Lesson Number 1 - Functions as Mathematical Models of Natural Phenomena
Objective(s) - students will be able
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1) To discuss and provide examples of how mathematics is used to describe nature, and natural disasters in particular
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2) To define and provide an example of a function and a mathematical model
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3) To provide examples of dimensions and the units used to measure them
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4) To apply the concepts of function, independent and dependent variable to describe relationships between natural phenomena using multiple representations of functions.
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5) To classify the relationship between pairs of variables associated with natural disasters as linear, quadratic, or exponential
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6) To pose questions about natural disasters that may be answered by using a mathematical model
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7) Describe the difference between a natural hazard and a natural disaster
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8) Define tsunami, volcano, earthquake, and tornado
Prerequisite skills
graphing two-variable data, evaluating expressions, translating verbal phrases and sentences to algebraic expressions and equations, solving equations, creating and interpreting tables and graphs on coordinate plane
Materials
Ruler or tape measure, thermometer, clock, scale
Worksheet or text with tables of different types of function, without the equations
Vocabulary - Terms & Symbols
function, domain, range, independent variable, dependent variable, f(x) = y, vertical line test, mapping diagram, earthquake, volcano, water wave, tornado, tsunami, dimension, unit
Strategy
Students should be prompted to discuss different types of phenomena that are measured. They should also discuss what types of instruments are used for the measurements. This discussion should result in defining
dimension
as in (Banks, 1998). As a class, we list different units that are used to measure the dimensions of mass, force, length, time and temperature. Review the definition of function. Use the student as function input metaphor - can't go to two places at once. Encourage students to come up with measureable quantities and measurement units. Prepare a measurement activity that involves a simple linear function. Prepare tables of different types of functions related to natural disaster for review of function types. Prompt students to classify each table as a type of function. Describe natural hazards and disasters, the different types of natural disasters, and their characteristics. Use the case studies to provide an example of each type of disaster to be studied. An overhead slide show would nicely show the results of the disasters. Ask questions about the characteristics of natural disasters, i.e. "are these two related? How so?" Provide encouragement and feedback.
Student tasks
Form small groups
List examples of quantities that we measure in the environment
Classify each type of quantity by its dimension, and give an example of units
Discuss how some pairs of quantities may be related
Find two quantities that have a functional relationship
Describe the functional relationship in words using "is a function of" and in symbols using a = f(b)
Present one of your functions to the class, with both verbal and symbolic representations
Engage in measurement activity involving length, time, and force, using ruler, clock, and scale A distance, rate, and time problem would suffice.
Classify a function represented with a table as linear, quadratic or exponential.
Answer elementary questions about the natural disasters discussed, define terms.
Questions
For each of the following pairs of natural phenomena, determine the dimension and possible units of each variable, and determine if there may be a functional relationship between the two. Then identify the independent and dependent variables, and write a sentence that describes the relationship.
time of day & temperature
month & temperature
location & weather
seismic wave (moment) amplitude & Richter scale magnitude
number of deaths & earthquake magnitude
number of deaths & building materials
Give examples of dimensions and units
What is the difference between a natural hazard and a natural disaster?
What questions can we ask about these natural disasters?
What type of mathematics would help us answer these questions?
Lesson Number 2 - Modeling Tsunami Height, Tornado Wind Speed, and Safe Distance from Volcanic Bombs with Quadratic and Root Functions
Objective(s) - students will be able
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1) To use mathematical models to describe observed relationships between characteristics of natural disasters when values of some variables are known
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2) To derive mathematical models from existing equations and relations (solve multivariate equations for one of the variables)
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3) To provide examples of the application of a quadratic or square root function to natural disasters
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4) To represent functions as equations, tables, graphs, and verbal descriptions
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5) To describe the domain and range of functions used to model natural disasters
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6) To draw and label a simple pictoral representation of a wave
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7) To calculate wavelengths, periods, depths, and velocities of near-shore ocean waves
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8) To describe the relationship between water depth and wave velocity of shallow water waves in at least two different ways
Prerequisite skills
Vertex form of quadratic function, translations of parabolas, solving equations using square roots
Materials
graphing calculator, graph paper, worksheets with student tasks, questions, and a place to write answers
Vocabulary - Terms & Symbols
Parabola, vertex, axis of symmetry, x-intercepts, maximum, minimum, Wave, water depth, wave velocity, run up, velocity
Strategy
A wave can be modeled by having two students hold a piece of rope in front of the class. One student propagates a wave, and the other students can be prompted to estimate the height of the wave. Another student propagates a larger wave, and again the students can estimate the height of the second wave. A discussion explaining the definition of amplitude follows. The amplitude here can be related to water depth instead of wave height. Provide one form each of the equations for tsunami and tornado. Prompt students to derive the other form of the relation. Provide students with specific calculation questions that can be solved using the presented equations. Provide encouragement and feedback
Student tasks
Describe a wave, and its characteristics.
Give examples of waves
Give examples of waves involved with natural disasters
Take notes on tsunami and tornado equations
Attempt to solve equations for the independent variable
Answer questions using either form of the given relations
Questions
Define a wave, tsunami, volcano and tornado
Find the wavelength of a tsunami with given velocity and period
Find the period of a tsunami with given velocity and wavelength
Find the velocity of a tsunami with given wavelength and period
Find the height of a tsunami with given velocity
Find the velocity of a tsunami with given water depth
Find the energy of a tsunami with given water depth and wavelength
Find the wavelength of a tsunami with given water depth and energy
Find the height of a tsunami with given wavelength and energy
What is a shallow water wave?
Find the safe distance from a volcano that shot a pyroclasic bomb with a given initial velocity
Find the initial velocity of a bomb that lands a given distance from the chute of a volcano
What is the difference between a tornado and a hurricane?
What scale is used to measure tornadoes?
Find the magnitude of a tornado with given wind speed
Find the wind speed of a tornado with a given magnitude
Compare the energies in two shallow water waves with two given sets wavelength and height
What is the relationship between these energies compared to their heights?
Which natural phenomena have quadratic relationships?
Write a sentence describing the relationship between tsunami height, wavelength and velocity
Write a sentence describing the relationship between initial velocity of volcanic bomb and the safe distance from the volcano
Write a sentence describing the relationship between tornado wind speed and magnitude
