### Fractions Resources

## Abbreviations:Resource Categories: |

Resource TitleResource DescriptionResource CategoryApplicable Grade Level(s)

[External Site] Web page in Spanish, providing multiple choice assessment items | Multiple choice items in Spanish to assess understanding of fractions on the number line | A | 3+ |

Module 1 Materials | This is the introductory unit for the workshop “Fractions on a Number Line”. It begins to develop the meaning of a fraction in relation to the number line. | L | 3-6 |

Module 2 Materials | This is the second unit in the workshop “Fractions on a Number Line”. It focuses on developing participants understanding of the number line and for locating fractions on a number line. | L | 3-6 |

[External Site] Example of lesson plan | Number Line journeys – lesson plans from Illuminations at NCTM | L | 4,5,6 |

Module 7 Materials | This is the seventh unit in the workshop “Fractions on a Number Line”. It focuses on using the number line to subtract like and unlike fractions. | L | 3-6 |

Module 9 Materials | This is the ninth unit in the workshop “Fractions on a Number Line”. It focuses on using the number line to divide fractions. | L | 3-6 |

Module 8 Materials | This is the eighth unit in the workshop “Fractions on a Number Line”. It focuses on using the number line to multiply fractions. | L | 3-6 |

Module 6 Materials | This is the sixth unit in the workshop “Fractions on a Number Line”. It focuses on using the number line to add like and unlike fractions. | L | 3-6 |

Module 4 Materials | This is the fourth unit in the workshop “Fractions on a Number Line”. It focuses on comparing and odering fractions on a number line | L | 3-6 |

Module 3 Materials | This is the third unit in the workshop “Fractions on a Number Line”. It focuses on developing an understanding of fraction equivalence built upon an understanding of fractions and the number line. | L | 3-6 |

Module 5 Materials | This is the fifth unit in the workshop “Fractions on a Number Line”. It focuses on assessment. | L | 3-6 |

Appendix for FNL Workshop | This is the appendix for the workshop “Fractions on a Number Line”. It includes support materials to assist in providing the workshop. | L | 3-6 |

Module 10 Materials | This is the tenth and final unit in the workshop “Fractions on a Number Line”. It reviews the rationale for using the number line as well as the use of the number line as a toolde for teaching fractions. | L | 3-6 |

[External Site] A source to create number lines | A website for teachers that creates number lines based on their own needs | P | K + |

[External Site] Decimal numbers on the Number line – a Geogebra applet | An online resource to practice placing decimal numbers on the number line | P | 5-6 |

[External Site] Applet for teachers that need to create a worksheet with number lines | This website create PDF files with worksheets for your students. Teachers can choose the level (natural numbers, integers, rationals, etc). The page created has several copies of the same number line. | P | 1+ |

[External Site] Decimal numbers on the number line – an applet | This applet enables playing with the number line and locate points on it based on the decimal number they represent | P | 5-6 |

[External Site] Another source for Number lines | Website with pre-made number lines | P | K+ |

[External Site] Online game about equivalent fractions and ordering fractions | EXCELLENT website with a game to engage students in identifying equivalent fractions and recognizing what fraction is smaller or larger than a given one. | P | 3-6 |

[External Site] Applet to represent operation of whole numbers on the number line | A meaningful way to show the meaning of the result of the four operations among whole numbers | P | 1-4 |

[External Site] Comparing fractions on the number line | Connect the representation of a fraction with area model and the number line. | P | 4-6 |

[External Site] Visual Fractions – applet for converting Mixed numbers to improper fractions | Online resource that enables practice and visualization of the conversion from mixed numbers to improper fractions | P | 3-6 |

[External Site] Dig it – A game to play online | An online game to identify fractions on the number line; the better their estimation, the better their chances to win. Players needs to register as a user. It is free | P | 2-6 |

[External Site] Visual Fractions – applet for Identification of fractions | Online resource that enables identification of fractions on the number line | P | 3-6 |

[External Site] Locating fractions on the number line – A Geogebra applet | An online resource that enable sudents to play with the number line | P | 3-6 |

[External Site] Video in Spanish – How to locate a simple fraction on the number linetion on | Video in Spanish explaining how to represent a fraction on a number line | P | 3,4 |

