Students will examine the structure of the three sequences presented in arith_sequence_day2_open. I give students the following directions:
Find the next three terms of each sequence. Then determine which of these sequences seems to be different from the others.
Once students have had an opportunity to work on the sequences and think about the patterns individually, I have them Think-Pair-Share.
In order to get a pulse on where the thinking is in the room I tell the students that we will call the black sequence #1, red is #2 and blue is #3. I have them do a non_verbal cue to show me which sequence they decided is different from the other two. Most students will be able to come up with the fact that the red sequence (#2) is different from the others because it has a common difference between its terms. This allows us to specifically name that sequence as arithmetic.
During this investigation students work with arithmetic sequences from a variety of perspectives. At this point, I am not going to show students the formula for finding the nth term of an arithmetic sequence (a(n)=d(n-1)+a(1)). Students will rely on using the common difference to find what I call the "constant adjustment" for each sequence. As long as they know the constant difference and one other term in the sequence they can find this constant by using inputs and outputs.
Examples of this are shown in the document: arith_sequences_day2.pdf. I feel that determining the explicit formula in this way leads to a deeper understanding of how the terms in the sequence are related to one another. The formula should be shown at some point in time. In the past I have learned that when in introduce the formula too early students blindly "plug and chug" without really knowing why.
This investigation offers students several opportunities to make sense of challenging problems (MP1). For example, in questions 3 and 4, students need to grapple with the concept of determining a common difference when the terms are not consecutive. However, I like the way this understanding is scaffolded. In question #3, students are given a table which can help them visualize the spacing between the terms in the sequence. This scaffold is removed in #4 and students need to reason quantitatively about "the difference between the terms" and the "number of jumps" to get them to a certain number in the sequence (MP2). By engaging in these investigations, students lay a foundation for understanding the rate of change of the linear function without that term explicitly being used.
This simple Ticket out the Door allows the students to show their understanding but also to be creative about their thinking. I ask students to do the following:
Make up your own arithmetic sequence and show at least the first five terms. Then, come up with an explicit formula that will represent your sequence.
I encourage students to check several terms in their sequence to ensure that their formula holds true for all of their terms. In this exit ticket, I can quickly spot check for (1) understanding of an arithmetic sequence and (2) Students ability to use the terms in a sequence to create an explicit formula.
I was happy to see that all students were able to come up with an arithmetic sequence for this ticket out the door. At the same time, all students were able to come up with the common difference for their sequence (of course, these two go hand in hand). Not all students were able to determine the constant that should be added to the formula to ensure that it works for all terms in the sequence.
The biggest confusion for students during this lesson was the input vs. the output. If a student saw that a1=5 they did not know whether to plug in the 5 or the 1 for n. For some students I began to write the terms in function notation (e.g. a(1)=5) or even list the terms in a table:
This helped more of the students to visualize the role that the sub-script was playing in this notation. However, this will be something that we certainly have to revisit in the next lesson.
This investigation is from Mathematics Teaching in the Middle School, March 2012, NCTM.