# What are some "deep" questions to explore in elementary school math?

My first grader is very advanced in math. Rather than doing more and more math and making school math even more boring for him, I recently decided to start going "deeper" rather than "faster."

Some of the questions we've explored recently are:

• Why do we use a numbering system? (Because otherwise we'd need an infinite number of names)
• Why is multiplication commutative? (Because you could switch the number of items and the number of people and you'd still need the same number of items in the 'rectangle')
• What might math look like without the number 0?
• Why are 2, 5, and 10 the only numbers whose divisibility tests only need the last digit? (because they're the factors of 10)
• Why is a number divisible by 9 if its digit sum is divisible by 9? (becuase every power of 10 has remainder 1 when divided by 9)

Are there any resources I can use to find more questions/explorations like this?

• Maybe you would get better answers at matheducators.stackexchange.com Commented Oct 26, 2023 at 9:38
• Speaking from personal experience, the book called "The Number Devil" by Hans Magnus Enzensberger could be a fun read for a math loving first grader. I loved that back when I was in early elementary school. It isn't very deep, but it does explore some cool topics. Commented Oct 26, 2023 at 10:01
• "What might math look like without the number 0?" Apart from exotic rings , every ring has a null-element , so I do not think that we can do reasonable math without $0$ , not even if we work in a different really useful number system. In base $b$-systems where $b>1$ is an integer we need $0$ and other systems have little to no merit. Commented Oct 26, 2023 at 14:25
• @Peter That's a very...modern perspective. I would argue that a lot of good math was done for thousands of years (e.g. Babylonian approximation of $\sqrt 2$) before zero gained full-fledged number status. Commented Oct 26, 2023 at 15:24
• The number $0$ , perhaps , but they needed the digit $0$. Commented Oct 27, 2023 at 16:03