Pre-Algebra (2014-15)
Unit 5: Understand the Connections Between Proportional Relationships, Lines and Linear Equations
Unit Overview
Students will begin by looking at rate of change for proportional and non-proportional relationships shown in tables and graphs. This will lead to determining the slope of a line. Students will be asked to explain how similar triangles explain why slope is the same between any two points on a non-vertical line. Students will graph proportional relationships, interpreting the unit rate as the slope of the graph. Students will find the equation y = mx for a line. The student will compare two different proportional relationships represented in different formats. Finally, the student will compare proportional relationships with non-proportional relationships and derive the equation y = mx + b.
Nevada Academic Content Standard(s):
Understand the connections between proportions relationships, lines, and linear equations. (Identified as a major cluster by SBAC.)
8.EE.B.5 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.
8.EE.B.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
Define, evaluate, and compare functions. (Identified as major cluster by SBAC.)
8.F.A.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an
algebraic expression, determine which function has the greater rate of change.
8.F.A.3 (part) Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.
Approximate Time Frame: 3-4 weeks
Terms:
Pre-Algebra Unit 05: Connections between Proportional Relationships, Lines, and Linear Equations Page 1 of 4
Revised September, 2014
Pre-Algebra (2014-15)
Unit 5: Understand the Connections Between Proportional Relationships, Lines and Linear Equations
ü rate of change / ü proportional relationship / ü independent variable / ü direct variationü slope / ü constant of proportionality / ü dependent variable / ü y=mx
ü rise/run / ü unit rate / ü similar triangles / ü y=mx+b
Prep for 8.F.B.4 / Constant Rate of Change
(proportional and non-proportional)
Slope / Ø EX 8-1 Graphing Linear Equations: Rates of Change (page 295)
Ø Pearson Video: Finding Rate of Change Using a Table
Ø Learn Zillion Lesson: Determining the constant rate of change
Ø Pearson Video: Finding the Rate of Change Using a Graph
Ø Khan Academy: Slope and Rate of Change
Ø EX 8-2 Slope of a Line (page 299)
Ø ML 8.4 The Slope of a Line (page 404)
Ø Khan Academy: What does the slope represent?
8.EE.B.6 / Use similar triangles to explain why the slope m is the same between any two distinct points on a line
SBAC Evidence:
Ø The student uses similar triangles to determine that the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane.
/ Ø Learn Zillion Video Lesson: Make Lines from Right Triangles
Ø Khan Academy Video Lesson: Slope and Triangle Similarity 2
Ø Slide Share Lesson: Similar Triangles and Slope
Ø NYCDOE: Slippery Slopes
Ø PBS Learning Media: Understanding Slope with Similar Triangles
Ø TEKS Lesson: Expressions and Equations
Ø NY Module 4, Lesson 16: The Computation of the Slope of a Non-Vertical Line
Ø MA Lesson 4: Analyzing Rates of Change Visually and Numerically
8.EE.B.5
8.EE.B.6 / Proportional Relationships
y = mx
Interpreting Unit Rate as Slope
SBAC Evidence:
Ø The student graphs proportional relationships.
Ø The student interprets the unit rate as the slope of the graph of a proportional relationship.
Ø The student finds the equation y = mx for a line.
/ Ø Learn Zillion Lesson Plan: Graphing Proportional Relationships
Ø Learn Zillion Video Lesson: Display all possibilities in proportional relationship
Ø Khan Academy Video: Identifying a Proportional Relationship
Ø Khan Academy Video: Proportional Relationships from a Table
Ø ML 8.6 Direct Variation (page 423)
Ø EX 8-5 Direct Variation (page 317)
Ø Learn Zillion Lesson Plan: Interpret Unit Rate as Slope
Ø Learn Zillion Video Lesson: Find a Unit Rate Using a Graph
Ø Learn Zillion Video Lesson: Find a Fractional Unit Rate by using a Graph
Ø Khan Academy Video: Constructing an Equation for a Proportional Relationship
Ø Learn Zillion Video Lesson: Derive y = mx Using Similar Triangles
Ø Howard County Lesson: NFL Football and Direct Variation
Ø Illustrative Math Problems: Click here
8.EE.B.5 / Compare proportional relationships in different formats
Ø The student compares two different proportional relationships represented in different formats.
/ Ø YouTube Lesson: Compare Rates (Slopes) in Different Forms
Ø Khan Academy Video: Comparing Proportional Relationships
Ø Learn Zillion Video Lessons: Understand Proportional Relationships by Relating Graphs and Equations
Ø Learn Zillion Lesson Plan: Compare Proportional Relationships
Ø Learn Zillion Video Lesson: Compare Proportional Relationships
8.EE.B.6
8.F.A.3 / Proportional relationships vs
non-proportional relationships
Slope-Intercept Form
SBAC Evidence:
Ø The student finds/derives the equation for a line.
Ø The student interprets the equation y = mx + b as defining a linear function with a graph that is a straight line / Ø ML 8.5 Slope-Intercept Form (page 412)
Ø EX 8-3 Using Slopes and Intercepts (page 305)
Ø Learn Zillion Video Lessons: Interpret the Equation y = mx + b
Ø Learn Zillion Video Lessons: Interpret the Equation y = mx + b as defining a linear Function
Ø Illustrative Math Task: Introduction to Linear Functions
Ø Video: Proportional vs Non-proportional Relationships
Ø YouTube: Introducing Desmos Online Graphing Calculator
Ø Learn Zillion Video Lesson: Derive y = mx + b Using Similar Triangles
Ø MAP: Lines, Slopes and Linear Equations
Pre-Algebra Unit 05: Connections between Proportional Relationships, Lines, and Linear Equations Page 1 of 4
Revised September, 2014
Pre-Algebra (2014-15)
Unit 5: Understand the Connections Between Proportional Relationships, Lines and Linear Equations
Pre-Algebra Unit 05: Connections between Proportional Relationships, Lines, and Linear Equations Page 1 of 4
Revised September, 2014