Curricula for Teaching About Fractions
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by: Gayle M. Millsaps & Michelle K. Reed
February 1998 (Updated July 2002)
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 fraction concepts.
Seeing Fractions
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
shape.
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 proportion problems.
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 arrays.
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 counters.
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 subse-quent
lesson extends the concept of frac-tion equivalence using the rate
series by having students look for number patterns in the information
they have already gathered about equivalent fractions.
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.
Summary
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.
Beginning fraction concepts
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 fractions.
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.
References
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:
Kendall/Hunt Publishing.
Cramer, K., Behr, M., Post, T., & Lesh, R. (1997b). Rational Number
Project: Fraction Lessons for the Middle Grades: Level 2. Dubuque, IA:
Kendall/Hunt Publishing.
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, 41(9), 556-558.
World Wide Web Resources
Math Archives: Topics in Mathematics
http://archives.math.utk.edu/topics/
Use the search engine on this page to find many fine resources about
teaching fractions.
AskERIC Lesson Plans: Mathematics
http://askeric.org/cgi-bin/lessons.cgi/Mathematics
Many lesson plans by topic in mathematics. Look under "Arithmetic."
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