Standard 3: Extend understanding of proportional relationships to two-variable linear equations.

Students are asked to interpret a specific solution and the y-intercept of a linear equation that describes a context.

Type: Formative Assessment

Students are asked to interpret the slope of a linear model.

Type: Formative Assessment

Students are asked to derive the general equation of a line with a y-intercept of (0, b).

Type: Formative Assessment

Students are asked to use similar triangles to explain why the slope is the same regardless of the points used to calculate it.

Type: Formative Assessment

Students are asked to derive the general equation of a line containing the origin.

Type: Formative Assessment

Students are asked to write a function to model a linear relationship given its graph.

Type: Formative Assessment

Students are asked to construct a function to model a linear relationship between two quantities given a table of values.

Type: Formative Assessment

Students are asked to determine the rate of change and initial value of a linear function given a table of values, and interpret the rate of change and initial value in terms of the situation it models.

Type: Formative Assessment

Students are asked to construct a function to model a linear relationship between two quantities given two ordered pairs in context.

Type: Formative Assessment

Students are asked to determine the rate of change and initial value of a linear function when given a graph, and to interpret the rate of change and initial value in terms of the situation it models.

Type: Formative Assessment

Students are asked to identify, describe and compare the slopes of two proportional relationships given the graph of one and the equation of the other.

Type: Formative Assessment

Students are given a graph of a proportional relationship and asked to determine the unit rate of the relationship and compare it to the slope of the graph.

Type: Formative Assessment

Students are asked to graph a proportional relationship, given a table of values, and find and interpret the slope.

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 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

In this discovery oriented lesson, students will explore the use of non-standard units of measurement. They will convert linear measurements within the metric system and convert measurements given in astronomical units (AU) into more familiar units, specifically meters and kilometers. The unit conversions will be completed with measurements that are expressed in scientific notation. Students will recall their prior knowledge of how to add and subtract numbers given in scientific notation. They will also use their knowledge of exponent rules to determine an efficient method for multiplying and dividing numbers expressed in scientific notation.

Type: Lesson Plan

Students investigate the amount of space that could be saved by flattening cardboard boxes. The analysis includes linear graphs and regression analysis along with discussions of slope and a direct variation phenomenon.

Type: Lesson Plan

Students will create an outline of a room and write equations of the lines that contain the sides of the room. This lesson provides an opportunity to review and reinforce writing equations of lines (including horizontal and vertical lines) and to apply the relationship between the slopes of parallel and perpendicular lines.

Type: Lesson Plan

In this activity, students investigate a set of bivariate data to determine if there is a relationship between concession sales in the park and temperature. Students will construct a scatter plot, model the relationship with a linear function, write the equation of the function, and use it to make predictions about values of variables.

Type: Lesson Plan

Students create scatter plots, calculate a regression equation using technology, and interpret the slope and y-intercept of the equation in the context of the data. This review lesson relates graphical and algebraic representations of bivariate data.

Type: Lesson Plan

The students will use NBA player data to determine if there is a correlation between the height of a basketball player and his free throw percentage. The students will use technology to create scatter plots, find the regression line and calculate the correlation coefficient.

Basketball is a tall man's sport in most regards. Shooting, rebounding, blocking shots - the taller player seems to have the advantage. But is that still true when shooting free throws?

Type: Lesson Plan

The Triangle Mid-Segment Theorem is used to show the writing of a coordinate proof clearly and concisely.

Type: Lesson Plan

In a 55 minute class, students use whiteboards, Think-Pair-Share questioning, listen to a quadrilateral song, and work individually and in groups to learn about and gain fluency in using the distance and slope formulas to prove specific polygon types.

Type: Lesson Plan

This lesson is an introduction to scatterplots and how to use a trend line to make predictions. Students should have some knowledge of graphing bivariate data prior to this lesson.

Type: Lesson Plan

Students will gather and use data to calculate a line of fit and the correlation coefficient with their classmates' height and hand size. They will use their line of fit to make approximations.

Type: Lesson Plan

This lesson provides the students with scatter plots, lines of best fit and the linear equations to practice interpreting the slope and y-intercept in the context of the problem.

Type: Lesson Plan

Students will make a scatter plot and then create a line of fit for the data. From their graph, students will make predictions and describe relationships between the variables. Students will make predictions, inquire, and formulate ideas from observations and discussions.

Type: Lesson Plan

This is a simple lesson used to describe the concept of slope to algebra students. 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 create scatter plots, and lines of fit, and then calculate the correlation coefficient. Students analyze the results and make predictions. This lesson includes step-by-step directions for calculating the correlation coefficient using Excel, GeoGebra, and a TI-84 Plus graphing calculator. Students will make predictions for the number of views of a video for any given number of weeks on the charts.

