Advanced Mathematical Decision Making Course … Mathematical Decision Making ... situations using a variety of quantitative measures and ... The Analyzing Numerical Data unit builds

Advanced Mathematical Decision Making (2010) Course and Unit Overviews

Charles A. Dana Center at The University of Texas at Austin 1

Advanced Mathematical Decision Making Course and Unit Overviews

The Course

AMDM is designed as a 12th-grade course to follow Algebra II, either as an alternative to Precalculus or as an elective to accompany or follow Precalculus. It builds on, reinforces, and extends what students have learned and covers a range of mathematics topics that are not part of most school mathematics programs. The course offers student activities in a range of applied contexts and helps students develop college and career readiness skills such as collaborating, conducting research, and making presentations, as called for in the Texas College and Career Readiness Standards.

AMDM was designed by a diverse group of mathematics and education professionals, organized by the Charles A. Dana Center at the University of Texas at Austin, growing directly out of work by the Texas Association of Supervisors of Mathematics (TASM). The course represents a grassroots effort to address a widespread need for a high-quality, relevant, engaging mathematics course that students take to satisfy high school graduation requirements and prepare for future success after high school.

Instructional Materials

The Dana Center, in collaboration with TASM and educators and other mathematics experts around the state, has developed AMDM course materials that provide comprehensive support for the course. Materials for teachers include Unit Overviews (included in this document) as well as detailed Section Planners that offer suggestions for setting up lessons and facilitating student engagement in the mathematics of the lesson. Student Activity Sheets are provided to structure activities, and teachers are provided with Teacher Versions of these activity sheets that provide guidance on the types of responses students might offer. Clearly, the use of these materials is optional for any school or teacher. We provide these course materials as a starting point so that teachers do not need to begin from scratch in developing their own materials, as is typical for a new course. Other materials may be used as teachers choose.

AMDM materials have been developed thanks to the generous funding of Greater Texas Foundation (greatertexasfoundation.org). Thus, we are able to offer the content of these materials at no charge for use with students in Texas. Accordingly, the Dana Center grants a nonexclusive license in perpetuity to the people of Texas to use the 2010 edition of AMDM course materials (reflecting improvements based on the 2009–2010 pilot) in Texas classrooms and homes for the education of our children. Teachers outside of Texas can also make arrangements to use these materials by contacting the Dana Center at [email protected]. We strongly recommend that teachers participate in professional development to support this course, such as that offered by the Dana Center (see the AMDM website: utdanacenter.org/amdm).



Course Overview

AMDM addresses a diverse set of topics, some of which are essential for future success and others that cover critical areas of mathematics not typically addressed in the high school mathematics program. AMDM is organized as follows:

• Unit I: Analyzing Numerical Data (Semester 1) • Unit II: Probability (Semester 1) • Unit III: Statistical Studies (Semester 1) • Unit IV: Using Recursion in Models and Decision Making (Semester 2) • Unit V: Using Functions in Models and Decision Making (Semester 2) • Unit VI: Decision Making in Finance (Semester 2) • Unit VII: Networks and Graphs (Semester 2)

The first semester of the course focuses on advanced numerical reasoning, statistics, and probability. As described in the unit overviews that follow, Unit III is a comprehensive treatment of how to make sense of statistical studies, including having students interpret published studies as well as design and conduct their own simple studies. This semester of in-depth study of statistical concepts is a critical building block for the units that follow in the second semester.

In the second semester, students experience a diverse set of topics. The precise sequence of these four units is less critical than in the first semester, as the units tend to be less dependent on each other. However, Units IV and V together comprise a nice development of mathematical modeling that introduces students to recursion and extends what they know about using functions as models. Unit VI provides students with in-depth experience on advanced financial topics well beyond what we have seen in high school consumer math courses. Unit VII provides a glimpse into the world of discrete mathematics through networks and graphs, representing topics rapidly increasing in importance in our technological world. Students learn how to use unique forms of mathematical techniques, models, and approaches to deal with situations in business and computer science, among other fields.

The unit overviews that follow in this document provide more detail about what the course looks like.

