| Code |
Description |
| MA.8.F.1.1: | Given a set of ordered pairs, a table, a graph or mapping diagram, determine whether the relationship is a function. Identify the domain and range of the relation.Clarifications:
Clarification 1: Instruction includes referring to the input as the independent variable and the output as the dependent variable.
Clarification 2: Within this benchmark, it is the expectation to represent domain and range as a list of numbers or as an inequality.
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| MA.8.F.1.2: | Given a function defined by a graph or an equation, determine whether the function is a linear function. Given an input-output table, determine whether it could represent a linear function.Clarifications:
Clarification 1: Instruction includes recognizing that a table may not determine a function. |
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| MA.8.F.1.3: | Analyze a real-world written description or graphical representation of a functional relationship between two quantities and identify where the function is increasing, decreasing or constant.Clarifications:
Clarification 1: Problem types are limited to continuous functions.
Clarification 2: Analysis includes writing a description of a graphical representation or sketching a graph from a written description.
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This cluster includes the following access points.
Vetted resources educators can use to teach the concepts and skills in this topic.
| Name |
Description |
| Functions, Functions Everywhere: Part 1: | What is a function? Where do we see functions in real life? Explore these questions and more using different contexts in this interactive tutorial.
This is part 1 in a two-part series on functions. Click HERE to open Part 2. |
| Math Models and Social Distancing: | Learn how math models can show why social distancing during a epidemic or pandemic is important in this interactive tutorial. |
| Cruising Through Functions: | Cruise along as you discover how to qualitatively describe functions in this interactive tutorial. |
| Summer of FUNctions: | Have some fun with FUNctions! Learn how to identify linear and non-linear functions in this interactive tutorial. |
| Driven By Functions: | Learn how to determine if a relationship is a function in this interactive tutorial that shows you inputs, outputs, equations, graphs and verbal descriptions. |
| Name |
Description |
| Population Trend: | Students are asked to describe the relationship between two quantities in a nonlinear function. |
| 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. |
| Jet Fuel: | Students are asked to analyze and describe the relationship between two linearly related quantities. |
| Graph the Ride: | Students are given a verbal description of the relationship between two quantities and are asked to sketch a graph to model the relationship. |
| Bacterial Growth Graph: | Students are given a verbal description of the relationship between two quantities and are asked to sketch a graph to model the relationship. |
| Recognizing Functions: | Students are asked to determine whether or not each of two graphs represent functions. |
| What Am I?: | Students are asked to describe a linear function, its graph, and the meaning of its parameters. |
| What Is a Function?: | Students are asked to define the term function and describe any important properties of functions. |
| Car Wash: | Students are asked to describe the domain of a function given its graph. |
| 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. |
| Linear or Nonlinear?: | Students are asked to identify a function as either linear or nonlinear and to justify their decision. |
| Explaining Linear Functions: | Students are asked to describe defining properties of linear functions. |
| 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 a Function?: | Students are asked to define the term function including important properties. |
| Tabulating Functions: | Students are asked to determine whether or not tables of ordered pairs represent functions. |
| Identifying Algebraic Functions: | Students are asked to determine if each of three equations represents a function. Although the task provides equations, in their explanations students can use other representations such as ordered pairs, tables of values or graphs. |
| Taxi Ride: | Students are asked to sketch a graph from a verbal description. |
| Bike Race: | Students are asked to evaluate three verbal descriptions and to state why each does or does not match a given graph. |
| Name |
Description |
| Functions: Domain and Range: | Students will identify if a graph represents a function and determine domain and range of the graphs. |
| From Tables to Graphs and Back!: | Students will match corresponding sets of tables, graphs and linear equations in order to deepen their understanding of multiple representations of the relationships between dependent and independent variables. |
| The Linear Function Connection: | The students will compare two linear functions that have been represented in different ways (equation, table, graph, verbal description). They will be able to find and compare the rate of change, or slope, of the function from any of the representations. |
| Functions: Are They Linear or Non-Linear?: | In this lesson, students will investigate 5 different functions to see if they are linear or non-linear. They will then analyze the functions in groups. After that they will present their results and reasoning. |
| The Speeding Ticket: Part 2 - Graphing Linear Functions: | This lesson allows the student to learn about dependent and independent variables and how to make the connection between the linear equation, a linear function, and its graph. The student will learn graphing relationships and how to identify linear functions. |
| Constructing and Calibrating a Hydrometer: | Students construct and calibrate a simple hydrometer using different salt solutions. They then graph their data and determine the density and salinity of an unknown solution using their hydrometer and graphical analysis. |
