DEVELOPING
R A T I O
CONCEPTS:
An Asian
Perspective
J A N E - J A N E L O, T A D W A T A N A B E,
AND JINFA CAI
T
HE FOLLOWING VIGNETTE ILLUSTRATES HOW
a Taiwanese textbook series envisions introducing the concept of ratio.
Textbook. There are two blocks in front of you.
One is 6 cm long and the other is 2 cm. How
many times as long is the 6 cm block compared
with the 2 cm block?
362
Some students use the 2 cm block as a measuring
unit to figure out that 6 cm is 3 units of 2 cm. Other
students reason with the two quantities directly and
come up with the equation 6 ÷ 2 = 3.
JANE-JANE LO , [email protected], teaches at
Western Michigan University, Kalamazoo, MI 490085248. Lo’s special interests include studying the development of multiplicative concepts and preparing future teachers. TAD WATANABE, [email protected], teaches at Penn
State University, University Park, PA 16802. His interests
include children’s multiplicative concepts and mathematics
education in Japan. JINFA CAI, [email protected], teaches at the University of Delaware, Newark, DE 19716. His
interests include cognitive studies of mathematical problem
solving and integration of assessment into the classroom.
Textbook. When comparing two quantities, one of
them has to be used as the base quantity. There
are two ways to relate the other quantity to the
base quantity. The first way is to find out how
much more the second quantity is than the base
quantity. For example, how many cm longer is
the 6 cm block than the 2 cm block?
Solution. 4 cm.
Textbook. The second way is to find out how
many times as long is the second quantity as the
base quantity. For example, 6 cm is 3 times
longer than 2 cm. Another way to represent this
relationship is to use the word bi. Write as 6 bi 2,
or 6:2. The result of this comparison, 3, is called
the “value of the ratio.”
The preparation of this article was supported, in part, by a
grant from the National Academy of Education. Any opinions
expressed herein are those of the authors and do not necessarily represent the views of the National Academy of Education.
A recent analysis of Asian curricular materials has
identified several key ideas that are emphasized in
the introductory lessons of ratio (Lo, Cai, and Watanabe 2001). These key ideas include distinguishing a
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
Copyright © 2004 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
multiplicative comparison from an additive comparison; identifying a base quantity and measuring unit
for comparison; distinguishing and relating the ratio
a:b, the division a ÷ b, and the value of ratio a/b; and
learning the importance of units in forming a meaningful ratio relationship. After the introduction of
ratio, two or three more lessons were devoted to the
ideas of equivalent ratios, simplified integer ratios,
and applications of ratio concepts. Some of these discussions are familiar to mathematics teachers in
North America, whereas others seem to be unique to
the Asian materials. In this article, we will elaborate
on these key ideas and give examples from textbook
series in China, Taiwan, and Japan (Division of
Mathematics 1996; National Printing Office 1999;
Tokyo Shoseki 1998). Our goal is not to evaluate
Asian materials but rather to provide an international
perspective that may help increase teachers’ experience and awareness when they strive to help students develop ratio concepts (Cai and Sun 2002).
Introduction of Ratio Concepts
Defining ratio as being a multiplicative relationship
Unlike typical U.S. textbooks that consider a:b and
a/b as two different ways to represent a ratio, Asian
textbooks clearly distinguish between ratio a:b as a
multiplicative relationship between two quantities
and the value of ratio as the quotient a/b of the division a ÷ b. In the previous example, the multiplicative relationship between the 6 cm block and the 2
cm block can be represented as 6:2. The result of
6 ÷ 2 , or 3, is called the value of the ratio 6:2, where
6 is called the front term of the ratio and 2 is called
the back term of the ratio. Conceptually, this idea is
equivalent to saying “6 is 3 times as many as 2.”
Note that the idea of using the second quantity as
the base for comparison can be linked directly to
measurement division (quotitive), even though the
term “measurement division” is not directly used in
Asian textbooks. For example, the teacher’s manual
of a Japanese textbook talks about conceptualizing
the value of the ratio of a:b as the relative value of a
when considering b as a base quantity.