AMDM Topic List (revised December 2009) (DM.1) Developing college and career skills. The student develops and applies

skills used in college and careers, including reasoning, planning, and communication, to make decisions and solve problems in applied situations involving numerical reasoning, probability, statistical analysis, finance, mathematical selection, and modeling with algebra, geometry, trigonometry, and discrete mathematics.

(DM.2) Analyzing numerical data. The student analyzes numerical data in everyday situations using a variety of quantitative measures and numerical processes.



(DM.3) Analyzing information using probability. The student analyzes and evaluates risk and return in the context of everyday situations.

(DM.4) Critiquing applications of statistics. The student makes decisions based on understanding, analysis, and critique of reported statistical information and statistical summaries.

(DM.5) Conducting statistical analyses. The student applies statistical methods to design and conduct a study that addresses one or more particular questions.

(DM.6) Communicating statistical information. The student communicates the results of reported and student-generated statistical studies.

(DM.7) Mathematical decision making in ranking and selection. The student analyzes the mathematics behind various methods of ranking and selection.

(DM.8) Modeling data. The student models data, makes predictions, and judges the validity of a prediction.

(DM.9) Modeling change and relationships. The student uses mathematical models to represent, analyze, and solve problems involving change.

(DM.10) Mathematical decision making in finance. The student creates and analyzes mathematical models to make decisions related to earning, investing, spending, and borrowing money.

(DM.11) Network modeling for decision making. The student uses a variety of network models represented graphically to organize data in quantitative situations, make informed decisions, and solve problems.

(DM.12) Modeling with geometric tools. The student uses a variety of tools and methods to represent and solve problems involving static and dynamic situations.



Unit I Analyzing Numerical Data

Mathematics Overview

The Analyzing Numerical Data unit requires approximately four weeks of instructional time. It focuses on deepening students’ understanding of proportional reasoning and basic numerical calculations—such as ratios, rates, and percents—by applying them to settings in business, media, consumer, and other areas. Working with familiar mathematical tools and learning some new ones, students improve their ability to solve problems by applying appropriate strategies.

Typically, in middle school, students learned about using ratios to describe direct proportional relationships involving number, geometry, measurement, and probability. The emphasis generally was on using ratios to describe and make predictions in proportional situations and on representing ratios and percents with concrete models, fractions, and decimals. Most middle school students have estimated—and found—solutions to application problems involving percent and proportional relationships, such as similarity, scaling, unit costs, and related measurement units. As a critical connection for their future work in high school, middle school students worked with proportional and nonproportional linear relationships and continued to estimate—and solve—application problems involving percents. As students progressed through Algebra I, Algebra II, and Geometry, or Integrated Mathematics I, II, and III, they likely continued to gain experience with proportional linear relationships.

The Analyzing Numerical Data unit builds upon students’ prior knowledge of ratio and focuses on helping students learn how to make decisions in everyday situations after analyzing information. Using contextual situations, students develop skills that they can apply outside the classroom. Because this first unit extends students’ previous knowledge in engaging contexts, the unit provides a solid foundation for teachers to set the stage for the year in terms of how the AMDM classroom will operate. Students begin to see that mathematics can be highly engaging and relevant, and they come to realize that they will have to figure out challenging problems without always being told exactly what to do first. Students begin the development of critical college and career readiness skills as they research and answer questions, present their solutions to the class, and provide feedback to others.

Student Expectations (DM.1) Developing college and career skills. The student develops and applies skills used in

college and careers, including reasoning, planning, and communication, to make decisions and solve problems in applied situations involving numerical reasoning, probability, statistical analysis, finance, mathematical selection, and modeling with algebra, geometry, trigonometry, and discrete mathematics.



The student is expected to:

(A) gather data, conduct investigations, and apply mathematical concepts and models to solve problems in mathematics and other disciplines;

(B) demonstrate reasoning skills in developing, explaining, and justifying sound mathematical arguments, and analyze the soundness of mathematical arguments of others; and

(C) communicate with and about mathematics orally and in writing as part of independent and collaborative work, including making accurate and clear presentations of solutions to problems.

