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Title: Know Your Limits
Description:
The limit is the fundamental notion of calculus. This underlying concept is the thread that binds together virtually all of the calculus studied by the student. In this lesson, the student's knowledge of limits will be extended using some common language and illustrated ideas through simple examples. The concept of limits will be used to describe functions with specific properties. Additionally, the concept of a limit function is further developed, and theorems for limits are analyzed and applied. Students will refer to these rules as they work through exercises.
Standard(s): [MA2013] PRE (912) 4: Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity. (Alabama)
Subject: Mathematics (9  12)
Title: Know Your Limits
Description: The limit is the fundamental notion of calculus. This underlying concept is the thread that binds together virtually all of the calculus studied by the student. In this lesson, the student's knowledge of limits will be extended using some common language and illustrated ideas through simple examples. The concept of limits will be used to describe functions with specific properties. Additionally, the concept of a limit function is further developed, and theorems for limits are analyzed and applied. Students will refer to these rules as they work through exercises.
Thinkfinity Lesson Plans
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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
Web Resources
Informational Materials
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Title: Karl's Calculus Tutor
Description:
Karl's Calculus Tutor is an online resource that contains a wide variety of practice problems and examples pertaining to topics in precalculus and calculus.
Standard(s): [MA2013] PRE (912) 4: Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity. (Alabama) [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)
Karl's Calculus Tutor
http://www.karlscalc...
Karl's Calculus Tutor is an online resource that contains a wide variety of practice problems and examples pertaining to topics in precalculus and calculus.
Interactives/Games
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Title: EpsilonDelta Game (Limit of a Function)
Description:
This is a game that simulates the epsilondelta definition of the limit of a function. Moveable sliders allow the values of "epsilon", "delta", and the limit of the function to be changed.
Standard(s): [MA2013] PRE (912) 4: Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity. (Alabama)
EpsilonDelta Game (Limit of a Function)
http://www.desmos.co...
This is a game that simulates the epsilondelta definition of the limit of a function. Moveable sliders allow the values of "epsilon", "delta", and the limit of the function to be changed.
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Title: Limit Simulator
Description:
This is an interactive tool which simulates the limit of a function.
Standard(s): [MA2013] PRE (912) 4: Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity. (Alabama)
Limit Simulator
http://illuminations...
This is an interactive tool which simulates the limit of a function.
Thinkfinity Learning Activities
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Title: Limits
Description:
This student interactive, from an Illuminations lesson, allows students to see a representation of three geometric series and the harmonic series.
Standard(s): [MA2013] DM1 (912) 2: Determine characteristics of sequences, including the Fibonacci sequence, the triangular numbers, and pentagonal numbers. (Alabama) [MA2013] PRE (912) 4: Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity. (Alabama) [MA2013] AM1 (912) 12: Calculate the limit of a sequence, of a function, and of an infinite series. (Alabama)
Subject: Mathematics Title: Limits
Description: This student interactive, from an Illuminations lesson, allows students to see a representation of three geometric series and the harmonic series. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12

