The Launch activity to begin this lesson works best with students in small groups, preferably in pairs. I hand each group of students one Entrance Slip and ask that they work together following the instructions and answering the questions. I tell the class that although they can literally draw any line through the plotted point, they should proceed with certain astuteness, instead of blindly drawing lines, because of the fact that they will have to find the equations to both lines. For example, they can construct a line by starting at the plotted point (2, 3) and use any slope to locate a second point, and so forth.
As I walk around the classroom, I prompt students toward the realization that point (2, 3), the intersection of the two lines, is the only coordinate pair that satisfies both equations. This I do without handing students too much information so that they can come up with this themselves. Once they do, I have students substitute the point into each of the two equations. This last step may seem unnecessary, but it actually is helpful because it emphasizes in their minds, the fact that the intersection is the solution to both equations.
Once students have concluded, I have a group go up to the document camera to show their work and explain how they drew their lines. I also have them state the answers to the questions in the entrance handout, aloud. When the group presenting is done, I ask the class if they have any questions and if anyone did anything differently. Chances are that some other group graphed their lines using another method, so I ask this group to share what they did with the whole class.
I conclude by questioning...."a line is a set of?..(points), and each point on the line is a?....(solution to its equation), therefore, what can you say about (2, 3), the point of intersection? (It's a solution to both equations)
I tell the class that two or more equations graphed on the same plane is a system of equations, and that the average person encounters systems of equations in everyday life.
I begin this segment of the lesson by handing a copy of System Applications Tasks to each pair of students. The groups are suppose to read each task carefully, write the equations that represent the situations in each problem, graph these, and determine the solution to the system. (See my reflection on ELL students and systems)
I tell the students to use rulers and pencils and to be as accurate and neat as possible when graphing the equations. I ask the students if they know why I stress this accuracy. I hope that students will respond with a statement like, "Because the solution will be determined at the point of intersection on both graphs, so its important to be graph carefully."
At the end of each task, I will ask several students to explain what the solution point represents in the problem. For example, in task one, students should be able to say something along the lines of "Point (4, 16) means that when my partner and I purchase 4 CDs, we both spend $16 dollars." I insist that they make concluding statements like this, in a clear manner, using language that is precise, but in students own words.
As I walk around the class assessing students, I make sure they answer Task 2b in a full sentence, indicating that the tomato plants reach the same height after Day 2 when their height will be 14 inches.
To end today's lesson, I ask each group to turn their Application Tasks sheet over while I project the Closure System graph on the white board. I inform the class that each equation represents a monthly phone plan and the equations of both plans are shown. Students should listen to the questions asked and write their answers on the back of the Applications Tasks worksheet. I allow a couple of minutes for students to discuss the questions with their partners before calling on a volunteer to answer.
Questions for Closure Task:
I expect students may choose one plan as better than the other without giving a valid reason for their statement. If so, I will insist that they justify their response. I want them to be good consumers, here. I will lead them into understanding that the better plan depends on if the user is a frequent speaker or not. Frequent speakers should use the 10 cent per minute plan, because after 80 minutes, it is cheaper.
These questions usually take under 10 minutes to answer, unless questions arise. I try to spend a few more minutes clearing up any points of confusion. I also request they write their answers on their worksheets and hand these back to me as they walk out of the room. Assessing these written responses will give me an idea of what questions I would need to pose on Day 2 of this lesson.
The Homework assignment contains 4 systems to solve by graphing. I tell the class that they should show all work if they were to graph the equations using tables of values. Many students prefer to use the slope and the y-intercept, especially when they see the equation written in "slope-intercept" form, and this is fine. I also tell students to make sure that they write the solution to the system in ordered pair form in the spaces provide.