Common Core State Common Core State Standards: New Standards: New

Opportunities for Student Opportunities for Student LearningLearning

Gail BurrillGail Burrill

Michigan State UniversityMichigan State University

burrill@msu.eduburrill@msu.edu

Comparing Quantities. Kindt et al, 2006

OverviewOverview

What to pay attention to that is different What to pay attention to that is different conceptuallyconceptually

ProgressionsProgressions

StatisticsStatistics

Mathematical PracticesMathematical Practices– Use appropriate tools strategicallyUse appropriate tools strategically– Look for and make use of structureLook for and make use of structure

Why Standards?Why Standards?

One school’s A is another school’s D.One school’s A is another school’s D.

Opportunity to learn for Opportunity to learn for allall students students

New York assessment and NAEPNew York assessment and NAEP

8th Grade Achievement on State Assessment v. NAEP (2009)

85%

35% 26%71%

0%

25%

50%

75%

100%

Grade 4 Proficient Grade 8 Proficient

New YorkNAEP

Source: National Assessment of Educational Progress downloaded 3/25/2011 from febp.newamerica.net/k12/NY

Massachusetts assessment and NAEPMassachusetts assessment and NAEP8th Grade Achievement on State Assessment v.

NAEP (2009)

57%43%39%

32%

0%

25%

50%

75%

100%

4th grade Proficient 8th grade Proficient

MassachusettsNAEP

Source: National Assessment of Educational Progress downloaded 3/25/2011 from febp.newamerica.net/k12/MA

Common Core State StandardsCommon Core State Standards

Standards - not a curriculumStandards - not a curriculum

Conceptual and proceduralConceptual and procedural

Content and mathematical practicesContent and mathematical practices

K-8 by grade level; 9-12 by conceptual themesK-8 by grade level; 9-12 by conceptual themes– Domains Domains

- Clusters- Clusters– StandardsStandards

Words matter- not a list of topicsWords matter- not a list of topics

Interpret the structure of expressionsInterpret the structure of expressions

1. Interpret expressions that represent a quantity in 1. Interpret expressions that represent a quantity in terms of its context.terms of its context.– a. Interpret parts of an expression, such as terms, factors, a. Interpret parts of an expression, such as terms, factors,

and coefficients.and coefficients.

– b. Interpret complicated expressions by viewing one or more b. Interpret complicated expressions by viewing one or more of their parts as a single entity. of their parts as a single entity.

2. Use the structure of an expression to identify ways 2. Use the structure of an expression to identify ways to rewrite it. to rewrite it.

Factor a quadratic expression Factor a quadratic expression to reveal the zeros of the to reveal the zeros of the function it definesfunction it defines.. High SchoolHigh School

CCSSO, 2010

Words matterWords matter

Understand solving an equationUnderstand solving an equation or inequality as a or inequality as a process of answering a question: which values from a process of answering a question: which values from a specified set, if any, make the equation or inequality specified set, if any, make the equation or inequality true? true? (Grade 6)(Grade 6)

Use properties of operations to Use properties of operations to generate equivalent generate equivalent expressionsexpressions..Understand that Understand that rewriting an expression in different rewriting an expression in different formsforms in a problem context can shed light on the in a problem context can shed light on the problem and how the quantities in it are related. problem and how the quantities in it are related. (Grade 7)(Grade 7)

Emphasis on VisualizationEmphasis on Visualization

Understand the absolute value of a rational Understand the absolute value of a rational number as its distance from 0 on the number line number as its distance from 0 on the number line (Grade 6)(Grade 6)

represent addition and subtraction [of rational represent addition and subtraction [of rational numbers] on a horizontal or vertical number line numbers] on a horizontal or vertical number line diagram diagram

Understand p + q as the number located a distance Understand p + q as the number located a distance |q| from p… |q| from p… (Grade 7)(Grade 7)

Use similar triangles to explain slope (Grade 8)

