# Conceptual Understanding of Fractions

Title Conceptual Understanding of Fractions

• [Current Contributors: June Choi, Gina Lee, Jenny Nguyen, Doris Ni, and Poonam Patel; Previous Contributors: Kaitlin Franks, Jane Kim, Lori Labuzetta, Cynthia Leon, and Cheryl Panado]

August 2008

## Cognition and Learning Background

Fractions are a major concept that students are first introduced to in second grade. This concept continues throughout their mathematical education. Teachers struggle to teach this concept effectively, while students struggle to gain a conceptual understanding. The goal of fraction instruction is for students to be able to visualize fractional amounts accurately.

#### Why Teach Conceptually

It is important for teachers to ensure that students have a firm grasp on the concepts of fractions. Once these concepts are established, students can better visualize and understand the value of a fraction. Students with a conceptual understanding will have fewer problems in the future when they are introduced to fraction operations and algorithms (especially when adding and subtracting with unlike denominators).

Brizuela (2005) explored the kinds of notations young children make for fractional numbers. Exploring childrens' notations reveals their thinking and knowledge of fractions. According to the study, most children in the study did not make conventional notations for fractions. The relationship between conceptual understanding and in making notations is a complex area. Brizuela argues that children’s notational competencies and conceptual understandings are intertwined. Thus, it is vital to integrate notations as students are exposed to learning and understanding of fractional numbers. Introducing fraction concepts to children at a young age builds their ability to conceptualize fractional numbers.

#### Theories of Cognition and Learning Frameworks

• Zone Proximal Development
Lev Vygotsky theorizes that every individual has a Zone of Proximal Development (ZPD) that varies for each concept. The ZPD is the difference between what people can achieve independently versus what they can achieve with guidance from a more knowledgeable source. When children are working within their ZPD, learning occurs. The teacher’s responsibility is to move a child from their current state of knowledge to the next level of development.
When students enter their classrooms, they bring with them knowledge of partitioning, fair share, and equality . The teacher’s challenge is to base fraction instruction on prior knowledge in order to make learning occur more efficiently. Teachers must first identify a child’s prior knowledge or current level of development. Instruction is then designed to guide children through their ZPD, to reach their potential level of development.
• Constructivism
Many theorists propose that meaningful learning occurs when students construct their own knowledge; this is referred to as constructivism. Social constructivism relies on a group of people collectively generating ideas and understandings of a concept, while individual constructivism focuses on individuals interacting with the content to make sense of it in their own way .
When introducing fractions, teachers should allow students to interact with materials to discover patterns and develop concepts before the teacher formally introduces the topic through direct instruction . Teachers should integrate a variety of cooperative learning groups as well as independent work in order to foster individual and social constructivism.

#### Fraction Proficiency & Success in Higher Level Math

In relation to achieving success in algebra, students need to gain fluency in understanding fractions early on in their educational endeavors. Since elementary algebra is built on a foundation of basic arithmetic concepts, a solid mastery of these concepts is essential towards generalizing their prior knowledge in symbolic representation of numbers (as cited in Brown & Quinn, 2007). According to a study that investigates the relationship between fraction proficiency and success in algebra, inability to successfully manipulate and perform basic operations on common fractions has led to negative performance in algebraic reasoning (Brown & Quinn, 2007). Suffice to say, promoting conceptual understanding of fractions leads to beneficial gains in higher level mathematics.

## Cognitive Obstacles and Common Misconceptions

#### Causes for Low Performances in Fractions:

• Complexity of fraction concepts (Bezuk & Cramer, 1989; Streefland, 1991)
• Incorrect knowledge of rote procedures (Hiebert & Wearne, 1985; Mack, 1990)
• Difficulties with identification of units (Hunting, 1983, 1986), teaching computation before understanding the meanings of fractions (Aksu, 1997)
• More emphasis on teaching procedures than on teaching conceptual meanings (Moss & Case, 1999)
• Interference from whole-number knowledge (Lukhele, Murray, & Olivier, 1999)
• Influence of multiple representations on initial learning of fraction ideas (Cramer, Post, & delMas, 2002)

