Making Sense of Fractions with GeoGebra:Representing Fractions Using Area and Length

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1 Making Sense of Fractions with GeoGebra:Representing Fractions Using Area and Length by Hea-Jin Lee and Irina Boyadzhiev This article discusses common misconceptions about fractions and offers classroom activity ideas using GeoGebra applets to the classroom teacher to mediate misconceptions of fractions and support students to enhance understanding of fractions. The article shares four GeoGebra applets. Introduction When fractions appear in the primary grades, students are introduced for the first time to the part-whole concept of a fraction and slowly learn all four arithmetic operations with fractions. In the secondary school, the interpretation of fractions is more sophisticated. By then, students are expected to be well versed in the applications of fractions such as percent, ratio, and rate. Regardless of the grade level, teachers observe children struggling with fractions and exhibiting misconceptions with fractions possibly due to the confounding meaning of fractions, ambiguous instruction, and/or misleading representations of fractions. McNamara and Shaughnessy (2010) suggest that students struggle with fractions, because a fraction has many meanings (part-whole, measurement, division, operator, and ratio) and is written in a different way from integers. Some criticize teachers for not focusing on a conceptual understanding of fractions. This could lead students to attempt to link the existing knowledge of whole numbers to fractions. Some educators blame inappropriate use of a model or manipulative in teaching fractions. A fraction is often modelled by using discrete objects (items that can be counted) or continuous models (length, area, and volume). Each model has pros and cons and has advantages to the others depending on the curriculum. Discrete objects provide an appropriate way to introduce unit fractions in the primary years but have limits. For example, one can only discuss how many, not how much, and it is unnatural to discuss 1/3 with two pencils as the whole. Continuous models require teachers careful and thoughtful planning and deliberation. A commonly observed mistake is to let a geometric shape be the whole, rather than the area of the shape. It leads to confusions, for instance, viewing a fraction as a geometric shape not a number. In that sense, the number line is much simpler for modelling operations with fractions but requires more time for younger students to get used to the number line model than the area model (Wu, 2011). The advocators of the number line claim that a fraction divided by n can be modelled by using the length model (number line) much simpler than using the area model (rectangle) or the set model (discrete objects). Also, the basic comparisons among fractions (equal, smaller, and bigger) can be explained easily by using the number line. The purpose of this paper is to support teachers to challenge common misconceptions about fractions and to strengthen student understanding of the concept and properties of fractions through the use of interactive GeoGebra applets. All GeoGebra applets shared in this paper use both the area model and the length model. GeoGebra GeoGebra is a free dynamic software for teaching and learning mathematics and science. It can be used with success at all levels of education. The software provides an intuitive environment for exploration and discovery of mathematical concepts in geometry, algebra, graphing, statistics and calculus. Since 2014 GeoGebra has been available in many formats. In addition to the desktop applications for Windows, Mac OS and Linux, GeoGebra provides tablet apps for Android, ipad and Windows, and a web application based on HTML5 technology. The teachers and students can make their own constructions in GeoGebra using the available tools and commands, or they can use applets created by others and available under the GNU General Public License at and also at many blogs and personal websites. GeoGebra can be downloaded for free at 2 Mathematics in School, January 2016 The MA website

