System of Equations -- Part I Graphing

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”.

1. 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.
2. 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.
3. 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.