Standard #: MAFS.8.EE.2.5 (Archived Standard)


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Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.



Remarks


Examples of Opportunities for In-Depth Focus

When students work toward meeting this standard, they build on grades 6–7 work with proportions and position themselves for grade 8 work with functions and the equation of a line.

General Information

Subject Area: Mathematics
Grade: 8
Domain-Subdomain: Expressions & Equations
Cluster: Understand the connections between proportional relationships, lines, and linear equations. (Major Cluster) -

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

Date Adopted or Revised: 02/14
Date of Last Rating: 02/14
Status: State Board Approved - Archived
Assessed: Yes

Test Item Specifications

    N/A

    Assessment Limits :

    Numbers in items must be rational numbers

    Calculator :

    Yes

    Context :

    Allowable



Sample Test Items (3)

Test Item # Question Difficulty Type
Sample Item 1

The graph of a proportional relationship is shown.

What is the amount of savings per week?

N/A EE: Equation Editor
Sample Item 2

The graph of a proportional relationship and an equation are shown.

   y space equals space 11 over 2 x space plus 3

What is the greater unit rate?

N/A EE: Equation Editor
Sample Item 3

A tub that holds 18 liters of water fills with 2 liters of water every 2.5 minutes.

A. Use the Add Arrow tool to create a graph that models the situation for the first 5 minutes.

B. At what rate is the tub filling with water? Drag symbols to the circle and numbers to the boxes to show the rate.

N/A GRID: Graphic Response Item Display


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Related Resources

Educational Software / Tool

Name Description
Free Graph Paper A variety of graph paper types for printing, including Cartesian, polar, engineering, isometric, logarithmic, hexagonal, probability, and Smith chart.

Formative Assessments

Name Description
Lines and Linear Equations This lesson is intended to help you assess how well students are able to:
  • Interpret speed as the slope of a linear graph.
  • Translate between the equation of a line and its graphical representation.
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.

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.

Interpreting Slope

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

Lesson Plans

Name Description
Slope Intercept - Lesson #2

This is lesson 2 of 3 in the Slope Intercept unit. This lesson introduces graphing non-proportional linear relationships. In this lesson students will perform an activity to collect data to derive y = mx + b and will use a Scratch program to plot the graph of the data, as well as check for proportional and/or linear relationships.

Slope Intercept - Lesson #1

This is lesson 1 of 3 in the Slope Intercept unit. This lesson introduces graphing proportional relationships. In this lesson students will perform an experiment to find and relate density of two different materials to the constant of proportionality and unit rate.

Seeing the Slope

This 5E lesson will build on students' prior knowledge of positive proportional relationships and graphing them. It will introduce students to negative values of slopes, which will lead to graphing negative proportional relationships. Students will discover properties of different values of slope and have the opportunity to practice graphing. This lesson is designed to be done in a 50-minute block.

Lines and Linear Equations 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.
Running and Rising on ALL Slopes!

In this lesson students will graph and compare two proportional relationships from different representations in contextual problems and be introduced to the slope as the unit rate.

Discovering Kepler's Law for the Periods of Planets

Students listen to a video that describes Kepler's determination that planetary orbits are elliptical and then will use data for the solar distance and periods of several of the planets in the solar system, then investigate several hypotheses to determine which is supported by the data.

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.

How Fast Can You Walk? (Graphing and Interpreting Slope)

This lesson requires that the students walk in the hallway along a path marked every five feet and record the total distance they traveled over 8 seconds. The students then use this point and the origin to graph a line of distance versus time. A class discussion then leads the students to understand that the slope of the line is their walking speed and this can be found using rise over run.

What's the Going Rate?

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.

Who goes faster, earns more, drives farthest?

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

Problem-Solving Tasks

Name Description
DVD Profits, Variation 1

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.

Sore Throats, Variation 2

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.

Who Has the Best Job?

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.

Coffee by the Pound

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.

Comparing Speeds in Graphs and Equations

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.

Peaches and Plums

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.

Equations of Lines

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

Find the Change

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.

Student Center Activity

Name Description
Edcite: Mathematics Grade 8

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.

Teaching Ideas

Name Description
Now That is a Dense Graph

Students will first measure and plot the total mass vs liquid volume in a graduated cylinder. They will then use slope and the mathematical formula for the plot to determine the density of the liquid, the density of a solid added to the liquid, and the mass of the graduated cylinder.

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.

Tutorial

Name Description
Linear Equations

This tutorial will help you to explore slopes of lines and see how slope is represented on the x-y axes.

Unit/Lesson Sequences

Name Description
Linear Functions and Slope 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.
Proportional Reasoning In this resource, a special kind of functional relationship is explored: the proportional relationship. Teachers may find the resource useful for professional development, especially the videos. Students develop proportional reasoning skills by comparing quantities, looking at the relative ways numbers change, and thinking about proportional relationships in linear functions.
This resource has four objectives. Students learn to differentiate between relative and absolute meanings of "more" and determine which of these is a proportional relationship, compare ratios without using common denominator algorithms, differentiate between additive and multiplicative processes and their effects on scale and proportionality, and interpret graphs that represent proportional relationships or direct variation.
Direct and Inverse Variation "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.

Video/Audio/Animation

Name Description
Slope

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

Virtual Manipulative

Name Description
Graphing Lines

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

Student Resources

Problem-Solving Tasks

Name Description
DVD Profits, Variation 1:

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.

Who Has the Best Job?:

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.

Coffee by the Pound:

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.

Comparing Speeds in Graphs and Equations:

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.

Peaches and Plums:

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.

Equations of Lines:

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

Find the Change:

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.

Student Center Activity

Name Description
Edcite: Mathematics Grade 8:

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.

Tutorial

Name Description
Linear Equations:

This tutorial will help you to explore slopes of lines and see how slope is represented on the x-y axes.

Video/Audio/Animation

Name Description
Slope:

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



Parent Resources

Problem-Solving Tasks

Name Description
DVD Profits, Variation 1:

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.

Sore Throats, Variation 2:

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.

Who Has the Best Job?:

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.

Coffee by the Pound:

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.

Comparing Speeds in Graphs and Equations:

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.

Peaches and Plums:

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.

Equations of Lines:

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

Find the Change:

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.



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