ERIC Identifier: ED433184
Publication Date: 1998-02-00
Author: Millsaps, Gayle M. - Reed, Michelle K.
ERIC Clearinghouse for Science Mathematics and Environmental Education
Curricula for Teaching about Fractions. ERIC Digest.
The study of fractions is foundational in mathematics, yet it is among the
most difficult topics of mathematics for school students. Students have
difficulty recognizing when two fractions are equal, putting fractions in order
by size, and understanding that the symbol for a fraction represents a single
number. Students also rarely have the opportunity to understand fractions before
they are asked to perform operations on them such as addition or subtraction
(Cramer, Behr, Post, & Lesh, 1997).
The "NCTM Curriculum and Evaluation Standards" (1989) promotes the use of
physical materials and other representations to help children develop their
understanding of fraction concepts. The three commonly used representations are
area models (e.g., fraction circles, paper folding, geoboards), linear models
(e.g., fraction strips, cuisenaire rods, number lines), and discrete models
(e.g., counters, sets).
Some may believe that children will automatically understand fraction
concepts simply as a result of using the various representations or
manipulatives. This is not necessarily the case (Thompson & Lambdin, 1994).
For instance, some students define a fraction as "a piece of pie to eat,"
because they have only seen fractions represented using circle diagrams (Niemi,
1996). Providing many kinds of representations can help students with this
problem, as long as teachers help students connect their understanding of
concepts to the different representations. This digest describes curriculum
guides which offer help to teachers wanting to extend student ideas of fraction
concepts, use manipulative materials in powerful ways, and help students connect
The "Seeing Fractions" unit (Corwin,
Russell, & Tierney, 1990) exposes students to several visual models for
fractions that represent the various contexts in which fractions are commonly
used-to denote part of the area of a shape (e.g., 3/5 of a pizza), part of a
group of things (e.g., six of the ten team members are girls), part of a length
(e.g., 2/3 of the distance from home to school), or a rate (e.g., three candy
bars for 50 cents). The unit is comprised of four modules, each representing one
of these contexts. In the first module, students explore the fraction
equivalencies of common fractions (halves, thirds, fourths, sixths, eighths,
twelfths, and twenty-fourths) using geoboard-based activities. Particular
emphasis is placed on encouraging students to generate many different examples
of a partitioning of a unit whole (such as into fourths) that are equal in area
but may not be congruent in shape. For example, half of a 4x4 geoboard may be
divided into two congruent squares and the other half into congruent triangles.
All pieces have an area of one-fourth of the whole, but are not all the same
In the second module, students explore rates as a setting for developing
procedures to compare fractions and to find equivalent fractions. Starting with
scenarios such as selling candies (five for two cents) students learn to create
equivalent fraction series (5/2 = 10/4 = 15/6 = ...) from a base rate (5/2).
Using this knowledge they are able to compare two fractions (5/2 is greater than
5/3 since in their series table 5/2 = 15/6 and 5/3 = 10/6) and to solve
In the third module, students relate fractions to division in the context of
sharing and as they develop strategies for adding fractions. An initial scenario
suggests students generate plans for sharing seven cookies among four people.
Students' strategies will likely fall into one of three categories: (1) give
everyone a cookie, divide the rest in half and give everyone half, then divide
the remaining cookie and give everyone a fourth; (2) divide all the cookies into
fourths and give everyone one fourth from each cookie; (3) give everyone one
cookie, then remove one fourth from each cookie and put it together so that each
person gets another 3/4 of a cookie. The amount of cookie(s) that one person
will receive can be represented by 1 + 1/2 + 1/4, 7/4, or 1 + 3/4, respectively.
Student strategies for solving the problem provide a context for exploring
addition of fractions as these representations must all yield the same amount.
In the fourth module, fraction strips are used to help students develop an
understanding of what is meant by partitioning a whole length into equal
fractional lengths, of estimating fractional lengths, and of measuring
distances. Activities include estimating lengths of objects with unmarked
strips, creating fraction rulers through strategies of equal partitioning such
as paper folding, and comparing fractions by organizing fraction rulers in
RATIONAL NUMBER PROJECT
"The Rational Number Project:
Fraction Lessons for the Middle Grades" (Cramer et al., 1997a, 1997b) comprises
two volumes of carefully researched lesson plans aimed at developing students'
number sense for fractions. Level 1 is written for students in grades 4-5 and
level 2 is written for grades 5-8. The model is based upon several key ideals.
First, to develop fraction number sense, students must spend time investigating
concepts of order, equivalence, unit, and addition and subtraction with
manipulative materials, such as fraction circles, counters, Cuisenaire rods, and
paper folding. Another aspect of the program is that each of the models used are
analyzed to see how they are alike and different, and efforts are made to
connect ideas across many different types of representations. This practice
corresponds to the Lesh Translation Model (see Cramer, Behr, Post, & Lesh,
1997a or b).