(DM.2) Analyzing numerical data. The student analyzes numerical data in everyday situations using a variety of quantitative measures and numerical processes.


(A) apply, compare, and contrast published ratios, rates, ratings, averages, weighted averages, and indices to make informed decisions;

(B) solve problems involving large quantities that are not easily measured;

(C) use arrays to efficiently manage large collections of data and add, subtract, and multiply matrices to solve applied problems; and

(D) apply algorithms and identify errors in recording and transmitting identification numbers.

Section Overviews

The Analyzing Numerical Data unit is divided into four sections as outlined below.

Section A: Estimating Large Numbers

Students use various numerical techniques to estimate large numbers in situations such as assessing the size of the crowd at a political rally and calculating the number of possible telephone numbers in the United States to see when the numbers will run out. Students also investigate various Fermi questions that ask them to estimate physical quantities.

Section B: Using Ratios

Students apply proportional reasoning with ratios, rates, and percents to real-world problems involving aspect ratios in movies shown on television, tires, and other applications.

Section C: Indices Using Weighted Sums and Averages

Students use averages and indices as a tool for thinking about which grading system is better for a hypothetical student, slugging averages in baseball and NFL quarterback ratings, Fan Cost Indices for attending a sporting event, and the Gunning Fog Index for measuring the readability of a piece of writing.



Section D: Validating Identification Numbers

Students learn how identification numbers such as Universal Product Codes (UPCs) and credit card numbers are created and how check digits are used to detect errors and prevent fraud.



Unit II Probability


The Probability unit requires approximately five weeks of instructional time. It focuses on the analysis of information using probability to make decisions about everyday situations. After determining the probability of various events, students expand their knowledge toward making decisions about the risks and mathematical fairness of these events.

In middle school, students learned about concepts of probability, how to apply these concepts in theoretical and experimental situations, and how to use these concepts to make predictions. The emphasis was on constructing sample spaces and tree diagrams, finding probabilities of simple events and their complements, and simulating events using models. As students progressed through high school courses, the emphasis shifts to using models to represent functional relationships with less focus around the probabilistic nature of decision making.

The Probability unit builds upon students’ prior knowledge of probability and focuses on how to make decisions in everyday situations after analyzing information. By using contextual situations, students develop skills that they can apply outside the classroom. In particular, students extend the range of situations they deal with to include those in which not all outcomes are equally likely, and they learn tools to account for weighting different possible outcomes in such situations.









(DM.3) Analyzing information using probability. The student analyzes and evaluates risk and return in the context of everyday situations.


(A) determine and interpret conditional probabilities and probabilities of compound events by constructing and analyzing representations, including tree diagrams, Venn diagrams, and area models, to make decisions in problem situations;

(B) use probabilities to make and justify decisions about risks in everyday life; and

(C) calculate expected value to analyze mathematical fairness, payoff, and risk.

Section Overviews

The Probability unit is divided into three sections as outlined below.

Section A: Determining Probabilities

Students construct and analyze representations of events, such as Venn diagrams and tree diagrams, to determine conditional probabilities, including situations where not all outcomes are equally likely. They determine probabilities of compound events to make decisions about the risks involved in a situation. Students investigate dependent and independent events. They also analyze and construct area models. This section ends with an activity in which students must analyze a weighted tree diagram to come up with a scenario that could describe the diagram.

Section B: Everyday Decisions Based on Probabilities

Students call on models of probability from earlier lessons to help them solve problems. They consider what information they know in determining the best way to organize the information. Students also learn how to represent data in Venn diagrams with more than two intersecting events. They analyze a probability situation that has changing factors and construct a model that helps them organize the data to answer questions.

Section C: Expected Value

Students learn about binomial probability and calculate expected values to analyze mathematical fairness, payoff, and risk in a variety of situations, including games and allowances. Students investigate the connection between probability and Pascal’s triangle, and they apply their understanding of expected values to make decisions.



