TEACHING MATHEMATICS WITH TECHNOLOGY: Order of Operations

TEACHING MATHEMATICS WITH TECHNOLOGY: Order of OperationsAuthor(s): James H. WiebeSource: The Arithmetic Teacher, Vol. 37, No. 3 (NOVEMBER 1989), pp. 36-38Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41193788 .

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TEACHING MATHEMATICS WITH TECHNOLOGY

Order of Operations NCTM's recently released Curriculum and

Evaluation Standards for School Mathematics (1989) recommends that calculators and computers be freely available to elementary school students for solving mathematical problems, exploring pat- terns and concepts, and investigating realistic applications. Most students, however, need some help in learning to use these tools, especially if they are using them with problems involving more than one operation, as many realistic applications do. They need to know that different calculators or computer software tools use different internal algorithms for finding answers and, thus, may give different answers to the same problems. They need to be taught how to enter multistep problems and evaluate the displayed result, that is, to do parallel men- tal computations. This article focuses on teaching elementary school students the order in which calculators and computer languages solve mathematical expressions.

A good way to introduce the idea that mathematical expressions with more than one operation are not necessarily solved from left to right is to bring two calculators into the classroom, one inexpensive, the other scientific. Have two students enter the sequence "4 + 5x3=" into each. The inexpensive calculator will display 27, whereas the scientific calculator will display 19 (see fig. 1).

The students will probably assume that the operation has been entered incorrectly into one of the calculators, so ask them to reenter the problem, perhaps several times. You could also ask some of your students to enter the following program line into the computer in either BASIC or Logo:

PRINT 4 + 5*3

Repeat the activity with another expression like "12 - 5 x 2." You might even want to heighten the suspense by stating that the answer obtained with

the inexpensive calculator is wrong. Of course, some students will be astonished at the suggestion that calculators might produce incorrect answers, leading to a discussion of the order of operations in mathematics.

The standard order of operations in mathematics, which is used by scientific calculators and most computer programming languages, is first to carry out all multiplications and divisions in the expression from left to right, then to carry out all addi- tions and subtractions from left to right. Notice how the following problem is evaluated:

Prepared by James H. Wiebc, California State University, Los Angeles, Los Angeles, CA 90032

36 ARITHMETIC TEACHER



NOVEMBER 1989 37

8-2x3 + 8-^4

Step 1:2 x3 = 6 Step 2: 8 + 4 = 2 Step 3: 8 - 6 = 2 Step 4: 2 + 2 = 4

Scientific calculators don't quite follow this same sequence, since an expression with twenty or thirty operations would require many memories. Actu- ally, when you enter a problem like "8 - 2 x 3" into a scientific calculator, the calculator stores the "8 -" operation in memory. Then, if the next operation is "+" or "-," the "8 - 2" operation is done immediately. If, however, the next operation is "x" or "+," the "2 x 3" operation is computed, and this result is combined with the "8 -" operation in memory. Watch the display on a scientific calculator as you enter //8-2x 3" and "8 - 2 + 3."

8-2x3

iOÎQi DjsBla* I Notice that the 8 8 calculator did not - 8 yet display the 2 2 results of "8 - 2" x 2 because the o o next operation is

2 [_^ 8-2 + 3

- Entry -*• - Display -^ I ,. . .. 7T7 - -*• - -^ Notice ,. . .. that the

° jj calculator dis- 2 played the results

2 2 of "8 - 2" here * * because the next ~ ~ operation is "+."

So, technically, the "true" mathematical order is not followed in memory; however, the results are derived as if this order were followed.

Inexpensive calculators are not sophisticated enough to hold the intermediate results in memory; therefore, they find answers as soon as the operation and the operands are entered - thus, the "er- roneous" results. In a problem like "8 - 2 x 3," the 8 - 2 operation is computed and displayed before the calculator receives the next operation.

If the expression involves exponents, scientific calculators, Logo, and BASIC evaluate the exponents from left to right prior to multiplications and divisions.

8 + 3x2^ PRINT 8 + 3 * 2 * 2

8+3x4 8+3*4

8+12 8 + 12

20 20

When we want to change this order of operations, we use parentheses. The expression inside of a pair of parentheses is considered a single number; therefore, the entire expression inside the parentheses is evaluated using the standard sequence of operations and the resulting number used in the rest of the expression as appropriate. When parentheses occur inside of parentheses, the innermost expressions are evaluated first.

3x (5 + 4) 24/(4* (2 + 1))

3 x 9 24/(4 * 3)

27 24/12

2

After students are familiar with these ideas, they can be reinforced by asking students to do a vari- ety of activities involving expressions with multiple operations. The results can then be checked using the PRINT command in BASIC or Logo or with scientific calculators. The following examples contain problems at several levels. Younger pupils should be given exercises only of the first and second type. Problems of type 4 can be rather difficult: you may wish to allow students to work in pairs or groups of three to solve them.

