Course: Algebra MP: Reason abstractively and quantitatively MP: Model with mathematics MP: Look for and make use of structure

Transcription

1 Modeling Cluster: Interpret the structure of expressions. A.SSE.1: Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. A.SSE.1 Cluster: Interpret the structure of expressions. Course: Algebra MP: Reason abstractively and quantitatively Knowledge: Define and recognize parts of an expression, such as terms, factors, and coefficients Reasoning: Interpret parts of an expression, such as terms, factors, and coefficients in terms of the context Reasoning: Interpret complicated expressions, in terms of the context, by viewing one or more of their parts as a single entity Cluster: Write expressions in equivalent forms to solve problems. A.SSE.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15 t can be rewritten as (1.15 1/12 ) 12t t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. A.SSE.4: Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. 56

2 A.SSE.3 Cluster: Write expressions in equivalent forms to solve problems. Course: Algebra MP: Reason abstractively and quantitatively MP: Look for and express regularity in repeated reasoning Knowledge: Explain the connection between the factored form of a quadratic expression and the zeros of the function it defines Knowledge: Explain the connection between the completed square form of a quadratic expression and the maximum or minimum value of the function it defines Knowledge: Explain the properties of the quantity represented by the quadratic expression Reasoning: Explain the properties of the quantity or quantities represented by the transformed exponential expression Reasoning: Complete the square on a quadratic expression to produce an equivalent form of an expression Reasoning: Choose and produce an equivalent form of a quadratic expression to reveal and explain properties of the quantity represented by the original expression Reasoning: Use the properties of exponents to transform simple expressions for exponential functions Demonstration: Choose and produce an equivalent form of an exponential expression to reveal and explain properties of the quantity represented by the original expression Demonstration: Choose and produce an equivalent form of a quadratic expression to reveal and explain properties of the quantity represented by the original expression Demonstration: Factor a quadratic expression to produce an equivalent form of the original expression A.SSE.4 Cluster: Write expressions in equivalent forms to solve problems. Course: Algebra Knowledge: Find the first term in a geometric sequence given at least two other terms Knowledge: Define a geometric series as a series with a constant ratio between successive terms Demonstration: Use the formula S + a (1-rn)/(1-r) to solve problems Demonstration: Derive a formula[i.e., equivalent to the formula S + a (1-rn)/(1-r)] for the sum of a finite geometric series (when the common ratio is not 1) 57

3 Cluster: Represent and solve equations and inequalities graphically. A.REI.11: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. A.REI.11 Cluster: Represent and solve equations and inequalities graphically. Course: Algebra Knowledge: Recognize and use function notation to represent linear and exponential equations Knowledge: Recognize that if (x 1, y 1 ) and (x 2, y 2 ) share the same location in the coordinate plane that x 1 = x 2 and y 1 = y 2 Knowledge: Recognize that f(x) = g(x) means that there may be particular inputs of f and g for which the outputs of f and g are equal Knowledge: Recognize and use function notation to represent linear, polynomial, rational, absolute value, exponential, and radical equations. Reasoning: Explain why the x-coordinates of the points where the graph of the equations y = f(x) and y = g(x) intersect are the solutions of the equations f(x) = g(x) Reasoning: Approximate/find the solution(s) using an appropriate method. For example, using technology to graph the functions, make tables of values or find successive approximations 58

4 Cluster: Interpret functions that arise in applications in terms of the context. F.IF.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F.IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. F.IF.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval Estimate the rate of change from a graph. F.IF.4 F.IF.5 Cluster: Interpret functions that arise in applications in terms of the context. Course: Functions Knowledge: Define and recognize key features in tables and graphs of linear and exponential functions: intercepts; intervals where the function is increasing, decreasing, positive, or negative, and end behavior Knowledge: Define and recognize key features in tables and graphs of linear, exponential, and quadratic functions: intercepts; intervals where the function is increasing, decreasing, positive, or negative, relative maximums and minimums, symmetries, end behavior and periodicity Knowledge: Identify the type of function, given a table or graph Knowledge: Identify whether a function is linear or exponential, given its table or graph Reasoning: Interpret key features of graphs and tables of functions in terms of the contextual quantities each function represents Demonstration: Sketch graphs showing the key features of a function, modeling a relationship between two quantities, given a verbal description of the relationship Cluster: Interpret functions that arise in applications Course: Functions in terms of the context. Knowledge: Identify and describe the domain of a function, given the graph or a verbal/written description of a function Knowledge: Identify an appropriate domain based on the unit, quantity, and type of function it describes Reasoning: Relate the domain of a function to its graph and to the quantitative relationship it describes, where applicable Reasoning: Explain why a domain is appropriate for a given situation 59

5 F.IF.6 Cluster: Interpret functions that arise in applications in terms of the context. Course: Functions Knowledge: Recognize slope as an average rate of change Reasoning: Estimate the rate of change from a linear or exponential graph Reasoning: Interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval Demonstration: Calculate the average rate of change of a function (presented symbolically or as a table) over a specified interval Cluster: Analyze functions using different representations. F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 60

6 F.IF.7 Cluster: Analyze functions using different representations. Course: Functions Reasoning: Determine the difference between simple and complicated polynomial functions Reasoning: Determine the difference between simple and complicated linear, quadratic, square root, cube root, and piecewise-defined functions Reasoning: Determine the differences between simple and complicated linear and exponential functions and know when the use of technology is appropriate Reasoning: Compare and contrast absolute value, step- and piecewisedefined functions with linear, quadratic, and exponential functions Reasoning: Compare and contrast the domain and range of absolute value, step- and piecewise-defined functions with linear, quadratic, and exponential functions Reasoning: Compare and contrast the domain and range of exponential, logarithmic, and trigonometric functions with linear, quadratic, absolute value, step- and piecewise-defined functions Reasoning: Analyze the difference between simple and complicated linear, quadratic, square root, cube root, piecewise-defined, exponential, logarithmic, and trigonometric functions, including step and absolute value functions Reasoning: Select the appropriate type of function, taking into consideration the key features, domain, and range, to model a real-world situation Reasoning: Relate the relationship between zeros of quadratic functions and their factored forms to the relationship between polynomial functions of degrees greater than two Demonstration: Graph exponential functions, by hand in simple cases or using technology for more complicated cases, and show intercepts and end behavior Demonstration: Graph polynomial functions, by hand in simple cases or using technology for more complicated cases, and show/label maxima and minima of the graph, identify zeros when suitable factorizations are available, and show end behavior Demonstration: Graph exponential, logarithmic, and trigonometric functions, by hand in simple cases or using technology for more complicated cases. For exponential and logarithmic functions, show: intercepts and end behavior; for trigonometric functions, show: period, midline, and amplitude Demonstration: Graph linear functions by hand in simple cases or using technology for more complicated cases and show/label intercepts of the graph Demonstration: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions, by hand in simple cases or using technology for more complicated cases, and show/ label key features of the graph 61

7 Cluster: Build a function that models a relationship between two quantities. F.BF.1: Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. 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 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. F.BF.2: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. F.BF.1 Cluster: Build a function that models a relationship between two quantities. Course: Functions MP: Look for and express regularity in repeated reasoning Knowledge: Define explicit function and recursive process Knowledge: Combine two functions using the operations of addition, subtraction, multiplication, and division Knowledge: Evaluate the domain of the combined function Reasoning: Given a real-world situation or mathematical problem, build standard functions to represent relevant relationships/quantities Reasoning: Given a real-world situation or mathematical problem, determine which arithmetic operation should be performed to build the appropriate combined function Reasoning: Given a real-world situation or mathematical problem, relate the combined function to the context of the problem Demonstration: Write a function that describes a relationship between two quantities by determining an explicit expression, a recursive process, or steps for calculation from a context 62

8 F.BF.2 Cluster: Build a function that models a relationship between two quantities. Course: Functions MP: Look for and express regularity in repeated reasoning Knowledge: Identify arithmetic and geometric patterns in given sequences Reasoning: Determine the recursive rule given arithmetic and geometric sequences Reasoning: Determine the explicit formula given arithmetic and geometric sequences Reasoning: Justify the translation between the recursive form and explicit formula for arithmetic and geometric sequences Demonstration: Generate arithmetic and geometric sequences from recursive and explicit formulas Demonstration: Given an arithmetic or geometric sequence in recursive form, translate into the explicit formula Demonstration: Given an arithmetic or geometric sequence as an explicit formula, translate into the recursive form Demonstration: Use given and constructed arithmetic and geometric sequences, expressed both recursively and with explicit formulas, to model real-life situations 63

9 Cluster: Model periodic phenomena with trigonometric functions. F.TF.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. F.TF.7: (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context. F.TF.5 Cluster: Model periodic phenomena with trigonometric functions. Course: Functions Knowledge: Define and recognize amplitude, frequency, and midline parameters in a symbolic trigonometric function Reasoning: Interpret the parameters of a trigonometric function (amplitude, frequency, midline) in the context of real-world situations Reasoning: Choose trigonometric functions to model periodic phenomena for which amplitude, frequency, and midline are already specified Demonstration: Explain why real-world or mathematical phenomena exhibit characteristics of periodicity F.TF.7 Cluster: Model periodic phenomena with trigonometric functions. Course: Functions MP: Look for and express regularity in repeated reasoning Reasoning: Find solutions to trigonometric equations using inverse functions and technology Reasoning: Interpret these solutions to trigonometric equations in context Demonstration: Use inverse functions to solve trigonometric equations 64

