Don't sell short the distributive propertyAuthor(s): RONALD V. McDOUGALLSource: The Arithmetic Teacher, Vol. 14, No. 7 (NOVEMBER 1967), pp. 570-572Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41185661 .

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Don't sell short the distributive property RONALD V. McDOUGALL University of Southern California, Los Angeles, California

Mr. McDougall is a graduate student at the University of Southern California and is working toward his master's degree.

V-* an you prove addition of fractions to your students? If your answer is yes, with the thought of dividing the proverbial pie into equal-sized sections and showing their sum, then you are not proving the con- cept but giving an example. Too frequent- ly, examples are passed off as proofs for many of our mathematical concepts.

If asked to explain addition of fractions, most of your students would say that if the denominators are alike, you add the numerators and put their sum over the denominator. The student will obtain the correct answer, but through mechanics, not through understanding. For rational num- bers of arithmetic we have no addition facts as we do for the whole numbers. We know that 5 + 7 is 12 and 2 + 4 is 6,

1 2 9 but how do we know that - + - is - ?

2 5 10 The explanation or proof of addition of like fractions is partially dependent on the distributive property. Students should be given an understanding of this simple proof:

- + - Given 7 7 I 1 If a and b are numbers of I II 1

al 2 • _ + 3 • - arithmetic, b Ф 0, - = a • -

(2 + 3) - Distributive property 7

(5) _ Addition fact 7

Reason 2 7

The proof of addition of unlike fractions would simply require the use of the multi- plicative property of one to express the fractions in equivalent forms as like frac- tions, and then we could proceed as in the above. This proof is applicable to sub- traction of fractions as well as addition.

Two often-stated objectives of the mod- ern approach to mathematics are to in- crease mental computational skills and to develop mathematical reasoning ability.1 The distributive property is one of the most effective instruments we have for achieving these worthwhile objectives. Let's explore a few more of the many applica- tions that are not only enlightening but downright fun.

How many times a day do you see a student grab a pencil to multiply two numbers? Quite frequently, I'm sure. With little instruction and a lot of drill, your students can become whiz-kids at mental math. For example, if you wish to multi- ply 11 x 43, it can be easily done with- out pencil and paper by having your students rename 11 as (10+1) and dis- tributing this multiplication over the indi- cated sum. It works with equal facility for subtraction. If you are multiplying 9x58, then 9 may be expressed as a difference, (10 - 1), and again the multiplication can be distributed over this indicated dif- ference. These problems and many more

1 John L. Marks, C. Richard Purdy, Lucien В. Kin- ney, Teaching Elementary School Math for Under- standing (2nd ed.; McGraw-Hill Book Co., 1965), pp. 22-28.

570 The Arithmetic Teacher

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can be done by your students with ease - and without the use of pencil and paper.

11 x 43 = (10 + 1) x 43 = (10 x 43) + (1 x 43) = 430 + 43 = 473.

9 X 58 = (10 - 1) X 58 = (10 X 58) - (1 X 58) = 580 - 58 = 522.

102 X 24 = (100 + 2) X 24 = (100 X 24) + (2 X 24) = 2400 + 48 = 2448.

We are fortunate in that this skill is one that lends itself to classroom fun. The skill of mental multiplication can be used in short spurts for variety in your daily lesson, while accomplishing reinforcement.

While teaching multiplication of two- digit numerals, what have you answered when confronted with the question, "Why do you have to move the second partial- product over one place to the left?" Once again the distributive property can be uti- lized to clarify this procedure of multipli- tion. In addition to clearly answering your student's question, you are reinforcing, at the same time, the ever-important under- standing of place value.

Example 1 45

X 32

Example 2 45 X 32 = (40 + 5) (30 + 2)

= (40 + 5) 30 + (40 + 5) 2 = 40»30 + 5»30 + 40*2 + 5-2

Show the students the indicated product of 45 and 32 in both the traditional form, Example 1, and the expanded form (the product of two indicated sums), Example 2. Referring to both forms, point out that when 5 is multiplied by 2, you are multi- plying 2 units times 5 units, and when multiplying 2 times 4, you are multiplying 2 units times 40 units. The following step, or the second partial-product, is the turn- ing point in this explanation. Here, point out that when multiplying the 3 of the factor 32 times the 5 of the factor 45, you are really multiplying 3 tens times 5 units. As the result is tens, this partial-product must be placed in the proper column or

November 1967

tens place. Frequent references should be made whenever multiplication occurs, point- ing out that when multiplying different units the product can fall in a variety of places. Further expansion of this concept can be demonstrated, if necessary, with the example of 4 tens times 3 tens, show- ing that tens multiplied by tens will yield hundreds. By this time not only should the original question be answered concern- ing the position of the partial-products, but the students should have a better un- derstanding of place value - through the use of the distributive property.

