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Skills available for Virginia high school math standards

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Virginia Curriculum Framework: Algebra I Virginia Curriculum Framework: Algebra I
Virginia Curriculum Framework: Algebra II Virginia Curriculum Framework: Algebra II
Virginia Curriculum Framework: Algebra, Functions and Data Analysis Virginia Curriculum Framework: Algebra, Functions and Data Analysis
Virginia Curriculum Framework: Geometry Virginia Curriculum Framework: Geometry
Virginia Curriculum Framework: Mathematical Analysis Virginia Curriculum Framework: Mathematical Analysis
Virginia Curriculum Framework: Probability and Statistics Virginia Curriculum Framework: Probability and Statistics
Virginia Curriculum Framework: Trigonometry Virginia Curriculum Framework: Trigonometry
Virginia Standards of Learning (2016): Algebra I Virginia Standards of Learning (2016): Algebra I
Virginia Standards of Learning (2016): Algebra II Virginia Standards of Learning (2016): Algebra II
Virginia Standards of Learning (2016): Algebra, Functions and Data Analysis Virginia Standards of Learning (2016): Algebra, Functions and Data Analysis
Virginia Standards of Learning (2016): Geometry Virginia Standards of Learning (2016): Geometry
Virginia Standards of Learning (2016): Mathematical Analysis Virginia Standards of Learning (2016): Mathematical Analysis
Virginia Standards of Learning (2016): Probability and Statistics Virginia Standards of Learning (2016): Probability and Statistics
Virginia Standards of Learning (2016): Trigonometry Virginia Standards of Learning (2016): Trigonometry
Virginia College and Career Ready Performance Expectations Virginia College and Career Ready Performance Expectations

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Descriptive Statistics

PS.1 The student will analyze graphical displays of univariate data, including dotplots, stemplots, boxplots, cumulative frequency graphs, and histograms, to identify and describe patterns and departures from patterns, using central tendency, spread, clusters, gaps, and outliers.
- PS.1.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
  - PS.1.a.1 Create and interpret graphical displays of data, including dotplots, stemplots, boxplots, cumulative frequency graphs, and histograms, using appropriate technology.
  - PS.1.a.2 Examine graphs of data for clusters and gaps, and relate those phenomena to the data in context.
  - PS.1.a.3 Examine graphs of data for outliers, and explain the outlier(s) within the context of the data.
  - PS.1.a.4 Examine graphs of data and identify the central tendency of the data as well as the spread.
  - PS.1.a.5 Explain the central tendency and the spread of the data within the context of the data.
PS.2 The student will analyze numerical characteristics of univariate data sets to describe patterns and departures from patterns, using mean, median, mode, variance, standard deviation, interquartile range, range, and outliers.
- PS.2.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
  - PS.2.a.1 Interpret mean, median, mode, range, interquartile range, variance, and standard deviation of a univariate data set in terms of the problem's context.
  - PS.2.a.2 Identify possible outliers, using an algorithm.
  - PS.2.a.3 Explain the influence of outliers on a univariate data set.
  - PS.2.a.4 Explain ways in which standard deviation addresses dispersion by examining the formula for standard deviation.
PS.3 The student will compare distributions of two or more univariate data sets, numerically and graphically, analyzing center and spread (within group and between group variations), clusters and gaps, shapes, outliers, or other unusual features.
- PS.3.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
  - PS.3.a.1 Compare and contrast two or more univariate data sets, numerically and graphically, by analyzing measures of center and spread within a contextual framework.
  - PS.3.a.2 Describe any unusual features of the data, such as clusters, gaps, or outliers, within the context of the data.
  - PS.3.a.3 Analyze skewness in conjunction with measures of center and spread in a contextual framework.
PS.4 The student will analyze scatterplots to identify and describe the relationship between two variables, using shape; strength of relationship; clusters; positive, negative, or no association; outliers; and influential points.
- PS.4.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
  - PS.4.a.1 Examine scatterplots of data, and describe skewness, and correlation within the context of the data.
    - Interpret a scatter plot (A1-MM.8)
  - PS.4.a.2 Describe and explain any unusual features of the data, such as clusters, gaps, or outliers, within the context of the data.
  - PS.4.a.3 Identify influential data points (observations that have a great effect on a line of best fit because of extreme x-values) and describe the effect of the influential points.
PS.5 The student will determine and interpret linear correlation, use the method of least squares regression to model the linear relationship between two variables, and use the residual plot to assess linearity.
- PS.5.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
  - PS.5.a.1 Calculate a correlation coefficient, r.
  - PS.5.a.2 Explain how the correlation coefficient, r, measures association by looking at its formula.
  - PS.5.a.3 Interpret the coefficient of determination, r², in a contextual framework.
  - PS.5.a.4 Use regression lines to make predictions, and identify the limitations of the predictions.
  - PS.5.a.5 Use residual plots to determine whether a linear model is satisfactory for describing the relationship between two variables.
  - PS.5.a.6 Describe the errors inherent in extrapolation beyond the range of the data.
  - PS.5.a.7 Use least squares regression to determine the equation of the line of best fit for a set of data.
  - PS.5.a.8 Interpret the slope and y-intercept of the least squares regression line in a contextual framework.
  - PS.5.a.9 Explain how least squares regression generates the equation of the line of best fit by examining the formulas used in computation.
PS.6 The student will make logarithmic and power transformations to achieve linearity.
- PS.6.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
  - PS.6.a.1 Apply a logarithmic transformation to data.
  - PS.6.a.2 Explain how a logarithmic transformation works to achieve a linear relationship between variables.
  - PS.6.a.3 Apply a power transformation to data.
  - PS.6.a.4 Explain how a power transformation works to achieve a linear relationship between variables.
PS.7 The student, using two-way tables and other graphical displays, will analyze categorical data to describe patterns and departures from patterns and to determine marginal frequency and relative frequencies, including conditional frequencies.
- PS.7.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
  - PS.7.a.1 Produce a two-way table as a summary of the information obtained from two categorical variables.
  - PS.7.a.2 Create and interpret graphical displays of categorical data including bar charts.
  - PS.7.a.3 Calculate marginal, relative, and conditional frequencies in a two-way table.
  - PS.7.a.4 Use marginal, relative, and conditional frequencies to analyze data in two-way tables within the context of the data.

