Indiana Academic Standards for Mathematics – Probability ......Indiana Academic Standards for...

Indiana Academic Standards for Mathematics – Probability and Statistics Standards Resource Guide Document

This Teacher Resource Guide, revised in July 2018, provides supporting materials to help educators successfully implement the Indiana Academic Standards for Probability and Statistics. This resource guide is provided to help ensure all students meet the rigorous learning expectations set by the academic standards. Use of this guide and the resources on the web page is optional – teachers should decide which resources will work best for their students. However, all guidance contained in this document and on the website has been chosen to best support effective teaching practices and promote the Mathematics Process Standards.

With an increased emphasis on content area literacy, academic vocabulary has been noted. Best practices should be utilized when teaching students academic vocabulary. Please see the Literacy Framework and the Science and Technical Subjects Content Area Literacy Standards for examples of best practices.

Examples have been removed from the document as they tend to limit interpretation and classroom application. Rather, success criteria, in the form of “I can” statements, have been included. According to Hattie (2017), success criteria is specific, concrete and measurable, describing what success looks like when a learning goal is reached. Additionally, success criteria contributes to teacher clarity, which has a 0.75 effect size! An effect size of 0.40 reportedly indicates one year of growth. Utilizing success criteria in the classroom allows students to monitor their own learning and increases motivation (Hattie, p. 57). It is important to note that the success criteria provided here are not intended to be limiting. Teachers may have additional success criteria for their students.

Guidance around vertical articulation has been provided in the last two columns. Knowing what was expected of students at previous grade levels will help teachers connect new learning to prior knowledge. Additionally, understanding what a student will be expected to learn in the future provides the teacher a context for the current learning. This information is not exhaustive; rather it is provided to give teachers a quick understanding of how the work builds from previous grade levels into subsequent courses. The Indiana Department of Education (IDOE) math team recommends teachers further study this vertical articulation to situate their course objectives in the broader math context.

If you have any questions, please do not hesitate to reach out to the IDOE math team. Contact information for the Elementary and Secondary Math Specialists can be found on the website: https://www.doe.in.gov/standards/mathematics. If you have suggested resources for the website, please share those as well.

Hattie, J., Fisher, D., Frey, N., Gojak, L. M., Moore, S. D., & Mellman, W. (2017). Visible learning for mathematics: What works best to optimize student learning, grades K-12. Thousand Oaks, CA: Corwin Mathematics.

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https://www.doe.in.gov/standards/mathematics


Data Analysis Probability and Statistics Mathematics

Standards Success Criteria Academic Vocabulary Looking Back MA.PS.DA.1: Create, compare, and evaluate

different graphic displays of the same data, using histograms, frequency polygons, cumulative frequency distribution functions, pie charts, scatterplots, stem-and-leaf plots, and box-and-whisker plots. Draw these with and without technology.

I can create histograms.

I can create frequency polygons.

I can create frequency distribution functions.

I can create pie charts.

I can create scatterplots.

I can create stem-and-leaf plots.

I can create box-and-whisker plots.

I can create and evaluate different graphic displays of the same data with the use of technology.

I can create and evaluate different graphic displays of the same data without the use of technology.

I can compare different graphical displays of the same data.

Histogram

Frequency polygons

Cumulative frequency distribution functions

Pie chart

Scatterplots

Stem and Leaf Plot

Box-and-whisker plots

Select, create, and interpret graphical representations of numerical data, including line plots, histograms, and box plots. (MA.6.DS.2)

Formulate statistical questions; collect and organize the data, display and interpret the data with graphical representations. (MA.6.DS.3)

Graph bivariate data on a scatter plot. (MA.AI.DS.2)

Read, interpret, and make decisions about data summarized numerically using measures of center and spread, in tables, and in graphical displays. (MA.AA.DSP.3)

MA.PS.DA.2: Compute and use mean, median, mode, weighted mean, geometric mean, harmonic mean, range, quartiles, variance, and standard

I can compute and use the mean, median, and mode.

I can compute and use weighted means.

Weighted mean

Geometric mean

Harmonic mean

Organize, graph, and compare univariate data of two or more different data sets using measures of center (mean and

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deviation. Use tables and technology to estimate areas under the normal curve. Fit a data set to a normal distribution and estimate population percentages. Recognize that there are data sets not normally distributed for which such procedures are inappropriate.

