making sense of decimal multiplication - marcia...

i(NCTM 2000). The Representation Process Standard explores the impor-tance of creating and using represen-tations to communicate mathematical ideas and interpret physical and math-ematical phenomena. What reason-ing and which representations might middle-grades students use to make sense of the procedures for multiply-ing decimals?

Students face several challenges as they transition from whole-number multiplication to decimal fraction mul-tiplication. As teachers, we should have a deep understanding of the math-ematics so that we can better support middle school students’ sense making.

Making Sense of Decimal MultiplicationTo build student confi dence,

relate decimals towhole-number operations,

geometry, and measurement.

Margaret M. Rathouz

25.7In the elementary grades, students learn procedures to compute the four arithmetic operations on multidigit whole numbers, often by being shown a series of steps and rules. In the mid-dle grades, students are then expected to perform these same procedures, with further twists. For example, to calcu-late 25.7 × 0.39, students are taught to compute 257 × 39. Then, the decimal point is moved three places to the left in the product. How can we help stu-dents understand why this works?

The Reasoning and Proof Process Standard suggests that students need to reason about the math that they are learning to make sense of the content

430 MatheMatics teaching in the Middle school ● Vol. 16, No. 7, March 2011

Copyright © 2011 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. 16, No. 7, March 2011 ● MatheMatics teaching in the Middle school 431Vol. 16, No. 7, March 2011 ● MatheMatics teaching in the Middle school 431

Making Sense of Decimal Multiplication

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Meanings and Models FoR MUltiPlicationOne way to assess students’ initial conceptions of numbers and opera-tions is to ask them to construct story problems that are appropriate for a particular arithmetic expression. Since students often arrive at middle school with an equal-groups understanding of multiplication, teachers might see this student-authored story problem to match the expression 3 × 4:

Three of my friends each have4 pets. How many pets do theyhave altogether?

A story such as this is not only a useful assessment tool but also an aid to help learners estimate quantities, justify arithmetic properties, and solve problems mentally by visualizing a scenario that uses a particular arith-metic expression.

From Equal Groups to MeasuresIf students are limited by an equal-groups interpretation, they may fi nd it challenging to write or recognize a correct and meaningful story problem when one or both numbers are deci-mal fractions (Graeber and Tanenhaus 1993). For example, how would a student write a word problem about

3.2 × 0.46? Would the same format be used, as in “3.2 friends each have 0.46 of a pet”? To reason about this problem, students need to think of two groupings (the items and the groups themselves) and that each grouping can be “sectioned” into pieces. In effect, instead of counting, they must think about measuring.

What types of measures help students attach meaning to decimal numbers? The most familiar measure involves money. For example, 0.46 could be thought of as 46 hundredths of a dollar, or 46 cents. Decimal frac-tions are commonly used to measure many other quantities, such as—

• distance (an odometer showed the bicycle traveled 3.4 miles),

• time (an athlete broke a record by 0.46 second),

• weight (the bulk food section sold 3.4 pounds of fl our), or

• volume (I pumped 0.46 of a gallon of gas).

Quantities measured in customary units (feet, yards, pounds, gallons, and so on) are more familiar to learners in the United States. The base-ten structure of the metric system provides a natural connection with decimal fractions.

Teachers who introduce metric

units of measure can make these available as referents to decimal frac-tions (Bonotto 2005; Hiebert 1989). For example, 3.2 can be thought of as 3 meters and 2 decimeters; 0.46 might be thought of as 460 grams or 0.46 (or almost one-half ) kilogram. Being familiar with amounts that are measured or expressed with decimal fractions enhances students’ work with decimal operations.

To give meaning to 3.2 × 0.46, a student can use these metric measures in a word-problem context:

It takes 3.2 meters of wood dowel to make a lightweight bridge. Each meter weighs 0.46 kg. How much does the entire bridge weigh?

Although it is clear that we have 3 of the 0.46-kg pieces, it is challenging to fi nd the weight of the additional 0.2-meter portion. Students often lose track of what the 0.2 is two tenths of. Does it refer to 0.2 of a meter, a kilogram, or a 0.46-kg piece? Rep-resentations such as that shown in fi gure 1 may help to distinguish the different types of units. Since each meter weighs 0.46 kg, it becomes clearer that a fraction of a meter will weigh less than 0.46 kg. Working with decimal estimation, as outlined in a section below, will help to reveal that one-tenth of 0.46 kg is 0.046 kg, and two-tenths must be double that amount, or 0.092 kg.

Additional diffi culties in under-standing can occur if the roles of the factors are switched, as in this problem:

A 1-meter length of wood weighs 3.2 kg. How much would 0.46 of a meter weigh?