CCSSO, 2010

Different expectationsDifferent expectations

Give examplesGive examples of linear equations in of linear equations in one variable with one solution, one variable with one solution, infinitely many solutions, or no infinitely many solutions, or no solutions. Show which of these solutions. Show which of these possibilities is the case by possibilities is the case by successively transforming the given successively transforming the given equation into simpler formsequation into simpler forms, until an , until an equivalent equation of the form equivalent equation of the form xx = = aa, , aa = = aa, or , or aa = = bb results (where results (where aa and and bb are different numbers). are different numbers). Grade 8Grade 8

Give examplesGive examples showing how extraneous showing how extraneous solutions may arise. solutions may arise. (High school)(High school)

CCSSO, 2010

Equivalent ExpressionsEquivalent Expressions

Albert wants to simplify the expression: 8(3–Albert wants to simplify the expression: 8(3–yy) + 5(3–) + 5(3–yy). ). Which of the following is equivalent to the expression Which of the following is equivalent to the expression

above?above? A. 39 – A. 39 – yy B. 13(3 – B. 13(3 – yy))

C. 40(3 – C. 40(3 – yy))D. 13(6 – 2D. 13(6 – 2yy))

Which expression is equivalent to 3(8x-2y+7)?A. 24x-2y+7B. 24x-6y+21C. 8x-6y+21D. 11x-5y+10

Equivalent ExpressionsEquivalent Expressions

Albert wants to simplify the expression: 8(3–Albert wants to simplify the expression: 8(3–yy) + 5(3–) + 5(3–yy) ) Which of the following is equivalent to the expression Which of the following is equivalent to the expression

above?above? 29%29% A. 39 – A. 39 – yy 40%*40%* B. 13(3 – B. 13(3 – yy))

7%7% C. 40(3 – C. 40(3 – yy))24%24% D. 13(6 – 2D. 13(6 – 2yy))

Michigan 2007, Gr. 8

(Florida 2006, grade 9)(Florida 2006, grade 9)

Which expression is equivalent to 3(8x-2y+7)?23%* A. 24x-2y+7

33% B. 24x-6y+2126% C. 8x-6y+2118% D. 11x-5y+10

Interpret the structure of expressionsInterpret the structure of expressionsInterpret expressions that represent a quantity in terms Interpret expressions that represent a quantity in terms of its context.of its context.– a. Interpret parts of an expression, such as terms, a. Interpret parts of an expression, such as terms,

factors, and coefficients.factors, and coefficients.– b. Interpret complicated expressions by viewing one b. Interpret complicated expressions by viewing one

or more of their parts as a single entity. or more of their parts as a single entity. Use the structure of an expression to identify ways to Use the structure of an expression to identify ways to rewrite it. rewrite it. Choose and produce an equivalent form of an Choose and produce an equivalent form of an expression to reveal and explain properties of the expression to reveal and explain properties of the quantity represented by the expression.quantity represented by the expression.★★Factor a quadratic expression Factor a quadratic expression to reveal the zeros of the to reveal the zeros of the function it definesfunction it defines.. High SchoolHigh School

CCSSO, 2010

Expressions and EquationsExpressions and Equations

Apply and extend previous understandings of arithmetic to Apply and extend previous understandings of arithmetic to algebraic expressions. (variables, integer, algebraic expressions. (variables, integer, absolute valueabsolute value))Reason about and solve one-variable equations and inequalities.Reason about and solve one-variable equations and inequalities. Represent and analyze quantitative relationships between Represent and analyze quantitative relationships between dependent and independent variables. dependent and independent variables. (Grade 6)(Grade 6) Use properties of operations to Use properties of operations to generate equivalent expressionsgenerate equivalent expressions..Solve real-life and mathematical problems using numerical and Solve real-life and mathematical problems using numerical and algebraic expressions and equations. algebraic expressions and equations. (Grade 7)(Grade 7)Work with radicals and integer exponents.Work with radicals and integer exponents.Understand the connections between Understand the connections between proportional relationships, proportional relationships, lines, and linear equationslines, and linear equations..Analyze and solve linear equations and pairs ofAnalyze and solve linear equations and pairs ofsimultaneous linear equationssimultaneous linear equations.(Grade 8).(Grade 8)

