

Unit 2 Expressions
and Linear Equations Grade
8 Math 


Unit
Length and Description: 35
days In
previous years, students have been working informally with onevariable
linear equations. This important line of development culminates in Unit 2
with the solution of general onevariable linear equations, including cases
with infinitely many solutions or no solutions as well as cases requiring
algebraic manipulation using properties of operations. Coefficients and
constants in these equations may be any rational numbers. In Unit 2, students also build on previous
work to make the
connection between proportional relationships, lines, and linear equations as
they develop ways to represent a line by different equations (y = mx + b, y – y1 = m(x – x1), etc.). Students use their knowledge of
unit rates and graphing to understand that the points (x, y) on
a nonvertical line are the solutions of the equation y = mx + b,
where m is the slope of the line as well as the unit rate of a
proportional relationship (in the case b = 0). Students build a critical foundation, in
preparation for future coursework, as they analyze and solve linear
equations and pairs of simultaneous linear equations, while working with a
variety of mathematical and reallife problems. The equation of a line
provides a natural transition into the idea of a function explored in Unit 5. 

Standards: 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 distancetime graph to a distancetime 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
nonvertical 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.
8.EE.C.7 Solve linear equations in one variable.
a. 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). b. Solve linear equations with rational
number coefficients, including equations whose solutions require expanding
expressions using the distributive property and collecting like terms. 8.EE.C.8 Analyze and solve pairs of simultaneous
linear equations. a.
Understand that solutions to a system of two linear equations in two
variables correspond to points of intersection of their graphs, because
points of intersection satisfy both equations simultaneously. b. Solve systems of two
linear equations in two variables algebraically, and estimate solutions by
graphing the equations. Solve simple cases by inspection. For example, 3𝑥+2𝑦=5 and 3𝑥+2𝑦=6 have no solution because 3𝑥+2𝑦 cannot simultaneously be 5 and 6. c. Solve realworld and
mathematical problems leading to two linear equations in two variables. For
example, given coordinates for two pairs of points, determine whether the
line through the first pair of points intersects the line through the second
pair. 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 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. For
example, the function 𝐴=𝑠^{2} giving the area of a square as a
function of its side length is not linear because its graph contains the
points (1,1), (2,4) and (3,9) which are not on a straight line.
8.F.B.4 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.
Focus Standards of
Mathematical Practice: MP.1 Make
sense of problems and persevere in solving them. MP.2 Reason
abstractly and quantitatively. MP.3
Construct viable arguments and critique the reasoning of others. MP.4 Model
with mathematics. MP.5 Use
appropriate tools strategically. MP.6 Attend
to precision. MP.7 Look for
and make use of structure. MP.8 Look for
and express regularity in repeated reasoning. Instructional
Outcomes: Full Development of the Major Clusters,
Supporting Clusters, Additional Clusters and Mathematical Practices for this
unit could include the following instructional outcomes: 8.EE.B.5 ·
I can determine the rate of change
by the definition of slope. ·
I can graph proportional
relationships. ·
I can compare two different
proportional relationships represented in different ways. (For example,
compare a distancetime graph to a distancetime equation to determine which
of the two moving objects has greater speed). ·
I can interpret the unit rate of
proportional relationships as the slope of a graph. 8.EE.B.6 ·
I can find the slope of a line
between a pair of distinct points. ·
I can determine the yintercept of
a line (interpreting unit rate as the slope of the graph is included in
8.EE). ·
I can analyze patterns for points
on a line through the origin. ·
I can derive an equation of the
form y=mx for a line through the origin. ·
I can analyze patterns for points
on a line that does not pass through or include the origin. ·
I can derive an equation of the form
y=mx + b for a line intercepting the vertical axis at b (the yintercept). ·
I can use similar triangles to
explain why the slope m is the same between any two distinct points on a
nonvertical line in the coordinate plane. 8.EE.C.7 ·
I can give examples of linear
equations in one variable with one solution and show that the given
example equation has one solution by successively transforming the equation
into an equivalent equation of the form x=a. ·
I can give examples of linear
equations in one variable with infinitely many solutions and show that
the given example has infinitely many solutions by successively transforming
the equation into an equivalent equation of the form a=a. ·
I can give examples of linear
equations in one variable with no solution and show that the given
example has no solution by successively transforming the equation into an
equivalent equation of the form b=a, where a and b are different numbers. ·
I can solve linear equations with
rational number coefficients. ·
I can solve equations whose
solutions require expanding expressions using the distributive property
and/or collecting like terms. 8.EE.C.8 ·
I can identify the solution(s) to a
system of two linear equations in two variables as the point(s) of
intersection of their graphs. ·
I can describe the point(s) of
intersection between two lines as the points that satisfy both equations
simultaneously. ·
I can define “inspection”. ·
I can solve a system of two
equations (linear) in two unknowns algebraically. ·
I can identify cases in which a
system of two equations in two unknowns has no solution. ·
I can identify cases in which a
system of two equations in two unknowns has an infinite number of solutions. ·
I can solve simple cases of systems
of two linear equations in two variables by inspection. 8.F.A.2 ·
I
can represent a linear equation algebraically using slope and yintercept. ·
I
can represent a linear equation graphically. ·
I
can represent a linear equation in a table. ·
I
can represent a linear equation using a verbal description. ·
I
can compare and contrast two linear equations with different representations. ·
I
can draw conclusions based on different representations of a linear equation.
8.F.A.3 ·
I can recognize that the graph of y=mx+b is a straight line. ·
I can recognize that the graph of y=mx+b is a straight line where m is the slope and b is the yintercept. ·
I
can recognize nonlinear equations by analyzing tables, graphs, and
equations. ·
I
can compare the characteristics of a linear and nonlinear equation using
various representations. 8.F.B.4 ·
I can recognize that slope is
determined by the constant rate of change. ·
I can recognize that the
yintercept is the initial value where x=0.
·
I can determine the rate of change
(slope) from two (x,y) values, a
verbal description, values in a table, or graph. ·
I can determine the initial value (yintercept) from two (x,y) values, a verbal description,
values in a table, or graph. ·
I can construct a model of a linear
relationship between two quantities. ·
I can relate the rate of change and
initial value to real world quantities in a linear equation in terms of the
situation modeled and in terms of its graph or a table of values. 

