MAKING CONNECTIONS: A CASE FOR PROPORTIONALITYAuthor(s): Kathleen Cramer and Thomas PostSource: The Arithmetic Teacher, Vol. 40, No. 6, FOCUS ISSUE: EMPOWERING STUDENTSTHROUGH CONNECTIONS (FEBRUARY 1993), pp. 342-346Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41195594 .

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MAKING CONNECTIONS: A CASE FOR

PROPORTIONALITY Kathleen Cramer and Thomas Post

reasoning is one form

Proportional of mathematical reasoning. Many aspects of our world operate ac-

cording to proportional rules. In the science classroom, proportionality surfaces when density is explored, when balance beams are used, and when any two equivalent rates are compared. In the mathematics class- room, proportionality surfaces when prop- erties of similar triangles are examined, when scaling problems are investigated, and when trigonometric functions are de- fined. The importance of proportional rea- soning is stressed in the Curriculum and Evaluation Standards for School Math- ematics (NCTM 1989, 82).

The ability to reason proportionally develops in students through grades 5 -8. It is of such great impor- tance that it merits whatever time and effort must be expended to assure its careful development. Students need to see many problem situations that can be modeled and then solved through proportional rea- soning.

Defining Proportionality The ability to solve missing-value prob- lems has been used to indicate that a student is a proportional reasoner (Karplus, Pulas, and Stage 1983; Noelting 1980). Missing- value problems are the typical tasks found in middle school mathematics textbooks wherein three or four values in two rate pairs are given and the fourth is to be found. The following is a missing-value story problem. The standard algorithm taught to solve this type of problem involves setting up a proportion and using the cross-product algorithm.

The formula for mixing a certain shade of blue paint is 2 parts blue paint and 3 parts white paint. At this rate, how much

Kathleen Cramer teaches at the University of Wis- consin, River Falls, WI 54022. She has studied the teaching and learning of rational numbers for a number of years. Thomas Post teaches at the Univer- sity of Minnesota, Minneapolis, MN 55455. He is the codirector of the National Science Foundation- sponsored Rational Number Project.

Proportional reasoning is more than learning the cross-product algorithm.

white paint is needed if 9 parts of the blue paint is used? (2/3 = 9/v; Ix = 27; л = 13.5)

Post, Behr, and Lesh (1988) believed that using solutions to missing-value prob- lems as the sole indicator of proportional reasoning is much too restrictive, since answers lend themselves to purely algorith- mic and possibly rote solutions. The factors involved in defining proportional reason- ing are more complex.

One way to document knowledge in mathematics is to describe the behaviors that depict understanding. We understand enough about what it means to be a propor- tional thinker to realize that it involves the following:

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• Knowing the mathematical characteris- tics of proportional situations

• Being able to differentiate mathematical characteristics of proportional thinking from nonproportional contexts

• Understanding realistic and mathemati- cal examples of proportional situations

• Realizing that multiple methods can be used to solve proportional tasks and that these methods are related to each other

• Knowing how to solve quantitative and qualitative proportional-reasoning tasks

• Being unaffected by the context of the numbers in the task

Understanding the mathematical char- acteristics of proportional situations is the most important part of this picture. One critical mathematical characteristic of pro- portional situations is the multiplicative relationship that exists among the quanti- ties that represent the situation. This multi- plicative relationship can be explored through tables, algebraic expressions, and coordinate graphs. By examining the math- ematical characteristics of proportional situ- ations, one sees the importance of making mathematical connections that will inevi- tably empower students to function intelli- gently when solving problems.

This article first explores the mathemati- cal characteristics of proportional situa- tions and then explains that this knowledge will enable students to solve a proportional- reasoning problem in several ways. The article concludes by analyzing the math- ematical connections that are made as mathematical characteristics are explored and applied.

Mathematical Characteristics of Proportional Situations As just stated, the critical component of all proportional situations is the multiplicative relationship that exists between the quanti-

342 ARITHMETIC TEACHER

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ties that represent the situation. Consider this proportion problem:

The scale on a map suggests that 1 centimeter represents an actual distance of 5 kilometers. The map distance between two towns is 8 centimeters. What is the actual distance?

A table can help highlight this relation- ship (see table 1).

The numerical relationship that exists between the two quantities of map distance and actual distance can be expressed in

two ways. If we multiply map distance by 5 km/cm, we find the corresponding number of kilometers for the actual distance. If we multiply the actual distance by 1/5 cm/km, we find the corresponding number of cen- timeters for the map distance. The constant factors 5 or 1/5 can be used to express either form of the numerical relationship algebra- ically: actual distance = 5 km/cm x map distance

or map distance = - cm/km x actual distance.

In all proportional situations, the nu- merical relationship between quantities can be expressed by a rule in the form у = mx9 where m is one of the constant factors relating the two quantities.

