We hope that teachers reading this chapter will be encouraged to use techniques involved in teaching experiments. Finding time to interview small numbers of students after instruction will be 2 worthwhile activity. Interviews are the major data source for research in mathematics education, and teachers will find interesting interview questions in the professional literature.
About the Authors
Kathleen Cramer is all associate professor in the College of Education at the University of Wisconsin-River Falls. Dr. Cramer teaches mathematics methods classes for undergraduate and graduate students. She is currently working on a National Science Foundation (NSF) grant revising the curriculum from RNP teaching experiments.
Thomas Post is a professor in mathematics education at the University of Minnesota. Besides teaching graduate and undergraduate courses in mathematics education, Dr. Post has been a co-director of the NSF-sponsored Rational Number Project since 1979.
Sarah Currier is on leave from her sixth-grade teaching responsibilities in the Minneapolis Public Schools. She is currently working oil a master's degree in mathematics education at the University of Minnesota. She particularly enjoys teaching geometry and incorporating cooperative learning techniques in her mathematics classes.
1. CRAMER, K., BEZUK, N., & BEHR, M. (1989). Proportional relationships and unit rates. Mathematics Teacher, 82(7), 537-544.
2. CRAMER, K., & LESH, R. (1988). Rational number knowledge of preservice elementary education teachers. In M. Behr & C. Lacampagne (Eds.), Proceedings of the Tenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 425-431). De Kalb, IL: Northern Illinois University.
3. CRAMER, K., POST, T, & BEHR, M. (1989). Interpreting proportional relationships. Mathematics Teacher, 82(6), 445-453.
*4. HELLER, P, POST, T, BEHR, M., & LESH, R. (1990). The effect of two context variables oil quantitative and numerical reasoning about rates. Journal for Research in Mathematics Education, 21 (5), 388-402.
5. HIEBERT, J., & LEFEVRE, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case for mathematics (pp. 1-27). Hillsdale, NJ: Erlbaum.
*6. HOFFER, A. (1988). Ratios and proportional thinking. In T. Post (Ed.), Teaching mathematics in grades K-8: Research based methods (pp. 285-313). Boston: Allyn & Bacon.
7. KARPLUS, R., KARPLUS, E., FORMISANO, M., & PAULSON, A. (1979). Proportional reasoning and control of variables in seven countries. In J. Lochhead & J. Clement (Eds.), Cognitive process instruction (pp. 47-103). Philadelphia: The Franklin Institute Press.
8. KARPLUS, R., PULOS, S., & STAGE, E. (1983). Proportional reasoning of early adolescents. In R. Lesh & M. Landau (Eds.), Acquisition of mathematical concepts and processes (pp. 45-89). New York: Academic Press.
9. KURTZ, B., & KARPLUS, R. (1979). Intellectual development beyond elementary school VII: Teaching for proportional reasoning. School Science and Mathematics, 79(5), 387 - 398.
10. LACAMPAGNE, C., POST, T, HAREL, C., & BEHR, M. (1988). A model for the development of leadership and the assessment of mathematical and pedagogical knowledge of middle school teachers. In M. Behr a, C. Lacampagne (Eds.), Proceedings of the Tenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 418-424). De Kalb), IL: Northern Illinois University.
11. NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
12. NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS. (1991). Professional standards for teaching Mathematics. Reston, VA: Author.
13. NOELTING, G. (1980), The development of proportional reasoning and the ratio concept. Part 1: The differentiation of stages. Educational Studies in Mathematics, 11 (3), (pp. 217-253).
14. NOELTING, G. (1980) The development of proportional reasoning and the ratio concept. Part II - Problem structure at successive stages: Problem-solving strategies and the mechanism of adaptive restructuring. Educational Studies in Mathematics, 11 (3), 331-363.
15. POST, T, BEHR, M., & LESH, R. (1988). Proportionality and the development of prealgebra Understanding. In A. Coxford (Ed.), Algebraic concepts in the curriculum K-12 (1988 Yearbook, pp. 78-90). Reston, VA: National Council of Teachers of Mathematics.
16. POST, T * , CRAMER, K., BEHR, M., LESH, R., & HAREL, G. (in press). Curriculum implications from research on the learning, teaching and assessing of rational number concepts: Multiple research perspectives. In T Carpenter & E. Fennema (Eds.), Learning, teaching and assessing rational number concepts: Multiple research perspectives. Madison: University of Wisconsin.
17. POST, T, & CRAMER, K. (1989). Knowledge, representation and quantitative thinking. In M. Reynolds (Ed.), Knowledge base for beginning teachers (pp. 221-232). Elmsford, NY. Pergamon Press.
18. TOURNAIRE, F., & PULOS, S. (1985). Proportional reasoning: A review of the literature. Educational Studies in Mathematics, 16(2), 181-204.
19. VERGNAUD, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 127-174). New York: Academic Press.
This activity may be used to pre-assess students’ knowledge related to testing for proportionality.
