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

General Information
Number: MA.8.AR.3
Title: Extend understanding of proportional relationships to two-variable linear equations.
Type: Standard
Subject: Mathematics (B.E.S.T.)
Strand: Algebraic Reasoning

## Related Benchmarks

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MA.8.AR.3.AP.1
Given a table, a graph, or equation, determine whether a linear relationship is proportional.
MA.8.AR.3.AP.2
Given a table or graph of a linear relationship, identify the slope.
MA.8.AR.3.AP.3
Given a table or graph of a linear relationship, identify from a list, the equation in slope-intercept form.
MA.8.AR.3.AP.4
Graph a two-variable linear equation from a table or an equation in slope-intercept form.
MA.8.AR.3.AP.5
Given a real-world context, identify the slope and y-intercept of a two-variable linear equation from a table, a graph, or an equation in slope-intercept form.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

## Formative Assessments

Stretching Statistics:

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

Type: Formative Assessment

Developmental Data:

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

Type: Formative Assessment

Deriving Lines - 2:

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

Type: Formative Assessment

Slope Triangles:

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

Deriving Lines - 1:

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

Type: Formative Assessment

Profitable Functions:

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

Type: Formative Assessment

Trekking Functions:

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

Type: Formative Assessment

Smart TV:

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

Construction Function:

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

Type: Formative Assessment

Drain the Pool:

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

Compare Slopes:

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

Proportional Paint:

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

Interpreting Slope:

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

Type: Formative Assessment

Innovative Functions:

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

Competing Functions:

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

This House Is Mine!:

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

## Lesson Plans

Beginning Linear Functions:

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

Designing a Skateboard Kicker Ramp:

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

My Candles are MELTING!:

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

Scatter plots, spaghetti, and predicting the future:

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

## Original Student Tutorials

Hailey’s Treehouse: Similar Triangles & Slope:

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

Constructing Functions From Two Points:

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

Functions, Sweet Functions:

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

It's a Slippery Slope!:

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

Scatterplots Part 6: Using Linear Models :

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.

Type: Original Student Tutorial

Scatterplots Part 5: Interpreting the Equation of the Trend Line :

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.

Type: Original Student Tutorial

Scatterplots Part 4: Equation of the Trend Line:

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.

Type: Original Student Tutorial

Constructing Linear Functions from Tables:

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

## Perspectives Video: Professional/Enthusiast

Slope and Deep Sea Sharks:

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

Type: Perspectives Video: Professional/Enthusiast

## Teaching Idea

Now That is a Dense Graph:

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

## Video/Audio/Animation

Slope:

"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

## Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

## Original Student Tutorials

Hailey’s Treehouse: Similar Triangles & Slope:

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

Constructing Functions From Two Points:

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

Functions, Sweet Functions:

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

It's a Slippery Slope!:

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

Scatterplots Part 6: Using Linear Models :

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.

Type: Original Student Tutorial

Scatterplots Part 5: Interpreting the Equation of the Trend Line :

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.

Type: Original Student Tutorial

Scatterplots Part 4: Equation of the Trend Line:

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.

Type: Original Student Tutorial

Constructing Linear Functions from Tables:

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

## Video/Audio/Animation

Slope:

"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

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.