This cluster includes the following access points.
Vetted resources educators can use to teach the concepts and skills in this topic.
| Name |
Description |
| Recursive Sequences: | Students are asked to find the first five terms of a sequence defined recursively, explain why the sequence is a function, and describe its domain |
| Which Sequences Are Functions?: | Students are asked to determine if each of two sequences is a function and to describe its domain, if it is a function. |
| Cell Phone Battery Life: | Students are asked to interpret statements that use function notation in the context of a problem. |
| What Is a Function?: | Students are asked to define the term function and describe any important properties of functions. |
| Writing Functions: | Students are asked to create their own examples and nonexamples of functions by completing tables and mapping diagrams. |
| Circles and Functions: | Students are shown the graph of a circle and asked to identify a portion of the graph that could be removed so that the remaining portion represents a function. |
| What Is the Value?: | Students are asked to determine the corresponding input value for a given output using a table of values representing a function, f. |
| Cafeteria Function: | Students are asked decide if one variable is a function of the other in the context of a real-world problem. |
| Identifying Functions: | Students are asked to determine if relations given by tables and mapping diagrams are functions. |
| Identifying the Graphs of Functions: | Students are given four graphs and asked to identify which represent functions and to justify their choices. |
| What Is the Function Notation?: | Students are asked to use function notation to rewrite the formula for the volume of a cube and to explain the meaning of the notation. |
| Graphs and Functions: | Students are asked to determine the value of a function, at an input given using function notation, by inspecting its graph. |
| Evaluating a Function: | Students are asked to evaluate a function at a given value of the independent variable. |
| Name |
Description |
| The Towers of Hanoi: Experiential Recursive Thinking: |
This lesson is about the Towers of Hanoi problem, a classic famous problem involving recursive thinking to reduce what appears to be a very large and difficult problem into a series of simpler ones. The learning objective is for students to begin to understand recursive logic and thinking, relevant to computer scientists, mathematicians and engineers. The lesson is experiential, in that each student will be working with her/his own Towers of Hanoi manipulative, inexpensively obtained. There is no formal prerequisite, although some familiarity with set theory and functions is helpful. The last three sections of the lesson involve some more formal concepts with recursive equations and proof by induction, so the students who work on those sections should probably be level 11 or 12 in a K-12 educational system. The lesson has a Stop Point for 50-minute classes, followed by three more segments that may require a half to full additional class time. So the teacher may use only those segments up to the Stop Point, or if two class sessions are to be devoted to the lesson, the entire set of segments. Supplies are modest, and may be a set of coins or some washers from a hardware store to assemble small piles of disks in front of each student, each set of disks representing a Towers of Hanoi manipulative. Or the students may assemble before the class a more complete Towers of Hanoi at home, as demonstrated in the video. The classroom activities involve attempting to solve with hand and mind the Towers of Hanoi problem and discussing with fellow students patterns in the process and strategies for solution.
