Common Core 8
Unit 3: Analyzing Functions and Equations
PARCC Exam Notes
Unit 3: Assessment Clarifications
Standard
M/S/A
8.F.A.1
M
PBA/
EOY?
PBA
Yes
Calc?
MP
Notes
No
2
Understand that a function is a rule that assigns to each input exactly one output.
i) Tasks do not involve the coordinate plane or the “vertical line test.”
ii) Tasks do not require knowledge of the concepts or terms domain and range.
iii) 20% of functions in tasks are non-numerical, e.g., the input could be a person and the output could be his or her month of birth.
No
2, 5
No
2
[Understand that] the graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
i) Functions are limited to those with inputs and outputs in the real numbers.
ii) Tasks do not require knowledge of the concepts or terms domain and range.
iii) 80% of tasks require students to graph functions in the coordinate plane or read inputs and outputs from the graph of a function in the
coordinate plane.
iv) 20% of tasks require students to tell whether a set of points in the plane represents a function.
Understand that a function is a rule that assigns to each input exactly one output.
i) Tasks do not involve the coordinate plane or the “vertical line test.”
ii) Tasks do not require knowledge of the concepts or terms domain and range.
iii) 20% of functions in tasks are non-numerical, e.g., the input could be a person and the output could be his or her month of birth.
No
2, 5
PBA
No
EOY
Yes
N/A
N/A
Yes
2, 5
PBA
Yes
Yes
1, 5
Yes
7
Yes
1, 5
Yes
7
EOY
Yes
8.F.A.2
8.EE.B.5
M
M
EOY
Yes
[Understand that] the graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
i) Functions are limited to those with inputs and outputs in the real numbers.
ii) Tasks do not require knowledge of the concepts or terms domain and range.
iii) 80% of tasks require students to graph functions in the coordinate plane or read inputs and outputs from the graph of a function in the
coordinate plane.
iv) 20% of tasks require students to tell whether a set of points in the plane represents a function.
N/A
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in table, or by verbal
descriptions).
i) Tasks have “thin context” or no context.
ii) The testing interface can provide students with a calculation aid of the specified kind for these tasks.
Graph proportional relationships, interpreting the unit rate as the slope of the graph.
i) Pool should contain tasks with and without context.
ii) The testing interface can provide students with a calculation aid of the specified kind for these tasks.
Compare two different proportional relationships represented in different ways.
i) Pool should contain tasks with and without context.
ii) The testing interface can provide students with a calculation aid of the specified kind for these tasks.
Graph proportional relationships, interpreting the unit rate as the slope of the graph.
i) Pool should contain tasks with and without context.
ii) The testing interface can provide students with a calculation aid of the specified kind for these tasks.
Compare two different proportional relationships represented in different ways.
i) Pool should contain tasks with and without context.
ii) The testing interface can provide students with a calculation aid of the specified kind for these tasks.
8.EE.B.6
8.F.A.3
8.F.B.4
8.F.B.5
8.EE.C.7a
M
M
M
M
M
PBA
Yes
(for
Sub-C)
Yes
2, 3, 7,
8
Connected to Evidence Statement 8.C.1.1
Base reasoning on the principle that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane.
Content Scope: Knowledge and skills articulated in 8.EE.6
i) Note especially the portion of 8.EE.6 after the semicolon.
Yes
2, 3, 5
EOY
Yes
Yes
2, 7
PBA
Yes
(for
Sub-C)
EOY
Yes
Yes
3, 6
No
2, 7
Connected to Evidence Statement 8.C.5.1
Apply geometric reasoning in a coordinate setting, and/or use coordinates to draw geometric conclusions.
Content Scope: Knowledge and skills articulated in 8.EE.6
i) Note especially the portion of 8.EE.6 before the semicolon.
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.
i) Tasks do not have a context.
ii) Given a non-vertical line in the coordinate plane, tasks might for example require students to choose two pairs of points and record the rise,
run, and slope relative to each pair and verity that they are the same.
iii) The testing interface can provide students with a calculation aid of the specified kind for these tasks.