Unit III Statistical Studies


The Statistical Studies unit requires approximately seven weeks of instructional time. It focuses on developing background statistical knowledge through the use of existing case studies and introducing students to the basic components of the design and implementation of statistical studies. After collecting and displaying data, students explore introductory techniques of statistical analysis. Students build the skills and vocabulary necessary to analyze and critique reported statistical information, summaries, and graphical displays; they prepare oral and written reports of these analyses. Throughout this unit, and as a culmination of the first semester’s work (depending on the school calendar), students work toward implementing their own statistical study, including all of the stages involved in designing the study, conducting the study, organizing and analyzing the data, and reporting the results. Specific steps toward that goal are included in the teacher notes where appropriate. The skills students develop at this stage can be used throughout the rest of the course to address problems and conduct projects specific to each unit

Typically in middle school, students learn to use statistical representations to analyze data. The emphasis in middle school tends to be the use of different graphical representations to display the same data (including line plots, line graphs, bar graphs, stem-and-leaf plots, and circle graphs); using mean, median, mode, and range as measures of center and spread; and collecting, organizing, displaying, and interpreting data. At that level, students are often expected to choose an appropriate display and justify their choice, and Venn diagrams are introduced. Students choose the appropriate measure of center and spread for a data set and justify their choice. Students may provide convincing arguments based on an analysis of data. They draw conclusions and make predictions by analyzing trends in scatterplots. Students may work with box-and-whisker plots and histograms. Sometimes, students are introduced to the concepts of sampling methods and these methods’ effects on validity, and they may even touch on misuses of statistical information. Statistics is generally not addressed in Algebra I, Algebra II, or Geometry, although in those states and districts using integrated mathematics standards, statistics is frequently woven into the mathematics program in these first three years.

The Statistical Studies unit builds upon students’ prior knowledge of statistics to ensure that they become more discerning consumers of statistics in everyday situations. They also develop skills to prepare them for the further use of statistics and statistical studies in their major field of study at the university level or in the workplace.




college and careers, including reasoning, planning, and communication, to make decisions and solve problems in applied situations involving numerical reasoning, probability, statistical analysis, finance, mathematical selection, and models with algebra, geometry, trigonometry, and discrete mathematics.





(DM.4) Critiquing applications of statistics. The student makes decisions based on understanding, analysis, and critique of reported statistical information and statistical summaries.


(A) identify limitations or lack of information in studies reporting statistical information, especially when studies are reported in condensed form;

(B) interpret and compare the results of polls, given a margin of error;

(C) identify uses and misuses of statistical analyses in studies reporting statistics or using statistics to justify particular conclusions, including assertions of cause and effect rather than correlation; and

(D) describe strengths and weaknesses of sampling techniques, data and graphical displays, and interpretations of summary statistics and other results appearing in a study, including reports published in the media.

(DM.5) Conducting statistical analyses. The student applies statistical methods to design and conduct a study that addresses one or more particular questions.


(A) determine the need for and purpose of a statistical investigation and what type of statistical analysis can be used to answer a specific question or set of questions;

(B) identify the population of interest, select an appropriate sampling technique, and collect data;

(C) identify the variables to be used in a study;

(D) determine possible sources of statistical bias in a study and how such bias may affect the ability to generalize the results;

(E) create data displays for given data sets to investigate, compare, and estimate center, shape, spread, and unusual features; and

(F) determine possible sources of variability of data, both those that can be controlled and those that cannot be controlled.



(DM.6) Communicating statistical information. The student communicates the results of reported and student-generated statistical studies.


(A) report results of statistical studies to a particular audience, including selecting an appropriate presentation format, creating graphical data displays, and interpreting results in terms of the question studied;

(B) justify the design and the conclusion(s) of statistical studies, including the methods used for each; and

(C) communicate statistical results in both oral and written formats using appropriate statistical and nontechnical language.

Section Overviews

The Statistical Studies unit is divided into three sections as outlined below.

Section A: Statistical Investigations

Students explore the research cycle and investigate the purposes of a variety of statistical investigations, with a particular focus on developing the research question and designing a study. They explore how to write a null hypothesis and an alternative hypothesis, as well as what makes up an experimental study. They identify the population of interest and the variables to be used in each study. Students then determine the appropriate sampling design, sampling technique, and statistical analysis for each research question. They also discuss data sources (including what constitutes primary data and secondary data) and the ethics of data collection, particularly with human subjects.

