Cluster 1: Define, evaluate, and compare functions. (Major Cluster)Archived

Students are asked to determine whether or not each of two graphs represent functions.

Type: Formative Assessment

Students are asked to describe a linear function, its graph, and the meaning of its parameters.

Type: Formative Assessment

Students are asked to provide an example of a nonlinear function and explain why it is nonlinear.

Type: Formative Assessment

This lesson unit is intended to help you assess how well students are able to interpret distance-time graphs and, in particular, to help you identify students who:

• Interpret distance-time graphs as if they are pictures of situations rather than abstract representations of them.
• Have difficulty relating speeds to slopes of these graphs.

Type: Formative Assessment

Students are asked to create their own examples and nonexamples of functions by completing tables and mapping diagrams.

Type: Formative Assessment

Students are asked to identify a function as either linear or nonlinear and to justify their decision.

Type: Formative Assessment

Students are asked to describe defining properties of linear functions.

Type: Formative Assessment

Students are asked to define the term function including important properties.

Type: Formative Assessment

Students are asked to determine whether or not tables of ordered pairs represent functions.

Type: Formative Assessment

Students are asked to determine the rates of change of two functions presented in different forms (an expression and a table) and determine which is the greater rate of change within a real-world context.

Type: Formative Assessment

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. 

Type: Formative Assessment

Students are asked to determine and interpret the initial values of two functions represented in different ways (equation and graph), and compare them.

Type: Formative Assessment

Students are asked to determine the rate of change of two functions presented in different forms (table and graph) and determine which is the greater rate of change within a real-world context.

Type: Formative Assessment

Students are asked to determine a specific value of two functions given in different forms (a graph and a verbal description) within a real-world context, and compare them.

Type: Formative Assessment

This lesson is designed to introduce students to the concept of slope. Students will be able to:

• determine positive, negative, zero, and undefined slopes by looking at graphed functions.
• determine x- and y-intercepts by substitution, or by examining graphs.
• write equations in slope-intercept form and make graphs based on slope/y-intercept of linear functions.

Type: Lesson Plan

Students are asked to determine a procedure for ranking healthcare plans based on their assumptions and the cost of each plan given as a function. Then, they are asked to revise their ranking based on a new set of data.

Type: Lesson Plan

Students will determine function rules that have been written on cards taped to their backs. They will suggest input values and peers will provide output values to help them determine their function. They will then graph their functions for additional practice.

Type: Lesson Plan

This lesson unit is intended to help you assess how well students are able to interpret speed as the slope of a linear graph and translate between the equation of a line and its graphical representation.

Type: Lesson Plan

This lesson introduces functions with the real-world example of the cost of tickets for a playoff game. Also, students will determine if tables, graphs, or sets of ordered pairs represent linear functions and explain their reasoning.

Type: Lesson Plan

This is an introductory lesson on functions. Students will review how to graph ordered pairs and learn how to use a function table. In addition, students learn the difference between functions and nonfunctions and how to distinguish between the two.

Type: Lesson Plan

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.

Type: Lesson Plan

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.

Type: Lesson Plan

This resource provides a lesson plan for teaching students how to recognize relations and functions. They will distinguish which relations are functions and which are not.

Type: Lesson Plan

In this lesson, shopping for cell phone plans provides a motivating context where students will compare and interpret the properties of two functions represented in different ways. The lesson starts with students examining and discussing an input/output table to identify rate of change and y-intercept. Students will each be assigned a group. Groups will solve two different problems involving cell phone plans. They must determine the rate of change, y-intercept, and the total cost after two years. Students will receive homework and have a summative assessment the next day.

Type: Lesson Plan

Students will define what a function is at the beginning of class. They will activate prior knowledge by playing "Four Corners." After that, there will be an investigation of what a function is and how it relates to the total questions on an assessment and the point value for each question. Then, students will analyze different assessments. and present their findings. For homework, students will have practice problems.

