the distributive property undersold

The Distributive Property Undersold

Jane M. WatsonDepartment of EducationUniversity of TasmaniaHobart, Tasmania7001 Australia

As pan of their training, all elementary schoolteachers are exposed to the three famous properties of thewhole numbers: the commutative property, the associa-tive property and the distributive property. But which iswhich? And for what operations are they true? Thethree properties are used constantly in informal arith-metic operations and yet there is some doubt thatteachers and students can state them precisely at amoment’s notice or realize their full potential to help inproblem solving.

These difficulties are probably most noticeable forthe distributive property of multiplication over addition.This is because it involves two operations and because itis usually stated twice to take account of the commuta-tive property of multiplication:

ax(b+c) = axb + axe(b + c) x a = bxa + ex a.

Learning the property by rote and thinking up exampleswith whole numbers to show that the property holds isoften as far as instruction goes. For example, ifa = 2, b = 3, and c = 4, then

2 x (3 + 4) = 2 x 7 = 14,2x3+2x4=6+8= 14,(3 + 4) x 2 = 7 x 2 = 14,3x2+4x2=6+8= 14.

Often the second form of the property is not stressedto the extent of the first because in examples like theabove the forms look so similar. As will be seen later,this may be a mistake. After a number of these examplesare given it is assumed that the student knows theproperty.

Within a week or so of "learning" the distributiveproperty, however, most students do not recognize asituation in which it can be used. That such situationsabound is evident to anyone who analyzes what they doin calculations in many later topics in the mathematicscurriculum. Those teachers, who have so assimilated theproperty that they do not even realize they are using it,may not stop to reinforce its usage and hence theunderstanding of its power to their students.

The following discussion provides examples wherethe distributive property is useful in solving problems, in

making calculations much more efficient to perform, orin justifying algorithms. The difficulty for instruction ofcourse is that it may not be appropriate to use theexamples for motivation at the time the property is firstintroduced. What is important, however, is to recognizethe use of the property when it occurs, point it out tostudents, and ask them to discuss why they have beenable to complete calculations using the property. It isunfortunate that most text books, both for children andfor preservice teachers, fail to stop and reinforce thedistributive property when it is applied in varied con-texts.

Intuitive Use in Grade 3

Although the distributive property would not beintroduced as such to children as young as Grade 3, thefollowing anecdote illustrates how it is used intuitivelyby children who are comfortable working with numbers.The need to calculate 7x7 arose when a Grade 3teacher and her class were discussing square numbers.The class had not begun to study the 7-times tables butthe 5-times tables were mounted on posters around theroom. The teacher expected no reply to 7 x 7 = A andwas surprised when an eight-year-old boy quicklyresponded "49". When asked how he worked it out, hereplied,<<! looked at the wall and found 5 sevens andthen needed 2 more." Although the child would not havebeen able to write the mathematics formally he had usedthe distributive property in his problem solving:

7x7 (5 + 2) x 75x7 + 2x735+1449.

To have this intuition at an early age is a great asset.to be reinforced and encouraged. It is the son of skillteachers try to develop with children who have troubleremembering their tables later in primary school.Perhaps it should be done more consciously as part of’talking* mathematics with younger children. Asresponses below show, a lively discussion is possible.

T: If I want twelve fours, how could I work itout?

School Science and Mathematics

Distributive Property Undersold

S\: Ten fours are 40 and two fours are 8, so it’s48.

S^: No. Six fours are 24, so 24 and 24 are 48.

S3: But, four fours are 16, so four fours andfour fours and four fours make 48.

The last suggestion leads to the next important pointabout the distributive property: its application to morethan two addends.

Extension to a Longer Sum

The distributive property usually appears in textbooks in the form presented earlier in this article withoutany discussion of its extensions. It is, of course, also truethat

a x (b + c + d) == axb + axe + axdand

(b + c + d) x a = bxa + cxa + dxa

for a sum of three numbers; and for n numbers,

a x (b\ �+- b^ 4-... + b^) = axb^ + axb^ +... + axbj^and

(b^ + b^ + ... + b^) xa = b^ x a + b^xa +... + b^xa.

Students can check this out with numerical examples aswas shown before and examples given below willdemonstrate how important this extension is for applica-tions in other areas.

