Direct and Inverse Variation

A direct variation is a situation in which two quantities — such as hours and pay, or distance and time — increase or decrease at the same rate. The ratio between the quantities is constant; that is, as one quantity doubles, the other quantity also doubles.

A mechanic who is paid hourly knows that working longer means making more money. That’s because his pay varies directly as the number of hours worked. As his hours increase, so does the amount of his paycheck.

A racecar driver knows that completing 100 laps before making a pit stop is better than completing only 80, because distance is directly proportional to time when driving at a constant speed. The longer she drives, the more distance she’ll cover.

Other examples:

• The circumference (C) of a circle varies directly as the diameter (d); that is, C = kd, where k = .
• The area (A) of an oil spill on land varies directly as the volume of oil spilled (V); that is, A = kV.

One quantity is directly proportional to another when the ratio of the two quantities is constant (the same). The constant is the constant of proportionalityand the ratio is a direct proportion.
(Source: SIMMS Integrated Mathematics: A Modeling Approach Using Technology; Level 1, Volume 2. Simon & Schuster Custom Publishing, 1996)

Two objects that vary directly are also said to be “directly proportional.” It should be noted that the graph of a direct variation, or a direct proportion, will always contain the origin (0, 0); conversely, if a line does not contain the origin, it is not a direct proportion.

Another widely-accepted definition is of direct variation is:
When two variables are so related that their ratio remains constant, one of them is said to vary directly as the other, or they are said to vary proportionately; i.e., when

, or y = cx,

where c is a constant, y is said to vary directly as x. The number c is the constant of proportionality (or factor of proportionality or constant of variation).
(Source: James, Robert C. and Glenn James. Mathematics Dictionary (5^thedition). New York: Chapman & Hall, 1992)

The National Council of Teachers of Mathematics (NCTM) states in Principles and Standards for School Mathematics (PSSM):

A major goal in [teaching Algebra 1] is to develop students’ facility with using patterns and functions to represent, model, and analyze a variety of phenomena and relationships in mathematics problems or in the real world. With computers and graphing calculators to produce graphical representations and perform complex calculations, students can focus on using functions to model patterns of quantitative change. Students should have frequent experiences in modeling situations with equations of the form y = kx, such as relating the side lengths and the perimeters of similar shapes. Opportunities can be found in many other areas of the curriculum; for example, scatterplots and approximate lines of fit can model trends in data sets.

The equation y = kx is the general equation for direct variation. This equation represents a linear function with slope k that passes through the origin. The PSSM also suggests that “[t]he study of patterns and relationships in the middle grades should focus on patterns that relate to linear functions, which arise when there is a constant rate of change.”

Direct variation can be studied through a process of mathematical modeling, in which students collect data and identify patterns. For instance, as in the Workshop 7 video, students might collect data that explores the relation between the volume of oil spilled to the area that the oil covers on land. The data will indicate that as one variable increases, the other variable increases proportionally. From a table or scatterplot of data, students can identify a linear equation that may be used to model the situation and allow predictions. According to the PSSM, “When students encounter a set of points suggesting a linear relationship, they can simply use a ruler to try several lines until they find one that appears to be a good fit and then write an equation for that line.” In addition, technology such as graphing calculators or computer algebra systems (CAS) can offer more precise methods of identifying a line of best fit.

In addition to revealing linear functions, direct variation equations provide a natural setting for understanding the concept of variables. In a classroom focused on symbolic manipulation, students may develop an implicit belief that variables are merely placeholders for numbers. For instance, in the equation 3x = 18, the variable x serves as a placeholder for the number 6. Consequently, the variable does not vary. In a situation of direct variation, however, such as the function y = 3x, the value of y varies as the value of x varies.

Direct variation appears throughout the Algebra 1 curriculum. Peggy Lynn, a high school teacher from West Yellowstone, MT, who appears in the Workshop 7 video, says, “[Students] will see the idea of a direct proportion when we work with similar shapes in geometry … and I’ll relate it back to a direct proportion and the slope, where their constant of proportionality is their scale factor.”

