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Order of Operations

Directions: Make the largest (or smallest) expression by using the whole numbers 0-9 in the boxes below.  Note: for 5th grade, remove the exponent to make it grade level appropriate.



How can we tell where it would be best to put the larger/smaller numbers?



Largest: 6 ÷ 1 (8 + 7)^9 · 5 – 0 = 1,153,300,781,250

Smallest: If the answer can be positive or negative, then putting a 0 in the first term and a 9 in the last term makes it -9.  If the answer must be positive, then 5 ÷ 9 (4 + 7)^0 · 2 – 1 = 1/9.

Source: Robert Kaplinsky with answer from Michael Fenton and his students.

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  1. I must be missing something. My students got about 1.15 trillion, and I agree with them. I think it comes down to interpretation of the second operation (reading left to right).

    Here are three options, two of which must be equivalent: https://www.evernote.com/shard/s5/sh/9d33a64c-f44b-415c-b505-921434224ad9/c6ad61b74f22237bbb7cd894b12d4c07/deep/0/MathType—Untitled-3.png

    My vote? Option (C) is the odd one out.

    If I’m wrong, can someone help me understand why?

    • I feel like B is the odd one out. When I see a(b), I see this as “a” “b’s”, making it equivalent to option C. I’m not sure why I think this and I can’t find anything to back up my logic. I feel like a similar conversation was had a few years ago on Twitter and someone (Cox) concluded that either could be correct but that, to avoid ambiguity, parentheses should be used.

    • I agree with option C being the odd one out. In option A you are going to follow up parenthesis and exponents with multiplying and dividing left to right. This means that you would divide 6 by 1 next which is the same thing you do next in option B since it is in parenthesis. In option C you are going to multiply by 1 before dividing 6 by a really big number! That’s going to give you a very small answer.

      • To me, a/b(c-d)
        implies a/[b(c-d)], whereas a/b*(c-d)
        implies (a/b)(c-d).

        Again, I have looked and can’t find anything to back up my logic…

    • So there is some logic to C. In fact, when there is implied multiplication like a/b(c-d) then for some there is some ambiguity as to what is meant. Personally I don’t think there is but the Internet and calculators say otherwise. I posted this (https://twitter.com/davidpetro314/status/908082923856953345) a few weeks ago on Twitter and found that when the multiplication is implied. On some calculators, this multiplication takes precedence (see the discussion that followed my post). Apparently this is an advanced feature. We don’t typically teach this in elementary or HS but apparently many engineers are aware of this “extra” order of operations rule. I have found very little about this but I did see one small reference to the fact that it has to do with b being a scale factor when in the form b(c-d) (http://blogs.ptc.com/2011/11/10/math-basics-order-of-operations-precedence-and-tricky-brainteasers/). I’m not sure I 100% agree but it is definitely a thing (especially when calculators have it as explicit rules https://twitter.com/MrBinfield/status/908367572634099713)

  2. Robert Kaplinsky

    Ah, I think you are right and that the answer I listed is wrong. I meant for it to be C but it turned out to be B. Nice catch! I will update the answer accordingly.

  3. Doesn’t the smallest possible (positive) answer have the same interpretation issue? (BTW: When I copy and paste 1 ÷ 6 (7 + 8)^9 · 2 into Wolfram Alpha I get an answer on the order of 1.28 x 10^10.)

    • Robert Kaplinsky

      Again, nice catch Gale. Thanks for the feedback. I tried it again and the smallest positive answer I was able to get is 1/4. Anyone have a better one?

  4. Great problem and easy to adapt! Just wanted to share what I’m planning to do for an intro to Exponential Rules (more of a review) in Algebra!
    I think I’m going to use just the binomial raised to the power and try to max/min it using only 1-3. Brings up a good discussion hopefully about when we can distribute the exponent and when we can’t! I might also adapt it for our other exponential rules 🙂

    Yay open middle!

    Side though, loving using just negative integers too!!!

  5. For working on this with kids I’m less interested in the right answer but in the reasoning and conversations about how to approach and what are the things to consider in placing the numbers. I’m happy with a really big answer and a really small answer and the reasoning. Although some will continue to find the smallest and largest and that’s lovely too

  6. type this into the online Google calendar and you get 30 instead of the large number. Excel only return a large number if you change the formula to include a * before the parenthesis.

  7. Where does it say that numbers can’t be re-used? I got just over 16 trillion, using a lot of 9’s. I’m going to try this problem with my 6th graders today and see what they get. I may include the rule that they can only use each number once, and then also see what the smallest positive number is that they can get. Fun problem! Great practice for students once they get the basics of the Order of Operations!

    • Robert Kaplinsky

      That’s a really great point Chris. I guess you could use numbers more than once with the way it is written. That being said, using all 9s or even a lot of 9’s could be very bad for making an expression with the greatest value as you have to divide by much of it.

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