helping children understand ratios

Helping Children Understand RatiosAuthor(s): Ana Helvia QuinteroSource: The Arithmetic Teacher, Vol. 34, No. 9 (May 1987), pp. 17-21Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194224 .

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Helping Children Understand Ratios By Ana Helvia Quintero

Temperature, velocity, acceleration, and density are but a few examples of scientific concepts that are expressed by a ratio. These topics are difficult for students to understand because they do not understand the mathemat- ical concept of a ratio.

Many studies have shown that the concept of ratio is difficult for children (e.g., Behr et al. 1983; Karplus, Pulos, and Stage 1983; Lovell and Butter- worth 1966; Lunzer and Pumfrey 1966). Yet, we find few activities in most school curricula that develop this concept. This article describes different levels of difficulties that children face when working with ratios and suggests a sequence of activities aimed at helping children develop this concept.

Level 1 A ratio is a relationship between two quantities. At first children focus on only one of the elements that form the ratio. For example, in studies I have conducted, children between nine and thirteen years of age were asked to predict the color of mixtures of different ratios of white and brown sugar. A group of children, usually the younger ones, always predicted that a mixture with more white sugar would be lighter than a mixture with less white sugar. Thus they predicted that 6 tablespoons of white sugar mixed

Ana Quintero teaches mathematics at the Uni- versity of Puerto Rico, Rio Piedras, PR 00931. She conducts research on students' difficulties with mathematics and projects aimed at im- proving teaching.

May 1987

with 8 tablespoons of brown sugar would be lighter than 4 tablespoons of white sugar mixed with 3 tablespoons of brown sugar because 6 is larger than 4.

The children were focusing on only the amount of white sugar, one of the elements of the ratio, without taking into account the amount of the other element of the ratio, the brown sugar. Children making this type of mistake need to learn about variables and their relationships. Some children may comprehend variables in some con- texts but need to understand that ratios depend on two variables.

The idea of a variable depending on other variables is rarely taught for- mally in school. In order to teach this concept, Karplus, Pulos, and Stage (1983) suggest the idea of presenting to the child problems whose solution depends on considering two or more variables. For example, students might be shown a pencil held upright against a desk top and asked what affects the length of its shadow when it is in the sunlight. In this problem students must consider the length of the pencil as a variable.

Level 2 Some children know that a ratio depends on two variables, yet they view the relationship in additive terms. When comparing the ratios 3/1 and 5/3, these children will say that they are equal because

3-1 = 5-3.

This error may be due tp the fact that the first comparisons children make are based on additive principles - for

example, who has more pencils? How many more?

To understand ratios students must move from an additive method of comparison to a multiplicative one. This move, however, doesn't occur in a single step. Different types of ratios present different types of difficulties to children.

The simplest type of ratio is the per-unit ratio - for example, 15 candies per bag, 20 children per teacher. In per-unit ratios the denominator of the ratio is 1. Errors arise from two sources when children deal with per- unit ratio: some children don't understand the language used to express the ratio, for example, the word per; others don't understand the concept of "per-unit ratio."

The activities that follow help to identify both types of difficulties and suggest ways of dealing with each one with an individual pupil.

Materials 1 . Set of concrete materials, like mar-

bles, Cuisenaire rods, paper money, and plastic cups

2. Set of cards with word problems that include per-unit ratios expressed in different ways: "in each,". "for," "for each," "per." (See fig. 1 .)

Procedure The child picks one of the cards, for example, card 1. You say, "If these are candies (pointing to some concrete material representing the candies) and these are boxes, how will you represent the situation described in this card?"

17



Fig. 1 Problems with per-unit ratios

Card 1 Card 2 Card 3

Mary bought 4 boxes of John has 3 sets of cars. Karen bought $4 worth of candies. There were 6 There are 5 cars per set. apples. 5 apples cost a candies in each box. dollar.

The procedure is repeated until we know the source of the student's difficulty-one of the expressions or the concept of ratio. A child who doesn't understand one of the expressions will represent correctly per-unit ratios expressed in some of the cards but not in

18

others. In this situation we must ex- plain to the child that all these words are equivalent ways of expressing the same concept. In a sense the child is taught the meaning of a new word.