[External Site] Three whiteboard resources to assist in the teaching and learning of decimals. | Online resources for the smartboard | P | 5-6 |

[External Site] Visual Fractions – applet for division of fractions | An online resource that demonstrates and enables practice of division of rtional numbers by using measuring | P | 5-6 |

[External Site] Convert mixed numbers to improper fractions and viceversa- Geogebra applet | An online resource based on Geogebra that – by means of sliders – enables the student to choose a fraction and convert it to another format | P | 3-6 |

[External Site] Applet to play with the number line and other representations of rational numbers | Good applet to practice “reading” fractions from points on the number Line and also to connect that representation to the area one. | P | 3-5 |

[External Site] Applet to practice place value | Place Value can be made relevant on this applet that enables ZOOM-IN of the number line | P | 1-6 |

[External Site] An online game about decimal numbers | A “detectives” games to be played online (find decimal numbers on the number line) | P | 5 |

[External Site] Create a Number line in your TI-73 calculator | Technical description of how to use your TI-73 with a Number Line application | P | 4-up |

[External Site] Addition and subtraction of integers on the number line – Applet | An applet that enables the visualization of the meaning of the sum and the difference of two integers | P | 6 |

[External Site] An integrated study of children’s construction of improper fractions on the teacher’s role in promoting that learning (Tzur,1999) | : In this constructivist teaching experiment with 2 fourth graders I studied the coemergence of teaching and children’s construction of a specific conception that supports the generation of improper fractions. The children’s posing and solving tasks in a computer microworld promoted a modification in their fraction schemes. They advanced from thinking about a unit fraction as a part of a whole to thinking about it as standing in a multiplicative relationship with a reference whole (the iterative fraction scheme). In this article I report an intertwined analysis of the children’s construction of this multiplicative relationship and an examination of the teacher’s adaptation of learning situations (tasks) and teacher-learner interactions to fit within the constraints of the children’s mathematical activity. | R | |

[External Site] Math Matters: Understanding the Math You Teach, Grades K-8 (Chapin & Johnson,2006) | This resource provides an in-depth study with 14 chapters covering number sense, computation, addition, subtraction, multiplication, division, fractions, decimals, percents, ratios, algebra, geometry, spatial sense, measurement, statistics, and probability. Over 100 activities give readers an opportunity to connect ideas, compare and contrast concepts, and consider how students understand the mathematics presented. | R | |

[External Site] Understanding the development of students’ thinking about length (Batista, 2006) | This article describes assessment tasks and a conceptual framework for understanding elementary students’ thinking about the concept of length. The framework is an important tool for improving instruction and formative assessment, as well as effectively diagnosing and remediating students’ difficulties in learning about length. | R | |

[External Site] Beyond Pizzas and Pies: Ten Essential Strategies for Supporting Fraction Sense (McNamara & Shaughnessy,2010) | This book supports teachers in addressing students’ common misconceptions about fractions as they guide their students in investigating and discussing fractions to expand and refine their understanding. In each chapter, the authors discuss one common dilemma that students have with fractions and present classroom strategies and activities for preventing and addressing it. Includes reproducibles. | R | |

[External Site] Using number sense to compare fractions (Bray & Abreu-Sanchez,2010) | Third-grade teachers found that giving particular attention to the use of real-world contexts, mental imagery, and manipulatives brought success to problem solving as students moved from using models to reasoning | R | |

[External Site] Using steffe’s advanced fraction schemes (McCloskey & Norton, 2009 | Recognizing schemes, which are different from strategies, can help teachers understand their students’ thinking about fractions. | R | |

[External Site] Sharing Teaching Ideas: Irrational Numbers on the Number Line: Perfectly Placed (Coffey,2001) | To improve understanding of irrational numbers, students create a number line from adding machine tape, using a square and a precisely-folded triangle as measuring devices. | R | 7+ |

[External Site] Multiplication and splitting: Their role in understanding exponential functions (Confrey,1998) | This paper focuses on exponentiation as repeated multiplication. It conjectures that that the concept of multiplication as repeated addition is NOT ADEQUATE to understand the exponentiation as repeated multiplication. It suggests that a partitive concept of multiplication (via splitting) may be more appropriate. | R | |

