Who hates the number 3?

I think the following activity is a fun way to think about how our number system works through considering what could change if one of the integers were deleted. Perhaps this could be a fun way to interact with elementary age students as they navigate counting and arithmetic, and how they think their understanding of the rules might change if something drastic happened, like if the number 3 disappeared! I bet elemetary school students could come up with some really cool ideas and thoughts and wonders.

The writing in black has to do with what is presented to students, and in red are my thoughts regarding this scenario and how students might consider how to proceed. Of course, if someone wants to use these ideas, feel free to change things up to better fit your classroom.

Prompt

Art’ill, a land where an evil mathemagician eliminated the number 3 from the number system. You live in a distant land and hear about this terrible scenario. What do you wonder about?

I’m wondering if that means that numbers like 13 and 23, etc. exist. I’m also wondering what that does to prime numbers. For example, 9 becomes a prime number, because 3 does not exist. Recall that prime numbers are those that can only be divided by 1 and themselves. However, 6 might become what we could call "semi-prime" or some new term to capture the situation where 6 is divisible by 2, but results in something undefined, which then could suggest that 6 really is not divisible by 2, which has some interesting implications and I’m not sure how that gets fixed. I also wonder what counting looks like, is it: 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14? Or do we eliminate all multiples of 3: 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16?

What would various arithmetic operations look like with this numbering system? What problems arise?

12+1=?? Seems problematic because unless 13 exists, then 12+1=14, but so is 12+2. These dilemmas would require the creation of new rules to satisfy the new situation, students would quite literally be creating new mathematical ideas.

What would happen if any multiple of 3 could not exist?

So 12+1 doesn’t exist. What other issues come up? 1+2=??? Also problematic… unless 1 is a multiple of 3 too (is it? I think it depends). So addition appears to be problematic in this situation. But multiplication should be okay, right?

End Prompt

I think it would just be so interesting to hear how students navigate this issue and pose questions and come up with solutions to get around the created blockades. How would students at different mathematics levels approach this problem similarly or differently?

UPDATE:

I did implement this in a course. I decided to use it because I had a hard time getting my undergraduate students to talk and participate in discussions around math. I guessed that one reason (though there could have been others) was due to some level of math anxiety/uncertainty. I wanted my students to approach math as not to focus solely on the answer. This particular question does not really have an answer so that required them to use their reasoning to make sense of the situation and propose what eliminating the number 3 would mean. I enjoyed seeing my students working in small groups and discussing together ideas they generated.