#### Complex Nature of Fractions vs. Rote Teaching and Learning Practices

• Complex nature of fractions
• fraction concepts and sophisticated fraction arithmetic can be confusing to students who are only familiar with whole-number systems
• students experience substantial conceptual leaps in symbolic representation, intrinsic meaning of a number, identification of a unit, and computational schemes when switching from whole numbers to fractions
• students tend to regard fractions as two or three whole numbers (Newstead & Murray, 1998; van Niekerk, Newstead, Murray, & Olivier, 1999)
• easily confusing the multiple roles that fractions play in real-life situations such as measuring, quotient, ratio, operator, and so on (Miemi, 1995)
• the hardest part of learning fractions is understanding that “what looks like the same amount might actually be represented by different numbers,” (Lamon, 1999, p. 22)
• the ability to hinder students’ learning fraction arithmetic
• students tend to apply a familiar algorithm (regardless of correctness) from whole number arithmetic in fraction arithmetic Lukhele et al., 1999; Niemi, 1995)

Student misconceptions are further noticed through the use of incorrect or inefficient strategies. It is often noticed that students with mathematical disabilities frequently utilize less mature strategies that their peers; they also “have more difficulty retrieving an efficient strategy, even when it has been taught to them” (as cited in Parmar, 2003). A specific example of an incorrect algorithm applied towards fractions is when students “invert the dividend and the divisor or make other related errors, which appear to be based on students’ rote learning of an algorithm without corresponding understanding” (Parmar, 2003). In addition, many students are perplexed with the concept of understanding that a fraction involves a part to whole relationship, which may further hinder their mathematical fluency and adeptness in working with fractions.

#### Three Main Explanatory Frameworks on Children's Interpretation and Understanding of Numerical Values of Fractions

1. Fraction consists of two independent numbers
2. Fractions as parts of a whole
3. Fractions can be smaller, equal, or even bigger than the unit
• The results agree with previous studies showing a large number of students adopt the second explanatory framework “Fractions as part of a unit,” which claim that students believe that a fraction with a numerator equal to the denominator is equivalent to the unit.
• Stafylidou and Vosniadou (2004) agree with previous research on the need for more practice on partitioning and measurement activities. These activities can help children develop the concept of a unit, which is best developed through activities that relate the unit to fractions.

#### Student Challenges

• Apply whole number knowledge to fractions
• Fractions are not arranged in a successive order as are whole counting numbers, such as 0, 1, 2, etc. Fractions with larger numbers do not imply larger quantities as they do in counting numbers, for example, > even through 3 > 2. The value of a unit varies.
• Depending on the situation and problem, the quantity of the unit varies.
• For example: One-half can be visually represented in different ways depending on the size of the unit.
• Rote memorization of procedures results in operational errors.
• Without a deep conceptual understanding of fractions, students incorrectly apply rote operational procedures when presented with addition problems.

#### Dealing with Student Errors

• Student errors should be used as a starting point for student inquiry into mathematics
• Discourse initiated around errors helped generate student inquiry into understanding student errors
• Builds on cognitive processes
• Teaches students’ to reflect on their work
• Sometimes it helps students to hear it from other students (in peer language)
• Provide students with diagrams

#### Studies Related to Common Misconceptions and Obstacles

Study 1: Research on fraction division has reported that teachers can confuse situations that call for dividing by a fraction with ones that call for dividing by a whole number or multiplying by a fraction (Armstrong & Bezuk, 1995; Ball, 1990; Borko, Eisenhart, Brown, Underhall, Jones, & Agard, 1992; Ma, 1999; Sowder, Philipp, Armstrong, & Schappelle, 1998). Based on the findings of this study, it is important that teachers are confident int heir teaching abilities and skills to properly explain and conceptually teach fractions so that students are not left with doubts on how to work with fractions. In this study, teachers provided students with diagrams to teach students division of fractions, but the diagrams were utilized ineffectively and students walked away from the lesson full of misconceptions and doubts. Teachers must know how to use available resources to enhance the lesson and not be use resources as a crutch in a procedural lesson.