I can compute and use geometric means and harmonic means.

I can compute and use measures of shape and spread such as range, quartiles, variance, and standard deviation.

I can use tables to estimate the area under the normal curve.

I can use technology to estimate the area under the normal curve.

I can fit a data set to a normal distribution.

I can estimate population percentages for a data set fit to a normal curve.

I can identify data sets that are not normally distributed.

Quartiles

Variance

Standard deviation

Normal distribution

Normal curve

Mean

Median

Mode

Range

Population percentages

median) and spread (range, inter-quartile range, standard deviation, percentiles, and variance). (MA.AII.DSP.3)

Analyze and compare univariate data of two or more different data sets using measures of center (mean, median, and mode), shape, and spread (range, interquartile range, standard deviation, percentiles, and variance) making use of technology. (MA.AA.DSP.4)

MA.PS.DA.3: Understand the central limit theorem and use it to solve problems.

I can discuss the central limit theorem.

I can use the central limit theorem to solve problems.

Central limit theorem Analyze and compare univariate data of two or more different data sets using measures of center, shape, and spread, making use of technology. (MA.AA.DSP.4)

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Record multiple observations of random events and construct empirical models of the probability distributions. (MA.AA.DSP.5)

Draw conclusions and make decisions in various probabilistic contexts. (MA.AA.DSP.6)

MA.PS.DA.4: Understand hypothesis tests of means and differences between means and use them to reach conclusions. Compute and use confidence intervals to make estimates. Construct and interpret margin of error and confidence intervals for population proportions.

I can use hypothesis tests of means to reach conclusions about data.

I can apply differences between means to draw conclusions about data.

I can compute confidence intervals.

I can use confidence intervals to make estimates.

I can construct the margin of error and confidence intervals for population proportions.

I can interpret the margin of error and confidence intervals for population proportions.

Hypothesis Test of means

Difference between means

Confidence intervals

Margin of error

Population proportions


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MA.PS.DA.5: Recognize how linear transformations of univariate data affect shape, center, and spread.

I can explore and discuss how linear transformations of univariate data affect shape, center, and spread.

Linear Transformation

Univariate data


Analyze and compare univariate data of two or more different data sets using measures of center (mean, median, and mode), shape, and spread (range, interquartile range, standard deviation, percentiles, and variance) making use of technology. (MA.AA.DSP.4)

MA.PS.DA.6: 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.

I can construct a two-way frequency table of data when two categories are associated with each object being classified.

I can interpret a two-way frequency table of data when two categories are associated with each object being classified.

I can use a two-way table as a sample space to determine if events are independent.

Two-way frequency tables

Conditional Probabilities

Sample space

Event

Independent events

Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. (MA.AI.DS.5)

Understand dependent and independent events, and conditional probability; apply these concepts to calculate probabilities. (MA.AII.DSP.5)

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I can use a two-way table to approximate conditional probabilities.

Record multiple observations of random events and construct empirical models of the probability distributions. Construct a theoretical model and apply the law of large numbers to show the relationship between the two models. (MA.AA.DSP.6)

MA.PS.DA.7: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

I can determine if a specified model is consistent with results from a given data-generating process.

I can predict the outcomes of a simulation in order to assign a correct model.

Simulation Record multiple observations of random events and construct empirical models of the probability distributions. Construct a theoretical model and apply the law of large numbers to show the relationship between the two models. (MA.AA.DSP.5)

Evaluate the validity of claims based on empirical probabilities and theoretical probabilities. (MA.AA.DSP.6)

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MA.PS.DA.8: Understand the meaning of measurement data and categorical data, of univariate and bivariate data, and of the term variable.

I can identify measurement data and categorical data.

I can distinguish between univariate and bivariate data.

I can describe the meaning of the term variable.

Categorical data

Measurement data

Univariate data

Bivariate data

Term variable

Choose, create, and critique, with technology, mathematical models (linear, quadratic and exponential) for bivariate data sets. (MA.AA.DSP.2)

Analyze and compare univariate data of two or more different data sets. (MA.AA.DSP.4)

MA.PS.DA.9: Understand statistics and use sampling distributions as a process for making inferences about population parameters based on a random sample from that population.

