ALEX | Alabama Learning Exchange

System of Equations -- Part I Graphing

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This lesson provided by:

Author:	Morgan Boyd
Organization:	Retirement

General Lesson Information

Lesson Plan ID:	35613
Title:	System of Equations -- Part I Graphing
Overview/Annotation:	Systems of Equations will be taught over a three-day period: the first day will include a lesson regarding equations that can be solved by graphing, the second day will include a lesson regarding equations that can be solved by substitution, and the third day will include a lesson regarding equations that can be solved by elimination. The students will graph two lines on the same coordinate axis and determine where the two lines cross. The teacher will be able to rock the teenage world with the website "DESMOS". Solving the equations graphically will enhance the graphing skill of the students. The lesson will explain all the ways to graph a line. This lesson results from the ALEX Resource Gap Project.

Associated Standards and Objectives

Content Standard(s):

Mathematics
MA2019 (2019)
Grade: 8

12. Solve systems of two linear equations in two variables by graphing and substitution.

a. Explain that the solution(s) of systems of two linear equations in two variables corresponds to points of intersection on their graphs because points of intersection satisfy both equations simultaneously.

b. Interpret and justify the results of systems of two linear equations in two variables (one solution, no solution, or infinitely many solutions) when applied to real-world and mathematical problems.

Unpacked Content

Evidence Of Student Attainment:

Students:

Graph a system of two linear equations, recognizing that the ordered pair for the point of intersection is the x-value that will generate the given y-value for both equations.
Recognize that graphed lines with one point of intersection (different slopes) will have one solution, parallel lines (same slope, different y-intercepts) have no solutions, and lines that are the same (same slope, same y-intercept) will have infinitely many solutions.
Use substitution to solve a system, given two linear equations in slope-intercept form or one equation in standard form and one in slope-intercept form.
Make sense of their solutions by making connections between algebraic and graphical solutions and the context of the system of linear equations.

Teacher Vocabulary:

System of linear equations
Point of intersection
One solution
No solution
Infinitely many solutions
Parallel lines
Slope-intercept form of a linear equation
Standard form of a linear equation

Knowledge:

Students know:

The properties of operations and equality and their appropriate application.
Graphing techniques for linear equations (using points, using slope-intercept form, using technology).
Substitution techniques for algebraically finding the solution to a system of linear equations.

Skills:

Students are able to:

generate a table from an equation.
Graph linear equations.
Identify the ordered pair for the point of intersection.
Explain the meaning of the point of intersection (or lack of intersection point) in context.
Solve a system algebraically using substitution when both equations are written in slope-intercept form or one is written in standard form and the other in slope-intercept form.

Understanding:

Students understand that:

any point on a line when substituted into the equation of the line, makes the equation true and therefore, the intersection point of two lines must make both equations true.
Graphs and equations of linear relationships are different representations of the same relationships, but reveal different information useful in solving problems, and allow different solution strategies leading to the same solutions.

Diverse Learning Needs:

Essential Skills:

Learning Objectives:
M.8.12.1: Define variables.
M.8.12.2: Recall how to estimate.
M.8.12.3: Recall how to solve linear equations.
M.8.12.4: Demonstrate how to graph solutions to linear equations.
M.8.12.5: Recall how to graph ordered pairs on a Cartesian plane.
M.8.12.6: Recall that linear equations can have one solution (intersecting), no solution (parallel lines), or infinitely many solutions (graph is simultaneous).
M.8.12.7: Define simultaneous.
M.8.12.8: Recall how to solve linear equations.
M.8.12.9: Recall properties of operations for addition and multiplication.
M.8.12.10: Discover that the intersection of two lines on a coordinate plane is the solution to both equations.
M.8.12.11: Define point of intersection.
M.8.12.12: Recall how to solve linear equations.
M.8.12.13: Demonstrate how to graph on the Cartesian plane.
M.8.12.14: Identify ordered pairs.
M.8.12.15: Recall how to solve linear equations in two variables by using substitution.
M.8.12.16: Create a word problem from given information.
M.8.12.17: Recall how to solve linear equations.
M.8.12.18: Explain how to write an equation to solve real-world mathematical problems.