Identifying the base quantity for comparison
Since the ratio is a way to compare two quantities
using the division operation and since division is
noncommutative, the order of the two terms for a
particular ratio is important. In other words, a:b and
b:a describe the multiplicative relationship between
quantities a and b from two perspectives. The value
of ratio a:b is not the same as the value of b:a, un-
Younger brother
Older brother
0
1
2
2 1/2
3
(5/2, 2.5)
Older brother
Younger brother
0
2/5 (0.4)
1
Fig. 1 Diagrams accompanying the discussion of 5:2 and 2:5
less a equals b. The Chinese teacher’s manual indicated the reciprocal relationship between a:b and
b:a but suggested that the reciprocal relationship
not be explicitly mentioned to students at the introductory stage to avoid possible confusion.
To highlight this idea, a Taiwanese textbook
posed two different questions comparing the number of cookies for two brothers when the younger
brother has 5 cookies and the other has 2 cookies.
The first question was this: “The number of cookies
the younger brother has is how many times the
older brother’s number?” The second question was
this: “The number of cookies the older brother has
is how many times the younger brother’s number?”
The solution to the first problem was 5 ÷ 2 = 5/2 =
2 1/2 = 2.5. Students can use 5:2 to represent this
ratio relationship. The solution to the second problem was 2 ÷ 5 = 2/5 = 0.4. Students can use 2:5 to
represent this ratio relationship. A pictorial representation similar to figure 1 was used to facilitate
understanding. Note that both fraction and decimal
notations can be used for the value of ratio.
We want to emphasize two cautions about forming
a ratio relationship:
1. After the discussion of ratio definitions, the
teacher’s manual in the Chinese textbook pointed
out two difficulties that students may encounter
when they relate ratio concepts to their daily experiences. First, not all related pairs of numbers form a
ratio relationship. For example, in Chinese spoken
language, the phrase “5 bi 3” is used to express the
scores of two teams in a sport event. However, in this
context, the focus of the comparison was on the addiV O L . 9 , N O . 7 . MARCH 2004
363
should be made with the same units to make them
meaningful. For example, in a Chinese textbook,
the following problem was posed:
Li Ming is 1 meter tall, and his dad is 173 cm tall.
Li Ming said that the ratio between his height
and his dad’s height is 1:173. Is 1:173 the best
way to describe the relationship between Li
Ming’s height and his dad’s height?
The width of the rectangle above is 20 cm and
the length is 30 cm. When considering 1 cm as
being the unit quantity, the ratio between the
width and length is 20:30. The value of the ratio
is 20 ÷ 30 = 20/30 = 2/3.
Through discussion, students are guided to form a
more meaningful ratio relationship if they either
convert 1 meter to 100 cm or convert 173 cm to 1.73
meter to form the ratio 100:173, or 1:1.73. This emphasis is important when considering the idea of
“value of ratio” as the relative size of the second
quantity when the base quantity is considered to be
1. In addition, this measurement context shows the
need to define equivalent ratios.
Conceptualization of equivalent ratios
When considering 5 cm as the unit quantity,
the ratio between the width and length is 4:6.
The value of the ratio is 4 ÷ 6 = 4/6 = 2/3.
When considering 10 cm as the unit quantity,
the ratio between the width and length is 2:3.
The value of the ratio is 2 ÷ 3 = 2/3.
Fig. 2 Conceptualizing equivalent ratios
tive relationship (“The number of team A has so
many more points than Team B”) rather than the
multiplicative relationship (“Team A’s points are so
many times the number of Team B’s points”). Teachers need to be aware of the potential confusion that
students may have about the use of language inside
and outside of mathematics classrooms. A similar
caution can be made about the English language,
since the phrase “a to b” is used both for ratio and for
a sports context in the United States.
Two ratios are defined as being equivalent if they represent the same multiplicative relationship. One natural implication of this definition is that the values of
two equivalent ratios have to be equal, that is, a:b = c:d
⇔ a ÷ b = c ÷ d. In both Chinese and Japanese textbooks, the principle of equivalent ratios, “Multiplying
or dividing the front term and the back term by the
same nonzero number will create equivalent ratios,”
was supported through examples and discussion of a
division principle: ak ÷ bk = a ÷ b when k ≠ 0, which
students have learned before. Furthermore, Asian
textbooks gave detailed illustrations to connect the
idea of equivalent ratio with the idea of changing
units. For example, a Taiwanese textbook identified a
ratio of 20:30 as being the relationship between the
width (20 cm) and the length (30 cm) of a rectangle.
