Math Models with Applications

one credit course

Introduction

Unit 01: Algebraic Models in Science, Engineering, and Social Sciences

Unit 02: Geometric Relationships in Architecture and Engineering

Unit 03: Shape and Size in Architecture and Engineering

Unit 04: Applications of Algebra and Geometry in Art

Unit 05: Applications of Algebra and Geometry in Music

Unit 06: Management of Personal Finances

Unit 07: Savings, Investments, and Insurance

Unit 08: Amortization of Loans

Unit 09: Probability

Unit 10: Statistical Data Analysis

Unit 11: Research and Marketing

Unit 12: Data Collection and Research Project

Texas Essential Knowledge & Skills (TEKS)

Introduction

(1) The desire to achieve educational excellence is the driving force behind the Texas essential knowledge and skills for mathematics, guided by the college and career readiness standards. By embedding statistics, probability, and finance, while focusing on fluency and solid understanding, Texas will lead the way in mathematics education and prepare all Texas students for the challenges they will face in the 21st century.

(2) The process standards describe ways in which students are expected to engage in the content. The placement of the process standards at the beginning of the knowledge and skills listed for each grade and course is intentional. The process standards weave the other knowledge and skills together so that students may be successful problem solvers and use mathematics efficiently and effectively in daily life. The process standards are integrated at every grade level and course. When possible, students will apply mathematics to problems arising in everyday life, society, and the workplace. Students will 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. Students will select appropriate tools such as real objects, manipulatives, paper and pencil, and technology and techniques such as mental math, estimation, and number sense to solve problems. Students will effectively communicate mathematical ideas, reasoning, and their implications using multiple representations such as symbols, diagrams, graphs, and language. Students will use mathematical relationships to generate solutions and make connections and predictions. Students will analyze mathematical relationships to connect and communicate mathematical ideas. Students will display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

(3) Mathematical Models with Applications is designed to build on the knowledge and skills for mathematics in Kindergarten-Grade 8 and Algebra I. This mathematics course provides a path for students to succeed in Algebra II and prepares them for various post-secondary choices. Students learn to apply mathematics through experiences in personal finance, science, engineering, fine arts, and social sciences. Students use algebraic, graphical, and geometric reasoning to recognize patterns and structure, model information, solve problems, and communicate solutions. Students will select from tools such as physical objects; manipulatives; technology, including graphing calculators, data collection devices, and computers; and paper and pencil and from methods such as algebraic techniques, geometric reasoning, patterns, and mental math to solve problems.

(4) In Mathematical Models with Applications, students will use a mathematical modeling cycle to analyze problems, understand problems better, and improve decisions. A basic mathematical modeling cycle is summarized in this paragraph. The student will:

(A) represent:

(i) identify the variables in the problem and select those that represent essential features; and

(ii) formulate a model by creating and selecting from representations such as geometric, graphical, tabular, algebraic, or statistical that describe the relationships between the variables;

(B) compute: analyze and perform operations on the relationships between the variables to draw conclusions;

(D) revise: confirm the conclusions by comparing the conclusions with the problem and revising as necessary; and

(E) report: report on the conclusions and the reasoning behind the conclusions.

Unit 01: Algebraic Models in Science, Engineering, and Social Sciences

(19 classes for the entire unit)

Students collect real data, study patterns, and use algebraic techniques to describe physical laws of science that are modeled by proportionality and inverse variation such as Hook’s Law, Newton’s Second Law of Motion, and Boyle’s Law. Exponential growth and decay functions are explored using technology to model problems in science and social studies, including radioactive decay. Students use quadratic functions to model motion by collecting data and determining quadratic functions to fit the data with and without technology. Students also use regression methods available through technology to generate linear and exponential functions and determine the strength of the model for making predictions using the correlation coefficient.

TEKS in this unit: M.1A, M.1B, M.1C, M.1D, M.1E, M.1F, M.1G, M.5A, M.5B, M.5C, M.9F

Unit 02: Geometric Relationships in Architecture and Engineering

(6 classes for the entire unit)

Students apply the Pythagorean Theorem, special right triangles relationships (30o– 60o– 90o or 45o– 45o– 90o), and trigonometric ratios to solve real-world problems. Students write representative equations using the appropriate theorem or trigonometric ratio, including the inverses of the trigonometric ratios, to find missing values and solve real-world problems in architecture and engineering.

