The Lion's Share - fractions and doubling

We enjoyed a great little exploration with fractions and doubling that you may want to do with your students. We used Matthew McElligott's book, The Lion's Share: A Tale of Halving Cake and Eating It, Too (2009.) In the story, an ant is invited to the lion king's dinner party. At the party, the other animals demonstrate horrendous manners. When the king passes the cake, each animal takes half of the cake he is passed. So, the elephant take half of the original cake and passes on a half. The hippo takes half of that piece (a quarter) and passes the remains to the gorilla who takes another half (an eighth of the original), passing the remainder on to each animal until a mere morsel is given to the ant. The tiny creature, embarrassed because he doesn't have enough to share, tells the king that he'll return the next day with a special cake for the king. Each animal, wanting to outdo the next, agrees to bring double the amount of cakes to the king. So the ant will bring one, the beetle two, the frog four, and so on.

I read about half the book, stopping at the point where the ant was left with a crumb. (Warning: the insides of the front and back covers show some of the fractional pieces of cake so you may want to cover them up.) I gave each student a square piece of paper and made a list of all the animals on the board. I asked, "How much would each animal get?" and challenged them to show the fractional pieces of cake with the paper. In the midst of cutting "cake" a student laughed and said, "This is going to go all the way up to 1/128. I figured it out." He soon discovered it got even smaller! (1/512)

After figuring out the fractional pieces of cake, I finished the story and asked, "If each animal doubles the number of cakes he is bringing to the king, how many cakes will the king receive?" (Work shown at right. 511 cakes.)

So, now a final question for you all...

Do you see a relationship between the two results? (1/512 and 511) Explain.

Spoiler alert: see chart at the bottom of the page...


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