Connected to Evidence Statement 8.C.3.1
Construct, autonomously, chains of reasoning that will justify or refute propositions or conjectures.
Content Scope: Knowledge and skills articulated in 8.F.3-2
i) Note especially the portion of 8.F.3 after the semicolon. Tasks require students to prove that a given function is linear or nonlinear.
Interpret the equation, y mx b as defining a linear function, whose graph is a straight line.
i) Tasks have “thin context” or no context.
ii) Tasks require students to approach linear equations from a functional perspective, for example by computing outputs from inputs or by
identifying equations that do or do not define one variable as a linear function of the other.
iii) Equations can be presented in forms other than y mx b . For example, the equation 2x
y
can be viewed as a function machine with
x the input and y the output - or as a function machine with y the input and x the output.
No
7
PBA
No
EOY
Yes
N/A
N/A
Yes
2, 4
PBA
No
EOY
Yes
N/A
N/A
No
2, 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).
i) Pool should contain tasks with and without contexts.
No
2, 5, 7
Yes
3, 6
Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
i) Pool should contain tasks with and without contexts.
Connected to Evidence Statement 8.C.2
Given an equation or system of equations, present the solution steps as a logical argument that concludes with the set of solutions (if any).
Content Scope: Knowledge and skills articulated in 8.EE.7a, 8.EE.7b, 8.EE.8b
PBA
Yes
(for
Sub-C)
Give examples of functions that are not linear and prove that they are not linear.
i) Tasks have “thin context” or no context.
ii) Tasks require students to demonstrate understanding of function nonlinearity, for example by recognizing or producing equations that do not
define linear functions, or by recognizing or producing pairs of points that belong to the graph of the function yet do not lie on a straight line.
iii) Tasks do not require students to produce a proof; for that aspect of standard 8.F.3, see Grade 8 PBA
iv) Tasks involving symbolic representations are limited to polynomial functions i.e. y = 3x2 + 2
N/A
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.
i) Pool should contain tasks with and without contexts.
ii) The testing interface can provide students with a calculation aid of the specified kind for these tasks.
N/A
8.EE.C.7b
M
PBA
Yes
No
6, 7
Yes
3, 6
8.EE.C.8a
8.EE.C.8b
M
M
EOY
Yes
No
6, 7
PBA
Yes
No
2, 5, 6,
7
Yes
2, 3, 5,
6, 7
EOY
Yes
No
2, 5, 6,
7
PBA
Yes
(for
Sub-C)
EOY
Yes
Yes
3, 6
No
1, 6, 7
Solve linear equations in one variable.
b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the
distributive property and collecting like terms.
i) Tasks do not have a context.
Connected to Evidence Statement 8.C.2
Given an equation or system of equations, present the solution steps as a logical argument that concludes with the set of solutions (if any).
Content Scope: Knowledge and skills articulated in 8.EE.7a, 8.EE.7b, 8.EE.8b
Solve linear equations in one variable.
b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the
distributive property and collecting like terms.
i) Tasks do not have a context
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.
i) Tasks do not have a context.
Connected to Evidence Statement 8.C.1.2
Base reasoning on the principle that the graph of an equation is two variables is the set of all its solutions plotted in the coordinate plane.
Content Scope: Knowledge and skills articulated in 8.EE.8a
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.
i) Tasks do not have a context
Connected to Evidence Statement 8.C.2
Given an equation or system of equations, present the solution steps as a logical argument that concludes with the set of solutions (if any).
Content Scope: Knowledge and skills articulated in 8.EE.7a, 8.EE.7b, 8.EE.8b
Analyze and solve pairs of simultaneous linear equations.
b. Solve systems of two linear equations in two variables algebraically.
i) 20% of tasks have a zero coefficient, e.g., as in the system s  3 t  2, t  6
4
ii) 20% of tasks have non-zero whole-number coefficients, and whole-number solutions.
iii) 20% of tasks have non-zero whole-number coefficients, and at least one fraction among the solutions.
iv) 20% of tasks have non-zero integer coefficients (with at least one coefficient negative).
v) 20% of tasks have non-zero rational coefficients (with at least one coefficient negative and at least one coefficient a non-integer).