Both level 1 and level 2 lessons emphasize developing the meaning for
fraction symbols before asking students to operate on them. Except for the topic
of multiplication of fractions and the introduction of a new, more complex model
for fractions (Cuisenaire rods) in level 2 lessons , both sets of lessons cover
the same topics.
Initial lessons in level 1 have students engage in modeling and naming
(verbally and symbolically) fractions less than 1 using area models such as
fraction circles, paper strips, and other shapes. Throughout these initial
lessons the concept of the flexibility of the unit is developed by using a
variety of non-standard shapes to represent one whole such as half of a circle.
Fraction equivalence and ordering are then introduced using area models before
students are asked to develop the same concepts using a discrete model such as
The next tier of lessons returns to the initial area models of fractions to
develop students' ability to reconstruct the whole given a fractional part and
to model and name fractions greater than 1. A subsequent lesson extends the
concept of fraction equivalence using the rate series by having students look
for number patterns in the information they have already gathered about
In the closing lessons, addition, subtraction, and multiplication (level 2
only) are introduced via students' modeling of stories. A special emphasis is
given to students' estimation of the sums, differences or products by
recognizing the approximate size of the fraction operands using their
internalized visual models of fractions.
NCTM ADDENDA SERIES
The NCTM Addenda Series booklet "Understanding rational numbers and proportions" (Curcio & Bezuk, 1994) for
grades 5-8 presents a collection of activities involving rational numbers and
proportions in a problem-solving context. Many of the activities stress the
application of rational numbers to real-world situations, the use of alternative
assessment techniques, and the integration of technology. The activities are
divided into three content clusters: exploring and extending rational number
concepts, applying rational number and proportion concepts, and making rational
number connections with similarity.
The key features of the curriculum materials
described here are implicit in the Lesh Translation Model presented above.
Students begin to construct a deeper understanding of fractions when they are
represented in a variety of ways and when there are explicit linkages to
everyday life and familiar situations involving the use of fractions. Other
resources that facilitate these linkages are grouped here by topic.
Leutzinger, L. P., & Nelson, G. (1980). Let's do it: Fractions with
models. 'Arithmetic Teacher," 27(9), 6-11. Presents beginning fraction concepts
using a circle-region or "pie" model.
McBride, J. W., & Lamb, C. E. (1986). Using concrete materials to teach
basic fraction concepts. "School Science and Mathematics," 86(6), 480-488.
Describes how to prepare inexpensive materials.
Van de Walle, J., & Thompson, C. S. (1984). Let's do it: Fractions with
fraction strips. "Arithmetic Teacher," 32(4), 4-9. Describes how to make
fraction strips and presents beginning activities.
MULTIPLICATION OF FRACTIONS
Sinicrope, R., & Mick, H.
W. (1992.) Multiplication of fractions through paper folding. "Arithmetic
Teacher," 40(2), 116-121. Uses an area model to show multiplication of
Cramer, K., & Bezuk, N. (1991). Multiplication of fractions: Teaching for
understanding. "Arithmetic Teacher," 39(3), 34-37. Uses geoboards, paper
folding, and counters to show multiplication of fractions concepts.
DIVISION OF FRACTIONS
Curcio, F. R., Sicklick, F., &
Turkel, S. B. (1987). Divide and conquer: Unit strips to the rescue. "Arithmetic
Teacher," 35(4), 6-12. Uses teacher-made strips to show division ideas.
Corwin, R. B., Russell, S. J., Tierney, C. C.
(1990). "Seeing fractions: A unit for the upper elementary grades." Sacramento,
CA: California Dept. of Education. (ED 348 211).
Cramer, K., Behr, M., Post, T., & Lesh, R. (1997a). "Rational Number
Project: Fraction Lessons for the Middle Grades: Level 1." Dubuque, IA:
Cramer, K., Behr, M., Post, T., & Lesh, R. (1997b). "Rational Number
Project: Fraction Lessons for the Middle Grades: Level 2." Dubuque, IA:
Curcio, F. R., & Bezuk, N. S. (1994). "Understanding rational numbers and
proportions." Reston, VA: NCTM. (ED 373 991)
NCTM. (1989). "Curriculum and evaluation standards for school mathematics."
Reston, VA: Author.
Niemi, D. (1996). A fraction is not a piece of pie: assessing exceptional
performance and deep understanding in elementary school mathematics. "Gifted
Child Quarterly," 40, 70-80.
Thompson, P. W., & Lambdin, D. (1994). Research into practice: Concrete
materials and teaching for mathematical understanding. "Arithmetic Teacher,"
WORLD WIDE WEB RESOURCES
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lesson plans by topic in mathematics. Look under "Arithmetic."