This opening activity (arith_sequence_day3_open.pdf) is all about students examining the structure of a sequence (MP7). A major goal of studying linear sequences and linear equations is learning to identity the pattern between the input and output variables. If a student can determine the rate at which a sequence is changing, then they know are well on their way to modeling the relationship.
The sequences in this Opening are scaffolded to develop this understanding. In the first sequence, students can simply list all of the missing terms to find the common difference, the rate of change. In the second, while it is still possible to list all of the missing terms, it becomes cumbersome (or tedious) form many students.
Put the first question up and let students work on it by themselves for 2-3 minutes. Then, let them share their ideas with a partner (MP3). Take some time to let students share their ideas with the class. There are a lot of ways to think about these types of problems. Because of this, students can learn a great deal from one another by hearing each other's ideas. As students are sharing, frequently check with the class to ensure they are following the student's (or pair of students') reasoning for finding the common difference. You can also ask students to find the explicit formula for each sequence as an extension.
Go through the same process with the second question, again emphasizing different ways of thinking about the same problem. Many students will start to look at the number of jumps from one term to the other. For example, in the second example this would be 8 jumps from the 2nd term to the 10th. They will then compare this to the distance they have to cover. In the second example the distance from 3 to 27 is 24. When students divide these values they can calculate the common difference as a ratio (i.e., 24/8=3). This ratio of rise to run is the rate of change or slope of a linear function.
This part of the lesson provides students an opportunity to practice working with arithmetic sequences. One of the difficulties that students face is becoming familiar with notation for describing sequences. This practice worksheet (arith_sequence_day3_practice.doc) provides a nice introduction to the notation. On this worksheet I let students get right to work. As questions arise we discuss them. In my experience, students can figure out a great deal of the notation on their own, even it is not completely intuitive.
This Ticket Out (arith_sequence_day3_close.pdf) will assess whether or not students can create a sequence based on the work they had done in class. I post the questions from the beginning of class and tell students to make up a similar question of their own. I remind students that the question on the left is simpler than the question on the right. They should choose to model their question after the one they feel is most appropriate for them.
I instruct students to put their question on one side of their paper and the solution to their question on the reverse. If time permits, students can switch papers and answer each other's questions.