Arithmetic Sequences and Series (Lesson 7.2)
Unit 7 - Day 3
Write explicit rules to describe sequences with a common difference
Generate a sum formula for arithmetic sequences using the idea of averages
Find missing terms of an arithmetic sequence
Solve for the term number in which a sequence reaches a particular sum.
Quick Lesson Plan
Today students look at Mallory’s running times during the month of June to explore the idea of arithmetic sequences. Students identify that her time increases by five minutes every day and use this to fill in her running log. While students may use a recursive pattern to find the first few values in the table, they should quickly recognize the need to make use of structure to find values for days later in June. We specifically ask for June 29th so students recognize that her running time on that day is exactly five less than her running time on the 30th. This idea of a constant (common) difference is critical to the rest of this lesson and ties in important ideas about a constant rate of change and linear functions.
Students use the idea of her average run time to find the sum of all 30 days. Another strategy is to realize that the days can be summed in any order and the sum of the first and last day is the same as the sum of the second and second to last day, is the same as the sum of the third and third to last day, and so on. This sum of 175 will occur 15 times since there are 15 pairings of days. Both of these strategies lead to the same sum formula, though written slightly differently. Be ready to build on student thinking and use the debrief to discuss both methods.
In the debrief, highlight the fact that there are two ways to write the explicit formula (and in fact, there are infinitely many, since we could use any day in June as our “anchor”). Make sure students understand the necessity of the (n-1) when starting with June 1st. I ask students when we want to start adding the five minutes, and they are able to reason that this is not until June 2nd, thus creating the need to “back track” our equation. When using the a(0) term, ask students if there was ever a day Mallory ran 10 minutes. Students should realize that no, this is not an actual running time but acts as a placeholder so that on June 1st the equation amounts to 15 minutes.