It's our first class of the new calendar year, hence the fireworks on today's opener. This is not the first time a lesson opener has been about patterns, however. After starting on familiar ground, today's lesson will quickly extend to formalizing some ideas about arithmetic sequences, which will in turn set this class up to work with linear functions for the next two weeks.
The task here is a little different than what we've done before. In fact, most students will find it easy compared to what they're used to doing. They just have to fill in a few blanks in each pattern. Just as I have all year, I call these patterns on this opener. By the end of the class however, we'll call them sequences, and even more specifically, we'll be able to talk about arithmetic sequences, and what makes them unique.
In order to get to that point, we need a reason to have such a name for something. Exercise c is not an arithmetic sequence. Students have an easy time filling in the blanks on the first two exercises, but the third one is different; I ask how so. Students notice that it's decreasing, and furthermore, that it's not changing by the same amount every time.
With the idea planted in everyone's mind that there are different operations that might characterize a sequence, I ask students to try to find multiple solutions to Exercise e. To get from 3 to 6, we might say that we added 3 or that we multiplied by 2. The next two terms depend on which of these we choose.
Of course, if we start by saying that Exercise e is an arithmetic sequence, it's a different story. Once we say that a sequence is arithmetic, we know more about it, and we need less information. This is an idea that we'll revisit in upcoming lessons, as students recognize that two points can define a unique line, even though there are all sorts of other functions that might pass through the same pair.
I'd like to start by saying that a few of the types of problems on today's arithmetic sequences handout are taken from Harold Jacobs' excellent Elementary Algebra textbook. Even though this book pre-dates the Common Core standards, I think that its approach fits perfectly with the goals of the CCS, and I highly recommend finding a copy if you're interested in strategies for developing deep algebraic thinking in your students.
For today's class, there is the handout called Notes: Arithmetic Sequences and a set of Prezi slides for today's lesson. Students work on the handout, collect notes and definitions, and work on problems until they have questions. I use the Prezi to navigate the handout (this is a real strength of Prezi, as opposed to traditional slides) and to share key notes. The handout is more than most students will be able to finish within a 43-minute class period.
The lessons that I want students to learn today are built into the problems they're solving, and there are a series of points I want kids to get. I think of today's work time as a "notes and work cycle." As I describe in this video, I can't be sure exactly when I'll lead a whole-class discussion, or even what it will be about. The key is to get students asking the questions that drive the lesson. In this narrative, I share some of my thinking about this assignment, and how I teach it, but every implementation of this lesson is different. Today, students will do as much as they can, and that's just the right amount!
#2: Defining an Arithmetic Sequence
When students see 2b, 2e, and 2j, they recognize patterns, and they get to put the definition of an arithmetic sequence to work. Students will say, “but there’s a pattern!” when they see b, c, e, or j. And there certainly is, but it’s the repeated adding that we’re looking for today.
Exercise 2f gets students to think precisely about what makes an arithmetic sequence what it is, and there can be useful debates among kids before we settle on this idea that if a sequence consists of one repeating number, there's a common difference of 0 and it is indeed arithmetic. See how this connects to the idea of a horizontal line with its slope of zero. Or taking it a step further, that it's possible for every term to be the same (horizontal line with slope zero), but that it's impossible for every term to be, say, the 8th term (a vertical line with undefined slope).
More generally, all of these problems are great context for students to develop number sense and to feel more confident with their arithmetic skills.
As kids complete this part of the assignment, I say, "I want you to keep this, so that if you can't quite remember exactly what an arithmetic sequence is, you'll be able to look at these notes and remember it. You'll be able to look at the definition, followed by examples of what are and are not arithmetic sequences."
#3 and 4: Using the Common Difference, and a Shortcut
In the context of this unit of study, today's work with arithmetic sequences is allows us to move more generally into linear functions later this week. None of today's work is brand new, but some of the formal vocabulary we're using today is. In addition to introducing the phrases arithmetic sequence and common difference, I show students the "shortest shortcut I know" for writing the algebraic rule for an arithmetic sequence. I note aloud that, "Everyone is comfortable with the idea of a common difference, but how do I figure out what constant term to add or subtract?" After giving students a chance to consider the strategies they already know, I introduce this new one: from the first term in the sequence, go one step backwards. Wherever you land, that's the constant to add to the end of the rule.
Sharing this trick serves two purposes: it gives students the simplest algorithm yet for writing a rule for an arithmetic sequence. Then, because this is a pretty memorable trick, I now have something to reference when we talk about y-intercepts, and how they're the point on a graph where x = 0, or in other words, "the term before the first term."
Students work together without too much trouble on #4 - they've been working on problems like these since the second day of school (see the homework). One by one, they continue to appreciate the idea that a rule helps us find the 100th term with far less than work than just continuing the pattern, even though, when it comes to finding the fifth term, either way works.
The last exercise is #4 is a sequence of fractions. When students think they're stuck, I say, "It's easier if you find a common denominator for these four fractions." It's pretty easy to see how confident a kid is with fractions by the way they react when I share that advice, and sometimes that's all they need. If they further help than that, I'm happy to make a mini-lesson out of it.
#5: Sequences of Sequences
The purpose of Exercise #5 is to give students a chance to practice writing rules for arithmetic sequences, but there's more going on than that. First of all, the instruction says to "use function notation," and this trips up a few students. There's always someone who asks, "What’s function notation?" I refer them back to what they've already done on the front of the page, and this gives us a moment to review precisely the language we're using here.
More importantly, there's the fact that these sequences are related. It's cool when students say something like, "I see a pattern, but I can’t describe it." Without explicitly saying it, these problems are laying foundations for us to talk about slope and y-intercepts. Exercises 5a through 5e share a y-intercept; 5f through 5j share a common difference or slope. We will revisit these sequences later, to see how they're related on a graph. This is another example of how I like to expose students to an idea before naming it and explicitly teaching it.
#6: Fill in the blanks
We started class with problems like these, and they're the focus of tomorrow's lesson, and I go in depth on my approach to teaching these problems there.
Today's exit slip is a quarter-sheet of paper that asks students to make a new arithmetic sequence by using their birth month and date as the first two terms in the sequence.
Two key things here. First, this is really a vocabulary quiz. I want to know if kids understand what a common difference is, and I want them to have further exposure to the phrase function notation.
Second, each student will have a different experience on this exit slip. The different kinds pairs of numbers students generate give way to all sorts of special cases: horizontal lines, common difference of 1, etc. When students see these, and that some are easier than others, it's cool, and that's why I don't mind if they chat with each other about the results. The act of categorizing sequences - ie "mine is easier to complete than yours" - is a mathematical task in itself.
It's also always so interesting to me when knowledge of an algorithm makes is easier to deal with messy numbers than those weirder cases. For example, this year I had a student born on February 2 wondering how the heck to write the rule for a sequence of repeating 2's while at the same time helping someone born on December 5 write a perfect rule.