A dozen pocket rocks. #mathphoto23 #dozen

A different bar model approach. Since the number of bananas didn’t change, 3 parts had to become 2 parts without changing the boxes - so each part is now 1.5 boxes. Making the kiwis into 3 parts of the same size showed the fraction made up by the 3 eaten kiwis.

A bar model approach:

#howmany #unitchat #ANNMath

I don't think there should be symmetry. The circles along the top determine the x's and the circles down the side determine the y's. Since the graphs of cos and sin are different, we get different graphs when we double the argument of one vs the other.

Rearranging the groups can help you see why the total stays the same. Make each new group by taking one from each of the original groups. If we get used to visualizing and playing with numbers, these ideas make sense. They may not be immediately obvious, but they become clear.

The standard algorithm hides the sense of what is going on. With partial products (especially if developed from area models), you can see why the product comes out the same regardless of the order of the factors.

Personally, I would worry less about making sense of the reciprocal and more about making sense of the context. Here's how I'd do it - the operations are the same as in multiplying by the reciprocal, but it is less abstract and you don't have to make sense of units of 1/minutes.

@RandyRaux @eappleton @mtrushkowsky