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.

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.