Classroom Activities
Cycle One, Lesson One
The first lesson, which takes place during the Preparation and Presentation phases of the unit’s first learning cycle, will focus on students’ general comprehension of exponential growth via examples of its occurrence in evolutionary biology. Students will watch a video depicting spliced news clips taken from around the advent of the COVID-19 pandemic.25 Students will then participate in a whole-class “pandemic” activity, in which one student is selected at random to be “Patient Zero.” Patient Zero will have the chance to infect three other students in a game of tag; then those three students will infect three more students each, etc. From there, we, as a class, will create a graph of the number of infections (frequency) over the number of “days” (time), so that students can see the curved, hockey stick pattern of a graph depicting exponential growth. We will then return to the video clip, during which there is an image of a graph illustrating the onset of the COVID-19 pandemic depicting exponential growth. Students will have a better contextual understanding of what this graph means and how it directly applies to the spread of COVID-19.
From there, students will be given a list of examples and non-examples of exponential growth. Students will work in partner pairs to determine whether each item is or is not an instance of exponential growth. These partner pairs will then combine with other partner pairs to compare their results. As a class, we will go through each example and will debate and discuss whether or not these examples are, in fact, depicting exponential growth. Upon completion of discussing these examples, students will, first in partner pairs and then independently, practice determining whether or not tables are showing exponential growth.
Cycle One, Lesson Two
In the second lesson, which takes place during the Generalization phase of the unit’s first learning cycle, students will be formally introduced to the formula for exponential growth, f(x) = abx (the formula for exponential growth without percent). Students will utilize this formula in partner pairs to translate given tables and word problems into equations. These partner pairs will then partner with others to determine if they got the same answers. Each newly formulated group will be instructed to put one translated equation on the board, so that the class can check their own and each other’s understanding of this concept. Once students are comfortable translating tables and word problems depicting exponential growth into equations, students will be tasked with another set of tables and word problems; however, this set will include linear growth patterns as well (which students have covered in a prior unit). Students will have to be able to differentiate between these two types of functions in order to successfully complete this task; this differentiation will facilitate students’ better understanding of exponential growth bias and will (hopefully) encourage students to apply this understanding to any future pandemic they may experience in their lifetimes.
Cycle One, Lesson Three
During the third lesson, which still takes place during the Generalization phase of the unit, students will be introduced to the exponential growth equation that deals with percent increase, f(x) = a (1 + r)x. Students will first practice translating a given percent into a decimal or fraction, so that they can confidently plug this value in for r. Upon successful understanding of how to determine and plug in the r-value, students will translate word problems depicting exponential growth with percent into equations; students will simplify these equations by adding 1 together with r (rather than keeping them as two separate entities). Once students are confident in utilizing the exponential growth formula with percent, students will be provided with an assortment of word problems which will depict both regular exponential and percent increase exponential functions as well as linear functions. Students will have to determine which formula to use based on the contents of the word problem. This will further protect students against harboring exponential growth bias toward what are undoubtedly incidences of exponential growth that they encounter in the future.
Cycle Two, Lesson One
The first lesson of cycle two sees a return to the Preparation phase. Students will be introduced to exponential decay through the concept of effective vaccination; during the first lesson, students will watch a news clip about the advent of the COVID-19 vaccine.26 Despite the resounding efficacy of the COVID-19 vaccines, the advent of new variants has not truly allowed for the exponential decay of the disease to occur; however, if the vaccine were to work perfectly and if 75% of the population were to get vaccinated, we would absolutely see a pattern of exponential decay at work.27 To best illustrate this concept, students will observe exponential growth of disease spread through a tree diagram; this same diagram will be continued after indicating where vaccination of the public occurred to show exponential decay of that same disease.
Similar to the happenings of the first lesson of cycle one, students will be given a list of examples and non-examples of exponential decay. Students will work in partner pairs to determine whether each item is or is not an instance of exponential growth. These partner pairs will then combine with other partner pairs to compare their results. As a class, we will go through each example and will debate and discuss whether or not these examples are, in fact, depicting exponential decay. Upon completion of discussing these examples, students will, first in partner pairs and then independently, practice determining whether or not tables are showing instances of exponential decay.
