Return to Mathematics IndexORDER OF OPERATIONSBernice E. Holloway Bellwood School District #88 1801 N. 36th Avenue Stone Park, IL 60165 1-708-344-9344OBJECTIVES:-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) -handoutSTRATEGIES: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.INFOWithout 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 35INFOIn 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