Children who have difficulty with the concept of per-unit ratio will have

difficulty representing the problem no matter how it is expressed. In a previous study (Quintero 1980), the rep- resentations in figure 2 were made from problems like "Mary bought 4 boxes of candies and there were 6 candies in each box."

Arithmetic Teacher

a. A representation that models the relationship as "6 candies in 4 boxes"



Fig. 3 Nine lollipops for 3 dollars is equivalent to 3 lollipops for 1 dollar.

¡¡i y i «

! ! ! ! ! !

iii ! ! !

| | | | | | | | I I I I

The activities used to determine if children understand the concept of per-unit ratio can be used to teach the concept to them. One could also use children's knowledge in other areas as a basis for teaching the concept of per-unit ratio. For example, children are familiar with the activity of give and take (e.g., 3 dimes for a comic book). They may also be familiar with some samples of homogeneous classes (e.g., 5 fingers per hand, 2 feet per person). These notions are easily developed into the concept of per-unit ratio.

Level 3 Ratios with denominators different from 1, for example, 9 lollipops per 3 dollars or 5 teaspoons of white suga *

for every 3 teaspoons of brown sugar, are quite difficult for many children. These ratios can be divided into integral ratios and nonintegral ratios. An integral ratio can be made easier by reducing it to an equivalent integral per-unit ratio. The ratio 9 lollipops per 3 dollars can be reduced to the inte-

gral per-unit ratio 3 lollipops per 1 dollar (see fig. 3).

Nonintegral ratios, like 5 teaspoons of white sugar for every 3 teaspoons of brown sugar, cannot be reduced to an integral ratio. If reduced to a unit ratio, it will be a fractional or decimal per-unit ratio; in this example, the ratio 5 to 3 is equivalent to 1 2/3 teaspoons of white sugar for 1 teaspoon of brown sugar. Since children understand integral ratios before nonintegral ones (Quintero and Schwartz 1982; Karplus, Pulos, and Stage 1983), integral ratios should be introduced first in the school curricu- lum.

In learning integral ratios, children should be guided to see their equiva- lence with per-unit ratios. This sug- gestion arises from observing children comparing integral ratios. The children who compared integral ratios correctly always reduced them to per- unit ratios before making the comparison. For example, if the children were asked which type of mixture of white and brown sugar will be lighter, one containing 9 teaspoons of white

sugar for every 3 teaspoons of brown sugar or one containing 8 teaspoons of white sugar for every 2 teaspoons of brown sugar, they would say, "In the first mixture there are 3 teaspoons of white sugar for every teaspoon of brown sugar, in the second there are 4 to 1. So the second one is lighter."

To help children see the equiva- lence of integral ratios and per-unit ratios, we can represent with concrete materials or with diagrams the integral ratio and how it can be transformed to a per-unit ratio, as shown in figure 3. Another example might be, "At a picnic there are 16 children for every 2 adults. How many children per adult?"

When the elements are continuous or quasi-continuous, a discrete interpretation of the quantities should be made. For example, see figure 4.

Level 4 Once students understand integral ratios, nonintegral ratios can be introduced. At first many students will use integral per-unit ratios to interpret

May 1987 19



nonintegral ratios. For example, when comparing "5 teaspoons of white sugar for every 2 teaspoons of brown sugar" with "8 teaspoons of white sugar for every 3 teaspoons of brown sugar," some students made the representation shown in figure 5.

Once they made that representation, they argued that "8 to 3 is lighter, since we have two teaspoons left." Among the students using this analysis were seventh graders who had studied fractions and decimals, yet they didn't use either of these concepts to solve the problem. These students had not been taught about per-unit interpretation of ratios. They used it spontaneously in their analysis.

In mathematics it is common for students to avoid computation with fractions and decimals. Yet with ratios, students9 avoidance of fractions and decimals goes deeper than mere computation. Students fail to use fractions and decimals in problems in- volving ratios because they do not see the relationship between these concepts. Recent NAEP data (e.g., Car-

penter et al. 1980) have pointed out that most thirteen-year-olds see different interpretations of rational numbers as separate, unrelated topics.

