Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Coordinate Plane
- Linear Expression
Purpose and Instructional Strategies
In grade 8, students graphed linear two-variable equations. In Algebra I, students graph the
solution set to a two-variable linear inequality. In later courses, students will solve problems
involving linear programming and will graph the solutions sets of two-variable quadratic
- Instruction includes the use of linear inequalities in standard form, slope-intercept form
and point-slope form. Include examples in which one variable has a coefficient of zero such as < −.
- Instruction includes the connection to graphing solution sets of one-variable inequalities
on a number line; recognizing whether the boundary line should be dotted (exclusive) or
solid (inclusive). Additionally, have students use a test point to confirm which side of the
line should be shaded (MTR.6.1).
- Students should recognize that the inequality symbol only directs where the line is shaded
(above or below) for inequalities when in slope-intercept form. Students shading
inequalities in other forms will need to use a test point to determine the correct half-plane
Common Misconceptions or Errors
- Students often choose to shade to wrong half-plane when graphing two-variable linear
- Students may think that the inequality symbol’s orientation always determines the side of
the line to shade.
- For example, students may say that inequalities with a less than symbol should be
shaded below the line while inequalities with a greater than symbol should be
shaded above the line. This typically happens after graphing multiple inequalities
in slope-intercept form. To address this, provides counterexamples to this such as
3 − 2 < 15 − 4 − 7 ≥ . Use these counterexamples to emphasize the
benefit of using a test point to confirm the direction of shading.
Strategies to Support Tiered Instruction
- Instruction includes opportunities to use a highlighter to identify the phrases “is less
than,” “is greater than,” “is less than or equal to,” and “is greater than or equal to” when
- Teacher provides instruction modeling how to correctly identify the solution set of a
linear inequality given in slope-intercept form. After graphing, students can circle the y intercept. If the inequality is in form < + or ≤ + , the solution set is the
half-plane that contains the -axis values below the -intercept. If the inequality is in
form > + or ≥ + , the solution set is the half-plane that contains the -
axis values above the -intercept.
- Instruction includes opportunities to graph the boundary line of a system of inequalities,
based on an inaccurate translation from word problem. To assist in determining the
boundary line for the system, students can create a graphic organizer like the one below.
- Instruction includes making the connection between the algebraic and graphical
representations of a two-variable linear inequality and its key features.
- For example, teacher can provide a graphic organizer such as the one below.
- Instruction includes opportunities to identify a test point to plug into an inequality. It is
usually easiest to use the origin (0,0) as it makes mental calculations easier. If the point
selected creates a true statement, their inequality is true and they should shade in the half-plane containing that point. If it creates a false statement, they should shade in the half-plane not containing that point. By using a test point, students avoid the mistake of
thinking that the direction of the inequality determines the shading.
- For example, the points (−4,3), (0,0), (3,2) and (3, −1) were used to
determine where to shade for the inequality 4 − 3 < 6 shown below.
Instructional Task 1 (MTR.7.1)
- Penelope is planning to bake cakes and cookies to sell for an upcoming school fundraiser.
- Each cake requires 1 cups of flour and each batch of cookies requires 2 cups of flour.
- Penelope bought 3 bags of flour. Each bag contains around 17 cups of flour.
- Part A. Assuming she has all the other ingredients needed, create a graph to show all the
possible combinations of cakes and batches of cookies Penelope could make.
- Part B. Create constraints for this given situation.
Instructional Item 1
- Graph the solution set to the inequality + 3 > −2( − 2).
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.