A Volume Problem

Consider the following problem from Bennett andNelson (1985, p. 314):

A house with ceilings that are 2.4 m highhas five rooms with the followingdimensions: 4 m by 5 m; 4 m by 4 m;6 m by 4 m; 6 m by 6 m; and6 m by 5.5 m. Which of the followingair conditioners will be adequate to coolthis house: an 18,000 Btu unit for 280m3; a 21,000 Btu unit for 340 m3; or a24,000 Btu unit for 400 m3?

Although the statement of the problem suggests acalculation of the floor area of the entire house multi-plied by the height of the ceiling, many students solvethis problem by working out the complete volume ofeach room and adding the results. There are morecalculations involved and they are more complex thanthe calculations made by considering the total floor areafirst:

; 48.0m3: 38.4m3= 57.6m3= 86.4m3= 79.2 m3309.6 m3.

4 m x 5 m x 2.4 m4 m x 4 m x 2.4 m6 m x 4 m x 2.4 m6 m x 6 m x 2.4 m6 m x 5.5 m x 2.4 m

Using the distributive property the calculation becomes

(4mx5m + 4mx4m + 6mx4m + 6 m x 6 m +6 m x 5.5 m) x 2.4 m

= (20 + 16 + 24 + 36 + 33) m2 x 2.4 m= 129 n^x 2.4m= 309.6m3.

The point to be made to students is that bothmethods lead to the correct answer, the reason being thedistributive property of multiplication over addition.Stating the property again in this context should help toreinforce the concepts involved:

(Area^ + Area^ + Area3 + Area4 + Area5) x height =Area^ x height + Area^ x height + Area3 x height +Area4 x height + Area5 x height.

If different students have obtained the correctanswer both ways, it may be worthwhile getting them tocompare answers and to "discover" again the distributiveproperty. If they do not remember its name, they can betold, in order to help recall next time. It may also beappropriate to mention that the distributive propertyholds in this application because the ceiling height isconstant throughout the house. Students may thendiscuss what would happen if there were different ceilingheights for some rooms in the house.

Another problem solving exercise related to theextended distributive property is to ask students to workout how many operations they save by using the"efficient" side of the distributive property equation forvarious values of n. Perhaps they can work out aformula depending on n.

Early Word Problems

Word problems are a common way of consolidatingthe application of all operations taught in the elementaryschool and Blume and Mitchell (1983) have made thepoint that they can also be used to focus attention on thedistributive property. Consider the following questions.

If Tom buys four apples at 250 each and fiveoranges at 250 each, how much will hespend?

If Alice travels for 3 hours today at 90 km/hrand 4 hours tomorrow at 90 km/hr, how farwill she travel?

Volume 93(6), October 1993


These problems can be interspersed with problems wherethe distributive property does not apply and discussioncan take place about how they are different from prob-lems where the prices and speeds vary. Debate can alsotake place between students who worked the givenproblems in the two different ways and the class candecide which is more efficient.

Area of a Trapezoid

As an application of finding the area of a triangle,children are often taught a method of finding the area ofa trapezoid using two triangles. Using the quadrilateralin Figure 1 with sides of length b^ and b^, parallel, andperpendicular height, h, between b^ and b^, the area ofthe trapezoid is the sum of the areas of the two trianglesshown.

Area trapezoid

Figure 1. Finding the area of a trapezoid

= AreaA^ + AreaA^= /2bih +y2b2h.

bl

h

b

it is that we are allowed to rewrite the expression thisway?"

A Pre-Algebra Exercise

Often as a part of pre-algebra problem solvingexercises, problems with boxes are given where it is thechild’s task to determine the value to be placed in thebox to make the equation true (repeated boxes containthe same number). One such problem is the following:

2 x [] + 3 x [] = 20.

Very often the only strategy used for the problem is trialand error. The usual error made is to substitute a value,say 1, in the first box and then conclude that the onlyway to make the sentence true is to put 6 in the secondbox. Many students do not recognize the application ofthe distributive property in this form because the quantityto be ’factored’ out is on the right (cf. comments made atthe beginning of this article on the two forms of thedistributive property). Since

(b + c) x a = bxa + cxa

in this case something = 2 xD + 3 x Dwhich of coursegives (2 + 3) xQ leading to 5 xQ = 20, soD = 4.

