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
ab^{ct} = 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). |