a number line can be a useful visual tool to help students progress

458 MatheMatics teaching in the Middle school ● Vol. 14, No. 8, April 2009

M any researchers and math-ematics educators claim that connections should be made

between new and prior knowledge, because knowledge is remembered better when it is well connected and more readily available in new situa-tions (Baddeley 1976; Bruner 1960; Hiebert and Carpenter 1992; Hilgard 1957; Skemp 1976). To facilitate these kinds of connections, the NCTM developed Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: “The decision to orga-nize instruction around focal points assumes that the learning of mathe-matics is cumulative, with work in the later grades building on and deepen-ing what students have learned in the earlier grades, without repetitious and

inefficient reteaching” (NCTM 2006, p. 5). Historically, students have had a difficult time transitioning from arith-metic to algebra, in part because of the effort needed to connect the two. When students possess a deep under-standing of numbers and the connec-tion between numbers and variables, they should be able to make a smooth transition from one to the other.

Research suggests that every effort should be made to connect students’ existing fraction knowledge to a quan-titative model so that fractions are recognized as numbers before engag-ing in formal algebra (Bracey 1996; Gelman, Cohen, and Hartnett 1989; Wu 2001). Wu (2001) defines both whole numbers and fractions as points on the number line. In so doing, there

arithmetictraveling from

A number line can be a useful visual tool to help students progress from arithmetic to

algebra in their quest to understand variables.

Joy W. Darley

Joy W. darley, [email protected], teaches mathematics content courses to preser-vice teachers at Georgia

Southern University in Statesboro. She is interested in rational number instruction and the impacts of this instruction on the understanding of algebraic concepts. P

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Copyright © 2009 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

Vol. 14, No. 8, April 2009 ● MatheMatics teaching in the Middle school 459

algebra

arithmetic

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will be a logical progression from the teaching of whole numbers to the teaching of fractions and ultimately to the teaching of rational expressions.

Since our goal is for students to better understand variables, we need to be certain that they understand numbers. Once the connection is made between the two, students will be more confident using variables. The follow-ing activities, which explicitly show connections between numbers and variables, use the number line as a tool to foster such connections. Instruction begins with using whole numbers and continues with fractions.

activity 1: Whole nuMbers and integersUse a number line that clearly defines the unit of reference (see fig. 1). For example, the arithmetic number line contains 0 and 1 as the unit of refer-ence, whereas the algebra number line contains 0 and x as the unit of reference. It is important for stu-dents to know that the exact distance between “0 and 1” and “0 and x” is not important at this time. The number 1 represents one of anything: one foot, one candy bar length, one meter, one pizza, or one set of objects. The vari-able x, however, represents an un-known quantity of anything, such as x feet, x candy bar lengths, x meters, x pizzas, or x sets of objects.

At this point, students can be asked, “Where is 2 on the first number line? Where is 3? How do you know?” After the students are adept at using 1 as the unit of reference on the arith-metic number line, ask these questions: “Where is 2x on the algebra number line? Where is 3x? How do you know? If x = 1, does this make sense? What if x = 2? What would this look like on a number line?” The desired connection should resemble figure 2.

Locating 2x or 3x on the algebra number line, given the location of 0 and x, is not a trivial task for many

students. After posing the previous question to a beginning algebra class, there was noticeable silence before the first incorrect guess was given. Even though students may understand that 3 • 2 is 2 + 2 + 2, understanding 3x as 3 • x, or x + x + x, does not always easily follow. In fact, many students commonly confuse 3x with x3, not recognizing that 3x represents three xs that have been added, whereas x3

represents three xs that have been multiplied. Another common miscon-ception is that the value of x is always 1. Students commonly confuse the “value” of x with the “coefficient” of

x. Emphasizing that x is the same as 1x, because it represents one of an un-known quantity, such as one 2 in the former example, should eliminate this confusion. Replacing x with several numbers would be helpful as well.

After students are adept at locating x, 2x, and 3x on the number line, and perhaps 4x for a sufficient number of xs, present the number line in figure 3a and ask students to locate −1 and −3. Have students describe how they determine these locations. After they are com-fortable identifying the locations with negatives, ask the class to find –x and –3x on the algebra number line. Suggest

Fig. 1 Students see a side-by-side comparison of both number lines.

Fig. 2 Students can visualize mixing whole numbers with variables.

Fig. 3 the next step involves using negative numbers.

(a) (b)


specific values for x, such as 2. Invite students to draw a number line and identify the locations of the pertinent values. The desired connection should resemble figure 3b. For the purpose of this activity, x represents a positive num-ber. The activity should be extended, at a later date, so that x represents a negative number to avoid the common misconception that a varible such as x is always positive.