PolynomialsPolynomials

Perform arithmetic operations on polynomialsPerform arithmetic operations on polynomials1. Understand that polynomials form a system 1. Understand that polynomials form a system analogous to the integersanalogous to the integers2. Know and apply the Remainder Theorem: 2. Know and apply the Remainder Theorem: 3. Identify zeros of polynomials and use the zeros to 3. Identify zeros of polynomials and use the zeros to construct a rough graph construct a rough graph

Use polynomial identities to solve problemsUse polynomial identities to solve problems4. Prove polynomial identities and use them to 4. Prove polynomial identities and use them to describe numerical relationships. describe numerical relationships. 5. (+) Know and apply the Binomial Theorem5. (+) Know and apply the Binomial Theorem

FunctionsFunctions

Define, evaluate, and compare functions.Define, evaluate, and compare functions.– LinearLinear

Use functions to model relationships betweenUse functions to model relationships betweenquantities. quantities. – Graphs, tables, symbols; describe from graph and Graphs, tables, symbols; describe from graph and

sketch from descriptions (Grade 8)sketch from descriptions (Grade 8)

Build a function that models a relationship Build a function that models a relationship between two quantitiesbetween two quantities

1. Write a function that describes a relationship between two 1. Write a function that describes a relationship between two quantities.quantities.– a. Determine an explicit expression, a recursive process, or steps for a. Determine an explicit expression, a recursive process, or steps for

calculation from a context.calculation from a context.– b. Combine standard function types using arithmetic operations. b. Combine standard function types using arithmetic operations. – c. (+) Compose functions. For example, if T(y) is the temperature in the c. (+) Compose functions. For example, if T(y) is the temperature in the

atmosphere as a function of height, and h(t) is the height of a weather atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.location of the weather balloon as a function of time.

2. Write arithmetic and geometric sequences both recursively 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and and with an explicit formula, use them to model situations, and translate between the two forms.translate between the two forms.★★8. Write a function defined by an expression in different but 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of equivalent forms to reveal and explain different properties of the function.the function.

Understand the concept of a function and use Understand the concept of a function and use function notation (language, notation, function notation (language, notation, sequences)sequences)

Interpret functions that arise in applications in Interpret functions that arise in applications in terms of the contextterms of the context– Calculate and interpret the average rate of change Calculate and interpret the average rate of change

of a function (presented symbolically or as a table) of a function (presented symbolically or as a table) over a specified interval.over a specified interval.

– Estimate the rate of change from a graph.Estimate the rate of change from a graph.

Connections to Functions and Modeling. Connections to Functions and Modeling. Expressions can define functions, and Expressions can define functions, and equivalent expressions define the same equivalent expressions define the same function.function.

Logarithms- confusion?Logarithms- confusion?

Graph exponential and logarithmic functions, Graph exponential and logarithmic functions, showing intercepts and end behavior, and showing intercepts and end behavior, and trigonometric functions, showing period, midline, and trigonometric functions, showing period, midline, and amplitude.amplitude.5. (+) Understand the inverse relationship between 5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.solve problems involving logarithms and exponents.4. For exponential models, express as a logarithm the 4. For exponential models, express as a logarithm the solution to absolution to abctct = d where a, c, and d are numbers and = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using the base b is 2, 10, or e; evaluate the logarithm using technology. technology. (High School)(High School)

GeometryGeometry

Solve real-world and mathematical problemsSolve real-world and mathematical problems

involving area, surface area, and volume. involving area, surface area, and volume. (Grade 6)(Grade 6)

Solve real-life and mathematical problemsSolve real-life and mathematical problems

involving angle measure, area, surface area,involving angle measure, area, surface area,

and volume. and volume. (Grade 7)(Grade 7)

