As students enter the classroom, they begin work on the Warm up. Because arithmetic sequences are the output of a linear function (with a domain restriction), we begin this lesson by practicing writing the equation of a line through two points. I expect my students will have mastered this skill in the Algebra 1 course at my school. I anticipate my students will need about 10 minutes to work through today's Warm up exercises.
While students work, I circulate around the room to assign a score to each student's homework according to my homework rubric. As I do this I note what parts of the assignment were difficult for students. When I have finished checking the homework I ask students to compare their answers to the assignment with a partner. I encourage them to discuss their strategies and try to make sense of their peer's ideas [MP3]. After a few minutes I will use Cold Calling to initiate a discussion of the recursive process that students used to solve the problem. We will discuss this for a while. I will not share the solution to the problem until I have shown students how to use a spreadsheet application to perform the recursion:
I ask students to open up a spreadsheet page on their TI NSpire calculators and use an overhead projector to walk them through the steps of solving the problem on a spreadsheet. Through this activity, students learn that spreadsheets can save lots of time when a task requires repetitive calculations [MP5].
Here we focus our attention as a class on the analysis of arithmetic patterns. We will work together to learn the formal methods of defining patterns explicitly and recursively. We build on the sequence strips activity from the previous lesson illustrating arithmetic progressions ([a), (d), and (f).
I ask students to work together to come up with a rule for each sequence that gives the term as a function of n, the term's position in the sequence [MP2, MP8]. As students work together, I offer support and hints as necessary. After 10 minutes I take a formative assessment - quick poll to check student progress. I ask students to submit their sequence rule, which gives us a collection of expressions to consider. We check each idea together to see if substituting 1 for n yields the correct initial term, 2 for n yields the correct second term, etc.
My intent is to employ the students' observations in a compare and contrast discussion of notations for describing sequences and linear functions. To ensure that we make strong connections to student’s Algebra 1 knowledge, it is important to help students recognize an arithmetic sequence as a linear function with a restricted domain, the positive counting numbers. During this segment of the lesson, students generally come up with slope-intercept form of an equation easily. Occasionally, one or more groups share the arithmetic sequence formula as well.
As we transition to the practice I’ll give students the opportunity to take notes on the standard formula used to generate the nth term in an arithmetic sequence. We'll review some examples of how to use this formula to find a term, the term number, or the common difference. We will also examine the graph of an arithmetic sequence and compare it to the graph of a linear function. I encourage my students to take note of the formulas in their notebooks and to sketch the illustrations and examples that I write on the board. Of course, I also want them to participate actively in the conversation that accompanies my presentation.
After our interactive note-taking session, students will work in table groups to practice with the arithmetic sequence formula. During this time I circulate with a clipboard to note problems that students are having both with the math content and with remaining on task. I stop to ask questions and give hints as necessary.
After students have worked for 10 or 15 minutes, I ask them to pair up and check answers. When they are finished I ask if anything needs to be resolved at the board. If so, I ask for student volunteers to work out a problem at the board or if there is confusion around a particular problem I solve it for them.
We wrap up today's lesson with students helping me generate a list of all the problem types from the group work. We discuss that fact that starting with different given information can alter the difficulty of the problem. For example, starting with the first 5 terms makes finding the explicit formula very straightforward, whereas starting with two non-consecutive terms makes the process more difficult.
Next, students take a minute to find the most difficult problems on their homework sheet and star them. I ask them to do this because when a student knows in advance that a problem is meant to be challenging they are often more persistent in trying to solve it [MP1].
The evening's homework, Arithmetic sequence WS B, is very similar to the classwork. I do this so that students have the opportunity to improve their fluency on the skills introduced through group work during class.