Today's opener is a number riddle with three variables. Some of my classes have seen problems like this before but others haven't. If they have, I'll stand back a bit to see if students can get through the problem on their own. If this is new to a class, I'll use this problem to make sure students can see the elegance of using substitution, when it works.
I give students the first five minutes to try the problem. I circulate, asking students to explain their thinking to me and to each other. If they're stuck, I tell them to start by defining variables, and I watch to see how comfortable they are doing that.
After those five minutes or so, I model how to solve this problem on the board. I think out loud and elicit each step from students. I'll say I need three variables, "Is it ok if I use x, y, and z? Do I have to use x, y and z?" Then I'll ask what each letter stands for. If someone says that x is the greatest number, I'll say, "Ok, that's fine, but does it have to be?" I want students to understand that sometimes defining variables can be arbitrary. "But once we decide that x is the greatest number, we have to stick to that," I'll say, and we move forward from there.
We take the problem one sentence at a time. As you can see in this photo, I write each constraint on the board, and then we translate each into an equation. I want students to be comfortable with the math; I also want them to see how to organize the page when they're solving a problem.
Once the equations are written, I try to see how quickly we can run through the solution by substitution. It's on the right in this photo of the board. Can you construct the steps that we took here? The idea is that I want students to feel like they've got this. When they see how efficiently this strategy can be run on a problem like this, they're all the more sold on the idea that this is a worthwhile tool.
I tell the class that I'm impressed by how much they know about substitution so far, and that I'm glad they're working hard to learn to use this tool. I return the Mastery Quizzes from the previous lesson so everyone knows where they stand. Then I say that we're going to look at a different sort of problem today.
Sweet Problems with Two Variables
As I distribute the Sweet Problems in Two Unknowns handout that we'll use today, I say that we've already started on today's work. "We looked at one problem about sweets at the end of yesterday's class," I say, in reference to the Pie Day problem. I tell everyone that they will see some similarities between that problem and today's work. The purpose of this work is introduce a context for solving linear systems by elimination.
This problem is adapted from one in NCTM's Mathematics Assessment Sampler 6-8. It is a great problem because the barrier to entry is very low, the values of the solutions are not too difficult to deal with, but they're not whole numbers, and there is a lot of room for kids to play with the problem. In my experience, students will naturally gravitate toward using the most advanced algebra they know: some will stick to guess and check, some will try to make substitution work, and others wonder how to graph it. My most important role is to stand back, look for the mathematics that my students develop, and then decide how to use that.
For example, some students will come up with the idea that a cupcake must cost 25 cents more that a candy apple. It's important not to rush toward congratulating a student on this idea, but to say, "Aha - that interesting! How did you come up with that?" Students will reveal the most useful ideas when prompted like that. Of course, if we create two equations and then subtract one from the other, we'll get the same result (beginning to develop that idea was the purpose of yesterday's pie problem), but kids aren't often thinking on that particular algebraic level when they come up with the idea. That's a fascinating thing about solving systems by combining equations and eliminating variables: unlike graphing and substitution, this method is rooted in some pretty intuitive moves. (In fact, students made a similar discovery two weeks ago. See the opener of this lesson.)
I give everyone about 20 minutes to work on this problem. I tell anyone who finishes the problem to get going on the other side of the handout. But it's important for everyone to really have time to come to a solution on their own, and in my experience almost everyone will. When I see that everyone is finished up on the first problem, I guide everyone through a solution by substitution. Some students have already done that, and others really wanted to see this. I work pretty quickly: define variables, write equations, isolate the "one candy apple" in the first equation, and then use substitution. Then I observe that substitution definitely works here, but not quite as "simply" as it did in today's opener. That's an observation that will carry us forward. Once we have another tool for solving systems, we'll want to be able to figure out which one is best. Being able to observe the relative strengths of one is a way to build toward that.
Apples and Oranges
After the cupcakes and candy apples, it's on to some healthier sweets. This problem is also adapted from the NCTM Assessment Sampler. I don't make any effort to finish this problem today. I talk to kids and encourage them to do as much as they can. I want them to be comfortable with a problem like this, and I've found that filling in this table helps them understand an important distinction. In the family of linear systems problems involving two items of different value, there are two sub-categories: problems in which the quantities are known and we're looking for two unknown values (often money, but also weight, speed) and problems in which values are known and we need to find quantities. One important purpose of this handout is to help students distinguish between the former (on the front) and the latter (on the back).
Problems #5 and #6 are challenging, and will help us dive into our study of elimination over the next few days.
You'll see that today's slides include some extensions and places we'll go next. I probably won't get to these today. They'll be ready for class two days from now, and we'll pick up right where we left off.
With just a few minutes left in class, I say, "There are better ways to solve problems like the candy apple and cupcake problem. We're going to spend the rest of the week studying a new method for solving systems of equations. You'll see it tomorrow, and then we'll use it for the rest of the week."
If we have time, we'll go back to the pie problem. It was introduced yesterday, and now we take another look. I ask students to compare this problem to today's work, as a way to solidify the distinction that I wrote about above. I hope that students can see that the pie problem is really very similar to the cupcakes and candy apples problem.
I say that tonight, everyone should finish as much as they can on today's apples & oranges problem.