Unit
Description:
By
the end of eighth grade students have learned to solve linear equations in
one variable and have applied graphical and algebraic methods to analyze and
solve systems of linear equations in two variables. This unit builds on these
earlier experiences by asking students to analyze and explain the process of
solving an equation. Students develop fluency writing, interpreting, and
translating between various forms of linear equations and inequalities, and
using them to solve problems. They master the solution of linear equations
and apply related solution techniques and the laws of exponents to the
creation and solution of simple exponential equations. All of this work is
grounded on understanding quantities and the relationships between them.
*Reason quantitatively and use units to
solve problems.
*Interpret the structure of expressions.
*Create equations that describe numbers or
relationships.
*Understand solving equations as a process
of reasoning and explain the reasoning.
*Solve equations and inequalities in one
variable.
*Solve systems of equations.
*Represent and solve equations and
inequalities graphically.
*Perform arithmetic operations on
polynomials.

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.

SSE
Seeing Structure in Expressions

A.
Interpret the structure of expressions.

ASSE.A.1.(ab)

Interpret expressions that
represent a quantity in terms of its context.^{★}
a. Interpret parts of an expression, such
as terms, factors, and coefficients.
b. Interpret complicated expressions by
viewing one or more of their parts as a single entity. For example,
interpret P(1+r)n as the product of P and a factor not depending on P.
*I
can, for expressions that represent a contextual quantity, define and
recognize parts of an expression, such as terms, factors, and coefficients.
*I
can, for expressions that represent a contextual quantity, interpret parts of
an expression, such as terms, factors, and coefficients in terms of the
context.
*I
can, for expressions that represent a contextual quantity, interpret
complicated expressions, in terms of the context, by viewing one or more of
their parts as a single entity.

ASSE.A.2

Use the
structure of an expression to identify ways to rewrite it. For example, see x^{4} y^{4} as (x^{2})^{2} (y^{2})^{2}, thus recognizing it as a difference of squares
that can be factored as (x^{2} y^{2})(x^{2} + y^{2}), or see 2x^{2}
+ 8x as (2x)(x) + 2x(4), thus recognizing it as a polynomial whose
terms are products of monomials and the polynomial can be factored as 2x(x+4).
*I
can identify ways to rewrite expressions, such as difference of squares,
factoring out a common monomial, regrouping, etc.
*I
can identify ways to rewrite expressions based on the structure of the
expression.
*I
can use the structure of an expression to identify ways to rewrite it.

APR
Arithmetic with Polynomials and Rational Expressions

A.
Perform arithmetic operations on polynomials.

AAPR.A.1

Understand that polynomials form a system
analogous to the integers, namely, they are closed under the operations of
addition, subtraction, and multiplication; add, subtract, and multiply
polynomials.
*I
can identify that the sum, difference, or product of two polynomials will
always be a polynomial, which means that polynomials are closed under the
operations of addition, subtraction, and multiplication.
*I
can define closure.
*I
can apply arithmetic operations of addition, subtraction, and
multiplication to polynomials.

CED Creating
Equations

A.
Create equations that describe numbers or relationships.

ACED.A.1

Create
equations and inequalities in one variable and use them to solve problems. Include equations arising from linear,
quadratic, and exponential functions.^{ }^{★}
*I
can solve linear equations in one variable.
*I
can solve inequalities in one variable.
*I
can describe the relationships between the quantities in the problem (for
example, how the quantities are changing or growing with respect to each
other); express these relationships using mathematical operations to create
an appropriate equation or inequality to solve.
*I
can create equations (linear and exponential) and inequalities in one
variable and use them to solve problems.
*I
can create equations and inequalities in one variable to model realworld
situations.
*I
can compare and contrast problems that can be solved by different types of
equations (linear and exponential).

ACED.A.2

Create equations in two
or more variables to represent relationships between quantities; graph
equations on coordinate axes with labels and scales.^{ }^{★}^{}
*I
can identify the quantities in a mathematical problem or realworld
situation that should be represented by distinct variables and describe
what quantities the variables represent.
*I
can create at least two equations in two or more variables to represent
relationships between quantities.
*I
can justify which quantities in a mathematical problem or realworld
situation are dependent and independent of one another and which operations
represent those relationships.
*I
can determine appropriate units for the labels and scale of a graph
depicting the relationship between equations created in two or more
variables.
*I
can graph one or more created equations on coordinate axes with appropriate
labels and scales.

ACED.A.3

Represent constraints
by equations or inequalities, and by systems of equations and/or
inequalities, and interpret solutions as viable or nonviable options in a
modeling context. For example, represent inequalities
describing nutritional and cost constraints on combinations of different
foods.^{ }^{★}^{}
*I
can recognize when a modeling context involves constraints.
*I
can interpret solutions as viable or nonviable options in a modeling
context.
*I
can determine when a problem should be represented by equations,
inequalities, systems of equations and/or inequalities.
*I
can represent constraints by equations or inequalities, and by systems of
equations and/or inequalities.

