Understanding Multiplication As One Operation

D I A N E A Z I M

Understanding Multiplication as One Operation

OME OF THE WONDER OF NUMBERS becomes apparent when we use numbers to perform calculations. Understanding multiplication with positive rational numbers is Snot a simple process. It requires reconceptualizing the meanings of multiplication with whole numbers to include numbers that are less than 1 or are mixed numbers. (Numbers in this article are all non-negative numbers.) To learn how multiplication works with fractions is to experience a whole new meaning for multiplication. In more than twenty years of teaching about multiplication with fractions, I have found that students often believe that multiplication works as two different operations––sometimes as an addition operation (increasing quantities by adding them re- peatedly) and sometimes as a division operation (decreasing quantities by dividing them into equal parts and taking one or more of these parts). The Recipe, Multiplication Fulcrum, and Photocopy Machine approaches presented in this article are methods I have shared with students to support the view that multiplication is one operation that has one fundamental meaning that may result in in- creased, decreased, or preserved quantities. This article also presents two methods invented by students to help them conceptualize multiplication.

DIANE AZIM, [email protected], is a mathematics teacher educator at Eastern Washington University, Cheney, WA 99004. She is interested in the mathematics content understanding of prospective teachers of grades K–8.

466 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL Copyright © 2002 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. The Recipe Method 4 × = STUDENTS CAN EASILY CONNECT MULTIPLICA- tion with doubling and tripling recipes. This con- 3 × = nection is helpful when students explore number patterns and investigate what happens to a recipe 2 1/4 × = when quantities are multiplied by numbers that are less than 1 and by mixed numbers. I ask students to × think about what would happen to the quantity of 2 = ingredients for one recipe (represented by one × bowl of ingredients) if it were multiplied by 4, 3, 2, 1 1/2 = 1, and 0. They draw their ideas on a sheet of paper, in this order, and leave space between their draw- 1 × = ings for other drawings. The students usually draw pictures that show 4, 3, 2, 1, and 0 recipes repre- 3/4 × = sented by 4, 3, 2, 1, and 0 bowls full of ingredients. Using their previous understanding of multiplica- 1/2 × = tion, they establish the fact that multiplying the recipe by a whole number produces exactly that 1/4 × = number of bowls of ingredients. I then ask students to figure out what the results 0 × = would be of multiplying the recipe by 1/4, 1/2, and 3/4 (numbers that are between 0 and 1) and by 1 1/2, recipe bowl 2 1/2, and 3 1/2 (numbers that are greater than 1 and between other whole numbers) if they adhere to the Fig. 1 Conceptualizing multiplication using recipes pattern established by the whole-number multipliers. What would happen to the bowl of ingredients in these 1 new recipes? Students use the pictures they have drawn to reach conclusions about the new multipliers. 0 1/4 1/2 3/4 1 1/2 2 2 1/2 3 3 1/2 4 With few exceptions, the students draw representa- × tions of 1/4, 1/2, 3/4, 1 1/2, 2 1/2, and 3 1/2 bowls Decrease Increase the values of other numbers the values fulcrum (except 0) (see fig. 1). They report that if they adhere to the of other whole-number-multiplication pattern, then multiplying numbers (except 0) the recipe by 1/4, 1/2, and 3/4 (numbers that are between 0 and 1) reduces the quantity of ingredients to Fig. 2 The Multiplication Fulcrum less than one whole recipe or bowl of ingredients. Mul- tiplying the recipe by mixed numbers (fractions that recipe all involve multiplication—multiplying by 2, are greater than 1) yields quantities that are greater 4, 1/2, 1/4, and 3/4—despite the fact that in some than 1 and lie between whole numbers. Multiplying the of these situations the quantity of ingredients is re- recipe by 1 1/2, for example, produces a quantity of in- duced. Students begin to view multiplication as one gredients that is greater than the quantity achieved by operation that increases, decreases, or preserves multiplying the recipe by 1 but less than the quantity the number of recipes depending on the size of the achieved by multiplying the recipe by 2. The result is multiplier (in relation to the number 1). exactly halfway between these two quantities. Students theorize that multiplying the recipe by numbers The Multiplication Fulcrum Method greater than 1 yields more than one recipe, multiplying the recipe by the number 1 preserves the one recipe ex- THE MULTIPLICATION FULCRUM TEACHING TOOL actly, and multiplying the recipe by numbers less than shows a number line balanced on a “fulcrum” (see 1 produces less than one whole recipe. Multiplying the fig. 2). The non-negative real numbers are repre- recipe by 0 produces the quantity of ingredients sented on the number line. The number 1 sits di- needed for no recipes, that is, no ingredients at all. rectly above the fulcrum; it is the pivot number in I use the recipe method to help students concep- multiplication. When other numbers are multiplied tualize multiplication as an operation in which one by the multiplier 1, their values are preserved. Multi- factor, the multiplier, operates or acts on the other pliers that are greater than 1 increase the values of factor, the multiplicand. Doubling, quadrupling, other numbers (the multiplicands). Multipliers that halving, quartering, and taking three-fourths of the are less than 1 decrease the values of other num-

