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10. Calculator Mystery:
Discovering the order of operations using calculators
Look in the Classroom
Mr. Atsalis knows that before students can successfully simplify algebraic expressions or solve equations, they need to have a complete understanding of the algebraic order of operations. As an initial investigation, Mr. Atsalis uses the Calculator Mystery to introduce the order of operations. During this investigation, his goals are to have his students develop an understanding of the algebraic order for the operations of addition, subtraction, multiplication, and division, as well as a realization that not all calculators follow this order.
To begin the activity, Mr. Atsalis divides the class into small groups and provides each group with a set of multi-operation expressions, such as 10 + 6 ÷ 2 - 2 x 3, which they are to evaluate mentally. He then provides each group with a classic four-function calculator and a scientific calculator to share as they check their individual results. Mr. Atsalis tells the students to enter the numbers and operation symbols as they appear in the expression from left to right and to press the = (Enter) key only after they have entered the entire expression. Once the class begins to review answers, there are obvious signs of discomfort in the room. Not only do many students disagree about their answers, but the calculators are also giving different answers.
S1: The answer is 18 - that's what I got, and the calculator gives the same.
S2: I thought it was 18 too, but my calculator gave me 7. Maybe it's broken.
S3: No, my calculator gave me 7, too-they can't both be broken.
Mr. A: What do you think is going on inside each calculator?
He directs his students to rework problems, check initial answers, and compare results with other group members. Mr. Atsalis circulates and asks questions that help the students clarify and communicate their mathematical reasoning.
S1: 10 + 6 is 16, then 16 divided by 2 is 8, then 8 minus 2 is 6, and 6 times 3 is 18.
S2: To get 7, my calculator could be taking 10 plus 3, that's 13, then minus 6.
Mr. A: So what's happening there? Where did the 3 come from?
S3: It might be finding 6 divided by 2 first-that's 3-and adding that to 10 to get 13.
S2: Yes, then it multiplies 2 times 3-that's 6-and subtracts 6 from 13 to get 7.
S1: So, that means our calculators are doing things in a different order.
Mr. A: What do you mean by a "different order"?
S2: Well, my calculator just does everything the way I put it in.
Mr. A: What's happening with the other calculator-the scientific calculator?
S2: Well, it's like it's doing the dividing and multiplying first, even when it doesn't come first.
After allowing a sufficient amount of time for this investigation, Mr. Atsalis leads a class discussion during which students share their findings and conjectures. An important conclusion emerges about the cause of the discrepancies: The students agree that, when simplifying a multi-operation expression, the order in which operations are completed makes a difference and leads to different answers. The class then makes conjectures about the way each calculator interprets key sequences and predicts the results each calculator will generate for new expressions. As the students check their predictions, they confirm or refine their conjectures and eventually solve the mystery.
With their interest piqued, Mr. Atsalis explains that, in order to avoid the confusion of getting different results when evaluating an expression like 1 + 2 x 3, mathematicians have developed a set of rules that describe which operation is to be done first. This is the set of rules used in the scientific calculator and in algebra. Over the course of the following days, Mr. Atsalis extends the investigation to include the use of parentheses and challenges the students to produce the same result on the scientific calculator as they get when they enter the numbers and symbols from left to right on the four function calculator.
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