In today's task, students look for the first time at an arithmetic sequence that is decreasing rather than increasing. This task also asks them to make sense of a word problem; the sequence is not as readily apparent as in previous tasks.
I begin class by asking students to articulate, once again, what an arithmetic sequence is. I ask them if they think it would still be an arithmetic sequence if the number in the out column was decreasing by the same amount each time. I take a variety of student responses and then we read through Something to Chew On together.
Before students get to work on the problem, I spend some time making sure they are clear on what the problem is asking them. I also remind them that they should plan to answer the question of "representing" using multiple representations, as usual!
Students get to work on Questions 1 through 3 in small groups. The decreasing arithmetic sequence is included in Question 1, so I want to make sure that students use as many representations for this problem as they can. They can then compare that work with the increasing sequence in Question 2. Question 3 is more of an extension problem, but many of my students will be ready for it. There are many different ways to solve Question 3, so I will be looking for students to present their various approaches.
We begin the whole group discussion by looking at the table for Question 1 in detail. It will be interesting to see what groups included a first differences column in their table. The main point of the discussion is to figure out if we are looking at an arithmetic sequence if the numbers are decreasing by the same amount in the out column. Next, we look at the graph. I ask students to compare and contrast this graph and table with previous arithmetic sequences we have seen. How are they similar and different?
Next, we compare the recursive and explicit functions for Questions 1 and 2. Again, I want students to note the differences and similarities between the two different problem situations.
If time permits, Question 3 is a good opportunity for students to share their mathematical thinking and problem solving skills, the main purpose of SMP 3: Construct viable arguments and critique the reasoning of others.
One of the main takeaways from this lesson is for students to begin to be able to recognize an arithmetic sequence in its different representations. For example, I want students to be able to look at an explicit equation and determine if it might represent an arithmetic sequence. In order to help students keep track of the characteristics of arithmetic sequences, I ask them to complete this identifying an arithmetic sequence table so they have a reference sheet for future work.
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