Many students will think that this problem no longer involves multiplica-tion, since the product is smaller than 3.2. Others have trouble interpreting the problem as repeated addition,

(a)

(b)

Fig. 1 At 0.46 kg per meter, the weight of 3.2 meters of wood can be seen as a multiplication problem, with the help of these representations.

Vol. 16, No. 7, March 2011 ● MatheMatics teaching in the Middle school 433

“What does it mean to add 3.2 to itself 0.46 of a time?” Whole-class discussions are helpful for students who need to air confusions and clarify their interpretations. Within these discussions, teachers can refer to diagrams like fi gure 1 and use lan-guage referring to whole-number mul-tiplication. They might say, in explana-tion, “Instead of fi nding 4 groups of 3 things, we are fi nding the amount in a part (0.46) of a group of 3.2 things.”

From Arrays to AreaOther models, such as rectangular area diagrams, have some appealing features that can help learners make sense of decimal multiplication (Graeber and Tanenhaus 1993). Students note the similarity between tiles arranged in an array and those in equal groups (e.g., the rows represent the groups, and the columns represent the number of items in each group). If an array grouping square units is used to tile a rectangular region, students begin to appreciate the source of the familiar l × w formula for the area of a rectangle (see fi g. 2). Using such models, educators may connect their students’ prior knowledge about multiplication as equal groups or arrays to an area meaning that extends to fractional factors.

Returning to the 3.2 × 0.46 prob-lem, a student might envision a rectan-gular wooden plank with dimensions 3.2 meters × 0.46 meters. The student might then draw a plank and visualize its area as approximately 1.5 m2 (see fi g. 3). The one-dimensional length and width (factors) are highlighted to distinguish them from the two-dimen-sional area (product) in this model. This feature, made explicit in the diagram, is often unclear in textbook diagrams and can lead to confusion between length units and area units (Izsak 2005).

Scalar Fraction of an AreaTo further scaffold learners’ use of area, teachers can introduce the idea

of a scalar fraction of an area with this story problem:

A farmer has 2.3 km2 of land. She planted beans on four-tenths (0.4) of her land. On how many square kilometers did she plant beans?

A grid diagram can represent the scenario and a numerical solution (see fi g. 4). Each large square represents 1 km2. Each row of 10 small squares (shown in purple) is 0.1 of a km2, and each small square (shown in green) is 0.01 of a km2. The total shaded part

shows the area planted in beans as 8 rows, or 8 tenths of a km2, or 0.8 km2, and an additional 12 small squares that are 12 one-hundredths of a km2, or 0.12 km2. The total area would be 0.8 + 0.12 = 0.92 km2. Within the grid diagram, proper size relationships are maintained among whole units, tenths, and hundredths, allowing students to visualize these orders of magnitude.

MaKing sense oF PRodUcts Using estiMationWhen students create representations in familiar contexts, they not only

Fig. 2 Students connect their prior knowledge about multiplication as equal groups or arrays to the concept of area, which can be extended to fractional factors.

Fig. 3 The one-dimensional measurements of length and width (factors) are distinguished from the two-dimensional area measurement (product) in this model.

Fig. 4 A grid diagram of 2.3 × 0.4 as 0.4 of 2.3 km2 preserves the proper size relationships among whole units, tenths, and hundredths and allows students to visualize these orders of magnitude.


provide meaningful referents to deci-mal numbers and operations but also use these referents as tools for estima-tion (Vance 1986). Consider how your students might use the interpretations of decimals and of multiplication introduced earlier to approximate the product of two decimal numbers:

Estimate a value that is close to the product 5.24 × 1.6, and explain why that value makes sense.

Four explanations appear in fi gure 5and represent an amalgam of actualresponses to a similar task given to preservice elementary school teach-ers. Consider different meanings of the numbers and operations used to fi nd reasonable approximations of the product 5.24 × 1.6.

Notice that the reasoning used in estimating products requires an

understanding of multiplication and the properties of multiplication, such as the distributive property. Building fl u-ency with the base-ten decimal system allows the calculations to be performed mentally (Rubenstein 2001). Estima-tion can assist in discovering properties and be used as a computational tool.

MaKing sense oF an algoRithM These are the steps in a commonly taught algorithm for multiplying decimals:

1. Remove the decimal points from all factors to make them whole numbers.

2. Multiply the whole numbers.3. Count the total number of digits to

the right of the decimal point in all the original factors.

4. Replace the decimal point in the

product by using the number of places found in step 3.

What is happening mathematically in this algorithm?

