The purpose of today's opener is to help students see the relationship between solving linear equations and solving linear inequalities. I give students a few minutes to get started, and I circulate to see how they're doing. Nearly all students can solve #1, a two-step equation, and most are confident with solving #3 as well. Some have already come to understand that there's little difference between solving equations and inequalities, and before we get to discussing the one difference between the two, I want to dispel some of the anxiety that surrounds those inequality signs by showing kids that this really is the same thing.
I don't go over the steps for solving the first equation. I just ask what everyone got, and record the answer, x = 6, on the board. I say, "Now, what this solution means is that if we replace the x with a six in this equation, then the equation will be true. That's what it means to solve an equation." Next, beside second question, I write x < 6, and say, "The steps to solve the inequality in number two are exactly the same. It's just that the solution will look like this." I point to my solution, pause, and continue, "How many numbers are equal to six?" We all agree that only six is equal to six. So in an equation there's just one solution. "But how many numbers are less than six?" Someone is always quick to shout "Six!" and then we recall that all of the negative numbers would work as well. So there are a lot of number that are less than six. You might even say an infinite amount. "The key here is that any number less than six will make this inequality true," I say, before asking for such a number. We take suggestions and substitute them in for x, noting that each makes the inequality true. This whole exchange takes maybe a minute.
Then we take a look at problems #3 and #4. There are two reasonable methods for solving the equation in #3, and I don't prioritize one over the other. Instead, I elicit suggestions from students, and it usually doesn't take too much effort to end up with the two solution strategies you see here. Of course, the right-hand example has the flaw of employing common sense instead of algebraic manipulation in the final step, which offers another great opportunity to start a conversation. It's neat to hear students express preferences for one or the other, and I think that this activity helps to open up the idea of algebra as a flexible language that can be molded in different ways.
When we move to the inequality version of the problem in #4, however, simply "bringing down" the less than sign yields two different results. The question I pose to the class is, "Which one is right?" If they have trouble following me, I try to get them to acknowledge that it can't be both, and it must be one. "There are numbers that are greater than 7 and numbers that are less than 7, but no numbers are both greater than and less than 7, right?"
This is where we check our work. I have students choose one of each sort of number, and substitute them into the original equation. We see which one works. If a class is getting this quickly here, I'll dig into the solution step of "turning around" the inequality sign when we multiply or divide by a negative number. In other classes that are just hanging on, the takeaway is that it's always worth it to check an answer to an inequality. Either way, I tell students to keep this in mind as we continue to work with inequalities.
We spend a short amount of time today digging back into the group problem solving that began in yesterday's class. My goal is for every class to advance by gaining one tidbit of knowledge today. Precisely what that one tidbit is depends on how the class did yesterday. I cover my introduction to this group problem solving in yesterday's lesson; if you haven't already, please take a look at that lesson for an overview of the work that continues here.
That lesson goes differently every time, so there are variety of points from which we may be starting today. There may still be some algebraic generalization to do with Vanessa's Raise, we may be ready for a clean start on Ed's Book, or if things went very quickly yesterday, students may already be started on that second problem. Whatever the case, the purpose of these two problems is to establish a structured form of guess and check, and then to use that form to develop equations that can be used to solve those problems.
In the "Ed's Book" problem, we can once again apply the "before and after" structure: there was the amount of the book that Ed had read before reading those 84 pages, and amount that he'd read after. If they need help, I'll help students set that up. Setting up guess and check is one possible takeaway for today's lesson.
For other kids an analysis of wrong guesses is what gets them going. It would be impossible to overemphasize the way that introducing such a line of thinking can draw students in. A structure like this serves struggling students by showing them how to be self-sufficient, and it furthers the thinking of advanced students by framing another linear relationship (ie. if my guess increases by this much, then the result changes like this.)
Some kids think we shouldn't guess and check if the goal is to create equations. I explain that guess and check is precisely what helps us to create those equations! What we're working toward is the ability to take the unknown - the number that we keep guessing at - and replace it with a variable. We can then take that variable and place it in an equation. Again, the precise pace of how this knowledge rolls out to students depends on the class. Some students were ready to dig into this idea yesterday, for others it might be the thing to examine right now, and for others, that will be the focus of tomorrow's lesson.
The latter half of today's class is spent on a Mastery Quiz for SLT 1.1. I want to see how well students can solve equations and inequalities, and show their steps, in a short amount of time. For those who have achieved mastery, this is a chance to show it. For those who are still working toward it, this is a chance to build a little urgency and facility with these skills, under the pressure of clock.
Here is the quiz: Mastery Quiz SLT 1.1. I project it on the screen. On the first page are the two learning targets I'll assess here: 1.1 and MP1. After instructing students to take out a sheet of paper and write a perfect heading, I say that I'm looking for and assessing two things with this quiz. I want to see if they can solve each equation and inequality correctly, and therefore demonstrate their level of mastery on SLT 1.1. I also want to see that they can clearly show their work, which demonstrates that they're persevering and making sense of each problem.
I ask if everyone is ready, then I put the first equation on the screen. This quiz moves from a Level 1 equation to a Level 7 equation over the course of 15 problems, I put them up one or two at a time, each with time limits. The allotted time for #'s 1-14 adds up to 18 minutes; to allow time for #15 and instructions, you'll want a little more time than that. If anyone finishes solving an equation early, I say that they can record some of the properties that justify their solution in the margin.
It's always interesting to see how different students respond to a race against time. For some, this is finally the thing that gets them to focus all of their energy on their work. For others, it's quite stressful, and when we debrief this quiz tomorrow, I'll want to help those students think about how panic probably doesn't help, and what we can do instead. In any case, this is a strategy to be used sparingly. During the quiz, there are always some students who say they love it, and some will say they hate it. In terms of skill development and test-anxiety-management, this structure yields positive results.