Prior Knowledge Skills:

Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
Demonstrate an understanding of an extended coordinate plane.
Draw a four-quadrant coordinate plane.
Draw and extend vertical and horizontal number lines.
Interpret graphing points in all four quadrants of the coordinate plane in real-world situations.
Recall how to graph points in all four quadrants of the coordinate plane.

Alabama Alternate Achievement Standards

AAS Standard:
M.AAS.8.12 Solve two-step linear equations where coefficients are less than 10 and answers are integers.

Local/National Standards:

Primary Learning Objective(s):

The student will be able to determine the solution to the system of equations by graphing.

The student will be able to explain if two lines will intersect, are parallel, or are the same line.

Additional Learning Objective(s):

The student will be able to graph linear equations using slope intercept form.

The student will be able to graph a linear equation using x- and y-intercepts.

Preparation Information

Total Duration:	31 to 60 Minutes
Materials and Resources:	Pencils Colored pencils (blue and red) Graph paper (Students may use their own paper; an attachment is included to make copies for the students if needed) Straight edge (ruler, Student ID, or Index Card) for all students Bell Ringer Graphing Lines for the interactive whiteboard (see attached document) Exit Slip Graphing copies for all students (see attachment document) Desktop computer for teacher use and interactive whiteboard Chromebook, iPad, or MacBook for all students so that they can access the website "DESMOS" Websites: https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-systems-topic/cc-8th-systems-graphically/a/systems-of-equations-with-graphing "DESMOS" Website: online graphing calculator https://cdn.kutasoftware.com/Worksheets/PreAlg/Systems%20of%20Equations%20by%20Graphing.pdf Print and make copies for all the students Activity for accelerated students, with answers: https://www.hoodriver.k12.or.us/cms/lib06/OR01000849/Centricity/Domain/788/_soe_word_problems_worksheet.pdf
Technology Resources Needed:	Desktop computer for teacher use and interactive whiteboard Chromebook, iPad, or MacBook for all students so that they can access the website "DESMOS" and use Google Classroom or email. Websites: "DESMOS" online graphing calculator https://cdn.kutasoftware.com/Worksheets/PreAlg/Systems%20of%20Equations%20by%20Graphing.pdf Print and make copies for all the students https://www.hoodriver.k12.or.us/cms/lib06/OR01000849/Centricity/Domain/788/_soe_word_problems_worksheet.pdf Print and make copies for accelerated students
Background/Preparation:	Teacher: The teacher will need preview the website "DESMOS". The website is a graphing calculator. The directions for the website are as follows: 1. In order to graph a line, click on the left side next to the number 1. This is the place to type in the first equation. 2. If the number is negative, press "-" before the number. 3. When typing fractions, you must use the right arrow to move the cursor off of the denominator. 4. As you are typing, the line will form on the graph. 5. Click on the number 2 to create the next line. The lines will be in different colors. The teacher needs to preview the website "Kuta Software". The website has many free worksheets that can be printed for all the students. The graphing worksheet for this lesson can be found at this link: https://cdn.kutasoftware.com/Worksheets/PreAlg/Systems%20of%20Equations%20by%20Graphing.pdf. The teacher will need to preview Google Docs as well as Google Classroom. The students will be able to attach their answers to the assignments. If Google Classroom is not used in your class then the answers will be done on graph paper. Student: The students will need to be able to graph linear equations. The students need to be able to change the form of a line. The students need to know how to graph lines with slope and x- and y-intercepts.

Procedures/Activities:

Before:

Bell Ringer: As students enter the room, the teacher will display the two equations that the students will graph on the same coordinate plane. The document is in the materials section called “Bell Ringer Graphing Lines”. The students will write on their own graph paper. The lines are graphed using the website DESMOS.com. The instructions are on the bell ringer.

During:

* The systems of equations solved by graphing will be taught in one class period. *

1. The teacher must tell the students that a system of equations has three possible solutions. They are as follows:

The lines will cross at a single point.
The lines will be parallel.
One line is a multiple of the other line.

2. The way that the teacher can explain the three ways is to look at the slope of the lines.

If they have different slopes, then the lines will cross at only one point. The point will be the solution because it will be on both lines.
If the lines have the same slope and different y-intercepts, then the lines are parallel. The system does not have a point that will work for both lines, thus, no solution.
If the lines have the same slope and y-intercept, then the lines are multiples. One line can be divided by a number and it will be exactly the same as the other line. The solution would be “all real numbers”. Every point that is on the first line will be on the second line.

