I took a break from reading math stuff today to do something hands-on.
I wrote an entry yesterday about an idea that I had read about tying math to simple machines, but I decided that if I were to use this in my class, I would simply offer it first as a challenge for the kids to build a box that magically scales its input to output. I would show them the one that I made, as a blackbox, (they wouldn't get to see what's inside), and then give them a week or two outside of class to meet in groups to brainstorm ideas and to do some research.
Bonus points for any group who brings me a working design (built/assembled, ideally with recycled material) by the end of the two weeks. And then we can discuss ratios involved as a class and relevance to other simple machines.
Sorry, my video is not great, but I had fun building this out of stuff laying around at home! (Paper clips, cardboard, masking tape, glue gun, and string.) You can't really see in the video, but the "SLOW" side takes in input more slowly than the other side spits out output. There are evenly spaced tick marks on the string to help with visualizing this. My hubby recommended putting the pieces of tape on the string to help with visualizing. If the ratio was 1, you'd expect the output length (from hole to ending position of tape) to be the same as the input length (from starting position of tape to hole).
It's magic! Or, just plain mathematics...