Traveling back and forth on the number line connecting arithmetic and algebra is a healthy practice that both deepens conceptual understand-ing and helps prevent misunderstand-ings. Before students work with frac-tions and rational expressions, they should benefit from revisiting whole numbers and integers and explicitly connecting this knowledge to algebra.

activity 2: FractionsJust as with integers, we begin our work with fractions on number lines that clearly define our unit of refer-ence (as shown in fig. 1). In this second activity, both the measurement and the sharing interpretations of fractions are used. Hence, we interpret the fraction a/b in this way:

1. a out of b equal parts in the inter-val [0, 1]

2. The size of the portion when a is divided into b equal parts (Wu 2002)

To address the measurement inter-pretation listed in item (1) above, ask questions such as the following:

• If Pat divided a one-mile track into three equal parts and ran two of them, what part of a mile did Pat run?

• Pat ran four of these equal parts. What part of a mile did she run?

• Where are 2/3 and 4/3 located on the arithmetic number line? How do you know?

Using the measurement interpreta-tion, students should take the inter-val from 0 to 1 on the number line and divide it into three equal parts. Students name the second part as the fraction 2/3 and the fourth part as 4/3. (See fig. 4.)

Once students are successful with the fractions, ask them the following questions:

• If Pat divided an x-mile track into three equal parts and ran two of them, what part of a mile did Pat run?

• If Pat ran four of these equal parts, what part of a mile did she run?

• Where are the rational expressions (2/3)x and (4/3)x located? How do you know?

• If the track was 3 miles long (x = 3), would both expressions make sense?

• Would both expressions be true for any value of x?

To address the algebraic version, a student ideally uses an extension of

the same fraction interpretation using x as the unit of reference in place of 1. As noted earlier, some students will have more difficulty with the algebra and may need to revisit the arithmetic many times before connecting with the algebra number line.

These questions help students understand the connection between numbers and variables, especially the relationship between fractions and rational expressions. Now they should see all of the above as simply points on a number line. Having this concrete representation promotes a deeper understanding of a variable as (merely) a symbol representing a number. This knowledge can lessen the fear of learning algebra.

Once students are proficient with the measurement interpretation of fractions, it is safe to explore the sharing interpretation, discussed in item 2 above. Making sense of the sharing interpretation of fractions as-sumes that one can locate a given frac-tion on a number line. To address the

Fig. 4 A side-by-side comparison illustrates the algebraic connection.

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Before students work with fractions, they should revisit whole numbers and integers.


sharing interpretation, start by asking this question:

If Jack has 1 candy bar and wants to divide it equally among three friends, how much of a candy bar would each friend get?

Encourage students to think about the size of each portion if 1 is divided into three equal parts. Be sure that stu-dents connect their responses to the number line, such as the example in figure 5a. If Jack has x candy bars and wants to divide them equally among three friends, how much would each friend get? To get students thinking, ask them to consider the size of each

portion if x is divided into three equal parts. Ask them if x/3 would make sense for any value of x and to make a conjecture about the location of (1/3)x and x/3 on the number line.

Next, ask this question:

If Jack has 2 candy bars and wants to divide them equally among three friends, how much of a candy bar would each friend get?

This question involves the size of each portion if 2 is divided into three equal parts, as shown in figure 5b. Have students express this question on the number line. If Jack has 2x candy bars and wants to divide them equally

Fig. 5 Dividing 1, 2, and 4 candy bars help students explore more variables.

(a)

(b)

(c)

among three friends, how much would each friend get? This question is asking about the size of each por-tion if 2x is divided into three equal parts. Students should consider if (2x)/3 would make sense for any value of x. Keeping in mind the number line in figure 5b, ask students to find locations of (2/3)x and (2x)/3 on the number line.

To extend this sharing interpreta-tion to improper fractions, pose this question:

If Jack has 4 candy bars and wants to divide them equally among three friends, how much of a candy bar would each friend get?

This question is asking students to think about the size of each portion if 4 is divided into three equal parts. On the number line, the results should resemble figure 5c. When moving to variables, the question becomes this: If Jack has 4x candy bars and wants to divide them equally among three friends, how much would each friend get? (What is the size of each portion if 4x is divided into three equal parts?) Would (4x)/3 make sense for any value of x? What is true about the location of (4/3)x

Traveling back and forth on the number line connecting arithmetic and algebra

is a healthy practice that both deepens conceptual understanding and

helps prevent misunderstandings

students use a number line to visualize dividing candy bars.


launching pad to introduce the ratio 2 to 1 highlights the cumulative nature of mathematics and is therefore beneficial in building more confident problem solvers.

the unit oF reFerenceWhy do many students, although they are skilled at both partitioning intervals and sharing, continue to have difficulties with fractions? One pos-sible source of confusion may be the unit of reference. The two Jack and the Candy Bar questions specifically address this issue:

1. If Jack has 1 candy bar and wants to divide it equally among three friends, how much of a candy bar would each friend get?

2. If Jack has 2 candy bars and wants to divide them equally among three friends, how much of a candy bar would each friend get?