Not only was the box on the right but the problemrequired the recognition of the property from the rightside of the equation, rather than the left. This points tothe suggestion that the property should be stated andstressed in both fashions when it is initially introduced:

It is then necessary to use the distributive property twiceto deduce the formula as usually learned by children. Inthe form

axb + axe = a x (b + c),V2 x bin + Vi x b^h = 16 x (bih + b^h)

and in the form

bxa +bih +

c x a

b^h == (b + c) x a,(bi+b2)h

which gives

Area trapezoid = Vi (b\ + b^)h .

Teachers often carry out these procedures withoutpausing for a breath. For students, however, suchoperations may be considered magic if they do notunderstand the distributive property. It is well to stopand reinforce an understanding of the distributiveproperty in this unfamiliar context; say, just to ask, "why

axb + axebxa + cxa

a x (b + c)(b + c) x a.

Place Value and "Long" Multiplication

The representation of numbers in the base-ten placevalue system has both multiplicative and additive aspectswhich make it possible to discuss the distributiveproperty when multiplication of whole numbers involv-ing at least one number greater than ten occurs. Since

312 = 300+10+2,

312x4 is really

312x4 (300 + 10 + 2) x 4(300x4) + (10x4) + (2x4)1200+40+81248.

The distributive property is the basis of all "long"multiplication algorithms but it is not always obvious to



students that this is the case. Even when written in theform:

1 ten 2 onesx 4 ones4 tens 8 ones,

3 hundred

12 hundreds

it is possible to discuss the application of the distributiveproperty.

The derivations of divisibility tests are goodexercises to set for advanced students as they use thedistributive property, as well as some of the otherproperties of the whole numbers. For example, to showa number is divisible by 3 if the sum of its digits isdivisible by 3, consider the base-ten place value expan-sion of the number and apply the distributive property.In the case of 4692.

4692 = 4 x 103 + 6 x 102 + 9 x 10 + 2= 4 x (999 -+� 1) + 6 x (99 + 1) + 9 x (9 +1) + 2= (4 x 999) + (4 x 1) + (6 x 99) + (6 x 1) + (9 x 9) +(9x1)+2

= (4 x 999) + (6 x 99) + (9 x 9) + 4 + 6 + 9 + 2,

using the distributive property three times. Since thebracketed products are divisible by 3, the remaining sumof four digits (the original four from 4692) will deter-mine if the number is divisible by 3. The same argumentworks for divisibility by 9.

Fractions

The distributive property of multiplication overaddition also holds for fractions but students are some-times surprised by this. Upon the introduction of theprocedures for addition and multiplication of fractions itis desirable to set exercises which demonstrate theproperty. For example, the following exercises could beset separately on an assignment with the request tocompare answers and comment.

1. Vi x (% + %) =D2. Vi x % + Vz \% = D3. Compare the answers to Questions 1 and 2

If they are the same, what property ofnumbers that we have studied before isbeing demonstrated? If they are not thesame, check your work!

Use of the distributive property can often simplifycalculations with fractions. Sometimes it is easier to addfirst and sometimes, to multiply.

Often students do not recognize the distributiveproperty when it is presented without explanation in thecontext of fractions. Many preservice elementaryteachers do not understand how the calculations for the

following problem from Bennett and Nelson (1985, pp.363-4) are performed.

The Rhind Papyrus is an ancient Egyptianmathematics text for solving practicalproblems. These problems are precededby a table for fractions with a numeratorof 2 and denominators which are oddnumbers from 5 to 101. Each of thesefractions is written as a sum of unitfractions with different denominators.Use the method suggested in the follow-ing equations to write ^ Vis, and %s as asum of unit fractions. Check your results.

% = 2/l(l)(5)] = % C/l + %) = V3 + Vl5%} = 2/[(3)(7)] = YlO 0/3 + VT) = Vl5 + V35

This is an excellent problem for upper level or giftedelementary school students. Of course they should beasked to discuss what property of numbers is beingdemonstrated in the examples.