VOL. 7, NO. 8 . APRIL 2002 467 bers. The multiplier 0 decreases the values of other 5 inches numbers until the result is a value of 0. The increasing, decreasing, and preserving work proportionally. 2 1/2 times as wide The value of any number when multiplied by 4 in- (250%, 2.5 × width) creases in proportion to the value of the number it- self. All numbers multiplied by 4 increase in value to four times their original values. All numbers multiplied by 1, 1/2, or 0 result in values that are equal to, width of one-half of, or 0 times (0 quantities of), their original original image values, that is, in proportion to their original values. (100%, 1 × width) The Multiplication Fulcrum brings into focus the 2 inches role of the multiplicative identity, 1. The enlarging and reducing influences of multiplication pivot around the number 1. This number separates the 1 inch multipliers that decrease other numbers’ values from the multipliers that enlarge other numbers’ values, that is, 1 preserves the identity of other numbers. 1/2 as wide Zero has an important position on the Multiplica- (50%, 1/2 × width) tion Fulcrum, too. As a multiplier, 0 is unique because it decreases the value of all other numbers Fig. 3 Conceptualizing multiplication by 1/2, 1, and 2 1/2 until they have the value 0. This property contrasts using the Photocopy Method with the role of 0 in additive situations, where 0 preserves the values of other numbers (whereas adding numbers that are greater or less than 0 to other numbers increases or decreases those numbers’ val- FRAMING THE FOUR FRAMED WHOLES ues). Zero is the additive identity; 1 is the multiplica- WHOLE tive identity. Zero, too, apart from its unique behav- Original Original Wholes ior as a multiplier, behaves uniquely in the role of Units Units multiplicand: 0 never increases or decreases in value 1 3 when multiplied by other multipliers. This property is noted on the Multiplication Fulcrum drawing. 2/3 2 2/3 The Photocopy Machine Method 1/3 2 1/3 4 THE OPERATION OF MULTIPLICATION FUNC- 2 tions in the same way that a photocopy machine Two-thirds of a does: it proportionally reduces, preserves, and en- rectangluar unit 1 2/3 3 larges quantities depending on the size of the multi- is framed to plier. When a person chooses to photocopy an image represent the whole 1 1/3 at 200 percent, all dimensions of that image are dou- that is being operated bled. Choosing 100 percent preserves the original on by the multiplier 4. 1 2 size of the image. Photocopying at 250 percent multi- (a) plies the lengths in the image by 2.5, or 2 1/2 (see 2/3 fig. 3). Similarly, choosing 50 percent reduces the lengths in the original image to one-half of their orig- 1/3 1 inal size, or multiplies them by 1/2. Choosing 25 percent multiplies all lengths in the original image by 1/4; choosing 75 percent multiplies all lengths in the Four wholes are needed. Four original image by 3/4, or reduces them to three- framed wholes are drawn and numbered (right). The result fourths of their original size. is 2 2/3 units (measured using An important distinction in discussing the photo- the units of the original whole). copy machine is that between the lengths and areas in images. The lengths are reduced or enlarged ac- (b) cording to the percents, used as multipliers. The areas in the image are reduced or enlarged by the Fig. 4 Representing 4 × 2/3 = 2 2/3 using the Frame Method squares of these multipliers; for example, for a 200