[External Site] Teaching and Learning Fraction Addition on Number Lines (Iszak et al,2008) | This article presents a case study of teaching and learning fraction addition on number lines in one sixth-grade classroom that used the Connected Mathematics Project Bits and Pieces II materials. The main research questions were (1) What were the primary cognitive structures through which the teacher and students interpreted the lessons? and (2) Were the teacher’s and her students’ interpretations similar or different, and why? The data afforded particularly detailed analyses of cognitive structures used by the teacher and one student to interpret fractions and their representation on number lines | R | |

[External Site] Making Sense of Fractions, Ratios, and Proportions (NCT, 2002 Yearbook) | NCTM’s 2002 Yearbook emphasizes that although fractions, ratios, and proportions are pivotal concepts in the middle school, their development and understandings begin in the elementary school. The companion booklet presents activities that illustrate some of the ideas in the yearbook and that go beyond the content of the yearbook itself. Teachers’ notes and handouts are designed to bring the yearbook to life in the classroom. | R | |

[External Site] Nonstandard Student Conceptions About Infinitesimals (Ely, 2010) | This article is a case study of an undergraduate calculus student’s nonstandard conceptions of the real number line. Interviews with the student reveal robust conceptions of the real number line that include infinitesimal and infinite quantities and distances. Similarities between these conceptions and thoseof G. W. Leibniz are discussed and illuminated by the formalization of infinitesimals in A. Robinson’s nonstandard analysis. These similarities suggest that these student conceptions are not mere misconceptions, but are nonstandard conceptions, pieces of knowledge that could be built into a system of real numbers proven to be as mathematically consistent and powerful as the standard system. | R | 10+ |

[External Site] Locating Proper Fractions on Number Lines: Effect of Length and Equivalence (Larson, 1980) | Compares student performance on fraction/number line tasks which varied length and equivalence. Associating proper fractions was easier on number lines of length one rather than length two; and where number of equivalent line segments in each unit segment was the same as the denominator, rather than twice that number. | R | |

[External Site] Traveling from Arithmetic to Algebra (Darley, 2009) | A conceptual knowledge of numbers, especially fractions, is an important link to a conceptual understandig of variables. This article suggests activities that explicitly show the connections between numbers and variables using the number line as the connecting device. | R | 6-7-8 |

[External Site] Identify fractions and decimals on a number line (Shaughnessy, 2011) | This article examines students’ understanding of the decimals numbers, analyzes their erroneous strategies. | R | 6,7,8 |

[External Site] The Empty Number Line: A Useful Tool or Just Another Procedure? (Bobis,2007) | This article explores the origins and potential benefits of the empty number line for the development of mental computation. It also provides a learner’s perspective of its use through the reflections of nine-year-old Emily. | R | 1-3 |

[External Site] Fractions (Wu,2010) | Abstract: The main points of this presentation (1) Every concept will have a definition: Fraction, sum of two fractions, product of two fractions, percent, ratio, etc. You do not need to know more about a concept that what is contained in the definition. There are no guesses in mathematics (2) Similarity between fractions and whole numbers will be emphasized throughout: Whole number facts provide the proper guidance for what we do with fractions (3) A reason will be given for every assertion: Everything will be explained. There will be no doubts or suspicions. | R | |

[External Site] A Conceptual Approach to Absolute Value Equations and Inequalities (Elis & Bryson, 2011) | Using number lines sweeps away the mystery of working with absolute values and empowers students to make connections between procedures and concepts. | R | Algebra + |

[External Site] Developing numbers sense on the number line (Bay, 2001) | A life-sized number line helps students visualize number relationships. | R | |

[External Site] Supporting Generative Thinking about the Integer Number Line in Elementary Mathematics (Saxe et al, 2010) | This report provides evidence of the influence of a tutorial “communication game” on fifth graders’ generative understanding of the integer number line. Students matched for classroom and pretest score were randomly assigned to a tutorial (n = 19) and control group (n = 19). The tutorial group students played a 13-problem game in which student and tutor each were required to mark the same position on a number line but could not see one another’s activities. To resolve discrepant solutions, tutor and student constructed agreements about number line principles and conventions to guide subsequent placements. Pre-/posttest contrasts showed that (a) tutorial students gained more than controls and (b) agreement use predicted gain. Analyses of micro-constructions during play revealed properties of student learning trajectories | R | |