Some students claim that the decimal points were moved in one direction before multiplying, so they must be moved back after multiply-ing to compensate. This compensationexplanation is, by itself, not mathe-matically meaningful (e.g., it does not work when adding decimal numbers). Initiate a conversation about what it really means to move the decimal point. Consider again 5.24 × 1.6. To start a discussion, the teacher might pose this question:

How are those two amounts [5.24 and 524] related to each other?

Students should be encouraged to use meanings of the numbers to justify their thinking, such as comparing $5.24 to $524.

To explain “moving the decimal point,” students evoke knowledge of multiplying and dividing by ten. When a base-ten number is multi-plied by 10, each digit is promoted

student a “I know the answer must be between 5 and 12 because 5 × 1 = 5 and 6 × 2 = 12.”

student B “One bag of potatoes weighs 5 kg. Two bags would weigh 10 kg. I want 1.6 bags, which is about halfway between, so somewhere around 7.5 kg.”

student c “I have fi ve buckets of water. Each bucket contains 1.6 liters of water, so that’s 1.6 + 1.6 + 1.6 + 1.6 + 1.6 = 8 liters . . . and about 0.4 of a liter more, since we have another almost quarter of a bucket [5.24 is almost 5 1/4]. So in all the buckets, there’s about 8.4 liters.”

student d “Think of a rectangular garden that is 5 meters and 24 cm long and 1.6 meters wide. What is the area?

You can see that the total area is going to be 5 m2 [blue] plus 2 greens are a little more than another m2, so 2.5 m2 more. The pink area is 0.24 m2 and the white box is about half of that [0.12 m2]. Altogether, we have about 5 + 2.5 + 0.24 + 0.12 = 7.86 m2.”

Fig. 5 Estimations of 5.24 × 1.6 are explained in students’ words. Fig. 6 When multiplying decimals procedurally, we want students to realize that the digits are moving, not the decimal point.


to a position that has a value of one higher power of ten.

Figure 6 shows that when a number with a 5 in the ones place is multiplied by 10, the 5 becomes worth 10 times as much, or 50. Likewise, the 2 tenths become 20 tenths, or 2, and the 4 hundredths become 40 hun-dredths, or 4 tenths, or 0.4. We want students to realize that the digits in the numeral are moving, not the deci-mal point. This idea can be demon-strated in the classroom with transpar-encies by putting the place-value slots on one (lower) transparency and the digits on an overlay (upper transpar-ency). To multiply by 10, all the digits are promoted one space left by sliding the upper transparency so that each digit moves up one place.

The following calculation is hidden inside the first part of the standard algorithm (multiplying 524 by 16):

(5.24 × 100) × (1.6 × 10) = 524 × 16 = 8384

It explains how whole numbers are produced from decimal factors. The resulting product is too big by a factor of 100 × 10, so we need to divide the product by 100 and again by 10. Here, we demote each numeric digit in 8384 by a power of 10 for each division by 10. We need to do that three times (two times for each factor of 10 in 100 and once for the one factor of 10 in 10) to find the product of 5.24 × 1.6 = 8.384.

Although this explanation tells more than the rule “multiply whole numbers and move the decimal point,” I have found that students are better able to justify why the values make sense when their explanations rely on various contexts and inter-pretations of the decimal factors and of multiplication. Which of these

processes would your students be able to generate or find most convincing? How do these connect to students’ earlier work with fractions or to later work with proportional reasoning?

Scaling Groups Up and Down One explanation for the decimal multiplication algorithm relies on the familiar equal-groups interpretation. The problem 5.24 × 1.6 could be understood as 5.24 bags that have 1.6 kg of potatoes in each bag. Be-cause fractional bags are difficult to imagine, we might consider 524 bags with 16 kg of potatoes in each bag, which is 524 × 16 = 8384 kg. But we only have 1.6 kg, which is one-tenth of the 16 kg we calculated, so 8384 kg is 10 times as big as it should be, or 838.4 kg. But we also only have 5 full bags and 24 hundredths of another bag, which is one-hundredth of

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524, so 838.4 kg is still 100 times as big as it should be. Demote the digits by two more positions (divide by 100) and arrive at the correct product of 8.384 kg. Using this context gives meaning to compensation.

Scaling Dimensions and AreaUp and DownUsing an area meaning of multiplica-tion, 5.24 × 1.6 could be explained as the area of a rectangular garden that has dimensions 5.24 m by 1.6 m. To make sense of the decimal multi-plication algorithm that begins with 524 × 16, consider a fi eld that is 100 times as long and 10 times as wide as the garden. Figure 7’s sketch repre-sents the scenario.

The area of the large 524 m × 16 mfi eld is 8384 m2. A thousand of the smaller 5.24 m × 1.6 m gardens fi t inside (100 across and 10 down), so the 8384 square meters must be divided by 1000 to get the area of the 5.24 m × 1.6 m garden.