Then the students were asked to use 5 cm as a unit to
measure the width and the length of the same rectangle. As a result, the width became 4 units (of 5 cm)
and the length became 6 units (of 5 cm), thus a ratio
of 4:6 can be used to represent the same width versus
length relationship. Last, the students were asked to
use 10 cm as a unit to measure the width and the
length of the same rectangle and obtain another ratio,
2:3. Thus, the relationship 20:30 = 4:6 = 2:3 was established and illustrated by diagrams similar to figure 2.
Discussion of Simplified Integer Ratios
EXERCISES ASKING STUDENTS TO CONVERT A GIVEN
2. The teacher’s manual indicated the importance of paying close attention to units when comparing two quantities. In particular, at the introductory level, the comparisons of two like quantities
364
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
ratio into a simplified integer ratio are another feature
of ratio discussion in Asian textbooks. Simplified integer ratios a:b mean that both a and b are integers and
that no common factor other than 1 is shared between
Examples: Simplify the following ratios.
1800 Yen
1. 12:9 = (12 ÷ 3):(9 ÷ 3) = 4:3
2. 0.9:1.5 = (0.9 × 10):(1.5 × 10) = 9:15
= (9 ÷ 3):(15 ÷ 3) = 3:5
3. 1/2:2/3 = 3/6:4/6 = 3/6 × 6:4/6 × 6 = 3:4
Fig. 3 Examples for finding simplified integer ratios
a and b. Another way to determine if two ratios are
equivalent is to convert both into simplified integer ratios, that is, a1:b1 = a2:b2 if and only if both a1:b1 and
a2:b2 are equivalent to the same simplified ratio a:b. All
three textbooks include examples like the ones in
figure 3 to help students apply this idea.
Several significant points can be made about this
type of exercise. First, it reinforces the idea that a
ratio is a relationship between two quantities and
that those two quantities can be represented in a
variety of numerical forms—integers, fractions, or
decimals. Second, it provides another method to
check the equivalence of two ratios that reinforces
the ratio concept (i.e., two ratios are equivalent if
after simplifying they both equal the same simplified
integer ratio). Third, it provides opportunities for
students to relate numbers to each other through
common multiples and factors. Lo and Watanabe
(1997) have found this kind of conceptualization essential to develop flexible proportional reasoning.
Application of Ratio Concepts
AFTER THE BASIC CONCEPTS OF RATIO AND
equivalent ratio were established, all three Asian text-
book series included examples and exercises that
ask students to apply the concepts of ratio in a variety
of contexts. There were two basic types of questions:
1. The first type gave a ratio relationship between
two quantities and the actual amount of one of those
two quantities, then asked students to use the ratio
relationship to find the actual amount of the second
quantity. The following is an example of this type of
question from the Taiwanese textbook:
The ratio between the number of boys and the
number of girls in a summer camp is 4:3. There
are 63 girls. How many boys are in the summer
camp?
This question may be classified as a missing-valueproportion problem because a proportional relationship
(equivalent ratio) is involved. However, it is easier to
solve than a typical proportion problem (“If a car uses 8
gallons of gasoline in traveling 160 miles, how many
miles could the car travel on 30 gallons of gasoline?”)
3 (older brother)
2 (younger brother)
Fig. 4 An example of a pictorial representation that facilitates solving a
part-of-whole problem
for the following two reasons: First, one major challenge of solving this sort of problem is to construct a
ratio relationship between two different measures: gallons and miles. In the summer-camp problem, a ratio relationship is stated explicitly in the question. Second, a
typical proportion problem involves some “changes” in
states—before and after. In these antecedent problems,
the ratio and the quantities are from the same situation.
