I would like to preface this post with my personal opinion – this is a silly argument that has gone viral. If teachers are teaching mathematical understanding and mathematical precision, and not a reliance on ‘calculators’, this ‘problem’ is not really a problem. It’s all about representation and being clear in what you want students to do, even in bare-naked math problems like this one. It shouldn’t be this hot-button topic if you are helping students understand mathematical operations and numbers. If students get different answers, it’s because the problem wasn’t clear….which actually, should lead to some great student discussion about mathematical precision.
The mathematical situation: 6/2(1+2).
Part of the problem here is that writing the problem in-line text, as the problem is represented above, does not allow for rational number representation in standard form (like ½), and many calculators have linear entry that looks like the above written problem. Some calculators use implied grouping, where anything under the division is considered a ‘group’ (so (2(1+2)), and some do not (only the first quantity, 2, is considered the denominator). If you were writing this problem on the board for students to solve, you would hopefully be more accurate and precise and use parentheses or the standard form of a rational number to avoid confusion. In a textbook, or on an assessment, same thing. But, as a result of the linear written form of this problem, i.e. 6/2(1+2), the ‘argument’ that people are touting is that the problem can be interpreted two ways:
- Either as 6 divided by 2, then multiplied by (1+2) or
- 6 divided by the grouping of “2 multiplied by (1+2)”.
My thoughts – 1) you shouldn’t be relying on the calculator to be the answer-giver (meaning students should notice/wonder and question if the division is the whole quantity, including the 2, or if its just the 2 as the divisor) before calculating; 2) If they get different answers, that should become a class discussion about precision and representation; 3) IF you are being a good teacher and focused on mathematical precision, you would not write the problem this way in either case, because you would and should use parentheses if you wanted either situation. So, what you SHOULD write, is:
- (6/2)(1+2) in which case, the parentheses are done first, so 3*3 = 9 solution, OR if you wanted the other situation, then you write..
- 6/(2(1+2) in which case, parentheses done first, so 6/(2(3))=6/6 = 1
So – it’s provider error (teacher, textbook, assessment) for not being precise in what you expect students to do. If you follow PEMDAS, with no parentheses, since it is linearly written, it can be interpreted as 6 divided by 2 times (1+2), so parentheses first giving 6 divided by 2 times 3. THEN, Divide and Multiply, as you see it, from left to right – so 6 divided by 2 is 3 and 3 x 3 is 9. But – with a calculator that uses implied grouping under a division symbol, the 2(1+2) is grouped. Conclusion – KNOW what you want students to do and write it that way (standard notation OR with parentheses), OR have a great discussion when they get different results that leads to the importance of precision in mathematics.
The real debate is over what calculators do when you enter the problem – which on most calculators, is linear entry, so as mentioned above, some calculators are doing implied grouping, where the 2 outside of the parentheses is considered ‘grouped’ with the parentheses under the division symbol, so it is interpreted as version 2 above (i.e. the entire quantity 2(1+2) is seen as one number). Others are not. And therein lies the argument – which calculator is correct…..which is not really the argument, since students should have clarity on that before they even get to the calculator (which, let’s be honest, you shouldn’t be using a calculator for this problem to begin with!!)
What’s the answer? Obvious one – use parentheses to clearly state what you want ‘grouped’ if writing/representing a problem linearly. Use standard notation of rational numbers so that fractions look like fractions and you can clearly see whether you are just dividing by the 2 or whether you are dividing by the entire quantity 2(1+2). If using a calculator, know what quantities are grouped, so teachers/texts/assessments need to clearly group or use standard notation to show what is in the divisor.
This brings me to the new fx-9750GIII graphing calculator, which has basically done the work for those who aren’t being precise in the way they present problems to students. The fx-950GIII has new order-of-operations functionality that basically puts the parentheses in for you if you don’t write the problem using standard notation. Basically, it is showing you how it is interpreting that division, so you can then analyze if that is truly what/ how the problem was meant to be solved. (Which again – should have been determined to begin with so this issue doesn’t even arise, but I am repeating myself……..)
Here is a short video showing you how the fx-9750GIII is helping teachers and students really analyze the order of operations if they enter a problem such as 6/2(1+2) linearly (instead of using the nice rational template or parentheses).
Be sure to visit Casio Cares: https://www.casioeducation.com/remote-learning
Here are quick links: