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Title: Graphical Analysis
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In this lesson, one of a multipart unit from Illuminations, students use iteration, recursion, and algebra to model and analyze a changing fish population. Students examine a graph of the data then use an interactive graphing tool to investigate three different situations.
Standard(s): [MA2013] AL2 (912) 33: Write a function that describes a relationship between two quantities.* [FBF1] [MA2013] PRE (912) 44: Understand statistics as a process for making inferences about population parameters based on a random sample from that population. [SIC1] [MA2013] PRE (912) 48: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. [SIC5]
Subject: Mathematics,Science Title: Graphical Analysis
Description: In this lesson, one of a multipart unit from Illuminations, students use iteration, recursion, and algebra to model and analyze a changing fish population. Students examine a graph of the data then use an interactive graphing tool to investigate three different situations. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
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Title: Birthday Paradox
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This Illuminations lesson demonstrates the birthday paradox, using it as a springboard into a unit on probability. Students use the TI83 graphing calculator to run a Monte Carlo simulation with the birthday paradox and engage in a graphical analysis of the birthdayproblem function.
Standard(s): [MA2013] (6) 25: Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. [6SP1] [MA2013] (6) 26: Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. [6SP2] [MA2013] (6) 27: Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. [6SP3] [MA2013] (6) 29: Summarize numerical data sets in relation to their context, such as by: [6SP5] [MA2013] (7) 17: Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. [7SP1] [MA2013] (7) 18: Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. [7SP2] [MA2013] (7) 21: Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ^{1}/_{2} indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. [7SP5] [MA2013] (7) 22: Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approximate relative frequency given the probability. [7SP6] [MA2013] (7) 23: Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. [7SP7] [MA2013] (7) 24: Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. [7SP8] [MA2013] AL2 (912) 46: (+) Use permutations and combinations to compute probabilities of compound events and solve problems. [SCP9] [MA2013] GEO (912) 42: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [SMD6] [MA2013] AL2 (912) 33: Write a function that describes a relationship between two quantities.* [FBF1] [MA2013] AL2 (912) 37: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [SMD6] [MA2013] PRE (912) 44: Understand statistics as a process for making inferences about population parameters based on a random sample from that population. [SIC1] [MA2013] PRE (912) 45: Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation. [SIC2] [MA2013] PRE (912) 46: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. [SIC3] [MA2013] PRE (912) 47: 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. [SIC4] [MA2013] ALT (912) 41: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [SMD6] [MA2013] DM1 (912) 12: Use combinatorial reasoning and counting techniques to solve applicationbased problems. (Alabama) [MA2013] PRE (912) 50: (+) 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. [SMD1] [MA2013] PRE (912) 51: (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. [SMD2]
Subject: Mathematics Title: Birthday Paradox
Description: This Illuminations lesson demonstrates the birthday paradox, using it as a springboard into a unit on probability. Students use the TI83 graphing calculator to run a Monte Carlo simulation with the birthday paradox and engage in a graphical analysis of the birthdayproblem function. Thinkfinity Partner: Illuminations Grade Span: 6,7,8,9,10,11,12
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Title: Stick or Switch?
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This lesson, from Illuminations, presents a version of a classic gameshow scenario. You pick one of three doors in hopes of winning the prize. The host opens one of the two remaining doors, which reveals no prize, and then asks if you wish to stick or switch. Which choice gives you the best chance to win? Students explore different approaches to this problem including guesses, experiments, computer simulations, and theoretical models.
Standard(s): [MA2013] (6) 25: Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. [6SP1] [MA2013] (7) 17: Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. [7SP1] [MA2013] (7) 18: Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. [7SP2] [MA2013] (7) 21: Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ^{1}/_{2} indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. [7SP5] [MA2013] (7) 22: Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approximate relative frequency given the probability. [7SP6] [MA2013] (7) 23: Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. [7SP7] [MA2013] (7) 24: Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. [7SP8] [MA2013] AL1 (912) 47: 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. [SCP2] [MA2013] AL2 (912) 40: 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. [SCP3] [MA2013] AL2 (912) 41: Construct and interpret twoway frequency tables of data when two categories are associated with each object being classified. Use the twoway table as a sample space to decide if events are independent and to approximate conditional probabilities. [SCP4] [MA2013] AL2 (912) 42: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. [SCP5] [MA2013] AL2 (912) 43: Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model. [SCP6] [MA2013] AL2 (912) 44: Apply the Addition Rule, P(A or B) = P(A) + P(B)  P(A and B), and interpret the answer in terms of the model. [SCP7] [MA2013] AL2 (912) 45: (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(BA) = P(B)P(AB), and interpret the answer in terms of the model. [SCP8] [MA2013] AL2 (912) 46: (+) Use permutations and combinations to compute probabilities of compound events and solve problems. [SCP9] [MA2013] GEO (912) 42: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [SMD6] [MA2013] AL2 (912) 33: Write a function that describes a relationship between two quantities.* [FBF1] [MA2013] AL2 (912) 37: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [SMD6] [MA2013] PRE (912) 44: Understand statistics as a process for making inferences about population parameters based on a random sample from that population. [SIC1] [MA2013] PRE (912) 45: Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation. [SIC2] [MA2013] PRE (912) 46: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. [SIC3] [MA2013] PRE (912) 47: 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. [SIC4] [MA2013] PRE (912) 48: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. [SIC5] [MA2013] ALT (912) 41: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [SMD6] [MA2013] DM1 (912) 12: Use combinatorial reasoning and counting techniques to solve applicationbased problems. (Alabama) [MA2013] PRE (912) 50: (+) 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. [SMD1]
Subject: Mathematics Title: Stick or Switch?
Description: This lesson, from Illuminations, presents a version of a classic gameshow scenario. You pick one of three doors in hopes of winning the prize. The host opens one of the two remaining doors, which reveals no prize, and then asks if you wish to stick or switch. Which choice gives you the best chance to win? Students explore different approaches to this problem including guesses, experiments, computer simulations, and theoretical models. Thinkfinity Partner: Illuminations Grade Span: 6,7,8,9,10,11,12
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Title: Sticks and Stones Demo
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This student interactive, from an Illuminations lesson, allows students to generate random throws for the game '' Sticks and Stones.'' In the game, three sticks are tossed and a player moves his or her marker according to how the sticks land.
Standard(s): [MA2013] (6) 25: Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. [6SP1] [MA2013] (7) 17: Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. [7SP1] [MA2013] (7) 18: Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. [7SP2] [MA2013] (7) 21: Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ^{1}/_{2} indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. [7SP5] [MA2013] (7) 22: Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approximate relative frequency given the probability. [7SP6] [MA2013] (7) 23: Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. [7SP7] [MA2013] (7) 24: Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. [7SP8] [MA2013] AL1 (912) 47: 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. [SCP2] [MA2013] AL2 (912) 40: 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. [SCP3] [MA2013] AL2 (912) 41: Construct and interpret twoway frequency tables of data when two categories are associated with each object being classified. Use the twoway table as a sample space to decide if events are independent and to approximate conditional probabilities. [SCP4] [MA2013] AL2 (912) 43: Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model. [SCP6] [MA2013] AL2 (912) 44: Apply the Addition Rule, P(A or B) = P(A) + P(B)  P(A and B), and interpret the answer in terms of the model. [SCP7] [MA2013] AL2 (912) 45: (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(BA) = P(B)P(AB), and interpret the answer in terms of the model. [SCP8] [MA2013] AL2 (912) 46: (+) Use permutations and combinations to compute probabilities of compound events and solve problems. [SCP9] [MA2013] GEO (912) 42: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [SMD6] [MA2013] AL2 (912) 33: Write a function that describes a relationship between two quantities.* [FBF1] [MA2013] AL2 (912) 37: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [SMD6] [MA2013] PRE (912) 44: Understand statistics as a process for making inferences about population parameters based on a random sample from that population. [SIC1] [MA2013] PRE (912) 45: Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation. [SIC2] [MA2013] PRE (912) 46: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. [SIC3] [MA2013] PRE (912) 47: 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. [SIC4] [MA2013] ALT (912) 41: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [SMD6] [MA2013] DM1 (912) 12: Use combinatorial reasoning and counting techniques to solve applicationbased problems. (Alabama) [MA2013] PRE (912) 50: (+) 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. [SMD1]
Subject: Mathematics Title: Sticks and Stones Demo
Description: This student interactive, from an Illuminations lesson, allows students to generate random throws for the game '' Sticks and Stones.'' In the game, three sticks are tossed and a player moves his or her marker according to how the sticks land. Thinkfinity Partner: Illuminations Grade Span: 3,4,5,6,7,8
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Title: Sticks and Stones
Description:
In this Illuminations lesson, students play Sticks and Stones, a game based on the Apache game Throw Sticks, which was played at multination celebrations. Students collect data, investigate the likelihood of various moves, and use basic ideas of expected value to determine the average number of turns needed to win a game.
