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10 + 2 + 10 + 5

2+2+2+2+2=10

The Common Core State Standards for Mathematics (CCSSM) call for an in-depth, integrated look at elementary school mathematical concepts. Some topics have been realigned to support an integration of topics leading to con-ceptual understanding. For example, the third-grade standards call for relating the concept of area (geometry) to multi-plication and addition (arithmetic). The third-grade standards also suggest that students use the commutative, associa-tive, and distributive properties of multi-plication (CCSSI 2010).

Traditionally, multiplication has been a major topic for third grade. Linked to repeated addition of equal-size groups, multiplication logically follows the study of addition. Introducing rectan-gular arrays to represent groups (rows) of equal size illustrates both numeric and geometric interpretations of mul-tiplication and naturally introduces the concept of area (CCSSI 2010). Rotating

the rectangular arrays illustrates the commutative property of multiplication (see fi g. 1), and determining the num-ber of tiles needed to build a multicolor rectangle allows students to demonstrate their understanding of conservation of area and to discover a geometric inter-pretation of the distributive property (see fi g. 2).

Although multiplication is typically a focus of third-grade mathematics (NCTM 2006), third-grade textbooks usually include few, if any, concepts of area or distribution and no geometric interpretation of the distributive prop-erty, which raises at least two ques-tions: (1) Are third graders ready for the reasoning needed to understand these concepts? And, if they are, (2) how can integrated exploration of these topics help students make connections that deepen their conceptual understanding of these topics and others already in the curriculum?

TheDistributivePropertyin Grade 3?

Are third graders ready to connect procedures to concepts of area conservation, distribution, and geometric interpretation?

By Christine C. Benson, Jennifer J. Wall , and Cheryl Malm

498 teaching children mathematics | Vol. 19, No. 8 www.nctm.orgCopyright © 2013 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.

A multicolored array allows for an exploration of the distributive property.

FIG

UR

E 2

y b g

r

r(y + b + g) = (r × y) + (r × b) + (r × g)

The problemTo answer the fi rst question and evaluate students’ understanding of multiplica-tion and area, we gave an open-ended problem (see fi g. 3) to a group of begin-ning third graders. The students had worked with decomposition strategies along with the associative and commuta-tive properties for addition; for example, thinking of 12 + 15 as 10 + 2 + 10 + 5 and seeing 2 tens for 20 and 2 + 5 for 7, to get 27. They also had used skip counting to add equal groups. Their teacher indi-cated, however, that they had not been introduced to multiplication, conceptu-ally or symbolically. To determine if their natural reasoning would support a more formal exploration of area and distribu-tion, we wanted to observe how students at this level of understanding would solve the given problem.

We gave students colored tiles, graph paper, and colored pencils to help them explore the problem. Their solutions illustrated several different approaches. Some students simply wrote the num-ber sentence 12 + 12 + 12 + 12 + 12 = 60. When asked, they explained that twelve tiles were in each row, and there were fi ve rows, so they added twelve fi ve times. These students illustrated the ability to

Three rows of tiles with fi ve tiles in each row

Five rows of tiles with three tiles in each row

3 × 5 = 15

5 × 3 = 15

Rectangular arrays encourage the discovery of the commutative property of multiplication.

FIG

UR

E 1

The problem that beginning third graders receive assesses their natural reasoning.

How many cards are needed?

A club wants to create a card section at the next football game by placing a card on each seat in a certain section. They are planning to use a section of seats that has 5 rows, and they plan to put 3 yellow cards, then 4 blue cards, and then 5 green cards down each row. How many cards altogether will be needed so that each seat in this section has a card on it? Write number sentences to show how you found your answer. Then write a sentence or two explaining your thinking.

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www.nctm.org Vol. 19, No. 8 | April 2013 499

visualize equal-size groups; they would be ready to link this to multiplication both con-ceptually and symbolically.

Other students, when asked how they determined the sum was equal to sixty, demonstrated a general understanding of the distributive property. These students explained that within the 5 twelves are 5 twos that would equal ten and 5 tens that would equal fi fty; then 50 + 10 = 60 (see fi g. 4). This decomposition naturally leads to symbolizing the problem as 5(2 + 10) = 5(2) + 5(10) = 10 + 50 = 60.

Another group of students exhibited a geometric interpretation of distribution. They explained by using color to separate the larger rectangle into smaller rect-angles. These students determined how many tiles were in each row of the smaller rectangles, used repeated addition to fi nd the total number of tiles in each smaller

rectangle, and then added those totals (see fi g. 5).

