Today's opener is yet another number pattern, with the first four figures in a dot pattern and a trio of the usual questions. Students are asked to find the number of dots in the fifth figure, to write a rule for the number of dots in the nth figure, and to show what they understand about the structure by sketching the 1000th figure in the pattern. I give them some time to complete all three tasks in their notes, and by now, most of them are demonstrating some confidence in completing all three. This gives me a chance to circulate and help students where they need it, and to make sure that notes are neat, helpful, and complete.
The twist is that I use a student-generated dot pattern, from the Make Your Own Dot Pattern lesson that happened a few days ago. With a camera on a smartphone, it's now pretty easy to capture student work and place it in a document. I make a different opener for each class by dropping a different photo onto each. It takes a few extra minutes, but when students see their own work in an opening problem, it increases engagement. Increasing student voice can mean that we contrive problems that relate to kids, or it can mean that we draw them into the work we're really doing here. In other words, although there is no "real-world" context to a dot pattern problem, the real work of students is presented in this problem, two kids are going to get to shout, "hey, that's my pattern!" and the increase in engagement among all students is clearly evident.
Today's assignment - Equivalent Line Segments - is inspired by problems from Harold Jacobs' Elementary Algebra textbook. The book is out of print, but if you care about finding ways to help students visualize and conceptualize essential algebraic ideas, I strongly recommend looking for a copy. The line diagrams on today's assignment are one of the visuals employed by Jacobs for getting students to think about first-degree expressions and linear equations. Take a look at this video for an overview of the assignment.
Earlier in the year, we looked at Guess and Check as a Bridge to Creating Equations. Today, we revisit that work. Guess and check is a completely viable option for solving today's problems, and it should be used until its limitations are obvious to students. Only when students experience the inefficiencies of a strategy firsthand can they really appreciate why algebraic modeling came to exist.
The guess and check part of today's lesson can be more or less structured. Some students need to see the series of guesses to understand what the repeated work looks like. It's useful to look at one example as a class, and to ask students to analyze how the solutions change as a function of the guess; today, I use problem #5 as an example in a brief mini-lesson. Comparing the results of different guesses is its own scaffold toward algebraic thinking and an understanding linear change. I leave the example on the board for students to reference as they move through problems 6 through 8.
Of course, I want students to create and solve equations to solve problems 4 through 8, but more importantly, I want them to engage with the work. If students see that they can write and solve equations on this assignment, they should. If they use guess and check, then it will be all the more exciting tomorrow when they see how graphical representations can be used to solve these problems.
As I note in the video overview to this assignment, there are three groups of problems on this assignment: #1-3, #4-8, and #9-13. What everything has in common is that it is scaffolding for what we're doing tomorrow, as we look at representations of each of these problems on a graph. Today, students will do as much as they can on this assignment. Tomorrow, they will see how lines and points work the coordinate plane. They will interpret the intersection of two lines as the solution to some of these problems, and they will see that a set of points that make a linear equation true will make a line.
Students work to complete as much of this assignment as possible. Whatever is unfinished will be tonight's homework, and I tell students that it's imperative that everyone brings this assignment to class tomorrow.
Students will get to problems #9 through 13 at different times, and when they do, I ask them to describe how these problems differ from the previous eight. They should see that now there are two variables, and that therefore there are multiple solutions to each problem. As I've mentioned in the previous section, I ask everyone to find at least four solutions to each problem.
Some students will complete the entire assignment. When they do, I ask them to plot their solutions to problem #9 on a graph. I provide graph paper, coach them through setting up the axes, and plotting their points. When they do, they should see that, when graphed, the set of points makes a line. Many students will respond with a knowing nod to me, saying something like, "Oh, I see what you're doing?" I'll respond by asking if they've seen this before, where they've seen it, and what this topic is called. "This is the main focus of tomorrow's lesson," I say. "Tomorrow, we're really going to see how the graphs of linear functions can give us an even better understanding of how these problems work."