468 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL FRAMING THE TWO-THIRDS OF THE TWO-THIRDS OF THE WHOLE FRAMED WHOLE FRAMED WHOLE (quantity un-named) (quantity named)

Original Original Wholes Original Wholes Units Units Units 4 4 3/3 4 3/3 3 2/3 3 1/3 3 3 3 2/3 2 2/3 2/3 2 1/3 2 2 2 1 2/3 1/3 1 1/3 1/3 1 1 1 2/3 1/3

Four original units are Two-thirds of the framed whole is The 4 original units are framed to represent drawn and numbered (right). The redrawn as 12 third-units. the whole that is being result is the quantity to and includ- Eight third-units represent operated on by the ing the 2/3 dividing line— 2/3 of the whole. The result multiplier 2/3. between 2 and 3 original units. is 2 2/3 (original) units. (a) (b) (c)

Fig. 5 Representing 2/3 × 4 = 2 2/3 using the Frame Method percent enlargement, the new area is 4 times the being acted on, but that whole may not be one com- original area and for a 50 percent reduction, the plete unit. It may be a whole of 2/3 of a unit, such as new area is 1/4 times (of) the original area. 2/3 of an hour. Although 2/3 of an hour is 2/3 of a The Recipe, Fulcrum, and Photocopy Machine whole unit, sometimes the 2/3 of an hour, not the com- approaches were created to support students in plete unit or hour, is the “whole” that is being acted on. constructing understanding about multiplication. Students might represent the expression 4 × 2/3, The following two approaches are methods of con- for example, by framing the “whole” of 2/3 unit in a ceptualizing multiplication that students have cre- rectangular frame (see fig. 4a). In figure 4a, 2/3 of ated when working with rational numbers. the original unit, which might be an hour, is represented using an area model, then framed in a bold The Frame Method box to demarcate it as the “whole” that the multiplier of 4 is acting on. Four of these framed wholes IN THE FRAME METHOD, A FRAME IDENTIFIES are needed, or the “whole” is needed four times. the multiplicand, which is the quantity that the multi- This idea is shown in figure 4b; four wholes of 2/3 plier is acting on under the multiplication operation. unit are stacked one on top of the other, and a count Students who find the Frame Method useful have of each of the four wholes is recorded on the right often experienced difficulty in realizing that one quan- side of the figure. On the left side of the figure, a cu- tity can be viewed as acting on the other in a multiplica- mulative record of the original units resulting from tion situation. To these students, clearly identifying the the stacking is kept. The product, or result of the quantity that is being acted on (the multiplicand) is im- multiplication, is 4 wholes of 2/3 unit. This product portant; this quantity becomes the one that they must be named using the original units; for example, “frame.” Recognizing, too, that a fraction or mixed the product of 4 × 2/3 is 2 2/3 units (2 2/3 hours). number, such as 2/3 or 4 1/4, may be the quantity Figure 5 shows a common version of the Frame being acted on is equally important. Students often de- Method for multiplying 2/3 × 4. A rectangular scribe this discovery using the word whole: a whole is model of 4 units, such as 4 acres, is drawn and

VOL. 7, NO. 8 . APRIL 2002 469 frame demarcates the “whole” that is acted on. It is enlarged, preserved, or reduced according to the multiplier quantity.

