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ages can then students may experience success in early middle school math.

down, particularly when numbers written in scien tific notation are included.

This sample below, figure 3, is from a seventh grade student currently enrolled in Pre - Algebra. Disregarding the purpose of the line on the paper, this student drew six evenly spaced lines to repre sent the location for each number. This shows that the student understands the order of the rational numbers, but has not transferred it to the number line. It is also important to note that the student converted both one - eighth and three - tenths into decimals to place them in order, which is common among middle school students. It is unclear if this student understands the purpose of the number line. Another seventh grade student used percentages for this problem, figure 4. They changed the frac tions and decimals to percentages than evenly spaced them along the number line. This student used their knowledge of finding ⅙ as a fraction and repeating addition to estimate the location of ⅚ on the number line. Although they identified and placed 50% correctly as the largest number. This leads to the idea of benchmarks to find if a number is closer to zero, half or whole.

Figure 3: Placing the Numbers Evenly Spaced Along the Number Line

Although students typically understand the basic concept of percentages due to grades, but compar ing them to a fraction and to a decimal can be problematic. In seventh grade, students learn about percentages less than five. Students experience confusion with percentages such as 50%, 5% or 0.5%. The number line is helpful for students to understand the sizes of these percentages. Building on their knowledge about benchmark numbers from elementary school, students can see the dis tance of the small percentages from zero as op posed to the distance from one - half to one - whole. Seventh grade students learn about negative ration al numbers. Students have difficulty understanding the concept of negative numbers. Using the num ber line is helpful for students to identify negative benchmarks, such as zero, negative wholes, halves, thirds, and fourths. Visually students see the dis tance between the numbers along a number line to help them compare numbers in a visual manner. This concept of negative rational numbers prepares students for eighth grade math that focus on ration al numbers greater than 100%. As students move through middle school the ex pectations are raised. Many students lose more ground due to their weak conceptual understanding about rational numbers. Most students believe that rational numbers are strictly between zero and one because many teachers focus within this unit. When students are expected to compare and order fractions, decimals, and percentages that are less than zero or greater than one, many students shut Number Lines in Upper Middle School

Figure 4: Sample of 7th grade student using per centages

The next example, figure 5, comes from an eighth grade student studying Math 8. This student first divided the number line into integers between neg ative two and one positive and subdivide the num ber line into benchmark fractions of one - fourth, one - half, and three - quarter sections between each integer. This shows a maturity in terms of his un derstanding the use of the number line. The student recognized the number line as a pictorial measure ment. Though the placement of each rational num

Virginia Mathematics Teacher vol. 47, no. 2

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