This section describes the main mathematical and scientific concepts that will be discussed in this unit. Terms that may be included in a vocabulary list are italicized. These concepts will be referred to in the individual lesson plans. The scientific concepts are related to the mathematical concepts that will be used to study the behavior of each natural phenomenon.
Mathematical Concepts
Measurement: Magnitude, Dimensions and Units
Since natural disasters include large
magnitudes
of energy and mass,
scientific notation
will be used to represent vary large and very small numbers. Discussion and use of
measurement units
will be included in comparisons of magnitude. The following provides some examples of different types of quantities or dimensions, and the units used to label them. Most physical phenomena can be described by using the following dimensions: mass M, force F, length L, time t, and temperature T (Banks, 1998). These dimensions could be made concrete for students by having them measure quantities using a scale, ruler or tape measure, clock, and thermometer. Students should be prompted to list the units they are familiar with for each dimension.
Other phenomena can be described by using derivations of these dimensions, most commonly a ratio of one of two of these dimensions. Some derived quantities can be calculated after students simultaneously make measurements with two different instruments. Calculating derived dimensions would be a good exercise for student group measurement activities. Some examples of derived quantities include the following: velocity equals length per unit time, pressure equals force per area, density equals mass per unit volume, energy equals Newton meters or joules, and power equals joules per second or watts (Banks, 1998). Some examples of units include the following: kg for mass, Newtons for force, meters for length, seconds for time, and degrees Celsius or degrees Kelvin for temperature. Some examples of derived units include the following: meters per second for velocity, Newtons per square meter, kilograms per cubic meter, Newton meters, Joules per second (Banks, 1998). All of the derived dimensions are simply
rates
.
Mathematics is used to describe nature in several ways. Numbers are used to describe the relative magnitudes of measured phenomena, and units are used to specify the dimension and perhaps the measurement instrument that was used. Measurements recorded with the same units are used to compare magnitudes of similar events. Mathematical functions are used to describe relationships between different variables, in this case, the characteristics of certain natural disasters. Mathematical models are used to summarize relationships between the characteristics of natural disasters. They are ultimately used to answer questions that humans have about natural disasters, and predict the results of events that have not occurred, but may be possible or even likely. Specific examples of mathematical models for natural disasters are discussed in the Natural Science Topics section.
As prerequisite knowledge, students should be familiar with the use of the following mathematical concepts. A
variable
is a symbol, commonly a letter, used to represent a quantity in an algebraic expression or equation. A variable may be used to denote many values, quantities, magnitudes or numbers. An
algebraic expression
consists of arithmetic operation(s) on number(s) and variable(s). An expression also can represent many values. An
algebraic equation
is two expressions linked by an equals sign. An equation may represent any number of values for the variable(s). A
function
is an equation with two or more variables, where one of the variables (the dependent variable) appears alone on one side of the equation. Also, for each value of the input variable, there is a maximum of one value of the output variable. A function represents a relationship between two sets of numbers, each of which measures the magnitude of a type of quantity or dimension.
Multiple Representations of Functions
A
mathematical function
is a relationship of two or more variables. One of the variables is the output, or
dependent variable
. The other variable(s) are known as input(s) or
independent variable(s)
. A
mathematical model
is a function that is used to describe a real situation (Connally et. al., 2001). In using a mathematical model, it is important that any variables have been defined as a quantity that can be measured, the dimension of each variable is known, and the
measurement units
are known. Mathematical functions, and thus mathematical models, can be represented in several different ways:
equations, tables, graphs
, and
verbal descriptions
. Examples of equations include y = kx, y = a(x - h)
2
+ k, y = Ae
bx
. Students should be familiar with vertically and horizontally oriented tables, as well as hand-drawn and calculator graphs. Many students have difficulty with verbally describing equations and functions. Students will be encouraged to verbalize functions using phrases such as "y varies directly with x", "y varies as the square of x", "y varies as the square root of x", or "y is a power function of x". When creating or using a mathematical model, students will be required to use at least two different representations of the model.
There are several types of functions that will be studied and applied in this unit, including the following: linear functions (including direct variation),
quadratic functions
(square root functions), and
exponential functions
(logarithmic functions). If the students are in a Precalculus or Algebra 2 course, it may be appropriate to discuss the definition of an
inverse function
. In this case, the derivation of the mathematical models used in this unit can be included as explanation or perhaps even as exercises. These derivations would necessitate
solving multivariate equations for one of the variables
, i.e. "isolating a variable", which some Algebra 1 students have proven capable of executing, while others have problems solving one variable equations. Regardless, Algebra 1 students should at least be introduced to the processes of solving a bivariate equation for either variable, e.g. taking the square root, log, or square of both sides of an equation. Perhaps these processes are best introduced by providing univariate examples with numbers on one side of the equation.