Study 2: Jigyel and Afamasaga-Fuata’i (2007) found that Australian students (equivalent to grade 4, 6, and 8 in the U.S.) perceive the numerator and denominator of a fraction as two unrelated whole numbers. Students were able to identify equivalent fractions when they were presented geometrically (particularly circle models compared to rectangular ones). However, misconceptions occurred when students were presented with equivalent fractions that were presented numerically (e.g. 4/6 and 2/3). Most students responded that 4/6 and 2/3 are not equal and that 4/6 is double of 2/3. The authors conclude that these misconceptions require a development and consolidation of students’ understanding fractions using multiple models that clearly represent the relationship between the numerator and the denominator. In addition, students need to understand the number of parts and size of parts when comparing fractions (Jigyel & Afamasaga-Fuata’i, 2007).

Study 3: According to a study on notations that young children form about fractions, a greater percentage of children in first grade interpreted fractions or halves as “little bits.” (Brizuela, 2005) Students have already been exposed to lessons regarding fractions and how to make notations for them. It may be such instruction that may lead students to certain misconceptions about fractions.

The best way to meet the needs of students is to categorize them into ability levels to understand their deficiencies and be able to teach how to conceptually deal with fractions.

## Pedagogical Tools and Strategies

#### Ineffective Strategies

(Meagher, 2002; Lamon, 2005)

• Emphasize procedures over concept development
• Adult-centered rather than child-centered approach
• Rational number instruction is limited to the part-whole component of rational numbers
• Not addressing students’ prior knowledge of fraction concepts
• Not addressing student misconceptions throughout instruction

#### Models and Visual Imagery

An important component of fraction concept development is the use of manipulatives or models (Cramer & Henry, 2002). Extended use of multiple manipulatives and models will result in a transfer to visual representations or mental images. Visual representations will lead to a more concrete understanding of fraction concepts (Cramer, Post, & Del Mas, 2002).

Types of Models
• Physical
• Area models: circular pie models, paper folding
• Length models: Cuisenaire rods, fraction strips, number line
• Set models: the whole is a set of objects, and the subset of the whole make up the fractional parts (e.g., counters, base ten blocks, colored beans)
• Pictorial
• Drawings
• Diagrams
• Real World
• Real-life situations

#### Develop Rational Number Concepts

Rational numbers encompass part-whole comparison, measures, operators, quotients, and ratios and rates (Lamon, 2005). These rational numbers are all represented in a symbolic notation of $\frac{a}{b}$. Students must understand that the fractional notation can mean many things beyond a part-whole relationship. Teachers should expose students to each branch of rational numbers because students need to be exposed to the broad range of meanings that the fraction notation symbolizes.

Types of Rational Numbers
• Part-whole comparison
• Example: Jake has 10 balls. 5 of the balls are baseballs or $\frac{1}{2}$ of the balls are baseballs.
• Measures
• Example: It takes Sue 30 minutes to drive to work or $\frac{1}{2}$ hour
• Operators
• Problems that involve increasing, decreasing, shortening, enlarging, and resizing. Example: Jessica had an image on her computer. She enlarged it by 2 (200%). However, she is unhappy with this size and would like it to be the original size again. What fraction of the resized image should she input in the computer in order to get the original sized image? $\frac{1}{2}$
• Quotients
• Example: There are two people evenly sharing a pizza. What portion of the pizza with they each get? In this problem, the rational number $\frac{a}{b}$ is formed with the meaning a pizzas are shared with b people. The answer would be $\frac{1}{2}$.
• Ratios and rates
• Example: For every 1 pant Timmy buys, he buys 2 shirts. The ratio of pants to shirts is 1:2 or $\frac{1}{2}$

#### Equal Partitioning and Equal Whole

According to Yoshida and Sawano (2002), students in the experimental group, where there was a great emphasis on equal partitioning and equal whole strategies, displayed significantly better comprehension of the equal-partitioning of fractions and the representation of fraction sizes compared to students in the control group, where traditional textbook instruction was implemented.

