How do viruses spread through a population? From ebola to bird flu, humans are surrounded by deadly viruses.
In this lesson, students use exponential growth and logarithms to model how a virus spreads through a population and evaluate how various factors influence the speed and scope of an outbreak.
REAL WORLD TAKEAWAYS
The world is full of viruses that are biologically programmed to spread. “Successful” viruses spread easily but kill very slowly so that they have more time in the infected host to contaminate others.
Humans can keep harmful viruses in check by: creating immunity in people without the virus or restricting the contact those with the virus have with others.
Write an equation to represent exponential growth; describe the effect of each parameter on the overall model
Solve exponential equations using a variety of methods, including logs
Appropriate most times as students are developing conceptual understanding.
Exponential Functions (Adv.)
Exponential Functions (Adv.)
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).
For exponential models, express as a logarithm the solution to <em>ab<sup>ct</sup></em> = <em>d</em> where <em>a</em>, <em>c</em>, and <em>d</em> are numbers and the base <em>b</em> is 2, 10, or <em>e</em>; evaluate the logarithm using technology.
Use the structure of an expression to identify ways to rewrite it. For example, see x<sup>4</sup> — y<sup>4</sup> as (x<sup>2</sup>)<sup>2</sup> — (y<sup>2</sup>)<sup>2</sup>, thus recognizing it as a difference of squares that can be factored as (x<sup>2</sup> — y<sup>2</sup>)(x<sup>2</sup> + y<sup>2</sup>).
Write a function that describes a relationship between two quantities. (a) Determine an explicit expression, a recursive process, or steps for calculation from a context. (b) Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. (c) (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Interpret the parameters in a linear or exponential function in terms of a context.
Model with mathematics.
Construct viable arguments and critique the reasoning of others.
Look for and make use of structure.