I did the following with 7 year olds several times when my children were in elementary school, and it might work with 5 year olds, too, *if* they know how to add. (Although maybe enough to know how to count.) The topic is **Triangle Numbers and Square Numbers**. First we played with triangle numbers $3,6,10,15,\ldots$. I drew them with dots on the blackboard, and the children, split into groups of 3 or 4, modeled them using M&Ms. Then we discussed how to get the next triangle number from the previous one, leading to the formula $T_n=1+2+3+\cdots$. (Of course, I didn't write this as a formula, but they seemed to have no trouble grasping the idea of putting another layer on the bottom of the triangle.) Next we turned to square numbers $4,9,16,25,\ldots$. Again, with pictures and M&Ms, they easily understood what a square number is. Then came the challenge. How to efficiently compute $S_n$, keeping in mind that although the children knew how to add, they did not know how to multiply. The solution, of course, is that $S_n=1+3+5+\cdots+(2n-1)$ is the sum of the first $n$ odd numbers. This becomes clear from the picture if you label the dots in shells. Here's a $5\times5$ picture using letters, but in the class I used colored dots, and the children made their own M&M models of a $4\times4$ square with the colors to illustrate the shells:
$$\begin{matrix} E&E&E&E&E\\ D&D&D&D&E\\ C&C&C&D&E\\ B&B&C&D&E\\ A&B&C&D&E\\ \end{matrix}\qquad 25=1+3+5+7+9$$

After all this fun, I posed the real question: *Are there any triangle numbers that are also square numbers?* So we made a short list of triangle numbers and a short list of square numbers and found that $36=T_8=S_6$. After this triumph, each group took 36 M&Ms and used them to transform $T_8$ into $S_6$, and then they got to eat the M&Ms.

To wrap things up, we tried to find another square-triangle number. Each group was tasked with making a list of either $S_n$ or $T_n$ by repeated addition, then we compared the lists. My recollection is that this was not always succuessful due to arithmetic errors, but that was okay. (The next one is $1225=T_{49}=S_{35}$, then $41616=S_{204}=T_{288}$.)

I've also talked about this subject to high school students (without the M&Ms), leading to Pell's equation and more-or-less proving that there are infinitely many square-triangle numbers. And also to college students, proving that the square-triangular numbers form a "1-parameter exponential family", i.e., that Pell's equation has a unique generator. This is one of the reasons that I like this problem so much, it can be studied at so many different levels.

**Update 2024:** When I recently did this activity with a grandson's second grade class, I wasn't allowed to bring in M&Ms. Not sure if it was the possible issue with food coloring (which I'm now more aware of, since I have a valued colleague who is quite allergic) or whether feeding children candy in school is frowned upon (also a reasonable concern). In any case, we used pennies instead of M&Ms, which worked fine.