• Equal Partitioning
• Comment on the process of equal parts during the teaching process. Have students particpate in a task that requires composing fractions from parts divided unequally. Afterwards, have the students compare the given task with fractions composed from parts divided equally and emphasize the possibility of comparing fractions by dividing the whole equally (Yoshida & Sawano, 2002).
• Equal Whole:
• Stress the importance of having equal wholes (same sizes) when comparing the order of 2/5 with 2/3 in the textbook, for example. Intentionally, in comparing two fractions, use incorrect figures in which the sizes of the wholes are unequal and have the students discover and emphasize that it would not be possible to compare fractions if the sizes of the wholes were different (Yoshida & Sawano, 2002)

#### Student Discourse

Empson (1999) found in her study of first graders that children who generated their own strategies and were involved in sharing, comparing, and justifying their mathematical thinking through teacher-guided discussion provided the foundation for developing fraction concepts.

• Frequent classroom discussions of fraction problems and using familiar resources to solve fraction problems help children make sense of fraction concepts.
• The interactions of fractions assist students in developing the necessary skills to give appropriate explanations and justifications of their fraction understandings.
• Children may further benefit in learning by coming up with their own strategies and actively participating in rich discussion with their peers to share multiple ways of solving a problem, thus facilitating student’s mathematical reasoning skills (Empson, 1999).

#### Integrated Mathematics Assessment (IMA)

Saxe, Gearhart, and Nasir’s (2001) study found that the professional development program, The Integrated Mathematics Assessment (IMA), was effective in enhancing teachers understandings of fractions, knowledge of students’ mathematical thinking, and students’ motivation. The program primarily focused on problem solving and conceptual understanding of fractions. Students who were taught by teachers participating in IMA achieved greater post test scores on the conceptual understanding of fractions compared to the two other groups of teachers, one group of teachers who only discussed strategies and the other group who received no professional development support and relied on textbooks. Teachers who participated in the IMA were taught principles from the following mathematical programs:

• Cognitively Guided Instruction program (Carpenter et al., 1989, as cited in Saxe, Gearhart & Nasir, 2001)
• Problem-Centered Mathematics Project (Cobb, Wood et al., 1991, as cited in Saxe, Gearhart & Nasir, 2001)
• The Educational Leaders in Mathematics Project (Simon & Schifter, 1993, as cited in Saxe, Gearhart & Nasir, 2001)

#### Manipulatives - Fraction Tiles

Martin and Schwartz (2005) stated that Physically Distributed Learning (PDL), or the adaptation and reinterpretion processes through active manipulation of fraction tiles, enhances students’ conceptual understanding of fractions compared to pie pieces.

• Teachers should not demonstrate how to use a manipulative to solve a math problem for children, but rather it is important to provide children with multiple opportunities to self-discover using materials by generating their own structures and interpretations.
• Having students actively engage with manipulatives is important as it promotes deeper learning of fraction concepts.
• Students should physically rearrange fraction tiles rather than pie pieces because it helps prepare them to "learn how to use new materials" (Martin and Schwartz, 2005).
• Principles of adaptation and reinterpretation in distributed learning, or rearranging fraction tiles, play an important role in developing students’ understanding of fraction concepts (Martin and Schwartz, 2005).

#### Manipulatives - Measurement Models

Pearn (2007) recommends using the measurement model, particularly paper folding, fraction walls, and number lines to develop students’ understanding of fractions, fractional language, and the flexibility between everyday language and fractional symbols. For instance, while students fold paper strips in fractional parts, teachers can lead a discussion based on the following questions that ask students to observe and predict the fractional parts:

• How many parts will there be? How many folds will there be? What do we call each part? Show me (one quarter) of the paper strip. Which is larger? How do you know?

Then, the paper strips are used to mark fraction walls showing a whole, half, third, fourth...tenths.