Type: Lesson Plan

See how sweet it can be to determine the slope of linear functions and compare them in this interactive tutorial. Determine and compare the slopes or the rates of change by using verbal descriptions, tables of values, equations and graphical forms.

Type: Original Student Tutorial

Have some fun with FUNctions! Learn how to identify linear and non-linear functions in this interactive tutorial.

Type: Original Student Tutorial

Learn how to determine if a relationship is a function in this interactive tutorial that shows you inputs, outputs, equations, graphs and verbal descriptions.

Type: Original Student Tutorial

Learn how to use the equation of a linear trend line to interpolate and extrapolate bivariate data plotted in a scatterplot. You will see the usefulness of trend lines and how they are used in this interactive tutorial.

This is part 6 in 6-part series. Click below to open the other tutorials in the series.

• Scatterplots Part 1: Graphing
• Scatterplots Part 2: Patterns, Associations and Correlations
• Scatterplots Part 3: Trend Lines
• Scatterplots Part 4: Equation of the Trend Line
• Scatterplots Part 5: Interpreting the Equation of the Trend Line

Type: Original Student Tutorial

Explore how to interpret the slope and y-intercept of a linear trend line when bivariate data is graphed on a scatterplot in this interactive tutorial.

This is part 5 in 6-part series. Click below to open the other tutorials in the series.

• Scatterplots Part 1: Graphing
• Scatterplots Part 2: Patterns, Associations and Correlations
• Scatterplots Part 3: Trend Lines
• Scatterplots Part 4: Equation of the Trend Line
• Scatterplots Part 6: Using Linear Models

Type: Original Student Tutorial

Learn how to write the equation of a linear trend line when fitted to bivariate data in a scatterplot in this interactive tutorial.

This is part 4 in 6-part series. Click below to open the other tutorials in the series.

• Scatterplots Part 1: Graphing
• Scatterplots Part 2: Patterns, Associations and Correlations
• Scatterplots Part 3: Trend Lines
• Scatterplots Part 5: Interpreting the Equation of the Trend Line
• Scatterplots Part 6: Using Linear Models

Type: Original Student Tutorial

Shark researcher, Chip Cotton, discusses the use of regression lines, slope, and determining the strength of the models he uses in his research.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

The goal of this task is to give students a context to investigate large numbers and measurements. Students need to fluently convert units with very large numbers in order to successfully complete this task. The total number of pennies minted either in a single year or for the last century is phenomenally large and difficult to grasp. One way to assess how large this number is would be to consider how far all of these pennies would reach if we were able to stack them one on top of another: this is another phenomenally large number but just how large may well come as a surprise.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

This task lets students explore the differences between linear and non-linear functions. By contrasting the two, it reinforces properties of linear functions.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Students design an experiment to model a leaky faucet and determine the amount of water wasted due to the leak. Using the data they gather in a table, students graph and write an equation for a line of best fit. Students then use their derived equation to make predictions about the amount of water that would be wasted from one leak over a long period of time or the amount wasted by several leaks during a specific time period.

Type: Problem-Solving Task

Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

Type: Student Center Activity

This video explains why a vertical line does not represent a function.

Type: Tutorial

This video demonstrates how to check if a verbal description represents a function.

Type: Tutorial

This video shows how to check whether a given set of points can represent a function. For the set to represent a function, each domain element must have one corresponding range element at most.

Type: Tutorial

In this video, you will determine if the situation is linear or non-linear by finding the rate of change between cooordinates. You will check your work by graphing the coordinates given.

Type: Tutorial

In this tutorial, you will practice using an equation in slope-intercept form to find coordinates, then graph the coordinates to predict an answer to the problem.

Type: Tutorial

In this video, you will practice writing the slope-intercept form for a line, given a table of x and y values.

Type: Tutorial

Students will learn how to find and graph the x and y intercepts from an equation written in standard form.

Type: Tutorial

Students will learn how to find the x and y intercepts from an equation in standard form.

Type: Tutorial

This tutorial shows students how to find the y inercept from a table.

