Solving Problems in Two or More Unknowns

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Objective

SWBAT practice solving problems with two or more unknown values. To start, they'll use Guess and Check, which will soon lead to more formal algebraic methods for solving systems of equations.

Big Idea

The reason systems of equations exist as an algebra topic is that they're a tool to be used when a problem has more than one unknown variable.

Opener: Ways to Make 50 Cents

10 minutes

I give students the first few minutes of class to work on the opener, which is on the first slide of today's lesson notes, then I use the document camera to give them a chance to share their work. As you can see in all three student work photos in the resources, we have a ways to go in each case.  I open up the floor for comments and questions, which usually leads to a lot of disorganized shouting of missing examples.  This, of course, mirrors the disorganization on the board, and gives me the chance to stand back, smile, and then point out that what I'm most concerned about is the organizational structure of everyone's work.  "I didn't give you this problem because I don't think you know how to make fifty cents," I say.  "I gave you this problem because it's challenge to get organized and to make sure our list is complete."

Teacher Note: Look at the photos of student work on the board to get an idea for how I start to help student organize their work.  I want to give them ideas, but not answers.  We're not going to finish this problem today, and that's the point.  I've given students something to chew on, they'll keep it in mind as we move through the rest of today's class, and we'll return to it tomorrow.

As will be a recurring theme in the coming weeks, I'm not too concerned about getting the right answer.  In fact, I don't share it unless kids come up with it on their own.  After at most ten minutes, which includes some discussion, I say that this will be the opener tomorrow, and we'll go a little deeper into it then.

I'll use this opener structure frequently over the next few weeks, providing a problem one day, not quite finishing, and revisiting it the next day.  When students are left to turn a problem over in their heads, good things happen.  I am often surprised to find which students approach with their solution.  By specifically de-emphasizing the solution, and not rushing into anything, I find that this is a great way to emphasize process over solution.

New Unit, New Learning Targets

3 minutes

As we transition from the opener - at least until tomorrow - I take a few minutes to introduce Unit 5, which is about Systems of Equations.  I say to students what I mentioned in the opener: a system of equations is a tool that can be used to solve problems with more than one unknown value.  "In most of the problems you've solved this year, there has been one variable, one unknown, one answer," I tell my students.  "Now we're moving beyond that.  Like the opener, which has three variables - numbers of quarters, dimes and nickels - and many solutions, the problems we look at now will have at least two variables.  In order to solve problems like these, you will need to use systems of equations."

Then I show students the first learning target of the unit.  The first two Student Learning Targets of Unit 5 are posted on the back wall of my classroom.  To get started, I project the first one at the front of the room:

5.1: I can use guess and check to solve problems with two or more unknown values.

I ask for a student to read the learning target, and then I ask if anyone has used this strategy before.  Of course, a bunch of hands shoot up.  I give students a chance to share their thoughts, and then conclude by saying, "if you're a thoughtful problem solver, then you've definitely used some form of Guessing and Checking."  My goal is to dignify - and eventually formalize - this important strategy.  To guess-and-check well requires number sense, common sense, and attention to patterns.  We'll see all of this over the new few days.

Collaborative Problem Solving: Ducks and Cows

25 minutes

I try not to say too much about what "Guess and Check" means.  Sure, it's self-explanatory enough, but there are best-practices that will develop over the next few days.  I make sure to say nothing of those practices now, though.  I want students to have room to have their own thoughts today.

New Structure: Collaborative Problem Solving

Here is the two-sided Collaborative Problem Solving handout that students will use regularly over the next few weeks.  At the top is space to record a problem.  On the front, there is room to solve that problem and to receive feedback from a classmate.  On the back is some space for "algebraic analysis."  We'll get to that later.

I distribute the handout and describe what's going to happen.  I'm going to put a new problem on the board.  Everyone is going to have six minutes to work silently (that's one minute to copy the problem, and five to work).  "After that time, everyone will have a chance to share their work, but to start, you have to try it on your own," I say.

Once that time has passed, students will pass their work to the right, and they will provide some feedback about their classmate's work.  After a few minutes of that, I ask for volunteers to share their work on the document camera.

The Problem: Ducks and Cows

The problem to be solved is on the second page of today's lesson notes.  It's a classic.  I think it's a great problem because it's easily approachable and because it's ridiculous.  This is not a problem that pretends to be "real world," and that can be a nice way to frame it with kids.  I'll often say something like, "I think we can be pretty sure that no farmer has ever gone out and actually wondered about this sort of thing, but this is certainly going to help you learn and think about some important algebraic ideas."

Another benefit of this problem is that it can be repeated with different numbers of animals as we learn other methods for solving systems, and that it's easily adapted to other animals, like the spiders and ants on tomorrow night's homework.

Debrief

After students work independently and then have time to discuss their work, I invite volunteers to share their work on the document camera.  Whatever the quality or accuracy of the work, it always starts a great conversation.  Here is what ideal use of guess and check might look like.  It's organized, and shows that the student solving the problem had a plan.  As this student explained her work to the class, she added the arrows while she explained her thoughts as she moved from one guess to the next.  The strategic thinking required to make efficient use of guess and check will come in handy as we formalize some algebraic methods for solving problems like this.  Seeing structure and patterns here is going to facilitate the rest of the work of our systems unit.  Along those lines, it's valuable to recognize the arithmetic sequences that appear in well-organized work. 

That will likely be it for today, depending on our pace.  If there is extra time, we may begin to discuss plotting possible answer pairs on the back of the handout.  For example, what are all the possible numbers of ducks and cows that would sum up to 12 total animals?  If there's just a little time, I might just give students a few minutes to get started on the homework.

A Good Joke Not to Be Missed

  • Q: How do you turn a duck into a cow?  
  • A: Give it two more legs!

This one always has kids in stitches.  The key is to deliver it in the mathematician's deadpan.

Homework: Two Problems

5 minutes

With five or so minutes left in class, I distribute tonight's homework.  It consists of two problems that can be solved by guess and check (or any algebraic method for solving systems).  To introduce the problems, I compare them to Ducks and Cows.  I ask students what we were trying to figure out in the previous problems.  We're not formally defining variables yet, but by having students say, "we were looking for the number of ducks and the number of cows," foundations are laid for that.  Then I ask how these problems are similar.  Students quickly recognize that instead of finding numbers of ducks and cows, they'll be finding numbers of dimes and quarters and numbers of two and three point baskets on these two problems.  I make a big deal of the idea that it's all really the same thing.

If we get here early, I can give kids a little time to work on this.

As we run out of time, I tell students to make sure to use some of what they did today as they work on this homework assignment.