Data Collection

PS.8 The student will describe the methods of data collection in a census, sample survey, experiment, and observational study and identify an appropriate method of solution for a given problem setting.
- PS.8.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
  - PS.8.a.1 Compare and contrast controlled experiments and observational studies and the conclusions one can draw from each.
  - PS.8.a.2 Compare and contrast population and sample, and parameter and statistic.
  - PS.8.a.3 Identify biased sampling methods.
  - PS.8.a.4 Describe simple random sampling.
  - PS.8.a.5 Select a data collection method appropriate for a given context.
    - Experiment design (A2-FF.12)
    - Experiment design (PC-Z.15)
PS.9 The student will plan and conduct a survey. The plan will address sampling techniques and methods to reduce bias.
- PS.9.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
  - PS.9.a.1 Distinguish between a population and a sample.
  - PS.9.a.2 Investigate and describe sampling techniques, such as simple random sampling, stratified sampling, and cluster sampling.
  - PS.9.a.3 Determine which sampling technique is best, given a particular context.
  - PS.9.a.4 Plan a survey to answer a question or address an issue.
  - PS.9.a.5 Given a plan for a survey, identify possible sources of bias, and describe ways to reduce bias.
  - PS.9.a.6 Design a survey instrument.
  - PS.9.a.7 Conduct a survey.
PS.10 The student will plan and conduct a well-designed experiment. The plan will address control, randomization, replication, blinding, and measurement of experimental error.
- PS.10.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
  - PS.10.a.1 Plan and conduct a well-designed experiment. The experimental design should address control, randomization, replication, blinding and minimization of experimental error.
  - PS.10.a.2 Identify treatments, levels, factors, control groups, and experimental units in an experimental design.
  - PS.10.a.3 Identify sources of bias and confounding, including the placebo effect.
  - PS.10.a.4 Identify a situation when a block design, including matched pairs, would reduce the effects of confounding variables.