TEKS in this unit: M.1A, M.1B, M.1C, M.1D, M.1E, M.1F, M.1G, M.6C, M.6D

Unit 03: Shape and Size in Architecture and Engineering

(13 classes for the entire unit)

Students use rigid transformations that maintain congruence (translation, reflection, rotation) and non-rigid, similarity transformations (dilations) to describe mathematical patterns and structure in architecture. Students analyze reflectional and rotational symmetry as demonstrated in architecture to imply balance within structures. Students use perspective drawings to represent three-dimensional relationships in two-dimensional representations using one- and two-point perspectives. Students use a two-dimensional net and its attributes and properties to analyze the surface area and volume of a three-dimensional figure. Students use scale factor(s) applied to one or two dimensions of a three-dimensional figure to demonstrate non-proportional change in surface area and volume. Students use a scale factor applied to all three dimensions of a three-dimensional figure to demonstrate proportional change in surface area and volume. Students generate scale models or drawings, using proportional and non-proportional change, for products in the fields of engineering and architecture.

TEKS in this unit: M.1A, M.1B, M.1C, M.1D, M.1E, M.1F, M.1G, M.6A, M.6B

Unit 04: Applications of Algebra and Geometry in Art

(12 classes for the entire unit)

Students use rigid transformations that maintain congruence (translation, reflection, rotation) and non-rigid, similarity transformations (dilations) to describe mathematical patterns and structure in art and photography. Students analyze reflectional and rotational symmetry as demonstrated in art and photography. Students use perspective drawings to represent three-dimensional relationships in two-dimensional representations using one- and two-point perspectives. Students use attributes and properties to analyze the surface area and volume of a three-dimensional figure. Students use scale factor(s) applied to one or two dimensions of a three-dimensional figure to demonstrate non-proportional change in surface area and volume. Students use a scale factor applied to all three dimensions of a three-dimensional figure to demonstrate proportional change in surface area and volume. Students generate scale models or drawings, using proportional and non-proportional change, for products in the field of art. Students determine the effects of cropping and sizing in photography. Students apply trigonometric ratios and the golden ratio to determine lengths and angle measure in right triangles found in art.

TEKS in this unit: M.1A, M.1B, M.1C, M.1D, M.1E, M.1F, M.1G, M.7A, M.7B, M.7D

Unit 05: Applications of Algebra and Geometry in Music

(10 classes for the entire unit)

Students investigate music using trigonometric functions (sine function) to model periodic behavior. Students connect periodic motion to sound, especially musical notes, using technology. Periodic models are analyzed to determine volume (amplitude) modeled by a vertical dilation and pitch (frequency) by a horizontal dilation. Students examine a periodic sound wave of a note to determine the period of the note and calculate the reciprocal of the period to determine the frequency. Students identify the period and frequency of a note as an inverse proportional relationship. Students collect real data using technology to model periodic motion of music and analyze the characteristics of the graph in terms of music. Students examine representative pieces of music to determine the relationship between rhythm, beats per note, and the number of beats per measure using the time signature. Students transpose (translated by proportions) frequency up or down for a given interval (e.g., a third, a fifth, an octave, etc.).

TEKS in this unit: M.1A, M.1B, M.1C, M.1D, M.1E, M.1F, M.1G, M.7A, M.7C

Unit 06: Management of Personal Finances

(14 classes for the entire unit)

Students use linear equations and functions to describe proportional and non-proportional linear relationships and apply them to represent real-world problems involving earnings and budgeting. Students explore methods employees are compensated (e.g., hourly pay, base salary plus commissions, straight commission, salary, etc.) and compare the methods of compensations using various representations. Students use IRS Tax tables to determine personal income taxes and amounts to be deducted from regular pay checks. Students calculate payroll deductions including taxes, pensions, and additional employee benefits, to determine net pay. Students analyze their personal budget based on their net earnings. Students explore types of bank services, including checking and savings options, overdraft protection policies, fees and charges, online banking options, and ATM and card availability and policies, etc. Students analyze banking options based on budgetary and banking needs. Students explore and study models of credit options in retail purchasing and compare relative advantages and disadvantages of each option. Students explain option choices based on income and spending scenarios. Throughout the unit, students develop an economic way of thinking, encouraging financial responsibility for both short-term and long-term goals.

TEKS in this unit: M.1A, M.1B, M.1C, M.1D, M.1E, M.1F, M.1G, M.2A, M.2B, M.2C, M.3B

Unit 07: Savings, Investments, and Insurance

(15 classes for the entire unit)

Students explore and compare options for life insurance, considering permanent and term policies, amount of coverage, premiums, and needs of different individuals based on income and family situation. Students explore and compare options for medical insurance considering coverage options, deductibles, premiums, and an individual's health needs, situation, and income. Students explore and compare options for homeowner's and renter's insurance, considering coverage, deductibles, premiums, and the different needs of individuals based on income and living situation. Students explore and compare options for automobile insurance, considering coverage options for state requirements for liability coverage, collision and comprehensive coverage, medical coverage, rental coverage, and premiums based on the value of the automobile being insured and the profile of the driver(s). Students define investment options, including the analysis of stocks, bonds, annuities, certificates of deposit, and retirement plans. Students explore and compare different investment options and the advantages and disadvantages, considering individual needs and time, earnings, fees, and accessibility of funds. Students explore savings options and compare simple and compound interest using multiple representations. Students demonstrate financial literacy and reasoning by presenting their preferences of insurance options, investments, and savings, supporting their preferences with tables, function models, and other influential factors.

