## Proportions and Ratios

In mathematics and physics, proportionality is a mathematical relation between two quantities. There are two different views of this “mathematical relation.” One is based on ratios, and the other is based on functions.

**I. Definition**

**1. Basic definition of proportional and ratio reasoning**

In mathematics and physics, proportionality is a mathematical relation between two quantities. There are two different views of this “mathematical relation.” One is based on ratios, and the other is based on functions.

**a. A ratio viewpoint**

In many books, proportionality is expressed as an equality of two ratios:

a/b = c/d

Given the values of any three of the terms, it is possible to solve for the fourth term.

**b. A functional viewpoint**

Consider the following equation for gravitational force:

F = Gm_{1}m_{2}/r^{2}

A scientist would say that the force of gravity between two masses is directly proportional to the product of the two masses and inversely proportional to the square of the distance between the two masses. From this perspective, proportionality is a functional relationship between variables in a mathematical equation. (http://en.wikipedia.org/wiki/Proportional_reasoning)

Proportional reasoning is associated with the formal operational stage of thought, according to Piaget’s theory of intellectual development.

**2. Definitions of proportional and ratio reasoning in research**

Proportional reasoning can be conceptualized in the following ways: identification of two extensive variables that are applicable to a problem; recognition of the rate of intensive variables whose constancy determines the linear function; and application of the given data and relationships to find (i) an additional value for one extensive variable (missing value problems) or (ii) comparison of two values of the intensive variable computed from the data (comparison problem). (Karplus et al., 1983)

**II. Simplified examples of proportional and ratio reasoning**

1. Question ID: 20800321100

A fifth grade class has 18 students. At lunch time, the teacher brings in 12 bottles of orange juice, which fully fill all students’ cups (no juice is left). How many cups can be filled with 16 bottles of orange juice?

a. 20 b. 24 c. 28

d. 30 e. 32 f. other

Answer: B

The question tells us that 12 bottles of juice fill 18 cups, so each cup holds 12/18 of a bottle of juice. This is a constant quantity. If there are 16 bottles of orange juice, a proportional relationship can be set up: 12/18 = 16/x. Solving for x yields 24.

2. Question ID: 20800241100

Below are drawings of a wide and a narrow cylinder. The cylinders have equally spaced marks on them. Water is poured into the wide cylinder up to the 4^{t}^{h} mark (see A). This water rises to the 6^{t}^{h} mark when poured into the narrow cylinder (see B).

Both cylinders are emptied (not shown) and water is poured into the wide cylinder up to the 6^{t}^{h} mark. How high would this water rise if it were poured into the empty narrow cylinder?

A. to about 8

B. to about 9

C. to about 10

D. to about 12

E. none of these answers is correct

Answer: B

Let the height of the water in the wide cylinder be *h _{w}*, the height of the water in the narrow cylinder be

*h*, the size of the wide cylinder be S

_{n}_{w}, and the size of the narrow cylinder be S

_{n}. The volume of water must be conserved, so h

_{w}S

_{w}= h

_{n}S

_{n}. This can be rearranged into a proportional relationship: . Since S

_{n}and S

_{w}are constants, the ratio between h

_{w}and h

_{n}is constant. In part A,

*h*= 4 and

_{w}*h*= 6. In part B,

_{n}*h*= 6 and

_{w}*h*=

_{n}*x*. This provides us with a new proportion: h

_{w}/h

_{n}= S

_{n}/S

_{w}. Solving for

*x*yields 9.

**III. Importance of proportional and ratio reasoning**

**1. The importance of proportional and ratio reasoning in learning**

Proportional reasoning is recognized as a fundamental reasoning construct necessary for mathematics and science achievement (McLaughlin, 2003). In scientific inquiry, we can define useful quantities through proportional reasoning. For example, we define density, speed, and resistance with ratios. Krajcik and Haney (1987) analyzed the American Chemical Society Exam and found that over 50% of the test involved tasks requiring proportional reasoning. This implies that proportional reasoning is the primary reasoning construct required for success in chemistry, and complete development of this skill is crucial for achieving understanding of the many formal concepts associated with the content. Akatugba and Wallace (1999) contend that almost every concept in physics requires a proficient understanding of proportional reasoning, and students who are not capable of this type of reasoning will have difficulty mastering the concepts.

Proportional reasoning is at the heart of middle grade mathematics.

Proportional reasoning is considered a milestone in students’ cognitive development. Proportional reasoning is associated with Piaget’s formal operational stage of thought. Many Piagetian and neo-Piagetian researchers identify the formal operational stage in subjects by having them perform tasks that require the use of ratios and proportions (Roth & Milkent, 1991).

**2. The importance of proportional and ratio reasoning in society**

Proportional reasoning is widely applied in everyday life. For example, gas mileage and unit price are ratios that may be grouped under the general notion of “rates” (Karplus et al. 1983). Consider going to the grocery store to buy flour. There are two choices: a 5 pound package that costs $2 and a 2 pound package that costs $1. Which is the better deal? Proportional reasoning will tell you that the 5 pound package costs less per pound.

**IV. Research on proportional and ratio reasoning**

http://en.wikipedia.org/wiki/Proportional_reasoning

Early adolescents' proportional reasoning on ‘rate’ problems (Robert Karplus, Steven Pulos, Elizabeth K. Stage, Educational Studies in Mathematics, Vol. 14, No. 3 (Aug., 1983), pp. 219-233)

The development of proportional reasoning and the ratio concept Part I — Differentiation of stages (Gerald Noelting, 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 (Gerald Noelting, 1980)

Proportional reasoning: A review of the literature (Francoise Tourniaire and Steven Pulos, 1985)

The influence of instruction on proportional reasoning in seventh graders (Warren T. Wollman, Anton E. Lawson, 1978)

Proportional Reasoning among 7th Grade Students with Different Curricular Experiences (David Ben-Chaim, James T. Fey, William M. Fitzgerald, Catherine Benedetto and Jane Miller, 1998)

Effect on development of proportional reasoning skill of physical experience and cognitive abilities associated with prefrontal lobe activity (Yong-Ju Kwon, Anton E. Lawson, Wan-Ho Chung, Young-Shin Kim, 2000)

Interpreting Middle School Students' Proportional Reasoning Strategies: Observations From Preservice Teachers (Ellen Hines, Mary T. McMahon, 2005)

Effect of Modeling Instruction on Development of Proportional Reasoning I/II (Shannon McLaughlin, 2003)

Didactical designs for students’ proportional reasoning: an “open approach” lesson and a “fundamental situation” (Takeshi Miyakawa and Carl Winsløw, 2009)

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