• Students will mark each of the folded parts with appropriate fraction units.
• Fraction walls help develop concepts of equivalent fractions by allowing students to compare fractions from zero to one.
• The author suggests that after work with folded paper strips, students can use Microsoft Word to make their own fraction walls using the Split Cells function.

After that, students can be provided number lines marked 0 to 1 and asked to estimate and write the given fraction units on the number line (which may change in the markings: 0 and 2, 1 and 2, 0 and 1/2 , etc.).

• Students need to provide reasoning for their estimation and placement of the fractions

The author also emphasizes explicit instruction of academic language of fractions during these activities.

• She recommends providing opportunities of practice where students express fraction units (with operations) in words and symbols.

In addition, Reys et al. (1999) recommends that teachers use benchmarks of zero, one-half, or one on the number line to help students develop their fraction number sense. Benchmarks allow student's to compare the relative sizes of fractions, through estimating, ordering, and placing them on a number line.

#### Graphical Partitioning Model (GPM) - A Web-Based Cognitive Tool (CT)

Kong's (2007) study on examining the learning outcomes based on the use of a cognitive tool for teaching fractions found that students were enthusiastic about using the cognitive tool as an educational tool. In addition, students found the cognitive tool to be an engaging resource. Thus, the cognitive tool has potential for further development in the learning of fractions and promoting collaborative learning in the classroom.

• The CT is a web-based learning tool for students to develop their concept of fractions.
• It aims to support student learning about fraction equivalence and addition and subtraction of fractions with like and unlike denominator (Kong 2007).
• Graphical Partitioning Model (GPM), a web-based CT, represents both the part-whole perspective and the measure perspective of fractions using a rectangular bar. (DIAGRAM)
• The teacher in the study claimed that the use of the GPM encouraged constructive teacher–student interactions and student–student discussions in the classroom setting.
• GPM addresses the diverse abilities of students and serves as a helpful tool for students to learn and understand the key mathematical concepts of fractions in an exploratory manner.

#### Part-Whole Theoretical Model

Kieren first established that the concept of fractions is not a single construct, but rather consists of five interrelated subconstructs: part-whole, ratio, operator, quotient, and measure (Charalambous and Pitta-Pantazi, 2007). However, Charalambous and Pitta-Pantazi's (2007) study on a theoretical model links the five subconstructs to the operations of fractions, fraction equivalence, and problem solving. The authors found the following benefits of using this theoretical model:

• The part-whole interpretation of fractions has a significant role in developing understanding of fractions.
• Part-whole subconstruct overlooks and relates to the other four subcontructs (ratio, operator, quotien, measure) DIAGRAM.
• Understanding of the different interpretations of fractions can improve students' performance on tasks involving the operations of fractions and fraction equivalence.

## Curricula and Technological Resources

These curricula and resources were chosen because they support the development of a conceptual understanding of fractions rather than procedural algorithms. Researchers recommend that teachers promote a quantitative understanding of fractions in their instructions. For example, students should understand that 5/6 + 7/8 is equal to about 2 because 5/6 is almost a whole and 7/8 is almost a whole. The following resources can be applied to build on student informal knowledge, resulting in a more meaningful conceptual understanding. The presented curricula resources can be useful for teachers in lesson planning of fractions. The provided technological resources can further aid students’ understanding in another interactive visual environment. Last, but not least, the literary resources can help students explore fraction ideas in a more open and informal context.

### Curricula Resources

#### National Council of Teachers of Mathematics (NCTM) Illuminations

Three units on fractions with 5 to 6 lessons each that teachers can adapt to their classrooms. They address various fraction models, such as regions, lengths, and sets, to develop a conceptual understanding of fractions.

#### Rational Number Project

As a cooperative research and development project, the Rational Number Project offers a book of sequential fraction lessons to help teachers facilitate students’ conceptual understanding of fractions. The proposed fraction lessons reflect the theoretical framework by Jean Piaget, Jerome Bruner, and Zoltan Dienes. Basing on these theoretical ideas, Richard Lesh, a Rational Number Project director, design an instructional model that illustrates how a teacher should organize his/her instruction to help students actively construct their learning. Thus, reflecting this instructional model, this book offers teachers instructional strategies that engage students in learning through the use of manipulatives and classroom discourse. Assessment tools are also available to provide teachers insight to students’ thinking for further planning of instruction (Cramer, Berh, Post, & Lesh, 1997).