A direct variation is a situation in which two quantities — such as hours and pay, or distance and time — increase or decrease at the same rate. The ratio between the quantities is constant; that is, as one quantity doubles, the other quantity also doubles.

A mechanic who is paid hourly knows that working longer means making more money. That’s because his pay varies directly as the number of hours worked. As his hours increase, so does the amount of his paycheck.

A racecar driver knows that completing 100 laps before making a pit stop is better than completing only 80, because distance is directly proportional to time when driving at a constant speed. The longer she drives, the more distance she’ll cover.

Other examples:

• The circumference (C) of a circle varies directly as the diameter (d); that is, C = kd, where k = .
• The area (A) of an oil spill on land varies directly as the volume of oil spilled (V); that is, A = kV.

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 (5^thedition). 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)

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.

The links below are to pages within stable sites and are current as of the date of publication of this workshop. Due to the ever-changing nature of the Web, it is possible that some links may change. Should you reach a non-working link, we recommend entering a couple words from its description into the site’s search function, or into a Web-based search engine.

Related Standards

NCTM Middle Grades Algebra Standard
http://standards.nctm.org/document/chapter6/alg.htm
This Web page describes what students should know and be able to do algebraically in grades 6-8, and offers suggestions for the type of classroom activities necessary to develop conceptual understanding.

NCTM High School Algebra Standard
http://standards.nctm.org/document/chapter7/alg.htm
This Web page describes what students should know and be able to do algebraically in grades 9-12, and offers suggestions for the type of classroom activities necessary to develop conceptual understanding.

Direct and Inverse Variation Lessons and Resources

Mystery Liquids
http://www.pbs.org/teachersource/mathline/
lessonplans/hsmp/mystery/mystery_procedure.shtm
In this lesson from the PBS Mathline project, students collect data for two “mystery liquids,” create graphs of the data, identify linear models, and interpret their results.

Variation Lesson
http://www.purplemath.com/modules/variatn.htm
This site provides sample questions, applications, and solutions dealing with variation.

Cat and Mouse: Math Problem in Direct and Inverse Variation
http://www.articlesforeducators.com/math/000008.asp
This article solves a classic problem using direct and inverse variation.

Sample Problems
http://www.glencoe.com/sec/math/algebra/algebra1/
algebra1_03/study_guide/pdfs/alg1_pssg_G039.pdf
http://www.glencoe.com/sec/math/algebra/algebra1/
algebra1_03/study_guide/pdfs/alg1_pssg_G091.pdf
These two files contain sample questions about direct and inverse variation.

Resources on Questioning Techniques

Effective Classroom Questioning
This brochure from the Center for Teaching Excellence at the University of Illinois at Urbana-Champaign provides a brief overview of questioning techniques, with tips on how to ask questions effectively in the classroom. Available online in the Instructional Resources section of http://www.oir.uiuc.edu/Did/.

Classroom Questions
http://pareonline.net/getvn.asp?v=6&n=6
This article, by Amy C. Brualdi, discusses good questions, bad questions, and how to ask questions that foster student achievement.

Classroom Questioning
http://www.nwrel.org/scpd/sirs/
Entering “classroom questioning” into this site’s search function brings up several excellent articles on the benefits of effective classroom questions and reviews of the research literature on the subject.

Web Sites Relating to Oil Spills

Exxon Valdez Oil Spill Trustee Council Home Page
http://www.oilspill.state.ak.us/
This site includes a history of the spill, a list of facts regarding the spill and the affected area, and information about the council that is attempting to restore the area.

NMFS Office of Exxon Valdez Oil Spill Damage Assessment and Restoration
http://www.fakr.noaa.gov/oil/default.htm
The National Marine Fisheries Service provides an overview of the research and restoration efforts surrounding the Exxon Valdez oil spill.

Environmental Protection Agency – Oil Spill
http://www.epa.gov/oilspill/exxon.htm
This site includes a brief history of the Exxon Valdez oil spill, as reported by the U.S. Environmental Protection Agency.