Type: Tutorial

Given two points on a line, you will find the slope and the y-intercept. You will then write the equation of the line in slope-intercept form.

Type: Tutorial

Students will learn how to graph a linear equation using a table. Students will not be required to graph from slope-intercept form, although they will convert the equation from standard form to slope-intercpet form before they create the table.

Type: Tutorial

This tutprial shows how to graph a line in slope-intercept form.

Type: Tutorial

It's helpful to represent an equation on a graph where we plot at least 2 points to show the relationship between the dependent and independent variables. Watch and we'll show you.

Type: Tutorial

Equations of the form y = mx describe lines in the Cartesian plane which pass through the origin. The fact that many functions are linear when viewed on a small scale, is important in branches of mathematics such as calculus.

Type: Tutorial

Linear equations of the form y=mx+b can describe any non-vertical line in the cartesian plane. The constant m determines the line's slope, and the constant b determines the y intercept and thus the line's vertical position.

Type: Tutorial

Functions can be thought of as mathematical machines, which when given an element from a set of permissible inputs, always produce the same element from a set of possible outputs

Type: Tutorial

This session on linear function and slope contains five parts, multiple problems and videos, and interactive activities geared to help students recognize and understand linear relationships, explore slope and dependent and independent variables in graphs of linear relationships, and develop an understanding of rates and how they are related to slopes and equations. Throughout the session, students use spreadsheets to complete the work, and are encouraged to think about the ways technology can aid in teaching and understanding. The solutions for all problems are given, and many allow students to have a hint or tip as they solve. There is even a homework assignment with four problems for students after they have finished all five parts of the session.

Type: Unit/Lesson Sequence

"Lesson 1 of two lessons teaches students about direct variation by allowing them to explore a simulated oil spill using toilet paper tissues (to represent land) and drops of vegetable oil (to simulate a volume of oil). Lesson 2 teaches students about inverse variation by exploring the relationship between the heights of a fixed amount of water poured into cylindrical containers of different sizes as compared to the area of the containers' bases." from Insights into Algebra 1 - Annenberg Foundation.

Type: Unit/Lesson Sequence

This tutorial will help you to graph a line using its slope and y-intercept, or to identify the slope and y-intercept from a linear equation written in slope-intercept form.

Type: Virtual Manipulative

This resource provides guided practice for writing and graphing linear functions.

Type: Virtual Manipulative

This manipulative will help you to explore the world of lines. You can investigate the relationships between linear equations, slope, and graphs of lines.

Type: Virtual Manipulative

This lesson is designed to introduce students to the vertical line test for functions as well as practice plotting points and drawing simple functions. The lesson provides links to discussions and activities related to the vertical line test and functions as well as suggested ways to integrate them into the lesson.

Type: Virtual Manipulative

In this activity, students adjust slider bars which adjust the coefficients and constants of a linear function and examine how their changes affect the graph. The equation of the line can be in slope-intercept form or standard form. This activity allows students to explore linear equations, slopes, and y-intercepts and their visual representation on a graph. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

This lesson is designed to introduce students to functions as rules and independent and dependent variables. The lesson provides links to discussions and activities that motivate the idea of a function as a machine as well as proper terminology when discussing functions. Finally, the lesson provides links to follow-up lessons designed for use in succession to the introduction of functions.

Type: Virtual Manipulative

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative

In this activity, students enter inputs into a function machine. Then, by examining the outputs, they must determine what function the machine is performing. This activity allows students to explore functions and what inputs are most useful for determining the function rule. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

This lesson is designed to introduce students to plotting points and graphing functions in the Cartesian coordinate system. The lesson provides links to discussions and activities that transition from functions as rules to the graphs of those functions. Finally, the lesson provides links to follow-up lessons designed for use in succession to an introduction to graphing.

Type: Virtual Manipulative

This interactive simulation investigates graphing linear and quadratic equations. Users are given the ability to define and change the coefficients and constants in order to observe resulting changes in the graph(s).

Type: Virtual Manipulative