Rational Number Operations on the Number Line

Abstract

(ProQuest: ... denotes formula omitted.)IntroductionMany teachers are familiar with the "colored chip" model for teaching integer concepts and integer operations. Briefly, different colored chips are used to represent positive or negative units that can then be combined in ways that model operations with integers. However, the CCSS-M make it clear that teachers should use the number line to represent integer operations as well; in fact Standard 7.NS.A.1 in Grade 7 states students: Clearly, addition and subtraction should be represented on the number line. But if teachers are only familiar with using the chip model for integer operations, then how do they make the transition to using the number line? This article will explore the connection between these two representations for addition and subtraction and make a strong case for using both as multiple representations of a challenging mathematical concept.There are several reasons for representing operations using the number line, not the least of which is that students must learn to perform operations with rational numbers, not just integers. Since the colored chip model can only reasonably illustrate integers and their operations, another model should be used that incorporates all rational numbers and is also mathematically sound.In this paper, we will investigate the basics of the colored chip and number line models for representing integers and their operations. We will then resolve the challenge of connecting the two models. Finally, we will argue that the number line model is an important one that should be taught alongside or in lieu of the chip model in order to deepen students' understanding of operations with integers, rational numbers, and real numbers in general.The Colored Chip Model: BasicsGenerally, the colored chip model for integers uses one colored chip to representa positive unit, and another colored chip to represent a negative unit. In this paper, we will use a circular yellow chip to represent a positive unit and a circular red chip to represent a negative unit. Readers familiar with the colored chip model will recall the notion of a "zero pair", that is, that a pair consisting of a positive and negative unit "cancels" out to make zero and therefore does not affect the value of the quantity shown.Addition with the Colored Chip ModelThe operation of addition with the chip model is rather simple, employing a "grouping together" interpretation of addition. This means that when an addition statement a + b is presented, we interpret the statement as meaning to start with quantity a, introduce quantity b ("add b'') and then determine the resulting value after this action, removing zero pairs that are formed. It is important to note here that addition is interpreted as an action, i.e. something "is done" or "happens" to the quantity a.Subtraction with the Colored Chip ModelRepresenting the operation of subtraction with the colored chip model takes more ingenuity. Typically, the subtraction sentence a - bis represented with a "take away" interpretation of subtraction: we begin with the minuend, a units, and attempt to take away the subtrahend, b units. It is important to note here that subtraction is again interpreted as an action: something is being done to the minuend. However, when there is not enough to take away, as in the case of -7 - 3 (i.e., you cannot take away 3 positives from 7 negatives), zero pairs come into play. The idea is to start with the minuend, represented here as 7 negatives, and then to introduce as many zero pairs as needed so that 3 positives can be removed. The result is then tallied up, and the difference of 10 negatives, or -10, is deduced. This is illustrated in Figure 2.Two more cases of subtraction are illustrated in Figure 3, in which including zero pairs is used when there are not enough to take away.By comparing number sentences and the results, teachers can use the colored chip models for addition and subtraction in tandem to illustrate the fact that a - b = a + (-b), that is, that subtraction is equivalent to adding the opposite. …