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 multi- plication 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 multiplica- tion 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 stu- dents often believe that multiplication works as two different operations––sometimes as an addition op- eration (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 stu- dents 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 under- standing 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 be- tween 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 increas- ing, 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 multi- plied 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 be- cause 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 pre- serves 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 per- cent 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.