Type: Lesson Plan

Students will interpret the meaning of slope and y-intercept in a wide variety of examples of real-world situations modeled by linear functions.

Type: Lesson Plan

This lesson uses real-world examples to practice interpreting the slope and y-intercept of a linear model in the context of data. Students will collect data, graph a scatter plot, and use spaghetti to identify a line of fit. A PowerPoint is included for guidance throughout the lesson and guided notes are also provided for students.

Type: Lesson Plan

This lesson helps students discover that in a reflection, the line of reflection is the perpendicular bisector of any segment connecting any pre-image point with its reflected image.

Type: Lesson Plan

In this lesson students learn how to informally determine a "best fit" line for a scatter plot by considering the idea of closeness.

Type: Lesson Plan

Students use real-world data to construct and interpret scatter plots using technology. Students will create a scatter plot with a line of fit and a function. They describe the relationship of bivariate data. They recognize and interpret the slope and y-intercept of the line of fit within the context of the data.

Type: Lesson Plan

Students will learn how the correlation coefficient is used to determine the strength of relationships among real data. Students use card sorting to order situations from negative to positive correlations. Students will create a scatter plot and use technology to calculate the line of fit and the correlation coefficient. Students will make a prediction and then use the line of fit and the correlation coefficient to confirm or deny their prediction.

Students will learn how to use the Linear Regression feature of a graphing calculator to determine the line of fit and the correlation coefficient.

The lesson includes the guided card sorting task, a formative assessment, and a summative assessment.

Type: Lesson Plan

This lesson will not only reinforce students understanding of slope and y-intercept, but will also ensure the students understand how it can be modeled in a real world situation. The focus of this lesson is to emphasize that slope is a rate of change and the y-intercept the value of y when x is zero. Students will be able to read a problem and create a linear equation based upon what they read. They will then make predictions based upon this information.

Type: Lesson Plan

In this lesson, students will learn how to analyze data and find the equation of the line of best fit. Students will then find the slope and intercept of the best fit line and interpret the meaning in the context of the data.

Type: Lesson Plan

Students will sketch and interpret the line of fit and then describe the correlation of the data. Students will determine if there’s a correlation between foot size and height by collecting data.

Type: Lesson Plan

Students will graph scatterplots and draw a line of fit. Next, students will write an equation for the line and use it to interpret the slope and y-intercept in context. Students will also use the graph and the equation to make predictions.

Type: Lesson Plan

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.

Type: Lesson Plan

This is an introductory lesson designed to help students have a better understanding of the interpretation of the slope (rate of change) of a graph.

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

This MEA provides students with opportunities to practice solving one-step equations while learning about density. Students will calculate density of regular and irregular objects.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

Type: Lesson Plan

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.

Type: Lesson Plan

"The Speeding Ticket" lesson uses real world application to create and solve linear equations and tables with one variable numerically, verbally, and algebraically. The student will also learn the relationship between the independent and dependent variables.

Type: Lesson Plan

In this lesson students will design a "Skateboard Kicker Ramp" to discover that slope of similar triangles is the same at any two distinct points.  Students will model with mathematics the concept of slope by looking at the pattern set by similar triangles.

Type: Lesson Plan

In this lesson, students will apply their knowledge to model a real-world linear situation in a variety of ways. They will analyze a situation in which 2 candles burn at different rates. They will create a table of values, determine a linear equation, and graph each to determine if and when the candles will ever be the same height. They will also determine the domain and range of their functions and determine whether there are constraints on their functions.

Type: Lesson Plan

Students will work in pairs and compose three different linear equations in slope intercept form. They will discover and describe how different values for the slope and y-intercept affect the graph. After graphing lines on graph paper, they will do a physical activity involving graphing.

Type: Lesson Plan

Students will be guided through the investigation of y = mx+b. Through this lesson, students will be able to determine whether lines are parallel, perpendicular, or neither by looking at the graph and the equation.

Type: Lesson Plan

Students will construct a scatter plot from given data. They will identify the correlation, sketch an approximate line of fit, and determine an equation for the line of fit. They will explain the meaning of the slope and y-intercept in the context of the data and use the line of fit to interpolate and extrapolate values.

Type: Lesson Plan

Students discover that the unit rate and the slope of a line are the same, and these can be used to compare two different proportional relationships. Students compare proportional relationships presented in table and graph form.

Type: Lesson Plan

Given a proportional relationship, students will determine the constant of proportionality, write an equation, graph the relationship, and interpret in context.

Type: Lesson Plan

Learn how similar right triangles can show how the slope is the same between any two distinct points on a non-vertical line as you help Hailey build stairs to her tree house in this interactive tutorial.

