An Intersection Point, and What Happens Around It

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SWBAT appreciate how graphing and substitution can be used in tandem to help deepen understanding of a problem.

Big Idea

Often the truest solution to a problem is that "it depends". Substitution is an efficient method for finding the precise intersection of two lines; a graph makes it clear what happens on either side of that point.

Opener: Solving by Substitution

5 minutes

Today's opener is on the first slide of today's lesson notes.  It's a quick solve-this-system-by-substitution problem.  Some kids have already got this in the bag, others don't.  If they do, I'm happy to see that, and if they don't, I keep that in mind.  Either way, we're not going to spend to long on this.  

I give students a few minutes to work, and then I ask for a volunteer to show us what they've got on the board.

On tonight's homework there are four "levels" of system solving, and this is an example of a "Level 2" problem.  Students will have the opportunity for further practice tonight, and algebra skills like this are the focus of tomorrow's class.  Before that, we're going to spend one more day on an application problem.

Commission-Based Salary Problem

33 minutes

Framing & Collaborative Problem Solving

Today's lesson picks up where yesterday's left off.  We're going to use both graphing and substitution to develop a full understanding of a linear systems problem, and we're going to investigate the kinds of questions that each method can help us answer.  We'll continue to apply the "Really Big Ideas" that were the focus of yesterday's lesson.  I provide an overview of my thinking in this narrative video.  

To get started, I distribute the Collaborative Problem Solving Handout.  Students have seen this structure before, so they know that I'm about to project a problem at the front of the room.  To frame our work, I say that I'm going to tell everyone the answer before I post the problem.  "The answer," I say mischievously, "is that it depends." Everyone groans, and someone invariably yells, "Oh, gee, thanks!" and if I'm lucky, someone might ask, "Depends on what?"  But now we're all in it, and kids are excited to see what this is all about.  Throughout the lesson, students are excited to return to this idea as they realize what I mean.

As before, the structure is as follows: I post the problem, give everyone five minutes to work on their own, and then a few minutes to examine and comment upon the work of a neighbor.  Then we get to work as a class to get through the problem.

In addition to the Collaborative Problem Solving handout, I make these two-sided first quadrant handouts available to anyone who might want to complete multiple drafts of their graph.

Likely Outcomes and Scaffolding

My experience is that kids work to define variables and write equations, but that they often need some help getting going on this.  I use the collaborative problem solving structure because, if it works, it's ideal.  It doesn't always catch, however, and that's what the fourth and fifth slides of today's lesson notes are for. 

One of the biggest hurdles to getting started on this problem is that few of my students understand how commission works.  So on the fourth slide, we define the variables: 

x = Emma's weekly sales

y = Emma's weekly pay

and then I ask, "If most people sell between $7000 and $10,000 worth of stuff each week, does it make sense to count by $1's when we consider her weekly sales?"  It's clear enough that it will probably make better sense to go by $1000's (or at least $100's, to start), and that's what we do in the table on slide #4.  I show students what it means to earn a 3.1% commission on $1000 in sales, and once they see that, they can fill in a few more rows of the table.  Immediately, students start to debate which option is the better deal, and they have their own ideas of what I meant when I said, "It depends."  When someone says, "Well, if she sucks at selling stuff, then the first plan is better," then I know that they're getting the gist of things.

Looking at a few examples takes us back to guess and check, which is great: it means that we're using all three tools we've seen thus far to tackle this problem.  That's what I love about this problem at this juncture - it synthesizes all the work of the last two weeks.  Seeing some examples helps to clarify what the variables are - that's a key feature of guess and check, after all - and once variables are defined, then we can write equations and use algebra.

Graphing vs. Substitution

From here, the process is much the same as it was on yesterday's break-even problem.  We write equations, talk about scale, and graph the problem.  

For our scaling options, as with other parts of this problem, we have to get used to the idea of there being a few different possible answers.  Counting by 400's, 500's, or 1000's on the x-axis are all well-justified.  Counting by 25's or 50's works on the y-axis, and that depends on the choice we've made for the x-axis.  I take the pulse of the kids to see what they think will work best.  We usually settle on 500's on the x-axis and 25's on the y-axis.  We discuss major and minor grid marks: the idea that "what I'm counting by" might be different from "the numbers I actually write".  As before, I model what I'm asking kids to do on the board.  The results look like this.  

As you can see in the photo, on my graph I plot the points that we found in the table, but after $7000, I extrapolate and sketch both lines.  It's virtually impossible to do this perfectly, and I do this to make a point: without using points as guidelines, our graphs probably won't be precise enough.  In this photo, my lines intersect a little beyond where x=9000, which is not accurate.  Students will call me on this, saying that their lines intersect right before $9000, or they'll use a table as evidence, to say that the intersection must happen in between 8000 and 9000 -- all of this is great.  "Whose graph is better?  Mine or yours?" and it's so exciting when they can say with certainty that it's theirs.

Of course, this is also an opportunity to illustrate the point of graphing versus substitution.  After solving the system algebraically, and seeing that indeed, the intersection point comes where x is a bit less that 9000, I label my point with exact coordinates, even though, on my graph these coordinates don't exactly match the graph.  

Then we go back to the big ideas.  What are the advantages of graphing?  Of substitution?  Why might someone do both?  What do we learn by having both?


After all of that, the question remains: Which job should Emma take?  We go back to the original answer: "It depends."  Depends on what?  Is she good at sales?  Is she confident?  What choice would you make?   Kids love to discuss this one.  It's a neat conversation because it allows them to take one position and appreciate the other.  I've seen a lot healthy recognition of two sides of an issue with this problem.

Work Cited

I adapted this problem from one in the excellent Park Math Curriculum.  Teachers at the Park School in Baltimore, MD collaborated to create an unbelievably rich problem-based math curriculum.  I'm not sure if it's available online, but they're nice folks, and were happy to share some of it when I asked.

Homework: Substitution

5 minutes

Tonight's homework is a good old-fashioned practice set.  I use Kuta's Infinite Algebra software to make worksheets to my liking, but textbooks and other practice options work just fine right here.

What I like about Infinite Algebra is that it allows me to customize the work for each section I teach, and to produce different versions for kids at different levels of mastery on SLT 5.3.  Everyone gets practice solving linear systems by substitution, on problems that are as challenging as they're ready to handle.  I've defined four levels of problems: 

  • Level 1: Two equations in which y is isolated, so students just have to set the two sides equal and solve.
  • Level 2: Either x or y is isolated in one equation, but not the other, so students must make a substitution step before solving. 
  • Level 3: Two equations in standard form.  At least one variable in one equation has a coefficient of 1, so it takes one step to isolate a variable.  From there, it becomes a "Level 2" problem.
  • Level 4: Two equations in standard form.  No variables have a coefficient equal to 1.  (Wait a minute.  Isn't substitution kind of annoying in this case?  There must be another way to solve a system like this...right?)

Most often, I'll make a worksheet with four problems at each level and tell students to get as far as possible, but like I say above, I might change that based where I've observed the class to be.  Or, I might make one version with Levels 1 through 3 and another with Levels 2 through 4, and then allow students to choose the one they'll take.