Lift Up Your Hearts!

At lunch today, Jonathan told me he wanted a quarter of a grilled cheese sandwich; he then amended that to "half of a quarter."  "So you want an eighth of a sandwich?" I inquired.  "Yes," he replied, and proceeded to ask Heather, as he had several times before, "What's half of an eighth?"  "A sixteenth."  What's half of a sixteenth?'  "A thirty-second."  Then followed a discussion of just what "one thirty-second" means.

After lunch we had some wonderful molasses cookies made by a friend.  There were just enough for each of us to have one, with one cookie left on the plate.  So I asked Jonathan what fraction of a cookie each of us would have if we shared the leftover cookie fairly.  This was confusing for him, so Jon simplified the question and began to lead Jonathan step by step to figuring out the answer.  Jonathan is adept at the concept of one half and one quarter, including the written form that he encounters in recipes.  However, this is a little hard to extend to one fifth, because there's no 2 in "half" and no 4 in "quarter."

After some drawing and patient discussion, Heather brought out their fraction circles for further exploration and clarification.  After Jonathan understood, at least to some degree, the concept of "one fifth" and how to write it, we went back to the original problem of sharing the cookie.  He knew he needed to cut it into five pieces, and began—as would many children—by cutting it into halves, then quarters.  "Now," he commented, "I cut this piece in half, and I'll have five pieces."

I then asked him which piece he would take for his own, hoping to force the observation that the pieces were not of equal size.  But he was 'way ahead of me.  "These two," he explained, pointing to the smaller pieces, "are for me and Noah.  The adults get the three big pieces."  After a brief examination of the one-fifth fraction circle pieces, with Jon explaining that to divide a circle into fifths you can't begin by cutting it in half, we enjoyed our fractional cookies.

It must be noted that while the adults and Jonathan were distracted by the fraction discussion, Noah had taken a more practical approach and put the entire cookie on his own plate.