“Number of a Number” Method

SOME STUDENTS THINK ABOUT MULTIPLICATION using a “number of a number” method. To these students, for example, 4 × 3 means four 3s and 4 × 1/3 means four 1/3s. The result is the total of the four 3s or four 1/3s, which is 12 or 1 1/3, respec- tively. These students think about multiplication as taking one number (or quantity) of another number (or quantity). These students are also able to extend their “number of a number” thinking to fractional number quantities of other numbers. To these students, framed using a bold outline to show that the 4 units 1/4 × 3 means “one-fourth of a three” (a number of are the whole that the multiplier of 2/3 is acting on 3s), 3/4 × 1/2 means “three-fourths of one-half,” (fig. 5a). Because the multiplier, 2/3, is less than 1, and 2 1/3 × 3 1/2 means “two and one-third of three less than one whole is needed, or exactly 2/3 of the and one-halves.” Although students do not gener- whole is needed. In other words, the whole is being ally use this language, their interpretations could be reduced. In figure 5b, the whole of 4 units is parti- translated as “3/4 of a ‘whole’ [or composite unit] of tioned vertically into three equal sections, and on 1/2” for 3/4 × 1/2, and “2 1/3 ‘wholes’ [or compos- the right side of the figure, these sections of 1/3 (of ite units] with 3 1/2 original units in each ‘whole’ the whole) are labeled. The result of the multiplica- [or composite unit]” for 2 1/3 × 3 1/2. tion, 2/3 of the whole, is represented by the two 1/3 Students can find exact numerical results for shaded sections below the 2/3 partition line. these meanings. Their reasoning might be summa- Figure 5b is a quantitative, rather than numeri- rized as follows: cal, representation of the multiplication. The product is a quantity between 2 and 3 original units; the • 1/4 × 3, or “one-fourth of a three,” equals 3/4 be- exact number of original units represented by these cause 1/4 of each 1 of the 3 is 1/4, and we want two 1/3 sections is not yet clear. To determine the three of those 1/4s: exact quantity of original units in the product, sub- divisions, such as those shown in figure 5c, are 1 1 1 1 3 often drawn. Each of the 4 original units in the × 3 =+ + = whole is subpartitioned into three 1/3 units to yield 4 4 4 4 4 a total number of subunits in the whole that is divis- ↑↑↑ ↑ 1 1 1 1 ible by 3. The subpartitioning yields twelve 1/3 ()()()() of 3 of 1 of 1 of 1 units, which are then partitioned into three sections 4 4 4 4 of four 1/3 units each. Each of these three sections is equivalent to 1/3 of the framed whole of 4 units. • 3/4 × 1/2, or “three-fourths of one-half,” equals The product of the multiplication of 2/3 × 4, which 3/8 because 1/4 of 1/2 is 1/8, and we want three is named in terms of the original units (e.g., acres) of those 1/4s: is eight 1/3 units. This amount is equivalent to 2 2/3 acres, as shown in the cumulative numbering 3 1 1 1 1 3 × =+ + = of original units on the left side of the figure. 4 2 8 8 8 8 Two aspects of the Frame Method to emphasize ↑↑↑ ↑ are the framing of the whole and the numbering of 3 1 1 1 1 1 1 1 ()()()() of of of of both the wholes and the original units. “You have a 4 2 4 2 4 2 4 2 whole, and you want more or less than the whole. But you have to keep track of how to count up the • 2 1/3 × 3 1/2, or “two and one-third of three and answer. My left number scale does this,” is a fairly one-halves”—conceptualizing 3 1/2 as one unit— common description of the reasoning of these stu- equals 8 1/6 because two 3 1/2s equal 7, and 1/3 dents. Students come to understand that in multipli- of 3 1/2 is 1 (1/3 of 3) and 1/6 (1/3 of 1/2). The cation, one quantity acts on another quantity. The result is then 7 + 1 + 1/6 = 8 1/6 (see fig. 6).