On leaving arithmetic and moving into the simple algebraic concepts introduced in the seventh and eighth grades, we see an application similar to addition of like fractions in the addition of like monomials. For example, addition of the monomials 2a and Aa can be easily accomplished with the assistance of the distributive property.

2a + Aa Given 2» a + 4* a Multiplication is understood

when no operational symbol appears between a constant and a variable

(2 + 4) a Distributive property (6) a Addition fact

ва Reason given in Step 2

. Some textbooks still treat the multipli- cation of two binomials as a purely me- chanical procedure, as in Example 3.

Example 3 2x + 3 3jc + 4 вх2 + 9x

+ 8jc + 12 ox2 + 17* + 12

Little explanation is given as to why you multiply in this fashion; it is apparently arbitrary. Applying the distributive prop- erty and other properties at our disposal, we can verify each step of this multiplica- tion, as seen in Example 4.

Once it is understood by the students that the procedure shown in Example 3 is really derived from the application of the distributive property, shown in the fourth example, then the use of the former pro- cedure is permissible. It would be unnec-

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Example 4

(2x + 3) (3jc + 4) Given (2jc + 3)3* + (2x + 3)4 Distributive property 2*#3;c + 3#3jc + 2*#4 + 3*4 Distributive property бх2 + 9x + 8jc + 12 Commutative and associative properties; multiplication

fact; multiplication is indicated when no operational symbol appears between a constant and variable

6** + (9 + 8)*+ 12 Distributive property бх2 + (17)* + 12 Addition fact бх2 + 17* + 12 Reason given in Step 4 for indicated multiplication

essary drudgery to require all of the steps of the proof whenever multiplying two binomials.

As the distributive property is used each year, it is easy to lose sight of our ob- jectives and emphasize mechanics rather than applications. We need to ensure an understanding of this property so that the student can apply the principles to any new situation. This is where its value lies.

Learning the laws or properties of mod-

ern mathematics can be fun! It can be stimulating, and it can be interesting. How- ever, this depends on how they are pre- sented. The importance of the distributive property, with its many applications throughout mathematics, cannot be empha- sized enough. Use these stimulating proofs in combination with increased stress on understanding and application. Your ef- forts will be rewarded by more interested and able students.

Books received

[Continued from ,p. 565]

Mathematics Reform in the Primary School, J. D. Williams (ed.). (UNESCO Institute for Education.) Paris: United Nations Edu- cational Scientific and Cultural Organization, 1967. Paper, 130 pp., $2.50. (Order from UNESCO Publications Center, New York.)

A report of a meeting in Hamburg, Ger- many, January 1966, to consider the "new" role of mathematics in the primary school. France, Germany, Hungary, Ireland, Swe- den, the United Kingdom, the United States of America, and the Union of Soviet So- cialist Republics were represented.

Modern Math Practice Book 1, Charles E. Shultz. Chicago: Kleeco Products, 1967. Paper, 26 pp., $4.25.

A chalk-markable, reusable supplementary text for use in Grades 1 and 2.

Modern School Mathematics, Structure and Use, Kindergarten-Grade 3 (Pupils and Teachers Edition), E. Duncan, L. Capps, M. Dolciani, W. Quast, and M. Zweng. New York: Hough- ton Mifflin Co., 1967. Paper, Grade K, $1.20; Grade 1-2, $2.40; Grade 3, $3.80.

Modern School Mathematics, Structure and Use, Grades 4-6, E. Duncan, et al. New York: Houghton Mifflin Co., 1967. Cloth, $3.80 each.

Modern School Mathematics, Structure and Method, Grades 7 and 8, M. Dolciani, W. Wooton, E. Beckenbach, and W. Chinn. New York: Houghton Mifflin Co., 1967. Cloth, Grade 7, $4.60; Grade 8, $5.20.

New Math for Parents and Pupils, W. Garfield Quast. New York: Arc Books, 1967. Paper, 129 pp., 95ф.

A very brief consideration of some topics in the mathematics curriculum of the elementary school.

The New Mathematics and an Old Culture, John Gay and Michael Cole. New York: Holt, Rinehart, & Winston, 1967. Paper, 100 pp., $1.95.

A research study of the learning of mathe- matics among the Kpella tribe of Liberia.

[Continued on p. 593]

572 The Arithmetic Teacher

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