[External Site] Young Mathematicians at Work: Constructing Fractions, Decimals, and Percents (Fostnot, C. et al. 2010) | This volume focuses on how children in grades 5-8 construct their knowledge of fractions, decimals, and percents. The book describes and illustrates what it means to do and learn mathematics, contrasts word problems with true problematic situations which support and enhance investigation and inquiry, provides strategies to help teachers turn their classrooms into math workshops, explores the cultural and historical development of fractions, decimals, and their equivalents and the ways in which children develop similar ideas and strategies, defines and gives examples of modeling, noting the importance of context, discusses calculation using number sense and the role of algorithms in computation instruction, describes how to strengthen performance and portfolio assessment, and focuses on teachers as learners by encouraging them to see themselves as mathematicians. | R | |

[External Site] Knowing and Teaching Elementary Mathematics: Teachers¹ Understanding of Fundamental Mathematics in China and the United States (Ma, 1999) | This book describes the nature and development of the “profound understanding of fundamental mathematics” that elementary teachers need to become accomplished mathematics teachers, and suggests why such teaching knowledge is much more common in China than the United States, despite the fact that Chinese teachers have less formal education than their U.S. counterparts. The studies described in this book suggest that Chinese teachers begin their teaching careers with a better understanding of elementary mathematics than that of most U.S. elementary teachers. Their understanding of the mathematics they teach and–equally important–of the ways that elementary mathematics can be presented to students, continues to grow throughout their professional lives. | R | |

[External Site] Representations of the Magnitudes of Fractions (Schneider & Siegler, 2010) | This paper reports about an experiment withg adults. They tested whether adults can use integrated, analog, magnitude representations to compare the values of fractions. | R | |

[External Site] Meanings for Fraction as Number-Measure by Exploring the Number Line (Psycharis,Latsi,Kynigos; 2009) | This paper reports on a case-study design experiment in the domain of fraction as number-measure. We designed and implemented a set of exploratory tasks concerning comparison and ordering of fractions as well as operations with fractions. Two groups of 12-year-old students worked collaboratively using paper and pencil as well as a specially designed microworld which combines graphical and symbolic notation of fractions represented as points on the number line. We used the students’ interactions with the available representations as a window into their conceptual understanding and struggles in making sense of fraction as number-measure. We report on the features of the available representations from an epistemological point of view, on the design of activities aiming at creating meaningful problem contexts for fractions as well as on the meanings generated by the students by some illustrative examples of their work indicating the potential of the activities and tools for expressing and reflecting on the mathematical nature of fraction as number-measure. | R | |

[External Site] Foundations for Success: The Final Report of the National Mathematics Advisory Panel (National Mathematics Advisory Panel,2008). | This Panel, diverse in experience, expertise, and philiosophy, agrees broadly that the delivery system in mathematics education—the system that translates mathematical knowledge into value and ability for the next generation—is broken and must be fixed. This is not a conclusion about any single element of the system. It is about how the many parts do not now work together to achieve a result worthy of this country’s values and ambition. On the basis of its deliberation and research, the Panel can report that America has genuine opportunities for improvement in mathematics education. This report lays them out for action.. | R | |

[External Site] Using a Lifeline to Give Rational Numbers a Personal Touch (Weidemann et al.,2001) | Middle school students and their parents construct a number line using positive and negative rational numbers to represent dates of events before and after the student’s birth. | R | |

[External Site] The Ann Arbor Workshop on Fractions (Tucker,nda). | The workshop was dominated by arguments about foundational concerns such as how and when to define a fraction. As is common in many academic discussions, participants were more interested in areas of disagreement than areas of agreement. The areas of disagreement were very difficult to resolve, because of different points of views and backgrounds that underlie how mathematicians and mathematics educators talk to each other about fractions. When a conscious effort to look for common ground was made, there turned out to be considerable agreement on many aspects of how classroom instruction about fractions in grades K-8 should be organized. | R | |