Know the formulas for the volumes of cones, Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-cylinders, and spheres and use them to solve real-world and mathematical problems. world and mathematical problems. (Grade 8)(Grade 8)

Grade 6: developing understanding of Grade 6: developing understanding of statistical thinkingstatistical thinking

Measures of center (mean, median)Measures of center (mean, median)

Variation (range, interquartile range, Variation (range, interquartile range, mean mean absolute deviationabsolute deviation))

Distributions (clusters, gaps, outliers, Distributions (clusters, gaps, outliers, symmetry)symmetry)

Context in which data collectedContext in which data collected

Grade 6: Summarize and describe Grade 6: Summarize and describe distributionsdistributions

6.SP.4. Display numerical data in dot plots, 6.SP.4. Display numerical data in dot plots, histograms, and histograms, and box plots.box plots.

6.SP.5. Summarize numerical data sets in 6.SP.5. Summarize numerical data sets in relation to their contextrelation to their context

Grade 7: Drawing inferences about Grade 7: Drawing inferences about populations based on samplespopulations based on samples

compare two data distributions and address compare two data distributions and address questions about differences between questions about differences between populationspopulations

begin informal work with random sampling to begin informal work with random sampling to generate data sets and learn about the generate data sets and learn about the importance of representative samples for importance of representative samples for drawing inferences.drawing inferences.

Grade 7: Use random sampling to draw Grade 7: Use random sampling to draw inferences about a population.inferences about a population.

7.SP.1. Understand that statistics can be used to gain 7.SP.1. Understand that statistics can be used to gain information about a population by examining a information about a population by examining a sample of the population; generalizations about a sample of the population; generalizations about a population from a sample are valid only if the sample population from a sample are valid only if the sample is representative of that population; is representative of that population; random sampling random sampling tends to produce representative samplestends to produce representative samples and support and support valid inferences.valid inferences.

7.SP.2. Use data from a random sample to draw 7.SP.2. Use data from a random sample to draw inferences about a population with an unknown inferences about a population with an unknown characteristic of interest.characteristic of interest. Generate multiple samples Generate multiple samples (or simulated samples) of the same size to gauge the (or simulated samples) of the same size to gauge the variation in estimates or predictionsvariation in estimates or predictions. Gauge how far . Gauge how far off the estimate or prediction might be.off the estimate or prediction might be.

Grade 7: Draw informal comparative inferences Grade 7: Draw informal comparative inferences about two populations.about two populations.

7.SP.3. Informally assess the degree of visual overlap of 7.SP.3. Informally assess the degree of visual overlap of two numerical data distributions with similar two numerical data distributions with similar variabilities, measuring the difference between the variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of centers by expressing it as a multiple of a measure of variability. variability.

7.SP.4. 7.SP.4. Use measures of center and measures of Use measures of center and measures of variability for numerical data from random samples variability for numerical data from random samples to draw informal comparative inferences about two to draw informal comparative inferences about two populations.populations.

Grade 8: bivariate dataGrade 8: bivariate data

8.SP.1. Construct and interpret scatter plots to investigate patterns 8.SP.1. Construct and interpret scatter plots to investigate patterns of association between two quantities. Describe patterns such as of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, clustering, outliers, positive or negative association, linear/nonlinear association.linear/nonlinear association.

8.SP.2. Model relationships between two quantitative variables 8.SP.2. Model relationships between two quantitative variables using straight lines. If linear, informally fit a straight line and using straight lines. If linear, informally fit a straight line and assess the model fit.assess the model fit.

8.SP.3. Use the equation of a linear model to solve problems with 8.SP.3. Use the equation of a linear model to solve problems with bivariate measurement data, interpreting m and b. bivariate measurement data, interpreting m and b.

