Part 1: Direct Variation
Direct variation is a critical topic in Algebra 1. A direct variation represents a specific case of linear function, and it can be used to model a number of real-world situations.
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.
- 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 proportionality and 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
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).
, or y = cx,
(Source: James, Robert C. and Glenn James. Mathematics Dictionary (5th edition). New York: Chapman & Hall, 1992)
Role in the Curriculum
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."