Standard(s): [MA2013] (3) 18: Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one and twostep "how many more" and "how many less" problems using information presented in scaled bar graphs. [3MD3] [MA2013] (5) 19: Make a line plot to display a data set of measurements in fractions of a unit (^{1}/_{2}, ^{1}/_{4}, ^{1}/_{8}).
Use operations on fractions for this grade to solve problems involving information presented in line plots. [5MD2] [MA2013] (6) 25: Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. [6SP1] [MA2013] (6) 26: Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. [6SP2] [MA2013] (6) 27: Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. [6SP3] [MA2013] (6) 28: Display numerical data in plots on a number line, including dot plots, histograms, and box plots. [6SP4] [MA2013] (6) 29: Summarize numerical data sets in relation to their context, such as by: [6SP5] [MA2013] (7) 17: Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. [7SP1] [MA2013] (7) 18: Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. [7SP2] [MA2013] (7) 21: Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ^{1}/_{2} indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. [7SP5] [MA2013] (7) 22: Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approximate relative frequency given the probability. [7SP6] [MA2013] (7) 23: Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. [7SP7] [MA2013] (7) 24: Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. [7SP8] [MA2013] AL1 (912) 42: 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. [SID2] [MA2013] AL1 (912) 43: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). [SID3] [MA2013] AL1 (912) 47: 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. [SCP2] [MA2013] AL2 (912) 40: 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. [SCP3] [MA2013] AL2 (912) 42: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. [SCP5] [MA2013] AL2 (912) 43: Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model. [SCP6] [MA2013] AL2 (912) 44: Apply the Addition Rule, P(A or B) = P(A) + P(B)  P(A and B), and interpret the answer in terms of the model. [SCP7] [MA2013] AL2 (912) 45: (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(BA) = P(B)P(AB), and interpret the answer in terms of the model. [SCP8] [MA2013] AL2 (912) 46: (+) Use permutations and combinations to compute probabilities of compound events and solve problems. [SCP9] [MA2013] GEO (912) 42: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [SMD6] [MA2013] AL2 (912) 33: Write a function that describes a relationship between two quantities.* [FBF1] [MA2013] AL2 (912) 37: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [SMD6] [MA2013] PRE (912) 44: Understand statistics as a process for making inferences about population parameters based on a random sample from that population. [SIC1] [MA2013] PRE (912) 45: Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation. [SIC2] [MA2013] PRE (912) 46: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. [SIC3] [MA2013] PRE (912) 47: 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. [SIC4] [MA2013] ALT (912) 41: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [SMD6] [MA2013] PRE (912) 50: (+) 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. [SMD1]
Subject: Mathematics Title: Sticks and Stones
Description: In this Illuminations lesson, students play Sticks and Stones, a game based on the Apache game Throw Sticks, which was played at multination celebrations. Students collect data, investigate the likelihood of various moves, and use basic ideas of expected value to determine the average number of turns needed to win a game. Thinkfinity Partner: Illuminations Grade Span: 3,4,5,6,7,8
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Title: Sports Rankings
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In this Science Update, from Science NetLinks, you'll hear how mathematicians at the Rochester Institute of Technology are suggesting changes to the ranking systems in sports. Project leader Darren Narayan says the goal is to factor in headtohead competition. Science Updates are audio interviews with scientists and are accompanied by a set of questions as well as links to related Science NetLinks lessons and other related resources.
Standard(s): [MA2013] (6) 29: Summarize numerical data sets in relation to their context, such as by: [6SP5] [MA2013] (7) 17: Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. [7SP1] [MA2013] AL2 (912) 33: Write a function that describes a relationship between two quantities.* [FBF1] [MA2013] PRE (912) 44: Understand statistics as a process for making inferences about population parameters based on a random sample from that population. [SIC1] [MA2013] PRE (912) 49: Evaluate reports based on data. [SIC6]
Subject: Mathematics, Physical Education Title: Sports Rankings
Description: In this Science Update, from Science NetLinks, you'll hear how mathematicians at the Rochester Institute of Technology are suggesting changes to the ranking systems in sports. Project leader Darren Narayan says the goal is to factor in headtohead competition. Science Updates are audio interviews with scientists and are accompanied by a set of questions as well as links to related Science NetLinks lessons and other related resources. Thinkfinity Partner: Science NetLinks Grade Span: 6,7,8,9,10,11,12
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Title: WhelkCome to Mathematics
Description:
In this fourlesson unit, from Illuminations, students make a conjecture, conduct an experiment, analyze data and work to a conclusion using rational functions to investigate the behavior of Northwestern Crows. Sea gulls and crows feed on various types of mollusks by lifting them into the air and dropping them onto a rock to break open their shells. Biologists have observed that northwestern crows consistently drop a type of mollusk called a whelk from a mean height of about 5 meters. Students investigate the relationship between the height of the drop and the number of drops.
