Teacher professional development and classroom resources across the curriculum
Teacher professional development and classroom resources across the curriculum
One of the key features of algebraic mathematics is the use of symbols instead of numbers. In algebra, we learn how to generalize and explore the rules of arithmetic by using variables that can stand for any number. We become less concerned with answers to specific problems and more concerned with the relationships between the entities and values under investigation. The advantage of this is that our analyses can be applied to a wider variety of situations than would be possible if we restricted ourselves to using specific numbers that apply only to a particular situation.
An example of this concern with relationships is the familiar slope-intercept form of the equation of a line: y = mx + b. Typical high school algebra courses reveal how this relationship can be applied to any number of situations. We can apply the equation to the cost of painting a house, for example, by letting y represent the total cost; x, the number of gallons of paint purchased; m, the price per gallon of paint; and b, the fixed cost of supplies such as brushes and buckets. The total cost can then be found by substituting real-world values for the variables and performing the indicated operations.
In general, a linear equation expresses a relationship between the two variables, x and y. These variables represent two values that are related in some way. In other words, changing one leads to a change in the other. The constants of the linear equation, m and b, help show specifically how x and y are related. These constants are determined by the conditions of the situation that we wish to understand and model with the equation.
In the house-painting example mentioned above, we saw that b represents a fixed, up-front cost. Graphically this value determines the placement of the line on the coordinate plane. Specifically, it identifies the point at which the line intersects the y-axis.
Many times in mathematics we have to choose what it is we care most about. In other words, in a given situation we must decide which quantities to de-emphasize and which to give our full focus. In our present discussion, we are going to ignore b for the time being, because what we are really interested in is how changes in one variable affect the other. In our painting example, the up-front cost becomes increasingly less important as more paint is purchased, so we should probably pay more attention to the price of paint than to those fixed, up-front costs. Knowing the price per gallon will enable us to determine exactly how our total cost changes as we use more or less paint. In the general case then, m is more interesting to us right now because it lets us calculate how a change in x will affect the value of y.
The number m compares the change in y to the change in x for a given line. We call this ratio of changes, the "slope." If we know two points on the line, (x1, y1) and (x2, y2), we can find the slope by taking the difference in y values and dividing it by the difference in x values. This slope ratio is commonly referred to as "rise over run."
Slope is a useful concept because it describes how two quantities change in relation to one another. The slope of a linear equation is constant; it never changes. While many real-world situations can be modeled with a linear equation, most cannot. For one thing, most real-world situations can't be modeled using just multiplication and addition. Equations involving powers of variables, such as the equation for the velocity of a falling object, don't lend themselves to the simple notion of a constant slope implied in the linear equation model. Let's look at how we can generalize the concept of slope to talk about such non-constant rates of change.
In our painting example, we might arrange a deal with the paint store that the more paint we buy, the less we pay per gallon. This means that while the total cost increases as we buy more paint, the rate at which the total cost changes actually decreases. Our slope is no longer constant; it, like y, depends on which x (amount of paint) we choose to consider. To better understand how real-life situations change, we need a more comprehensive concept of slope.
Notice that if we attempt to find the slope between two points on a curve, we end up with a straight line that doesn't correspond with the curve very well. Furthermore, notice how the slope between two points on a curve changes depending on which two points are selected.
Considering just a few examples also makes it clear that generally the further the two selected points are away from each other, the worse the correlation between the slope of the line and what is actually happening to the curve over the chosen interval.
If we could somehow have a notion of slope between two points on a curve that are not very far apart at all, we could practically eliminate the discrepancy between the line determined by those points and the path of the curve. Such a conceptual tool could help us understand mathematically all sorts of curves and the situations they represent. To do this, we can shrink our view as far as we wish and consider the slope between two points that are extremely close to one another on the curve.
On a curve, imagine a point whose horizontal position is x. Now imagine a second point on the curve that is some very small horizontal distance, Δx, from x. This point's horizontal position is x+Δx. The slope between these two points is represented by this expression:
This is the familiar "rise over run" expression indicating the rate of change between these two very slightly separated points. If we let their horizontal separation, Δx, approach zero, we will have an expression for the "instantaneous" rate of change for that section of the curve. Note that we cannot make the separation equal to zero, because division by zero is undefined. We can, however, talk about the slope as Δx "gets arbitrarily close" to zero. This quantity, called a derivative, is the generalized notion of slope that we need to deal with many complicated (i.e., "curvy") real-world models.
The derivative is a powerful mathematical tool because it allows us to describe in great detail not only how quantities change in relation to each other, but also how their changes change. We can now account for the vast amount of real-world phenomena that do not conform to the simple, linear notion of a constant slope.
We don't have the space in this text to explore how to find derivatives of specific functions, but we'll need to use some of them later as we attempt to mathematically model synchronization. The following table gives a few basic functions and their derivatives.
The derivative is one of the key ideas in differential calculus, which can be thought of as the mathematics of change. Calculus uses the concepts of infinite processes and infinitesimal steps to describe how changing quantities (e.g., those that grow, shrink, move, or proliferate) vary.
Ancient Egyptian thinkers, trying to compute the volumes of various solids, made the first strides toward this understanding. Greek mathematicians, such as Eudoxus and Archimedes, carried on this legacy by developing the "method of exhaustion," which involved dealing with infinite processes. As the West descended into the so-called Dark Ages, Indian, Arab, and Persian mathematicians flourished, making great strides toward an understanding of derivatives. By the late 1600s, European mathematicians were building upon the techniques of past thinkers, using calculus-like methods to understand physical processes. It was at this point that the traditionally-held "fathers of calculus," Isaac Newton and Gottfried Leibniz, simultaneously put centuries' worth of pieces together, and added many significant contributions of their own, to form a coherent whole called "the calculus."
Calculus provides us with the mathematical tools to deal with rates of change in a sensible manner. But as with any discipline, the tools are only as effective as the skill of the one who wields them. Using the tools of calculus to model real-world situations requires the ability to see a dynamic situation and recognize the relevant quantities and rates of change involved, and how they relate to each other. With a grasp of the elements and relationships in play, we are better prepared to express what is happening using equations that we can analyze to make predictions about the future and to find new understandings of our world.