Standards Database Logo
Home | Browse | Search | Purpose | History | Process | Acknowledgment| Reference

 


 


Topic: Modeling 

Common Core Mathematics 9-12

 Cluster  Reason quantitatively and use units to solve problems.
  Grade 9-12
   N.Q.1.Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
   N.Q.2.Define appropriate quantities for the purpose of descriptive modeling.
   N.Q.3.Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
 Cluster  Write expressions in equivalent forms to solve problems
  Grade 9-12
   A.SSE.3.Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
   A.SSE.4.Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.
 Cluster  Create equations that describe numbers or relationships
  Grade 9-12
   A.CED.1.Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
   A.CED.2.Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
   A.CED.3.Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
   A.CED.4.Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.
 Cluster  Represent and solve equations and inequalities graphically
  Grade 9-12
   A.REI.11.Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
 Cluster  Interpret functions that arise in applications in terms of the context
  Grade 9-12
   F.IF.4.For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
   F.IF.5.Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
   F.IF.6.Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
 Cluster  Analyze functions using different representations
  Grade 9-12
   F.IF.7.Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
 Cluster  Build a function that models a relationship between two quantities
  Grade 9-12
   F.BF.1.Write a function that describes a relationship between two quantities.
   F.BF.2.Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
 Cluster  Construct and compare linear, quadratic, and exponential models and solve problems
  Grade 9-12
   F.LE.1.Distinguish between situations that can be modeled with linear functions and with exponential functions.
   F.LE.1.a.Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
   F.LE.1.b.Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
   F.LE.1.c.Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
   F.LE.2.Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
   F.LE.3.Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
   F.LE.4.For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e;evaluate the logarithm using technology.
 Cluster  Model periodic phenomena with trigonometric functions
  Grade 9-12
   F.TF.5.Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
 Cluster  Define trigonometric ratios and solve problems involving right triangles
  Grade 9-12
   G.SRT.8.Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
 Cluster  Explain volume formulas and use them to solve problems
  Grade 9-12
   G.GMD.3.Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
 Cluster  Apply geometric concepts in modeling situations
  Grade 9-12
   G,MG.1.Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
   G.MG.2.Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
   G.MG.3.Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
 Cluster  Summarize, represent, and interpret data on a single count or measurement variable
  Grade 9-12
   S.ID.1.Represent data with plots on the real number line (dot plots, histograms, and box plots).
   S.ID.2.Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
   S.ID.3.Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
   S.ID.4.Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
 Cluster  Summarize, represent, and interpret data on two categorical and quantitative variables
  Grade 9-12
   S.ID.5.Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
   S.ID.6.Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
   S.ID.6.a.Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
   S.ID.6.b.Informally assess the fit of a function by plotting and analyzing residuals.
   S.ID.6.c.Fit a linear function for a scatter plot that suggests a linear association.
 Cluster  Interpret linear models
  Grade 9-12
   S.ID.7.Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
   S.ID.8.Compute (using technology) and interpret the correlation coefficient of a linear fit.
   S.ID.9.Distinguish between correlation and causation.
 Cluster  Understand and evaluate random processes underlying statistical experiments
  Grade 9-12
   S.IC.1.Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
   S.IC.2.Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
 Cluster  Make inferences and justify conclusions from sample surveys, experiments, and observational studies
  Grade 9-12
   S.IC.3.Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
   S.IC.4.Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
   S.IC.5.Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
   S.IC.6.Evaluate reports based on data.
 Cluster  Understand independence and conditional probability and use them to interpret data
  Grade 9-12
   S.CP.1.Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
   S.CP.2.Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
   S.CP.3.Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
   S.CP.4.Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
   S.CP.5.Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
 Cluster  Use the rules of probability to compute probabilities of compound events in a uniform probability model
  Grade 9-12
   S.CP.6.Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.
   S.CP.7.Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
   S.CP.8.(+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
   S.CP.9.(+) Use permutations and combinations to compute probabilities of compound events and solve problems.
 Cluster  Calculate expected values and use them to solve problems
  Grade 9-12
   S.MD.1.(+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.
   S.MD.2.(+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
   S.MD.3.(+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.
   S.MD.4.(+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?
 Cluster  Use probability to evaluate outcomes of decisions
  Grade 9-12
   S.MD.5.(+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
   S.MD.5.a.Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fastfood restaurant.
   S.MD.5.b.Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
   S.MD.6.(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
   S.MD.7.(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).