Secondary Math I: Linear and Exponential Functions
This module follows a previous module in which students were introduced to arithmetic and geometric sequences through the use of situations—such as story contexts or diagrams of growing dot patterns—that can best be represented as discrete arithmetic sequences with a constant difference between terms, or as discrete geometric sequences with a constant ratio between terms. As this module begins, students should be proficient in using the initial term, along with a constant difference or constant factor, to write recursive or explicit expressions to represent the terms in an arithmetic or geometric sequence. Students have also modeled these sequences using tables and graphs, and they typically note the constant difference or the constant growth factor on their tables of data with additional annotations.
In this module, students extend their understanding of the behavior of arithmetic and geometric sequences to include continuous linear and exponential functions. Linear functions, defined as a relationship between two quantities that grow by equal differences over equal intervals, and exponential functions, defined as a relationship between two quantities that grow by equal factors over equal intervals, can arise from situations that are either continuous or discrete. Students are required to attend to both aspects of a relationship defined between two quantities by a context and determine if the relationship is continuous or discrete and if the relationship is linear or exponential. In this module, students come to understand that situations in which one quantity grows or decays by a constant percent rate of change per unit interval relative to another quantity is a continuous exponential function. Students also grapple with the meaning and usefulness of recursive notation when describing a continuous situation.
The tasks in the module are designed and sequenced in “Learning Cycles” following the Comprehensive Mathematics Instruction Framework developed by the Brigham Young University Public School Partnership. In this framework, develop understanding tasks are designed to surface student thinking about key ideas, strategies, and representations of the module; solidify understanding tasks examine and extend those emerging ways of thinking; and practice understanding tasks refine those ideas for fluency and transfer to other contexts and applications.
We annotated student work based on the following criteria & standards:
Students can distinguish between situations that can be modeled with linear functions and with exponential functions (HSF-LE.A.1).
Students can construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs—including reading these from a table (HSF-LE.A.2).
Students observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity growing linearly (HSF-LE.A.3).
Students interpret the parameters in a linear or exponential function in terms of a context (HSF-LE.B.5).
Students look for and make use of structure (SMP.7).
Two different tasks from the module are annotated in this work: 4.1 Piggies and Pools: Connecting the Dots, a Develop Understanding Task and 4.5 Getting Down to Business, a Solidify Understanding Task. Piggies and Pools is designed to surface ideas about discrete and continuous linear and exponential functions by presenting students with contexts that may be continuous or discrete, as well as linear or exponential. Getting Down to Business revisits these ideas, specifically focusing on the distinguishing features between linear and exponential functions, including the fact that exponential growth eventually overtakes linear growth.
More specifically, Task 4.1 Piggies and Pools: Connecting the Dots, A Develop Understanding Task, is meant to surface ideas about discrete and continuous linear and exponential functions for students. Previously, students have worked with arithmetic and geometric sequences, with the expectation that students should be proficient with notation for describing recursive and explicit equations. In this task, students are beginning to distinguish between arithmetic and geometric sequences as discrete linear and exponential functions, and linear and exponential functions that are continuous. A major focus in this task is on distinguishing between linear and exponential functions as well as the discrete or continuous nature of the context. Students are also working toward an understanding of percent change as a growth factor in an exponential function. The standards of focus are HSF-LE.A.1, HSF-LE.A.2, and SMP.7.
Task 4.5, Getting Down to Business, A Solidify Understanding Task, is the fifth task in the module. Many of the standards in this task are solidifying ideas that surfaced in Task 1 of the module. Previously, students have worked with arithmetic and geometric sequences. The expectation is that students should be proficient with notation in recursive and explicit equations, but students in this task are working toward an understanding of percent change as a growth factor. This piece of student work was selected from a special education classroom. In this task, a major focus is on distinguishing between linear and exponential functions as well as the discrete or continuous nature of the context. There is also a focus on percent growth rate in exponential functions. The standards of focus are HSF-LE.A.3, HSF-LE.A.2, HSF-LE.B.5, HSF-LE.A.1c, and SMP.7.
The collection of work in the samples represent the thinking of six different students, three of which come from a special education classroom. Each student is represented by his or her work on task 4.1 and again on task 4.5, for comparison. An additional sample of student work is included from task 4.5, in order to present a broader range of responses.
Some student work annotations include “wonderings” about significant shifts in student thinking that seemed to occur almost spontaneously while working on the tasks. Such wonderings appear as questions in the annotations, or as cues that highlight particularly surprising features in student work. Such annotations may promote further thought and reflection for teachers reviewing these materials. Of particular interest is that students often erase a large section of work and replace it with something that is more accurate or insightful. The reasons for these shifts in perspective cannot be precisely determined, but in some cases the annotations provide possibilities to spur discussion.
Three particular observations are worth noting, as they relate to instructional choices for special education students:
Special education students who could successfully write both recursive and explicit equations for exponential growth situations when the growth factor was a whole number struggled to write similar equations when the growth factor was given as a percent. Typically, students calculated the percent change for the first unit interval, but then used this value as a constant difference, treating the exponential situation as a linear one.
Special education students struggled with the larger, more realistic number choices and descriptions of quantities in task 4.5, such as “5 million dollars” or “0.5 million per year.”
In annotating this set of student work, we found that the revisions that take place in student work is insightful. Unfortunately, all students typically replace their original thinking with their revised thinking, rather than keeping a record of both. Fortunately, we were able to read some of the erased original thinking in student work in order to tease out the story of how students naturally revise their work when they encounter some kind of dilemma or discrepancy. This seems like important work to keep track of, and we would encourage students to not erase or obscure their original thinking when they decide to revise their work.