|
| Representing Polynomials: | This lesson unit is intended to help you assess how well students are able to translate between graphs and algebraic representations of polynomials. In particular, this unit aims to help you identify and assist students who have difficulties in recognizing the connection between the zeros of polynomials when suitable factorizations are available, and graphs of the functions defined by polynomials as well as recognizing the connection between transformations of the graphs and transformations of the functions obtained by replacing f(x) by f(x + k), f(x) + k, -f(x), f(-x). |
| Functions: Domain and Range: | Students will identify if a graph represents a function and determine domain and range of the graphs. |
| Functions and Everyday Situations: | This lesson unit is intended to help you assess how well students are able to articulate verbally the relationships between variables arising in everyday contexts, translate between everyday situations and sketch graphs of relationships between variables, interpret algebraic functions in terms of the contexts in which they arise and reflect on the domains of everyday functions and in particular whether they should be discrete or continuous. |
| How much is your time worth?: | This lesson is designed to help students solve real-world problems involving compound and continuously compounded interest. Students will also be required to translate word problems into function models, evaluate functions for inputs in their domains, and interpret outputs in context. |
| Freeze: | In this lesson students will learn how to write equations in function notation when given a real-world scenario. Students will work in groups to determine an equation for a given scenario, as well as, write a scenario for a given equation. |
| Exponential Graphing Using Technology: | This lesson is teacher/student directed for discovering and translating exponential functions using a graphing app. The lesson focuses on the translations from a parent graph and how changing the coefficient, base and exponent values relate to the transformation. |
| Name |
Description |
| Your Father: | This is a simple task touching on two key points of functions. First, there is the idea that not all functions have real numbers as domain and range values. Second, the task addresses the issue of when a function admits an inverse, and the process of "restricting the domain" in order to achieve an invertible function. |
| Yam in the Oven: | The purpose of this task is to give students practice interpreting statements using function notation. It can be used as a diagnostic if students seem to be having trouble with function notation, for example mistakenly interpreting f(x) as the product of f and x. |
| Using Function Notation II: | The purpose of the task is to explicitly identify a common error made by many students when using the "identity" f(x + h) = f(x) + f(h). |
| Using Function Notation I: | This task addresses a common misconception about function notation. |
| The Random Walk: | This task requires interpreting a function in a non-standard context. While the domain and range of this function are both numbers, the way in which the function is determined is not via a formula but by a (pre-determined) sequence of coin flips. In addition, the task provides an opportunity to compute some probabilities in a discrete situation. The task could be used to segue the discussion from functions to probability, in particular the early standards in the S-CP domain. |
| The Parking Lot: | The purpose of this task is to investigate the meaning of the definition of function in a real-world context where the question of whether there is more than one output for a given input arises naturally. In more advanced courses this task could be used to investigate the question of whether a function has an inverse. |
| Domains: | The purpose of this task to help students think about an expression for a function as built up out of simple operations on the variable and understand the domain in terms of values for which each operation is invalid (e.g., dividing by zero or taking the square root of a negative number). |
| Cell Phones: | This simple task assesses whether students can interpret function notation. The four parts of the task provide a logical progression of exercises for advancing understanding of function notation and how to interpret it in terms of a given context. |
| The High School Gym: | This task asks students to consider functions in regard to temperatures in a high school gym. |
| The Customers: | The purpose of this task is to introduce or reinforce the concept of a function, especially in a context where the function is not given by an explicit algebraic representation. Further, the last part of the task emphasizes the significance of one variable being a function of another variable in an immediately relevant real-life context. |
| Random Walk II: | These problems form a bridge between work on functions and work on probability. The task is better suited for instruction than for assessment as it provides students with a non-standard setting in which to interpret the meaning of functions. Students should carry out the process of flipping a coin and modeling this Random Walk in order to develop a sense of the process before analyzing it mathematically. |
| Points on a graph: | This task is designed to get at a common student confusion between the independent and dependent variables. This confusion often arises in situations like (b), where students are asked to solve an equation involving a function, and confuse that operation with evaluating the function. |
| Pizza Place Promotion: | This tasks asks students to use functions to predict the price of a pizza on a specific day and find which day the pizza would be cheapest according to a promotion. |
| Parabolas and Inverse Functions: | This problem is a simple de-contextualized version of F-IF Your Father and F-IF Parking Lot. It also provides a natural context where the absolute value function arises as, in part (b), solving for x in terms of y means taking the square root of x^2 which is |x|.This task assumes students have an understanding of the relationship between functions and equations. |
| Interpreting the Graph: | The purpose of this task is to help students learn to read information about a function from its graph, by asking them to show the part of the graph that exhibits a certain property of the function. The task could be used to further instruction on understanding functions or as an assessment tool, with the caveat that it requires some amount of creativity to decide how to best illustrate some of the statements. |
| Name |
Description |
| What is a Function?: | This video will demonstrate how to determine what is and is not a function. |
| Real-Valued Functions of a Real Variable: | Although the domain and codomain of functions can consist of any type of objects, the most common functions encountered in Algebra are real-valued functions of a real variable, whose domain and codomain are the set of real numbers, R. |
| Domain and Range of Binary Relations: | Two sets which are often of primary interest when studying binary relations are the domain and range of the relation. |
| MIT BLOSSOMS - Fabulous Fractals and Difference Equations : | This learning video introduces students to the world of Fractal Geometry through the use of difference equations. As a prerequisite to this lesson, students would need two years of high school algebra (comfort with single variable equations) and motivation to learn basic complex arithmetic. Ms. Zager has included a complete introductory tutorial on complex arithmetic with homework assignments downloadable here. Also downloadable are some supplemental challenge problems. Time required to complete the core lesson is approximately one hour, and materials needed include a blackboard/whiteboard as well as space for students to work in small groups. During the in-class portions of this interactive lesson, students will brainstorm on the outcome of the chaos game and practice calculating trajectories of difference equations. |
Vetted resources students can use to learn the concepts and skills in this topic.