Sample Reinforcement Exercises

Type 7; Identify the order of operations in expressions with multiple operations.

Example: Write a 1 next to the first operation done, a 2 next to the second operation, and so on:

A. 12 + 5-8 + 4 + 2 B. 4x2 + 2 + 6x8

Type 2: Evaluate expressions containing multiple operations.

Example: Write the answers for each of the following. Check your answer using the computer.

Continued on next page



Metric Trend Thousands of United States businesses have quietly moved to metric. "Almost all blueprints and specs worldwide are entirely on metric now," says spokes- person Gilbert Nolde of Caterpillar, In- corporated. "We'll make a component in France, install it in a machine in Brazil or Japan, and ship the assembled product to the U.S. It became ridiculous for us to make machines and parts in two ver- sions: Every time we came out with a new model, we converted to metric. Now we are saving tens of millions of dollars by avoiding double inventory costs and operating all our thirty-two do- mestic plants on one system."

The Omnibus Trade Bill passed in 1988 requires all federal agencies to use metric rather than conventional units in their procurements, grants, and business activities by 1991, with some exceptions.

"Major weapons systems have operating lives well into the next century, by which time the U.S. will be predomi- nantly a metric-based nation," says John Tascher, metric coordinator for the De- fense Department. (Source: Forbes 143 [26 June 1989]: 106-7)

38 ARITHMETIC TEACHER

Reference

National Council of Teachers of Mathematics, Commission on Standards for School Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: The Council, 1989. m

Type 4: Arrange numbers, operations, and parentheses to obtain a given value.

Example: Use the given numbers in any order with any operations and with parentheses to obtain the number. Write your expression in the space. Check your answers using the computer.

A. Use 5, 4, and 1 to get 21 . B. Use 6, 5, 3, and 2 to get 32.

Continued from previous page

Computer Your answer answer

A. 5 + 6 x 7 B. 8-4x5 + 6 C. 5x4-9 + 3-1

Type 3. Place parentheses so that the given answer is correct.

Example: Place parentheses in each expression to make it true:

A. 1+5x4 + 3 = 36 B. 12 + 4 + 2x3-2 = 2 C 12 + 4 + 2x3-2 = 4

What Are the Standards? ■V VI The department "Implementing the Standards," as well m^ yÊ as several articles in this journal, refers to NCTM's pub- KS^^ Jk^i lication Curriculum and Evaluation Standards for School ËU^^^ÊÊtà, Mathematics, usually referred to as the Standards. Is- H¿^>y^7 j'' sued in the spring of 1989, this document is the product ^^M^V^™ of a long-term effort of a special commission established by the Board of Directors of NCTM in 1986. The publication established a set of standards for mathematics curricula in North American schools (K- 12) and for evaluating the quality of both the curriculum and students' achievement.

The Standards establishes a broad framework to guide reform in school mathematics in the next decade. It sets forth a vision of excellence in mathematics education by suggesting new emphases and priorities for curricula. NCTM encourages all those interested in the quality of school mathematics to collaborate in using the Standards as the basis for improving mathematics teaching and learning in our schools.

The Editorial Panel of the Arithmetic Teacher: Mathematics Education through the Middle Grades strongly supports NCTM' s Standards and con- siders the Arithmetic Teacher an important vehicle for helping to clarify and promote this bold vision of a high-quality mathematics program for all students. We welcome manuscripts related to the Standards, as well as your comments and suggestions for the Standards section of "Readers' Dialogue."

Copies of the Standards are available for $25, order number 396, from NCTM, 1906 Association Drive, Reston, VA 22091. You may use the order form in the "Reviewing and Viewing" section of the journal. Members receive a 20 percent discount. Please inquire about discounts for quantity orders.

ZZ^Zl PROJECTS TO ENRICH SCHOOL Um MATHEMATICS (former title: Student ^9ßP Um Merit Awards), edited by Leroy Sachs. Spark interest in your students! Now available for two levels - Level 2 for middle school and Level 3 for secondary school. Each unit provides chal- lenging projects and enrichment material requir- ing from ten to thirty hours of independent study and writing. You will find numerous hints, draw- ings, references, and ideas for further investiga- tions, as well as teacher notes, with key information and solutions. Level 2 (middle school): 1988; 96 pp.; #388; $7. Level 3 (secondary school): 1988; 128 pp.; #389; $10. See the NCTM Mate- rials Order Form in the back of this issue.



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TEACHING MATHEMATICS WITH TECHNOLOGY: Order of Operations