2. The second type of question in the Asian textbooks gave the ratio relationship between two quantities and the sum of the two quantities, then asked
students to use the ratio relationship to find the actual amount of each of the two quantities. For example, the following question was included in the
Japanese textbook series:
Two brothers shared 1800 Yen. The ratio between the older brother’s money and the
younger brother’s money was 3:2. How much
was the older brother’s share?
To prepare students for more complex proportion
problems, two methods of solution for each type of
problem were suggested in the student version of
the textbooks. One method helped students connect
ratio and fraction concepts through multiplicative
comparison, thus converting a ratio problem into a
problem involving multiplying by a fractional
amount. The other method required the direct application of the principle of equivalent ratios.
For the sharing-of-money problem, the Japanese
textbook series ask the following sequence of questions to encourage students to think about these
two solution methods:
1. The older brother’s money was what fraction of
the total amount of money?
2. Write down a computation sentence that will determine the older brother’s share.
3. Solve the problem using the following equation:
3:5 = x:1800.
4. What was the younger brother’s share?
The diagram in Figure 4 was used to help students
conceptualize the first two questions.
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365
From figure 4, one could reason that if the older
brother’s money comprised three units and the
younger brother’s money comprised two units, then
the total amount of 1800 Yen was equivalent to 5 units.
So the older brother’s money was 3/5 of the total
amount of money. Thus, the answer for question 2 was
1800 × 3/5, and students could figure out the older
brother’s share of 1080 Yen from this computation.
Question 3 above suggested a second strategy
that required directly applying equivalent ratios.
Since the ratio between the amount of money that
the older brother had (x Yen) and the amount of
total money (1800 Yen) could be expressed as the
ratio 3:5, one could solve this problem using the
principles of equivalent ratios: Because 1800 is
360 times 5, x must be 360 times 3, which results
in the answer of 1080 Yen. The younger brother’s
share could then be solved with either approach.
Using both methods helps students see how the
ideas of multiplicative comparison, fractions (or
decimals), ratios, simplified ratio, and equivalent
ratios are connected.
Conclusion
THE CONCEPTS OF RATIO AND PROPORTION ARE
among the most important topics in school mathematics, especially at the middle school level. However, studies have repeatedly shown that most middle school students have difficulties with these
concepts (NCTM 2000). This article included ideas
and examples used by Asian textbooks to teach the
concepts of ratio that are fundamental to the develop366
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
ment of proportional reasoning. In Asian textbooks,
the concepts were carefully introduced through an
emphasis on multiplicative comparison, the link to
measurement (quotitive) division, the identification
of base quantity, and the distinction between ratio
and nonratio pairs of quantities. The idea “value of
ratio” was introduced to firmly establish the ratio’s
identity as a relationship based on multiplicative
comparison rather than just another way to write a
fraction. Rather than move directly into the concepts
of proportion, Asian textbooks spent time developing
the idea of equivalent ratios and simplified integer
ratios and discussing how these ratio-related concepts could be used to solve problems in everyday
contexts. Typically, pictorial representations were
used and multiple solution methods were discussed
to help students relate ratio concepts to other previously learned concepts such as measurement (quotitive) division, fractions, and divisors. Furthermore,
exercises and examples were carefully chosen to link
the ratio concepts to previous studies on fractions
(including fractions greater than 1) and decimals.
We believe that these approaches all aim to develop
proportional reasoning, which is essential in solving
proportion problems.
In general, Asian textbook series do not include
units in mathematics sentences as part of the written computation. We can probably argue the advantages and disadvantages of such a practice, but it
goes beyond the focus of this article. Nevertheless,
the Asian materials we analyzed did treat units carefully and systematically. The examples of comparing Li Ming’s height with his father’s height as well
as using the units flexibly to generate equivalent ratios discussed earlier in this article illustrate this
emphasis. Furthermore, both the textbook series
and the teacher’s manuals routinely remind students to think about the meanings of the quantities
and the units used to quantify these quantities involved in computation. The goal is to prepare students for more complex contextual problems when
multiple computations are required to determine
unknown quantities. The examination of curriculum
and instructional practice in other nations provides
a broader point of view on how topics can be
treated. We hope that such an international perspective can add to U.S. teachers’ background when
they try to address the issues and challenges facing
students’ learning of ratio and proportion.
References
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