During today's lesson, I will formally introduce students to the definitions of a geometric sequence and recursive rule. These are two pretty big concepts to squeeze into the same lesson, but they're not completely foreign to students. Foundations have been laid for both in previous lessons, and for each, there is complementary idea against which we can compare these new ones: students will be able to compare geometric sequences to the arithmetic sequences with which they're already confident, and they'll be able to compare the recursive rules for sequences to the function rules with which they have a wealth of experience.
To begin the lesson, today's opener gives students the chance to compare an arithmetic sequence to a geometric one. The first term of both sequences is 10. One is an arithmetic sequence with a common difference of 25, and the other is a geometric sequence with common ratio of 2. I ask students which one will exceed 50, 100, and 200 first. This is the first time that students are seeing the words geometric sequence, but they're already familiar with the idea. In previous lessons, they've seen sequences that grow by repeated multiplication.
I give students a few minutes to work through the problem, and I encourage everyone to discuss it among their groups. Everyone should see that the arithmetic sequence exceeds 50 by its third term, while the geometric sequence takes until the fourth term to get there. Both surpass 100 in between the fourth and fifth terms, so for now we can say that they get there around the same time, although I ask the class to note that if we looked a little more closely, it might be easier to see which one passes 100 first (we'll save that for an upcoming lesson). As for 200, the geometric sequence blows past it well in front of the arithmetic sequence, which helps me summarize the last two lessons for the class: "One of the most important things I wanted you to see with the Payment Plans over the last two lessons was that repeated multiplication will always end up growing more quickly than repeated addition," I say. "It may take a little while before it happens, but it will."
With the idea of a geometric sequence so named, it's time to introduce kids to recursive definitions for sequences. I post the words "Function Rule" and "Recursive Rule" on the board, and I ask for a volunteer to make up a "pretty simple" arithmetic sequence to use in this lesson. I write what I'm told above the names of the rules. We quickly move through writing a function rule for this new sequence, because that's something we've been working on since the first days of school. I annotate the rule a little bit; my main goal is to make sure everyone recognizes where the common difference appears in this function rule.
Recursive Definitions, for the First Time
Once that's done, I point to the term Recursive Rule and say, "I'm going to show you another way to write the rule for a sequence. It's new, so it might look a little complicated at first, but in a way, I think it's actually easier than writing a function rule." I move quickly through giving this first example: giving the first term and a rule for finding the "next term". Here is what the notes look like. The notation is by far the most difficult thing for students to understand. The idea of a recursive rule, that it identifies a starting place and a rule for finding what comes next is straightforward. The hurdle kids have to clear before they can understand that is the subscript notation. I stay with this for a few moments, first by asking a few questions like, "If n is 50, what is n minus one?" then by writing some examples on the board. Once kids start reading the n-1 as the "the term before n," then we're really rolling.
I also compare writing recursive rules to what we did yesterday on Excel. Students should notice that the formulas we wrote in yesterday's spreadsheet were also recursive definitions: we input the first term, and then a rule for get each successive term. I don't sweat too much whether everyone understands this perfectly, because today's gallery walk will be another opportunity for kids to practice writing recursive rules.
Example with a Geometric Sequence
Next, leaving up that first example, I ask the class to give me a "pretty simple geometric sequence" and I write it on the side board. I say, "When it comes to geometric sequences like this one, I think recursive definitions are very helpful. As you'll see today, there are certain kinds of sequences that are much easier to model with a recursive rule than with a function rule."
We establish the first term once again, and then I show students how to generate the rule for the next term. The notes look like this. As for the function rule, I leave it as a coming attraction. I challenge students to try to figure out what the rule would be, and I say that we'll focus on that during tomorrow's opener.
Now it's time to practice. Students will practice writing rules for arithmetic and geometric sequences by working through today's gallery walk. To set up, I post each of these 15 pages around my classroom. I also make equal numbers of each of the Sequences Gallery Walk Note Catchers that you can find in the resources at right. Each note catcher lists the letters of 8 of the 15 sequences that are posted around the room. With students working in groups of 3 or 4, I ensure that each group has exposure to all 15 sequences by giving one copy of each version of the note catcher to each group (I just double up on one of the three for groups of 4).
Movement is one key to today's lesson. Students have to get up to see the numbers in the sequences they've been assigned. Check out my narrative video to see how I've set up the gallery walk, as well as a few reasons why this structure beats practice worksheets despite its simplicity. In general, a gallery walk will almost always make classroom management issues irrelevant, and it will allow essential conversations to happen naturally.
In the three columns of the note catcher, students must describe the sequences, write a function rule, and write a recursive rule for each sequence. If they're not sure what it means to "describe the sequence," I tell them to state whether it's an arithmetic or geometric sequence, and to determine how it's changing (if they know what I mean by common difference and common ratio, I'll use those words). The function rule column should be pretty straightforward for the arithmetic sequences, but as I've mentioned above, I haven't explicitly taught how to such a rule for geometric sequences. I'll give hints about how to do this if I see that kids are successful on everything else. Otherwise, I tell them to get everything else done before we focus on that tomorrow.
I give students until there are about five minutes left in class to do as much as they can. While they work, I have a great time joining in on conversations, answering questions, and running impromptu mini-tutoring sessions wherever students need me.
We'll spend a little more time on this Gallery Walk during tomorrow's lesson. If you're interested in seeing more examples of student work, I've posted a few in the next lesson.
As I note in the video, by locating different sequences in different parts of the room, we can help kids make a space in their memory for these concepts. If that sounds crazy, I recommend putting the book Moonwalking With Einstein on your summer reading list. In it, Joshua Foer recounts his experience training on the competitive memory circuit. As it turns out, one of the more popular strategies for making random memories stick is to assign each to a physical location in a place you know well. Here, instead of arithmetic sequences and geometric sequences occupying the same 8.5 x 11 inch pieces of real estate, we're able to point to different parts of the room as we consider each sequence. Furthermore, if running an advisory is one of your teaching responsibilities beyond the math classroom, I highly recommend playing memory games with your kids. I've had a great time constructing memory palaces full of all sorts of crazy objects with my advisees - building disciplined thinking habits and funny inside jokes in the process.
To debrief today's lesson, I ask everyone to circle up with a little less than five minutes left. First, I give everyone a prompt: "What is one word you've just got to remember from today's lesson?" There have been several - both new words like geometric and recursive, plus ideas that students continue to make their own: sequence, function, arithmetic, or common difference. Just thinking about these words, just hearing each person in the class say one, can help solidify knowledge. It's great when a kid says, "that was my word!" in response to someone else - hopefully, that means it's a word they won't forget!
After that whip around, we have a round of Appreciations. "I saw of people helping each other out today," I say. "If a classmate - in your group or not - helped you understand something today, you can appreciate them now."
As the first week of the new marking period winds down, I take this opportunity to appreciate the work I've seen this week. It's now the second half of freshman year for my students, and this is always a time when I start to see a new side of my kids. I name a few of the best habits I've seen this week, and I congratulate kids on their growing list of successes.