

Unit 5 Grade
8 Math 

Unit
Length and Description: 24
days The concept of a function started being
developed during 6^{th} grade when students took their first look at
the relationship between two quantities and attempted to describe the way the
quantities changed in relation to one another. In the 7^{th} grade,
students studied direct proportions algebraically and geometrically including
identifying the constant of proportionality. That work culminates in this
unit with the concept of a function with a strong emphasis on linear
functions. Unit 5 repositions previous
work with tables and graphs by putting them in a new context of input/output
rules. Students should leave this unit with a strong conceptual understanding
of functions, the ability to compare functions represented in different forms
(algebraically, graphically, numerically in tables, or by verbal
descriptions), the understanding that the equation y=mx+b defines a line, the ability to
identify functions that are linear and ones that are not, and the ability to
construct a function to model a linear relationship between two
quantities. Sufficient coverage of
nonlinear functions should be addressed to avoid giving students the
misleading impression that all functional relationships are linear. 

Standards: 8.F.A.1 Understand that a function is a rule
that assigns to each input exactly one output. The graph of a function is the
set of ordered pairs consisting of an input and the corresponding output.
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. 8.F.B.5 Describe
qualitatively the functional relationship between two quantities by analyzing
a graph (e.g., where the function is increasing or decreasing, linear or
nonlinear). Sketch a graph that exhibits the qualitative features of a
function that has been described verbally. 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.F.A.1 ·
I can determine if an equation represents a function. ·
I can apply a function rule for any input that produces
exactly one output. ·
I can generate a set of ordered pairs from a function
and graph the function. 8.F.A.2 ·
I can identify functions algebraically including slope
and yintercept. ·
I can identify functions using graphs. ·
I can identify functions using tables. ·
I can identify functions using verbal descriptions. ·
I can compare and contrast two functions with different
representations. ·
I can draw conclusions based on different
representations of functions. 8.F.A.3 ·
I can recognize that a linear function is graphed as a
straight line represented as an equation in the form y = mx + b. ·
I can recognize the equation y=mx+b is the equation of a function whose graph is a straight
line where m is the slope and b is the yintercept ·
I can provide examples of nonlinear functions using
multiple representations (tables, graphs, and equations). ·
I can compare the characteristics of linear and
nonlinear functions 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 function to model a linear relationship
between two quantities. ·
I can relate the rate of change and initial value to
real world quantities in a linear function in terms of the situation modeled
and in terms of its graph or a table of values. 8.F.B.5 ·
I can sketch a graph given a verbal description of its
qualitative features. ·
I can interpret the relationship between x and y values by analyzing a graph. ·
I can
analyze a graph and describe the functional relationship between two
quantities using the qualities of the graph. 

Enduring
Understandings: · Our world is filled with functions. By learning
how to represent, construct, and analyze functions we gain a better
understanding of how our world works. ·
Verbal descriptions, tables, equations, and
graphs can be used to represent linear and nonlinear functions. 
Essential
Questions: ·
What is a function? ·
How can you determine if a relation is a
function? ·
How can you represent a function? ·
How does a change in the independent variable
affect the dependent variable? ·
What types of relationships can be represented
as functions? ·
How can you model relationships between
quantities? ·
How can you use words, tables, equations, and
graphs to represent linear and nonlinear functions? ·
How do you define, evaluate, and compare
functions? 