No
5, 6, 7
Analyze and solve pairs of simultaneous linear equations.
b. Estimate solutions [to systems of two linear equations in two variables] by graphing the equations.
i) Tasks present students with technology that allows them to (1) graph a point based on coordinates of their choosing; (2) graph a line based on
the equation (3) zoom in if the student wishes to do so, rescaling the axes automatically.
ii) 20% of tasks have a zero coefficient, e.g., as in the system s  3 t  2, t  6
4
iii) 20% of tasks have non-zero whole-number coefficients, and whole-number solutions.
iiv 20% of tasks have non-zero whole-number coefficients, and at least one fraction among the solutions.
v) 20% of tasks have non-zero integer coefficients (with at least one coefficient negative).
vi) 20% of tasks have non-zero rational coefficients (with at least one coefficient a non-integer).
Analyze and solve pairs of simultaneous linear equations.
8.EE.C.8c
M
No
7
PBA
Yes
(for
Sub-C)
Yes
1, 2, 3,
6, 7
EOY
Yes
Yes
1, 5, 6,
7
b. Solve simple cases [of systems of two linear equations in two variables] by inspection. For example, 3x
y
3x
y
solution because 3x
y cannot simultaneously be 5 and 6.
i) Tasks have whole-number or integer coefficients, one coefficient in either or both equations possibly zero.
ii) One-third of tasks involve inconsistent systems, where the inconsistency is plausibly visible by inspection as in the italicized example given in
the standard 8.EE.8b.
iii) One-third of tasks involve degenerate systems (infinitely many solutions), where the degeneracy is plausibly visible by inspection, as for
example in 3x
y
x
y
.
iv) One-third of tasks involve systems with a unique solution and one coefficient zero, where the solution is plausibly visible by inspection, as
for example in y
x y
v) Tasks assess solving by inspection, for example by listing several systems and asking the student for the solution of any freely chosen one of
them by inspection.
Connected to Evidence Statement 8.C.4.1
Present solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equals signs appropriately (for example,
rubrics award less than full credit for the presence of nonsense statements such as 1+4=5+7=12, even if the final answer is correct), or identify or
describe errors in solutions to multi-step problems and present corrected solutions.
Content Scope: Knowledge and skills articulated in 8.EE.8c
i) See ITN Appendix F, section A, “Illustrations of innovative task characteristics,” sub-section 6, “Expressing mathematics,” sub-section
“Illustrative tasks that require students to express mathematical reasoning,” the problem of the two shepherds.
Analyze and solve pairs of simultaneous liner equations.
c. Solve real-world and mathematical problems leading to two linear equations in two variables.
i) Mixture problems are no more than 20% of tasks.
ii) For an example of an illustrative task, see ITN Appendix F, section A, “Illustrations of innovative task characteristics,” sub-section 6,
“Expressing mathematical reasoning,” sub-section “Illustrative tasks that require students to express mathematical reasoning,” the problem of the
two shepherds.
Other Integrative Tasks may be linked to Unit 3
Evidence
Statement
Reach
Back?
PBA/
EOY?
Calc?
MP
Notes
8.EE.C.Int.
1
N/A
EOY
Yes
4, 6, 7
8.D.1
N/A
PBA
Yes
4, 1, 2,
5, 7
8.D.2
PBA
Yes
4, 1, 2,
5, 7
8.D.3
7.RP.A
7.NS.3
7.EE
7.G
7.SP.B
N/A
Solve word problems leading to linear equations in one variable whose solutions require expanding expressions using the distributive property
and collecting like terms.
i) For an example of an illustrative task, see 2009 CCRS: “If a bar of soap balances 3/4 of a bar of soap and 3/4 of a pound, how much does
the bar of soap weigh?” At least 80% of tasks should involve contextual word problems (a noncontextual word problem could be “the sum of
two times a number and 8 is 16”).
