Understand that sequences are a special type of function whose domain is the positive integers.
Write an explicit rule for arithmetic sequences using the common difference and any term in the sequence.
Apply understanding of how arithmetic sequences grow to determine the common difference, find missing terms and reason about arithmetic sums.
Quick Lesson Plan
In today’s activity, students look at Mallory’s training schedule during the month of July 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 July. We specifically ask for July 30th so students recognize that her running time on that day is exactly five minutes less than her running time on the 31st. This idea of a constant (common) difference is critical to the rest of this lesson and both reviews and previews ideas about constant rate of change and linear functions. We also specifically ask for her running time on July 16th since this marks the exact midpoint of the month. Students may use this value in questions 3 and 4. Students may also average the first and last term directly as a way to find her average run time. Be prepared that students may also try to add all 31 terms and divide by 31.
Questions 5 and 6 are all about getting students to see that any term in the sequence can be used as an “anchor point” for the sequence. Instead of using repeated subtraction to find an initial term and then repeated addition to find the desired term, it is intuitive to simply add on or subtract from the given term. The debrief should focus on how students decided how many “copies” of 0.2 to add or subtract. This of course is the (n-k) portion of the explicit formula and represents how many terms the desired term is away from the given term.
How many more minutes will Mallory run 3 days from now?
What’s the average of the numbers 5, 6, 7, 8, and 9? What’s the average of the numbers 3, 5, 7, 9, 11?
How many days have passed since July 18th? Why does that matter?
Is David going to run more or less than 7.6 miles on this day?
In this unit we focus on functions exhibiting exponential growth or decay. That of course begins with a conversation about the rate of change of exponential functions and to study this, we compare and contrast linear and exponential functions. However, we back up even further and begin by studying change in arithmetic and geometric sequences which are special types of functions whose domain is the positive integers. The focus on absolute change in sequences (how do the terms change from one to the next?) is a great introduction to the rate of change in more general functions (how do the outputs change with respect to the inputs?)
Lesson 4.1 focuses on arithmetic sequences, or sequences with a common difference. One important goal of the lesson is to show students that explicit rules can be written given any term in the sequence, not just the 0th term or the 1st term. This is analogous to point-slope form for linear functions.
Though the AP Precalculus course framework does not explicitly mention partial sums of sequences, the Calc Medic curriculum does address them informally for the purpose of differentiating the features of arithmetic and geometric sequences. Knowing how a sequence grows is paramount to determining a strategy for finding a partial sum. In the next lesson, we do this in reverse. Knowing how geometric sequences grow prohibits us from using the same strategy as we did for arithmetic sequences to find a partial sum (i.e. finding the “middle” value and multiplying by the number of terms). Note that we will not ask students explicitly to find the nth partial sum or use the partial sum notation.