The graph of the rule у = 5л:, with у = actual distance and x = map distance shows another mathematical characteristic of pro- portional situations (see fig. 1). The graph of у = 5x is a straight line, climbs from left to right (has positive slope), and passes through the origin. The graph of у = 1/5* has similar characteristics. In all propor- tional situations, the points of the graph lie on a straight line. In real-world settings these lines always have positive slope. The points from table 1 are highlighted on the graph along with two points, (1.5, 7.5) and (2.5, 12.5), which are not recorded on the table. Note that these points fall on the line. All rate pairs for the given proportional situation fall on the line.

Graphing linear equations is a topic de- veloped in the middle grades. Students graph variations of the general linear equa- tion у = mx + b, explore different definitions of slope, describe the characteristics of line graphs, and relate the characteristics of line graphs to the general equation (m is slope and b is j-intercept). By applying our knowledge of straight-line graphs to graphs of proportional situations, we see that for the ruley = mx, m is the slope of the line. The slope of the line of proportional situations is always the constant factor relating the two quantities. For our map example the constant factor 5, which can be used to define the relationship between map dis- tance and actual distance as v = 5л' is the slope of the line for the graph of у = 5л' We also know that у = mx crosses the origin because in this equation "ft" is 0.

Another interesting characteristic of pro- portional situations can be shown by re- cording the different rate pairs (actual dis- tance/map distance) found in table 1 as fractions: 5/1, 10/2, 15/3, 20/4, and 25/5. All these fractions have a value of 5. Again, we see the presence of the constant factor 5. The reciprocal rate pairs, 1/5, 2/10, 3/15, and so on, all have a value of 1/5. This special characteristic of equivalent rate pairs that exist in all proportional situations enables one to use the cross-product algo- rithm.

The following list summarizes the math- ematical characteristics of proportional situ- ations:

FEBRUARY 1993 343

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1. A constant multiplicative relationship exists between two quantities and can be

expressed in two ways. 2. All rate pairs describing a given pro-

portional situation are equivalent. The same statement is true of the reciprocal of these rate pairs. These two constants of propor- tionality define the multiplicative relation- ship.

3. The rule that expresses the multiplica- tive relationship is always y = mx, where m is one of the constants of proportionality.

4. Graphically, all points for a propor- tional situation fall on a straight line pass- ing through the origin. In real-world set-

tings these lines always have positive slope. All rate pairs for the particular situation fall on the line.

5. The slope of the line is the m in the

equation y = mx and is one of the constant

multiplicative factors relating the two quan- tities y and x.

Proportionality can be explored

through tables and algebraic sentences.

The importance of understanding the mathematical characteristics of proportional situations can be highlighted by the follow-

ing two problems:

1 . If you travel to a foreign country, you exchange dollars for the currency used there. In England you could exchange $3 for 2 pounds. How many pounds could you exchange for $21?

2. Sue and Julie were running equally fast around a track. Sue started first. When she had run nine laps, Julie had run three laps. When Julie had completed fifteen laps, how many laps had Sue run?

Superficially the two problems look alike. Each contains three pieces of information with one unknown. They look like missing- value problems found in a sixth- or seventh-

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grade mathematics text. Thirty-two out of thirty-three preservice students solved both problems by setting up a proportion and using the cross-product algorithm to reach an answer. The cross-product algorithm is an appropriate strategy for the money- exchange problem because that context represents a proportional situation. It is not an appropriate strategy for the running-laps problem because that problem is not a proportional situation and depicts an addi- tive, and not a multiplicative, situation.

An understanding of why procedures

work and under what conditions proce- dures can be applied are objectives that are often lacking in mathematics instruction. When one has a superficial understanding of a concept, it is easy to apply memorized rules in the wrong situation.

If we look closely at the running-laps problem, checking it against our list of characteristics of proportional situations, we can see the importance of having a deeper understanding of this context. Be- ing able to look beyond the superficial characteristics of proportional situations

344 ARITHMETIC TEACHER

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will enable students to make appropriate decisions as to when and if a procedure can be applied.

Constructing a table helps to identify the numerical relationship between the two quantities (see table 2).

The numerical relationship between the number of Julie's laps and the number of Sue's laps can be expressed algebraically: Sues laps = Julies laps + 6. The numerical relationship is a constant sum, not a con- stant factor.

The graph in figure 2 is a straight line with positive slope, but the line crosses the y-axis at (0, 6) and not at the origin. If the situation were proportional, then the graph would have intersected the origin. If we express the rate pairs as fractions - 9/3, 10/4, 1 1/5, 12/6 - we see that they do not have the same value. In proportional situa- tions, all rate pairs are equivalent. The proportion algorithm based on equivalent

rates, therefore, is not the appropriate solu- tion strategy for this example.

Having a deeper understanding of the mathematical characteristics of proportional situations enables students to solve prob- lems in multiple ways. As Post, Behr, and Lesh ( 1 988) state, "Equipping students with a variety of perspectives and solution strat- egies fosters not only better understanding but also a more confident and flexible approach to problem solving."

