ORDER OF OPERATIONS
Bernice E. Holloway Bellwood School District #88
1801 N. 36th Avenue
Stone Park, IL 60165
1-708-344-9344
OBJECTIVES:
-to use grouping symbols and the standard order of operations to
simplify numerical expressions.
-to use the order of operations to evaluate variable expressions.
-to use the calculator and computer to solve numerical expressions.
MATERIALS:
-computers
-calculators(generic brand)
-paper
-pencils
-chalk
-chalkboard
-eraser
-banners(computer printouts)
-handout
STRATEGIES:
A. To bring out the comparison of punctuation marks in a sentence with
grouping symbols in a numerical expression (signs with the expressions
Robin Lee Travis and I love computers;Slow Children Playing;Save
Rags and Waste Paper.), ask if students can explain why the banners are
ambiguous.
INFO
Without commas, the sentence Robin Lee Travis and I love computers,
implies that two people love computers. Depending on where commas are
inserted, the sentence can state that three or four people love
computers.
Sample Activities:
1) ask students to simplify 10+2*3-1 to get as many different
answers as they can (use calculators and computers to compare
answers).
2) discuss the need for a standard order of performing operations
so that there is no ambiguity about the value of such
expressions.
3) discuss the steps of the standard order of operations and show
how they would be used to simplify the expression above.
10+2*3-1
10+6-1 multiplication
16-1 addition (from left to right)
15 subtraction (from left to right)
4) show how parentheses could be used to give different meanings
to the same expression.
(10+2)x3-1 (10+2)x(3-1)
12 x3-1 12 x 2
36 -1 24
35
INFO
In expressions with more than one operation, grouping symbols such
as parentheses or division bars are often used to indicate the
order in which to do the operations. These grouping symbols can
change the meaning of an expression, just as commas or other
punctuation marks can change the meaning of a sentence. Whenever
the order of operation is not indicated by grouping symbols, there
is a standard order of operations to be followed.(Do exponents,
multiplication/division, addition/subtraction from left to right.)
In mathematics, more than in some other forms of written
expression, ambiguity must be eliminated. Otherwise, different
people may assign different meanings to the same symbols, and
communication is faulty. Ambiguity is eliminated using grouping
symbols and the order of operations rule.
In examples #1 and #2, the expressions do not have grouping
symbols, the standard order of operations is used.
#1 13-4x2-3 #2 2x3^2-4
13-4x2-3 2x3^2-4
13- 8 -3 2x 9 -4
5-3 18-4
2 14
In examples #3 and #4, notice that the two expressions have the
same numbers and the same operations, but the results are
different because of grouping symbols.(Do operations within
parentheses, exponents, multiplication/division, addition or
subtraction from left to right.)
#3 (8+5)x3 #4 8+(5x3)
(8+5)x3 8+(5x3)
13x3 8+15
39 23
B. To give additional practice using the correct order of
operations, have students:
1) replace the variable in each row or column to make a true
equation in puzzle #1 (see handout).
2) write the operations sign (+,-,x,/) in each row or column to
make a true equation in puzzle #2 (see handout).
C. To check progress of students have them complete the Grouping
Symbols-Review (see handout and below).
IV. COMMENTS/INFO.
The use of the calculator is so common to us that we tend to take
certain things for granted...only with the wide use of personal
computers are we being forced to reevaluate the function, the
appropriate use, and the correct method(s) of teaching students
certain mathematical concepts using both machines.
3
It should be pointed out to students that people communicate with
computers by using programs. Programs tell the computer what to do.
However, it is not always necessary for a person to be able to write
a program in order to use a computer. Programs can be written in such
a way that an operator can use them by answering a series of questions
that are written into the program. Nevertheless, the best way to learn
what a computer can and cannot do is to learn a little about
programming.
To program arithmetic calculations in BASIC, you use the following
symbols:
+ addition
- subtraction
* multiplication
/ division
( ) parentheses
^ raised to a power(exponent)
BASIC follows the order of operations. Sample BASIC program:
10 PRINT 21*34+35/7
20 END
RUN
719 (answer)
Grouping Symbol-Review
Select each answer from the choices in parentheses. Write the answer in the
blank.
1) ab means a______________b. (plus, divided by, times)
a
2) _ means a______________b. (plus, divided by, times)
b
3) a____________b means a is not equal to b. (=,<>,.)
4) Parentheses are an example of a ______________.(grouping symbol, value,
variable)
Simplify each expression with the calculator. Translate each expression into
BASIC. Use the computer to check answers.(Remember to type PRINT before the
numerical expression.)
5) 7+(12-3)_____________________ 6) (18-3) ____________
-----
(3+2)
7) (7x3)-(5x4)__________________ 8) 10-(3+4)___________
9) 24- (63/(6+3))_______________ 10) 36/12+6____________
-------
8-5
11) 15-5x2+8/4__________________ 12) 20(12-8)-30/(10+5)___________
Simplify the expression on each side of the ----?----.
Make a true statement by replacing the ? with the symbol = or <>. Check your
answers using the computer.(If the computer prints 1, your answer is true; if your
answer is false, the computer will print 0.) Remember the numerical expressions
must be in BASIC.
13) 16+3 ? 9+3 _______ 14) (8-3x2) ? (8-3)x2_________
---- ---- --- -------
8+4 4-1
4
15) 3(5+2) ? 3x5+2 ______ 16+4 ? 8+4x3____________
----- 16) 1+ ---- ----- ----
3+2 8-2-2
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