10 Cluster: Define trigonometric ratios and solve problems involving right triangles. G.SRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. G.SRT.8 Cluster: Define trigonometric ratios and solve problems involving right triangles. Course: Geometry Knowledge: Recognize which methods could be used to solve right triangles in applied problems Knowledge: Solve for an unknown angle or side of a right triangle using sine, cosine, and tangent Reasoning: Apply right triangle trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems Cluster: Use coordinates to prove simple geometric theorems algebraically. G.GPE.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. G.GPE.7 Cluster: Use coordinates to prove simple geometric theorems Course: Geometry algebraically. Knowledge: Use the coordinates of the vertices of a polygon to find the necessary dimensions for finding the perimeter Knowledge: Use the coordinates of the vertices of a triangle to find the necessary dimensions (base, height) for finding the area Knowledge: Use the coordinates of the vertices of a rectangle to find the necessary dimensions (base, height) for finding the area Reasoning: Formulate a model of figures in contextual problems to compute area and/or perimeter Cluster: Explain volume formulas and use them to solve problems. G.GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. G.GMD.3 Cluster: Explain volume formulas and use them to solve problems. Course: Geometry Knowledge: Utilize the appropriate formula for volume, depending on the figure Reasoning: Use volume formulas for cylinders, pyramids, cones, and spheres to solve contextual problems 65

11 Cluster: Apply geometric concepts in modeling situations. G.MG.1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). G.MG.2: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). G.MG.3: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). G.MG.1 Cluster: Apply geometric concepts in modeling situations. Course: Geometry Reasoning: Given a real world object, classify the object as a known geometric shape use this to solve problems in context Reasoning: Focus on situations well modeled by trigonometric ratios for acute angles Demonstration: Use measures and properties of geometric shapes to describe real world objects G.MG.2 Cluster: Apply geometric concepts in modeling situations. Course: Geometry Knowledge: Define density Reasoning: Apply concepts of density based on area and volume to model real-life situations G.MG.3 Cluster: Apply geometric concepts in modeling situations. Course: Geometry Knowledge: Describe a typographical grid system Reasoning: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios) 66

12 S.ID.1: Represent data with plots on the real number line (dot plots, histograms, and box plots). Cluster: Summarize, represent, and interpret data on a single count or measurement variable. S.ID.2: Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. S.ID.3: Interpret differences in shape, center and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). S.ID.4: Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. S.ID.1 Cluster: Summarize, represent, and interpret data on a single count or measurement variable. Demonstration: Represent data with plots on the real number line, using various display types by creating dot plots, histograms, and box plots S.ID.2 Cluster: Summarize, represent, and interpret data on a single count or measurement variable. Reasoning: Choose the appropriate measure for center (mean, median) and spread (interquartile range, standard deviation) based on the shape of a data distribution Demonstration: Use appropriate statistics for center and spread to compare two or more data sets S.ID.3 Cluster: Summarize, represent, and interpret data on a single count or measurement variable. Knowledge: Define the context of data sets as meaning the specific nature of the attributes under investigation Reasoning: Interpret differences in shape, center, and spread in the context of the data sets Reasoning: Describe the possible effects the presence of outliers in a set of data can have on shape, center, and spread in the context of the data sets 67

13 S.ID.4 Cluster: Summarize, represent, and interpret data on a single count or measurement variable. Knowledge: Describe the characteristics of a normal distribution Reasoning: Recognize that there are data sets for which such a procedure is not appropriate Demonstration: Use the mean and standard deviation of a data set to fit it to a normal distribution Demonstration: Use a normal distribution to estimate population percentages Demonstration: Use a calculator, spreadsheet, and table to estimate areas under the normal curve Cluster: Summarize, represent, and interpret data on two categorical and quantitative variables. S.ID.5: Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. S.ID.6: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models. b. Informally assess the fit of a function by plotting and analyzing residuals. c. Fit a linear function for a scatter plot that suggests a linear association. S.ID.5 Cluster: Summarize, represent, and interpret data on two categorical and quantitative variables. Knowledge: Recognize the differences between joint, marginal, and conditional relative frequencies Knowledge: Summarize categorical data for two categories in twoway frequency tables Reasoning: Interpret relative frequencies in the context of the data Reasoning: Recognize possible associations and trends in the data Demonstration: Calculate relative frequencies including joint, marginal, and conditional 68

