Today's lesson picks up right where yesterday's lesson left off, starting with this opener. During yesterday's lesson, I showed my classes how to write recursive rules for geometric sequences, but students did not see how to write explicit function rules. Some students figured it out on their own, but many need a push in the right direction, so that's how we kick things off today.
The first example in today's opener is pretty straightforward: each term is a power of 3, starting from
t(1) = 3^1 = 3
so the rule will be
t(n) = 3^n
To get there I backtrack to arithmetic sequences for a moment. "In an arithmetic sequence, we're adding repeatedly," I say. "How does that appear in a function rule?" It's not a new idea to my kids, but we're looking at it through a new lens, so I take a moment to make sure that everyone is ok with the idea that we indicate repeated addition with multiplication in a function rule. "So what is the mathematical notation for repeated multiplication?" I ask. Kids are usually excited to pull exponents out of their mind attics, which helps us build momentum. What is most new here is the idea that the variable n - the number of the term - can be the exponent. But in the context of the first example, it makes enough sense that kids feel confident.
Where exponential functions - and the rules for geometric sequences - get tricky is when the common ratio is not a factor of the first term. Actually, writing recursive rules for such sequences is pretty straightforward, and that's why I started with that yesterday. It's the trick of using n - 1, which was introduced yesterday in the recursive definitions, that is again the key today.
As a solution to the second opening exercise, I simply show students how this works by writing
t(n) = 3^(n-1)
next to the second example. By working through what it means, term by term, kids can see where this comes from. Here also is a contextual justification for why the "zero power" of any number is 1.
Once kids are comfortable enough with this rule (I won't go as far as to say that everyone is satisfied right away, but comfortable enough will do as we get started), we compare the third example to the second. I want kids to notice on their own that we've just multiplied every term in the second example by 5. "This means that we just have to multiply this rule by 5," I say, pointing to the second rule, "to get the rule for the third sequence."
I move quickly, knowing that that this quick lesson is just what some kids need to feel like they can complete yesterday's work, and that others will need a little more focused help to use what I've just shown them. Now it's back to the Gallery Walk, where we'll see how well students understand what they've seen here. As they practice, I'll be able to provide tutoring as needed.
My ideal is that everyone is pretty much done with the Gallery Walk by now, and they're just filling in some gaps by trying to write function rules for the geometric sequences. The reality is that there are a wide range of results so far, so all of my students need this time to get more done on their Gallery Walk Note Catchers. Please see yesterday's lesson for a description of what we're doing right now. This time is simply an extension of yesterday's work time.
I tell everyone to do as much as they can, and not to sweat it if they can't finish everything. As much as I want students to show me what they've mastered, their results at the end of today's class will show me exactly where I should go next.
With a little more than 20 minutes left in today's class, I ask everyone to return to their groups. Everyone is at a different point in their work on the gallery walk. Now, with whatever amount of work they have complete, students will get together in their groups and work through this Sequences Gallery Walk Debrief.
Now I want to emphasize group work, and I know that sometimes tacking the word "quiz" onto a task can make 9th graders take things just a little more seriously, so that's what I do. I tell everyone that they should definitely work together to discuss their results, but that if they come to a disagreement about any answer, they're not required to write the same thing on each individual quiz.
The learning target I'm assessing when I do grade this is whether or not students can differentiate between arithmetic sequences and geometric sequences. Their first task is to categorize the 15 sequences from the gallery walk as arithmetic, geometric, or neither, and this task yields great conversations in groups.
Next, students must identify a sequence that was "easy to understand" and one that was "difficult" and justify their reasons for doing so. Writing about how and why different problems are more or less difficult than others gives students a chance to elucidate what they know so far, and it's valuable evidence that I'll be able to use as I plan my next steps.
On the back of this assignment, groups must make up their own examples of different sequences. Check out the student work in the section following this one to learn a little more about the debrief quiz and what I learn about my kids when I collect this assignment.
Another aspect I love about this assignment is that it doesn't take too long a glance to see how much each of my students knows and is able to do. One flip through the work I collect today, and I'll know how I have to focus my attention next.
As this student points out, for example, "The ones that are empty I did not understand." Of course: if there are blanks on the Gallery Walk Note Catcher, that's where the extra help has to start. The arrangement of filled and blank squares on the note catcher gives me a quick and easy overview.
Looking more closely at both the note catchers and the group debrief quiz, I can see who is able to classify sequences as arithmetic, geometric, or neither, and who needs a little help on that.
The back of the debrief quiz is similar to the note catcher. If there are blanks they're obvious, and I'm not too surprised when I see that a student has the arithmetic sequences down, but not the geometric ones. Minor misunderstandings, like this student's too-specific recursive rules are easy to spot and fix, while bigger gaps in knowledge reveal which kids need a more focused intervention. And of course, it's always nice to see who is really getting it.