M.AMM: Advanced Mathematical Modeling

Advanced Mathematical Modeling

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Advanced Mathematical Modeling applies mathematics to real-world decision making, with probability, statistical reasoning, and data analysis as the primary tools. Students evaluate statistical studies, interpret margins of error, use conditional probability, and calculate expected value to assess risk. Matrices, large-quantity estimation, function modeling, and social decision-making contexts give the course its applied character.

M.AMM.1Demonstrate reasoning skills in developing, explaining and justifying sound mathematical arguments and analyze the soundness of mathematical arguments of others.M.AMM.10Determine and interpret conditional probabilities and probabilities of compound events by constructing and analyzing representations, including tree diagrams, Venn diagrams, two-way frequency tables and area models, to make decisions in problem situations.M.AMM.11Use probabilities to make and justify decisions about risks in everyday life.M.AMM.12Calculate expected value to analyze mathematical fairness, payoff and risk.M.AMM.13Identify limitations or lack of information in studies reporting statistical information, especially when studies are reported in condensed form.M.AMM.14Interpret and compare the results of polls, given a margin of error.M.AMM.15Identify uses and misuses of statistical analyses in studies reporting statistics or using statistics to justify particular conclusions, including assertions of cause and effect versus correlation.M.AMM.16Describe 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.M.AMM.17Identify the population of interest, select an appropriate sampling technique and collect data.M.AMM.18Identify the variables to be used in a study.M.AMM.19Determine possible sources of statistical bias in a study and how such bias may affect the ability to generalize the results.M.AMM.2Communicate 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.M.AMM.20Create data displays for given data sets to investigate, compare, and estimate center, shape, spread and unusual features.M.AMM.21Determine possible sources of variability of data, both those that can be controlled and those that cannot be controlled.M.AMM.22Report 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.M.AMM.23Communicate statistical results in both oral and written formats using appropriate statistical and nontechnical language.M.AMM.24Solve problems involving large quantities that are not easily measured.M.AMM.25Use arrays to efficiently manage large collections of data and add, subtract, and multiply matrices to solve applied problems.M.AMM.26Determine or analyze an appropriate model for problem situations - including linear, quadratic, power, exponential, logarithmic and logistic functions (e.g., stopping distance, period of a pendulum, population growth, Richter Scale, and/or Fujita Tornado Scale).M.AMM.27Determine or analyze an appropriate cyclical model for problem situations that can be modeled with trigonometric functions (e.g., predator-prey models, tide heights, diurnal cycle, and/or music).M.AMM.28Determine or analyze an appropriate piecewise model for problem situations (e.g., postal rates, phase change graphs, sales tax, and/or utility usage rates).M.AMM.29Solve problems using recursion or iteration (e.g., fractals, compound interest, population growth or decline, and/or radioactive decay).M.AMM.3Gather data, conduct investigations and apply mathematical concepts and models to solve problems in mathematics and other disciplines.M.AMM.30Collect numerical bivariate data; use the data to create a scatter plot; determine whether or not a relationship exists; if so, select a function to model the data, justify the selection and use the model to make predictions.M.AMM.31Solve problems involving scheduling or routing situations that can be represented by a vertex-edge graph; find critical paths, Euler paths, Hamiltonian paths, and minimal spanning trees (e.g., Konigsberg bridge problem, mail vs. Fed Ex delivery routes, kolam drawings of India, traveling salesman problem, and/or map coloring).M.AMM.32Construct, analyze, and interpret flow charts in order to develop and describe problem solving procedures.M.AMM.33Apply and analyze various ranking algorithms to determine an appropriate method for a given situation (e.g., fair division, apportionment, and/or search engine results).M.AMM.34Analyze various voting and selection processes to determine an appropriate method for a given situation (e.g., preferential vs. non-preferential methods, and/or weighted voting).M.AMM.35Create and use two- and three-dimensional representations of authentic situations using paper techniques or dynamic geometric environments for computer-aided design and other applications.M.AMM.36Solve geometric problems involving inaccessible distances.M.AMM.37Use vectors to represent and solve applied problems.M.AMM.38Use matrices to represent geometric transformations and solve applied problems.M.AMM.4Determine, represent and analyze mathematical models for loan amortization and the effects of different payments and/or finance terms (e.g., Auto, Mortgage, and/or Credit Card).M.AMM.5Determine, represent and analyze mathematical models for investments involving simple and compound interest with and without additional deposits. (e.g., Savings accounts, bonds, and/or certificates of deposit.)M.AMM.6Determine, represent, and analyze mathematical models for Inflation and the Consumer Price Index using concepts of rate of change and percentage growth.M.AMM.7Research and analyze personal budgets based on given parameters (e.g., Fixed and discretionary expenses, insurance, gross vs. net pay, types of income, wage, salary, commission), career choice, geographic region, retirement and/or investment planning, etc.).M.AMM.8Research and analyze taxes including payroll, sales, personal property, real estate and income tax returns.M.AMM.9Use the Fundamental Counting Principle, Permutations and Combinations to determine all possible outcomes for an event; determine probability and odds of a simple event; explain the significance of the Law of Large Numbers.M.AMM.DCCDeveloping College and Career SkillsM.AMM.FFinanceM.AMM.GGeometryM.AMM.MModelingM.AMM.NNetworksM.AMM.PProbabilityM.AMM.SStatisticsM.AMM.SDMSocial Decision Making
Example Problems
If you flip three fair coins, what is the probability that you'll get at least two heads?
Ramkin is going to flip a fair coin 1,200 times.
What is the best prediction for the number of times that the coin will land heads up?
A fair coin is flipped 5 times. Let be the number of heads. What is the expected value of ?
A restaurant chain wants to make a one-sample z interval to estimate the proportion of customers who are satisfied with a new menu item. They want the margin of error to be no more than at confidence.

What is the smallest sample size required to obtain the desired margin of error?
The parks department will repaint 5 of 50 trail markers. They number the markers and use the random digit table printed below to choose a simple random sample.

Which markers are in the sample?
Goblins

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