Solve multi-step contextual word problems with degree of difficulty appropriate to Grade 8, requiring application of knowledge and skills
articulated in the Evidence Statements on the PBA (excludes Reasoning Evidence Statements).
Tasks may have scaffolding if necessary in order to yield a degree of difficulty appropriate to Grade 8.
Solve multi-step contextual problems with degree of difficulty appropriate to Grade 8, requiring application of knowledge and skills
articulated in 7.RP.A, 7.NS.3, 7.EE, 7.G, and 7.SP.B
Tasks may have scaffolding if necessary in order to yield a degree of difficulty appropriate to Grade 8.
PBA
Yes
4, 1, 2,
5, 7
8.D.4
N/A
PBA
Yes
4, 1, 2,
Micro-models: Autonomously apply a technique from pure mathematics to a real-world situation in which the technique yields valuable
results even though it is obviously not applicable in a strict mathematical sense (e.g., profitably applying proportional relationships to a
phenomenon that is obviously nonlinear or statistical in nature).
Content Scope: Knowledge and skills articulated in the Evidence Statements on the PBA (excludes Reasoning Evidence Statements).
Tasks may have scaffolding if necessary in order to yield a degree of difficulty appropriate to Grade 8.
Reasoned estimates: Use reasonable estimates of known quantities in a chain of reasoning that yields an estimate of an unknown quantity.
5, 7
8.C.6
7.RP.A
7.NS.A
7.EE.A
PBA
Yes
3, 6
Content Scope: Knowledge and skills articulated in the Evidence Statements on the PBA (excludes Reasoning Evidence Statements).
Tasks may have scaffolding if necessary in order to yield a degree of difficulty appropriate to Grade 8.
Construct, autonomously, chains of reasoning that will justify or refute propositions or conjectures.
Content Scope: Knowledge and skills articulated in 7.RP.A, 7.NS.A, 7.EE.A
Tasks may have scaffolding if necessary in order to yield a degree of difficulty appropriate to Grade 8.
Grade 8 Sub-Claim A Performance Level Descriptors applicable to Unit 3
Proportional
Relationships and
Linear Equations
8.EE.5-1
8.EE.5-2
8.EE.6-1
8.F.3-1
Solving Linear
Equations
8.EE.7
8.EE.C.Int. 1
Simultaneous Linear
Equations
8.EE.8a
8.EE.8b-1
8.EE.8b-2
8.EE.8b-3
Grade 8 Math: Sub-Claim A
The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice.
Level 5: Distinguished Command
Level 4: Strong Command
Level 3: Moderate Command
Level 2: Partial Command
Graphs linear relationships in the form
Graphs linear relationships in the form
Graphs linear relationships, in the form
Graphs linear relationships, in the form
y=mx+b, including proportional
y=mx+b, including proportional
y=mx+b, including proportional
y=mx+b, including proportional
relationships.
relationships.
relationships.
relationships.
Interprets the unit rate as the slope of the
graph of a proportional relationship and
applies these concepts to solve realworld problems.
Interprets the unit rate as the slope of the
graph of a proportional relationship and
applies these concepts to solve realworld problems.
Interprets the unit rate as the slope of the
graph of a proportional relationship and
applies these concepts to solve realworld problems.
Compares two different proportional
relationships represented in different
ways.
Compares two different proportional
relationships represented in different
ways.
Compares two different proportional
relationships represented in different
ways.
Interprets y=mx+b as defining a linear
function.
Interprets y=mx+b as defining a linear
function.
Uses similar triangles to show that the
slope is the same between any two
distinct points on a non-vertical line in
the coordinate plane
Solves complex mathematical and realworld linear equations in one variable,
with rational number coefficients,
including those that require use of the
distributive property and combining of
like terms.