Probability

PS.11 The student will identify and describe two or more events as complementary, dependent, independent, and/or mutually exclusive.
- PS.11.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
  - PS.11.a.1 Define and give contextual examples of complementary, dependent, independent, and mutually exclusive events.
  - PS.11.a.2 Given two or more events in a problem setting, determine whether the events are complementary, dependent, independent, and/or mutually exclusive.
PS.12 The student will determine probabilities (relative frequency and theoretical), including conditional probabilities for events that are either dependent or independent, by applying the Law of Large Numbers concept, the addition rule, and the multiplication rule.
- PS.12.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
  - PS.12.a.1 Calculate relative frequency and expected frequency.
  - PS.12.a.2 Determine conditional probabilities for dependent, independent, and mutually exclusive events.
PS.13 The student will develop, interpret, and apply the binomial and geometric probability distributions for discrete random variables, including computing the mean and standard deviation for the binomial and geometric variables.
- PS.13.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
  - PS.13.a.1 Develop the binomial and geometric probability distributions within a practical context.
  - PS.13.a.2 Calculate the mean and standard deviation for the binomial and geometric variables.
    - Mean, variance, and standard deviation of binomial distributions (PC-Y.11)
  - PS.13.a.3 Use the binomial and geometric distributions to calculate probabilities associated with experiments for which there are only two possible outcomes.
PS.14 The student will simulate probability distributions, including binomial and geometric.
- PS.14.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
  - PS.14.a.1 Design and conduct a simulation of a binomial distribution.
  - PS.14.a.2 Design and conduct a simulation of a geometric distribution.
  - PS.14.a.3 Calculate probabilities resulting from simulations of binomial and geometric distributions.
PS.15 The student will identify random variables as independent or dependent and determine the mean and standard deviations for random variables and sums and differences of independent random variables.
- PS.15.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
  - PS.15.a.1 Compare and contrast independent and dependent random variables.
  - PS.15.a.2 Determine the mean (expected value) and standard deviation for a random variable and linear transformation of a random variable.
  - PS.15.a.3 Determine the mean (expected value) for sums and differences of random variables.
  - PS.15.a.4 Determine the standard deviation for sums and differences of independent random variables.
PS.16 The student will identify properties of a normal distribution and apply the normal distribution to determine probabilities.
- PS.16.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
  - PS.16.a.1 Identify the properties of a normal distribution.
  - PS.16.a.2 Describe how the standard deviation and the mean affect the graph of the normal distribution.
    - Find values of normal variables (PC-Y.15)
  - PS.16.a.3 Calculate and interpret the ??-score of a given data value from a normal distribution.
    - Find z-values (A2-EE.13)
    - Find z-values (PC-Y.14)
  - PS.16.a.4 Determine the probability of a given event, using the normal distribution.
  - PS.16.a.5 Use a graphing utility and a table of Standard Normal Probabilities to determine probabilities.