8.SP.4. 8.SP.4. Recognize patterns of association in bivariate categorical Recognize patterns of association in bivariate categorical data by displaying frequencies and relative frequencies in a data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from summarizing data on two categorical variables collected from the same subjects. Use relative frequencies for rows/columns to the same subjects. Use relative frequencies for rows/columns to describe possible association between two variables.describe possible association between two variables.

Two Way TablesTwo Way Tables

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Statistics Nspired Conditional Probability

High School StatisticsHigh School Statistics

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

S-ID.1. Represent data with plots on the real number line (dot S-ID.1. Represent data with plots on the real number line (dot plots, histograms, and box plots).plots, histograms, and box plots).S-ID.2. Use statistics appropriate to the shape of the data S-ID.2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread distribution to compare center (median, mean) and spread (interquartile range(interquartile range, standard deviation) , standard deviation) of two or more of two or more different data sets.different data sets.S-ID.3. Interpret differences in shape, center, and spread in the S-ID.3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of context of the data sets, accounting for possible effects of extreme data points (outliers).extreme data points (outliers).S-ID.4. S-ID.4. Use the mean and standard deviation of a data set to fit Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population it to a normal distribution and to estimate population percentages.percentages. Recognize that there are data sets for which such Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.and tables to estimate areas under the normal curve.

Summarize, represent, interpret data on two Summarize, represent, interpret data on two categorical and quantitative variablescategorical and quantitative variables

S-ID.5. S-ID.5. Summarize categorical data for two categories in 2-way Summarize categorical data for two categories in 2-way frequency tables. Interpret relative frequencies in the context of the frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, conditional relative frequencies). data (including joint, marginal, conditional relative frequencies). Recognize possible associations and trends in the data.Recognize possible associations and trends in the data.S-ID.6. Represent data on two quantitative variables on a scatter S-ID.6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.plot, and describe how the variables are related.– a. Fit a function to the data; use functions fitted to data to solve 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 problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.quadratic, and exponential models.

– bb. Informally assess the fit of a function by plotting and . Informally assess the fit of a function by plotting and analyzing residuals.analyzing residuals.

– c. Fit a linear function for a scatter plot that suggests a linear c. Fit a linear function for a scatter plot that suggests a linear association.association.

Interpret linear modelsInterpret linear models

S-ID.7. Interpret the slope (rate of change) and the intercept S-ID.7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.(constant term) of a linear model in the context of the data.

S-ID.8. Compute (using technology) and S-ID.8. Compute (using technology) and interpret the interpret the correlation coefficient of a linear fit.correlation coefficient of a linear fit.

S-ID.9. Distinguish between correlation and causation.S-ID.9. Distinguish between correlation and causation.

Understand and evaluate random processes Understand and evaluate random processes underlying statistical experimentsunderlying statistical experiments

S-IC.1. Understand statistics as a process for making S-IC.1. Understand statistics as a process for making inferences about population parameters based on a inferences about population parameters based on a random sample from that population.random sample from that population.

S-IC.2. S-IC.2. Decide if a specified model is consistent with Decide if a specified model is consistent with results from a given data-generating process, e.g., results from a given data-generating process, e.g., using simulation.using simulation.

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

S-IC.3. S-IC.3. Recognize the purposes of and differences Recognize the purposes of and differences among sample surveys, experiments, and among sample surveys, experiments, and observational studies; explain how randomization observational studies; explain how randomization relates to each.relates to each.S-IC.4. Use data from a sample survey to estimate a S-IC.4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of population mean or proportion; develop a margin of error through the use of simulation models for random error through the use of simulation models for random sampling.sampling.S-IC.5. Use data from a randomized experiment to S-IC.5. Use data from a randomized experiment to compare two treatments; use simulations to decide if compare two treatments; use simulations to decide if differences between parameters are significantdifferences between parameters are significant..S-IC.6. Evaluate reports based on data.S-IC.6. Evaluate reports based on data.

