Part 2: Inverse Variation
Inverse variations are excellent vehicles for investigating nonlinear functions. A number of real-world phenomena are described by inverse variations, and they are typically the first functions that students encounter that do not cross either axis on a graph.
An inverse variation is a situation in which one quantity increases while another quantity decreases -- such as the number of diners and serving size for a given amount of food, or speed and travel time for a given distance. The product of the quantities remains constant; that is, as one quantity doubles, the other quantity is cut in half.
A caterer who takes a watermelon to a picnic knows that each person will receive more watermelon if there are fewer attendees, but each person will receive less watermelon if there are more attendees. That's because the amount of watermelon for each person varies inversely as the number of attendees. The more people, the less each person gets.
A truck driver knows that driving at 75 miles per hour will get her to her destination faster than driving at 65 mph, because time is inversely proportional to speed. As her speed increases, her travel time decreases.
- The length (l) varies inversely as the width (w) for a rectangle of constant area (A); that is, A = lw.
- The depth (h) of oil in a cylinder varies inversely as the area of the cylinder's base (B); that is, as the cylinder becomes narrower, the oil becomes deeper, or V = Bh.
Inverse variation: When the ratio of one variable to the reciprocal of the other is constant (i.e., when the product of the two variables is constant), one of them is said to vary inversely as the other; that is, when
or xy = c,
y is said to vary inversely as x.
(Source: James, Robert C. and Glenn James. Mathematics Dictionary (5th edition). New York: Chapman & Hall, 1992)
Two objects that vary inversely are also said to "vary indirectly" or to be "inversely proportional."
Alternative definition: One quantity is inversely proportional to another when the product of the two quantities is constant. An inverse proportion can be described by an equation of the form xy = k, where k is the constant of proportionality. The equation of an inverse proportion can also be written in the form .
(Source: SIMMS Integrated Mathematics: A Modeling Approach Using Technology; Level 1, Volume 2. Simon & Schuster Custom Publishing, 1996)
Role in the Curriculum
Inverse variation provides a rich curricular complement to direct variation. As teacher Peggy Lynn says in the Workshop 7 video, "I like teaching these two topics in the same context because of their relationship to each other." The constant of proportionality in a direct variation represents a quotient; by contrast, the constant of proportionality in an inverse variation represents a product. "Division and multiplication go hand in hand, so the students can relate to that," Peggy says.
Direct variation and inverse variation are related topics, and it makes sense to study them in parallel. Because they have striking differences, the contrast allows students to gain a deeper understanding of various functions. "Students should have experience in modeling situations and relationships with nonlinear functions," according to the PSSM. Inverse variation allows students to consider nonlinear functions. The graph of an inverse variation never crosses the x-axis or the y-axis, nor does it pass through the origin. "[Students] should connect their experiences with linear functions to their developing understandings of proportionality, and they should learn to distinguish linear relationships from nonlinear ones," the PSSM states.
When teaching inverse variation - as with direct variation and other activities involving mathematical modeling - asking students to gather data helps to spark their interest. Students are required to think more when investigating a phenomenon using a hands-on approach, though they often don't realize they're learning because they're having fun. In addition, when students are exposed to "messy data," they must make thoughtful decisions in order to identify functions that fit the data well enough to be useful in making predictions. Making sound mathematical decisions is the basis of effective modeling, so providing opportunities for students to make choices helps to develop their analytical abilities.
For real-world explorations involving inverse variation, it will be necessary to collect enough data to make the nonlinear pattern obvious. Too few points may result in a pattern that appears to be linear. Once sufficient data have been collected, students can use tables and graphs to represent the data.
Finally, students should compare direct variation with indirect variation, illuminating the differences and highlighting the similarities. For instance, they might describe the relationship between the general equations
y = kx and . They should recognize that the constant of proportionality in the direct variation is a quotient of the variables, while the constant of proportionality in the inverse variation is a product. Or they might consider the graphs, since a direct variation is linear and passes through the origin, while an inverse variation is a curve with no x- or y-intercepts. Making these comparisons will allow students to understand the differences within a family of functions.