Interesting Math for 3-graders

Solution 1:

$3\times8$ means $8+8+8$.

$8\times3$ means $3+3+3+3+3+3+3+3$.

Why should those both be the same number?

Why should $a\times b$ always be the same as $b\times a$?

Solution 2:

The Seven Bridges of Königsberg is a nice one. From memory, third-grade is around the time when kids like to try those problems of "can you draw a house without raising your pen and only drawing each line once", etc. which is essentially what this problem is. The solution, I believe, is simple enough for them to understand.

Solution 3:

Discuss the birthday paradox if there are at least 23 people in the room. In fact, ask the teacher in advance if he/she knows from student records if two students share the same birthday (an illustration with the students in the room won't go over well if you try it and nobody shares a birthday).
Look at the sum of odd numbers: 1, 1+3, 1+3+5, 1+3+5+7,... until someone notices a pattern. This is basically your idea of the sum of an arithmetic progression, but made concrete in an easily grasped way.
The 3x+1 problem. This will show them an example of an elementary unsolved problem that they can experiment with.
Examples of patterns that break down: if a sequence starts off as 1, 2, 4, 8, 16, what is the next term? I would hope some of the students will recognize these as powers of 2. The point is to show them several examples of counting problems that all start off this way but the next term is not 32. See examples 1, 5, and 6 in my answer at https://mathoverflow.net/questions/52101/longest-coinciding-pair-of-integer-sequences-known.

Solution 4:

Let them try to create maps that need as few colours as possible. The rule is that two counties are neighbours if their borders meet in more than a finite number of points and neighbours should not have the same colour. Hopefully, this might lead to an interesting discussion about the 4-colour theorem.

Here are example "maps".

Here is the Wikipedia page.

Solution 5:

Teach 'em how to play sprouts.