3. As the teacher works the examples on the interactive whiteboard, the teacher will need to graph the systems of equations. The teacher will need to make sure that the students have their graph paper and colored pencils.

Examples:

2x + y = 3 x – 2y = 4

The slopes are -2 and ½, thus, the lines will intersect and have a solution. The point in common will be the solution in the form (x, y).

Y = -x +1 3x + 4y = -12

The slopes are negative, but the y-intercepts are 1 and -3. The lines are parallel.

Y = -2x – 3 4x + 2y = -6

The second line can be divided by two. The result would be the same line. The solution would be “all real numbers”.

After the last example, the teacher will show the entire lesson from the website, “Khan Academy”. The link is https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-systems-topic/cc-8th-systems-graphically/a/systems-of-equations-with-graphing.
The website will review graphing skills and solving systems of equations by graphing. The teacher can solicit questions from the students to make sure that they are engaged and writing the material in their in their notebooks. If the students understood the material, then the last section can be skipped.
The teacher will demonstrate how to use the website, DESMOS.com. The instructions for the website is as follows:
1. The keyboard at the bottom of the screen will allow you to type the equations on the left side of the screen.
2. Each equation will have a different color for the line.
3. After the equations are graphed, the student will share or copy the web address. This task is executed by clicking the box with the arrow. The icon is located at the top of the graph on the right. This will allow the student to copy the web page address.
4. Next, the student will paste the address in the Google Doc under the equations.
5. After completing the assignment, the students will submit their answers to Google Classroom, which is similar to adding an attachment to an email. If the teacher is not familiar with Google Classroom, then the students can send their answers as an attachment in the email.
6. If technology is not available, then the student can graph the equations on graph paper using blue and red pencils for each line.
7. The teacher will use “Kuta Software” as a resource for worksheets. The worksheets will already have the coordinate plane. The student will graph the lines using the blue and red pencils. (https://cdn.kutasoftware.com/Worksheets/PreAlg/Systems%20of%20Equations%20by%20Graphing.pdf)
8. The teacher can choose to use the technology-based option or the traditional paper/pencil option.
9. The teacher will hand out the worksheets from Kuta Software. The teacher will need to copy sheets for all the students.
10. The teacher can place students in collaborative groups of four students. The intervention will allow the students more time to complete the assignment, working with a peer-tutor, or one-on-one with the teacher. The teacher will monitor the students by walking around the room. The teacher will complete an informal assessment at this time by asking students questions about their graphs and solutions.
11. The teacher will need place the answers to the problems on the interactive whiteboard. The answers are on the “Kuta Software” sheets pages 3 and 4.

*The graphing part is concluded for day one of the 3 parts. The next day will be substitution.*

After:

The student will complete an exit slip. This will be an opportunity for a formal assessment. The exit slip, “Exit Slip Graphing”, is in the attachments section. The teacher will need to copy enough sheets for all the students. The students will turn in the exit slip as they leave the room. The accelerated students will use the worksheet from https://www.hoodriver.k12.or.us/cms/lib06/OR01000849/Centricity/Domain/788/_soe_word_problems_worksheet.pdf. These students will be graphing systems of equations with three lines.

Assessment

Assessment Strategies

Informal:

The teacher will solicit questions from students during the Khan Academy program. The teacher will be able to observe the students working after the lesson.

Formal:

The teacher will use the exit slip as the formal assessment.

Acceleration:	The accelerated students can be used as peer-tutors during the independent practice. The website, https://www.hoodriver.k12.or.us/cms/lib06/OR01000849/Centricity/Domain/788/_soe_word_problems_worksheet.pdf, will be copied and assigned.
Intervention:	The teacher will place the students in groups and assign a peer-tutor. The teacher will also monitor the students' work while they are working on the independent exercises. The teacher can adjust the time and/or shorten the assignment.

View the Special Education resources for instructional guidance in providing modifications and adaptations for students with significant cognitive disabilities who qualify for the Alabama Alternate Assessment.

ALEX Lesson Plan

System of Equations -- Part I Graphing