A fourth group of students used the smaller, single-color rectangles to count the number of columns in each of the smaller rectangles so they could skip-count by fi ves. Although students could have used skip counting by fi ves in the large rectangle to fi nd the total number of tiles (e.g., 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 60), most students using this method indicated that they visualized the problem as three smaller rectangles. They explained that there were three groups of yellow tiles (or 15), four groups of blue tiles (or 20), and fi ve groups of green tiles (or 25). All these students did not neces-sarily add 15 + 20 + 25 to get the total num-ber of tiles, reverting instead to counting all the tiles to answer the question; but they did exhibit a basic understanding of

Student strategies could be used to develop the distributive property.

FIG

UR

E 4 Students used repeated

addition for each of the smaller rectangles.

FIG

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E 5

23 × 14 = 322

FIG

UR

E 6 Area models of multiplication allowed students to see equal-size groups.

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conservation of area. Students’ exhibition of intuitive understanding supports their readiness for an integrated exploration of these topics.

Nearly all students demonstrated an understanding of conservation of area, that is, that the area of the large rectangle is equal to the sum of the areas of each of the smaller rectangles. From this, many students understood that five rows of twelve tiles (5 × 12) is the same as fi ve rows of three tiles plus fi ve rows of four tiles plus fi ve rows of fi ve tiles, or 5(3 + 4 + 5), which illustrates the distributive property. When students can add, skip-count, and conserve area, a concrete understanding of the distributive property is intuitive:

The precursors of…[the] distributive propert[y]…are already there as natural logic, the child’s natural habits of mind and the building blocks of higher math-ematics. (Goldenberg, Mark, and Cuoco 2010, p. 555)

Finding evidence that such understanding was in place, we proceeded to the second question.

The conceptsTo answer the question of how to integrate the exploration of these topics, consider the following examples, sequenced to illustrate the connections among multiplication, area, the distributive property, and algo-rithms currently in most textbooks. When doing these activities, be sure that students

23 × 14 = 322

FIG

UR

E 7

The area box model without base blocks provided a transition from the concrete representation of the distributive property with the base blocks to a more abstract representation in this model.

FIG

UR

E 8

20 + 3

10 200 30

4 80 12

23 × 14 = 322

+

The area model with base blocks enabled students to make connections to the area model with unit tiles. 6464614149191

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www.nctm.org Vol. 19, No. 8 | April 2013 501

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The expanded notation method allowed students to see the distributive property numerically.

FIG

UR

E 9

(20 + 3) × (10 + 4) =

(20 × 10) + (3 × 10) + (20 × 4) + (3 × 4) =

200 + 30 + 80 + 12 =

322

The expanded notation vertically enabled students to transition to the standard algorithm while still connecting the concepts learned in the previous models.

FIG

UR

E 1

0

20 + 3

× 10 + 4 12 ( 4 × 3) 80 ( 4 × 20) 30 (10 × 3)

200 (10 × 20) 322

are active participants at each step, not just observers (NRC 2001).

Area model with unit tilesAs discussed earlier, building rectangles to illustrate equal-size groups (i.e., rows, each with the same number of tiles) shows mul-tiplication as the area of a rectangle. Here, the area of the rectangle, or the number of square tiles it takes to cover that rectangle, is the product of its dimensions. This is an effective model for single-digit factors, but it is cumbersome with large numbers (see fi g. 6). So, to be more effi cient and reinforce place value, we switch to base-ten blocks.

Area model with base blocksWe still make a rectangle using the factors as dimensions, and the area of the rectangle

is still the product. But we use the blocks to group the tens and use the physical dimensions of the factors represented by the blocks to confi rm the dimensions of the blocks that represent the product. Geomet-rically, this is the distributive property. For example, 4 × 12 is illustrated as four groups of ten and four groups of two, showing 4 × 12 = 4(10 + 2) = 4(10) + 4(2) = 40 + 8 = 48.

For some factors, like 23 × 14, students may need to regroup to get the fi nal answer (see fi g. 7). Working with physical blocks before working with pictures allows stu-dents to manipulate the blocks, exchang-ing ten units for a rod or ten rods for a fl at. However, students soon just draw pictures, especially on homework. But like before, as the values of the numbers increase, the physical objects and drawings become cumbersome, and we look for something more effi cient.

