enVision Mathematics Topic 2
8th Grade; September – October (6 weeks); 1st/2nd Quarter
Embedded Files
Topic Title(s):
Analyze and Solve Linear Equations
Prepared Graduates:
MP1. Make sense of problems and persevere in solving them.
MP3. Construct viable arguments and critique the reasoning of others.
MP4. Model with mathematics.
MP6. Attend to precision.
MP7. Look for and make use of structure.
Standard(s):
2. Algebra and Functions
The highlighted evidence outcomes are the priority for all students, serving as the essential concepts and skills. It is recommended that the remaining evidence outcomes listed be addressed as time allows, representing the full breadth of the curriculum.
Students Can (Evidence Outcomes):
8.EE.B. Expressions & Equations: Understand the connections between proportional relationships, lines, and linear equations.
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. (CCSS: 8.EE.B.5)
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. (CCSS: 8.EE.B.6)
8.EE.C. Expressions & Equations: Analyze and solve linear equations and pairs of simultaneous linear equations.
Solve linear equations in one variable. (CCSS: 8.EE.C.7)
Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). (CCSS: 8.EE.C.7.a)
Solve linear equations with rational number coefficients, including equations with variables on both sides and whose solutions require expanding expressions using the distributive property and collecting like terms. (CCSS: 8.EE.C.7.b)
Make connections between representations of linear growth. (Entrepreneurial Skills: Inquiry/Analysis)
Record information about constant rates of change in graphs and equations. (Entrepreneurial Skills: Literacy/Writing)
Make sense of and compare proportional relationships represented in different forms. (MP1)
Compare, contrast, and make claims with proportional relationships based on properties of equations, tables, and/or graphs. (MP3)
Explore the structure of proportional relationships expressed as equations or graphs for methods of comparison. (MP7)
Solve problems involving linear equations and systems of linear equations. (Entrepreneurial Skills: Critical Thinking/Problem Solving)
Model real-world problems with linear equations and systems of linear equations, with variables defined in their real-world context. (MP4)
Solve equations and systems of equations and express solutions with accuracy that makes sense in the real-world context modeled by the equations. (MP6)
Recognize the structure of equations that produce one, infinitely many, or no solution. (MP7)
Inquiry Questions
How is the unit rate of a proportional relationship related to the slope of its graphical representation?
Why are similar triangles effective for describing slope geometrically?
What is meant by a “solution” to a linear equation?
How is it possible for an equation to have more than one solution? How is it possible for an equation to have no solution?
Coherence Connections
This expectation represents major work of the grade.
In Grade 7, students recognize and represent proportional relationships, calculate the constant of proportionality, and graph proportional relationships on the coordinate plane, recognizing that they always pass through the origin.
In previous grades, students reason about and solve one-step and two-step, one-variable equations and inequalities, use properties of operations to generate equivalent expressions, and solve real-world and mathematical problems using numerical and algebraic expressions and equations.
In Grade 8, this expectation connects with analyzing and solving linear equations and pairs of simultaneous linear equations, with defining, evaluating, and comparing functions, and with understanding congruence and similarity using physical models, transparencies, or geometry software.
In Grade 8, this expectation connects with understanding the connections between proportional relationships, lines, and linear equations and with investigating patterns of association in bivariate data.
In high school, students compare multiple representations of inverse proportional relationships.
In high school, students abstract and generalize about linear functions and how they compare and contrast to nonlinear functions. Students also reason about and solve systems of equations that include one or more nonlinear equations.
Academic Vocabulary & Language Expectations:
Slope, y-intercept, slope-intercept form
Assessments:
Instructional Resources & Notes:
enVision Mathematics Topic 2
Let's Investigate! Tasks
Let's Investigate! It All Balances Out (TE) (relates to Lesson 2-2)
Let's Investigate! Rise to the Top (TE) (relates to Lesson 2-6)
Tier 1 Intervention & Supports (i-Ready Tools for Instruction):
Tier 1 Intervention: Compare Proportional Relationships, Slope and Unit Rate, The Equation of a Line, Solving Linear Equations, Solve Equations with the Variable on Both Sides
Coherence Map/Concept Progressions: 8.EE.B.5, 8.EE.B.6, 8.EE.C.7.b
enVision Mathematics 6-8 & Number Worlds Connections (for SVVSD Special Education teachers only)
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