long term goals?
• How does being financially literate protect an individual’s financial stability?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Equations
• Patterns/Rules
• Rates
• Solve

Functions

• Non-Linear Functions

Personal Financial Literacy

• Borrowing
• Credit
• Financial Records
• Interest
• Spending

Associated Mathematical Processes

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

Algebraic formulas and equations are used to generate tables that model a series of payments for loan amortizations.

• Why is an amortization table beneficial when investigating a loan for a financed purchase?
• What is the monthly payment formula and how is it used?
• What components of a loan are displayed on an amortization table?
• How is computer software such as a spreadsheet used to generate amortization tables?
• How can formulas be used to generate components of an amortization table when using a spreadsheet or other computer software?
• What comparisons can be made from an amortization table?

In order to make financially responsible decisions and set long term goals, it is critical to analyze and compare financing options.

• How does the use of technology aid in investigating and comparing financed purchases?
• How does being financially literate in terms of financed purchases protect an individual’s financial stability?
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Patterns/Rules
• Rates
• Solve

Functions

• Non-Linear Functions

Personal Financial Literacy

• Borrowing
• Credit
• Credit Reports
• Interest

Associated Mathematical Processes

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

Amortization models can be used to investigate financing options, including home and automobile loans.

• How is a graphing calculator used to generate amortization models?
• How are online amortization generators used to create amortization models?
• What components of a loan can be determined from an amortization model?
• How can amortization models be used to compare the effects of different components of the loan?
• How can amortization models be used to compare different loan options?

In order to make financially responsible decisions and set long term goals, it is critical to analyze and compare financing options, including purchasing or renting a home and purchasing or leasing an automobile.

• How does the use of technology aid in investigating and comparing financed purchases such as homes and automobiles?
• Under what conditions might it be better to rent a home?
• Under what conditions might it be better to buy a home?
• Under what conditions might it be better to lease an automobile?
• Under what conditions might it be better to buy an automobile?
• How does being financially literate in terms of financing options protect an individual’s financial stability?
• How does being financially literate in terms of buying or renting a home protect an individual’s financial stability?
• How does being financially literate in terms of buying or leasing an automobile protect an individual’s financial stability?

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think the variable i in the monthly payment formula, AP •  , 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

Show this message:

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

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

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards (select CCRS from Standard Set dropdown menu)

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

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

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

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

Texas Instruments – Graphing Calculator Tutorials

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)
• Blue text: Supporting information / Clarifications from TCMPC (Specificity)
• Blue text in italics: Unit-specific clarification
• Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
M.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
M.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

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

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
M.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

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

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
M.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

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

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
M.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

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

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
M.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

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

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
M.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

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

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
M.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

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

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
M.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
• , 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):

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