

Unit 5 Modeling
– Equations & Functions Algebra I 

Unit
Length and Description: 20
days In this unit, students learn to anticipate
the graph of a quadratic function by interpreting various forms of quadratic
expressions. Students expand their experience with functions to include more
specialized functions—absolute value, step, and those that are
piecewisedefined. (Mathematics Appendix A, p.25) ·
Construct and compare
linear, quadratic, and exponential models and solve problems. ·
Build new functions
from existing functions. ·
Build a function that
models a relationship between two quantities. ·
Interpret functions
that arise in applications in terms of a context. ·
Create equations that
describe numbers or relationships. 

Standards: 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. 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.^{★} FIF.B.4 For a function that models a relationship
between two quantities, interpret key features of graphs and tables in terms
of the quantities, and sketch graphs showing key features given a verbal
description of the relationship. Key features include: intercepts; intervals
where the function is increasing, decreasing, positive, or negative; relative
maximums and minimums; symmetries; end behavior; and periodicity.^{★} FIF.B.5 Relate the domain of a function to its
graph and, where applicable, to the quantitative relationship it describes.
For example, if the function h(n) gives the number of personhours it takes
to assemble n engines in a factory, then the positive integers would be an
appropriate domain for the function.^{★}^{ } FIF.B.6 Calculate and interpret the average rate of
change of a function (presented symbolically or as a table) over a specified
interval. Estimate the rate of change from a graph.^{★} FBF.A.1 Write a function that describes a
relationship between two quantities. ^{★}
FLE.A.1 Distinguish between situations that can be
modeled with linear functions and with exponential functions.^{★}
FLE.A.2 Construct linear and exponential functions,
including arithmetic and geometric sequences, given a graph, a description of
a relationship, or two inputoutput pairs (include reading these from a
table).^{★} 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: 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 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 FIF.B.4 ·
I
can define and recognize the key features in tables and graphs of linear and
exponential functions: intercepts; intervals where the function is
increasing, decreasing, positive, or negative, and end behavior ·
I
can identify whether the function is linear or exponential, given its table
or graph ·
I
can interpret key features of graphs and tables of function in the terms of
the contextual quantities the function represents ·
I
can sketch graphs showing key features of a function that models a
relationship between two quantities from a given verbal description of the
relationship FIF.B.5 ·
I
can, given the graph or a verbal/written description of a function, identify
and describe the domain of the function ·
I
can identify an appropriate domain based on the unit, quantity , and type of
the function it describes ·
I
can relate the domain of the function to its graph and, where applicable, to
the quantitative relationship it describes ·
I
can explain why a domain is appropriate for a given situation FIF.B.6 ·
I
can recognize slope as an average rate of change ·
I
can calculate the average rate of change of a function (presented
symbolically or as a table) over a specified interval ·
I
can estimate the rate of change from a linear or exponential graph ·
I
can interpret the average rate of change of a function (presented
symbolically or as a table) over a specified interval FBF.A.1a ·
I
can define “explicit function” and “recursive process” ·
I
can write a function that describes a relationship between two quantities by
determining an explicit expression, a recursive process, or steps for
calculation from a context FLE.A.1a ·
I
can recognize that linear functions grow by equal differences over equal
intervals ·
I
can recognize that exponential functions grow by equal factors over equal
intervals ·
I
can distinguish between situations that can be modeled with linear functions
and with exponential functions to solve mathematical and realworld problems ·
I
can prove that linear functions grow by equal differences over equal
intervals ·
I
can prove that exponential functions grow by equal factors over equal
intervals FLE.A.1b ·
I
can recognize situations in which one quantity changes at a constant rate per
unit (equal differences) interval relative to another to solve mathematical
and realworld problems FLE.A.2 ·
I
can recognize arithmetic sequences can be expressed as linear functions ·
I
can recognize geometric sequences can be expresses as exponential functions ·
I
can construct linear functions, including arithmetic sequences, given a
graph, a description of a relationship, or two inputoutput pairs (include
reading these from a table) ·
I
can construct exponential functions, including geometric sequences, given a graph,
a description of relationship, or two inputoutput pairs (include reading
these from a table) ·
I
can determine when a graph, a description of a relationship, or two
inputoutput pairs (include reading these from a table) represents a linear
or exponential function in order to solve problems 

Enduring
Understandings: ·
Algebraic
representations can be used to generalize patterns in mathematics. ·
Relationships
between quantities can be represented symbolically, numerically, graphically,
and verbally in the exploration of real world situations. ·
Mathematical
rules that reflect recurring patterns facilitate efficiency in problem
solving. ·
When
analyzing linear and exponential functions, different representations may be
used based on the situation presented ·
There are
multiple algorithms for finding a mathematical solution and those algorithms
are frequently associated with different contexts. ·
Functions
can be used to model reallife situations. ·
There is
an important distinction between solving an equation and solving an applied
problem modeled by an equation. 
Essential
Questions: ·
What
characteristics of problems would determine how to model the situation and
develop a problem solving strategy? ·
How
can patterns, relations, and functions be used as tools to best describe and
help explain relationships between quantities? ·
How
can I use patterns to establish relationships that will help make decisions
in reallife situations? ·
How
can a new function be created from an existing function? ·
How
do parameters introduced in the context of the problem affect the symbolic,
numeric and graphical representations of a quadratic function? ·
How
are patterns of change related to the behavior of functions? 
