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Many students have little or no understanding of the magnitude of a rational number and thus every operation has no meaning. Rational numbers are part of many higher level mathematics concepts. This weak knowledge of rational number arithme tic impacts advance learning of mathematics and science. Students ’ understanding is weak which sets them up for more struggles in the workforce beyond high school or college. In both the U.S. and the U.K., 5th graders ’ knowledge of fractions predicts 10th graders ’ over all math achievement. Their knowledge of algebra, when controlling for knowledge of whole number arithmetic, IQ, working memory, parental educa tion and income, and a host of other relevant varia bles. (Siegler et al., 2010) As children develop their sense of number, one of the early tools that they use is a number line. Chil dren learn to count on, compare and even add and subtract numbers on a number line. Somewhere along the way, as procedures become more com plex, many teachers can easily fall into the trap of moving away from visual representation of num bers to focus more on procedural fluency through the use of algorithms. While the algorithms are beneficial for fluency, they do not promote con ceptual understanding. If visual representations and manipulatives are only used in primary grades for whole number identification and calculation, the transition to problem solving with fractions can be a struggle for many students. The procedures that students use and the meaning of the numbers completely change and if there is no conceptual understanding of what fractions and mixed num bers represent. These algorithms become robotic repetition for students and can quickly cause them to become confused and disinterested in mathemat ics. The number line is a great introduction to frac tions, as it is a connection between whole numbers. As one reads a number line, the space between whole numbers is connected with a line, represent ing the numeric significance between each whole number. Even as early as kindergarten, students are Number Lines in Upper Elementary School

learning to fair share objects into two equal groups. The number line can easily help children make connections between a physical object, such as a brownie or cookie, and an equal distance between two whole numbers on a number line. This visuali zation connects to a concrete representation and helps show the importance of halves by partition ing the number line into two equal parts between the whole numbers. One of the foundational under standings of fractions is the value of a unit frac tion. As students continue to learn about fractions in elementary school, this idea of using number lines to fair share or partition the line into equal seg ments also helps the students when comparing fractions and mixed numbers. As students become more comfortable with common fractions such as halves and fourths, as they learn to use those as benchmarks when comparing rational numbers. As students start to learn how to represent whole num bers on a number line, they start to see patterns that they can use to compare numbers without having to use a pictorial representation. Such patterns for whole numbers could include looking for the num ber that goes out to the greatest place or finding the larger digit of two 2 - digit numbers in the tens place. The value of denominators of fractions is different from whole numbers. As the denominator gets larger the smaller the piece and it has less val ue. This change can be very difficult for students to grasp, and using these common benchmark frac tions on a number line can help students compare fractions and mixed numbers with greater accura cy. The more familiar they become in partitioning a number line to represent a fraction or mixed number, the more it can help them estimate wheth er a fraction is closer to zero, one half or to one whole. In addition, students in the upper elementary grades begin to study decimals. While decimals represent parts of a whole like fractions, their sym bolic representation is very different. While deci mals bring out a variety of new challenges for stu dents, number lines are still a representation that can help students make sense of many ‘ rules ’ that they had previously learned about whole numbers that may no longer apply to rational numbers. One

Virginia Mathematics Teacher vol. 47, no. 2

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