| Title |
Description |
| Your Father: | This is a simple task touching on two key points of functions. First, there is the idea that not all functions have real numbers as domain and range values. Second, the task addresses the issue of when a function admits an inverse, and the process of "restricting the domain" in order to achieve an invertible function. |
| Yam in the Oven: | The purpose of this task is to give students practice interpreting statements using function notation. It can be used as a diagnostic if students seem to be having trouble with function notation, for example mistakenly interpreting f(x) as the product of f and x. |
| Using Function Notation I: | This task addresses a common misconception about function notation. |
| The Random Walk: | This task requires interpreting a function in a non-standard context. While the domain and range of this function are both numbers, the way in which the function is determined is not via a formula but by a (pre-determined) sequence of coin flips. In addition, the task provides an opportunity to compute some probabilities in a discrete situation. The task could be used to segue the discussion from functions to probability, in particular the early standards in the S-CP domain. |
| The Parking Lot: | The purpose of this task is to investigate the meaning of the definition of function in a real-world context where the question of whether there is more than one output for a given input arises naturally. In more advanced courses this task could be used to investigate the question of whether a function has an inverse. |
| Domains: | The purpose of this task to help students think about an expression for a function as built up out of simple operations on the variable and understand the domain in terms of values for which each operation is invalid (e.g., dividing by zero or taking the square root of a negative number). |
| Cell Phones: | This simple task assesses whether students can interpret function notation. The four parts of the task provide a logical progression of exercises for advancing understanding of function notation and how to interpret it in terms of a given context. |
| The High School Gym: | This task asks students to consider functions in regard to temperatures in a high school gym. |
| The Customers: | The purpose of this task is to introduce or reinforce the concept of a function, especially in a context where the function is not given by an explicit algebraic representation. Further, the last part of the task emphasizes the significance of one variable being a function of another variable in an immediately relevant real-life context. |
| Random Walk II: | These problems form a bridge between work on functions and work on probability. The task is better suited for instruction than for assessment as it provides students with a non-standard setting in which to interpret the meaning of functions. Students should carry out the process of flipping a coin and modeling this Random Walk in order to develop a sense of the process before analyzing it mathematically. |
| Points on a graph: | This task is designed to get at a common student confusion between the independent and dependent variables. This confusion often arises in situations like (b), where students are asked to solve an equation involving a function, and confuse that operation with evaluating the function. |
| Pizza Place Promotion: | This tasks asks students to use functions to predict the price of a pizza on a specific day and find which day the pizza would be cheapest according to a promotion. |
| Parabolas and Inverse Functions: | This problem is a simple de-contextualized version of F-IF Your Father and F-IF Parking Lot. It also provides a natural context where the absolute value function arises as, in part (b), solving for x in terms of y means taking the square root of x^2 which is |x|.This task assumes students have an understanding of the relationship between functions and equations. |
| Interpreting the Graph: | The purpose of this task is to help students learn to read information about a function from its graph, by asking them to show the part of the graph that exhibits a certain property of the function. The task could be used to further instruction on understanding functions or as an assessment tool, with the caveat that it requires some amount of creativity to decide how to best illustrate some of the statements. |
| Title |
Description |
| What is a Function?: | This video will demonstrate how to determine what is and is not a function. |
| Real-Valued Functions of a Real Variable: | Although the domain and codomain of functions can consist of any type of objects, the most common functions encountered in Algebra are real-valued functions of a real variable, whose domain and codomain are the set of real numbers, R. |
| Domain and Range of Binary Relations: | Two sets which are often of primary interest when studying binary relations are the domain and range of the relation. |
| MIT BLOSSOMS - Fabulous Fractals and Difference Equations : | This learning video introduces students to the world of Fractal Geometry through the use of difference equations. As a prerequisite to this lesson, students would need two years of high school algebra (comfort with single variable equations) and motivation to learn basic complex arithmetic. Ms. Zager has included a complete introductory tutorial on complex arithmetic with homework assignments downloadable here. Also downloadable are some supplemental challenge problems. Time required to complete the core lesson is approximately one hour, and materials needed include a blackboard/whiteboard as well as space for students to work in small groups. During the in-class portions of this interactive lesson, students will brainstorm on the outcome of the chaos game and practice calculating trajectories of difference equations. |
Vetted resources caregivers can use to help students learn the concepts and skills in this topic.