Multiplication of mixed numerals is another situa-tion in which the distributive property can come intoplay. For example.

3 x 21/2 = 3 x (2 +l/2) = 3 x 2 + 3 x Vi = 6 +% = 7 Vz.

In the following more complex example, the propertycan be used twice to achieve an answer:

y/i x 214 = (3+Y2) x (2+16) = (3+16) x 2 + (Wi) x Vz.= 3x 2 + Vz x 2 + 3 x 16 + Vz x Vi=6+l+3/2+l/4 =8%.

The advantage of using the distributive property overusing improper fractions is that smaller numbers areinvolved in the calculations. Of course the distributiveproperty must be understood to be applied correctly!

Decimals

As place-value representations of fractions, decimalsprovide opportunities similar to the previous two sectionsfor the reinforcement of the distributive property. Theexample in Figure 2 is from the National Council ofTeachers of Mathematics* (NCTM) Standards (1989, p.88) and shows that an area model can be used to furtherillustrate the property as it applies to the process ofmultiplying decimals. In justifying

1.2x1.3=1.56

by counting the squares, children can express theproblem in several ways. One child might suggestworking across the figure to produce:

Volume 93(6), October 1993


1.2 x 1.3 =L2x(l+.3)==1.2x1 + 1.2 x.3=1.2+.36= 1.56.

Figure 2. An area model/or multiplication of deci-mals

1.3

1.2

1.2 x 1.3 = 1.56

Another might suggest working down the figure to show1.2 x 1.3 =(l+.2)x 1.3

=1x1.3 + .2 x 1.3=1.3+.26= 1.56;

while a third might see the four easily distinguishedsections of the area model in terms of

1.2x1.3 =(l+.2)x(l+.3)=1x1 + 1 x .3 + .2x1 + .2 x .3= 1 + .3 + .2 + .06 .

There is great opportunity for class discussion andcomparison of solutions.

Algebra

It is not the purpose of this article to look in depth atthe use of the distributive property in higher mathemat-ics. Suffice it to say that the property is taken for grantedand used constantly by teachers at the high school leveland above.

A couple of examples are given, in passing, in orderto justify to elementary school teachers the importance ofchildren understanding the distributive property. Allexpanding and factoring of algebraic expressions areapplications of the distributive property. For example,expanding

x^(xy + y2 + 2) = x^y + x^2 + 2x2

and factoringx2 + 3x = x(x + 3)

andx2 + 5x + 6 = (x + 3)(x + 2)

all illustrate a generalized distributive property. If theproperty has been well ingrained at an early age, suchproblems as these will be handled more easily. Theimportance of this is demonstrated well by Miller andEngland (1989) in a project which asked students towrite about algebra. The writings showed that theapplication of memorized rules and properties concern-ing the distributive property and factoring a polynomialwas not being stressed enough during instruction.

Conclusion

These examples only scratch the surface of theapplications of the distributive property of multiplicationover addition. It is hoped that they will serve as cues forelementary school teachers, who will then begin torecognize examples of their own in the classroom. Ifapplications are brought to the notice of studentswhenever they occur, there will be a greater likelihood ofthe property becoming properly assimilated for futureuse.

The type of development suggested here supportsmany of the objectives for children’s mathematicallearning stated in the NCTM*s first four standards:

develop and apply strategies to solve a widevariety of problems;acquire confidence in using mathematics mean-ingfully (p. 23);reflect on and clarify their thinking about math-ematical ideas and situations (p. 26);use . .. properties, and relationships to explaintheir thinking;believe that mathematics makes sense (p. 29); andrelate various representations of concepts orprocedures to one another (p. 32).

References

Bennett, A.B., Jr. and Nelson, L.T. (1985). Mathemat-ics: An informal approach (2nd ed.). Boston:Allyn and Bacon.

Blume, G.W. and Mitchell, C.E. (1983). Distributivity:A useful model or an abstract entity? SchoolScience and Mathematics, 83, 216-221.

Miller. L.D. and England, D.A. (1989). Writing to learnalgebra. School Science and Mathematics, 89,299-312.

National Council of Teachers of Mathematics. (1989).Curriculum and evaluation standards for schoolmathematics. Reston, VA: Author.


the distributive property undersold

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