Inferential Statistics

PS.17 The student, given data from a large sample, will determine and interpret appropriate point estimates and confidence intervals for parameters. The parameters will include proportion and mean, difference between two proportions, difference between two means (independent and paired), and slope of a least-squares regression line.
- PS.17.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
  - PS.17.a.1 Construct confidence intervals to estimate a population parameter, such as a proportion or the difference between two proportions; a mean or the difference between two means; or slope of a least-squares regression line.
  - PS.17.a.2 Select a value for the confidence level of a confidence interval.
  - PS.17.a.3 Interpret confidence intervals and confidence levels in the context of the data.
    - Interpret confidence intervals for population means (A2-FF.11)
    - Interpret confidence intervals for population means (PC-Z.14)
  - PS.17.a.4 Explain the importance of random sampling for confidence intervals.
  - PS.17.a.5 Explain how changes in confidence level and sample size effect width of the confidence interval and margin of error.
  - PS.17.a.6 Calculate point estimates for parameters and discuss the limitations of point estimates.
PS.18 The student will apply and interpret the logic of an appropriate hypothesis-testing procedure. Tests will include large sample test for proportion, mean, difference between two proportions, difference between two means (independent and paired); chi-squared tests for goodness of fit, homogeneity of proportions, and independence; and slope of a least-squares regression line.
- PS.18.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
  - PS.18.a.1 Use the chi-squared test for goodness of fit to decide whether the population being analyzed fits a particular distribution pattern.
  - PS.18.a.2 Use hypothesis-testing procedures to determine whether or not to reject the null hypothesis. The null hypothesis may address proportion, mean, difference between two proportions or two means, goodness of fit, homogeneity of proportions, independence, and the slope of a least-squares regression line.
  - PS.18.a.3 Compare and contrast Type I and Type II errors.
  - PS.18.a.4 Explain how and why the hypothesis-testing procedure allows one to reach a statistical decision.
PS.19 The student will identify the meaning of sampling distribution with reference to random variable, sampling statistic, and parameter and explain the Central Limit Theorem. This will include sampling distribution of a sample proportion, a sample mean, a difference between two sample proportions, and a difference between two sample means.
- PS.19.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
  - PS.19.a.1 Describe the use of the Central Limit Theorem for drawing inferences about a population parameter based on a sample statistic.
  - PS.19.a.2 Describe the effect of sample size on the sampling distribution and on related probabilities.
  - PS.19.a.3 Use the normal approximation to calculate probabilities of sample statistics falling within a given interval.
    - The Central Limit Theorem (PC-Y.17)
  - PS.19.a.4 Identify and describe the characteristics of a sampling distribution of a sample proportion, mean, difference between two sample proportions, or difference between two sample means.
    - Distributions of sample means (A2-EE.15)
    - Distributions of sample means (PC-Y.16)
PS.20 The student will identify properties of a t-distribution and apply t-distributions to single-sample and two-sample (independent and matched pairs) t-procedures.
- PS.20.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
  - PS.20.a.1 Identify the properties of a t-distribution.
  - PS.20.a.2 Compare and contrast a t-distribution and a normal distribution.
  - PS.20.a.3 Use a t-test for single-sample and two-sample data.

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Descriptive Statistics

PS.1 The student will analyze graphical displays of univariate data, including dotplots, stemplots, boxplots, cumulative frequency graphs, and histograms, to identify and describe patterns and departures from patterns, using central tendency, spread, clusters, gaps, and outliers.

PS.1.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

PS.1.a.1 Create and interpret graphical displays of data, including dotplots, stemplots, boxplots, cumulative frequency graphs, and histograms, using appropriate technology.

PS.1.a.2 Examine graphs of data for clusters and gaps, and relate those phenomena to the data in context.

PS.1.a.3 Examine graphs of data for outliers, and explain the outlier(s) within the context of the data.

PS.1.a.4 Examine graphs of data and identify the central tendency of the data as well as the spread.

PS.1.a.5 Explain the central tendency and the spread of the data within the context of the data.

PS.2 The student will analyze numerical characteristics of univariate data sets to describe patterns and departures from patterns, using mean, median, mode, variance, standard deviation, interquartile range, range, and outliers.

PS.2.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

PS.2.a.1 Interpret mean, median, mode, range, interquartile range, variance, and standard deviation of a univariate data set in terms of the problem's context.

PS.2.a.2 Identify possible outliers, using an algorithm.

PS.2.a.3 Explain the influence of outliers on a univariate data set.

PS.2.a.4 Explain ways in which standard deviation addresses dispersion by examining the formula for standard deviation.

PS.3 The student will compare distributions of two or more univariate data sets, numerically and graphically, analyzing center and spread (within group and between group variations), clusters and gaps, shapes, outliers, or other unusual features.

PS.3.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

PS.3.a.1 Compare and contrast two or more univariate data sets, numerically and graphically, by analyzing measures of center and spread within a contextual framework.

PS.3.a.2 Describe any unusual features of the data, such as clusters, gaps, or outliers, within the context of the data.

PS.3.a.3 Analyze skewness in conjunction with measures of center and spread in a contextual framework.

PS.4 The student will analyze scatterplots to identify and describe the relationship between two variables, using shape; strength of relationship; clusters; positive, negative, or no association; outliers; and influential points.

PS.4.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

PS.4.a.1 Examine scatterplots of data, and describe skewness, and correlation within the context of the data.

PS.4.a.2 Describe and explain any unusual features of the data, such as clusters, gaps, or outliers, within the context of the data.

PS.4.a.3 Identify influential data points (observations that have a great effect on a line of best fit because of extreme x-values) and describe the effect of the influential points.