To build background knowledge, students are introduced to case studies. They determine whether given studies are observational or experimental and learn about identification of participants, assignment of treatments, and the placebo effect. Students also identify various sampling techniques used. Students analyze results, including margin of error in opinion polls; they then identify strengths and weaknesses in a study and its presentation of results.

Students complete the initial phases of their unit-level research study. They develop their question/hypothesis and choose their sample technique.

Section B: Analyzing Data

Students are introduced to categorical and quantitative data, and then the focus is narrowed to quantitative data and then to univariate data. Students identify the variable of interest, interpret a variety of graphical displays (particularly histograms), and estimate center, spread, shape, outliers, and unusual features. Students explore histograms in depth, analyzing the effect of changing the bin size (also known as interval width). They analyze the appropriateness and usefulness of the chosen measure of center and the graphical display. Students identify limitations, lack of information, and possible misinterpretations in media reports. They then prepare



more appropriate reports for a given audience. Students collect sets of data and create a variety of data displays. They describe the distribution by estimating center, shape, spread, and unusual features. Students also compare and contrast multiple data sets. Throughout this section, they communicate their analyses orally and/or in writing, using appropriate statistical language as well as nontechnical language.

Students develop an abstract format of their research project. They begin writing survey questions for their research project (if applicable) and try out the questions to see if the responses are what they were expecting. Students finish designing their study or project, including the data collection plan, and pilot their survey questions or observation instruments.

Section C: Sources of Variability

Students build on the skills practiced and information gathered during the first two sections of this unit to investigate possible sources of variability in the data, including biased sampling methods (such as nonrepresentative sampling and undercoverage) and biased statistics, as well as natural and induced variability. They search for various possible sources of statistical bias (such as response bias, nonresponse bias, and observer effect) and examine the effects of statistical bias on the generalizability of results. Students also explore the importance of designing surveys and/or observation instruments as they finalize their own study and presentation of their results.

Students complete the data collection for their project or study. They focus on the data analysis and final preparation for their presentation. Students share their work with the class.



Unit IV Using Recursion in Models and Decision Making


The Using Recursion in Models and Decision Making unit requires approximately four weeks of instructional time. It focuses on analyzing data and finding rules to model the data. By looking at recursive models for bivariate data and relationships, students expand their set of tools for data analysis. While teachers (and some students) might associate the term bivariate with statistical analysis, in reality this term can refer to any relationship between two variables or quantities. Thus, nearly all high school work with algebraic modeling involves bivariate relationships.

In previous courses, students learned about various function families, including linear and exponential functions. This unit builds on students’ knowledge of these functions and focuses on recursive rules that model data exhibiting exponential and linear patterns. In this way, students reinforce their understanding of the concepts associated with linear and exponential functions while building a new way to think about modeling these types of data. By introducing a leveling-off value, exponential growth can be extended to the study of logistic growth patterns.

Students add a new type of function to their library of functions as they analyze cyclical data. The sine function is developed through explorations of data that exhibit periodic behavior and through investigations of the concept of a wrapping function. More work with the sine function follows in Unit V, “Using Functions in Models and Decision Making.”











(A) determine whether or not there is a linear relationship in a set of bivariate data by finding the correlation coefficient for the data, and interpret the coefficient as a measure of the strength and direction of the linear relationship; and

(B) collect numerical bivariate data; use the data to create a scatterplot; select a function to model the data, justify the selection, and use the model to make predictions.



(A) determine or analyze an appropriate growth or decay model for problem situations, including linear, exponential, and logistic functions;

(B) determine or analyze an appropriate cyclical model for problem situations that can be modeled with trigonometric functions;

(D) solve problems using recursion or iteration, including those involving population growth or decline and compound interest.

Section Overviews

The Using Recursion in Models and Decision Making unit is divided into four sections as outlined below.

Section A: Relationships in Data

Students focus on bivariate data, identifying the variables of interest. They also analyze representations of bivariate statistics and make assertions of cause-and-effect relationships rather than correlations. Students continue to analyze data that follow a linear pattern using recursively defined rules and compare those rules to explicit function rules.

