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Mathematics Item Specifications
Grade HS, Target G
Claim 1: Concepts and Procedures
Students can explain and apply mathematical concepts and carry out mathematical procedures with
precision and fluency.
Content Domain: Algebra: Reasoning with Equations and Inequalities
Target G: Represent and solve equations and inequalities graphically.
Tasks for this target will require students to interpret a line or curve as a solution set of an equation in
two variables, including tasks that tap student understanding of points beyond the displayed portion of
a graph as part of the solution set. Some of these tasks should explicitly focus on non-integer solutions
(e.g., give three points on the graph of y = 7x + 2 that have x-values between 1 and 2).
Other tasks for this target will require students to approximate solutions to systems of equations
represented graphically, including linear, polynomial, rational, absolute value, exponential and
logarithmic functions (often paired with 9-12.F Target B).
Other tasks for this target will require students to graph solutions to linear inequalities and systems of
linear inequalities in two variables. In some of these tasks, students may be given points, sets of points,
or regions and asked to determine whether the indicated point(s) or regions are part of a solution set.
Standards: A-REI.10, A-REI.11, A-REI.12
Allowable item types*: SR, CR, TE
DOK target(s): 1, 2
Evidence required:
1. The student is able to understand that the set of all points that
make up the graph of a linear or a non-linear equation in two
variables is the solution set of that equation.
2. The student is able to approximate the solutions represented
graphically to systems of equations in two variables.
3. The student is able to graph solutions to linear inequalities in
two variables.
4. The student is able to graph solutions to systems of linear
inequalities in two variables.
Allowable stimulus The coordinate plane, sets of coordinate points, table of x-y- values,
materials: linear equations in the coordinate plane, non-linear equations in the
coordinate plane, linear equations, non-linear equations, systems of
Samples are from SBAC Content Specification Appendix (Link)
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linear and nonlinear equations in the coordinate plane, linear
inequalities, linear inequalities in the coordinate plane, systems of linear
inequalities, systems of linear inequalities in the coordinate plane,
shaded regions in the coordinate plane
Allowable disciplinary linear inequality with two variables, system of equations, system of
vocabulary: inequalities, polynomial function, rational function, absolute value
function, exponential function, logarithmic functions, solution set,
coordinate plane (or coordinate grid), half-plane
Allowable manipulative
materials:
Task models: 1. SR
Prompt Features: The student is prompted to identify points located on
a linear or non-linear graph in the coordinate plane or to identify the
graph of an equation in two variables that contains a set of given points.
Stimuli: The student is presented with a linear or non-linear graph in the
coordinate plane and asked to identify solutions that are located on the
graph. The solutions to be identified may or may not be visibly apparent
on the stimulus (i.e., the student may need to interpolate to find noninteger solutions or extrapolate to identify solutions beyond the visible
stimulus). The student may also be presented with a set of coordinate
points, either in set notation or as a table of values, and asked to identify
a linear or nonlinear graph that is represented by those points. The
student may also be presented with a set of multi-part items that require
yes/no or true/false responses that are scored according to a rubric.
1. CR
Prompt Features: The student is prompted to find a solution set of
coordinate points or complete a table of values for a linear or non-linear
coordinate graph. The prompt may require the student to find a set of
non-integer solutions, a set of solutions with x-values not visible on the
graph, or a set of solutions with y-values not visible on the graph.
Stimuli: The student is presented with a linear or non-linear graph in a
coordinate plane and asked to find solutions that are located on the
graph. The solutions to be identified may or may not be visibly apparent
on the stimulus (i.e., the student may need to interpolate to find noninteger solutions or extrapolate to identify solutions beyond the visible
stimulus).
1. TE
Prompt Features: The student is prompted to graph the solution set for
a linear or a non-linear equation in two variables.
Stimuli: The student is presented with a linear or a non-linear equation
in two variables.
Interaction: The student uses a tool to plot points representing part of
the solution set of the equation and draws a line to connect the points.
Samples are from SBAC Content Specification Appendix (Link)
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2. SR
Prompt Features: The student is prompted to identify the approximate
solution to a system of equations in two variables represented
graphically on a coordinate plane.
Stimuli: The student is presented with a system of equations graphed in
the coordinate plane. The system may include two linear graphs, two
non-linear graphs, or one linear and one non-linear graph. Non-linear
graphs could include polynomial, rational, absolute value, exponential,
and logarithmic functions.
3. SR
Prompt Features: The student is prompted to identify the graphical
region representing the solution set of a linear inequality in two
variables. Or the student is prompted to identify a set of points
representing the solution set of a linear inequality in two variables.
Stimuli: The student is presented with a linear inequality in two
variables. Or the student may be presented with a set of multi-part items
that require yes/no or true/false responses that are scored according to
a rubric.
3. CR
Prompt Features: The student is prompted to graph the region
representing the solution set of a linear inequality in two variables. Or
the student is prompted to list a set of coordinate points representing
the solution set of a linear inequality in two variables.
Stimuli: The student is presented with a linear inequality in two variables
and a blank coordinate plane. Or the student may be presented with a
linear inequality in two variables and asked to identify a set of
coordinate points that are part of its solution set. Or the student is
presented with a graph in the coordinate plane of a linear inequality with
no shaded region drawn and asked to shade the region, or half-plane,
representing the solution set.
3. TE
Prompt Features: The student is prompted to graph the solution set for
a linear inequality in two variables.
Stimuli: The student is presented with a linear inequality in two
variables.
Interaction: The student uses a tool to draw either a solid line or a
dotted line to represent the graph of a linear inequality in two variables
and shade the half-plane that represents the solution set to the
inequality.
4. SR
Prompt Features: The student is prompted to identify the graphical
region representing the solution set of a system of linear inequalities in
two variables. Or the student is prompted to identify a set of points
representing the solution set of a system of linear inequalities in two
Samples are from SBAC Content Specification Appendix (Link)
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variables.
Stimuli: The student is presented with a system of linear inequalities in
two variables. Or the student may be presented with a set of multi-part
items that require yes/no or true/false responses that are scored
according to a rubric.
4. CR
Prompt Features: The student is prompted to graph the region
representing the solution set of a system of linear inequalities in two
variables. Or the student is prompted to list a set of coordinate points
representing the solution set of a system of linear inequalities in two
variables.
Stimuli: The student is presented with a system of linear inequalities in
two variables and a blank coordinate plane. Or the student may be
presented with a system of linear inequalities in two variables and asked
to identify a set of coordinate points that are part of its solution set. Or
the student is presented with a graph of a system of linear inequalities in
two variables with no shaded region drawn and asked to shade the
region representing the solution set.
4. TE
Prompt Features: The student is prompted to graph the solution set for
a system of linear inequalities in two variables.
Stimuli: The student is presented with a system of linear inequalities in
two variables.
Interaction: The student uses a tool to draw either a solid line or a
dotted line to represent the graph of each linear inequality and shades
the intersection of the half-planes that represent the solution set to the
system of inequalities.
Key non-targeted
constructs:
Target-specific attributes:
It will be a challenge to assess blind or low-vision students on this target.
Accessibility concerns: It will be a challenge to assess physically disabled students on the TE
items of this target.
Sample items: 012
*SR = selected-response item; CR = constructed-response item; TE = technology-enhanced
item; PT = performance task;
Samples are from SBAC Content Specification Appendix (Link)