#### New Zealand Maths

The Numeracy Professional Development Projects present a book on teaching fractions, decimals, and percents that focuses on students' mental strategies. The activities are designed to meet the learning needs of students at different strategy stages (i.e., emergent, counting all, advanced additive part-whole,...,advanced proportional part-whole). Each activity composes of suggested instructional approaches that help develop students' strategies between and through the phases of Using Materials, Using Imaging, and Using Number Properties. (Numeracy Professional Development Projects, 2006).

Van de Walle, J. A. (2007). Elementary and Middle School Mathematics: Teaching Developmentally 6th ed.. United States of America: Pearson Education, Inc.

• Chapters 16 addresses key learning issues in developing fraction concepts and provides activities as a guide to instructional planning.

Lamon, S. J. (2005). Teaching Fraction and Ratios for Understanding: Essential Content Knowledge and Instruction Strategies for Teachers.

• This book provides background knowledge on fractions and rational numbers. The end of each chapter includes activities that address various components of fractions and rational numbers.

Burns, M. (2001). Teaching Arithmetic: Lessons for Introducing Fractions, grades 4-5. Sausalito, CA: Math Solutions Publications.

• This book offers well-designed lessons with numerous details, sample student dialogue, and Blackline Masters. Lessons present introductory ideas of fraction concepts, including looking at one-half as a benchmark.

Litwiller, B. (Editor). Making Sense of Fractions, Ratios, and Proportions (2002 Yearbook). National Council of Teachers of Mathematics.

• NCTM's yearbook offers a variety of articles that provide insights to students' thinking about fractions, rations, and proportions. The Classroom Companion booklet suggests different activities that teachers can use in their classrooms.

#### Fraction PowerPoint Presentations

The following websites offer various PowerPoint Presentations that teachers can use to instruct students on fraction concepts. It is advised that teachers should be conscious at selecting these PowerPoint slides according to the provided theoretical frameworks and to meet the learning needs of their own students.

### Technological Resources On-Line

Visual Fractions: http://www.visualfractions.com/

• This website allows students to see a variety of fractions represented in multiple ways (circles, bars, etc.). This resource has segments focused on identifying fractions, comparing fractions, and operating with fractions (addition, subtraction, multiplication, and division). Games are also included as possible extension activities.

No Matter What Shape Your Fractions Are In: http://www.math.rice.edu/~lanius/Patterns/

• Going beyond the basic circle and/or bar fraction representations, this website uses pattern blocks to explore fractions. This contains lessons and activities where fractions are represented in different and unique ways.

• Once students become familiar with operating on fractions, this website can be used to reinforce students’ skills and knowledge. This website presents students with virtual flashcards, in which students must type in the answer (numerator, then denominator) to various fraction problems.

National Council of Teachers of Mathematics Illuminations Fraction Games:

National Library of Virtual Manipulatives: http://nlvm.usu.edu/en/nav/index.html

• This website contains teacher lesson plans for students in Pre-kindergarten through 12th grade. This resource contains applets and lessons involving: Numbers & Operations, Algebra, Geometry, Measurement, and Data Analysis & Probability. Lessons allow students to work with virtual base-ten blocks, abacuses, charts, and counters.

Fraction Track from National Library of Virtual Manipulate: http://standards.nctm.org/document/eexamples/chap5/5.1/index.htm

• In this game, students position fractions on number lines with different denominators. They will have the opportunity to develop their conceptual understanding of fractions, including equivalent-fraction concepts.

Fraction Bars from National Library of Virtual Manipulate: http://nlvm.usu.edu/en/nav/frames_asid_203_g_1_t_1.html

• Showing a similar model to Cuisenaire rods (except that bars are segmented), this virtual manipulation allow students to create and compare multiple bars (representing fraction units) of any size from 1 to 10 in any color.