Renaming Decimals as FractionsRewriting decimal numbers as frac-tion equivalents is another way to explain why the multiplication algo-rithm works. For example, 5.24 can be written as 524/100, and 1.6 as 16/10. In this example,

524100

1610

× ,

the numerators are multiplied (524 × 16) (this explains multiplying the whole numbers), and the product is placed in the numerator. Then the denominators are multiplied (100 × 10), and that product is placed in the denominator. The result represents the division of the numera-tor (8384) by 1000, which corre-sponds to moving the decimal point three places to the left or, more meaningfully, demoting each digit by three powers of 10.

Changing the Size of UnitsAn approach that connects with renaming decimals as fractions is to change the size of units to one-tenth, one-hundredth, or one-thousandth of the original so that the same numeri-cal amounts are represented but with different unit names.

In the garden example, instead of changing the dimensions of the 5.24 m × 1.6 m garden, we can write the same dimensions as whole num-bers by changing the units of measure:

524 cm × 160 cm = 83,840 cm2.

Measuring 524 cm is equivalent to 524 hundredths (524/100) of a meter, and 160 cm is equivalent to 160 hundredths (160/100) of a meter. The resulting whole-number product (524 × 160) represents the number of cm2 in the garden. As 10,000 of these (cm2) area units fi t inside a 1 m2 area, we would need to divide 83,840 by 10,000 to yield the area in square meters, 8.384 m2. This type of explanation is supported by earlier work with measurement in the metric system and foreshadows work with scaling and proportional reasoning.

deePening UndeRstandingWhen middle school students repre-sent and reason about decimal num-bers and multiplication, they are less likely to memorize and quickly forget

isolated mathematical procedures. Instead, they are able to construct a connected web of relationships among mathematical ideas and processes. By introducing referents for decimal numbers, learners connect multiple representations for the same amount, such as decimal notation (0.347), currency (almost 35 cents), metric measurement (0.347 meter, or 347 millimeters), and fraction notation (a little more than 1/3). Through discus-sion of several robust interpretations of multiplication (equal measures, area, and scalar multiplication), students construct meanings for the opera-tion that make sense with decimals and fractions. Simultaneously, they enhance their multiplicative thinking and understanding of geometry and measurement. Other mathematical concepts (estimation, base-ten place value, and the distributive property) are interwoven as students develop their solution methods.

Ultimately, the meanings of decimal numbers and multiplication, as well as the concepts developed in mental computation, are used as tools by the students to discover generaliza-tions, communicate their ideas about decimal multiplication, and justify why rules work. By engaging in these processes, students have opportuni-ties to build connections to other mathematical topics (such as propor-tional reasoning) and to other content

Fig. 7 An area model of multiplication and fi tting of gardens relates powers of ten to place value operations.



areas (science and social studies). In forming these links, students learn mathematical content with deeper understanding.

ReFeRencesBonotto, Cinzia. “How Informal Out-

of-School Mathematics Can Help Students Make Sense of Formal In-School Mathematics: The Case of Multiplying by Decimal Numbers.” Mathematical Thinking and Learning7, no. 4 (2005): 313−44.

Graeber, Anna O., and Elaine Tanenhaus. “Multiplication and Division: From Whole Numbers to Rational Numbers.” In Research Ideas for the Classroom; Middle Grades Mathematics, edited by Douglas T. Owens, pp. 99−117. Reston, VA: National Council of

Teachers of Mathematics, 1993.Hiebert, James. “The Struggle to Link

Written Symbols with Understand-ings: An Update.” Arithmetic Teacher36, no. 7 (March 1989): 38−44.

Izsak, Andrew. “You Have to Count the Squares: Applying Knowledge in Pieces to Learning Rectangular Area.” Journal of the Learning Sciences 14, no. 3 (2005): 361−403.

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

Rubenstein, Rheta N. “Mental Mathematics beyond the Middle School: Why? What? How?” Mathematics Teacher 94, no. 6 (September 2001): 442−45.

Vance, James H. “Estimating Decimal Products: An Instructional Sequence.”

In Estimation and Mental Computation, 1986 Yearbook of the National Council of Teachers of Mathematics (NCTM), edited by Harold L. Schoen and Marilyn J. Zweng, pp. 127−34. Reston, VA: NCTM, 1986.

Margaret M. Rathouz,[email protected], an assistant pro-fessor of mathematics education at the University

of Michigan−Dearborn, is interested in students’ reasoning about and representa-tions of number and operations. Although this article lists one author, several col-leagues were involved in the design and implementation related to this work. In particular, the author is indebted to Rheta Rubenstein for thinking about and discuss-ing the work presented here.

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