Exploring an Arithmetic Sequence - Dots
Lesson 1 of 10
Objective: SWBAT develop representations for arithmetic sequences including graphs, tables, recursive and explicit equations.
This lesson sets the stage for students to develop conceptual understanding of arithmetic and geometric sequences. A key piece of this lesson is to get students in the habit of always looking at a sequence using multiple representations. Students will be cued to represent the pattern they see as a table, graph, recursive, and explicit equation.
I start today's class by looking at the Growing Dots pattern on the board together. I give students a couple of minutes to examine the pattern and then ask them to share out how they see it. I am looking for different descriptions of the pattern here so students get the idea that there is more than one way to see the pattern.
Next students get to work on Questions 2 through 4. I ask them to start on Question 2 individually and when they are finished can shift to working together for Questions 3 and 4. Because my students may not automatically look at a variety of representations, I might remind them that it might be interesting to see what this pattern looks like in a table or a graph.
As I circulate around the room, I watch for students who recognize the recursive pattern easily but have trouble finding the number of dots at 100 minutes without extending their table. I might ask them to look across their table and see if they notice a pattern that way. I might also ask them how they could represent this idea of adding 4 over and over again (or a set number of times).
If some students find the explicit equation quickly, I ask them to represent the pattern in as many ways as they can.
For the discussion section of this lesson, I want to have students share out as many representations of the Growing Dots pattern as possible. As students work in the Investigation section, I identify different students that I'd like to share out their work in the Discussion section. I might take a picture of the work and project it or have the student come to the board and share out.
I like to start by having a student share out his/her table. At this point, I think this is the most accessible representation to most students and most students have made a table by this point. I ask students to share what patterns they see in the table. Most will be able to identify the common difference and we'll note that to the right of the Out column. I ask students to connect this "adding of 4" each time to the pattern. Next, I try to elicit from students that this pattern continues in the Out column and therefore we could find any number of dots for a specified time if we just kept continuing the table. At this point I introduce the idea of a recursive pattern. I think it's an effective strategy to have students start writing recursive functions in words. I think if we introduce function notation right away, some students will immediately be lost. I try to elicit a sentence from students like: The number of dots = The number of dots in the previous figure plus 4.
Once I see that students can understand this pattern as a way of looking down the out column of the table, I'm ready to introduce the function notation. Students sometimes struggle with the f(t-1) piece as a representation of the previous figure so we spend some time making sense of this. Students are usually clear on the rest of the function notation.
Next, we take a look at a graph. Again, I ask students where they see this addition of 4 happening. We talk about how the extension of the graph would tell us the number dots needed for any number of minutes.
The conversation about the graph may be a lead in to the explicit equation or we may address the equation separately. Ideally, I want students to be able to see a pattern across the table (as opposed to the recursive pattern down the out column). I ask for suggestions for how students see the pattern and then we write an equation together. Again, I want to make sure to connect to the equation to the graph and to the pattern itself, asking students to make connections.
I finish today's lesson by telling students that this pattern is a particular kind of pattern called an arithmetic sequence. I let students know that what makes it an arithmetic sequence is that there is a constant difference between consecutive terms. We take one more look at the original table to see this pattern. I also let students know that moving forward, we will continue to look at tables, graphs, recursive and explicit equations to represent these kind (and some different kinds) of patterns.
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