Patterns and Abstractions
Lesson 5 of 12
Objective: SWBAT represent number patterns with rules and verbal expressions with both symbols and algebra.
Today's opener (Opener Sept4 Patterns) looks just like the first homework assignment of the year, which was assigned last week. (See Unit 1, Lesson 2) As students enter the room, I rally them to get started on these problems, even though I know that they'll need some help. When I looked at student work on the third day of class, I saw that very few of my new students are coming in with this skill. For today's opener, I provide the first few terms of four arithmetic sequences, and I ask students to find the 5th, 10th, and 100th terms. It seems like a simple enough opener, but it will turn into today's mini-lesson.
After welcoming them in and taking attendance (it's the 5th day of school, and I expect to know everyone's name by now), I've given students a few minutes to get started. While they work, I draw a chart around the opener. As I do so, I recite a line that I've already said during our Number Trick work: "When I'm trying to understand something better, one thing that always helps me is to make a chart." It looks like this: Opener with Blank Chart. Now we're going to work together to fill it in.
What's nice about these pattern problems is that they have a low entry point: almost everyone feels comfortable finding the 5th term in each sequence, which shows me that most of my students can recognize a pattern and add/subtract integers. I tell them that it's a great strategy to start by doing whatever they can: if it's easy to find the 5th term in every pattern, then do it. I ask for those 5th terms, allow for clarifying questions, and fill in the chart like this: Opener 5th terms.
Next we talk about that first pattern. This is another pretty low bar, and again, I'm encouraged to see that nearly all of my students feel comfortable counting by 3's. More than two-thirds of my students can pretty quickly see that the 10th term in the first pattern is 30, and it's not a huge leap to see why the 100th term will be 300. I fill those in, then I ask for someone to compare the first pattern to the second. Again, it's no great leap to see that each number in the second pattern is three less than the number above it. Soon we have this: Opener 5th terms and thinking.
"Now," I say, "let's try to generalize what's happening here." I don't stop to check if kids know the word generalize, there will be time for that, and I wonder if the context will be enough. "In the first pattern, the first term is 3, the second term is 6, the third term is 9..." and I continue, until saying, "so if I'm looking for the 100th term, I multiply 100 by 3 to get 300. What if was looking for the 1000th term?" All can answer. I say a few more numbers, and then say, "So whatever term I'm looking for, I can just multiply by 3 and find the number. If this was a number trick, I would say pick a number, and multiply by 3." I write 3n on the board, and give students a chance to think about that. Then I say, "So if the rule for the first pattern is 3n, then the second one is just 3 less than that." I write 3n - 3, and I ask for thumbs up or down. Usually it's all thumbs up right now, even though I know they're going to need some practice.
So we continue by considering how we'd describe each pattern in words. "In the first pattern, we're adding 3 every time, so I wrote 3n. In the second one, we're adding three every time, so I wrote 3n again. What's happening in the next two patterns?" I elicit verbal descriptions of each pattern, then we all figure out what the rule might look like. For the third one, some students will surface the idea that if we're subtracting 3 each time, that must be division. I love that idea, and although I want to really dig into it, I choose not to jeopardize the majority of kids who aren't wondering about such a finer point, and for now I just say that, rather, it just means we'll multiply by -3. The time for that idea will come in a few weeks. Finally, we have another mini-debate about whether the fourth pattern is increasing or decreasing by 5 each time. I draw a number line and suggest that students do the same, which leads to the consensus that we're adding 5. I write 5n, which brings us to here: Opener Developing Rules 1.
I want to make it feel obvious that these rules are incomplete. I point to the first term in the third pattern and ask for the product of -3 and 1. "Whatever it is, it's not 100," I say. Same with -3 times 2: I point to the 97 and ask if this is the right product. I continue along the list, taking the time to point to each number and giving kids a chance to catch what I'm talking about. As I move from number to number, kids get the idea, and soon the question is raised. "So what are we going to do about it?" I wait a few beats to continue. "I'd like to show you another chart," I say.
