Lesson Plan Solving Systems by Graphing

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 below1. 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.
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
the homework and ask questions about the assignment.
Norm Ebsary Multimedia #2 Lesson Plan: Solving Systems by Graphing
Key questions to be asked:
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
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
Assignments for students: A total of 10 problems, to be solved by
graphing and classifying Independent, Dependent, or Inconsistent.
Norm Ebsary Multimedia #2 Lesson Plan: Solving Systems by Graphing
Assessments: in 6 levels as follows1. 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
3. Warmup- Warmup is based on previous lesson, so the ease in
which students complete the Warmup correctly is observed and
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
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
Y = .10x + 300
Radio Barn Weekly + Commission
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!
Norm Ebsary Multimedia #2 Lesson Plan: Solving Systems by Graphing
Appendix A: Answer keys to assignments and assessments
Figure 3: Lesson Warm- Up
Norm Ebsary Multimedia #2 Lesson Plan: Solving Systems by Graphing
Figure 4: Lesson Warm- Up solution key
Gr aphing Syst ems of
1. Graph and solve the system.
4x + y = –1
–x + 3y = 10
(–1, 3)
Classify each system by graphing. Tell how many solutions
there are.
5x + 3y = 10
–x – 0.6y = –2
infinitely many
12x – 18y = 9
–6x + 9y = 13
no solutions
4x + 5y = –10
3x – 8y = 15
one solution
Figure 5: Lesson Quiz with solution key
Norm Ebsary Multimedia #2 Lesson Plan: Solving Systems by Graphing
Graphing Systems of Equations
38. inconsistent
39. dependent
44. (continued)
c. The Pooch Pad
would be cheaper for
a 7-day stay.
45. (continued)
After 10 minutes the
numbers of flyers
will be equal.
45. x = minutes,
y = flyers;
y = 6x + 80
y = 4x + 100
46. Answers may vary.
Sample: y = x + 3
40. independent
41. inconsistent
42. inconsistent
47. Answers may vary.
Sample: y = –4x + 8
43. dependent
48. Answers may vary.
44. a. c = 20d + 30
c = 25d
Sample: y = 2x + 3
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
the same line, and
the system would be
dependent and
50. An independent
system has one
solution. The slopes
are different, but the
y-intercepts could be
the same. An
inconsistent system
has no solution. The
slopes are the same,
and the y-intercepts
are different.
50. (continued)
A dependent system
has an infinite
number of solutions.
The slopes and
y-intercepts are the
51. Answers may vary.
Sample: 3x + 4y = 12
52. Answers may vary.
Sample: y = –
53. Answers may vary.
–10x + 2y = 4
5x – y = –2
54. They are the same
equation written in
different forms.
55. a. p: independent,
n: dependent
b. n = –1600p + 14,800
c. n = –6000 + 32,000
Figure 7: Homework solution key
Norm Ebsary Multimedia #2 Lesson Plan: Solving Systems by Graphing
Graphing Systems of Equations
55. (continued)
d. About
(3.91, 8545);
profits are
maximized if
about 8545
widgets are sold
for about $3.91
56. C
57. G
58. B
59. H
60. [2] The slope of 2x – 5y = 23 is 2 and the slope of
3y – 7x = –8 is 7 . Since the slopes are not equal,
the lines are not parallel and they do not
coincide. Therefore, the lines intersect; the
system has exactly one solution and is consistent.
[1] does not include explanation
61. [4] Answers may vary. Sample:
(a) A second equation is 4x – 6y = 10, or any
equation of the form 2ax – 3ay = 5a.
(b) A second equation is 2x – 3y = 6 or any
equation of the form 2ax – 3ay = 5b,
where a =/ b.
[3] minor error in either part (a) or (b)
Figure 8: Homework solution key
Figure 9: Quiz on Linear Systems
Norm Ebsary Multimedia #2 Lesson Plan: Solving Systems by Graphing
Figure 10: Chapter Test on Linear Systems
Norm Ebsary Multimedia #2 Lesson Plan: Solving Systems by Graphing
Appendix B: Supplement activities lesson
Downloaded from UnitedStreaming.com a video on Linear Equations with
emphasis on graphing. The tile of the video is
“Adventurous_World_of_Algebra__The__Program_3__Linear_Equations” and has a runtime of approximately 45 minutes. The supporting cast is called the “Standard
Deviants”, that provide an effective yet humorous at times lesson on Linear
Figure 11: Vide from UnitedStreaming.com on Linear Algebra
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