Creating Linear Equations in One Variable
Lesson 1 of 8
Objective: SWBAT create linear equations in one variable, and use them to solve problems.
Not all students have completed this project yet (and therefore on time), but I take those that are ready. Yesterday, I made a big deal of the fact that there won't be class time for this today. Even thought this write up says I give it 3 minutes, that's really a maximum. The opener is already up, and I'm just circulating to get the project from students.
Submission rates have dropped dramatically from the last project as we begin to focus on quality, but we'll talk about that tomorrow. After school tomorrow, I'm hosting a help session for students who want to finish this work. Next week, students will have time to choose to finish this work.
Today, we officially begin our work on the standard
A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
For now we're only going to work on the linear part of this standard; the other functions will come later in the year. I say officially because we're not starting from scratch. We've been building toward this for weeks. To demonstrate the transition we're making, I use this opener today. It bridges the gap between number tricks and creating equations.
There are a few ways students may think about this problem, and I can count on seeing all of them in every class.
- There are students who have taken to heart the idea of working backwards, and who can solve this problem quite efficiently by doing just that. These students should be commended for their fine command of that strategy, and because they'll be done earliest, offered another challenge like, "What if you changed all those 3's to 4's - then what would the original number be?" or even beyond that, "What if you changed all those 3's or 4's to any number - can you generalize?"
- There are students who will try to write the algebraic representation of this trick, and if they're successful, they'll end up with an equation whose solution is the answer to this question. These students should also be commended for engaging in the important work that's happening right now. They are a step ahead today, because creating equations is the focus of our next few days of work.
- There are students who will employ a guess and check strategy, a subset of whom will do so quite vocally. These students should also be commended for having a good idea of where to start, and I am prepared to ask them questions like, "If you started with 9 and got 8, then what's an appropriate number to guess next? Why?" or "Can you spot any relationships between the starting and ending number.
I pay close attention to the strategy that each student uses, because this will inform the kind of help I offer each student over the next few days. On a whole class level, it's often the guessers and checkers that form the majority, so I guide the discussion in that direction. "Guess and check will be an important tool as we learn to create equations," I say. "It's always a viable option when you're not sure where to start on a problem, as long you make educated, rather than random, guesses about what to do next."
Group Problem Solving
With new groups established, we review the student learning target:
SLT 1.4: I can create equations and inequalities in one variable and use them to solve problems.
Here's how that works. I ask for a volunteer to read the SLT aloud. Once someone reads it, I ask students to shout out the most important words in the SLT. I underline each word they say: equations, inequalities, variable, solve, problems. When someone says create, I circle it. "That's the verb," I say. "That's what you're going to be doing now. Notice that this SLT doesn't just say you have to get the right answer. It says you have to create equations. That's what we're going to work on over the next few days." We also note that the SLT says one variable, which means that, for now, each problem we solve will have one unknown.
In order to practice this new learning target, I say that we have two problems to solve. I enjoy telling each class that we're going to "work on a problem called Vanessa's Raise, which is about a woman named Vanessa who gets a raise, and another problem called Ed's Book, which is about a guy named Ed who is reading a book." In a world rich with contrived introductory algebra problems - all of which serve a clear purpose in an instructional sense, but might bend the laws of a kid's everyday reality - this kind of humor draws students in by making it ok to suspend disbelief and play with the math.
I use the fourth slide of today's notes to introduce the structure for today's class. It is one that will grow more common as the year goes on. I explain that I'm going to grade everyone right now by observation. "What I'm looking for is that each you is working to help your group make sense of each problem, and that you're helping your group persevere," I say. We discuss what this might look like: asking questions, keeping everyone involved, explaining your thoughts, writing things down, and not giving up are all possibilities. "The answers to these problems won't be obvious right away," I say, "so make sure you're doing your best to make sense of whatever you can."
I distribute one copy of Vanessa's Raise (from CME Algebra 1) to each table, and I project it on the screen at the front of the room. Everyone is working in groups of 4, although there might a group or two of fewer students if that's how the numbers work out.
I say, "I'm going to give you 10 minutes to work on this as a group. Persevere, and try to figure out whatever you can. In 10 minutes, I'll give you a hint about how to solve this problem.
The Hint: How to Guess and Check
Great conversations happen in these first 10 minutes, and I often see a new and excellent side of many students in this context. Usually, I don't have too many groups who can say with confidence that they've solved the problem, however, and even fewer are able to create an equation to represent it. That's why I take this opportunity to introduce a highly-structured form of guessing and checking, that will help students develop the ability to create equations.
I say that I'm going to give a hint by showing everyone a wrong answer. This gets everyone's attention. I say that I'd like to guess an answer to this problem and check it. "When I guess, I'd like to choose a realistic number that's easy to work with it," I say. "Is it reasonable to say that Vanessa might make $10 per hour?" We decide that it is, and I continue, "I'm choosing this number because it's easy to work with. I think it's easy to multiply by 10." If students hesitate, I might compare this to multiplying by $13.75 an hour, and they get the idea.
I also introduce the idea of "before" and "after", which I'll usually find in one form or another on the work of a student or two. If students have had this idea, I'll give them credit by saying that I got the idea using these words from those students, and I'll write those two words on the board. So before and after what, then? Students consider this, and realize that Vanessa has been offered a raise. She was making one amount before the raise, and another after. "This is something that I hope will happen to all of you," I say. "And when it does, I'd like for you to be able to analyze the situation, just like Vanessa does here." I record my guess and check like you can see here, and you can imagine how, after another couple of guesses, it will lead to the correct answer. Once this structure is set up, kids are excited to try their own guesses. My job is to circulate and make sure that they're still discussing the problem, recording their attempts in their notes, and having good ideas about what to try next.
Two More Points to Make, if Time Allows
- When I see that students are on the right track, I might take the opportunity to further develop functional reasoning by analyzing a pair of wrong answers to see how guess and check works. By comparing inputs and outputs, students can see what a reasonable next guess might be.
- I also like to emphasize the power of what Vanessa is doing in this problem. I ask everyone how excited they would be if their boss offered them a raise of $2 per hour and 8 hours per week. Does that sound exciting? Then we expand it out: how does $200 per week sound? How does $800 per month sound? How does $9600 per year sound? Changing these contexts can really emphasize the power of thinking like this, and might better inform someone as to just how excited they should be. It's a fun conversation to have.
Debrief and Homework
The debrief is so important when we're doing group work like this, and this is what the class will look like more and more as the year goes on, so it's important to establish it now.
Two key points so far:
- What role did critique play in your group? We just finished a project about this. So what can we make of it?
- It's ok to be wrong! In fact, you can learn a lot from it. That's why I shared by wrong answer - and I think that was exactly the thing that helped some groups solve the problem.
Homework is in the textbook and looks like the work we did today. You're given words, and have to write and equation or an inequality. We'll see how this goes.
To close out, I remind students of their homework tonight. It's a series of number sentences to be translated into equations, and it it great practice for all that we're doing next. I emphasize how useful this assignment will be for anyone who gives it a try.