

INTERPRETING PROPORTIONAL RELATIONSHIPS By KATHLEEN A. CRAMER, THOMAS R. POST, and MERLYN J. BEHR Teacher's Guide Introduction: Proportional reasoning is one form of mathematical reasoning. It involves a sense of covariation, multiple comparisons, and the ability to remember and process several pieces of information. Proportional reasoning is very much concerned with inference and prediction and involves both qualitative and quantitative methods of thought. The fact that many aspects of our world operate according to proportional rules makes proportionalreasoning abilities extremely useful in the interpretation of realworld phenomena (Post, Behr, and Lesh 1988). The importance of proportional reasoning is also stressed in the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989):
The activities in these lessons present students with situations involving proportional and nonproportional reasoning. For each problem they construct a table based on the particular situation, generalize a rule to describe those data, and then plot the data on coordinate axes. From these situations students learn to identify characteristics of those involving proportional and nonproportional relationships. Specifically t that proportional relationships scribed using a multiplicative rule always represented by straightline going through the origin. The completion of this activity, particularly by students working in small cooperative groups, can also contribute to the attainment of the general goals reflected in the standards, namely, becoming a mathematical problem solver, learning to communicate mathematically, learning to reason mathematically, learning to value mathematics, and becoming confident in ability to do mathematics. Grade levels: 68 Materials: Cuisenaire rods, graph and activity sheets 14 for each Students will need the results from sheet 1 when working on sheet 2. Objectives: (1) Students generate data by translating information from a verbal problem, record results in tabular form, and write a rule for the number patterns discovered in the table. (2) Students plot a set of data points and connect the points to form a graph of the related rule. (3) Students learn to discriminate proportional from nonproportional relationships on the basis of formulas and graphs used to describe these situations. Directions: For each of the activity sheets, the teacher should introduce the problem in a largegroup setting and discuss the solution plan. Students can complete the solution plan independently or in small groups. They should initially discuss the followup questions in small groups and then in a large group. Each activity may take more than one class period. Sheet 1: On this sheet, the tabular method is introduced through a physical experiment in which students collect and organize data using a table, look for patterns, and find a rule. This information is then used to find various missing values. The students collect data for two situationsone involving proportional relationships and one involving nonproportional relationships. These tables will be used again on sheet 2. Sheet 2 : On sheet 2 the graphical interpretation is presented by graphing data collected from the first sheet and then comparing and categorizing the graphs generated. The teacher should explain that proportional relationships, which form a special class of problems, always have straightline graphs passing through the origin. Only one of the two experiments on sheet 1 represents a proportional relationship. The teacher should also point out that the rules for proportional relationships involve only multiplication or division. At this point, the students have two ways to distinguish situations involving proportional and nonproportional relationships (l) examining their graphs and (2) observing characteristics of the related formula. The activities on sheets 3 and 4 reinforce these important distinctions. Sheets 3 and 4 : On these sheets, students are asked to determine if given problem situations involve proportional or nonproportional relationships. They repeat the steps used for the previous activities: (a) organize data in a table, (b) write a rule to describe the relationships in the data, (c) use the rule to solve missingvalue problems, and (d) graph the data. From the formulas and the graphs students then determine whether the situation involves a proportional relationship. After completing these activities the teacher should help students consolidate the relevant information about proportional relationships: (a) the graphs of proportional relationships form a straight line through the origin and (b) the formula or rule used to describe the class of data points in proportional relationships involves only multiplication or division. Readers should notice that no mention has been made of the most frequently used algorithm for solving problems involving proportional relationships  cross multiply and divide. This algorithm badly distorts all rational consideration of the underlying concepts in a proportional relationship. As such, it should not be introduced until students have had many experiences with proportionality in settings similar to those described here. For additional information on this point, see Post, Behr, and Lesh (1988). An alternative to the crossmultiplyanddivide algorithm, the unitrate approach for solving proportional relationships, will be presented in a future activity. The links will open a new window with the activity sheets in .pdf format Sheet
1
Answers: Sheet 1: Solution plan:




Questions: 1. 62 cm^{2}; T = 4 x l + 2. 2. Refer to the table given for the solution plan; L = 4 x l. 3. Each formula involves multiplying the length by the constant 4. 4. The formula for total surface area involves the addition of a constant nonzero value; the formula for lateral surface area involves no such addition.
Sheet 2: Solution plan:


Questions: 1. Both graphs are straight lines; each line has the same steepness (slope). 2. The graph of lateral surface area through the origin; the graph of to face area crosses the vertical axis above the origin.
Sheet 3: Solution plan: One possible given. Other scores are possible.




Questions: 1. L = 20  W. 2. 8,5,9 3. Since last year's record is to be beaten, the number of wins must be greater than twelve.




5. No. The situation does not involve a proportional relationship because (a) the formula uses an operation other than multiplication or division and (b) the graph does not go through the origin.
Sheet 4: Solution plan (students may choose to add number pairs beyond those indicated in the table):




Questions: 1.
Rockwood and Webster: 10 km 2. 1 cm on the map represents 2.5 km of actual distance. 3. D = 2.5 X d or an equivalent form. 4. Since d = 14, D = 2.5 x 14 = 35 km. 5. 

Note: Some students
may have graphed the inverse relation represented by the formula 6. Yes. The problem involves a proportional relationship because (a) the formula uses only multiplication or division and (b) the graph is a straight line through the origin.
REFERENCES National Council of Teachers of Mathematics. Commission on Standards for School Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: The Council, 1989. Post, Thomas R., Merlyn J. Behr, and Richard Lesh. "Proportionality and the Development of Prealgebra Understandings." In The Ideas of Algebra, K12, edited by Arthur F. Coxford and Albert P. Schulte, 7890. Reston, Va.: The Council, 1988. This paper is based in part on research supported by the National Science Foundation under grants DPE, 8470077 and TEI8652341 (The Rational Number Project). Any opinions, findings, and conclusions expressed are those of the authors and do not necessarily reflect the views of the National Science Foundation. Edited by: This section is designed to provide mathematical activities in reproducible formats appropriate for grades 712. This material may be reproduced by classroom teachers for use in their own classes. Readers who have developed successful classroom activities are encouraged to submit manuscripts, in a format similar to the "Activities" already published, to the editorial coordinator for review. Of particular interest are activities on the Council's curriculum standards, its expanded concept of basic skills, problem solving and applications, and the uses of calculators and computers. 
