HCPSS Curriculum Framework
Common Core Mathematics 8
Unit 3: Analyzing Functions and Equations
Overview (Big Ideas):
In this unit, students will grasp the concept of a function as a rule that assigns to each input
exactly one output. They understand that functions describe situations where one quantity
determines another. They can translate among representations and partial representations of
functions (noting that tabular and graphical representations may be partial representations), and
they describe how aspects of the function are reflected in the different representations.
Students will also use linear equations and systems of linear equations to represent, analyze, and
solve a variety of problems. Students recognize equations for proportions (y/x = m) as special
linear equations (y = mx + b), understanding that the constant of proportionality (m) is the slope,
and the graphs are lines through the origin. They understand that the slope (m) of a line is a
constant rate of change, so that if the input or x-coordinate changes by an amount A, the output or
y-coordinate changes by the amount mA. Students also use a linear equation to describe the
association between two quantities in bivariate data (such as arm span vs. height for students in a
classroom). At this grade, fitting the model, and assessing its fit to the data are done informally.
Interpreting the model in the context of the data requires students to express a relationship
between the two quantities in question and to interpret components of the relationship (such as
slope and y-intercept) in terms of the situation.
Students strategically choose and efficiently implement procedures to solve linear equations in
one variable, understanding that when they use the properties of equality and the concept of
logical equivalence, they maintain the solutions of the original equation. Students solve systems
of two linear equations in two variables and relate the systems to pairs of lines in the plane; these
intersect, are parallel, or are the same line. Students use linear equations, systems of linear
equations, linear functions, and their understanding of slope of a line to analyze situations and
solve problems. (CCSS Grade 8 page 53)
Enduring Understandings:
1. Functions provide a tool for describing how variables change together. Using a function in
this way is called modeling, and the function is called a model.
2. Functions can be represented in multiple ways—in algebraic symbols, graphs, verbal
descriptions, tables, and so on—and these representations, and the links among them, are
useful in analyzing patterns of change.
3. Functions are single-valued mappings from one set—the domain of the function—to
another—its range.
4. Functions apply to a wide range of situations. They do not have to be described by any
specific expressions or follow a regular pattern. They apply to cases other than those of
“continuous variation.” For example, sequences are functions.
5. For functions that map real numbers to real numbers, certain patterns of covariation, or
patterns in how two variables change together, indicate membership in a particular
family of functions and determine the type of formula that the function has.
6. A rate of change describes how one variable quantity changes with respect to another—
in other words, a rate of change describes the covariation between two variables.
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7. A function’s rate of change is one of the main characteristics that determine what kinds
of real-world phenomena the function can model.
8. One important way of describing functions is by identifying the rate at which the variables
change together. It is useful to group functions into families with similar patterns of change
because these functions, and the situations that they model, share certain general
characteristics.
9. Members of a family of functions share the same type of rate of change. This
characteristic rate of change determines the kinds of real-world phenomena that the
functions in the family can model.
10. Linear functions are characterized by a constant rate of change. Reasoning about the
similarity of “slope triangles” allows deducing that linear functions have a constant rate
of change and a formula of the type f (x)  mx  b for constants m and b.
11. Two or more expressions may be equivalent, even when their symbolic forms differ. A
relatively small number of symbolic transformations can be applied to expressions to yield
equivalent expressions.
12. Variables have many different meanings, depending on context and purpose.
13. Using variables permits writing expressions whose values are not known or vary under
different circumstances.
14. Using variables permits representing varying quantities. This use of variables is particularly
important in studying relationships between varying quantities.
15. The equals sign can indicate that two expressions are equivalent. It is often important to find
the value(s) of a variable for which two expressions represent the same quantity. Finding the
value(s) of a variable for which two expressions represent the same quantity is know as
solving an equation.
16. An inequality is another way to describe a relationship between expressions; instead of
showing that the values of two expressions are equal, inequalities indicate that the value of
one expression is greater than (or greater than or equal to) the value of the other expression.
17. In solving an inequality, multiplying or dividing both expressions by a negative number
reverses the sign that indicates the relationships between the two expressions.
18. The equals sign can be used in defining or giving a name to an expression or function rule.
Source:
Cooney, T. J., Beckmann, S., & Lloyd, G.M. (2010). Developing essential understanding of
functions grades 9-12. Reston, VA: The National Council of Teachers of Mathematics, Inc.
Lloyd, G., Herbel-Eisenmann, B., & Star, J.R. (2011). Developing essential understanding of
expressions, equations, and functions for teaching mathematics in grades 6-8. Reston, VA:
The National Council of Teachers of Mathematics, Inc.
Essential Questions:
o What is a function? Describe what it means for a situation to have a functional
relationship.
o What is the relationship between the input and output of a function?
o In what ways can different types of functions be used and altered to model various
situations that occur in life?
o What units, scales and labels must be applied to accurately represent a linear function in
the context of a problem situation?
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o Can students represent a function using real world contexts, algebraic equations, tables,
and with words?
o What are the advantages of representing the relationship between quantities
symbolically? Numerically? Graphically?
o Are students able to compare the properties of multiple functions, given a linear function,
and determine which function has the greater rate of change?
o Can students construct a function to model a linear relationship between two quantities,
and determine the rate of change and initial values of the functions?
o How can proportional relationships be used to represent authentic situations in life and
solve actual problems?
o In what way(s) do proportional relationships relate to functions and functional
relationships?
o Are students able to calculate the slope of a line graphically, apply direct variation,
differentiate between zero slope and undefined slope, and understand that similar right
triangles can be used to establish that slope is a constant for a non-vertical line?
o Do students have the knowledge to solve multi-step equations using simple cases by
inspection, one solution, infinitely many solutions, or no solution?
o Are students able to solve systems of linear equations numerically, graphically, or
algebraically using substitution or elimination?
o Do students have the ability to discuss efficient solution methods when solving a system
of equations?
o Can students use a system of equations to solve real-world problems and interpret the
solution in the context of the problem?
Curriculum Standards:
8.F.A Define, evaluate, and compare functions.
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.
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.EE.B Understand the connections between proportional relationships, lines, and linear
equations.
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 distance-time graph to a distance-time equation to determine which of
two moving objects has greater speed.
6. Use similar triangles to explain why the slope m is the same between any two distinct points
on a non-vertical 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.F.A Define, evaluate, and compare functions.
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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
points (1, 1), (2, 4), and (3, 9), which are not on a straight line.
8.F.B Use functions to model relationships between quantities.
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.
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.
8.EE.C Analyze and solve linear equations and pairs of simultaneous linear equations.
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. 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 solutions because 3x  2y cannot
simultaneously be 5 and 6.
c. Solve real-world 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.
Common Misconceptions:
o Students may mistakenly believe that a slope of zero is the same as “no slope” and then
confuse a horizontal line (slope of zero) with a vertical line (undefined slope).
o Students may interchange the meanings of x (independent variable) and y (dependent
variable), particularly when graphing the line of an equation.
o Students often struggle with the difference between combining like terms on one side of
an equation and eliminating a variable on one side of an equation.
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under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.