REI Reasoning with
Equations and Inequalities

A. Understand solving equations as a process of
reasoning and explain the reasoning.

AREI.A.1

Explain
each step in solving a simple equation as following from the equality of
numbers asserted at the previous step, starting from the assumption that
the original equation has a solution. Construct a viable argument to
justify a solution method.
*I
can demonstrate that solving an equation means that the equation remains
balanced during each step.
*I
can recall the properties of equality.
*I
can explain why, when solving equations, it is assumed that the original
equation is equal.
*I
can determine if an equation has a solution.
*I
can use units as a way to understand problems and to guide the solution of
multistep problems.
*I
can choose an appropriate method for solving the equation.
*I
can justify solution(s) to equations by explaining each step in solving a
simple equation using the properties of equality, beginning with the
assumption that the original equation is equal.
*I
can construct a mathematically viable argument justifying a given, or
selfgenerated, solution method.

B. Solve equations and inequalities
in one variable.

AREI.B.3

Solve
linear equations and inequalities in one variable, including equations with
coefficients represented by letters.
*I
can recall properties of equality.
*I
can solve multistep equations in one variable.
*I
can solve multistep inequalities in one variable.
*I
can determine the effect that rational coefficients have on the inequality
symbol and use this to find the solution set.
*I
can solve equations and inequalities with coefficients represented by
letters.

C. Solve Systems of equations.

AREI.C.5

Prove that,
given a system of two equations in two variables, replacing one equation by
the sum of that equation and a multiple of the other produces a system with
the same solutions.
*I
can recognize and use properties of equality to maintain equivalent systems
of equations.
*I
can justify that replacing one equation in a twoequation system with the
sum of that equation and a multiple of the other will yield the same
solutions as the original system.

AREI.C.6

Solve
systems of linear equations exactly and approximately (e.g., with graphs),
focusing on pairs of linear equations in two variables.
*I
can solve systems of linear equations by any method.
*I
can justify the method used to solve systems of linear equations exactly
and approximately focusing on pairs of linear equations in two variables.

D. Represent and solve equations and
inequalities graphically.

AREI.D.10

Understand
that the graph of an equation in two variables is the set of all its
solutions plotted in the coordinate plane, often forming a curve (which
could be a line).
*I
can recognize that the graphical representation of an equation in two
variables is a curve, which may be a straight line.
*I
can explain why each point on a curve is a solution to its equation.

AREI.D.12

Graph the solutions to
a linear inequality in two variables as a halfplane (excluding the
boundary in the case of a strict inequality), and graph the solution set to
a system of linear inequalities in two variables as the intersection of the
corresponding halfplanes.
*I
can identify characteristics of a linear inequality and system of linear
inequalities, such as: boundary line (where appropriate), shading, and
determining appropriate test points to perform tests to find a solutions
set.
*I
can explain the meaning of the intersection of the shaded regions in a
system of linear inequalities.
*I
can graph a line, or boundary line, and shade the appropriate region for a
two variable linear inequality.
*I
can graph a system of linear inequalities and shade the appropriate
overlapping region for a system of linear inequalities.

NQ Quantities

A.
Reason quantitatively and use units to solve problems.

NQA.1

Use units
as a way to understand problems and to guide the solution of multistep
problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays.^{ }^{★}
*I
can calculate unit conversions.
*I
can recognize units given or needed to solve problems.
*I
can use given units and the context of a problem as a way to determine if
the solution to a multistep problem is reasonable (e.g.,
length problems dictate different units than problems dealing with a
measure such as slope).
*I
can choose appropriate units to represent a problem when using formulas or
graphing.
*I
can interpret units or scales used in formulas or represented in graphs.
*I
can use units as a way to understand problems and to guide the solution of
multistep problems.

NQA.2

Define
appropriate quantities for the purpose of descriptive modeling.^{ }^{★}
*I
can define descriptive modeling.
*I
can determine appropriate quantities for the purpose of descriptive
modeling.

NQA.3

Choose a
level of accuracy appropriate to limitations on measurement when reporting
quantities.^{ }^{★}
*I
can identify appropriate units of measurement to report quantities.
*I
can determine the limitations of different measurement tools.
*I
can choose and justify a level of accuracy and/or precision appropriate to
limitations on measurement when reporting quantities.
*I
can identify important quantities in a problem or realworld context.


Enduring
Understandings:
·
Changing the way that a function is represented does
not change the function, although different representations highlight
different characteristics, and some may only show part of the function.
·
Algebraic and numeric procedures are interconnected and
build on one another. Integration of various mathematical procedures builds a
stronger foundation of finding solutions.
·
The different parts of expressions can represent
certain values in the context of a situation.
·
Linear equations and inequalities can be modeled with
technology.
·
Mathematical models can both clarify and distort the
meaning of data.
·
Making an informed decision often involves comparing
and contrasting linear relationships by solving systems of equations.
·
Integration of various mathematical procedures builds a
stronger foundation of finding solutions.

Essential
Questions:
·
How can I use patterns
to establish relationships that will help make decisions in reallife
situations?
·
How do parameters
introduced in the context of the problem affect the symbolic, numeric and
graphical representations of a quadratic function?
·
How can verbal,
numerical, graphical and analytical representations be used to analyze and
solve problems?
·
How can the solution of
a system of equations be used to make decisions and predictions?
·
How might technology be
used to model linear systems or inequalities? Which technology should I use?
How do I decide?
·
How do changes in
equations lead to changes in graphs?
·
What makes alternative
algebraic algorithms both effective and efficient?
