Introduction: About This Lesson
The purpose of today's lesson is to generalize some of the mathematics that students have seen during the first two lessons of Unit 3. In the first two lessons, we've looked closely at real world data: first about the value of a car over time, and then about how auto loans work. In each of these situations, we have constructed different kinds of models. In the Auto Depreciation lesson, students constructed linear and exponential models of data they collect online. In the following lesson, Auto Loans, among what students saw was how to write recursive formula definitions in an Excel spreadsheet. I didn't use the word recursive yesterday, however. I showed students an idea and allowed them to use it.
Today, we'll abandon context entirely as we investigate sequences. Students will be able to distinguish between arithmetic and geometric sequences, which is, for our purposes, analogous to being able to distinguish between linear and exponential functions. They will then see how to write explicit functions and recursive rules for each kind of sequences, as well as seeing the advantages and limitations of each. Throughout the lesson, I will refer back to the work of the previous two days to help students connect this generalized sort of mathematics to the applied work they've already done.
As students arrive today, the opener is posted on the second slide of the lesson notes. Students are prompted to write a function rule for each of four sequences. The twist here is that although I know students are comfortable writing rules for linear functions, I know that few of them have learned how to write them for exponential functions. So the first step is for kids to recognize that. I'm hoping that kids can come up with their own solutions to letters (a) and (c), and then to notice that letters (b) and (d) are different. I also hope that most kids can explain what's happening in each of these sequences - (b) is doubling at each step, and (d) is decreasing by half each time - even if they're not totally sure how to write the rules.
Now for both sorts of sequences, teachers have to make a decision: do these sequences start on "Term 0" or "Term 1"? That's an important distinction to make, because the results are different either way. I choose to show students that these sequences begin at "Term 1". With that in mind, we can say that the solutions are as follows:
As you can see by considering these solutions, there are good reasons to take either approach.
Following a discussion of these solutions, I move to slide #3 for definitions of arithmetic and geometric sequences. I say, "You probably already know how to write rules for arithmetic sequences, and writing rules for geometric sequences is really similar in structure. For both, you have to think about where the sequence starts, and what is happening at each step.
Finally, I relate this back to our work of the previous two lessons, by saying that an arithmetic sequence is essentially the same as a linear function, and a geometric sequence equates to an exponential function. I want students to recognize that they've had some experience here, and that we're just formalizing from what they know.
Quick Checks and New Ideas
Everyone gets a chance to check their understanding of these definitions when I post slide #4 of the lesson notes. Embedded in the definitions of arithmetic and geometric sequences are the terms "common difference" and "common ratio". This slide gives students a chance to use those words in context as well.
Then, we move on to slide #5, where students get another chance to practice writing function rules. There's a little more to it than that however, as each example from (b) through (d) highlights a new idea.
Sequence (b) alternates between positive and negative values, as only a geometric sequence could. If students are confused at first, they are very satisfied when they recognize the consequences of using a negative common ratio. This example highlights a limitation to the similarities between geometric sequences and exponential functions. I describe for students how the former is discrete data and the latter is continuous, a distinction that we might relate to the last two lessons. When we think about how a car loses value over time, that change is continuous, and therefore appropriate to model with a function. Paying off a loan is different: you don't pay tiny amounts constantly, you make payments at regular intervals. That's discrete data, best modeled by a sequence.
Sequence (c) introduces the idea that a common ratio can indeed be a ratio instead of just a whole number. This one is challenging for students. They can see that numbers are decreasing, and not by the same amount every time, and they all have a hunch that this is a geometric sequence, but most have a hard time determining what the ratio will be. Here I get to share another trick. I refer back to sequence (a), which didn't give anyone too much trouble. It was pretty clear that (a) is an arithmetic sequence with the common difference -50. By subtracting one term from the one that follows it, we can arrive at that value. Common ratios work the same way: we can divide a term by the one before it to calculate a common ratio. I show students that it works on (b) before setting them to it on (c). Everyone realizes that we're going to need to reduce the fraction 540/810, and if it takes a few moments to review how to do that, so it goes. When we all agree that it reduces to 2/3, we'll test that common ratio on the next term. "Is 540 times two-thirds 360?" I ask. I remind students that to multiply a whole number by a fraction, you can multiply by the numerator and divide by the denominator in any order. I challenge everyone to do so without a calculator.
Then it's on to sequence (d). This one is neither arithmetic or geometric, although it is famous enough. Just like they did on the opener, many students will be able to explain what is happening in this, the Fibonacci sequence, even if they can't write a rule. When it comes to this sequence, it's pretty hard to write an explicit function rule. For that, we're going to need a new tool.
Introduction to Recursive Definitions
And that's when I get to show kids how to define a rule with a recursive definition. I use example (d), the Fibonacci sequence, first. I ask for volunteers to describe as clearly as they can what's happening here. Students are able to come to the idea that each term is the sum of the two terms before it. "But where do the first two terms come from?" I ask. That one always gets everybody, and I allow a few moments of thought before saying what most students think you're not allowed to say in math: they're just there because we say so. This sequence simply supposes that we start with a first term of 1 and second term of 1. Then, each successive term can be the sum of the two before it.
And that's what I love about teaching recursive formulas: the definitions are powerfully common-sense, but the notation presents a high barrier to entry. It just feels confusing to kids because the notation is new - the multiple parts, the use of subscript - but as we guide them to see how it works and what it means, kids are thrilled at their recognition that they understand what's going on.
From these lecture notes and opportunities for guided practice comes today's primary activity: the Sequences Gallery Walk! Why a gallery walk? Because it gets kids moving and talking, exactly where they need to be.
It would be just as easy to print 15 examples on worksheet and have kids practice. Instead, students move around the room, where I've posted each of the 15 sequences in this file, each printed on a separate sheet of paper. When they arrive at each sequence, they write it in the corresponding row of their note catcher, before they proceed to describe the sequence and write a function rule and recursive rule to match.
In the "description" column, I tell students to identify whether each is an example of an arithmetic sequence, a geometric sequence, or neither, and when appropriate, to give the common ratio or common difference.
There are a few sequences that are not arithmetic or geometric. (C) is the sequence of "triangular numbers", and (H) is like the Fibonacci sequence, except that it begins with two numbers other than 1.
Here's the great thing that happens when you use a gallery walk instead of a one-page set of practice exercises. Kids get up and move around, which helps activate their thinking. Movement assists in circulation, and being in different parts of the room makes working on each problem more memorable. Furthermore, it makes it easier for me to help students who are working on the same problem. Students will find themselves standing next to colleague who is stumped by the same problem, and they'll start to talk about. When they realize that need my help, I can move to that spot in the room, and instantly see who is there, then offer my help. If other kids have skipped over a particular problem, they'll see me near it, helping other folks, and they'll return to get the help they need. I always look forward to teaching via a gallery walk, because I can always look forward to a series of little teaching moments that exactly what they needed to be.
As students finish their gallery walk, I provide this debrief assignment, which poses a series of questions aimed at getting students to summarize what they've seen, and then drawing on their new knowledge.
Most students will finish their gallery walk with too little time left to complete this today, and in that case this handout becomes homework.
With five minutes left, no matter how much everyone has accomplished, I post the last slide of the lesson notes, which describes today's exit task. I use this exit ask to check for understanding of the vocabulary.