| Name |
Description |
| Battery Charging: | This task has students engaging in a simple modeling exercise, taking verbal and numerical descriptions of battery life as a function of time and writing down linear models for these quantities. To draw conclusions about the quantities, students have to find a common way of describing them. |
| Bike Race: | The purpose of this task is for students to interpret two distance-time graphs in terms of the context of a bicycle race. There are two major mathematical aspects to this: interpreting what a particular point on the graph means in terms of the context and understanding that the "steepness" of the graph tells us something about how fast the bicyclists are moving. |
| Foxes and Rabbits: | This task emphasizes the importance of the "every input has exactly one output" clause in the definition of a function, which is violated in the table of values of the two populations. Noteworthy is that since the data is a collection of input-output pairs, no verbal description of the function is given, so part of the task is processing what the "rule form" of the proposed functions would look like. |
| Function Rules: | This task can be played as a game where students have to guess the rule and the instructor gives more and more input output pairs. Giving only three input output pairs might not be enough to clarify the rule. Instructors might consider varying the inputs in, for example, the second table, to provide non-integer entries. A nice variation on the game is to have students who think they found the rule supply input output pairs, and the teachers confirms or denies that they are right. Verbalizing the rule requires precision of language. This task can be used to introduce the idea of a function as a rule that assigns a unique output to every input. |
| Modeling with a Linear Function: | The primary purpose of this task is to elicit common misconceptions that arise when students try to model situations with linear functions. This task, being multiple choice, could also serve as a quick assessment to gauge a class' understanding of modeling with linear functions. |
| Tides: | This is a simple task about interpreting the graph of a function in terms of the relationship between quantities that it represents. |
| Riding by the Library: | In this task students draw the graphs of two functions from verbal descriptions. Both functions describe the same situation but changing the viewpoint of the observer changes where the function has output value zero. This small twist forces the students to think carefully about the interpretation of the dependent variable. This task could be used in different ways: To generate a class discussion about graphing. As a quick assessment about graphing, for example during a class warm-up. To engage students in small group discussion. |
| Velocity vs. Distance: | In this task students interpret two graphs that look the same but show very different quantities. The first graph gives information about how fast a car is moving while the second graph gives information about the position of the car. This problem works well to generate a class or small group discussion. Students learn that graphs tell stories and have to be interpreted by carefully thinking about the quantities shown. |
| US Garbage, Version 1: | In this task, the rule of the function is more conceptual. We assign to a year (an input) the total amount of garbage produced in that year (the corresponding output). Even if we didn't know the exact amount for a year, it is clear that there will not be two different amounts of garbage produced in the same year. Thus, this makes sense as a "rule" even though there is no algorithmic way to determine the output for a given input except looking it up in the table. |
Vetted resources students can use to learn the concepts and skills in this topic.
| Title |
Description |
| Functions, Functions Everywhere: Part 1: | What is a function? Where do we see functions in real life? Explore these questions and more using different contexts in this interactive tutorial.
This is part 1 in a two-part series on functions. Click HERE to open Part 2. |
| Math Models and Social Distancing: | Learn how math models can show why social distancing during a epidemic or pandemic is important in this interactive tutorial. |
| Cruising Through Functions: | Cruise along as you discover how to qualitatively describe functions in this interactive tutorial. |
| Summer of FUNctions: | Have some fun with FUNctions! Learn how to identify linear and non-linear functions in this interactive tutorial. |
| Driven By Functions: | Learn how to determine if a relationship is a function in this interactive tutorial that shows you inputs, outputs, equations, graphs and verbal descriptions. |
| Title |
Description |
| Bike Race: | The purpose of this task is for students to interpret two distance-time graphs in terms of the context of a bicycle race. There are two major mathematical aspects to this: interpreting what a particular point on the graph means in terms of the context and understanding that the "steepness" of the graph tells us something about how fast the bicyclists are moving. |
| Foxes and Rabbits: | This task emphasizes the importance of the "every input has exactly one output" clause in the definition of a function, which is violated in the table of values of the two populations. Noteworthy is that since the data is a collection of input-output pairs, no verbal description of the function is given, so part of the task is processing what the "rule form" of the proposed functions would look like. |
| Function Rules: | This task can be played as a game where students have to guess the rule and the instructor gives more and more input output pairs. Giving only three input output pairs might not be enough to clarify the rule. Instructors might consider varying the inputs in, for example, the second table, to provide non-integer entries. A nice variation on the game is to have students who think they found the rule supply input output pairs, and the teachers confirms or denies that they are right. Verbalizing the rule requires precision of language. This task can be used to introduce the idea of a function as a rule that assigns a unique output to every input. |
