SWBAT find a solution that fits two linear conditions.

How many basketball shots were made at practice? Students begin to explore systems of equations intuitively in the context of tables and graphs.

10 minutes

I beegin class by working on the first section of the Free Throws activity together. I ask students for some possible combinations of shots that Camila and Jennifer could have made that would total 25 shots altogether. I also keep track of shots using an In/Out table with Jennifer as the In column and Camila as the Out column. I ask students to write an equation that represents this work. If they struggle, I ask, "What did you do to test the numbers you came up with for Question 1?"

Next, we will work together to make a class graph. I tell students to put the number of shots Jennifer makes on the x-axis because it might be helpful for them to look at the situation like that later, but I also make sure students understand that in this situation, neither variable is dependent on the other in a traditional sense.

If I decide to start class more quietly, you could give the first three questions from the activity as a Warm-Up.

30 minutes

Next, I let students get to work on Questions 4 through 8 of Free Throws individually or in pairs. I remind them that the second section has nothing to do with the first (for now). That is, Jennifer and Camila do not have to combine to make a total of 25 shots; Camila has to make three more than Jennifer regardless of how many shots are made.

Things I watch for as students work:

- I encourage students to keep track of their results in an In/Out table as they did in the first section. Again, the number of shots Jennifer makes should be the In value and the number of shots Camila makes should be the Out value.

- Students may struggle to write the appropriate equation. I like to guide them here by asking about specific numbers. I might ask, "What if Jennifer made 10 shots? How many would Camila make? Let's try that again, what if Jennifer made 20 shots, how many would Camila make? Now, what if Jennifer made x shots? How many would Camila have to make?" Then ask, "Ok, so write in words what you do to J in order to get C." This is the essence of Question 5 and should be a helpful intermediary step for students. This should help them see that C = J + 3 rather than J = C + 3 which is what students often want to write because they think about Camila making more shots so they believe the + 3 should go with Camila. Help them to see why the second equation doesn't work.

As students get to Question 9, I try not to help or guide them too much and let them wrestle with the problem themselves. There are many different ways to find the solution and I will be looking for students who use different methods that I can highlight during the whole group discussion.

20 minutes

As we discuss today's task, I make sure everyone is on the same page about two equations:

**c + j = 25 **

**c = j + 3**

If students are confused by the second equation, I will ask for a student volunteer to help explain.

Next, we move on to how different students solved Question #9. We'll keep up the discussion until students find several of the following methods:

- Students who used a table of values to find a common point.

- Students who used the graph of one equation to find a point that would satisfy the other equation (likely looking at the graph of c + j = 25 to find a place where c was three more than j).

- Students who used guess-and-check. You can allow a student to share out how s/he solved using guess-and-check, but try to guide the discussion toward using some of the representations we have been learning about in class in order to find the solution.

- Students who graphed both lines and found the intersection (save this one for last).

If a student can share out about graphing both lines on the same set of axes (either on the graphing calculator, or by hand), I will take the time to ask him/her some questions about this method. I want to elicit from students that each line represents all the points that satisfy that equation, so if you are looking for a point that satisfies both equations, it should be on both lines. I try to avoid a more traditional systems of equations approach that tells the students to look for the point where the lines cross. Instead, I allow students to think conceptually about the problem as they examine different points that would fit both equations.

Once the solution has been made known to the class, I have students think about the ordered pair in the context of each situation, then demonstrate plugging it into the corresponding equation. I might say, "If Jennifer makes 11 free throw shots and Camila makes 14, does that work for the first day of practice?" Generally, my students will realize that 11 + 14 is 25, but I typically show them the math with the equation as well. Then, we do the same for the second part where Camila makes three more shots than Jennifer. It is also helpful to show students how to check this point on their graphing calculator by entering the equations and using the Trace feature to find the appropriate point.

In today's reflection, I really want to emphasize the idea that the answer to today's task is a simultaneous solution. I want students to think about what the answer means in terms of the multiple representations they have been learning about, rather than in the specific context of a system of equations. I might ask them to complete an exit ticket in response to the following prompt:

**Write about the answer (11 shots for Jennifer and 14 shots for Camila) in the context of graphs or tables. How does it relate to the tables or graphs?**