2 Common Misconceptions about Fractions Commonly observed misconceptions with fractions are due to the lack of understanding of a fraction as a single value, equal-sized parts, and value of fractions. Also, students often struggle in comparing fractions, applying properties of whole numbers, and representing fractions (Cramer and Whitney, 2010; McNamara and Shaughnessy, 2010). A Fraction as a Single Value: Students often consider the numerator and the denominator independent parts of the fraction not as a single value. Recognizing a fraction as one number is hard for young learners, because two numbers are used to write a fraction. Activities such as placing a fraction on the number line and read one third instead of one out of three or one over three can help students develop the notion of a single value. Equal-sized Parts: The importance of the equalsized parts is not emphasized enough in teaching the meaning of the fraction as part-whole. Students often concentrate on the number of parts, rather than the equal-sized parts. For example, in the figure above students might see the hexagon as 3/4 green and 1/4 red, instead of 1/2 green and 1/2 red. Comparing Fractions: Students think the fraction 1/5 is smaller than 1/10, because it has a smaller number (denominator). In order to correct the misconception, some teachers introduce a rule, such as fractions are the reverse, the bigger the denominator, the smaller the fraction. However, such a rule can lead students to further confusion. Students might overgeneralize and conclude that 1/2 is more than 2/3. This notion should be taught conceptually with elaborate explanations, contexts, and visuals. Misapplications of Whole Numbers: Students do not fully understand why fractions do not behave like integers when they use rules to compute, including the four operations. For example, students would compute 1/2 + 1/2 = 2/4. Visuals and interactive explorations can help students see that their solution does not make sense. Representing Fractions: Students believe that every fraction has a unique representation. For example, students do not see that 1/2, 2/4, 3/6, are the same fraction. Using a number line or visuals can help to develop the concept of equivalent fractions. Value of Fractions: Students often believe that fractions are always smaller than 1, possibly because textbooks and teachers overemphasize the part-whole meaning of fractions and use the same model repeatedly to introduce fraction concepts. This misconception can be corrected by using various models (region, length, area, or set), by introducing different meanings of fractions (part-whole, measurement, division, operator, and ratio), and by using different tools (paper, rods, technology, counters, number line, blocks). Whole-number knowledge both supports and inhibits the development of fraction concepts. Teachers and students need to understand that it is important to build on prior knowledge, which includes focusing on how fractions and whole numbers are the same and how they are different. This is a developmental process that will take time and multiple experiences. Another cause of some misconceptions is that, too often, part-whole is the only way that fractions are taught. The students need experiences and opportunities to think more broadly about fractions. Fractions represent regions, lengths, or quantities, so teaching and learning about fractions needs to include models across these three categories, providing contexts and manipulatives that fall in each. Developing fraction concepts such as partitioning and equivalence is essential to all advanced fraction work and success in algebra and advanced mathematics. These misconceptions are not yet registered in students permanently, so students can be helped to understand fractions correctly through purposefully designed lessons and instructional materials, including technology. GeoGebra Apps Addressing Common Misconceptions GeoGebra applets shared in this paper are interactive and provide directions for students and teachers to be able to manipulate without additional lessons on GeoGebra. More applets can be found at the author s website Relating a Fraction as Area and a Number on the Number Line This applet can be used as an introduction to relating an area (a unit square) with the number line. While playing with the applet, the students are asked to form several different fractions, including some less than one, whole numbers, and mixed numbers. After the student selects the number of parts in the whole, the unit segments on the number line are divided in the same number of parts as the square. As the student enters the numerator, the corresponding fraction of the square is coloured and is projected on the number line. Mathematics in School, January 2016 The MA website 3

3 student can see the dynamic effect on the fraction as the numerator increases/decreases or as the denominator increases/decreases another difficult concept. Comparing Fractions and Equivalent Fractions The goal in all the examples here is to keep the area representation and the number line representation of a fraction tied together. Improper Fractions and Whole Numbers When fractions are introduced, the examples are often related to a pizza or a pie. In this case it is hard for a student to imagine taking ten pieces of the pizza when the pizza is cut in only six parts. In this applet, students can see the relationship between improper fractions and mixed numbers. If the number of parts taken (numerator) exceeds the number of parts in the whole (denominator), a new unit square appears and the number moves in a natural way along the number line. This GeoGebra applet can be used to compare two fractions visually. Students can see the fractions at the same time as numbers, as parts of the area of the unit square, and as points on a number line. Initially the students are presented with two identical orange unit squares. They are asked to enter two fractions. The fractions are represented as differently coloured parts of the squares red for one and blue for the other. By dragging the slider segments with length equal to the length of the coloured part of each square are moved toward the number line. They are aligned there and their lengths compared. With this activity the students experience a different level of abstraction from part of the whole to just one attribute (the length of the coloured part) to a number on the number line. This transition very often is missed in traditional models of fractions. Using the applet the students can compare fractions with equal numerators (1/5 and 1/10) or fractions with the same denominators (1/5 and 2/5), before moving to a more advanced level, such as comparing 3/5 and 2/7. The teacher can choose some examples of equivalent fractions to give the students some experience in seeing different fractions representing the same number on the number line. Equivalent Fractions and Least Common Denominator When the number of parts in the whole is one, the fraction is a whole number. By dragging the sliders the Before studying the addition of fractions the students need to develop a very solid understanding of equivalent fractions. When introducing the idea of least common denominator, the denominators are often chosen as relatively prime numbers (for simplicity). In this case the least common denominator (LCD) is the product of 4 Mathematics in School, January 2016 The MA website

