options: For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Distinguish between situations that can be modeled with linear functions and with exponential functions.
problem_activity: Activity
This is the first of two activities where students define sequences with equations and use their equations to answer questions about the context (MP2). The population values were purposefully chosen in order for students to focus on creating representations (like tables and graphs) and not on calculating “best fit.”
Making spreadsheet and graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Arrange students in groups of 2. Display the table for all to see.
| years since 1990 | Population                   \(A\) | Population                   \(B\) |
|------------------|------------------------------------|------------------------------------|
|        0         |               23,000               |               3,125                |
|        1         |               29,000               |               3,750                |
|        2         |               35,000               |               4,500                |
|        3         |               41,000               |               5,400                |
|        4         |               47,000               |               6,480                |
Ask students, "What are some ways you could figure out if the sequences represented by the populations are arithmetic, geometric, or neither?" (I could use a spreadsheet to calculate if the terms have a common difference or a growth factor.) Give students quiet work time and then time to share their work with a partner. Select 2–3 groups to share their ideas with the class while recording them for all to see.
Student Facing
The table shows two animal populations growing over time.
| years since 1990 | Population                   \(A\) | Population                   \(B\) |
|------------------|------------------------------------|------------------------------------|
|        0         |               23,000               |               3,125                |
|        1         |               29,000               |               3,750                |
|        2         |               35,000               |               4,500                |
|        3         |               41,000               |               5,400                |
|        4         |               47,000               |               6,480                |
Are the sequences represented by Population
\(A\)
and Population
\(B\)
arithmetic or geometric? Explain how you know.
Write an equation to define Population
\(A\)
.
Write an equation to define Population
\(B\)
.
Does Population
\(B\)
ever overtake Population
\(A\)
? If so, when? Explain how you know.
Student Response
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Anticipated Misconceptions
If students incorrectly identify the type of sequence from the table, encourage these students to graph the points or, if they have already made an equation, check that their equation generates the sequence correctly.
Activity Synthesis
The goal of this discussion is for students to understand how different representations of functions are useful in different ways. For example, using technology to make a graph from the definition of the
\(n^{\text{th}}\)
term of each sequence can lead to the solution efficiently while testing values or trying to write an equation to solve may be more time consuming or challenging.
Select students to share what kind of sequence is represented by Population A along with their recursive or
\(n^{\text{th}}\)
term definitions. Then do the same for population
\(B\)
. Record for all to see any representations made by students that they used to write their equations, such as extra rows in the table, graphs, or spreadsheets. If not brought up in student explanations, ask how each representation shows different features of the sequence, for example, the growth factor or rate of change.
Conclude the discussion by inviting students to share how they determined whether Population B would ever overtake Population A. Display any representations used by students, such as graphs, equations, or spreadsheets, for all to see during explanations.
Representing, Conversing, Listening: MLR 7 Compare and Connect
. Create a need for students to use language to make sense of representations of sequences, and use their representations to find a term that meets a condition (the year when Population B overtakes Population A). As students prepare their work for display, look for approaches that extend the sequences in a recursive fashion, and approaches that make use of the definition for the
\(n^{\text{th}}\)
term. When students are comparing the approaches and representations used, prompt for and amplify comments about what is easier to see and understand in each, and comments about what might make each more complete or clear. Call students’ attention to the different ways students reasoned about when Population B overtakes Population A, and to the different ways these methods are made visible in each representation.
Design Principles: Maximize Meta-awareness, Support Sense-making


correct_option_index: 2