Four Fours Problem

I’m not sure where I found this problem, but it was probably part of Jo Boaler’s MOOC, “How to Learn Math”, which was great by the way, but unfortunately I didn’t have the time to complete it.

I gave the problem on the first day of school to Forms 3 -5.  The students were asked to use four 4s and any mathematical symbols to yield the numbers 1 through 20.   I had given them a sheet with 20 numbered boxes to place their final answers, but encouraged them to just play around with the four 4s and various operations on scratch paper first.  I advised them to come up with the answers in random order not to just start with 1, then 2 etc.  It is definitely easier to tackle that way.  For example, a student might start with 4 + 4 + 4 + 4 and put that in the 16 box.   They had a few minutes to get started in class and left with the instructions to complete as many as possible by the following day.  (Note:  I haven’t come up with a way to get 11 and 13 without using 4!  My students weren’t familiar with factorial so I did share that with them as a possible operation to use, but didn’t mention for which numbers.  The students also needed to use 4! for 11 and 13, so if anyone has an answer that doesn’t use 4! for 11 and 13, I would love to see it!)

This is a great problem for two reasons, it doesn’t really matter what type of math student you are as long as you show persistence!  Yes, it was easier for some students than others, but with persistence all students could come up with all or most of the numbers.

The next day, I asked students to share an answer that they were particularly proud of.  A student would come up to the board and write up their four 4s and various operations WITHOUT sharing the answer they had in mind.  At this point, we treated it like a “number talk”  where class members would share their answers without the right answer being revealed until everyone agreed on the same answer and how to arrive at it.  This was a great review on order of operations!  Students were forced to defend their answers.

One of my favorite moments was when a less confident student initially said “Eleven” as his answer, then another more confident student offered “Fourteen”.  The student that said “eleven” raised his hand and “OK, I see it now, I change my answer to 14”, a third student then showed how the answer was actually eleven!  and yes the answer was 11.   The confident student then changed her mind and agreed with 11. The first student realized that he should have stuck with his original answer and not been swayed by the confident student’s answer!

We later looked at those problems that would work for any number “n ” if the 4 were replaced by “n”  For example 4/4 + 4/4 and n/n + n/n will both equal 2.

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7 thoughts on “Four Fours Problem

  1. I also encountered this four fours problem in Jo Boaler’s “How to Learn Math” course. You found a great way to incorporate the task into your class to encourage persistence, allow for student discussion, and review essential concepts (order of operations) all at the same time. Thanks for sharing your success story!

  2. I just did this with my students. One of my classes was stuck on the number 11. Students kept thinking they had it, only to find out they did not. A student who I have been struggling to reach so far this year, put up his hand and wanted me to check his work. He thought he had 11, but wasn’t sure. He was correct, and I was so proud! The whole class cheered for him. I am hoping that it was a turning point for this student and his relationship with math.

  3. GREAT post.

    I started toying around with possibilities for 11 and 13. We could brute force attack the problem to find all possible outcomes. We would need to define which operations are permissible (+, -, x, / ) as well as whether or not we treat the 4’s as distinct, e.g., if the 4’s were all different colors. We also need to be explicit on commutativity, associativity, etc.

    I wrote up a shotgun expression assuming four operations and associativity. We could account for commutativity by multiplying by a factor or two.

    Sum (k = 0 to k = 4) (4 choose k) * 4^3 where k = the number of 4’s associated together

    Again, I know this isn’t very rigorous and doesn’t do ALL the counting, but it’s a start!

  4. I love the four fours problem! It’s easy to jump into and explore, and it can also really challenge kids to be creative. I like the way you ask students to explain the expression that they discovered that they are most proud of.

    When I do the four fours with students, I allow a few other operations. We can use two fours to write forty-four, or use a decimal point to write “4.4” or “.4”. If we’re allowed to do that, we could write “(4/.4)+(4/4)” and “(4-.4)/.4 + 4”. I allow the square root as an operation, too, and that comes in handy sometimes: “44/4 + sqrt(4)”. Maybe some new operations will spark your students to find new solutions!

    And, in a bit of shameless self-promotion, I’ll mention that Challenge 05 over at collaborativemathematics.org has a similar feel to the four fours, in the sense that the goal is to build an expression out of some given building blocks. If your students liked the four fours, they might like “pieces of eight” ! 🙂

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