Area box model without base blocksAt this stage, we no longer use base blocks but still use a rectangular box model, which we no longer draw to scale (see fi g. 8). We split the dimensions of the rectangle into tens and ones, and it is important to make sure students understand how the areas of the sides of this new rectangular prism relate back to the base-blocks rectangles. Students should do several examples both ways, side by side, so they see this important connection of something concrete to some-thing more abstract (Reys et al. 2009, p. 188).

The partial-products method provided the link between the expanded notation methods and the standard algorithm.

23× 14

12 8030

200322

( 4 × 13)( 4 × 20)(10 × 3)(10 × 20)

FIG

UR

E 1

1

502 teaching children mathematics | Vol. 19, No. 8 www.nctm.org

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Transition to expanded notationOnce students are comfortable using the box method, we begin a transition to some-thing more abstract by adding expanded notation to the box method. Again, students do both methods side by side for a while to make sure they see the connections (see fig. 9). Using the box methods and expanded notation, we can illustrate the distributive property both geometrically and numerically. Eventually, students no longer need to draw the pictures, although some may continue to visualize the box in their minds.

Expanded notation—verticallyNow that students understand what multi-plication is and why it is important that we consider place value when multiplying, we begin to transition to the standard Ameri-can algorithm (an effi cient way of multiply-ing). We continue to use expanded notation, but we rearrange how we write it, so that it starts to look like the standard algorithm (see fi g. 10). The partial products that are listed match the representations seen in the base-block models, and the connecting line segments help students see the relationship to the arcs in fi gure 9.

Partial productsThis is the same thought process we used previously, but we do not write the factors in expanded notation (see fi g. 11). However, we still consider a digit in the tens place to represent that many tens (20 for example, not just 2).

The standard American algorithm Continuing to use the distributive property, we multiply the units digit in the second factor by both digits in the fi rst factor, all in the same step (rather than writing it in two steps) (see fi g. 12). Again, doing both methods side by side for a few problems is important so that students make the con-nection and understand that the steps in the algorithm are based on using the distribu-tive property.

Students will no longer see the standard algorithm as a meaningless list of steps that must be completed in a certain order.

Instead, they will understand how it con-nects to the basic multiplication of single-digit numbers in the area model and to the distributive property.

ExtensionsOlder students (or young students who need a greater challenge) could use the same models but could change the base. Instead of grouping by tens, they could group by nines, for example. Consider these problems (see fi g. 13):

12nine × 4nine and 23nine × 14nine

When students make (or draw) the rectan-gles, they see that the pictures look the same as they did when using base-ten blocks (see fi g. 7), but the rods are shorter; and to get the fi nal answer, they group by nines instead of tens. When we have students multiply 12 × 4 and 23 × 14 in bases 5, 7, and 8, they see that the rectangular pictures look the same every time. Only the size of the grouping changes when simplifying to get the fi nal answer. This means, and we have students verify, that all the other methods follow as well, as long as we group accord-ing to the base we are in. What students can learn later depends on what they have learned before (CCSSI 2010, p. 5), so teach-ers of older students can use this pattern to help students make sense of algebra as generalized arithmetic.

Area model with algebra tilesWhen comparing fi gure 7 and fi gure 13 to fi gure 14, we notice that the size of the base does not matter in the picture; so we can replace each base number rod with b for

23× 14

92 (Notice that this is 12 + 80)

230 (Notice that this is 200 + 30)

322

FIG

UR

E 1

2

The standard algorithm is an effi cient means of multiplying multidigit numbers.

www.nctm.org Vol. 19, No. 8 | April 2013 503

The area models for multiplication in base nine are identical to the area models for multiplication in base ten.

FIG

UR

E 1

3

(10nine + 2nine ) × 4nine =40nine + 8nine =

48nine =

(20nine + 3nine ) × (10nine + 4nine ) =

200nine + 30nine + 110nine + 13nine = 353nine

base, and all of the multiplying concepts, including the distributive property, remain the same. Additionally, in standard algo-rithms for arithmetic, we align our columns so that we add quantities of the same place value; whereas in algebra, we combine like terms (such as shapes). Although there are many similarities, there are differences as well. There is no regrouping—for instance, ten bs do not make a b2. Also, in arithmetic, we can add hundreds and tens, for example; but in algebra, we cannot combine unlike terms. Thus the “answer” becomes an alge-braic expression with a term for each type of block used in the diagram. Although base

blocks (or algebra blocks) become cumber-some for three-digit numbers (polynomials with terms having exponents greater than one), students should work with such exam-ples to verify that the distributive property continues to hold.

