More with Solving Two Variable Equations

This Opening Activity will require that students reason quantitatively about the number of students that could attend a party. I structure the beginning of class in way very similar to the previous lesson.

1. I give each pair of students some time to determine possible solutions.
2. I post a graph on the board. 
3. I have one student from each pair come up and post their solutions to the question.  

This time, however, when students mark an ordered pair on the graph, I will ask them to label the coordinates as well.  Here are some questions that I will use to help guide the discussion and help students to notice the structure in this linear equation:

• Is there an equation we can use to model this situation? I will Guide students towards seeing that x + 2y = 30 would model the situation appropriately since each freshman (x) is only bringing $1 and each sophomore (y) is bringing $2.
• So if x + 2y = 30 is our equation, why didn't anyone mark the coordinate (5, 12.5)? This is an important point, so I will give students time to process at this point. When students share out, point the conversation towards the fact that the equation is a model, but in the context of the problem we cannot have decimal values because our variables stand for number of students.
• Yesterday all of our coordinates added up to the same number.  But look at our coordinates today (2, 14), (4, 13), (6, 12)...Why is the sum not consistent?  It may take some thought but eventually students will come to the understanding that in the Groupon question, both parts were equal (x + y =25).  In this question the sum of x and twice y will give the same sum of 30 because that is the equation.  I plan to show students using the coordinates (that is why we had students list coordinates) that this is the pattern.  Explain to them that all points on the line will have the same relationship (even ones like (5, 12.5) which are not in the context of this problem).

As we begin to work on the investigation I will ask students to do a Think-Pair-Share and list as many important ideas as they can think of dealing with solving two variable equations based on what they have seen so far (see Collaboration: Building Communication and Student Ownership for an example of how this works in my classroom). Students will probably come up with many ideas, but they will fall within the following categories:

1. Two variable equations have many solutions.
2. The solutions can be represented by coordinates on a line (or curve) in the coordinate plane.
3. All of the ordered pairs that make up the line (or curve) must be solutions to the equation.

I explain to my students that they are going to continue to investigate these ideas by exploring some more linear functions that don't necessarily have a context (that is, their domain is all the real numbers).  I give students about 10-15 minutes to complete the investigation.  We may ither summarize the results as a whole class, or I may let groups share what they discovered.   In either case, here are some points of emphasis for this investigation:

1c) I will look to see if students are finding an organized way to list ordered pairs.  For example, start with a y-value and then find the corresponding x value.  Choose y-values in some logical order (y=1, y=2, y=3...).  Students will then realize that they can choose any y-value and find the corresponding x value.

1e) A line drawn (with arrows on both ends) illustrates the fact that we don't have to (nor can we) list all of the solutions we can use a line to show that there are an infinite number and the follow the pattern of coordinate pairs on the line.

2b) There is only 1 x-value that will make this equation true, namely, x= -2.  Since we are graphing the solution in the coordinate plane, we can use any y-value as long as the x-value remains x=-2.

2a/2c) Have students discuss the values they chose for x and y.  Students will notice that some choices lead to easier calculations and there is nothing wrong with making your life easier from time to time!  For example, in 2a the choice of y=1 or y=-1 leads to integer answers for x while y=0 does not.  Since students are trying to graph their solutions (1, -2) is easier to graph than (0, -3/2).

This Ticket out the Door requires students to match equations to graphs.  I allow students to work on this activity in pairs due to the understanding that can be developed through their conversations. Students have not yet been exposed to graphing lines by slope-intercept or another method.  So, they will rely on finding solutions to each equation in order to match the equation to the graph (MP8).

I encourage my students to choose x and y values that fall in the 10 x 10 grid for efficiency. Also, students will be developing a great deal of conceptual understanding from this activity. I will listen to and make note of student conversations during this closing activity.  This will help me to get a sense for how you to formalize the concepts of rate of change, intercepts, and linear functions in future lessons.