Analyzes and solves mathematical and
real-world problems leading to pairs of
simultaneous linear equations
graphically, algebraically and by
inspection.
Understands the relationship between the
graphic representation and the algebraic
solution to the system.
Verifies a solution utilizing multiple
Interprets the unit rate as the slope of the
graph of a proportional relationship.
Makes some comparisons between two
different proportional relationships
represented in different ways.
Solves mathematical and real-world
linear equations in one variable, with
rational number coefficients, including
those that require use of the distributive
property or combining of like terms.
Solves linear equations in one variable,
with rational number coefficients,
including those that require use of the
distributive property and combining like
terms.
Solves linear equations in one variable,
with rational number coefficients,
including those that require use of the
distributive property or combining like
terms.
Analyzes and solves mathematical and
real-world problems leading to pairs of
simultaneous linear equations
graphically, algebraically and by
inspection.
Analyzes and solves mathematical
problems leading to pairs of
simultaneous linear equations
graphically and algebraically.
Solves mathematical problems leading to
pairs of simultaneous linear equations
graphically or by inspection.
Understands the relationship between the
graphic representation and the algebraic
solution to the system.
Functions
8.F.1-1
8.F.1-2
8.F.2
8.F.3-2
Modeling with
Functions
8.F.4
8.F.5-1
8.F.5-2
methods to prove accuracy.
Understands that a function is a rule
assigning to each input exactly one
output which can be graphed as a set of
ordered pairs.
Understands that a function is a rule
assigning to each input exactly one
output, which can be graphed as a set of
ordered pairs.
Understands that a function is a rule that
assigns to each input exactly one output
and can be graphed as a set of ordered
pairs.
Compares properties of two functions
represented in different ways.
Compares properties of two functions
represented in different ways.
Compares some of the properties of two
functions represented in different ways.
Identifies and proves functions as nonlinear.
Identifies functions that are non-linear.
Understands that a function is a rule that
assigns to each input exactly one output
and can be graphed as a set of ordered
pairs.
Grade 8 Math: Sub-Claim B
The student solves problems involving the Additional and Supporting Content for grade/course with connections to the Standards for Mathematical Practice.
Level 5: Distinguished Command
Level 4: Strong Command
Level 3: Moderate Command
Level 2: Partial Command
Constructs a function to model a linear
Constructs a function to model a linear
Constructs a function to model a linear
Constructs a function to model a linear
relationship between two quantities
relationship between two quantities
relationship between two quantities
relationship between two quantities in a
described with or without a context.
described with or without a context.
without a context.
table or a graph.
Given a description of a relationship or
two (x,y) values in a table of values or a
graph, determines the rate of change and
initial value of the function.
Given a description of a relationship or
two (x,y) values in a table of values or a
graph, determines the rate of change and
initial value of the function.
Analyzes, describes and contextualizes
the functional relationship between two
quantities.
Sketches a graph of a function when
given a written description.
Given two (x,y) values in a table of
values or a graph, determines the rate of
change and initial value of the function.
Determines the rate of change and initial
value of the function from a table or
graph that contains the initial value.
Analyzes and describes the functional
relationship between two quantities.
Analyzes the graph of a linear function
to describe the functional relationship
between two quantities.
Analyzes the graph of a linear function
to describe the functional relationship
between two quantities.
Sketches a graph of a function when
given a written description.
Sketches the graph of a function when
given a written description.
PARCC Abbreviation Key:
PBA: Performance-Based Assessment
EOY: End of Year Assessment
M/S/A: Indicates whether this standard is considered Major content, Supporting Content, or Additional Content
MP: Standards for Mathematical Practice
* Indicates a modeling standard
PARCC Sub-Claims:
Sub-Claim A: Information on how PARCC will assess major content
Sub-Claim B: Information on how PARCC will assess additional and supporting content
Sub-Claim C: Information on how PARCC will assess reasoning
Sub-Claim D: Information on how PARCC will assess modeling