As a prerequisite, students should be able to describe similarities and differences between types of functions, including comparison of inputs and outputs and the shape of the graphs. Direct variation and linear functions are the same in that they both have constant differences of the dependent variable. They are different because direct variation can be modeled by a proportion. Another difference is that a direct variation graph crosses the origin of the coordinate plane. Linear and quadratic functions differ in that there is no constant rate of change in a quadratic, best seen with a table. A linear function has no maximum or minimum, has no axis of symmetry. The range of an exponential function is the same as the general quadratic. Many students have difficulty describing the domain and range of continuous function. By using models containing continuous function to study natural disasters, these sets may be thought of as discrete, measured quantities. This may be aided by using data from tables in (Abbott, 2004), and discussion of magnitude scales for all types of natural disasters.
Mathematics can be used to solve problems involving natural disasters through the use of mathematical models. Questions humans have about natural disasters can be answered using
interpolation
or
extrapolation
from a table or graph, or through the use of a mathematical model per problem situation, and solving the resulting equation(s).
Natural Science Concepts
A
natural hazard
is a situation in the natural environment that exhibits "clear signs of danger" (Abbott, 2004). A
natural disaster
is a naturally occurring event that exceeds the ability of a region to "rescue and care for it's people, to clean up the destruction, and to begin reconstruction" (Abbott, 2004). Clearly, a natural hazard exists before it becomes a natural disaster. Also, not every natural hazard becomes a natural disaster; for instance, an earthquake that occurs in a sparsely populated area would not constitute a natural disaster, since it would not destroy much property or hurt many people. The following describes several types of natural disasters whose characteristics, causes, and results will be discussed and explored in this unit.
Waves
Since both tsunami and earthquakes involve different types of waves, a discussion about waves is a necessary prerequisite to the discussions of tsunami and earthquakes. Since Algebra 1 students have probably not studied periodic functions, it may be useful to graph a sine wave on an overhead graphing calculator, and label the wavelength and wave height, and briefly discuss period and amplitude. Discussion may be prompted by having students list the types of waves with which they are familiar, e.g. ocean waves, sound waves.
Tsunami
A
tsunami
is a high-speed sea wave of seismic origin created by "an underwater earthquake, landslide, or volcanic eruption"(Johnston, 2001). A
shallow water wave
is a water wave in which the wavelength is larger than the water height, or ocean depth (Banks, 1998). Since humans are concerned with a tsunami primarily at the shore, where the water is not deep, tsunami are explored as a shallow water wave. That is, people are most concerned with the height of a wave as it hits shore. Tsunami will be discussed in the context of water depth, wave velocity, period, wavelength and energy.
The wavelength, velocity and period of a shallow
ocean wave are related by the direct variation equation L = CT, where L = wavelength, C = velocity, and T = period (Bryant, 2005). This is an example of a linear relation between velocity and period. Period or velocity may also be expressed as a ratio of the other two quantities. Students may be asked to find any one of the three variable quantities, provided the other two.
A square root function that models tsunami velocity as a function of water depth is
v = (gD)
1/2
, where g = 9.8m/s
2
and D = water depth. Alternatively, this can be a quadratic model to find the depth of water if the wave velocity is known: D = v
2
/ g (Abbott, 2004, Banks, 1998). The energy of a water wave, particularly a tsunami, can be modeled as a function of wave height and wavelength, using the following quadratic equation: E W = 0.125pgH
2
L, where E W = wave energy in joules, p = density of water, g = 9.8m/s
2
, H = wave height, L = wavelength (Abbott, 2004). Alternatively, wave height can be modeled as a square root function of wave energy and wavelength: H = 2(2E W / (pgL)
1/2
). Students will use this relation to find both water depth and velocity. They should also be asked to compare the energy of shallow waves of different heights and wavelengths, in order to differentiate between linear and quadratic relationships. This is probably most easily found through using a tabular representation of the relation, then comparing the respective energies.