Standard(s): [MA2013] AL2 (912) 22: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. [ACED3] [MA2013] AL2 (912) 29: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [FIF5] [MA2013] AL2 (912) 33: Write a function that describes a relationship between two quantities.* [FBF1] [MA2013] ALT (912) 22: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. [ACED3] [MA2013] PRE (912) 16: 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. Determine odd, even, neither.)* [FIF4] (Alabama) [MA2013] ALT (912) 29: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [FIF5] [MA2013] ALT (912) 33: Write a function that describes a relationship between two quantities.* [FBF1] [MA2013] PRE (912) 44: Understand statistics as a process for making inferences about population parameters based on a random sample from that population. [SIC1] [MA2013] PRE (912) 45: Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation. [SIC2] [MA2013] PRE (912) 46: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. [SIC3] [MA2013] PRE (912) 49: Evaluate reports based on data. [SIC6] [MA2013] PRE (912) 4: Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity. (Alabama) [MA2013] PRE (912) 18: c. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. [FIF7d] [MA2013] AM1 (912) 12: Calculate the limit of a sequence, of a function, and of an infinite series. (Alabama)
Subject: Mathematics,Science Title: WhelkCome to Mathematics
Description: In this fourlesson unit, from Illuminations, students make a conjecture, conduct an experiment, analyze data and work to a conclusion using rational functions to investigate the behavior of Northwestern Crows. Sea gulls and crows feed on various types of mollusks by lifting them into the air and dropping them onto a rock to break open their shells. Biologists have observed that northwestern crows consistently drop a type of mollusk called a whelk from a mean height of about 5 meters. Students investigate the relationship between the height of the drop and the number of drops. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
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Title: Northwestern Crows
Description:
In this lesson, one of a multipart unit from Illuminations, students use rational functions to investigate the feeding behavior of Northwestern Crows. Biologists have observed that northwestern crows consistently drop a type of mollusk called a whelk from a mean height of about 5 meters. Students investigate the relationship between the height of the drop and the number of drops by viewing a video clip; then, they plan a classroom experiment to model the dropping of whelks in order to collect and analyze data.
Standard(s): [MA2013] AL2 (912) 21: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [ACED2] [MA2013] AL2 (912) 29: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [FIF5] [MA2013] AL2 (912) 33: Write a function that describes a relationship between two quantities.* [FBF1] [MA2013] ALT (912) 21: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [ACED2] [MA2013] ALT (912) 22: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. [ACED3] [MA2013] PRE (912) 16: 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. Determine odd, even, neither.)* [FIF4] (Alabama) [MA2013] ALT (912) 29: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [FIF5] [MA2013] ALT (912) 33: Write a function that describes a relationship between two quantities.* [FBF1] [MA2013] PRE (912) 44: Understand statistics as a process for making inferences about population parameters based on a random sample from that population. [SIC1] [MA2013] PRE (912) 45: Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation. [SIC2] [MA2013] PRE (912) 46: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. [SIC3] [MA2013] PRE (912) 49: Evaluate reports based on data. [SIC6] [MA2013] PRE (912) 4: Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity. (Alabama) [MA2013] PRE (912) 18: c. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. [FIF7d] [MA2013] PRE (912) 50: (+) 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. [SMD1]
Subject: Mathematics,Science Title: Northwestern Crows
Description: In this lesson, one of a multipart unit from Illuminations, students use rational functions to investigate the feeding behavior of Northwestern Crows. Biologists have observed that northwestern crows consistently drop a type of mollusk called a whelk from a mean height of about 5 meters. Students investigate the relationship between the height of the drop and the number of drops by viewing a video clip; then, they plan a classroom experiment to model the dropping of whelks in order to collect and analyze data. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
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Title: Conduct an Experiment
Description:
In this lesson, one of a multipart unit from Illuminations, students use rational functions to investigate the feeding behavior of Northwestern Crows. Biologists have observed that northwestern crows consistently drop a type of mollusk called a whelk from a mean height of about 5 meters. Students investigate whether the crows minimize their work by dropping whelks as they do. The amount of work depends upon the height of the drop and the number of times the crow has to fly to this height. To answer the question, students explore the relationship between the height of the drop and the number of drops needed.
