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VO L. 8, NO . 4 . DECEMBER 2002 193

S A N J A Y R A M B H I A

O

NE OF THE MANY IMPORTANT IDEAS THAT

we teach in the mathematics and prealgebra

curricula is the concept of order of operations.However, it is a concept that many studentsconsistently forget from year to year. Students in-variably solve problems from left to right, regard-less of the hierarchy associated with the opera-tions. This article outlines a new approach toteaching this important concept.

Traditional and New Approaches

TO BEGIN TEACHING THE ORDER OF OPERATIONS

in my eighth-grade prealgebra classes, I write a rel-atively easy problem on the board:

14 + 6(3) = _______

Approximately half the students answer 60 and theother half answer 32. I hand out calculators to twoof the students. One is a scientific calculator andthe other, a basic-level calculator; both the answersof 60 and 32 are confirmed. I then present a wordproblem and ask for answers:

Joan has $14, and she collects $6 from three of herfriends. How much money does she now have?

Most of the students correctly see that she shouldhave $32. Returning to the original problem, moststudents now agree that the answer to the problemis 32.

I use this strategy to introduce the concept oforder of operations and the importance of under-standing the proper use of calculators when doingmathematics. I am sure that many teachers aroundthe country approach this topic similarly. My goalfor my students, however, is to go much furtherwith order of operations than this simple exercise.

My goal is to have students solve problems like thefollowing, which I have used on tests for extracredit:

and

5[(14 18)(3 + 5)] 7(12 18) + (4 6) 3 = _____

5(3)2(10 2)

8 4

2+

=7 ______

3[(20 16)(5 3)] 436

42 5(6) +

Teacher to Teacher

A New Approachto an Old Order

SANJAY RAMBHIA, [email protected], is currently

teaching at Farmington High School in Farmington,

CT 06032. Previously, he taught eighth-grade mathe-

matics at Griswold Middle School in Rocky Hill,

Connecticut, where he served as department supervisor.

This material may not be copied or distributed electronically or in any other format without written permission from NCTM.Copyright 2002 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.


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194 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL

When I first started teaching, I introducedorder of operations as I was taught the concept,using the phrase Please Excuse My DearAuntSally. This familiar phrase helps students remem-ber to do operations in parentheses before expo-nents and multiplication and division before addi-

tion and subtraction. Many students come tobelieve, however, that the phrase also means thatmultiplication is done before division and that ad-dition is more important than subtraction. For thisreason, students are also taught an exception tothe phrase; that is, that operations of multiplica-tion and division (or addition and subtraction) areperformed in the order that they appear, from leftto right. In essence, this exception complicates anotherwise easily remembered and appliedmnemonic device.

To simplify the concept, I began to teach orderof operations using a table format, as shown intable 1. I explain that the higher up the chart anoperation is, the more important it is and that itmust be done first. If operations appear on thesame level, then they are of equal importance and,therefore, must be done as they appear (from leftto right) in the problem. I include more than justparentheses in level 1 to explain that multiplegrouping symbols must be done from the insideout to obtain a single answer. As we proceedthrough the year, I add other grouping symbols tolevel 1, such as absolute value bars and the frac-tion bar.

If I want my students to be able to solve a com-plex problem, such as

I begin by having them analyze simple problems,such as the one in the introduction:

14 + 6(3) = _____

How many operations are in the problem? Whatoperation should be done first, and what operation

should be done second? How many parts does thisproblem have? What separates the problem intoparts? We work through these questions as welook at the problem. I then show students howthey can separate the problem into parts by draw-ing lines:

14 + 6(3) =14 + 18 = 32

A slightly more difficult problem would be thisequation:

6(3 + 5) 4(8) + 5(6 + 1) = _____

Some students feel overwhelmed by such problemsuntil I remind them to separate the problem intoparts. They start to understand that because addi-tion and subtraction are done last, those operationsare the keys to breaking down the problem. Specifi-cally, the addition or subtraction signs that are notenclosed in grouping symbols partition the prob-lem. The problem above, for example, has threeparts, as shown below:

6(3 + 5) 4(8) + 5(6 + 1) =6(8) 4(8) + 5(7) =48 32 + 35 =48 32 + 35 = 51

How many parts are in the following problem?

Three parts is correct. This problem can be sepa-rated into parts by looking for addition and sub-traction signs that are not inside groupingsymbols.

To solve this problem, compute each section inde-pendently. Start with the first section.

Move to the second section.

______

9(7 2) 3[(4 1) (12 8)]8(5 2)

6+ + + +

=

9(9)

81

______9(7 2) 3[(4 1) (12 8)]8(5 2)

6+ + + +

=

9(7 2) 3[(4 1) (12 8)]8(5 2)

6______+ + + +

=

9(7 2) 2[(4 1) (12 8)]8(5 2)

6______,+ + + +

=

TABLE 1Order of Operations

Level 1 { [ ( ) ] }all grouping symbols

Level 2 exponents

Level 3 multiplication and division

Level 4 addition and subtraction


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VO L. 8, NO . 4 . DECEMBER 2002 195

Then, calculate the third section.

Finally, perform the final operations.

81 27 + 4 = 58

Conclusion

WHEN STUD ENTS LEARN THAT CO MPLEX

mathematics problems can be broken down into

much simpler problems, the feeling of being over-

whelmed is diminished. Organizing the order of op-

erations into a table format allows students to re-

member and apply the hierarchical rules of order of

operations more proficiently than traditional meth-

ods. I have had more students successfully apply

the concept of order of operations when I combine

both methods in my teaching.

______

9(7 2) 3[(4 1) (12 8)]8(5 2)

6+

+ + +

+

+

+

=

=

=

=

9(9) 3[5 4]8(3)

6

3[9]24

6

81 27 4

+

81

______

9(7 2) 3[(4 1) (12 8)]8(5 2)

6+ + + +

=

9(9) 4]

3[9]

81 27

+

81

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