Standards:
ASSE.A.1 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.
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}).
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.
ACED.A.1 Create equations and
inequalities in one variable and use them to solve problems. Include
equations arising from linear and quadratic functions, and simple rational
and exponential functions.^{ }^{★}
ACED.A.2 Create equations in two or more variables to represent relationships
between quantities; graph equations on coordinate axes with 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.^{ }^{★}^{}
ACED.A.4 Rearrange formulas
to highlight a quantity of interest, using the same reasoning as in solving
equations. For example, rearrange Ohm’s law V = IR to highlight resistance
R.^{ }^{★}
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.
AREI.B.3 Solve linear
equations and inequalities in one variable, including equations with
coefficients represented by letters.
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.
AREI.C.6 Solve systems of
linear equations exactly and approximately (e.g., with graphs), focusing on
pairs of linear equations in two variables.
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).
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.
NQ.A.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.^{ }^{★}
NQ.A.2 Define appropriate
quantities for the purpose of descriptive modeling.^{ }^{★}
NQ.A.3 Choose a level of
accuracy appropriate to limitations on measurement when reporting quantities.^{
}^{★}
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:
ASSE.A.1
·
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
·
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.
AAPR.A.1
·
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
ACED.A.1
·
I can solve linear and
exponential 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
·
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 equation on a coordinate axes with
appropriate labels and scales
ACED.A.3
·
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
ACED.A.4
·
I can define a “quantity of interest” to mean any number or algebraic
quantity (e.g. 2(a/b) = d, in which 2 is the quantity of interest showing
that d must be even; πr^{2}h/3 = V_{cone}
and πr2h = V_{cylinder} showing that V_{cylinder} = 3* V_{cone})
·
I can rearrange formulas to highlight a quantity of interest, using the
same reasoning as in solving equations. (e.g. π * r^{2} can be
rewritten as (π *r) *r which makes the form of this expression resemble
b*h)
AREI.A.1
·
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 classify expression by structure and develop strategies to assist
in classification
·
I can explain the properties of the quantity represented by the
quadratic expression
·
I can choose and produce an equivalent form of a quadratic expression
to reveal and explain properties of the quantity represented by the original
expression
·
I can explain the properties of the quantity represented by the
expression
·
I can create equations and inequalities in one variable and use them to
solve problems.
·
I can rearrange formulas to highlight a quantity of interest, using the
same reasoning as in solving equations.
·
I can 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.
·
I can construct a viable argument to justify a solution method.
·
I can solve linear equations and inequalities in one variable,
including equations with coefficients represented by letters.
·
I can solve quadratic equations in one variable.
·
I can solve quadratic equations by taking square roots.
·
I can explain why the sum or product of two rational numbers is
rational.
·
I can explain why the sum of a rational number and an irrational number
is irrational.
·
I can explain that the product of a nonzero rational number and an
irrational number is irrational.
·
I can use units as a way to understand problems and to guide the
solution of multistep problems.
·
I can choose and interpret units consistently in formulas.
·
I can define appropriate quantities for the purpose of descriptive
modeling.
·
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
AREI.B.3
·
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
AREI.C.5
·
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
·
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
AREI.D.10
·
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
·
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.A.1
·
I can calculate unit conversions
·
I can recognize units given or need 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
NQ.A.2
·
I can define descriptive modeling
·
I can determine appropriate quantities for the purpose of descriptive
modeling
NQ.A.3
·
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