4 all denominators. Many students apply this rule even when the denominators have a common factor. For example, with denominators 6 and 4 they choose a common denominator of 24, rather than 12. The GeoGebra applet on the right can be used for practising these concepts. The two uncut unit squares are positioned on both sides of the number line. The students are prompted to enter two fractions, for example, 5/6 and 1/4 in the input boxes. They should notice that the denominator has the important role of determining the size of the parts in the whole. This applet can be used to address various concepts: The Meaning of the LCD: By clicking the Add Parts button the squares are cut simultaneously in the same number of parts. The green light appears by a square when the new dividing lines coincide with the initial dividing segments. Finding the factors for expanding or reducing fractions: The students can see the number of equalsized parts in the initial divisions of the square as the factors used in expanding or reducing fractions. Equivalent Fractions: By continuing the process of dividing the two squares into smaller and smaller parts, the students can see that there are other common denominators and that a fraction can be written in many equivalent ways. Equivalent Fractions One Common Misconception Another misconception associated with the equivalent fractions is that adding or subtracting the same number to the numerator and denominator will produce equivalent fractions. The following applet can address the misunderstanding. The bottom part of the screen is the reference area. First, students are asked to enter a new proper fraction and see it as a part of a unit square. The top part is for experimenting with the fraction. Click on the Multiply button. This will bring an input box and an operation button. Enter a number and click the multiplication sign. Repeat the same in the numerator. Drag the Compare the Fractions slider to compare the original with the new fraction (right figure). Click on Add button to experiment adding the same number to the numerator and the denominator (left figure). Repeat with multiplying/adding several numbers in the top part of the applet, and with different fractions (lower part of the applet). By playing with this applet, students can see that multiplying the same number to the numerator and the denominator produces an equivalent fraction (right figure), but adding the same number (left figure) does not keep the same value. Conclusion Fractions are part of our daily life. We use the fraction concepts and language in the house, at restaurants, in stores, etc. Students use fraction concepts to communicate and describe objects without any problems until fractions are introduced as a mathematical concept in school. That makes us wonder why is that? and what are we not doing right? Studies on fractions suggest that teachers often teach only the procedures for manipulating fractions. Fraction concepts should be taught by emphasizing number sense, focusing on the Mathematics in School, January 2016 The MA website 5

5 meaning of fractions, providing varied models and contexts, emphasizing that fractions are numbers, and dedicating time for understanding each concept. For the purpose of this article, the authors intentionally used area and length models. By using number lines, teachers can send the message that fractions are numbers. As can be seen in the sample applets in the article, we can teach improper fractions and equivalence conceptually by using GeoGebra applets, while students are still building their understanding of what fractions are. GeoGebra applets can be used by students for selfstudy, not only for instructional demonstration by teachers. Teachers can use GeoGebra applets to assess student thinking strategies and have students do activities individually to assist/enhance their understanding of fractions. Operations with fractions should come after developing basic concepts of fractions fully. Teaching fraction computation should begin with contextual tasks (story problems) that connect fraction computation to whole number computation. Also, estimation and informal methods should be allowed in the development of strategies, and each of the operations should be explored using various models. References Clarke, D. M., Roche, A. and Mitchell, A Practical Tips for Making Fractions Come Alive and Make Sense, Mathematics Teaching in the Middle School, 13, 7, pp Cramer, K. A. and Whitney, S. R Learning Rational Number Concepts and Skills in Elementary School Classrooms. In Lester, F. K. (Ed.) Teaching and Learning Mathematics: Translating Research for Elementary School Teachers, National Council of Teachers of Mathematics, Reston, VA, pp McNamara, J. C. and Shaughnessy, M. M Beyond Pizzas & Pies: Ten Essential Strategies for Supporting Fraction Sense (Grades 3 5), Math Solutions, Sausalito, CA. Melis, E. and Goguadze, G Representation of Misconceptions. In Sampson, D. G., Spector, J. M. and Isaias, P. (Eds), Proceedings of the IADIS International Conference on Cognition and Exploratory Learning in Digital Age (CELDA), IADIS Press, Spain, pp Wu, Hung-His 2011 Teaching Fractions According to the Common Core Standards. Keywords: Teaching Fractions; GeoGebra; Misconceptions with fractions. Authors Hea-Jin Lee and Irina Boyadzhiev, College of Education and Human Ecology, Ohio State University-Lima, Galvin Hall 460B 4240 Campus Drive, Lima OH., USA. Lee.1129@osu.edu; boyadzhiev.1@osu.edu Looking for ideas? The Mathematical Association, 259 London Rd, Leicester LE2 3BE Registered Charity No VAT GB Extension material Problem solving Fun word problems Enriching the curriculum Challenging your pupils Look no further. Include PMC in your mathematics resources. 6 Mathematics in School, January 2016 The MA website