Arithmetic And Geometric Sequences Practice Homework Assignments
Students work in groups of three to complete a sequence puzzle that reinforces what we have learned about the numeric and algebraic representations of sequences. The goal of this activity is for students to practice matching up symbolic and numeric representations of arithmetic and geometric sequences. [MP2]
I discuss why I use groups of three and the formative benefits of this activity in my sequence dominoes reflection video. When this activity is used the first time, it will be necessary to print 6-10 sets of sequence dominoes, one for each group. Printing sets on different colors of card stock makes it easier to keep the sets of pieces together. Each group receives a set of 24 cards of the same color that have either a sequence formula on one side and a few terms of a sequence shown. The students are told to match up the pieces, domino-style, so that they have a trail that starts at the card labeled "start" and ends at the one labeled "finish."
When students have finished, I project an image of the completed puzzle on the board and ask groups to check their work. We discuss briefly which of the pairings were most difficult [MP2 and MP3] and any strategies students had for making those easier.
This lesson parallels that of the previous day. Here, we focus our attention on geometric patterns and learn the formal methods of defining geometric number patterns explicitly and recursively. In a prior lesson, we sorted sequence strips into piles according to the math used to move from one term to the next. In this lesson, we focus on the geometric sequences from this activity. Using the sequence strips labeled (b), (g), (h) and (j), I ask students to work together to come up with a rule that expresses the term in terms of n, the term's position in the sequence [MP7]. As students work together, I offer support and hints as necessary.
When students have had time to develop these rules, I ask them to share their ideas using a quick poll on the TI NSpire Navigator System. When we have a collection to look at, 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. This leads to a discussion of sequence notation and how it is similar to and different from exponential function notation. [MP3] Students need to connect this new notation to what they have learned in Algebra 1, so it is important to help students understand that the geometric sequence is an exponential function whose only input values are the positive counting numbers.
After the group discussion of student findings, I write the formulas on the board and we work a few examples of using the formulas to find the nth term, the term number, or the common ratio. We also examine the graph of a geometric sequence and compare it to the graph of an exponential function. Students take note of the formulas, illustrations and examples that I write on the board and actively participants in the discussion.