I can describe the purpose and uses of statistics.

I can use sampling distributions as a process for making inferences about population parameters.

I can make inferences about populations based on a random sample from that population.

Sampling Distribution

Population parameters

Inference

Random sample

Population

Understand that statistics can be used to gain information about a population by examining a sample of the population. (MA.7.DSP.1)

Use data from a random sample to draw inferences about a population. (MA.7.DSP.2)

Understand that statistics and data are non-neutral and designed to serve a particular interest. (MA.AI.DS.6)

MA.PS.DA.10: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of

I can use data from a sample survey to estimate a population mean.

Sample survey

Population mean

Make inferences and justify conclusions from sample surveys. (MA.AA.DSP.1)

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simulation models for random sampling.

I can use data from a sample survey to estimate a population proportion.

I can develop a margin of error through the use of simulation models for random sampling.

Population proportion

Margin of error

Simulation models

Random sampling

MA.PS.DA.11: Find linear models by using median fit and least squares regression methods to make predictions. Decide which among several linear models gives a better fit. Interpret the slope and intercept in terms of the original context. Informally assess the fit of a function by plotting and analyzing residuals.

I can find a linear model using median fit regression in order to make predictions.

I can find a linear model using least squares regression in order to make predictions.

When given several linear models for a bivariate data set, I can determine which model gives a better fit.

I can interpret the meaning of the slope and intercept of a linear model in a given context.

I can plot and analyze residuals in an effort to informally assess the fit of a function to data.

Median fit regression

Least squares regression

Linear model

Bivariate data

Slope

Intercept

Residuals

For scatter plots that suggest a linear association, informally fit a straight line, and describe the model fit by judging the closeness of the data points to the line. (MA.8.DSP.2)

Use technology to find a linear function that models a relationship for a bivariate data set to make predictions; interpret the slope and y-intercept. (MA.AI.DS.3)

Use technology to find a linear, quadratic, or exponential function that models a relationship for a bivariate data set to make predictions. (MA.AII.DSP.2)

Choose, create, and critique, with technology,

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mathematical models (linear, quadratic and exponential) for bivariate data sets. (MA.AA.DSP.2)

MA.PS.DA.12: Evaluate reports based on data by considering the source of the data, the design of the study, the way the data are analyzed and displayed, and whether the report confuses correlation with causation. Distinguish between correlation and causation.

I can evaluate reports based on data by considering the source and design of the study.

I can consider the way data are analyzed and displayed in order to evaluate a report based on data.

I can determine if a report based on data confuses correlation and causation.

I can identify various types of causation in order to distinguish between correlation and causation.

Correlation

Causation

Distinguish between correlation and causation. (MA.AI.DS.4)

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. (MA.AII.DSP.1)

Make inferences and justify conclusions from sample surveys, experiments, and observational studies. (MA.AA.DSP.1)

Interpret the correlation coefficient for linear models. (MA.AA.DSP.2)

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Experimental Design Probability and Statistics Mathematics

Standards Success Criteria Academic Vocabulary Looking Back MA.PS.ED.1: Formulate questions that can be

addressed with data. Collect, organize, and display relevant data to answer the questions formulated.

I can ask questions that can be answered with data.

I can collect, organize, and display relevant data to answer questions.

Design simple experiments or investigations to collect data to answer questions of interest, and make inferences from sample results. (MA.AI.DS.1)

Read, interpret, and make decisions about data summarized numerically using measures of center and spread, in tables, and in graphical displays, (MA.AA.DSP.3)

MA.PS.ED.2: Use election theory techniques to analyze election data. Use weighted voting techniques to decide voting power within a group.

I can use election theory techniques to analyze election data.

I can use weighted voting techniques to decide voting power within a group.

Election theory

Weighted voting

MA.PS.ED.3: Construct simulated sampling distributions of sample proportions and use sampling distributions to identify which proportions are likely to be found in a sample of a given size.

I can create simulated sampling distributions of sampling proportions.

I can use sampling distributions to identify which proportions are likely to be found in a sample of a given size.