Fraction Pieces from National Library of Virtual Manipulate: http://nlvm.usu.edu/en/nav/frames_asid_274_g_2_t_1.html?open=activities

• Students can use this open-ended manipulation to explore fraction pieces through a circular model and a rectangular model. Students learn the different fraction units making up a whole.

Fractions—Comparing from National Library of Virtual Manipulate: http://nlvm.usu.edu/en/nav/frames_asid_159_g_2_t_1.html

• Through using visuals of circular pieces, students find common denominators and compare two fractions. Then, they locate these fractions on a number line and identify a new fraction between these two.

Fractions—Visualizing from National Library of Virtual Manipulate: http://nlvm.usu.edu/en/nav/frames_asid_103_g_1_t_1.html

• In this applet, students use the given rectangular area model to subdivide the area into sections and color in the appropriate sections to corresponding the given fraction.

• This game allows students to explore the concept of equal in quantity, not necessarily in size.

MIND Research Institute: Jiji Cycle: http://www.mindresearch.net/media/edu/demoFolder/demo/games/JiJiCycle/jijicycle.html

• This math game offers a visual way to develop the player’s conceptual understanding of fraction through estimating a fraction unit on a number line (through the use of an animated scenario about a penguin named Jiji). Students can choose to estimate fractions and fraction addition through a focus on spatial temporal (language free) or symbolic (language integration).

• This applet connects an area model with number line. Students create area models for two fractions indicated on the number line and then create a new fraction between the first two.

Who Wants Pizza?: http://math.rice.edu/~lanius/fractions/index.html

• This is a series of interactive, straightforward lessons on fractions that include visual and written explanation on fraction concepts as well as practice problems with immediate feedbacks. The lessons cover topics on the meaning of fractions, equivalent fraction, adding fractions, and multiplying fractions.

Fraction Games: http://www.gamequarium.com/fractions.html

• This website provides various interactive games for all grade levels to use that reinforce multiple fraction concepts, such as learning halves, comparing fractions with like denominators, equivalent fractions, simplifying fractions, as well as adding, subtracting, multiplying and dividing fractions.

Fresh Baked Fractions: http://www.funbrain.com/fract/index.html

• Students can choose from four difficulty levels (easy, medium, hard, super brain) on identifying the fraction that is not equivalent from the others.

• This is a Connect Four version on fractions. Students are able to place a piece on the board by answering questions about simplifying fractions. There are also more challenging questions that involve converting fractions to decimals and percents, and algebra with fractions.

• This game allows students to compare two fractions to see which one is larger or if they are the same. They can also choose from four different ways to see what the fraction looks like by slicing a pizza, looking at the number of people in a group, filling up a jug, and dividing up a chocolate bar.

• This game has two different levels (easy and hard) and offers different colored dolphins that students can choose from. The goal of the game is to choose the biggest fraction to move the dolphin the farthest from the other dolphins in a time limit.

Pizza Party: http://www.primarygames.com/fractions/start.htm

• A fun website that teaches fractions by using pizzas that is targeted towards younger students who are just beginning to grasp basic fraction concepts.

Fraction Bar from New Zealand Maths: http://www.nzmaths.co.nz/LearningObjects/FractionBar/index.swf

Technology Resources for Purchase:

MathKeys: Unlocking Fractions and Decimals, Grades 3-5

• This for-purchase software includes bar models, counter models, and a circle model for developing fraction concepts.

### Literary Resources

The Doorbell Rang by Hutchins

• Sharing requires partitioning so everyone would have equal amounts of cookie.

Gator Pie by Mathews

• A group of alligators attempt to split a pie so that everyone gets a piece.

Discovering Math: Fractions by David Stienecker

• This book has different activities and games that help students develop conceptual understanding of fractions (i.e. Dividing Shapes, Fraction Blocks, Folding Fractions)

The Man Who Counted: A Collection of Mathematical Adventures by Malba Tahan

• A man uses his remarkable mathematical skills to settle conflicts and give wise advice. Through his journey, he shares insights from the minds of great mathematicians.

