Experience First:

In “Geri’s Greeting Cards” students use the context of greeting cards to look at both a linear and exponential function, connecting them to their corresponding sequence (arithmetic and geometric, respectively), and comparing them to each other. 

 

In questions 1-3, students see that the number of packages Geri is stocked is increasing by 35 packages every hour. The common difference in the packages over each half-hour interval points to a constant rate at which Geri is stocking cards.

 

In question 4, students are given information about how the number of greeting cards sold annually changes over time, this time as a fixed percent decrease. Students may use a variety of strategies for question 4, beginning with calculating 2.8% of 6.5 and subtracting this from 6.5, all the way to immediately calculating 0.975(6.5). In question 5, students will see how a fixed percentage change corresponds to a common ratio. Instead of memorizing a rule about using (1+r) or (1-r) as the growth rate, we want students to see that finding a ratio between two consecutive terms will always give the common ratio/constant proportion.

 

In question 6, students will complete a table of values, working forwards and backwards and in question 7 they will again make explicit the key difference between linear and exponential functions.

 

Question 9 has students reflect on the two functions they wrote in this lesson and how they are the same or different. This conversation will be an important part of the debrief (see notes in the “Formalize Later” section.)

​

Monitoring Questions:

• Is the common difference you saw of +35 also the constant rate of change? Why or why not?

• [Question 2] What kind of function is this?

• What does it mean when a quantity is decreasing by 2.8% every year?

• What does this ratio [in question 5] tell you?

• How is the ratio [in question 5] related to the percent decline that was given to you?

• How do you go backwards and figure out what happened before 2020?

• How is this equation similar to the explicit rule for a geometric sequence? What’s the same? What’s different?

• Does x represent the same quantity in P(x) an G(x)?

• What else is the same about functions P and G?

• What else is different about the functions P and G?

​

Formalize Later:

Today’s lesson transitions students from arithmetic and geometric sequences to linear and exponential functions. These are more alike than different! The pattern of growth and the general form of the equation are identical (but traditionally use different variables). The difference is the domain. While the domain of sequences is the positive integers, the domain of a linear or exponential function is all real numbers (unless restricted in some way by the context).

 

Note that there is a slight change in vocabulary when transitioning to functions. We do not refer to the common difference of a line, for example. We refer to the constant rate of change, or the slope. For exponential functions, there is a plethora of vocabulary words for the multiplier. Common ratio is typically used for sequences, though can apply to more general functions as well. Oddly, the phrase is often changed to constant ratio (perhaps to mirror constant rate of change). The AP Precalculus framework often uses constant proportion. Growth/decay factor and constant multiplier are also used. All of these phrases mean the same thing and can mostly be used interchangeably. 

 

When debriefing question 9, we suggest making a large list on poster paper or on the whiteboard. Write down everything students say, even if it seems trivial. Keep using the question “what else?” to garner additional (and deeper) responses. Be prepared that some students will focus on surface features such as that P(x) represents the number of packages and G(x) represents number of annual greeting cards, or that the values themselves are different. The heart of the discussion should be comparing their formulas and growth rates.