| Title |
Description |
| Your Father: | This is a simple task touching on two key points of functions. First, there is the idea that not all functions have real numbers as domain and range values. Second, the task addresses the issue of when a function admits an inverse, and the process of "restricting the domain" in order to achieve an invertible function. |
| Yam in the Oven: | The purpose of this task is to give students practice interpreting statements using function notation. It can be used as a diagnostic if students seem to be having trouble with function notation, for example mistakenly interpreting f(x) as the product of f and x. |
| Using Function Notation I: | This task addresses a common misconception about function notation. |
| The Random Walk: | This task requires interpreting a function in a non-standard context. While the domain and range of this function are both numbers, the way in which the function is determined is not via a formula but by a (pre-determined) sequence of coin flips. In addition, the task provides an opportunity to compute some probabilities in a discrete situation. The task could be used to segue the discussion from functions to probability, in particular the early standards in the S-CP domain. |
| The Parking Lot: | The purpose of this task is to investigate the meaning of the definition of function in a real-world context where the question of whether there is more than one output for a given input arises naturally. In more advanced courses this task could be used to investigate the question of whether a function has an inverse. |
| Domains: | The purpose of this task to help students think about an expression for a function as built up out of simple operations on the variable and understand the domain in terms of values for which each operation is invalid (e.g., dividing by zero or taking the square root of a negative number). |
| Cell Phones: | This simple task assesses whether students can interpret function notation. The four parts of the task provide a logical progression of exercises for advancing understanding of function notation and how to interpret it in terms of a given context. |
| The High School Gym: | This task asks students to consider functions in regard to temperatures in a high school gym. |
| The Customers: | The purpose of this task is to introduce or reinforce the concept of a function, especially in a context where the function is not given by an explicit algebraic representation. Further, the last part of the task emphasizes the significance of one variable being a function of another variable in an immediately relevant real-life context. |
| Random Walk II: | These problems form a bridge between work on functions and work on probability. The task is better suited for instruction than for assessment as it provides students with a non-standard setting in which to interpret the meaning of functions. Students should carry out the process of flipping a coin and modeling this Random Walk in order to develop a sense of the process before analyzing it mathematically. |
| Points on a graph: | This task is designed to get at a common student confusion between the independent and dependent variables. This confusion often arises in situations like (b), where students are asked to solve an equation involving a function, and confuse that operation with evaluating the function. |
| Pizza Place Promotion: | This tasks asks students to use functions to predict the price of a pizza on a specific day and find which day the pizza would be cheapest according to a promotion. |
| Parabolas and Inverse Functions: | This problem is a simple de-contextualized version of F-IF Your Father and F-IF Parking Lot. It also provides a natural context where the absolute value function arises as, in part (b), solving for x in terms of y means taking the square root of x^2 which is |x|.This task assumes students have an understanding of the relationship between functions and equations. |
| Interpreting the Graph: | The purpose of this task is to help students learn to read information about a function from its graph, by asking them to show the part of the graph that exhibits a certain property of the function. The task could be used to further instruction on understanding functions or as an assessment tool, with the caveat that it requires some amount of creativity to decide how to best illustrate some of the statements. |
