For much of the last two weeks, I have not rushed to solutions for the daily opening problem, but that won't be the case today. I want to make sure that students see a neat and complete solution by substitution.
At this point, many of my students can solve this problem on their own, and a few need some help. I give everyone a few minutes to work, and I encourage groups to discuss the problem. If I see that one student in a group is successful while another struggles, I'll refer them to each other.
If I see that each group has developed an accurate solution, I'll skip the full-class discussion of this problem today. If we do decide to put a solution up on the board, I'll try to move through it quickly. I want kids to feel like they're getting substitution, and that it's an intuitive, efficient tool for solving problems like this. Moving quickly helps to develop that sense.
No Grades, Here's Why
As students worked on the opener, I returned the SLT 5.2 Mastery Quiz from the previous lesson. In most of my classes, I saw that students hadn't fully mastered SLT 5.2 yet, so these quizzes are ungraded. Instead, we can use these problems to continue to move toward mastery and to necessitate the use of substitution, and that's what we'll do today. I elaborate on my thinking about grading and using this quiz in this narrative video.
Specifically, I can see that my students are pretty comfortable graphing lines in slope-intercept form, but that some still have trouble with lines in other forms. For the problem-solving application that was the second half of the quiz, kids definitely still need some practice.
Quiz Debrief: Graphing vs. Substitution
As I describe in the video, the problems on this quiz are designed to illustrate the usefulness of graphing and substitution as complementary tools for solving linear systems. On the third slide of today's lesson notes, I frame the quiz debrief by asking students how many intersections are on their graph, and which of these are "easier" and "more difficult" to find by graphing.
As you can see in this image, some of the lines on the front of Friday's quiz intersect at lattice points, but others do not:
Graphing lines in slope-intercept form makes it pretty easy to find those lattice points, but it only allows us to approximate the non-integer coordinates of other intersection points. Students have enough background knowledge to understand that substitution can provide a better way to find the exact coordinates of an intersection point.
We look at the graphs, and work together to categorize equation pairs. For example, y = -2x + 10 and x - y = 5 intersect neatly at (5,0), so that one is easy. But graphing x - y = 5 and 5x + 15y = 90 can only help us approximate a solution to that system, so substitution will be necessary.
I ask if everyone agrees that we'll need to use algebra. I provide one example before giving the class a few minutes to work out the other two non-lattice solutions. As students work, they're able to assess for themselves the accuracy of the approximations they made on the quiz. I circulate to troubleshoot.
When they're done, I reveal the solution image above, and ask everyone how they did.
Now that we've practiced solving the systems that comprised the first half of the quiz, we can apply these skills to a context and dig a little deeper into the uses of graphing and substitution.
The second half of yesterday's quiz was to solve graphically the problem on the eighth slide of today's lesson notes. Almost all of my students struggled with this problem. I say that we're going to work through this problem together, but that first I'd like to make a few observations about graphing and substitution as solution methods. I reveal these "Really Big Ideas" one at a time on the fifth through seventh slides of the lesson notes.
(I should also note that I use today's agenda to build toward this part of class. When kids arrive and they see how I've written "REALLY BIG IDEAS!" on the agenda, they want to find out what I mean by that. Some kids asked about this at the start of class. Now we jump in.)
I summarize the ideas of "easy" vs. "difficult" and "approximate" vs. "precise" solutions that were developed in the previous section. Then I make the claim that, while substitution can get us a solution to a system, we need a graph to get a more complete understanding of a situation, and to see what happens around that point of intersection. "I'd like to illustrate each of these points by working through the break-even problem from yesterday's quiz," I say.
We run the algebra first. I ask students to help me define variables and write equations for each part of the problem, and that gets us this far. Now, to write and solve the equation
5x + 130 = 9.5x
doesn't take much. Just like that, we have the solution:
x = 130/4.5 = 28.89
"So that's that," I say, "if you want to find the answer. But what if Felicity wants to be able to make plans for selling more than 28.89 t-shirts?" I pause, because no one ever really talks about 28.89 t-shirts. "What if she wants to know how much debt she'll have after selling 10 t-shirts, or how much profit she'll make after selling 50?" That's what graphs are for.
As we've done previously, we spend some time deciding how to scale each axis, and then we talk about how to express a slope of 5 when the x-axis is counting by 1's while the y-axis counts by 10's. All the while, I do the work I'm asking kids to do on the board. I try to continuously model craftsmanship, and to give students an idea of what a neat, useful graph can look like. As you can see in the image, we came to the understanding that every two shirts cost $10 to produce, so we can move "over" two spaces (two shirts) and "up" one ($10) to get successive points.
Similarly, it helps to note that Felicity will sell 5 shirts for $47.50 or 10 for $95, and a few points like that will be enough the sketch the other line. The "in-between-ness" of these numbers further illustrates the idea that graphing by hand can't yield a perfectly precise solution on this graph, but that it will get us close. Once both lines are graphed, we discuss how well the intersection matches our algebraic solution. If "x is in-between 28 and 29" and "y is in between 270 and 280," then we've done pretty well for ourselves.
At this point, there usually isn't much time left, so I elicit observations about the graph, and I ask questions like those noted above: "How much debt will Felicity have after selling $10 t-shirts?" We see that the distance between the two lines provides at least an approximate solution. As for selling 50 t-shirts, we'd need to rescale the x-axis, and I can briefly assess student sentiments by seeing how confident they feel that they'd be able to do that.