How would you explain commutativity of multiplication of naturals to kids?

Question

I would do it like this, and you?

Put in a rectangular basket apples in such a way that you have three rows and in every row 4 apples, then you have 3x4 apples in the basket. Stay where you are and rotate the basket by 90 degrees. Now rows become columns and columns rows so now you have 4 rows with 3 apples in each row so you have 4x3 apples. Since during rotation the number of apples does not change we have 3x4=4x3.

Now turn 3 into m and 4 into n by saying: "whatever the number of rows of apples with whatever number of apples in each row by rotating we do not change the number of apples inside so that is why multiplication is commutative".

Answer 1

3 people bringing 5 apples bring the same amount of apples as 5 people bringing 3 apples.

I think your rotation example is good for later, but first contact might be easier without the added complexity.

Answer 2

I’ve tutored several grade school kids learning multiplication and what has worked pretty well is giving them physical objects to work with so they can see what is happening. Pretty much analogous to your rotation scenario but with a different approach.

For example, I would give the student 15 beans and ask them to express 3x5 using them. They would then (maybe with a little guidance) arrange them in 5 groups of 3. Then after regrouping the beans ask them to express 5x3 in which they’ll do 3 groups of 5. Then to address the idea of commutativity make note that there are only 15 beans and we were able to arrange as 5x3 and 3x5 so that 5x3=3x5.