Section B: Recursion in Exponential Growth and Decay

Students explore data that follow an exponential pattern using the idea of a common ratio between consecutive values. They find recursive rules to model the data and make connections between the recursive rule and the explicit function rule of the exponential relationship.

Section C: Recursion Using Rate of Change

Students review modeling with recursively defined functions and extend their modeling capability to Newton’s Law of Cooling. They explore data that follow an exponential pattern using the idea of a common ratio between consecutive values. A maximum value is introduced to set up the exploration of logistic models. Students explore logistic models by looking at a difference equation.



Section D: Recursion in Cyclical Models

Students use cyclical functions based on the trigonometric ratios to model cyclical physical phenomena in solving problems. They analyze the characteristics, including period and amplitude, to model simple cyclical data.



Unit V Using Functions in Models and Decision Making


The Using Functions in Models and Decision Making unit requires four weeks of instructional time. It focuses on analyzing data and finding mathematical functions (rules) to model real-world data and contexts with functions. Here students expand their set of tools for data analysis, building on their previous work with continuous and piecewise-defined functions. They also build on their work in Unit IV, “Using Recursion in Models and Decision Making,” connecting recursive rules and explicit function rules.

In earlier studies, students likely learned about a variety of functions, including linear, quadratic, exponential, rational, and possibly step functions. In this unit, students work with rules in business and natural contexts to create models for a variety of situations. They test these models against data and common sense to answer questions and solve problems. In this way, students enhance their ability to use the power of mathematical modeling, to understand the limitations of modeling, and to use data and modeling to deal with complex problems in the world in which they live.









(A) determine whether or not there is a linear relationship in a set of bivariate data by finding the correlation coefficient for the data, and interpret the coefficient as a measure of the strength and direction of the linear relationship; and

(B) collect or use numerical bivariate data; use the data to create a scatterplot; select a function to model the data, justify the selection, and use the model to make predictions.





(A) determine or analyze an appropriate growth or decay model for problem situations, including linear, exponential, and logistic functions;

(B) determine or analyze an appropriate cyclical model for problem situations that can be modeled with trigonometric functions;

(C) determine or analyze an appropriate piecewise model for problem situations;

Section Overviews

The Using Functions in Models and Decision Making unit is divided into three sections as outlined below.

Section A: Regression in Linear and Nonlinear Functions

Students analyze the finite first differences to determine that a linear function is a good model for an appropriate set of data. They analyze the correlation coefficient of the data to determine the strength of the linear model. Additionally, students explore data that follow an exponential pattern using the idea of a common ratio between consecutive values. They find recursive rules to model the data and make connections between the recursive rule and the function of the exponential rule. Students also analyze data that are logistical in nature, studying the unique pattern of rate of change in the data.

Section B: Cyclical Functions

Students use cyclical functions based on a sinusoidal curve to model business cycles and cyclical physical phenomena. These models are based on an understanding of business contexts and physical principles. Students check their models against existing data. They discuss the various types of limitations that occur in models, especially problems with extrapolating outside the data with models that fit the data but do not adhere to business principles or natural laws.

Section C: Step and Piecewise Functions

Students investigate and model several applications of step and piecewise functions. They use scatterplots to assess the validity of a model and the function rule to determine values of the function at particular points in time. Students use these values to make predictions and decisions about a variety of problem situations.



Unit VI Decision Making in Finance


The Decision Making in Finance unit requires five to six weeks of instructional time. It focuses on the financial decisions that surround borrowing, loaning, and investing money and how the time value of money affects such decisions. While some of these topics may be familiar to teachers and students, the mathematics behind them can be challenging. Thus, these contexts provide rich opportunities for critical thinking and problem solving. This unit goes well beyond typical “consumer math” skills that might be addressed in middle school or high school. It asks students to use sophisticated mathematical models to deal with problems in these familiar situations.

In earlier units, students studied the mathematical structure involved in such decision making: f(t) = abt , the general exponential function. They use this function as the basis for more complex functions that model change in a variety of financial situations. The overall goal of this unit is to provide future citizens with mathematical and financial tools they can use to plan wisely and use credit knowledgeably.







