Mathematics is a powerful tool for solving problems in the world around us. Using very abstract models we are able to describe and predict the sometimes very complex behaviors of people, markets, diseases and physical objects. It is often difficult for students of mathematics to grasp some of these abstract concepts without concrete examples. In particular, it can be a challenge to motivate students without showing some relevance to their own lives. I would like to capitalize on the students' natural curiosity about their own brains to motivate them to learn mathematics.
As a teacher of mathematics at Metropolitan Business Academy (MBA) in New Haven I have used real examples from the seminar. They include comparing the reaction time of a giraffe and a mouse. What is the relation between the number of neurons and brain diameter? How much louder is a jet taking off than a vacuum cleaner? Why do some musical notes sound pleasant while others do not? Relevant mathematical models, and their representations, will be used in answering these questions.
In addition to business careers, many students at MBA will pursue careers in the health professions. Additionally, everyone is curious how the brain works, especially as it pertains to them. The students will use examples taken from the seminar to create mathematical models and then represent the data and their models. These relevant examples will help motivate my students to understand this very abstract subject matter.
The New Haven curriculum for Algebra II and Precalculus includes units on the family of functions. Consequently the curriculum unit developed here could be adopted for either course, although this unit will be devoted to the Precalculus curriculum in order to include logarithm and periodic functions.
Students coming into Algebra II, in particular, as well as Precalculus often do not have a grasp on the properties of various functions and the type of change modeled by each. Furthermore, given a mathematical model, or equation, there frequently is difficulty in displaying data and mathematical models for precise, effective, quick analysis. This unit will develop the students ability to select an appropriate function in order to best represent the data.
The unit will reinforce the notion of data, mathematical models, and graphs as representations of change. The uniquely mathematical perspective of change is the rate of change. In mathematical terms rate of change may be depicted as slope in a graph. By working with data and models, and then making visual representations of that change, it is intended that students will have a better understanding of this concept. Additional skills in rates, ratios and proportions will also be bolstered in depicting and analyzing the data.
Each member in the family of functions will be developed using data from the seminar. The functions included will be linear functions, power functions, quadratic functions, polynomial functions, exponential functions, logarithm functions and periodic functions.
The linear function in slope intercept form is, y = mx + b where m represents slope and b is the y intercept. This function will be used to compare the time it takes for nerve conduction to travel from the foot to the brain of the giraffe and the mouse.
The power function, in function notation, takes the form f(x) = x
, where a is a constant real number. This function will be used to show the relationship between total number of neurons and brain diameter. The quadratic function, a specific type of polynomial function, might also model the same sort of change as above, but takes the form f(x) = ax
+ bx + c, where a, b, c are constant real numbers. The polynomial function utilizes a combination of operations on variables and constants with non-negative whole number exponents. Of these, the third degree polynomial will probably prove most useful.
The logarithm function generally may be stated in any base. The natural base of e is used in many growth functions while this unit will utilize base 10. The conventional function notation is f(x) = ln(x) for natural base and f(x) = Log(x) for base 10. This function, in base 10, will be utilized in measuring sound levels in decibels.
Finally, periodic functions will be demonstrated. These are the trigonometric functions sine, cosine, and tangent. For modeling purposes the sine function will mostly be utilized in the form f(x) = a*sin(b*x + c)+d , where a is amplitude, b is the frequency, which determines period, c determines phase shift, and d determines vertical shift. This function will be used to determine the frequency of sound and the activation of the auditory system.
Using concrete and interesting data the intent is to have students better understand rate of change, how best to model and represent that change by choosing appropriate mathematical models and graphs, and apply skills involving rate, ratio, and proportion. Students will also be introduced to music theory.