Jennifer in MamaLand posted about Cuisenaire Rods for Big(ger) Kids, showing how she and her daughter used them to solve a problem of area. The "which one has more wood" idea is classic Miquon, and it does work!
Crayons/Dollygirl and I had the rods out yesterday too, for some work on mixed numbers (in the last book of Key to Fractions). After several pages of very easy stuff, the book suddenly demanded that she divide, and show the remainder as a fraction. For instance, 64 ÷ 7 = 9 1/7. Division is not one of Crayons' strong points anyway, and she was mystified as to why you would want to have something other than an "ordinary" remainder. The best illustration I could come up with was the idea of somebody running laps around a track of a given length, running a certain length, and then figuring out how many whole laps and how many fractional laps they had completed. We set up a demonstration with some building blocks (they were all over the floor already--don't ask) and a Ty Beanie Bopper, who enthusiastically ran around the track.
But the real breakthrough came when we set up a length of Cuisenaire rods, and let "Footie" run past those. For the question above, we set up a long row of black rods (which normally represent 7). I asked how many black rods he would have to run past to get to 64. Past one...puff puff...past two...and so on, all the way to 9. Where did he stop? Just past the 9th set of 7. Actually ONE unit past it. So he ran past 9 and 1/7 sets.
For some reason, that made perfect sense to Crayons, and she was able to finish the page without any more trouble.
As a postscript: today's page took that idea into the "new" realm of changing an imperfect fraction into a mixed number by using division. For instance 15 / 4 becomes 15 ÷ 4, which is 3 3/4. This was still a bit puzzling, so I drew several "chocolate bars," divided them in quarters, and had Crayons colour in 15 quarters. How many chocolate bars did she get? 3 3/4. Then we compared that to the next question: 13 / 3. Would you rather have 15 / 4 chocolate bars, or 13 /3? We drew more chocolate bars, divided them into thirds, and coloured those too. Oh--that means you would get 4 1/3 chocolate bars! So if anybody offers you 13 / 3 chocolate bars--choose those.
(The concept of "more wood" may be classic Miquon, but "more chocolate" works just as well.)