Seeing Structure in Dot Patterns and Linear Functions
Lesson 4 of 10
Objective: SWBAT look for and make use of structure as they construct linear models that correspond to a series of dot patterns.
Today's opener draws a line connecting an important assignment from earlier in the year (Part 2 of the Number Line Project) to the Hour of Code that students completed yesterday. With this code block written on today's agenda and a blank copy of Part 2 of the Number Line Project projected on the board, I invite students to explain what will happen.
I encourage kids to find that assignment from earlier in the year, and to take a look at it. Enough of my students are keeping organized binders that there will be at least one completed addition table for each group to look at.
I let kids talk and think about it for a few minutes, then I hold up a handful of markers and ask who would like to show us what they're thinking on the board. It's not too hard for someone to pick a starting place and draw arrows representing the code. When that's done, I ask if this reminds anyone of any of the coding problems from yesterday's class. I'm hoping that someone recognizes the similarity between this and some of the pathways that zombies had to be programmed to walk.
Then I try to get more specific. "Now pick a starting number on your addition table," I say. "After each time you run this loop, which number will you be on?" I want students to see that each time they move up two and right one, the number increases by 3. Seeing the structure of a pattern is the goal of today's lesson, and this visual gives us a great place to start.
Usually I hand out the learning targets on the first day of a unit, but I've shaken things up a bit this week. Rather than saying what we're doing and then setting to it, we've done some work already. Now, on the fourth day of this mini-unit, will we look at the learning targets and set a course for the next few weeks.
I distribute this organizer, which serves two purposes. It lists the Student Learning Targets that I want students to try to master between now and the end of the marking period, and it gives students a place to record the assignments that they complete as evidence of each. I show students where to record the work of our first two days this week - the two "Math Without Words" puzzles that I assessed under Mathematical Practice #1 - and I return that work. I also show students where to find today's work, under Mathematical Practice #7.
As a way to debrief the work of earlier in the week, I've posted this extension on the board. I show students that the solution to the problem, as some of them have found, is 34. In each class, some students have noticed a relationship between the numbers in this pattern. Up to now, I haven't explicitly said that this number pattern is "right," but now I do. More importantly, I now ask why it exists. "My challenge to you is to create a visual explanation of why this pattern works, in relation to the groups of dots."
Cultivating Mathematical Practices
So far this year, there have been plenty of assignments aimed at getting students to "look for and make use of structure," but today is the first time that I'm explicit in making Mathematical Practice #7 the focal point of the lesson. So in addition to using the learning target tracker to identify that today's work is aligned to that learning target, I've written MP7 on the board, and I ask for a volunteer to read it. Then I give students the chance to identify key words in the learning target. Structure is the one word that really stands out. Today, I hope that students will see connections between the the structure of different representations of patterns.
Before distributing today's Dot Patterns handout, I reprise the message of earlier in the week. "On this handout, there are no words. Your first task is to define the problem," I explain. "Once you define each problem, you can solve it." As was the policy two days ago, I say that I'll take questions in five minutes, but that I want everyone to see what they can figure out first. It's satisfying to see how much more receptive students have grown to this approach in just a few days.
It's easier for kids to define what these problems are asking them to do, the solution pathways are more obvious, and students have seen arithmetic sequences earlier this year. All of this is designed to help students see clear growth in their problem solving skills after the challenges of solving those two Math Without Words puzzles earlier in the week. As my students and I dig a little more specifically into the content today, I hope that they will employ the mathematical practices they've been developing this week.
Mini-Lesson: What Does a Solution Entail?
After five minutes, the questions students ask are more specific and direct than they were earlier in the week. I don't give away any solutions; I do answer all clarifying questions. Soon, all students understand that on each of these five problems, they are given the first four figures in a dot pattern, and they have to figure out what a certain figure will look like. On the first problem, where they're asked for the 5th figure, it's easy enough to draw what that will look like. But it's less obvious what's going to happen on the problems that ask for the 50th, 100th, or 200th figure. Eventually a student will ask, "Do we really have to draw the 200th figure?" Or, "What do I have to write on the paper?" I say, "I'm going to show you how to get a 4 on this assignment."
