Reasoning with Equations and Inequalities; Represent and solve equations and inequalities graphically

NC.M1.A-REI.10 Understand that the graph of a two variable equation represents the set of all solutions to the equation.

Interpreting Functions

Understand the concept of a function and use function notation.

NC.M1.F-IF.1 Build an understanding that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range by recognizing that:

• if f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x.

• the graph of f is the graph of the equation y = f(x).

NC.M1.F-IF.2 Use function notation to evaluate linear, quadratic, and exponential functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

NC.MI.F-IF.3 Recognize that recursively and explicitly defined sequences are functions whose domain is a subset of the integers, the terms of an arithmetic sequence are a subset of the range of a linear function, and the terms of a geometric sequence are a subset of the range of an exponential function.

Interpreting Functions

Interpret functions that arise in applications in terms of the context.

NC.M1.F-IF.4 Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities, including: intercepts; intervals where the function is increasing, decreasing, positive, or negative; and maximums and minimums.

NC.M1.F-IF.5 Interpret a function in terms of the context by relating its domain and range to its graph and, where applicable, to the quantitative relationship it describes.

NC.M1.F-IF.6 Calculate and interpret the average rate of change over a specified interval for a function presented numerically, graphically, and/or symbolically.

Interpreting Categorical and Quantitative Data

Interpret linear models.

NC.M1.S-ID.7 Interpret in context the rate of change and the intercept of a linear model. Use the linear model to interpolate and extrapolate predicted values. Assess the validity of a predicted value.

NC.M1.S-ID.8 Analyze patterns and describe relationships between two variables in context. Using technology, determine the correlation coefficient of bivariate data and interpret it as a measure of the strength and direction of a linear relationship. Use a scatter plot, correlation coefficient, and a residual plot to determine the appropriateness of using a linear function to model a relationship between two variables.

NC.M1.S-ID.9 Distinguish between association and causation.

Summarize, represent, and interpret data on a single count or measurement variable.

NC.M1.S-ID.1 Use technology to represent data with plots on the real number line (histograms, and box plots).

NC.M1.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. Interpret differences in shape, center, and spread in the context of the data sets.

NC.M1.S-ID.3 Examine the effects of extreme data points (outliers) on shape, center, and/or spread.

Interpret functions that arise in applications in terms of the context.

NC.M2.F-IF.4 Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities, including: domain and range, rate of change, symmetries, and end behavior.

Making Inference and Justifying Conclusions

Understand and evaluate random processes underlying statistical experiments.

NC.M2.S-IC.2 Use simulation to determine whether the experimental probability generated by sample data is consistent with the theoretical probability based on known information about the population.

Conditional Probability and the Rules for Probability

Understand independence and conditional probability and use them to interpret data.

NC.M2.S-CP.1 Describe events as subsets of the outcomes in a sample space using characteristics of the outcomes or as unions, intersections and complements of other events.

NC.M2.S-CP.3b Develop and understand independence and conditional probability.

a. Use a 2-way table to develop understanding of the conditional probability of A given B (written P(A|B)) as the likelihood that A will occur given that B has occurred. That is, P(A|B) is the fraction of event B’s outcomes that also belong to event A.

b. Understand that event A is independent from event B if the probability of event A does not change in response to the occurrence of event B. That is P(A|B)=P(A).

NC.M2.S-CP.4 Represent data on two categorical variables by constructing a two-way frequency table of data. Interpret the two-way table as a sample space to calculate conditional, joint and marginal probabilities. Use the table to decide if events are independent.

NC.M2.S-CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

Conditional Probability and the Rules for Probability

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

NC.M2.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 context.

Create equations that describe numbers or relationships.

NC.M3.A-CED.1 Create equations and inequalities in one variable that represent absolute value, polynomial, exponential, and rational relationships and use them to solve problems algebraically and graphically.

NC.M3.A-CED.2 Create and graph equations in two variables to represent absolute value, polynomial, exponential and rational relationships between quantities.

NC.M3.A-CED.3 Create systems of equations and/or inequalities to model situations in context.

Geometric Measurement & Dimension

Explain volume formulas and use them to solve problems.

NC.M3.G-GMD.3 Use the volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems.

Geometric Measurement & Dimension

Visualize relationships between two-dimensional and three-dimensional objects.

NC.M3.G-GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Modeling with Geometry

Apply geometric concepts in modeling situations.

NC.M3.G-MG.1 Apply geometric concepts in modeling situations

• Use geometric and algebraic concepts to solve problems in modeling situations:

• Use geometric shapes, their measures, and their properties, to model real-life objects.

• Use geometric formulas and algebraic functions to model relationships.

• Apply concepts of density based on area and volume.

• Apply geometric concepts to solve design and optimization problems.

Making Inference and Justifying Conclusions

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

NC.M3.S-IC.3 Recognize the purposes of and differences between sample surveys, experiments, and observational studies and understand how randomization should be used in each.

NC.M3.S-IC.4 Use simulation to understand how samples can be used to estimate a population mean or proportion and how to determine a margin of error for the estimate.

NC.M3.S-IC.5 Use simulation to determine whether observed differences between samples from two distinct populations indicate that the two populations are actually different in terms of a parameter of interest.

NC.M3.S-IC.6 Evaluate articles and websites that report data by identifying the source of the data, the design of the study, and the way the data are graphically displayed.

COMPETENCY GOAL 1: The learner will analyze data and apply probability concepts to solve problems.

Objectives 1.01 Create and use calculator-generated models of linear, polynomial, exponential, trigonometric, power, and logarithmic functions of bivariate data to solve problems.

a) Interpret the constants, coefficients, and bases in the context of the data.

b) Check models for goodness-of-fit; use the most appropriate model to draw conclusions and make predictions.

Summarize and analyze univariate data to solve problems.

a) Apply and compare methods of data collection.

b) Apply statistical principles and methods in sample surveys