Type: Original Student Tutorial

Learn to construct a function to model a linear relationship between two quantities and determine the slope and y-intercept given two points that represent the function with this interactive tutorial.

Type: Original Student Tutorial

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

Learn what slope is in mathematics and how to calculate it on a graph and with the slope formula in this interactive tutorial.

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

Learn to construct linear functions from tables that contain sets of data that relate to each other in special ways as you complete this interactive tutorial.

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

Unlock an effective teaching strategy for teaching solving quadratic equations by completing the square in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

In this task, students are asked to determine the unit price of a product under two different circumstances. They are also asked to generalize the cost of producing x items in each case.

Type: Problem-Solving Task

This task provides a unique application of modeling with mathematics. Also, students often think that time must always be the independent variable and so may need some help understanding that one chooses the independent and dependent variable based on the way one wants to view a situation.

Type: Problem-Solving Task

Students graph proportional relationships and understand the unit rate as a measure of the steepness of the related line, called the slope. Students will also treat slopes more formally when they graph proportional relationships and interpret the unit rate as the slope of the graph.

Type: Problem-Solving Task

This task asks the student to graph and compare two proportional relationships and interpret the unit rate as the slope of the graph. Students are also asked to write an equation and graph each scenario.

Type: Problem-Solving Task

Students will answer questions about unit price of coffee, make a graph of the information, and explain the meaning of constant of proportionality/slope in the given context.

Type: Problem-Solving Task

This task could be put to good use in an instructional sequence designed to develop knowledge related to students' understanding of linear functions in contexts. Though students could work independently on the task, collaboration with peers is more likely to result in the exploration of a range of interpretations.

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 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

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.

Type: Problem-Solving Task

This task would be especially well-suited for instructional purposes. Students will benefit from a class discussion about the slope, y-intercept, x-intercept, and implications of the restricted domain for interpreting more precisely what the equation is modeling.

Type: Problem-Solving Task

The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.

Type: Problem-Solving Task

In this resource, real-world bivariate data is displayed in a scatter plot. The equation of the linear function which models the relationship between the two variables is provided, and it is graphed on the scatter plot. Students are asked to use the model to interpret the data and to make predictions.

Type: Problem-Solving Task

This task provides the opportunity for students to reason about graphs, slopes, and rates without having a scale on the axes or an equation to represent the graphs. Students who prefer to work with specific numbers can write in scales on the axes to help them get started.

Type: Problem-Solving Task

In this problem-solving task students are challenged to apply their understanding of linear relationships to determine the amount of chicken and steak needed for a barbecue, which will include creating an equation, sketching a graph, and interpreting both. This resource also includes annotated solutions.

Type: Problem-Solving Task

Students are asked to create and graph linear equations to compare the savings of two individuals. The purpose of the table in (a) is to help students complete (b) by noticing regularity in the repeated reasoning required to complete the table.

Type: Problem-Solving Task

This task asks students to reason about the relative costs per pound of two fruits without actually knowing what the costs are. Students who find this difficult may add a scale to the graph and reason about the meanings of the ordered pairs. Comparing the two approaches in a class discussion can be a profitable way to help students make sense of slope.

Type: Problem-Solving Task

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Type: Problem-Solving Task

In this problem-solving task students are challenged to apply their understanding of linear relationships to determine the amount of chicken and steak needed for a barbecue, which will include creating an equation, sketching a graph, and interpreting both. This resource also includes annotated solutions.

Type: Problem-Solving Task

This problem-solving task involves constructing a linear function and interpreting its parameters in a mail delivery context. It includes annotated solutions.

Type: Problem-Solving Task

This problem-solving task asks students to find a linear function that models something in the real world. After finding the equation of the linear relationship between the depth of the water and the distance across the channel, students have to verbalize the meaning of the slope and intercept of the line in the context of this situation. Commentary and illustrated solutions are included.

Type: Problem-Solving Task

This task asks the student to understand the relationship between slope and changes in x- and y-values of a linear function.

Type: Problem-Solving Task

This activity challenges students to recognize the relationship between slope and the difference in x- and y-values of a linear function. Help students solidify their understanding of linear functions and push them to be more fluent in their reasoning about slope and y-intercepts. This task has also produced a reasonable starting place for discussing point-slope form of a linear equation.

Type: Problem-Solving Task

In this activity, the density of ethanol is found by graphical means. In the second part, the density of sodium thiosulfate is found, also by graphical means. The values found are then analyzed statistically.

Type: Teaching Idea

"Slope" is a fundamental concept in mathematics. Slope of a linear function is often defined as " the rise over the run"....but why?

Type: Video/Audio/Animation