On the board, I write, "What does the answer entail?", followed by a checklist of what I want students to include in their solution to each problem. By the time we get here, most students are already satisfied with their drawing of the 5th figure on the first problem, and they can easily count or extend a pattern to see that this figure will consist of 13 dots. I say, "Like many of you have already done on the first problem, I'm looking for a drawing and the number of dots." I make sure that everyone agrees on what these will be. Then I say that I also want to see a brief explanation and an algebraic rule for each problem.
To move in that direction, I ask for someone to describe what's happening in the first pattern, and I student descriptions to move the class toward constructing the rule. For example, one student might point out that "two dots are being added each time," while another notes that "each side is going up by one" and this is great. This is seeing structure, and although it's rather simple here, we'll soon move on to more complex problems. I ask who is right, and everyone sees that whether we talk about adding "two dots each time" or "one dot to each side," we're talking about the same thing.
I make a horizontal table in which g represents the number of the figure (or group) and d(g) represents the number of dots in group g. This is the second time this year that I've introduced function notation. Most students remember what the notation means, and using it in context like this helps them make sense of why such notation exists. Now we just have to remember where the 2 we've identified goes in the algebraic rule. If students need some help here, I hold up a hand and raise each finger as I say, "What's a better way to say two plus two plus two plus two plus two?" I want students to recall that multiplication can be defined as repeated addition, and so we're going to have the product 2g in the rule. "But that's not enough," I say as I point to the table of values. "Because 2 times 1 isn't 5 and 2 times 2 isn't 7. So what else do we have to add here?" Soon we have the rule: d(g) = 2g+3.
Then comes the crux of the lesson - and a key point of the year so far - in getting at structure and seeing structure in expressions. We look at the parameters in our newly-minted function rule, and we look to see where they can be found in the dot patterns. It's a remarkable moment for kids. I ask where we can see "2g" and where we can see "+3" in each dot figure. As shown in this photo, I circle the pairs of dots that are added to the top of each figure in one color, and the three at the base of every figure in another. Then, I use the same color-coding to show how the structure of each dot figure is related to the structure of the rule.
"This is what I hope you'll be able to do on each of these problems," I say. "So yes, you do have to draw the 100th figure on problem #4, but you don't have to draw every dot. You can just figure out a rule, and then use the rule to help you describe what the 100th figure will look like." The subtext here is that writing a rule - which many kids take to be a pretty complicated task - is actually easier that drawing the 100th figure. The rule is the tool that save us from drawing all those dots. For your reference as you plan to teach this lesson, here is the work of a student who has done a good job drawing generalized diagrams (although if I'm being really picky I'd have a few tiny pointers here), and here is the work of another who is close will need just a few revisions.
Now students are off and running. This assignment really runs itself from here, and I really enjoy hearing what kids have to say about the structure of each pattern. It's neat to figure out which part of each pattern is staying the same, and which part is changing. Problem #2 allows us to talk about how the constant term doesn't have to be at the base of the figure, and Problem #4 challenges students to see the structure when a rule ends with subtraction rather than division. I'll leave it to you to decide how you frame these conversations with your own students.
A Quick Note on Supplies
It's very useful to have the following tools available to kids today. Both provide a great scaffold and entry point for different kinds of learners.
- Crayons, colored pencils or markers: I tell students, "crayons are not just for coloring - they can help you to make sense of problems and see structure!"
- Mini-whiteboards and markers at each table: These supplies really help students start some conversations with each other about the work. When students are stuck and can't decide if they're on the right track, I say, "Write something! The worst thing that can happen is that you're wrong, you erase it, and it's gone."
With a few minutes left in class, I really want to hammer home the importance of finding connections between the generalized sketch of a figure and an algebra rule. If students need to see it explicitly, I draw the 100th figure from the first pattern like this.
Then, I'll choose another problem from the remaining four and ask for a few volunteers to sketch the 1000th figure in that pattern. What ensues are a few informal, but useful conversations - and maybe even a little argument - about how the pattern is structured. I hope that this will help to send students off to finish strong on this assignment, as they complete it for homework tonight.