# Lesson Plan Solving Systems by Graphing

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```					      Norm Ebsary Multimedia #2 Lesson Plan: Solving Systems by Graphing
Lesson Plan Solving Systems by Graphing

Introduction to the Lesson: Systems of linear equations will have 1
solution (independent), no unique solution (dependent), and no solution
(inconsistent). Students will learn to graph, analyze, and solve systems
of equations by using slope- intercept, standard, and point- slope forms
of equations. Students will learn how to find 1 solution, no unique
solution, or no solution by graphing system of equations. Emphasis will be
on graphing systems of equations, with students leveraging their graphing
skills as well as using GeoGebra to capture exact solutions.

Explanation of Math Involved: The primary math involved in solving
systems by graphing is graphing. By graphing these linear equations, the
solution needs to be common to both equations. By graphing these linear
equations, students in addition will have to perform substitution to verify
the solution is valid for both equations.

Use 3- Steps as shown below-
1. Graph both equations using slope- intercept format.
a. Plot y intercept (b).
b. Count- out slope by rise and run, plot 2nd point.
c. Connect 2 points with straight line.
2. Determine the solution, if it exists:
a. One solution- point of intersection.
b. Infinite solution- coincident, equivalent equations.
c. No solution- parallel lines, slopes of both lines are equal.
3. Validate solution, by performing check.
a. Substitute x and y coordinates found (if exists) back into
original equations.
Solve equations to ensure correctness, that equations balance properly.

Instructional Methods: Visual based using Interactive Electronic text,
PowerPoint, and GeoGebra with interactive kinesthetic exercises. So
students will follow teacher with classroom examples- using handout of
graph paper. After graphing examples in class, students will have
classroom assignment using GeoGebra- exploring solutions of 2 linear
equations manipulating slope (m) and y intercept (b) of each equation.

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Norm Ebsary Multimedia #2 Lesson Plan: Solving Systems by Graphing
Step by step procedure: A Daily Agenda is organized, to keep the
students aware of the class lesson that day. The lesson is organized as
shown below in the PowerPoint, the students see this as they enter the
classroom and it is reviewed at the start of the class by the teacher.

Figure 1: Classroom Lesson Daily Agenda
The step by step procedure will follow the Daily Agenda shown above:
1. Warmup- See Appendix A for 6 graphing problems.
2. Lesson Introduction- based on PowerPoint and electronic textbook,
students review linear equations with slope- intercept, standard,
and point- slope forms, then are introduced to lesson concepts with
systems of equations and how to solve by graphing.
a. Students do 3 classwork examples using graph paper
b. Students work in teams of 2 to 3 for GeoGebra activity
3. Lesson Quiz assesses students comprehension of the lesson.
4. Homework start- students have the 15 minutes or more to start

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Norm Ebsary Multimedia #2 Lesson Plan: Solving Systems by Graphing
1. Lines with the same slope have how many solutions?
2. How can a system of equations have 1 Solution?
3. How can a system of equations have infinite Solutions?
4. If a system of equations has no solutions, what do these equations
have in common?

Description of class activities:
1. Warmup activity- 3 problems on graphing slope- intercept,
standard, and point- slope form of linear equations.
2. Review graphing and introduce systems of equations.
3. Classroom examples- 3 different examples where students explore
1 solution, infinite solutions, and no solutions.
4. GeoGebra interactive exercise, exploring linear solutions by
changing slope and y intercept values using sliders.
5. Lesson Quiz- checks comprehension and provides additional
assessment.
6. Homework start, begin homework assignment and help answer final
doubts and questions.

Closure to the lesson: Students working on homework assignment, as a
way of assessing their understanding of the class lesson. Students end
the lesson by practicing their systems graphing through homework
assignment.

Assignments for students: A total of 10 problems, to be solved by
graphing and classifying Independent, Dependent, or Inconsistent.

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Norm Ebsary Multimedia #2 Lesson Plan: Solving Systems by Graphing
Assessments: in 6 levels as follows-
1. Classwork activity- observations on how students complete
exercise activities for length of time and correctness of content.
2. Homework- review homework for completeness and correctness of
content.
3. Warmup- Warmup is based on previous lesson, so the ease in
which students complete the Warmup correctly is observed and
noted.
4. Lesson Quiz- at the end of lesson to check for understanding.
5. Quiz- section short assessment using system of equations by
graphing, 10 problems total.
6. Test- chapter test provides measure of how students comprehend
the graphing of system equations and how this interacts with
earlier and later sections of the chapter.
Plan for providing feedback: for classwork activity and Warmup, the
feedback will be interactive as student’s progress through the
assignment- I will be walking around the classroom interacting with
students and providing direct feedback on my observations. For
homework, I also provide direct feedback while grading the assignments
for completeness and correctness as I take attendance and students are
working on their Warmup assignments. For Quiz and Tests, students
receive their corrected versions back in a timely manner and problems
are reviewed in class.