Standard(s): [MA2013] AL2 (912) 21: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [ACED2] [MA2013] AL2 (912) 29: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [FIF5] [MA2013] AL2 (912) 33: Write a function that describes a relationship between two quantities.* [FBF1] [MA2013] ALT (912) 21: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [ACED2] [MA2013] PRE (912) 16: 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. Determine odd, even, neither.)* [FIF4] (Alabama) [MA2013] ALT (912) 29: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [FIF5] [MA2013] ALT (912) 33: Write a function that describes a relationship between two quantities.* [FBF1] [MA2013] PRE (912) 44: Understand statistics as a process for making inferences about population parameters based on a random sample from that population. [SIC1] [MA2013] PRE (912) 45: Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation. [SIC2] [MA2013] PRE (912) 46: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. [SIC3] [MA2013] PRE (912) 49: Evaluate reports based on data. [SIC6] [MA2013] PRE (912) 4: Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity. (Alabama) [MA2013] PRE (912) 18: c. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. [FIF7d] [MA2013] PRE (912) 50: (+) 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. [SMD1] [MA2013] AM1 (912) 12: Calculate the limit of a sequence, of a function, and of an infinite series. (Alabama)
Subject: Mathematics,Science Title: Conduct an Experiment
Description: In this lesson, one of a multipart unit from Illuminations, students use rational functions to investigate the feeding behavior of Northwestern Crows. Biologists have observed that northwestern crows consistently drop a type of mollusk called a whelk from a mean height of about 5 meters. Students investigate whether the crows minimize their work by dropping whelks as they do. The amount of work depends upon the height of the drop and the number of times the crow has to fly to this height. To answer the question, students explore the relationship between the height of the drop and the number of drops needed. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
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Title: Sports Rankings
Description:
In this Science Update, from Science NetLinks, you'll hear how mathematicians at the Rochester Institute of Technology are suggesting changes to the ranking systems in sports. Project leader Darren Narayan says the goal is to factor in headtohead competition. Science Updates are audio interviews with scientists and are accompanied by a set of questions as well as links to related Science NetLinks lessons and other related resources.
Standard(s): [MA2013] (6) 29: Summarize numerical data sets in relation to their context, such as by: [6SP5] [MA2013] (7) 17: Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. [7SP1] [MA2013] AL2 (912) 33: Write a function that describes a relationship between two quantities.* [FBF1] [MA2013] PRE (912) 44: Understand statistics as a process for making inferences about population parameters based on a random sample from that population. [SIC1] [MA2013] PRE (912) 49: Evaluate reports based on data. [SIC6]
Subject: Mathematics, Physical Education Title: Sports Rankings
Description: In this Science Update, from Science NetLinks, you'll hear how mathematicians at the Rochester Institute of Technology are suggesting changes to the ranking systems in sports. Project leader Darren Narayan says the goal is to factor in headtohead competition. Science Updates are audio interviews with scientists and are accompanied by a set of questions as well as links to related Science NetLinks lessons and other related resources. Thinkfinity Partner: Science NetLinks Grade Span: 6,7,8,9,10,11,12
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Title: Online Algebra Problem Solver
Description:
Online Algebra Solver allows students to create equations related to a variety of areas of math and solve them online.
Standard(s): [MA2013] AL1 (912) 12: Create equations and inequalities in one variable, and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. [ACED1] [MA2013] AL1 (912) 13: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [ACED2] [MA2013] AL1 (912) 17: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. [AREI3] [MA2013] AL1 (912) 18: Solve quadratic equations in one variable. [AREI4] [MA2013] AL1 (912) 20: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. [AREI6] [MA2013] AL2 (912) 4: Solve quadratic equations with real coefficients that have complex solutions. [NCN7] [MA2013] AL2 (912) 20: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. [ACED1] [MA2013] AL2 (912) 21: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [ACED2] [MA2013] AL2 (912) 33: Write a function that describes a relationship between two quantities.* [FBF1] [MA2013] ALT (912) 4: Solve quadratic equations with real coefficients that have complex solutions. [NCN7] [MA2013] ALT (912) 31: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [FIF8] [MA2013] AL2 (912) 8: (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. [NVM7] [MA2013] AL2 (912) 9: (+) Add, subtract, and multiply matrices of appropriate dimensions. [NVM8] [MA2013] ALT (912) 8: (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. [NVM7] [MA2013] ALT (912) 9: (+) Add, subtract, and multiply matrices of appropriate dimensions. [NVM8]
Online Algebra Problem Solver
http://www.algebrao...
