What's In an Intersection?

Today's opener picks up where yesterday's lesson left off.  The minimum students should understand from yesterday's work on the Chloe and Zeke problem is that we're given two clues (eventually, we'll call these constraints) about the relationship between the ages of two people.  We'll use this problem to build some algebraic ideas.

This lesson happens in the computer lab.  As students arrive, they see their task on the front board.  They should go to desmos.com, open up the graphing calculator, and create a table of values for each constraint.  Most students have already created such tables in their notes, but mine start at an extreme value and increase from there.

Students have seen how to make tables in Desmos before, but I know that I'll have to review it with a few of them.  It doesn't take much to set kids on their way.  When they plot the points correctly, of course, they'll see the beginnings of two lines that move toward each other.  This is an opportunity once again to review the idea that a graph is a set of points (see A-REI.10).  It's also an opportunity to connect back to guess and check.  We can talk about the "closeness" of the lines: they start far apart at the y-axis, then move closer together.  This is similar to the way that our guesses might start off far apart, but then get closer as we keep checking.

When I see that everyone is on track, I do the same work on the screen, and we'll be able to talk about it.  In order to help us transition to using the equations that describe these sets of points, I use a white-board marker to connect the dots I've plotted, just to be completely clear about what's happening.  In just a moment, we're going to use the technology to do the same thing, but I find that drawing these lines really helps students see what's going on.

Everyone is noticing that these points make a line.  I've connected them on the board.  The "closeness" (one could say "proximity" but I'm choosing a student-friendly word here) of one line to another matches what we would see in a series of guesses and checks.  It's important to point this out.  I hope that students have started to notice and generalize what happens to the "check" as the "guess" changes -- this is slope.  I don't call it that, because that would spoil the fun, and given time, students do come to that understand on their own.  

With all of these scaffolds in place, it's time for the mini-lesson.  "If there are lines here," I say, "then there must be equations to match them.  So what are they?"  Students should have their Collaborative Problem Solving Collaborative Handouts handouts for the two problems they've seen over the last few days.  Today is all about the back: the algebraic analysis part.

Take a look at the "Algebraic Analysis" slide in today's lesson notes to see the steps that students will see several times in the next week or so.  We want to be able to write equations to represent each part of the problem.  Most of my kids know how to do this, but they lack confidence.  I lead a conversation, and I model how to take notes on the board.  I show students how to define variables, and then how to use these variables to write equations.  As students guide the conversation, we might discuss the possibility that there are multiple solutions.  Students will point out that each part of this problem has many possibilities, but then when we check most possibilities in both equations, we'll see that it's hard to find something that works for both.

The big reveal happens when we type the new equations into Desmos.  Kids get excited about two things - both of which are precisely the point of today's lesson.  First, that the lines we by adding these equations to the graph go directly through the points we've already graphed.  This is a fundamental understanding that students need to get from Algebra 1, and as teachers, it's easy to forget that it's not obvious to everyone.  This gives them a chance to experience it.  Second, the lines do indeed intersect, and not only that, the intersection means something!  Because we've already solved this problem at least once, students recognize the coordinate values of the intersection point immediately.

And that's the big takeaway: all of this equation-writing and line-graphing is for naught if we can't figure out what the intersection point tells us.  Interpretation of solutions will continue to be theme as we study systems of equations.