[External Site] The Number Line as a Representation of Decimal Numbers: A Research with Sixth Grade Students (Michaelidou et al.,2004) | Examination of 12-year olds’ understanding of decimal numbers, including their representation on the number line | R | 6 |

[External Site] Developing effective fractions instruction for kindergarten through eighth grade: A practice guide (Siegler,Carpenter,Fennell,Geary,Lewis,Okamoto,Thompson,Wray,2010) | This practice guide presents five recommendations intended to help educators improve students’ understanding of fractions. Recommendations include strategies to develop young children’s understanding of early fraction concepts and ideas for helping older children understand the meaning of fractions and the computations involved. The guide also highlights ways to build on students’ existing strategies to solve problems involving ratios, rates, and proportions. | R | |

[External Site] Fractions, Decimals, and Rational Numbers (Wu,2008). | This is a slightly revised version of the report, written in the early part of 2007 at the request of the Learning Processes task Group of the National Mathematics Advisory Panel (NMP), on the curricular aspects of the teaching and learning of fractions. | R | |

[External Site] Adding It Up: Helping Children Learn Mathematics (Kilpatrick,Swafford,Findel, Eds; National Research Council,2001) | This book explores how students in pre-K through 8th grade learn mathematics and recommends how teaching, curricula, and teacher education should change to improve mathematics learning during these critical years. | R | |

[External Site] Modeling students’ mathematics using steffe’s fraction schemes (Norton & McCloskey,2008) | Descriptions and examples of Steffe’s fractions schemes, which characterize common and developing conceptions of fractions among children, help teachers understand students’ ways of operating. Several student examples are included. | R | |

[External Site] Fraction Number Line Tasks and the Additivity Concept | Research article about the sum of fractions on the number line | R | 3-6 |

[External Site] Locating negative decimals on the number line: Insights into the thinking of pre-service primary teachers (Widjaja et all, 2011) | This paper explores misconceptions of the number line which are revealed when pre-service primary teachers locate negative decimals on a number line. Written test responses from 94 pre-service primary teachers provide an initial data source which is supplemented by group responses to worksheets completed during a lesson and individual interviews. Two main misconceptions leading to incorrect placement of negative decimals on a number line are identified. One relates to having separate number ‘rays’ for positive and negative numbers, which are aligned according to context. The other (with several variations) results from creating the negative part of the number line by amalgamating translated positive intervals. These misconceptions explain a large percentage of wrong answers. The most important implication for education at school, as well as in teacher education, is that the teaching of negative numbers and of the number line must not be confined to integers, as is frequently the case, but must also include negative fractions and decimals | R | 6, 7, 8 |

[External Site] Irrational numbers on the number line – where are they? (Sirotic & Zaskis,2007) | This paper reports an investigation into the understanding of irrational numbers by future secondary school teachers. It focuses on the representation of irrational numbers as points on a number line. | R | 6 + |

[External Site] Teaching fractions in elementary school: A manual for teachers (Wu,1998) | It is to be emphasized that (1) the primary audience of this article is teachers of grades 5-8, not students, and (2) this approach to fractions is definitely not for grades 204 where fractions first make their tentative appearances. The overriding concern here is that, after children’s informal encounter with fractions in the early grades, they reach the point in grade 5 where their haphazard knowledge needs consolidation and their initial foray into abstract mathematics (fractions) needs some structure for support. This article is designed to help teachers face up to the challenge of leading students to the next phase of mathematical achievement. | R | |

[External Site] Activities for Students: Algebra on the Number Line (McCabe et al., 2010) | Authors provide activities that help students understand distance and the number line. | R, P | 6,7,8 |

[External Site] Using, Seeing, Feeling, and Doing Absolute Value for Deeper Understanding (Ponce, 2008) | Using sticky notes and number lines, a hands-on activity is shared that anchors initial student thinking about absolute value. The initial point of reference should help students successfully evaluate numeric problems involving absolute value. They should also be able to solve absolute value equations and inequalities that are typically found in algebra textbooks. | R, P | Algebra + |