| Modeling with a Linear Function: | The primary purpose of this task is to elicit common misconceptions that arise when students try to model situations with linear functions. This task, being multiple choice, could also serve as a quick assessment to gauge a class' understanding of modeling with linear functions. |
| Tides: | This is a simple task about interpreting the graph of a function in terms of the relationship between quantities that it represents. |
| Riding by the Library: | In this task students draw the graphs of two functions from verbal descriptions. Both functions describe the same situation but changing the viewpoint of the observer changes where the function has output value zero. This small twist forces the students to think carefully about the interpretation of the dependent variable. This task could be used in different ways: To generate a class discussion about graphing. As a quick assessment about graphing, for example during a class warm-up. To engage students in small group discussion. |
| Velocity vs. Distance: | In this task students interpret two graphs that look the same but show very different quantities. The first graph gives information about how fast a car is moving while the second graph gives information about the position of the car. This problem works well to generate a class or small group discussion. Students learn that graphs tell stories and have to be interpreted by carefully thinking about the quantities shown. |
| US Garbage, Version 1: | In this task, the rule of the function is more conceptual. We assign to a year (an input) the total amount of garbage produced in that year (the corresponding output). Even if we didn't know the exact amount for a year, it is clear that there will not be two different amounts of garbage produced in the same year. Thus, this makes sense as a "rule" even though there is no algorithmic way to determine the output for a given input except looking it up in the table. |
Vetted resources caregivers can use to help students learn the concepts and skills in this topic.
| Title |
Description |
| Battery Charging: | This task has students engaging in a simple modeling exercise, taking verbal and numerical descriptions of battery life as a function of time and writing down linear models for these quantities. To draw conclusions about the quantities, students have to find a common way of describing them. |
| Bike Race: | The purpose of this task is for students to interpret two distance-time graphs in terms of the context of a bicycle race. There are two major mathematical aspects to this: interpreting what a particular point on the graph means in terms of the context and understanding that the "steepness" of the graph tells us something about how fast the bicyclists are moving. |
| Foxes and Rabbits: | This task emphasizes the importance of the "every input has exactly one output" clause in the definition of a function, which is violated in the table of values of the two populations. Noteworthy is that since the data is a collection of input-output pairs, no verbal description of the function is given, so part of the task is processing what the "rule form" of the proposed functions would look like. |
| Function Rules: | This task can be played as a game where students have to guess the rule and the instructor gives more and more input output pairs. Giving only three input output pairs might not be enough to clarify the rule. Instructors might consider varying the inputs in, for example, the second table, to provide non-integer entries. A nice variation on the game is to have students who think they found the rule supply input output pairs, and the teachers confirms or denies that they are right. Verbalizing the rule requires precision of language. This task can be used to introduce the idea of a function as a rule that assigns a unique output to every input. |
| Modeling with a Linear Function: | The primary purpose of this task is to elicit common misconceptions that arise when students try to model situations with linear functions. This task, being multiple choice, could also serve as a quick assessment to gauge a class' understanding of modeling with linear functions. |
| Tides: | This is a simple task about interpreting the graph of a function in terms of the relationship between quantities that it represents. |
| Riding by the Library: | In this task students draw the graphs of two functions from verbal descriptions. Both functions describe the same situation but changing the viewpoint of the observer changes where the function has output value zero. This small twist forces the students to think carefully about the interpretation of the dependent variable. This task could be used in different ways: To generate a class discussion about graphing. As a quick assessment about graphing, for example during a class warm-up. To engage students in small group discussion. |
| Velocity vs. Distance: | In this task students interpret two graphs that look the same but show very different quantities. The first graph gives information about how fast a car is moving while the second graph gives information about the position of the car. This problem works well to generate a class or small group discussion. Students learn that graphs tell stories and have to be interpreted by carefully thinking about the quantities shown. |
| US Garbage, Version 1: | In this task, the rule of the function is more conceptual. We assign to a year (an input) the total amount of garbage produced in that year (the corresponding output). Even if we didn't know the exact amount for a year, it is clear that there will not be two different amounts of garbage produced in the same year. Thus, this makes sense as a "rule" even though there is no algorithmic way to determine the output for a given input except looking it up in the table. |