Sampling distributions

Sampling proportions

Sample

Record multiple observations of random events and construct empirical models of the probability distributions. Construct a theoretical model and apply the law

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of large numbers to show the relationship between the two models. (MA.AA.DSP.5)

MA.PS.ED.4: Use simulations to explore the variability of sample statistics from a known population and to construct sampling distributions.

I can use simulations to explore the variability of sample statistics from a known population.

I can construct sampling distributions.

Simulation

Variability

Sample statistics

Population

Sampling distributions

Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for the variability in the answers. (MA.6.DS.1)

MA.PS.ED.5: Model and solve real-world problems using the geometric distribution or waiting-time distribution, with or without technology.

I can model real-world problems using geometric distribution with and without technology.

I can use a geometric distribution to solve real-world problems.

I can model real-world problems using waiting-time distribution with and without technology.

I can use waiting-time distribution to solve real-world problems.

Geometric distribution

Waiting-time distribution

Represent real-world problems using exponential functions in one or two variables and solve such problems with technology. (MA.AA.EL.3)

MA.PS.ED.6: Model and solve real-world problems involving patterns using recursion and iteration, growth and decay, and compound interest.

I can model real-world problems involving patterns using recursion and iteration.

Recursion

Iteration

Compound interest

Write arithmetic and geometric sequences both recursively and with an explicit formula;

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I can solve real-world problems involving recursion and iteration.

I can model real-world problems involving growth, decay, and compound interest.

I can solve real-world problems involving growth, decay, and compound interest.

use them to model situations. (MA.AII.EL.1)

Model real world situations involving arithmetic sequences and understand that they can be defined both recursively and with an explicit formula. (MA.AA.LF.1)

Model real world situations involving geometric sequences and understand that they can be defined both recursively and with an explicit formula. (MA.AA.EL.1)

Given real-world contexts, identify the percent rate of change in exponential functions written as equations. (MA.AA.EL.5)

MA.PS.ED.7: Understand and apply basic ideas related to the design, analysis, and interpretation of surveys and sampling, such as background

I can discuss basic ideas related to the design, analysis, and interpretation of surveys and sampling.

Bias

Sampling

Random sampling

Distinguish between random and non-random sampling methods, identify

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information, random sampling, causality and bias.

I can apply basic ideas related to the design, analysis, and interpretation of surveys and sampling.

I can elaborate on the role of background information, random sampling, causality, and bias in surveys and sampling.

Causality possible sources of bias in sampling, describe how such bias can be controlled and reduced, and evaluate the characteristics of a good survey and well-designed experiment. (MA.AI.DS.1)

Make inferences and justify conclusions from sample surveys, experiments, and observational studies. (MA.AA.DSP.1)

MA.PS.ED.8: Understand how basic statistical techniques are used to monitor process characteristics in the workplace.

I can discuss how basic statistical techniques are used to monitor process characteristics in the workplace.

Statistical techniques Understand that statistics and data are non-neutral and designed to serve a particular interest. (MA.AI.DS.6)

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. (MA.AII.DSP.1)

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Probability Probability and Statistics Mathematics

Standards Success Criteria Academic Vocabulary MA.PS.P.1: Understand and use the addition

rule to calculate probabilities for mutually exclusive and nonmutually exclusive events.

I can determine if events are mutually exclusive.

I can describe the addition rule used to calculate probabilities of mutually exclusive and nonmutually exclusive events.

I can apply the addition rule to find probabilities for mutually exclusive and nonmutually exclusive events.

Addition rule

Mutually Exclusive

Understand and use appropriate terminology to describe independent, dependent, complementary, and mutually exclusive events. (MA.8.DSP.4)

MA.PS.P.2: Understand and use the multiplication rule to calculate probabilities for independent and dependent events. Understand that two events A and B are independent if the probability of A and B occurring together is

I can informally decide if events are independent or dependent.

I can apply the idea that the probability of two independent

Multiplication rule

Independent events

Dependent events

Understand and use appropriate terminology to describe independent, dependent, complementary, and mutually exclusive events. (MA.8.DSP.4)

MA.PS.ED.9: Understand the differences among various kinds of studies and which types of inferences can legitimately be drawn from each.

I can show the differences among various kinds of studies.

I can identify which types of inferences can legitimately be drawn from each type of study.