Cycle Two, Lesson Two
In the second lesson, which takes place during the Generalization phase of the unit’s second learning cycle, students will form the understanding that the exponential growth formula, f(x) = abx, is also used for exponential decay; they will learn through explicit instruction that the b-value, in the case of exponential decay, must be between the values of 0 and 1 and, therefore, must be written either as a fraction or as a decimal.
Students will utilize this formula in partner pairs to translate given tables and word problems into equations. These partner pairs will then partner with others to determine if they got the same answers. Each newly formulated group will be instructed to put one translated equation on the board, so that the class can check their own and each other’s understanding of this concept. Once students are comfortable translating tables and word problems depicting exponential decay into equations, students will be tasked with another set of tables and word problems; however, this set will include tables and word problems that depict linear growth and exponential growth as well, so that students can learn to effectively differentiate between each type of function.
Cycle Two, Lesson Three
During the third lesson, which still takes place during the Generalization phase of the unit, students will be introduced to the exponential decay equation that deals with percent decrease, f(x) = a(1 - r)x. Students will translate word problems depicting exponential decay with percent into equations; students will simplify these equations by subtracting the r-value from 1 (rather than keeping them as two separate entities). Once students are confident in utilizing the exponential decay formula with percent, students will be provided with an assortment of word problems which will cover all types of functions touched on thus far: linear growth, exponential growth, exponential growth with percent, exponential decay, and exponential decay with percent. Students will have to determine which formula to use based on the context of the given word problem and appropriately plug values into said formula in order to successfully prove their understanding of these functions.
Final Assessment Lessons
In the lessons that cover the Application phase of the unit, students will have two tasks to complete: (1) researching and presenting on a graph they’ve found depicting either exponential growth or decay and (2) taking a summative assessment to show what they’ve learned about exponential (and linear) functions.
Research and Presentation
For the first task, students will research and identify a real-world instance of an exponential function (with teacher assistance to ensure the information is coming from a reliable source). Students will work in groups of three to create posters that (A) identify and define various aspects of the graph shown, including the x and y-axes and the interpretation of a single point shown on the graph, (B) explain (in words) the story the graph is telling, and (C) explain at least 3 reasons as to why the graph is or may be important to understand. These students will be assigned the following roles:
- Researcher/Presenter 1: This student (who also serves as presenter 1) is in charge of the computer during the research portion of the task. They are taking direction from their classmates (as well as themselves) to inform what type of exponential relationship they want to explore. Additionally, this student will present the information listed in part A of the poster.
- Poster Writer/Presenter 2: This student creates an artistically appealing poster with the input of the group that provides the above-mentioned information (A-C) to the reader; they will present the information from section B of the poster.
- Poster Illustrator/Presenter 3: This student creates an illustration with the input of the group of what the graph is about; this illustration will be included on the poster; they will present the information from part C of the poster.
The presentations given as part of the students’ final assessment will likely improve achievement in Communicating Reasoning (as laid out by CAASP), as it will encourage students’ “ability to put together valid arguments to support [their] own mathematical thinking…” Furthermore, students’ use of critical thinking in part C to determine why the graph may be important will likely bolster student success in Problem Solving and Modeling & Data Analysis, as it will facilitate their “ability to analyze real-world problems, or build and use mathematical models to interpret and solve problems” as they relate to exponential functions. (The rubric for the students’ posters and presentations will be provided in this section of this unit).
Summative Assessment
For the second task, the summative assessment, students will be evaluated on their understanding of exponential (and linear) functions in terms of the following:
- Identifying instances of exponential or linear growth or decay in graphs
- Translating x/y tables into linear or exponential equations
- Translating word problems into linear or exponential equations
- Evaluating exponential functions
(The final assessment for this unit will be provided in this section).
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