Goals: (Some Algebra 1 goals are included)
• Write the equation of and graph linear relationships given the slope and one
point on the line.
(Algebra 1, 3.06)
• Investigate and de termine the effects of changes in slope and intercepts on
the graph and the equation
of a line. (Algebra 1, 3.07)
• Use systems of linear equations or inequalities in two variables to solve
problems. (Algebra 1, 3.09)
• Write the equation of a line parallel to a given line through a given point.
(Geometry, 3.02)
• Use systems of two or more equations to solve problems. Solve by: Elimination
and/or substitution,
graphing, and matrices. (3.12)
Materials and Equipment Needed:
• Copy of handout for each student
• Graphing calculator
• 2 sheets of graph paper for each student
• Ruler
• Paper for note taking during class
Activity One: Find linear equations that define the sides of a parallelogram
given only one vertex
and find the remaining vertices.
1. Students read Question 1 on the handout (from indicator 1.13 A) and have
students sketch a
parallelogram on the top half of the graph paper.
2. Using Geometer’s Sketchpad, show a parallelogram described in the question.
By moving the
parallelogram, illustrate the number of possible parallelograms that fit the
description of the problem.
3. Discuss what information each person needs and must decide as he/she defines
their parallelogram.
Illustrate using GSP and the graphing calculator.
• Discuss form of possible equations (use point-slope form of the linear
equation) for the two
sides that contain the given vertex at (-3,4). y − 4 = m(x + 3) and selecting
two values for
m . Show graphs on the calculator. What happens if you select ≤ or ≥ for the
linear
inequalities?
• Make decision about whether to select a point as the vertex opposite the given
point or to
select a point on one of the sides defined by lines found above as a vertex
adjacent to the
given point.
• Based on this decision, discuss the next steps of defining the parallelogram
using GSP to
illustrate. What moves as you change slopes? What changes as you select
different points as
vertices?
• What is expected in the answer? Emphasize the answers should be linear
inequalities.
• Show result on paper, on graph paper, and on the calculator. Show how shading
inequalities
works on the graphing calculator. Shading one at a time works well in making
sure you have
the inequality. Showing several at once is overwhelming.
4. In order to check mathematics as we proceed, students will need to have the
same parallelogram.
Although students should be doing in dividual work , it is good to have them
paired in order to check
their work.
• Have students graph the point (-3,4) on the lower part of their graph paper.
Call this point P1.
• Ask a student to give a slope through the point (-3,4) that we will all use.
Graph that line and
write its equation. Call it Line1.
• Ask another student to select a new point on line Line1. Call the new point
P2.
• Decide whether the parallelogram will be above or below line Line1.
• Ask a student to give a slope through point P2. Graph that line and write its
equation. Call
the new line Line2.
• Discuss whether the parallelogram will be above or below line Line2.
• At this point you can either find a line through P1 and parallel to Line2 or
you can select a
new point P3 on Line2 and find the line through P3 and parallel to Line1.
• Proceed until all students have the same parallelogram.
5. Give students an opportunity to write all the inequalities that describe the
parallelogram.
6. Use the flex cam to see what students are doing. Have several pairs of
students show their work and
progress.
7. Once equations are complete, discuss how to find vertices of the
parallelogram that have not yet been
determined. Use both substitution and elimination methods of solving
simultaneous equations .
8. Show some parallelograms on GSP using the coordinates of the vertices and the
graphing calculator
using the linear inequalities. Show how to find points of intersection on the
calculator.
Follow-Up Problem
Write the system of inequalities that describes a triangular region with one
vertex at (7, 5) and another on
the x-axis. None of the sides can be horizontal or vertical. Verify your result
on your graphing calculator.
There are many correct answers to this question. A possible way for students to
check work is to have
work in pairs and have one student graph the systems of inequalities of the
other student to determine if
the system meets the criteria of the question.
Student Handout
Linear Inequalities with a Parallelogram
Algebra II
1. Write a system of inequalities that defines the region enclosed by a
parallelogram in
the second quadrant with one vertex at (-3,4). The sides cannot be vertical or
horizontal. (This problem is taken from the Algebra II Indicators on the NCDPI
website.)
2. Find the vertices of the parallelogram described by
this system of inequalities.
Follow-Up
Linear Inequalities
Algebra 2
1. Write the system of inequalities that describes a
triangular region with one vertex at
(7, 5) and another on the x-axis. None of the sides can be horizontal or
vertical. Verify
your result on your graphing calculator. (This problem is taken from the Algebra
II Indicators on the NCDPI
website.)