Online Algebra Solver allows students to create equations related to a variety of areas of math and solve them online.
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Title: Relations and Functions Puzzle
Description:
An "AhBach" style worksheet/puzzle dealing with evaluating functions and matching equations and graphs. Question solutions yield the needed letters to decipher a message.
Standard(s): [MA2013] AL1 (912) 12: Create equations and inequalities in one variable, and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. [ACED1] [MA2013] AL2 (912) 33: Write a function that describes a relationship between two quantities.* [FBF1] [MA2013] ALT (912) 33: Write a function that describes a relationship between two quantities.* [FBF1]
Relations and Functions Puzzle
http://mathbits.com/...
An "AhBach" style worksheet/puzzle dealing with evaluating functions and matching equations and graphs. Question solutions yield the needed letters to decipher a message.
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Title: Online Algebra Problem Solver
Description:
Online Algebra Solver allows students to create equations related to a variety of areas of math and solve them online.
Standard(s): [MA2013] AL1 (912) 12: Create equations and inequalities in one variable, and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. [ACED1] [MA2013] AL1 (912) 13: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [ACED2] [MA2013] AL1 (912) 17: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. [AREI3] [MA2013] AL1 (912) 18: Solve quadratic equations in one variable. [AREI4] [MA2013] AL1 (912) 20: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. [AREI6] [MA2013] AL2 (912) 4: Solve quadratic equations with real coefficients that have complex solutions. [NCN7] [MA2013] AL2 (912) 20: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. [ACED1] [MA2013] AL2 (912) 21: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [ACED2] [MA2013] AL2 (912) 33: Write a function that describes a relationship between two quantities.* [FBF1] [MA2013] ALT (912) 4: Solve quadratic equations with real coefficients that have complex solutions. [NCN7] [MA2013] ALT (912) 31: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [FIF8] [MA2013] AL2 (912) 8: (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. [NVM7] [MA2013] AL2 (912) 9: (+) Add, subtract, and multiply matrices of appropriate dimensions. [NVM8] [MA2013] ALT (912) 8: (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. [NVM7] [MA2013] ALT (912) 9: (+) Add, subtract, and multiply matrices of appropriate dimensions. [NVM8]
Online Algebra Problem Solver
http://www.algebrao...
Online Algebra Solver allows students to create equations related to a variety of areas of math and solve them online.
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Title: Adjustable Spinner
Description:
This student interactive, from Illuminations, allows students to create their own spinners and examine the outcomes given a specified number of spins. Students learn that experimental probabilities differ according to the characteristics of the model.
Standard(s): [MA2013] (6) 25: Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. [6SP1] [MA2013] (6) 29: Summarize numerical data sets in relation to their context, such as by: [6SP5] [MA2013] (7) 17: Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. [7SP1] [MA2013] (7) 18: Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. [7SP2] [MA2013] (7) 21: Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ^{1}/_{2} indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. [7SP5] [MA2013] (7) 22: Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approximate relative frequency given the probability. [7SP6] [MA2013] (7) 23: Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. [7SP7] [MA2013] GEO (912) 42: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [SMD6] [MA2013] AL2 (912) 33: Write a function that describes a relationship between two quantities.* [FBF1] [MA2013] AL2 (912) 37: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [SMD6] [MA2013] PRE (912) 44: Understand statistics as a process for making inferences about population parameters based on a random sample from that population. [SIC1] [MA2013] PRE (912) 45: Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation. [SIC2] [MA2013] PRE (912) 46: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. [SIC3] [MA2013] ALT (912) 41: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [SMD6]
Subject: Mathematics Title: Adjustable Spinner
Description: This student interactive, from Illuminations, allows students to create their own spinners and examine the outcomes given a specified number of spins. Students learn that experimental probabilities differ according to the characteristics of the model. Thinkfinity Partner: Illuminations Grade Span: K,PreK,1,2,3,4,5,6,7,8,9,10,11,12