Answer keys to assignments and assessments: See Appendix A

Supplement activities lesson: See Appendix B

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Norm Ebsary Multimedia #2 Lesson Plan: Solving Systems by Graphing
Connections section: There is opportunity to utilize systems of equations
for real- life applications. Several examples are provided in the
homework assignment, for example problem #30 below:

Y = .15x + 200        Radio Barn Weekly + Commission
Y = .10x + 300         Woofer, Etc. Weekly + Commission

Figure 2: GeoGebra System of Equation for Homework Example
So the solution by graphing the system is x = 2000, or 2000 units of
sales- both salaries are equal. Here is the check:

Y = .15(2000) + 200 = 500
Y = .10(2000) + 300 = 500
Both salaries are equal, at \$500 for sales of 2000 units. This
shows a real- life connection, that students are always asking
“When am I going to use this?”, well here it is!

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Norm Ebsary Multimedia #2 Lesson Plan: Solving Systems by Graphing
Appendix A: Answer keys to assignments and assessments

Figure 3: Lesson Warm- Up

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Norm Ebsary Multimedia #2 Lesson Plan: Solving Systems by Graphing

Figure 4: Lesson Warm- Up solution key

Gr aphing Syst ems of
Equations
4x + y = –1
1. Graph and solve the system.
–x + 3y = 10

(–1, 3)

Classify each system by graphing. Tell how many solutions
there are.
5x + 3y = 10             12x – 18y = 9             4x + 5y = –10
2.                        3.                        4.
–x – 0.6y = –2             –6x + 9y = 13            3x – 8y = 15
dependent;                  inconsistent;           independent;
infinitely many             no solutions            one solution

Figure 5: Lesson Quiz with solution key

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Norm Ebsary Multimedia #2 Lesson Plan: Solving Systems by Graphing

Graphing Systems of Equations
38. inconsistent             44. (continued)               45. (continued)
c. The Pooch Pad              After 10 minutes the
39. dependent                       would be cheaper for       numbers of flyers
a 7-day stay.              will be equal.
40. independent
45. x = minutes,              46. Answers may vary.
41. inconsistent                 y = flyers;                   Sample: y = x + 3
y = 6x + 80
42. inconsistent                 y = 4x + 100              47. Answers may vary.
Sample: y = –4x + 8
43. dependent
44. a. c = 20d + 30                                                                 7
Sample: y = 2x + 3
c = 25d

b. The cost would
be the same for a
6-day stay.

Figure 6: Homework solution key

Graphing Systems of Equations
49. No; they would be          50. (continued)             53. Answers may vary.
the same line, and             A dependent system          Sample:
the system would be            has an infinite              –10x + 2y = 4
dependent and                  number of solutions.         5x – y = –2
consistent.                    The slopes and
y-intercepts are the    54. They are the same
50. An independent                 same.                       equation written in
system has one                                             different forms.
solution. The slopes       51. Answers may vary.
are different, but the         Sample: 3x + 4y = 12    55. a. p: independent,
y-intercepts could be                                         n: dependent
the same. An               52. Answers may vary.
inconsistent system                            5x          b. n = –1600p + 14,800
Sample: y = –      +7
has no solution. The                           2
slopes are the same,                                       c. n = –6000 + 32,000
and the y-intercepts
are different.

Figure 7: Homework solution key

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Norm Ebsary Multimedia #2 Lesson Plan: Solving Systems by Graphing

Graphing Systems of Equations
55. (continued)           60. [2] The slope of 2x – 5y = 23 is 2 and the slope of
5
d. About                      3y – 7x = –8 is 7 . Since the slopes are not equal,
(3.91, 8545);                               3
profits are                the lines are not parallel and they do not
maximized if               coincide. Therefore, the lines intersect; the
about 8545                 system has exactly one solution and is consistent.
widgets are sold
for about \$3.91        [1] does not include explanation
each.
61. [4] Answers may vary. Sample:
56. C                             (a) A second equation is 4x – 6y = 10, or any
equation of the form 2ax – 3ay = 5a.
57. G                             (b) A second equation is 2x – 3y = 6 or any
equation of the form 2ax – 3ay = 5b,
58. B                                where a = b.
/

59. H                         [3] minor error in either part (a) or (b)

Figure 8: Homework solution key

Figure 9: Quiz on Linear Systems

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Norm Ebsary Multimedia #2 Lesson Plan: Solving Systems by Graphing

Figure 10: Chapter Test on Linear Systems

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Norm Ebsary Multimedia #2 Lesson Plan: Solving Systems by Graphing
Appendix B: Supplement activities lesson
emphasis on graphing. The tile of the video is
time of approximately 45 minutes. The supporting cast is called the “Standard
Deviants”, that provide an effective yet humorous at times lesson on Linear
Algebra

Figure 11: Vide from UnitedStreaming.com on Linear Algebra

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