Volcanoes
A
volcano
is "a cone shaped mountain or hill, formed by the accumulation of hardened magma" with a "hole . . . from which lava and/or hot ash and gases erupt from deep underground" (Johnston, 2001). Volcanoes are commonly located near
subduction zones
between
tectonic plates
(Abbott, 2004). During an eruption,
pyroclasitic
bombs may be ejected from the volcano (Scheidegger, 1975, Abbott, 2004). A relation between the maximum distance a bomb may shoot and the initial velocity as it leaves the chute is given by the model D = v 0
2
/ g, where D = maximum distance, g = 9.8m/s
2
and v 0 = initial velocity (Scheidegger, 1975). This model is simplified in that it assumes the bomb is ejected from the volcano at a 45
o
This function is quadratic, and may be used to find the distance is the velocity is known. More commonly, this relation has been used in the square root form v 0 = (gD)
1/2
to estimate initial velocities of the bombs. Students will be asked to use this relation to find both the maximum distance of the projectile and the initial velocity, in different situations.
Earthquakes
An
earthquake
is shaking of the earth caused by
seismic waves
(Johnston, 2001), of which there are several types.
Seismic waves
are waves that travel from the
focus
of an earthquake (Johnston, 2001). The
epicenter
of an earthquake is the point on the earth's surface directly above the earthquakes
focus
, or where the actual sliding of a
fault
occurred (Abbott, 2004). Earthquakes will be discussed in the context of different magnitude scales, amount of energy released, and frequency of occurrence.
Most people are familiar with earthquakes as being classified according to the
Richter scale
, which was developed in the 1930s (Abbott, 2004). However, in the 1970s Hiroo Kanamori developed a more precise measurement of earthquakes, based on the physical properties of an earthquake, as opposed to seismic measurements (Abbott). The
seismic moment
is the product of rock rigidity, length of fault, relative movement of fault, and the magnitude (Scheidegger, 1975). The
moment magnitude scale
describes the moment magnitude as a function of seismic moment using the logarithmic model M W = 2/3 log 1 0 (M 0) - 6, where M W = moment magnitude and M 0 = seismic moment. Alternatively, the moment can be made a function of moment magnitude using the exponential function M 0 = 10
6
10
3/2Mw
. Students will be asked to find both the seismic moment and the moment magnitude by using this relation.
The total energy released in an earthquake can be related to the Richter magnitude using another exponential function: E = 10
11.8+1.5M
, where E = energy in ergs and M = Richter magnitude (Bercovici & Brandon). Alternatively, Richter magnitude can be a function of the amount of energy released: M = (2/3)log 1 0(E) - 7.86. Students will convert between Richter magnitude and energy.
Like most natural disasters, earthquake magnitude and frequency are inversely related, i.e. larger magnitude earthquakes occur much less frequently than smaller earthquakes (Scheidegger, 1975). The relationship between earthquake magnitude and the frequency of occurrence (recurrence interval) is modeled by a logarithmic equation: log (N) = 5.3-0.93M, where N = the cumulative number of earthquakes and M = minimum Richter magnitude (Bolt, 2004). This can easily be written so that N is an exponential function of M: N = 10
5.3
- 10
0.93M
. Similarly, magnitude can be a logarithmic function of the cumulative number of earthquakes: M = 5.7 - ( log 1 0(N) / 0.93 ). This relation will enable students to calculate recurrence intervals of earthquakes of given Richter magnitudes.
Tornadoes
A
tornado
is a very fast wind vortex, usually about 20 feet wide, extending downward from a cloud. The wind speeds in a tornado are among the fastest wind speeds measured (Johnston, 2001, Scheidegger, 1975, Abbott, 2004). The common scale used to measure tornadoes is the Fujita scale, which is listed in Abbott, p295. The wind velocity and Fujita scale number are related by the function v = 6.3(F + 2)
3/2
, where v = wind velocity, F = Fujita scale number. This is the velocity as a square root function of the Fujita scale number. This relation can also express the Fujita scale number as a quadratic function of the wind velocity, although it involves a cube root: F = ( v
2 / 36.7 )3/2
- 2. Students will use this relation to find both velocity and Fujita scale numbers.
Population growth
Students should be presented with a table of global population values for different years, and then graph a discrete data set to appreciate the rapid growth rate of the global human population. An exponential function that describes population as a function of time is p = p 0(1 + r)
t
, where p = population, p 0 = initial population, r = percent growth rate, and t = time, usually in years (Connally et. al., 2001). In this case, r would be negative if there existed a decline in population. Otherwise, r can be found as a ratio of the birth rate to the death rate. If we let 1 + r = b, the base of the exponential expression, the function becomes p = p 0b
t
. An alternative version of this relation describes the time it would take for the population to reach a given amount p: t = logb( p / p0). Students will use both functions to predict the future global population, provided the current growth rate or 1.3% (Abbott, 2004). This relation will also be used to find out how long it will take the global population to reach a certain level.
