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of the best connections that students can make re garding decimals on a number line is with tenths. Once a number line is divided into ten equal seg ments, students can give that a fractional value as well as a decimal value. By naming the first parti tion past zero as one tenth, students can show it in a fraction representation, 1/10 or a decimal repre sentation 0.1. The fact that both numbers are read the same also helps support the connection be tween these two numerical representations. As stu dents start to see the places of each tenth on the number line, they can relate them to the benchmark fractions. For example, 0.2 is closer to zero than it is to one or ½. It also helps students develop the number sense of decimals and ties the decimal val ues into fractional values. This representation is extremely helpful to students when they are com paring and ordering fractions and decimals. By fifth grade, students have a variety of strategies to compare the values of these rational numbers such as converting fractions to decimals and vice versa and using benchmark fractions and decimals to compare and order the numbers of a given set. This is extremely important as students make the transi tion to middle school and begin learning about per centages and the connections they have to fractions and decimals. Number lines are helpful for this. Below are examples of fifth grade students placing a list of fractions and decimals on a number line. One can see that there are some good thoughts, but the sense of place value in certain aspects of their thinking, has been lost in these representations. In figure 1, this fifth - grader explains that they changed the fractions into decimals to put the num bers in least to greatest order and then converted back to their original form to place them on the number line. This student did find some success with certain numbers in the list. In particular, their interpretation of the tenths was accurate as 3/10 - properly placed on the third of ten partitions be tween zero and one. They also placed 0.7 on the seventh partition between zero and one. They mis read 1/8 as 1/5 and correctly placed it on the num ber line as the decimal value of 0.2. 5/6 is the greatest number in this list and they were unsure of how to convert this to a decimal number. They knew that it was closer to one, but placed it on the fifth partition because the numerator was 5.

In figure 2, this student also chose to convert the fractions to decimals, but did not recognize the par titions as tenths and instead, put the numbers in order along the first six partitions. This student

Figure 1: Converting to Decimals and Evenly Plac ing on the Number Line

communicated that they were unsure about 5/6, but they knew that it had to be closer to a whole be cause the numerator is almost the same as the de nominator. Had they taken into consideration the value of each place on the number line and placed 0.7 on the seventh partition, placing 5/6 on the next partition would not have been entirely accurate, but would have been closer. This idea of using bench marks and partitions as a way to place fractions and decimals in order can be very helpful, but stu dents need time to explore this representation for it to have meaning for them.

Number Lines in Early Middle School

In sixth grade, students move from fractions and

Figure 2: Placing the Numbers on the Number Line in Order of Next Partition, Not by Proper Spacing

decimals to percentages. The number line in mid dle school is a great tool to learn rational number vocabulary. If students have a good understanding of fractions, decimals, and the common bench marks on the number line, then including percent

Virginia Mathematics Teacher vol. 47, no. 2

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