From Guess and Check to Graphing Systems
Lesson 6 of 20
Objective: SWBAT to solve a system of linear equations by graphing, and to solve problems by creating and graphing linear equations.
There are four learning targets to master during this five-week Systems of Equations Unit, and I loosely organize each week around those four targets:
- Week 1, SLT 5.1: I can use guess and check to solve problems with two or more unknowns.
- Week 2, SLT 5.2: I can solve a system of equations by graphing, and I can interpret the intersection of two graphs in terms of a context.
- Week 3, SLT 5.3: I can use algebraic substitution to solve a system of equations.
- Week 4, SLT 5.4: I can solve a system of equations by elimination.
- Week 5 is for mixed problem solving, review, and mastery.
We got started on graphing last week, and this week I expect students to master it. Similarly, we'll start moving toward substitution later this week, before it takes center stage next week. Each week, students will master a strategy before the limitations of that strategy are used to motivate us toward the next.
Hopefully, all students have demonstrated mastery of Guess & Check, but a few kids still have a little work to do on that front, so today's lesson scaffolds for that. As students arrive, there's a new seating arrangement posted on the board. Students are grouped by how well their work from Friday's Mastery Session demonstrates that they've got Guess & Check down, and after the opener, groups will get different work based on those results.
My goal is to give all students a sense of success on SLT 5.1 before really getting into all the algebra that a systems unit entails. From there, it's all about using what we know about guess and check to digging deeply into how graphs work.
Before moving on to mixed practice, today's class opens with this graphing exercise. By the end of Friday's Mastery Session, some of my students had time to work on some graphing problems, but many didn't get that far, and everyone can use this practice. For anyone who did get that far, I return their graphing work from Friday and say that this opener can be done on the same sheet of graph paper. Everyone else gets a fresh sheet of graphing paper to use here.
This is an "opener" that serves as a mini-lesson or as guided practice. I hope that students are confident enough to graph these lines on their own, but many will need a push in the right direction. In general, a systems unit is an opportunity to review and apply skills that were covered previously, like graphing lines and all the steps of solving linear equations. As I'll show later, students will get a chance to raise their mastery grades on prior learning targets.
There are four lines to graph, and in a neat little nod back to the handshake problem, I ask students how many intersections they expect to see in a graph of four different lines. Just like four teammates will exchange a total of six high-fives at the end of a game, there will be six intersections here. These lines are carefully chosen so that all but one of the intersection points consist of an integer pair, and so that there are intersections on both the x- and y-axes. (Later, those non-lattice points will provide one motivation for using algebraic substitution to solve for a variable, rather than graphing.)
Like I did at the end of last Thursday's class, I use the document camera to model how I go about completing this task. As students work, I carefully set up my axes, which helps my less confident students get an idea for what quality craftsmanship can look like. Then, I take on a gentle tone of showing students that they already know how to graph lines. I encourage students to try to stay a few steps ahead of me, but this task is geared mainly toward students who need the help. Going one line at a time, I point to the y-intercept in an equation, plot it, and use the slope to find more points. My work is color-coded, and colored pencils are available so my students can do the same.
When it's modeled for them, students take pride in producing such good-looking work, and these notes will make a nice page in student notebooks. They'll be able to reference these notes during tomorrow's quiz, which is just like this opener.
With the week framed like that, students finish up their graphs and I return all work that I received on Friday. I tell students what I've written about above, that today's task is based on what I've seen in their work on SLT 5.1. I post the learning targets on the third slide of today's lesson notes, and say that this week we're moving from guess and check into solving problems in two unknowns by graphing.
For Students Who Demonstrated Mastery of SLT 5.1:
These groups receive this "From Guess and Check to Graphing" handout, and I generally leave them to figure out the instructions by working together in groups. I expect that everyone will be able to define variables. The first sentence of each problem is easy enough to translate into an equation, and to graph, but the second is more challenging. In the first problem, for example, students can take the sentence, "Michael has five more dimes than quarters," and write the equation q + 5 = d.
Here, some students will have trouble deciding between q + 5 = d and d + 5 = q. If that happens, I remind them to make a table with a few possibilities to satisfy the sentence, and then to match it to the equation. This instruction is stated more explicitly in the more-scaffolded work for the other half of the class, and as I want to show here, there's interaction between the two concurrent assignments.
The second sentence of each problem is a bit more difficult to handle, and usually comes down to whether students have seen it before or not. While I work with the groups who are practicing Guess & Check (see below), I give students a little time to discuss the sentences like "The total amount of money he has is $8.20." If they can't come up with the equation on their own, kids usually just have to see it once: 0.25q + 0.10d = 8.20. As for graphing that line, the options are to figure out a few points and plot them, or to change the equation into slope intercept form. These are generally my stronger students, and I let them grapple with that for a while.
But here's the real key: Guess & Check has already scaffolded for this. When students use Guess & Check, they're listing possibilities, and each possibility can be a point on the graph. So while all of this is happening, I'm helping the other half of the class to master Guess & Check. What I don't tell the kids is that by thinking about Guess & Check, they can get a better idea of what points should be on their graphs. That's a connection I'm hoping they'll be able to make on their own.
For Students in Need of Mastery on SLT 5.1:
These groups receive "What Does Mastery of SLT 5.1 Look Like" handout. It consists of the same four problems from Friday's Mastery Session, and it's the same four problems that half of the class is currently attempting to solve by graphing. On this handout, I explicitly provide a structure for keeping track of each guess and check. This table, of course, is also a scaffold toward graphing, so it serves a dual purpose. This gives my weaker students a chance to organize and analyze their work, and to build confidence. Then, as I provide an explicit example of what it looks like on the board, I'm also providing a tool for the graphing groups to use as they grapple with the problem of how to structure their equations and create these graphs.
This is too much work for any group to finish in just 25 minutes, but the activity lays foundations for the transition we're making this week: from listing possibilities and checking them, to thinking of those lists as points on a graph, to connecting the dots and seeing how the resulting graphs can be used to solve a problem.
Tonight's homework is another Infinite Algebra assignment, but textbook problems can do here as well. I provide a set of six "classic" problems in two unknowns:
- Two sum and difference problems, like "The sum of two numbers is 20 and the difference is 7, find the two numbers."
- Two digit reversal problems, like "I'm thinking of a two digit number. The sum of the digits is 11. If you reverse the digits, the new number is 45 less than the original number. What is my number." (Here the two variables are the digits.)
- And two of what I'll call "apples and oranges problems", where the apples and oranges can be any pair of similar items with different cost, and we're given the total cost of two different quantities of items.
Most kids are right in between using informal and formal methods for solving some of the classic problems in two variables. Kuta's Infinite Algebra software helps me make customize the problems by playing with the difficulty levels of the numbers involved, but here a textbook can work as well. The key is that I want to introduce these "classic problems" and let students experiment with their approaches. I might alter the level of difficulty for each class, depending on what they've shown me so far.