TEKS in this unit: M.1A, M.1B, M.1C, M.1D, M.1E, M.1F, M.1G, M.4A, M.4B, M.4C

Unit 08: Amortization of Loans

(10 classes for the entire 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.

TEKS in this unit: M.1A, M.1B, M.1C, M.1D, M.1E, M.1F, M.1G, M.3A, M.3C, M.3D

Unit 09: Probability

(13 classes for the entire unit)

Students determine the number of ways an event may occur using combinations, permutations, and the Fundamental Counting Principle. Students perform experiments, demonstrating empirical probability, to predict the likelihood of an event occurring from the outcomes of the experiment. Students use the same events, calculating theoretical probability, to predict the likelihood of an event occurring using formulas and mathematical calculations without conducting an experiment. Students then compare the experimental outcomes to the calculated theoretical probability. After conducting multiple experiments, students realize as the number of trials in the experiment increases, the experimental probability of an event approaches the theoretical probability of the same event, known as the Law of Large Numbers. Students define binomial distribution and use the formula, with and without technology, to generate probabilities of various events. They perform various experiments using binomial distribution and compare the theoretical model to the experimental results to determine the reasonableness of the theoretical model. Students define geometric distribution and use the formula, with and without technology, to generate probabilities of various events. Students use the results of experiments and theoretical models to make general comparisons of theoretical to empirical probability.

TEKS in this unit: M.1A, M.1B, M.1C, M.1D, M.1E, M.1F, M.1G, M.8A, M.8B, M.8C

Unit 10: Statistical Data Analysis

(12 classes for the entire unit)

Students use prior knowledge and experience with various graphs, including line graphs, bar graphs, circle graphs, histograms, scatterplots, dot plots, stem-and-leaf plots, and box and whisker plots. Students use various sources (e.g., internet, print media, etc.) to find various types of graphs and interpret the graphs to draw conclusions and determine the strengths or weaknesses of the data and the graphs. Students analyze numerical data using measures of central tendency (mean, median and mode) and variability (range, interquartile range or IQR, and standard deviation) in order to make inferences with normal distributions. Mathematical Models with Applications introduces examining data using standard deviation. Throughout the unit, students examine various data sets and analyze various measures to describe the center, spread, and shape of the data distributions in order to make inferences and predictions regarding real world situations.

TEKS in this unit: M.1A, M.1B, M.1C, M.1D, M.1E, M.1F, M.1G, M.9A, M.9B

Unit 11: Research and Marketing

(12 classes for the entire unit)

Students define different types and methods of research. Students identify and differentiate purposes for surveys, experiments, and observational studies. Students predict the appropriate type of research to answer questions. Students generate questions that would require research, define the purpose of the research, and choose a method of research appropriate to the problem situation. Students define population mean and population proportion and discuss how to use data to find each. Students use either given data sets or collected data sets to find population means and proportions. Students are introduced to methods marketing professionals use to influence consumers. Students analyze examples of graphs and sets of statistics to discuss the validity of the claims. Students use the internet or print media to find examples of marketing using statistics or graphs and analyze the graphs and statistics to justify or refute the validity of the marketing claims.

TEKS in this unit: M.1A, M.1B, M.1C, M.1D, M.1E, M.1F, M.1G, M.9C, M.9D, M.9E

Unit 12: Data Collection and Research Project

(11 classes for the entire unit)

Students determine a meaningful hypothesis or question for which data can be collected and analyzed. Students determine the type of data best used to analyze the hypothesis or question and the appropriate method to collect the data. Students represent and analyze data using tables and appropriate graphs. Students calculate numerical measures of central tendency and measures of variability and distribution as appropriate for the collected data. Students draw conclusions from the graphical and numerical analysis in terms of the hypothesis or question. Students organize results in a written report that includes a description of the type data and collection methods, data tables, appropriate graphical representations, numerical calculations, and conclusions in terms of the hypothesis or question. Students present a summarization of the hypothesis/question that includes visual displays, graphical and numerical analysis, and conclusions through an oral and/or multimedia presentation.

TEKS in this unit: M.1A, M.1B, M.1C, M.1D, M.1E, M.1F, M.1G, M.10A, M.10B

Texas Essential Knowledge & Skills (TEKS)

TEKS - Math - Math Models.pdf

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