I tell everyone that I'd like to introduce an algorithm for finding a pattern rule. In the coming weeks, I'll dig deeper into this word, but for now I just say that it's like a mathematical recipe. "This recipe, this algorithm, has two steps," I say. "First, you figure out what you're counting by, and we've already done that." As I speak, I start putting a new chart together. I fill in some rows, and soon it looks like this: U1L5 Algorithm Write a Rule 1. "We have the numbers here" I say, pointing to the Step 1 column, "but we need to get the numbers here." I point to the last column. "The question is, how do we get from here to here?" I write the question as I say it: U1L5 Algorithm Write a Rule 2. I give students a few chances to share their ideas, and in some classes, kids figure it out. If they're stuck, I rephrase the question: "How far is -3 from 100?" This is a number line question. Many students can answer it, but some aren't sure. If it looks like I need to, I draw a number line with 0, -3 and 100. "How far is -3 from 0?" I ask. "How far is 0 from 100? What does that make total?" We repeat the process from -6 to 97 and -9 to 94, each time getting 103. Now we're onto something, and hopefully the table is helping: U1L5 Algorithm Write a Rule 3.
Now we're able to take this rule back to the chart, and fill in the rest of the third rule, which we'll then use to find the 10th and 100th terms (Opener Developing Rules 2). To the extent that we need to and that time allows, we quickly repeat the process for the fourth pattern and finishing the chart: U1L5 Opener Complete Chart.
I tell students how important it is that they have this in their notes. If they feel like they already know it, I say, these notes will help them remember. If they feel like this is hard, then writing these notes will help them begin to make sense. We'll try an opener like this again tomorrow, and it will be good for both me and my students to see how it goes.
Why This Now?
As I reference above, I gave students a few patterns as their first homework assignment, because I find that patterns are a quick litmus test for how much kids know about linear modeling, and when I saw their work, I saw that their skills were weak in this area. We will return to this idea later, as we get into linear functions, but I want to start with it now. I want to see how kids do with these ideas, and there are great connections between these sorts of patterns problems and the abstract reasoning that we're doing on the Number Trick Project.
Putting the opening pattern problems aside and moving into a homework check, we shift gears back into symbols and algebra. We're moving back and forth between patterns and number tricks, and I hope that students will ask what one has to do with the other. If they do, then we'll have a good conversation on our hands. If they don't, then we'll continue building from both sides until they meet at a cornerstone of abstract generalization. Or general abstraction. I'm only half joking about this. To throw around big words for their own sake is a poor use of intellect, but I think that these two words can actually give kids more power over their thoughts, so I start their year by giving them some experiences in both. By the middle of the year I'd like both words to have a tangible meaning to each of my students, the moment anyone says them.
For the homework check, I simply share the solutions with students. Their buy-in to this assignment was pretty high yesterday. After some of them struggled with the first part of the Number Trick project, this seemed a little easier, and all were eager to get it done right. So although I will circulate and look to see who has how much done, I'm much more interested in each of them seeing how they've done.
My answer key is not perfect. Although the algebra all checks out, there are other varieties of symbols that could be used on some of these expressions. I encourage students to advocate for their own ideas if theirs don't match mine. It's important to gently dispel the myth that mathematics is all about one right answer, especially when the "answer" means "the symbolic representation that makes the most sense to one particular person."
I also encourage students to discuss their work with each other, and to ask questions of me. Some questions I'll answer on an individual basis, while for others it's worthwhile to get the whole class involved. I know this is just a "homework check," but amazing conversations happen here.
Tonight's homework is to finish Part 1 of the Number Trick Project. Some students are already done with the first two tasks. Others are feeling more prepared to try them now. The third task has been impossible until now, because I haven't given kids the problem. The task is, given a symbolic representation, to write a number trick in words. I put the document on the board, then I draw a number trick (like this: NTP Part 1 Task 3.jpg) for students to copy.
If there is time remaining, I give them a few minutes to get started on this assignment.