Fraction Fun by David A. Adler

• Fraction concepts are introduced by looking at a pizza pie that is divided, studied, compared, and eaten. Coins are weighed to determine how many make one ounce, and what the fractional value of each coin is.

Fraction Action by Loreen Leedy

• Balancing drawings of geometric shapes divided into sections and familiar concepts of half of a sandwich and a pie cut into four pieces makes it easy for children to visualize the meanings of various fractional quantities.

Apple Fractions by Jerry Pallotta

• Playful elves demonstrate how to divide apples into halves, thirds, fourths, and more.

Hershey's Fractions by Jerry Pallotta

• A Hershey's Milk Chocolate Bar, made up of 12 little rectangles, is used to teach fractions

Eating Fractions by Bruce McMillan

• Real-life photos illustrate the budding of fractions and food

Jump, Kangaroo, Jump by Stuart J. Murphy

• Kangaroo and his friends divide in (fractional) teams at field day.

Polar Bear Math: Learning about Fractions from Klondike and Snow by Ann Whitehead Nagda

• Young readers learn about fractions through following baby polar bears, Klondike and Snow, in their development from newborns to mature bears.

How Many Ways Can You Cut a Pie? by Jane Belk Moncure

• A squirrel decides of ways to cut a pie to give to her animal friends when she wins the pie contest.

## Annotated References

Brooks, J. & Brooks, M. (1999). Becoming a constructivist teacher. In search of understanding: the case for constructivist classrooms. Virginia: Association for Supervision and Curriculum Development.

Brown, G. & Quinn, R. J. (2007). Investigating the relationship between fraction proficiency and success in algebra. Australian Mathematics Teacher, 63(4), 8-15.

Brizuela, B. (2005) YOUNG CHILDREN’S NOTATIONS FOR FRACTIONS. Educational Studies in Mathematics 62: 281–305.

Charalambous C. & Pitta-Pantazi D. (2006). Drawing on a theoretical model to study students' understandings of fractions. Educational Studies in Mathematics, 64(3), 293-316.

Cramer, K & Bezuk, N. (1989). Teaching about fractions: what, when, and how? In National Council of teachers of Mathematics 1989 Yearbook: New Directions for Elementary School Mathematics (pp 156-167): Virginia.

Cramer, K & Henry, A. (2002). Using manipulative models to build number sense for addition of fractions. National Council of Teachers of Mathematics 2002 Yearbook: Making Sense of Fractions, Ratios, and Proportions (pp. 41-48). Virginia: National Council of Teachers of Mathematics.

Cramer, K., Post, T., del Mas, R. (2002). Initial fraction learning by fourth- and fifth-grade students: A comparison of the effects of using commercial curricula with the effects of using the rational number project curriculum. Journal for Research in Mathematics Education 33 (2), 111-144.

Empson, S. B. (1999). Equal sharing and shared meaning: the development of fraction concepts in a first-grade classroom. Cognition and Instruction, 17(3), 282-342.

Jigyel, K., & Afamasaga-Fuata'i, K. (2007). Students' conceptions of models of fractions and equivalence. Australian Mathematics Teacher, 63(4), 17-25. Retrieved from http://www.aamt.edu.au/

Kong, S. (2008) The development of a cognitive tool for teaching and learning fractions in the mathematics classroom: A design-based study. Computers & Education 51, 886–899.

Lamon, S. (2005). Teaching fraction and ratios for understanding. New Jersey: Lawrence Erlbaum Associates.

Mack, N. (1990). Learning fractions with understanding: Building on informal knowledge. Journal for Research in Mathematics Education 21(1), p16-32.

Martin, T. & Schwartz, D. L. (2005). Physically distributed learning: adapting and reinterpreting physical environments in the development of fraction concepts. Cognitive Sciences 29(2005), 587-625.

Mayer, R. (2003). Learning and instruction. New Jersey: Merrill Prentice Hall.

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