14 S.ID.6 Cluster: Summarize, represent, and interpret data on two categorical and quantitative variables. MP: Look for and express regularity in repeated reasoning Knowledge: Represent data on a scatter plot (2 quantitative variables) Knowledge: Fit a given function class (e.g., linear, exponential) to a data set Knowledge: Represent the residuals from a function and the data set it models, numerically and graphically Reasoning: Using given scatter plot data represented on the coordinate plane, informally describe how the two quantitative variables are related Reasoning: Determine which function best models scatter plot data represented on the coordinate plane, and describe how the two quantitative variables are related Reasoning: Use functions fitted to data to solve problems in the context of the data Reasoning: Informally assess the fit of a function by analyzing residuals from the residual plot Reasoning: Fit a linear function for a scatter plot that suggests a linear association Cluster: Interpret linear models. S.ID.7: Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. S.ID.8: Compute (using technology) and interpret the correlation coefficient of a linear fit. S.ID.9: Distinguish between correlation and causation. S.ID.7 Cluster: Interpret linear models. MP: Look for and express regularity in repeated reasoning Reasoning: Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data 69

15 S.ID.8 Cluster: Interpret linear models. MP: Look for and express regularity in repeated reasoning Knowledge: Define the correlation coefficient Reasoning: Interpret the correlation coefficient of a linear fit as a measure of how well the data fit the relationship Demonstration: Using technology, compute the correlation coefficient of a linear fit S.ID.9 Cluster: Interpret linear models. Knowledge: Define positive, negative, and no correlation and explain why correlation does not imply causation Knowledge: Define causation Reasoning: Distinguish between correlation and causation Cluster: Understand and evaluate random processes underlying statistical experiments. S.IC.1: Understand statistics as a process for making inferences about population parameters based on a random sample from that population. S.IC2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model? S.IC.1 S.IC.2 Cluster: Understand and evaluate random processes underlying statistical experiments. Knowledge: Explain that statistics is a process for making inferences about population parameters, or characteristics Knowledge: Explain that statistical inferences about population characteristics are based on random samples from that population Cluster: Understand and evaluate random processes underlying statistical experiments. Knowledge: Use various, specified data-generating processes/ models Knowledge: Recognize data that various models produce Knowledge: Identify data or discrepancies that provide the basis for rejecting a statistical model Reasoning: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. 70

16 Cluster: Make inferences and justify conclusions from sample surveys, experiments, and observational studies. S.IC3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. S.IC4: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. S.IC5: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between two parameters are significant. S.IC6: Evaluate reports based on data. S.IC.3 Cluster: Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Knowledge: Recognize the purpose of surveys, experiments, and observational studies in making statistical inferences and justifying conclusions and explain how randomization relates to each of these methods of data collection Knowledge: Recognize the differences among surveys, experiments, and observational studies in making statistical inferences and justifying conclusions and explain how randomization relates to each of these methods of data collection S.IC.4 Cluster: Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Knowledge: Define margin of error Knowledge: Explain the connection of margin of error to variation within a data set or population Reasoning: Interpret the data generated by a simulation model for random sampling in terms of the context the simulation models Reasoning: Develop a margin of error, assuming certain population parameters/characteristics, through the use of simulation models for random sampling Demonstration: Use a simulation model to generate data for random sampling, assuming certain population parameters/characteristics Demonstration: Use data from a sample survey to estimate a population mean or proportion 71

17 S.IC.5 Cluster: Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Knowledge: Using an established level of significance, determine if the difference between two parameters is significant Reasoning: Choose appropriate methods to simulate a randomized experiment Reasoning: Establish a reasonable level of significance Demonstration: Use data from a randomized experiment to compare two treatments S.IC.6 Cluster: Make inferences and justify conclusions from sample surveys, experiments, and observational studies. MP: Look for and express regularity in repeated reasoning Knowledge: Define the characteristics of experimental design (control, randomization, and replication) Reasoning: Evaluate experimental study design, how data was gathered, and what analysis (numerical or graphical) was used Reasoning: Draw conclusions based on graphical and numerical summaries Reasoning: Support with graphical and numerical summaries how appropriate the report of data was 72