(DM.10) Mathematical decision making in finance. The student creates and analyzes mathematical models to make decisions related to earning, investing, spending, and borrowing money.


(A) determine, represent, and analyze mathematical models for various types of income calculations to determine the best option for a given situation;

(B) determine, represent, and analyze mathematical models for expenditures, including those involving credit, to determine the best option for a given situation; and



(C) determine, represent, and analyze mathematical models and appropriate representations for various types of loans and investments to determine the best loan or investment plan for a given situation.

Section Overviews

The Decision Making in Finance unit is divided into four sections as outlined below. A graphing calculator and a spreadsheet application or similar technology are required in this unit.

Section A: Future Value of an Investment

Students study the future value of an investment. The exponential function that represents the future value of an investment with interest compounded annually and monthly is developed. Students also compare and contrast the nominal interest rate with the annual percentage rate (APR). After completing this section, students can answer questions about the difference between simple and compound interest in given situations.

Section B: Present Value of an Investment

Students develop the formula for the present value of an investment from the future value of an investment rule developed in Section A. After completing this section, students can compare investment scenarios using the concept of present value of an investment.

Section C: Building an Investment

Students develop and study the formula for determining the future and present value of an investment such as an annuity. They also study the effect of increasing the interest rate on the present value of an annuity. Together, the tools in Sections A, B, and C equip students to make knowledgeable decisions about investments as they think about what is involved in planning for long-term financial security.

Section D: Using Credit

Students study different real-world scenarios involving credit. They investigate the formula for determining the monthly payment to retire a debt at a fixed rate in a fixed number of monthly payments. Students compare and contrast different credit card offers. They also study credit card statements to fully understand the concept of minimum payment, the length of time to pay off a credit card debt using the minimum payment, and the APR of such minimum payments. Additionally, students study methods of comparing car loans using a variety of tools. They compare and contrast the bank or credit union loans, automobile dealer loans, and car purchases that have a cash-back feature.



Unit VII Networks and Graphs


The Networks and Graphs unit requires approximately four weeks of instructional time. It focuses on the creation of models that represent real-world contexts involving networks and graphs and the use of these networks and graphs to investigate real-world scheduling problems. In this unit, students extend their ability to solve abstract and concrete problems.

Although networks and graphs have geometrical connections (in that they are drawn in two dimensions with points, lines, and curves), the mathematical reasoning required to create, understand, and use them is new to most students.







(DM.11) Network modeling for decision making. The student uses a variety of network models represented graphically to organize data in quantitative situations, make informed decisions, and solve problems.


(A) solve problems involving scheduling or routing situations that can be represented by a vertex-edge graph, and find critical paths, Euler paths, or minimal spanning trees; and

(B) construct, analyze, and interpret flow charts in order to develop and describe problem solving procedures.



Section Overviews

The Networks and Graphs unit is divided into four sections as outlined below.

Section A: Circuits, Paths, and Graph Structures

Students use graphs and the definitions of circuits and paths to study a situation like the Königsberg Bridge problem to determine if certain conditions can be satisfied. They use theorems and algorithms to solve such problems. Students create graph structures to use in determining the best methods for scheduling tasks and making assignments. These structures are a bit more complicated and generally more applicable.

Section B: Spanning Trees

Students represent situations with tree diagrams and then look at ways of determining the spanning trees that solve questions arising from the situation. Some algorithms for finding spanning trees are presented and used without proof.

Section C: Graph Coloring

Students consider problems that can be resolved by coloring graphs. They create graphs from a description of a situation and then determine if the graphs can be colored in specific ways using theorems and algorithms.

Section D: Program Evaluation and Review Technique (PERT) Charts

Students study the scheduling of projects using the Program Evaluation and Review Technique. They work with information about a project, including tasks and their time constraints along with interrelationships between and among tasks. Optionally, freely available PERT software is employed to determine the scheduling pattern for the project.

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Advanced Mathematical Decision Making Course … Mathematical Decision Making ... situations using a variety of quantitative measures and ... The Analyzing Numerical Data unit builds