Problem Set: Creating Equations
Lesson 3 of 8
Objective: SWBAT to create equations and inequalities in one variable and use them to solve problems.
Today is the third day in which we engage in some group problem solving. Take a look at my previous two lessons for a description of how I set up group problem solving and use the guess and check strategy to move students toward creating equations.
Today's class opens with another ten minutes of group problem-solving time, as I instruct students to join their groups and try to finish up the "Ed's Book" problem. At a minimum, groups should have established a structure for guessing and checking a solution to this problem, and some may even have the solution. After about five minutes, my goal is to explicitly show students how to move from guessing and checking to creating equations. Some groups may have already achieved this goal, and others will need these notes for what comes next.
Once that happens, there are several terrific ways to think about this problem:
- Guessing and checking, we may come to the realization that for every 6 pages our guess increases, the difference between 1/2 and 2/3 of the book increases by 1 page.
- Solving the equation, we may see that taking the equation
(1/2)x + 84 = (2/3)x
and multiplying through by 6 - which is the LCD for 1/2 and 2/3 - yields
3x + 504 = 4x
- Or, by placing 1/2 and 2/3 on a number line, we may recognize that the distance between 1/2 and 2/3 is 1/6, meaning that those 84 pages Ed just read were 1/6 of the book.
All of these approaches are related, of course, and it's up to you and me as teachers to assess the understanding of each individual class and choose which of these interpretations will make the most sense to kids. In a perfect world, all classes would be able to assess the relationships between all three approaches, and when this happens, it's teaching gold. Other times, it might be best to really focus on the one approach that makes the most sense to kids, and then to allow them to try other problems while demonstrating the alternatives to smaller groups.
Once we have a strategy and a solution to the "Ed's Book" problem, I return the Mastery Quizzes that students took yesterday. This part of class will often be great - as quiz review can often be - because kids are so tuned in. We get to further debrief on the idea of racing the clock and test-anxiety. "It doesn't help to panic when I say you've got two minutes to solve a problem," I say. "Many of you got nervous about the idea of being timed, but then finished solving an equation well before the time limit."
I choose two equations that many students struggled with, and by referencing a few errors made by many students, I try to show the class that my feedback is real and responsive. It always draws students in, to specifically address an error so immediately.
We actually have some fun, because students are beginning to embrace the idea of mastery-based grading, and they know that this is an opportunity to learn. I can tell how well a class understands the grading system by their reactions to a quiz like this. If they are frustrated, then my work of building a growth mindset and showing them how to better next time continues; if they are ready to examine their errors and look forward to next time, I know that my work is starting to pay off.
Students will ask other questions. If we ran out of time a little early yesterday, they may even want to see the equations that they didn't get to see. I always take a little time to honor these requests. Today's class is lightly structured, and I've allotted some flexible time here so students can make these calls. If they're interested in seeing some examples, that's worth our time.
If not, there's a problem set to assign, and they can get started.
As a week-ending class, this one is loosely structured because each of my sections are in a different place. No matter how we spend our time, however, I want all classes to receive this problem set, so no matter what we've done, I call everyone to attention with somewhere between 10 and 25 minutes left.
I distribute the Creating Equations Problem Set to all tables. I project it on the screen and provide the instructions, which students must write on their own handouts. As I do so, I explain that this is the first assessment of SLT 1.4, and another assessment of Mathematical Practice #1. When I grade it, I'll look for exactly what I've written: for MP1, I want students to "demonstrate perseverance by trying every problem," and to "demonstrate making sense by making sense." We'll talk briefly about what evidence of perseverance looks like, and we talk about the qualities of work that makes sense: how presentation matters, and how guesses and solutions should be reasonable. There should not be anyone with a negative age, nor should there be any million-page books.
For SLT 1.4, the goal is for everyone to be able to create equations. "This is an important, but sometimes challenging, skill to learn," I tell everyone. "During the next week, you're all going to work on this. For now, try to write equations whenever you can. But if you can't, show how you can guess and check to solve each problem."
In some sections, there is time to start working, in others we've spent enough time looking at other things so they'll take this home. Either way, this will set the stage for Monday.
Next week, this problem set will occupy a lot of our attention. I go over specific problems during the next few lessons. Please look ahead for detailed descriptions of how I use some of these problems.