470 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL 1 1 1 1 1 1 1 2 ×= 3 = 3 + 3 + [ 1 + ] 7 + 1 = 8 3 2 2 2 6 6 6 ↑↑↑↑↑ 1 1 1 1 1 1 1 (2 o f 3 ’ s)( one 3 )( one 3 ) [ ( of 3 ) ( of ) ] 3 2 2 2 3 3 2 ↑↑↑↑ 1 1 [two 3 ’ s] []one- third of 3 2 2

Fig. 6 Breaking down 7 + 1 + 1/6 = 8 1/6

In other words, multiplication may be involved in In this method, multiplication is viewed as one op- situations in which the result is more than or less eration in which a given number (the multiplicand) is than the original quantity given. taken a designated number (the multiplier) of times. • Situation 1: Alison bought 7 3/8 pounds of wal- Summary nuts at $6 per pound. How much money did she spend on walnuts? THE FOCUS OF THIS ARTICLE IS ON STUDENTS’ • Situation 2: Alison bought 5/8 of a pound of wal- constructing representations for multiplication with nuts at $6 per pound. How much money did she rational numbers, as recommended in the Represen- spend on walnuts? tation Standard for grades 6–8 in the Principles and Standards for School Mathematics (NCTM 2000). The Helping students develop methods to under- meanings of rational numbers in this article are pri- stand the invariance of multiplication supports them marily the operator (multiplier) and measure or quan- in discovering the amazing flexibility of the (con- tity (multiplicand) meanings, three of the eight mean- ceptual) tool of multiplication in the realm of ratio- ings outlined in the Number Sense Standard for nal numbers. grades 6–8 (NCTM 2000). The methods in this article support an understanding of multiplication in which Bibliography one quantity proportionally alters (or preserves) the size of another quantity. These methods require both Confrey, Jere, and Guershon Harel. “Introduction.” In quantitative and numerical reasoning (Thompson The Development of Multiplicative Reasoning in the 1994; Sowder et al. 1998). The move from quantitative Learning of Mathematics, edited by Guershon Harel to numerical reasoning, from the reasoning repre- and Jere Confrey, pp. vii–xxviii. Albany, N.Y.: State sented in figure 5b to the reasoning represented in University of New York Press, 1994. figure 5c, may be difficult for students. Yet when the Greer, Brian. “Multiplication and Division as Models of goal is for students to develop a conceptual under- Situations.” In Handbook of Research on Mathematics standing of multiplication, they need to conceptualize Teaching and Learning, edited by Douglas A. Grouws, the operation quantitatively to devise meaningful pp. 276–95. New York: Macmillan, 1992. methods for determining numerical results. National Council of Teachers of Mathematics (NCTM). Supporting students in constructing meaning Principles and Standards for School Mathematics. Res- for multiplication that involves all rational num- ton, Va.: NCTM, 2000. bers—that is, giving multiplication one meaning Sowder, Judith, Barbara Armstrong, Susan Lamon, Mar- that explains how it increases and decreases the tin Simon, Larry Sowder, and Alba Thompson. “Edu- values of other numbers—may enable students to cating Teachers to Teach Multiplicative Structures in recognize a multiplication structure in a story the Middle Grades.” Journal of Mathematics Teacher problem or real-life situation involving multipliers Education 1 (1998): 127–55. that are greater and less than 1. They may be able Thompson, Patrick. “The Development of the Concept of to discern that the following situations, for exam- Speed and Its Relationship to Concepts of Rate.” In ple, can be modeled by multiplication because The Development of Multiplicative Reasoning in the they understand that multiplication is one invari- Learning of Mathematics, edited by Guershon Harel ant operation with one meaning that incorporates and Jere Confrey, pp. 179–234. Albany, N.Y.: State Uni- both integer and rational numbers (Greer 1992). versity of New York Press, 1994. C

VOL. 7, NO. 8 . APRIL 2002 471