Talking to AB about multiplying rational numbers. She understands the commutativity of multiplication of integers perfectly well. But I had forgotten that commutativity in the rational setting is actually conceptually harder! That four sixes is six fours you can conceptualize by thinking of a rectangular array, or something equivalent to that. But the fact that seven halves is the same thing as seven divided by two doesn’t seem as “natural” to her. (Is that even an instance of commutativity? I think of the first as 7 x 1/2 and the second as 1/2 x 7.)
It gets worse. I got completely stymied trying to explain to my nephew why 2/5 x 3/7 was the same as 2/7 x 3/5. Began to doubt it myself!
Another example I saw on Twitter recently: it feels counterintuitive that if you make something 10% smaller, then 10% bigger it ends up smaller than it started, and if you make it 10% bigger then 10% smaller it *also* ends up smaller.
But that’s just 0.9*1.1=1.1*0.9.
Another example that comes up every now and again on Twitter is the counterintuitive fact that “x% of y” is the same as “y% of z”. Calculating 4% of 75 seems difficult, but 75% of 4 is obviously 3. But these are (x/100)*y and (y/100)*x.
It is an instance of commutativity, and it is indeed non-trivial. There is a “rectangular array” point of view: arrange seven discs or squares in a row, and cut them horizontally in two. On the one hand, each side of the cut represents half of seven. On the other hand, each side contains seven halves.
Of course eventually (grade 5?) you construct the rationals as the field of fractions of the integers and get commutativity in the rationals from that of the integers.
I like Lior’s suggestion! I will try it and report back.
I feel like this is weirdly easier with real numbers than with rationals.
When my kids were younger I used snap cubes to help them see how some fraction problems that are similar to what you are discussing could be converted back to the rectangle ideas that come up with integers – this is a fraction division example, which is just what I found first:
We loved this video — thanks!
Being comfortable with changing the gauge could help with understanding associativity and commutativity. In Terry Rudolph’s example above, re-define “1=35” and then use the “rectangular array” picture.