18 Cluster: Understand independence and conditional probability and use them to interpret data. S.CP.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ). S.CP.2: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. S.CP.3: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S.CP.4: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. S.CP.5: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. S.CP.1 Cluster: Understand independence and conditional probability and use them to interpret data. Knowledge: Define unions, intersections and complements of events Reasoning: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events S.CP.2 Cluster: Understand independence and conditional probability and use them to interpret data. Knowledge: Categorize events as independent or not using the characterization that two events A and B are independent when the probability of A and B occurring together is the product of their probabilities S.CP.3 Cluster: Understand independence and conditional probability and use them to interpret data. Knowledge: Know the conditional probability of A given B as P(A and B)/P(B) Knowledge: Interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B 73

19 S.CP.4 Cluster: Understand independence and conditional probability and use them to interpret data. Knowledge: Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities Knowledge: Build on work with two way tables from Algebra 1 Unit 3 S-ID.5 to develop understanding of conditional probability and independence Reasoning: Interpret two-way frequency tables of data when two categories are associated with each object being classified S.CP.5 Cluster: Understand independence and conditional probability and use them to interpret data. Knowledge: Recognize the concepts of conditional probability and independence in everyday language and everyday situations. Reasoning: Explain the concepts of conditional probability and independence in everyday language and everyday situations. Cluster: Use the rules of probability to compute probabilities of compound events in a uniform probability model. S.CP.6: Find the conditional probability of A given B as the fraction of Bs outcomes that also belong to A, and interpret the answer in terms of the model. S.CP.7: Apply the Addition Rule, P(A or B) = P(A) + P(B) P(Aand B), and interpret the answer in terms of the model. S.CP.8: (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B A) = P(B)P(A B), and interpret the answer in terms of the model. S.CP.9: (+) Use permutations and combinations to compute probabilities of compound events and solve problems. S.CP.6 Cluster: Use the rules of probability to compute probabilities of compound events in a uniform probability model. Knowledge: Find the conditional probability of A given B as the fraction of B s outcomes that also belong to A Reasoning: Interpret the answer in terms of the model S.CP.7 Cluster: Use the rules of probability to compute probabilities of compound events in a uniform probability model. Knowledge: Use the Additional Rule, P(A or B) = P(A) + P(B) - P(A and B) Reasoning: Interpret the answer in terms of the model 74

20 S.CP.8 Cluster: Use the rules of probability to compute probabilities of compound events in a uniform probability model. Knowledge: Use the multiplication rule with correct notation Reasoning: Apply the general Multiplication Rule in a uniform probability model P(A and B) = P(A)P(B A) = P(B)P(A B) Reasoning: Interpret the answer in terms of the model S.CP.9 Cluster: Use the rules of probability to compute probabilities of compound events in a uniform probability model. Knowledge: Identify situations that are permutations and those that are combinations Demonstration: Use permutations and combinations to compute probabilities of compound events and solve problems Cluster: Calculate expected values and use them to solve problems. S.MD.1: (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions. S.MD.2: (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. S.MD.3: (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated, find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes. S.MD.4: (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? S.MD.1 Cluster: Calculate expected values and use them to solve problems. Knowledge: Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space Demonstration: Graph the corresponding probability distribution using the same graphical displays used for data distributions 75

21 S.MD.2 Cluster: Calculate expected values and use them to solve problems. Reasoning: Interpret expected value as the mean of the probability distribution Demonstration: Calculate the expected value of a random variable S.MD.3 Cluster: Calculate expected values and use them to solve problems. Demonstration: Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated Demonstration: Find the expected value of a random variable for a probability distribution S.MD.4 Cluster: Calculate expected values and use them to solve problems. Reasoning: Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically Demonstration: Find the expected value of a random variable for a probability distribution Cluster: Use probability to evaluate outcomes of decisions. S.MD.5: (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant. b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident. S.MD.6: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). S.MD.7: (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). 76

22 S.MD.5 Cluster: Use probability to evaluate outcomes of decisions. Reasoning: Find the expected payoff for a game of chance Reasoning: Evaluate and compare strategies on the basis of expected values S.MD.6 Cluster: Use probability to evaluate outcomes of decisions. MP: Look for and express regularity in repeated reasoning Knowledge: Compute Theoretical and Experimental Probabilities Knowledge: Recall previous understandings of probability Reasoning: Use probabilities to make fair decisions Reasoning: Extend to more complex probability models. Include situations such as those involving quality control, or diagnostic tests that yield both false positive and false negative results S.MD.7 Cluster: Use probability to evaluate outcomes of decisions. Knowledge: Recall previous understandings of probability Reasoning: Analyze decisions and strategies using probability concepts Reasoning: Extend to more complex probability models. Include situations such as those involving quality control, or diagnostic tests that yield both false positive and false negative results 77