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

S-CP.1. Describe events as subsets of a sample space S-CP.1. Describe events as subsets of a sample space (outcomes) using characteristics of outcomes ( unions, (outcomes) using characteristics of outcomes ( unions, intersections, complements of other events -“or, and, not”).intersections, complements of other events -“or, and, not”).S-CP.2. Understand that two events A and B are independent S-CP.2. Understand that two events A and B are independent if the probability of A and B occurring together is the product if the probability of A and B occurring together is the product of their probabilities, and use this to determine independence.of their probabilities, and use this to determine independence.S-CP.3. S-CP.3. Understand the conditional probability of A given B Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as P(A and B)/P(B), and interpret independence of A and B S-CP.4. Construct and interpret two-way frequency tables of S-CP.4. Construct and interpret two-way frequency tables of data when two categories are associated with each object data when two categories are associated with each object being classified. Use 2-way table to decide if events are being classified. Use 2-way table to decide if events are independent and to approximate conditional probabilitiesindependent and to approximate conditional probabilities. . S-CP.5. Recognize and explain conditional probability and S-CP.5. Recognize and explain conditional probability and independence in everyday language and everyday situations.independence in everyday language and everyday situations.

Mathematical PracticesMathematical PracticesMake sense of problems and persevere in solving themReason abstractly and quantitativelyConstruct viable arguments and critique the reasoning of othersModel with mathematicsUse appropriate tools strategicallyAttend to precisionLook for and make use of structureLook for and express regularity in repeated reasoning

Common Core Standards (CCSSO) 2010

Construct viable arguments & Construct viable arguments & critique the reasoning of otherscritique the reasoning of others

make conjectures and build a logical progression of ideasuse stated assumptions, definitions and previously established results in constructing arguments break things into cases and use counterexamples reason inductively about data, making plausible arguments that take into account the context evaluate arguments, decide whether they make sense, and evaluate arguments, decide whether they make sense, and ask useful questions to clarify or improve the argumentsask useful questions to clarify or improve the argumentsdistinguish correct reasoning from that which is flawed distinguish correct reasoning from that which is flawed and explain any flawsand explain any flaws

CCSSO, 2010

Common Core Mathematical Common Core Mathematical Practices: Practices:

OpportunitiesOpportunities for revisiting our own for revisiting our own practice and practice and

OpportunitiesOpportunities for more students to for more students to learn what mathematics is really learn what mathematics is really

about- sense making and reasoningabout- sense making and reasoning

ReferencesReferencesCouncil of Chief State School Officers (CCSSO). Common Core State Standards. Council of Chief State School Officers (CCSSO) and (National Governor’s Association (NGA)

Florida Department of Education (2006). FCAT Mathematics Florida Department of Education (2006). FCAT Mathematics Released Items, Grade 9.Released Items, Grade 9.

Kindt, M., Abels, M., Meyer, M., Pligge, M. (1998). Kindt, M., Abels, M., Meyer, M., Pligge, M. (1998). Comparing QuantitiesComparing Quantities. From Mathematics in Context. . From Mathematics in Context. Directed by Romberg, T. & deLange, J. Chicago IL: Directed by Romberg, T. & deLange, J. Chicago IL: Encyclopedia BritannicaEncyclopedia Britannica

Michigan Department of Education. (2007). Released items Michigan Department of Education. (2007). Released items mathematics grades 6, 7, 8; fall. mathematics grades 6, 7, 8; fall. www.www.michiganmichigan.gov/mde/0,1607,7-140-22709_31168_31355-95470--,00.htm.gov/mde/0,1607,7-140-22709_31168_31355-95470--,00.htmll

National Assessment of Educational Progress (2009). National Assessment of Educational Progress (2009). downloaded 3/25/2011 from febp.newamerica.net/k12/MA

National Assessment of Educational Progress (2009). National Assessment of Educational Progress (2009). downloaded 3/25/2011 from febp.newamerica.net/k12/MA

Statistics Nspired Conditional Probability. (2011). Texas Instruments Educational Technology. Online http://education.ti.com/calculators/timathnspired/http://education.ti.com/calculators/timathnspired/