Guess and Check is a Bridge to Creating Equations
Lesson 5 of 8
Objective: SWBAT employ the guess and check strategy as a method for learning more about problems, on their way to creating equations that represent those problems.
Opener: Examples for 1.4
Today's opening problems are examples of that can be used for SLT 1.4, which comes from CCS A-CED.1:
SLT 1.4: I can create equations and inequalities in one variable and use them to solve problems.
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 some classes, these will serve as review problems without too much effort on my end. For other classes, these two problems may take up most of today's lesson. Either of these scenarios is fine. My goal is for everyone to learn something this week. If kids can already do this, I know that we can move on to other challenges. If they cannot, then this is precisely where my attention must be.
Whichever of these two scenarios pans out in each class, I want to use this opener as an opportunity to show students how to organize a series of guesses and checks, and then to use that to develop an equation that represents a problem. My notes are summarized in this image. On the right, we start by making a table that records our guesses and each step in checking them (those guesses are in blue ink). After we get near the solution - by which point, some students have discovered it on their own - we generalize to algebra (that's in red at the top of the chart, where I've erased our first guess).
When we have an expression that represents the whole problem, we translate "you get 100" into "= 100", which yields an equation to be solved. At this point, I hope that some students have already used inverse operations to solve the problem. There are at least a few (and hopefully more) such students in every class. We can then note that the steps for solving the equation are those inverse operations.
The second problem is like the Ed's Book problem, so the solution strategy is similar, but with one important difference. In Ed's Book, Ed reads from 1/2 to 2/3 of the way through the book, and while he does so, the length of the book does not change. Here, on the other hand, Fran goes from being 1/3 of her uncle's age to 1/2, but the reference point changes. As Fran's age advances, so does Stan's. How is this expressed when it comes to writing an equation?
After giving students some time to try the two openers on their own, I ask for their attention to frame the week's work. I review the agenda and use today's class notes and explain that the Unit 1 exam is coming up on Friday. The exam will be organized by Student Learning Target. I explain that exams differ from quizzes. While quizzes are always open-notebook affairs, exams are not.
Students may bring one "Cheat Sheet," however. They may write whatever information they'd like on a sheet of letter-sized paper - yes, they may use both sides - and they may have that page at the ready during the exam. Of course, I'm just tricking them into studying. Over the years, it has become clear that students who prepare their Cheat Sheets will actually need them the least. It's the act of writing all that stuff down that has the most power. It's fun to watch students realize this over the course of the school year.
For now, they just feel like I'm giving them a great idea, and many students are excited: "Mister, I'm going to put so much on my Cheat Sheet, wait until you see it!" they'll say. Perfect, perfect.
With class framed as a chance to prepare for the end of the marking period, it then follows one of the two narratives that I've summarized in the opener. We may spend a lot of time studying those opening problems, or we may have moved through them pretty quickly.
If we need to dig into the first problem, we do. Then, I tell students that problems #3-5 on the Creating Equations Problem Set are just like that problem. I tell students to practice each problem by moving from guess and check to writing an equation. I continue to share the idea that guess and check is a viable option, and that if you can write an equation then try it.
These kinds of riddles are amazing in their power to get kids involved. The thing about guess and check is that once kids see how this chart works, no matter what their skill level, they can resist throwing out a guess. The feedback, and the gratification of getting closer to the answer, is instant. This means that students are engaged. Once that happens, they soon realize, "hey, there must be a better way. I'm just doing the same steps over and over again." That's when they have the opportunity to learn algebra.
If students already feel confident with problems like these, we breeze through the opener and it's time to get back to work. Just like yesterday, they should choose the one SLT on which they'd like to improve, then work accordingly.
As I asked them to do yesterday, today's exit slip is for students to write what they've done today, what they can do tonight, and what they can do this week to demonstrate the most mastery possible by the end of the week. It's the act of writing this down that I'm most interested in. Explicitly stating goals and the steps to get there is part of developing an ethic of hard work and a growth mindset. If I see that students are getting that idea, I won't even collect what they've written. If students are demonstrating a variety of levels of motivation, I'll collect the exit slips so I can see exactly what each of them are thinking.