Applying Knowledge of the Mathematical Characteristics of Proportional Situations A student can use his or her knowledge of the mathematical characteristics of propor- tional situations to solve the following task in several ways.

Problem: Complete the table. Are the data related proportionally? Define the rule for the data predicting у when x is given.

j 3 6 9 12 15 ~x 2 4 6 8 10 16 19 Possible solutions: If the number of pairs

are related proportionally, then the rate pairs should form a set of equivalent frac- tions. The fractions 3/2, 6/4, 9/6, 12/8, and 1 5/ 1 0 all have a value of 3/2, so the data are proportionally related. This fact validates the use of the cross-product-algorithm so- lution strategy to find the corresponding y-values for x = 16-3/2 = y/16, 2y = 48, and у = 24; and for x = 19-3/2 = y/19, 2y = 57, and v = 57/2. Since this is a proportional situation, the constant rate,

Many aspects of our world operate

according to proportional rules.

3/2, is the constant factor for the rule relating the data in the table; у = (3/2)uc. Note that 2/3, the reciprocal of 3/2, is the other constant factor relating x and y. The equation x = (2/3)y would be used if one was given a y-value and x was unknown.

A solution strategy using the graphing characteristics is possible. The graph in figure 3 connects the data points from the table by a straight line. The graph is a straight line with positive slope, and when it is extended it passes through the origin. One can conclude that the data are propor- tionally related.

Since all data points will fall on the line, the line can be extended upward and the corresponding y- value for x = 16 can be found as shown in figure 4. The graph is less helpful for finding the corresponding y-value for x = 19, since it crosses betweeen integral values. The rule for describing the data would be helpful.

To find the rule first find the slope of the line, which can be found from the graph. It is the ratio of the vertical distance between

FEBRUARY 1993 345

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two points on the graph to the horizontal distance. The right triangle seen in figure 4 shows the slope to be 3/2. Note that for any two points selected the slope will have a value equal to 3/2. The slope of the line is the m in the equation, so the rule is y - (3/2)дг. The corresponding value for x = 1 9 can be found using the formula у = (3/2)(19).

Multiple solution strategies give stu- dents the power to choose a strategy that best fits the data; the resources they have available, such as a graphing calculator; and their personal preferences. In general it is always a good idea to view a concept from multiple perspectives.

Mathematical Connections What connections have been made as we explored the mathematical characteristics of proportional situations? First, we repre- sented proportional contexts in a table to examine number patterns. We then trans- lated the numerical relationship relating the two quantities into algebraic sentences. We translated the algebraic representation to a graphical representation. Examining the relationships among different repre- sentations is important. Different represen- tations highlight different aspects of the situation, each fostering insights and inter- connections to the other. NCTM' s curricu- lum standards (1989) emphasize the im- portance of making connections among tables, algebraic generalizations, and graphical representations. Proportionality presents a good way to make these connec- tions.

To continue our exploration of the math- ematical characteristics of proportional situ- ations, we used our knowledge of line graphs, slope, and ̂-intercept to make ad- ditional generalizations about graphs of proportional situations. Students need to understand that they can often use one form of a mathematical idea to help them understand another idea. This "persistent attention to recognizing and drawing con- nections among topics will instill in stu- dents an expectation that the ideas they learn are useful in solving other problems and exploring other mathematical concepts" (NCTM 1989, 85).

Understanding proportionality by using several representations enables students to evaluate problem situations critically and

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to determine whether the context is propor- tional or nonproportional. Empowering stu- dents means not only equipping them with strategies to solve problems but helping them understand underlying concepts so they can apply strategies appropriately at a later stage.

In conclusion, by exploring the math- ematical characteristics of proportional situ- ations, students learn a variety of very important mathematical strategies for solv- ing proportional-reasoning problems.

Different representations of problems serve as dif- ferent lenses through which students interpret the problems and the solutions. If students are to become mathematically powerful, they must be flexible enough to approach situations in a variety of ways and recog- nize the relationships among different points of view (NCTM 1989,84).

References

Karplus, Robert, Steven Pulas, and Elizabeth Stage. "Proportional Reasoning and Early Adolescents." In Acquisition of Mathematics Concepts and Pro- cesses, edited by Richard Lesh and Marsha Landau. New York: Academic Press, 1983.

National Council of Teachers of Mathematics. Cur- riculum and Evaluation Standards for School Mathematics. Reston, Va.: The Council, 1989.

Noelting, Gerald. The Development of Proportional Reasoning and the Ratio Concept, Part 1 - the Differentiation of Stages, Educational Studies in Mathematics II, 217 -53. Boston: Reidell Publish- ing Co., 1980.

Post, Thomas, Merlyn Behr, and Richard Lesh. "Pro- portionality and the Development of Prealgebra Understandings." In The Ideas of Algebra, K-I2, 1988 Yearbook of the National Council of Teach- ers of Mathematics, edited by Arthur F. Coxford and Albert P. Shulte, 78-90. Reston, Va.: The Council, 1988. •

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