Inference Make inferences and justify conclusions from sample surveys, experiments, and observational studies. (MA.AII.DSP.1)

Looking Back

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the product of their probabilities, and use this characterization to determine if they are independent.

events is the product of the probability of the individual events.

I can retell the multiplication rule used to find probabilities of independent and dependent events.

I can use the multiplication rule to find the probabilities of independent and dependent events.

Represent sample spaces and find probabilities of compound events (independent and dependent). (MA.8.DSP.5)

Understand dependent and independent events; apply these concepts to calculate probabilities. (MA.AII.DSP.5)

Record multiple observations of random events and construct empirical models of the probability distributions. Construct a theoretical model and apply the law of large numbers to show the relationship between the two models. (MA.AA.DSP.6)

MA.PS.P.3: Understand the multiplication counting principle, permutations, and combinations; use them to solve real-world problems. Use simulations with and without technology to solve counting and probability problems.

I can determine when the multiplication counting principle is appropriate to use.

I can decide when a counting situation represents a combination or a permutation.

Multiplication counting principle

Permutations

Combinations

Develop the multiplication counting principle and apply it to situations with a large number of outcomes. (MA.8.DSP.6)

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I can use the multiplication counting principle, permutations, and combinations to solve real-world problems.

I can use simulations to solve counting and probability problems.

I can use technology to solve counting and probability problems.

Understand the multiplication counting principle, permutations, and combinations; apply these concepts to calculate probabilities. (MA.AII.DSP.6)

Determine the nature and number of elements in a finite sample space to model the outcomes of real-world events using the multiplication counting principle, permutations, and combinations. (MA.AA.DSP.7)

MA.PS.P.4: Calculate the probabilities of complementary events.

I can calculate the probabilities of complementary events.

I can define complementary events.

Complementary events Understand and use appropriate terminology to describe independent, dependent, complementary, and mutually exclusive events. (MA.8.DSP.4)

MA.PS.P.5: Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

I can calculate the expected value of a random variable.

I can define the expected value of a random variable as the mean of the probability distribution.

Expected value

Random variable

Probability distribution

Record multiple observations (or simulated samples) of random events and construct empirical models of the probability distributions. (MA.AII.DSP.4) & (MA.AA.DSP.5)

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MA.PS.P.6: Analyze decisions and strategies using probability concepts. Analyze probabilities to interpret odds and risk of events.

I can examine decisions and strategies using probability concepts.

I can investigate probabilities to interpret odds.

I can analyze probabilities to interpret the risk of events.

Odds

Risk

Evaluate the validity of claims based on empirical probabilities and theoretical probabilities, including those derived from dependent and independent events. Draw conclusions and make decisions in various probabilistic contexts. (MA.AA.DSP.6)

Understand dependent and independent events, and conditional probability. (MA.AII.DSP.5)

Understand the multiplication counting principle, permutations, and combinations. (MA.AII.DSP.6)

MA.PS.P.7: 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.

I can assign a numerical value to each event in a sample space in order to define a random variable for a quantity of interest.

I can graph a probability distribution using similar graphical displays for data distributions.

Sample space

Random variable

Probability distribution

Data distribution

Record multiple observations of random events and construct empirical models of the probability distributions. (MA.AA.DSP.5)

MA.PS.P.8: Develop a probability distribution for a random variable defined for

I can develop a probability distribution for a random variable

Probability distribution Record multiple observations (or simulated

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a sample space in which theoretical probabilities can be calculated; Compute and interpret the expected value of random variables.

defined for a theoretical sample space.

I can compute the expected value of random variables.

I can interpret the expected value of random variables.

Theoretical probability

Sample space

Random variable

Expected value

samples) of random events and construct empirical models of the probability distributions. (MA.AII.DSP.4) & (MA.AA.DSP.5)

Construct a theoretical model and apply the law of large numbers to show the relationship between the two models. (MA.AII.DSP.4)

MA.PS.P.9: Derive the binomial theorem by combinatorics. Use combinatorial reasoning to solve problems.

I can arrive at the binomial theorem by combinatorics.

I can use combinatorial reasoning to solve problems.

Binomial theorem

Combinatorics

Binomial coefficients

Combinatorial reasoning

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