Give the Is It Proportional sheet (M-7-3-2_Is It Proportional and KEY.docx) to all students. Ask them to take 5 minutes to write down some descriptions how they can determine if different representations show a proportional relationship or not. Students should include examples, if possible. Next, have students share ideas and examples with a partner. After about 5 more minutes, the class may reconvene. One member from each group should share their ideas on determining proportionality. Encourage discussion and debate.
“In this lesson, we are going to determine whether two quantities are proportionally related. We will look at written ratios, verbal descriptions, equations, tables, and graphs. The intent of the lesson is for you to be able to look at any form of a relation and determine if it represents a proportion. For each example given, we will determine if it represents a proportional relationship. We will also justify our thinking.”
Give students time to provide answers, discuss, and ask any questions, prior to confirming each answer. This part of the lesson is intended for whole class discussion and participation.
“Let’s look at some written ratios and decide whether or not they are proportional.”
“Now we’ll look at some verbal descriptions and decide whether or not they describe a proportional relationship.”
- Example 5: 12 apples: $4.00
3 apples: $1.00
- “This describes a proportional relationship. The ratio of is equivalent to the ratio of . Each ratio has a value of 3.”
- Example 6: 3 tanks of gas for every 1200 miles driven
7 tanks of gas for every 2800 miles driven
- “This describes a proportional relationship. The ratio is equivalent to the ratio . Each ratio has a value of .”
- Example 7: 4 pizzas for 16 people
9 pizzas for 42 people
- Example 8: 35 proposals to 7 employees
105 proposals to 21 employees
- “This describes a proportional relationship. The ratio is equivalent to the ratio . Each ratio has a value of 5.”
“Now, we’re ready to determine whether equations represent proportional relationships.”
- Example 9:
- “This equation represents a proportional relationship because it has a constant rate of change and a y-intercept of 0. In other words, no amount is added to or subtracted from the term containing the constant rate of change, 7x. Another way to recognize that this equation represents a proportional relationship is to see that it is in the form of y = kx, where k is the constant of proportionality (in this case, 7).”
- Example 10:
- “This equation does not represent a proportional relationship because the y-intercept is not 0. The y-intercept is 4, indicating the graph crosses the y-axis at the point, (0, 4), not (0, 0). This equation is not in the form of y = kx, but rather in the form y = mx + b, meaning a constant term has been added to or subtracted from the term with the x.”
- Example 11:
- “This equation does not represent a proportional relationship because the
y-intercept is not 0. The y-intercept is −2, indicating the graph crosses the
y-axis at the point, (0, −2), not (0, 0). This equation is not in the form of
y = kx, but rather in the form y = mx + b, meaning a constant term has been added to or subtracted from the term with the x.”
- “This equation does not represent a proportional relationship because the
- Example 12:
- “This equation represents a proportional relationship becauseit has a constant rate of change and a y-intercept of 0. In other words, no amount is added to or subtracted from the term containing the constant rate of change, .”
“Now, we’re ready to look at some tables of values to determine whether they represent proportional relationships.”
“Look at this table and determine if it represents a proportional relationship. How can you tell?”
- “This table is easy to interpret. We are given the y-intercept, or point at which the graph crosses the y-axis; so we know just by looking at the first row of the table (0, 0) that the relationship satisfies one requirement of proportionality: the y-intercept is zero. As the x-values increase by 1, the
y-values increase by a constant rate of 3. This satisfies the other requirement of proportionality: a constant rate of change. Thus, we may declare that this table represents a proportional relationship.”
- “We can check our decision by making sure that the ratios of all x-values to corresponding y-values are equivalent. We may write the following: . This statement is true. Each ratio has a value of . We have now confirmed our decision that this table represents a proportional relationship.”
Many students will simply check to see if there is a constant rate of change present in the table and then declare the relationship to be proportional. It is important that they understand the table must represent the ordered pair (0, 0). The y-intercept must be at zero. Otherwise, the table simply represents a linear equation that is not proportional. This is an important distinction to make: all proportions are linear, but not all linear equations are proportional. If students are unsure, they should check the equivalence of ratios of x-values to corresponding y-values.
- “Notice the x-values in this table are not consecutive. For this one, it will be easier to simply compare ratios of x-values to y-values. Let’s compare and . Are these ratios equal?” (No) “So, we can declare that this table does not represent a proportional relationship. We do not need to look any further.”
“Graphs are very easy to check for proportionality. There are only two questions we must ask ourselves. 1) Is the graph a straight line? 2) Does the graph cross the y-axis at the point (0, 0)? In other words, does the linear graph pass through the origin? If it does, the graph represents a proportional relationship. If it does not, the graph does not represent a proportional relationship. It is that simple.”
- “This graph does not represent a proportional relationship. It does not pass through the origin, or the point, (0, 0). In other words, the y-intercept is not zero. It IS a straight line, but it must also pass through the origin to qualify as a proportional relationship. This one fails the test of proportionality.”