What IS the answer to 8 ÷ 2(2+2)?
8÷2(2+2)=?
In past days, Twitter has exploded with this question, with some people arguing that the answer is 1 and others arguing it is 16. And these are adults who are trying to apply BEDMAS or PEMDAS (depending on where they went to school).
My first response is what a great question in that it is sparking a math discussion:
Many people are arguing the answer was 1 – they see 2(2+2) as a unit that equals 8 so
8 ÷ 8 = 1
Others rewrote the expression as 8 ÷ 2 x (2+2) then and get an answer of 16
8 ÷ 2 x (4) =
4 x 4 = 16
Using BEDMAS literally, one does division and then multiplication after doing the operation in the parentheses.
But if one follows PEMDAS literally, the answer is again 1
8 ÷ 2 x (4) =
8 ÷ 8 = 1
So which is right – we shouldn’t get a different answer based on which we notate a question or which acronym we use.
Based on my prior studies, I could easily explain the difference between BEDMAS and PEMDAS. Both of these mnemonics hide the fact that division and multiplication have the same prominence. After all, multiplication and division of inverse operations and 5 X 1/4 = 5 ÷ 4. (See Questioning the Order of Operations (Dupree, 2016) for other misconceptions that arise from using these mnemonics.) Realising that multiplication and division have the same prominence, if we rewrite the initial equation as 8 ÷ 2 x (2+2), then the answer is 16.
But is there a mathematical difference between 8 ÷ 2 x (2+2)and 8 ÷ 2(2+2)? People who I asked saw 2(2+2) as a unit with implied brackets (2(2+2)). I went searching for the mathematically correct answer and could not find anything that indicated that 8 ÷ 2 x (2+2) is different from 8 ÷ 2(2+2)? Photomath and Wolfram Alpha rewrote the equation: 8 ÷ 2 x (2+2) and 8/2(2+2) respectfully, but these programs are only as good as their programming.
So which is right? I would argue that the expression is ambiguous and should be rewritten to make the intent clearer. As the expression currently stands, it is equivalent to the phrase, “What are we having for dinner Mother? This expression is grammatically correct but could be asking Mother a question or asking if Mother is on the menu. We have the tools to rewrite the sentence to make the intended meaning clearer. Indeed, guidelines for writing tell us to revise our prose to avoid ambiguity.
And if one looks at the history of order of operations, avoiding ambiguity and clarity were the purposes for developing current conventions for order of operations (Peterson, 2000). When algebra was developing, authors would begin their work with a discussion of the conventions they used. Over time mathematicians developed standard algebraic notations including agreed-upon order of operations.
But why a convention that says move left to right. The current order of operations simplifies the writing of expressions like ax2+ bx + c.
Without prominence of multiplication (and division) aver addition, the quadratic expression would need to be written as (a(x2)) + (bx) + c (Peterson, 2000). Make that a cubic or quartic expression, and the number of brackets would become very confusing.
And because mathematics is a system and algebra and arithmetic need to follow the same rules, the order of operations for arithmetic need to be the same as the order of operations for algebra.
After revisiting the history of order of operations and their purpose, I am rethinking my original assessment of this question. Do we really need complicated questions about order of operations with arithmetic?
Dupree (2016) argues that if we are highlighting the algebraic nature of arithmetic—building understanding of general properties of mathematics such as distributive and commutative properties then discussions about order of operations arise naturally. However, if we just introduce order of operations as an arbitrary convention, then we add to that frequent perception that math has nothing to do with the real world.
Introduced in the context of algebraic thinking, order of operations are a convenient convention and help us work with algebraic expressions.
So is 8÷2(2+2) = ? a good question? Not if it is used to assess people’s understanding of order of operations. I feel that it is deliberately confusing which is counter to the reasons that order of operations were originally developed.
However, as seen on twitter this week, this question has a lot of value in generating discussions about order of operations and clear mathematical communication.
References
Dupree, K (2016) Questioning the Order of Operations. Mathematics Teaching in the Middle School. October 2016, Vol. 22, Issue 3
Peterson, Dr (2000) History of the Order of Operations. The Math Forum. Ask Dr Math.