PS.5 The student will determine and interpret linear correlation, use the method of least squares regression to model the linear relationship between two variables, and use the residual plot to assess linearity.

PS.5.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

PS.5.a.1 Calculate a correlation coefficient, r.

PS.5.a.2 Explain how the correlation coefficient, r, measures association by looking at its formula.

PS.5.a.3 Interpret the coefficient of determination, r², in a contextual framework.

PS.5.a.4 Use regression lines to make predictions, and identify the limitations of the predictions.

PS.5.a.5 Use residual plots to determine whether a linear model is satisfactory for describing the relationship between two variables.

PS.5.a.6 Describe the errors inherent in extrapolation beyond the range of the data.

PS.5.a.7 Use least squares regression to determine the equation of the line of best fit for a set of data.

PS.5.a.8 Interpret the slope and y-intercept of the least squares regression line in a contextual framework.

PS.5.a.9 Explain how least squares regression generates the equation of the line of best fit by examining the formulas used in computation.

PS.6 The student will make logarithmic and power transformations to achieve linearity.

PS.6.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

PS.6.a.1 Apply a logarithmic transformation to data.

PS.6.a.2 Explain how a logarithmic transformation works to achieve a linear relationship between variables.

PS.6.a.3 Apply a power transformation to data.

PS.6.a.4 Explain how a power transformation works to achieve a linear relationship between variables.

PS.7 The student, using two-way tables and other graphical displays, will analyze categorical data to describe patterns and departures from patterns and to determine marginal frequency and relative frequencies, including conditional frequencies.

PS.7.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

PS.7.a.1 Produce a two-way table as a summary of the information obtained from two categorical variables.

PS.7.a.2 Create and interpret graphical displays of categorical data including bar charts.

PS.7.a.3 Calculate marginal, relative, and conditional frequencies in a two-way table.

PS.7.a.4 Use marginal, relative, and conditional frequencies to analyze data in two-way tables within the context of the data.

Data Collection

PS.8 The student will describe the methods of data collection in a census, sample survey, experiment, and observational study and identify an appropriate method of solution for a given problem setting.

PS.8.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

PS.8.a.1 Compare and contrast controlled experiments and observational studies and the conclusions one can draw from each.

PS.8.a.2 Compare and contrast population and sample, and parameter and statistic.

PS.8.a.3 Identify biased sampling methods.

PS.8.a.4 Describe simple random sampling.

PS.8.a.5 Select a data collection method appropriate for a given context.

PS.9 The student will plan and conduct a survey. The plan will address sampling techniques and methods to reduce bias.

PS.9.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

PS.9.a.1 Distinguish between a population and a sample.

PS.9.a.2 Investigate and describe sampling techniques, such as simple random sampling, stratified sampling, and cluster sampling.

PS.9.a.3 Determine which sampling technique is best, given a particular context.

PS.9.a.4 Plan a survey to answer a question or address an issue.

PS.9.a.5 Given a plan for a survey, identify possible sources of bias, and describe ways to reduce bias.

PS.9.a.6 Design a survey instrument.

PS.9.a.7 Conduct a survey.

PS.10 The student will plan and conduct a well-designed experiment. The plan will address control, randomization, replication, blinding, and measurement of experimental error.

PS.10.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

PS.10.a.1 Plan and conduct a well-designed experiment. The experimental design should address control, randomization, replication, blinding and minimization of experimental error.

PS.10.a.2 Identify treatments, levels, factors, control groups, and experimental units in an experimental design.

PS.10.a.3 Identify sources of bias and confounding, including the placebo effect.

PS.10.a.4 Identify a situation when a block design, including matched pairs, would reduce the effects of confounding variables.

Probability

PS.11 The student will identify and describe two or more events as complementary, dependent, independent, and/or mutually exclusive.

PS.11.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

PS.11.a.1 Define and give contextual examples of complementary, dependent, independent, and mutually exclusive events.

PS.11.a.2 Given two or more events in a problem setting, determine whether the events are complementary, dependent, independent, and/or mutually exclusive.

PS.12 The student will determine probabilities (relative frequency and theoretical), including conditional probabilities for events that are either dependent or independent, by applying the Law of Large Numbers concept, the addition rule, and the multiplication rule.

PS.12.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to