So, to help students deal with nonintegral ratios, we should develop activities that relate different interpretations of rational numbers. Let's dis- cuss one such activity.

Fractions can be interpreted as a part-to-whole comparison, a ratio, an indicated division, an operation, or a measure. Behr, Post, Silver, and Mierkiewicz (1980) suggest that parti- tioning, specifically the part-whole interpretation, should be the starting point in the teaching of fractions. Other interpretations should be based on, and related to, the part-whole interpretation.

For example, children understand better the use of fractions as a model of ratios when they see ratios as an example of part-and-whole comparison. This was true of my daughter. When she was in the second grade, she was asked to identify the fraction that was represented in the following situations:

• • and A • O O and A

O O • O O

For this task she needed to represent a ratio as a fraction. She had difficulty with it until I showed her how to interpret the ratio as a part- and-whole comparison. For this I pre- sented the previous situations in the following way:

O O i O O «

The black balls are half of the whole.

I - i - ¡ - I

• o o mmm^ i • i o i o j • o o

mmm^ ! • ! o ! o i I I - I - I

The black balls are one-third of the whole.

Once she related this new situation to the part-whole interpretation of a fraction, she saw how the same sym-

20 Arithmetic Teacher

Fig. 4 Eight gallons of gasoline for every 2 hours is equivalent to 4 gallons for every hour.



bol, for example, 1/2, could be used to represent two different situations:

€ Situation 1

Part of plane regions

• • O O

Situation 2 Part of sets or arrays

Conclusion Concepts that are difficult to learn and to teach should become targets for teachers' and researchers' efforts in developing effective ways of teaching. It is therefore important to identify students' immature or incorrect conceptualizations of these ideas and

what is preventing students from developing these conceptualizations into mature understanding. This article identifies some of the stumbling blocks in the development of the concept of ratio and presents several ideas on how to deal with these difficulties in school.

References Behr, Merlyn J., Richard Lesh, Thomas R.

Post, and Edward A. Silver. "Rational- Number Concepts." In Acquisition of Math- ematics Concepts and Processes, edited by Richard Lesh and Marsha Landau, pp. 91-126. New York: Academic Press, 1983.

Carpenter, Thomas P., Mary K. Corbitt, Henry S. Kepner, Jr., Mary M. Lindquist, and Rob- ert E. Reys. "Solving Verbal Problems: Re- sults and Implications from National Assess- ment." Arithmetic Teacher 28 (September 1980):8-12.

Goodstein, Madeline P., and William W. Boelke. "A Prechemistry Course on Propor- tional Calculation." Columbus, Ohio: ERIC

May 1987 21

Clearinghouse for Science, Mathematics and Environmental Education, 1980.

Herron, J. Dudley, and urayson A. Wheatley. "A Unit Factor Method for Solving Propor- tion Problems." Mathematics Teacher 71 (January 1978): 18-21.

Karplus, Robert, Steven Pulos, and Elizabeth Stage. "Proportional Reasoning of Early Ad- olescents." In Aquisition of Mathematics Concepts and Processes, edited by Richard Lesh and Marsha Landau, pp. 45-90. New York: Academic Press, 1983.

Lovell, K., and I. B. Butterworth. "Abilities Underlying the Understanding of Proportion- ality." Mathematics Teaching 37 (1966):5-9.

Lunzer, E. A., and P. D. Pumfrey. "Under- standing Proportionality." Mathematics Teaching 34 (1966):7-12.

Quintero, Ana H. "The Role of Semantic Un- derstanding in Solving Multiplication Word Problems." Ph.D. diss., Massachusetts Insti- tute of Technology, 1980.

Quintero, Ana H., and Judah L. Schwartz. "The Development of the Concept of Ratio on Children." Working Paper 15, Division for Study and Research in Education. Cam- bridge, Mass.: Massachusetts Institute of Technology, 1982. 9

Fig. 5 Errors in finding equivalent expressions for nonintegral ratios



helping children understand ratios

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