I start class by having students read the first question from Get to the Point! aloud. We only focus on Question #1 to start and I let students know that they need to have a written explanation of what window company they would recommend and why. I also tell them they will need to have as much evidence to support their answers as they can and they should include a table, graph, and equation if possible. Like I said before, my students have already seen problems similar to this one, and this is a good opportunity for them to spiral back to that learning.
Next, I let students get to work and circulate around the room to check on progress. It is interesting to see if students start with the table or the graph and what that means about how they understand the problem. Some students will be confused about making a recommendation because the problem does not state how many windows Carlos and Clarita have. I let students know they may be making different recommendations based on different number of windows and this is something they have to sort out. I also keep an eye out for representations that can be shared out when we get to the whole group discussion part of the lesson. If a student finishes early, I have him/her start putting some work up on the SmartBoard that we can look at together as a whole class.
Once students have written recommendations to Carlos and Clarita about which company they should go with to have their windows cleaned, I bring the class back together to take a look at the representations they created to support their arguments. I like to have different students share out their work. I might have one student share out his/her graph and a different student share out a table and an algebraic representation.
I like to start by looking at the graph. This gives me the opportunity to review some key ideas with students that they have previously learned, like where they can see the slope and the starting point in the graphs and the equations. I also talk a lot about the simultaneous solution and how the point (5, 100) is the only value for x and y that will work simultaneously in both equations. Students can explain how they know that Pane-less is the more cost effective option if you have less than 5 windows and Sunshine is a better choice financially if you have more than 5 windows.
Next we look at the table. Again, students can see how one company starts out costing less, but then beyond a certain point (5 windows), starts to cost more. The key part of the table for today's lesson is what if Carlos and Clarita have x (or n) number of windows. I ask students what math they have to do to determine the cost for each of the companies. It's usually fairly easy for them to move to equations for each company here.
I now talk about how if both of the companies charge the same amount at some point, then both of the equations can be set equal to each other. This concept requires a leap in thinking for some students. I try to help them along by saying something like, "well at some point we know that both companies charge the same price for the same amount of windows. At this point, both of their y values (or out values) will be the same. If both y's are the same, can we take the expressions on the other side of the equations and set them equal to each other and solve."
Once we solve for x in this way, I go back and talk again about how the ordered pair (5, 100) is a simultaneous solution for both equations. And, of course, it is the point where the two lines cross on the graph.
With the exception of setting the two equations equal to each other, most of the above should have been a review for students. Now we move to the importance of being able to solve systems of equations exactly. In my class, we read Question #2 together as a class and take a look at our graph with all four constraints lines on it (from earlier in the unit). We figure out that Carlos wants to know exactly where the start up costs and space lines cross. On our graph, we can see that this point is not clear. We have a general idea of how many cats and dogs would fit, but we need an exact answer. Students sometimes need a reminder that we're taking the inequalities of the situation and making them equations to find the exact point of where they cross. For me, this is one of the main ideas of the unit, so I make it a point to revisit it frequently.
Now we take a look at the two equations and I ask students what's different between this system of equations and the previous set we looked at (in the previous set, both equations were already solved for y). I ask students what we can do to make these equations more like the ones we've seen before so we know what to do with them. In this case, some students may recommend just solving for one variable and plugging it into the second equation. This is great! Or, students may begin by solving both equations for the same variable as if it were the y in the previous example. For these numbers, either option will work, but I try to highlight both methods to students (as they'll start solving for just one variable soon).
Lastly, we look at the third problem together. The third problem is a little trickier because none of the variables are easy to solve for (without creating fractions). Try to get students to point out that both equations have 20y (or 20x if they've put dogs on the x axis). I try to get students to see that if they solve both equations for 20y, then because the coefficients are the same, they can again set the equations equal to each other. In other words, no matter what y comes out to be, the left sides of both equations will always be the same.
In my experience, solving systems of equations using substitution takes some time for students to get the the hang of. The students I work with, in particular, usually need some time to practice (and sometimes it's helpful to generate a list of steps with certain students). I find homework to be a useful and practical way for students to spend some time practicing solving by substitution. Questions #5-8 on Systems 2.6.pdf is a good place to start. The questions also ask them to check their work by graphing, which I think is a great way to solidify the idea of a simultaneous solution. (Be warned that the graphs do not all use neat and clean numbers). Of course, there are lots of worksheets out there for practicing substitution.
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