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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