In the first question, 1 explicitly refers to 1 candy bar. Consequently, most students both illustrate and get the correct answer, 1/3, whether they

and (4x)/3 on the number line?A conceptual knowledge of the

measurement interpretation of a fraction not only leads to a deeper understanding of the sharing inter-pretation but also lends itself to a richer foundation for understand-ing fractions as ratios. For example, a student who interprets (2/3)x as being two-thirds of some quantity x will be able to better visualize the following problem:

There are 18 students in Mrs. Jones’s class. Two-thirds are boys. How many are boys, and how many are girls?

To make sense of this problem, a stu-dent may simply replace the unit of reference x with 18 as shown in figure 6. The thought process in-volved in understanding the previous problem should enhance students’ knowledge of a similar problem in-volving ratios:

The ratio of boys to girls in Mrs. Jones’s class is 2 to 1. There are 18 students in the class. How many are boys, and how many are girls?

Using the fractions 2/3 and 1/3 as a

Fig. 6 expressing a ratio on a number line and as parts and a whole

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12 boys and 6 girls

12 boys and 18 − 12, or 6, girls

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Cake.” Phi Delta Kappan 78 (October 1996): 170.

Bruner, Jerome. The Process of Education. New York: Vintage Books, 1960.

Darley, Joy. “Ninth Graders’ Interpreta-tions and Use of Contextualized Models of Fractions and Algebraic Properties: A Classroom-Based Ap-proach.” Ph.D diss., University of South Carolina, 2005.

Gelman, Rochel, Melissa Cohen, and Patrice Hartnett. “To Know Mathe-matics Is to Go beyond Thinking That ‘Fractions Aren’t Numbers.’ ” Proceed-ings of the Eleventh Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. New Bruns-wick, NJ: Rutgers, 1989.

Hiebert, James, and Thomas Carpenter. “Learning and Teaching with Under-standing.” In Handbook of Research in Mathematics Teaching and Learning, edited by Douglas A. Grouws, pp. 65−97. New York: Macmillan, 1992.

Hilgard, Ernest. Introduction to Psychol-ogy. 2d ed. New York: Harcourt Brace, 1957.

National Council of Teachers of Math-ematics (NCTM). Principles and Stan-dards for School Mathematics. Reston, VA: NCTM, 2000.

———. Curriculum Focal Points for Pre-kindergarten through Grade 8 Math-ematics: A Quest for Coherence. Reston, VA: NCTM, 2006.

Skemp, Richard. “Relational Understand-ing and Instrumental Understand-ing.” Arithmetic Teacher 26 (November 1976): 9−15.

Wu, Hung-Hsi. “How to Prepare Stu-dents for Algebra.” American Educator 25 (Summer 2001): 10−17.

———. “Challenges in the Mathemati-cal Education of Teachers: Why Is the Preparation of Mathematics Teachers So Difficult?” National Summit on the Mathematical Education of Teachers. www.cbmsweb.org/NationalSummit/Plenary_Speakers/wu.htm. 2002. ●

recoMMendationsTraveling back and forth from arith-metic to algebra on the number line is a practice that should promote a solid foundation for learning algebra. In particular, conceptual knowledge of numbers, especially fractions, is an important link to a conceptual un-derstanding of variables. We should neither assume that our students are aware of the “arithmetic to algebra” connections nor assume that they understand the underlying arith-metic concepts. Instead, we should patiently travel the road in both directions with them, asking many questions along the way, to better prepare our students for thinking algebraically.

bibliograPhyBaddeley, Alan. The Psychology of Memory.

New York: Basic Books, 1976.Bennett, Albert. Fraction Bars: Step by

Step Teaching Guide. Fort Collins, CO: Scott Resources, 2003.

Bracey, Gerald. “Fractions. No Piece of

draw number lines or candy bars to visualize the question. In this case, 1/3 implies both 1/3 candy bar and 1/3 group, or total pieces. However, in the second question, the unit of reference is not explicitly stated. Notice that figure 7 shows Sam’s and Vanessa’s work, both beginning algebra stu-dents. They each drew correct pictures for the second question. Sam gave the correct answer, 2/3, implying 2/3 candy bar. When Vanessa was asked the meaning of 1/3, she answered, “One-third candy bar.” When asked if that answer would make sense, she replied, “No, I meant 1/3 pieces.” If the unit of reference was the number of pieces in her drawing, Vanessa’s an-swer would be correct. However, the question asks, “How much of a candy bar would each get?” not “What part of all the candy did each get?” Oppor-tunities such as this allow rich discus-sion concerning the unit of reference. For example, when students present 2/3 as an answer, teachers should ask, “Two-thirds of what?”

Fig. 7 Sam’s correct number line (a) and Vanessa’s incorrect unit of reference (b) allow the teacher to assess their understanding.

(a)

(b)

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a number line can be a useful visual tool to help students progress

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