Enduring
Understandings: ·
Linear equations in
one variable can have one solution, infinitely many solutions, or no
solutions. ·
Equations offer an
efficient way to organize and solve problems, and an effective way to express
relationships. ·
Mathematical models are powerful
tools we can use to represent realworld problems. ·
Mathematical situations and
structures can be translated and represented abstractly using variables,
expressions, and equations. ·
An equation can be written for two
quantities that vary proportionally. ·
Unit rates can be explained in graphical
representations, algebraic representations, and in geometry through similar
triangles. ·
The unit rate for a data set that represents a
proportional relationship can be interpreted as slope when the data is
graphed on a coordinate plane. ·
The slope m
is the same for any two distinct points on a nonvertical line graphed on the
coordinate plane.. ·
Graphs of linear equations that intersect the
yaxis at any point other than the origin (0, 0) do not represent
proportional relationships. ·
The points (x, y) on a nonvertical line are the
solutions of the equation y = mx + b. 
Essential
Questions: · If two different graphs model the same information, will they have
the same meaning? · What has to be present to produce the same results? · How can situations be expressed with symbols in different ways, but
have the same meaning? · What are the conditions that make circumstances equal? · How can I communicate mathematical information and ideas more
effectively? · How do
we understand and represent linear relationships, inverse relationships, and
various nonlinear relationships? · What
is the meaning of slope? ·
How can we transfer data and
information between multiple representations? (e.g. graphs, tables,
equations, descriptions, etc.) ·
What is the difference between a ratio and a unit
rate? ·
How can proportional relationships be used to
represent authentic situations in life and solve actual problems? ·
In what way(s) do proportional relationships
relate to functions and functional relationships? ·
What do the points on a line represent? ·
What does the slope of a line represent? ·
What does the point of intersection of two
simultaneous equations represent? 