Helping students make connections like those illustrated here allows them to deepen their conceptual understanding (Baek 2008; Clements 1999; CCSSI 2010; NRC 2005) that the distributive property is not an algorithm but a property, a characteristic that holds throughout mathematics—arithmetic, geometry, algebra, and other branches as well. When they begin to recognize it in

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different contexts, this familiarity will increase their understanding, build their confi dence, and increase their success in algebra and beyond (Carpenter, Franke, and Levi 2003; Kaput 1999).

Making the connectionsSome educators question the appropriate-ness of including the distributive property at the third-grade level in the Common Core State Standards for Mathematics, wondering whether students are ready to understand it. We found that the distribu-tive property is naturally logical to most students if we fi rst allow students to think through problems, view them from mul-tiple perspectives—numerically and geo-metrically—and then connect other models and more efficient procedures to those original models. Unfortunately, we often jump from one method to the next without making connections between the methods or to the underlying ideas that hold them all together. The value of each process or model described in this article will be undermined if students do not understand how each is connected to the previous one. And at some point, we must give our stu-dents a name for this idea—the distributive property—that keeps coming up over and over again. We must then continue to use that phrase whenever it appears in differ-ent contexts. We know we are teaching this effectively when, as students are faced with “new” problems, we hear them say, “Oh, that’s just the distributive property again. I know how to do that.”

REFERENCESBaek, Jae Meen. 2008. “Developing Algebraic

Thinking through Explorations in Multi-plication.” In Algebra and Algebraic Thinking in School Mathematics, 2008 Yearbook of the National Council of Teachers of Mathematics (NCTM), edited by Carole Greenes and Rheta Rubenstein, pp. 141–54. Reston, VA: NCTM.

Carpenter, Thomas P., Megan Loef Franke, and Linda Levi. 2003. Thinking Mathemat-ically: Integrating Arithmetic and Algebra in Elementary School. Portsmouth, NH: Heinemann.

Clements, Douglas H. 2000. “‘Concrete’ Manipulatives, Concrete Ideas.” Contem-porary Issues in Early Childhood 1 (1): 45–60.

Common Core State Standards Initiative. 2010. Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Offi cers. http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf.

Goldenberg, E. Paul, June Mark, and Al Cuoco. 2010. “An Algebraic-Habits-of-Mind Perspective on Elementary School.” Teaching Children Mathematics 16 (9): 548–56.

The area model for multiplication of algebraic terms resembles the area models used to multiply integers.

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b

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b2 b b

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b

1b b

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506 teaching children mathematics | Vol. 19, No. 8 www.nctm.org

Kaput, James J. 1999. “Teaching and Learn-ing a New Algebra.” In Mathematics Classrooms That Promote Understanding, edited by Elizabeth Fennema and Thomas Romberg, pp. 133–55. Mahwah, NJ: Lawrence Erlbaum Associates.

National Council of Teachers of Mathematics (NCTM). 2006. Curriculum Focal Points for Prekindergarten through Grade 8 Math-ematics: A Quest for Coherence. Reston, VA: NCTM.

National Research Council (NRC). 2001. Add-ing It Up: Helping Children Learn Math-ematics, edited by Jeremy Kilpatrick, Jane Swafford, and Bradford Findelle. Math-ematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washing-ton, DC: National Academies Press.

———. 2005. How Students Learn: History, Mathematics, and Science in the Class-

room, edited by M. Suzanne Donovan and John D. Bransford. Committee on How People Learn, A Targeted Report for Teachers. Division of Behavioral and Social Sciences and Education. Washington, DC: National Academies Press.

Reys, Robert E., Mary M. Lindquist, Diana V. Lambdin, and Nancy L. Smith. 2009. Help-ing Children Learn Mathematics. 9th ed. Hoboken, NJ: John Wiley and Sons.

Christine C. Benson, cbenson@nwmissouri.edu;Jennifer J. Wall, jwall@nwmissouri.edu; and Cheryl Malm, cgmalm@nwmissouri.edu, are colleagues at Northwest Missouri State University in Maryville. They teach content and mathematics methods to preservice teachers, emphasizing inquiry learning, conceptual understanding, and making connections between branches of mathematics and between mathematics and other content areas.

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