Cookie-sharing/ Chapter 2

This is only one of the most interesting and exciting of all the work I've done with young people. Why? Because young people find it difficult to cut a cookie into more that 2 equal pieces, so they solve this kind of problem in often completely unexpected ways, and unlike the 'normal' way. This leads to very interesting discussions about how many pieces to cut it into, how many pieces make a whole cookie, what to name the pieces, how to write the name of the pieces, will we ever finish, how do we know that the answer is correct..??

And yet this starts with such a simple idea that involves sharing with others (in this day and age, that's a great thing), and gets into important mathematics- like division, fractions, adding fractions, multiplying fractions, equivalent fractions, binary arithmetic, infinite repeating decimals, and infinite series, of course, and even multiplying infinite series!.. And most young people can and should do it! This happened about 3 years before Don wrote and published his book Calculus By and For Young People (ages 7, yes 7 and up) and became Chapter 2.

The beginning of cookie-sharing:

[Note: Don doesn't use circles, pies, pizzas or the like, for the cookies, because using these rectangular cards makes the cutting of the pieces and naming them much simpler. Simple is good. Of course you can search annemily who uses circles to do very fine work with + and x fractions, contrary to what Don says!]

Brad, who had finished 2^nd grade, was sent by his Mom to work with Don during that summer. At one point, Don gave Brad the problem: share 6 cookies between 7 people. The cookies were 3'x5' cards and he was given scissors to cut them. You should probably stop here and try this yourself. Don walked away to work with other students, and when he rejoined Brad, there were 7 piles of cookie crumbs before him! Something inside Don said 'you need to listen very carefully to Brad as he describes what he did!' Don could feel something exciting was going to come from this, but it was going to take great effort on both our parts!

Brad said with only 6 cookies, he didn't have enough to share between 7 people, so he cut each cookie into 2 equal pieces. This gave him 12 pieces, which he knew each to be ' of a cookie.

He could then share between 7 people. So he gave each person ' of a cookie.

[Don reads this as one-twoth, not to be funny, but this will help in trying to name the fractions later, when he asks the student- how many of these pieces make a whole cookie- if the answer is 7, then the name of the piece is 1/7, one-seventh of a cookie- very important and something not many students are aware of even in middle school].

Each person now had ' cookie

Brad had 5-' pieces left over. He then cut each left-over ' into 2 equal pieces and ended up with 10 pieces.

What was the name of each piece? How many of each make a whole cookie? Well 2 make a ' of a cookie, so how many make a whole cookie? 4. So what's the name of each piece? ' of a cookie, yes! If needed, Don would use the pieces to help him see this. [Notice the name of the piece one-fourth is consistent with one-twoth, and one-eighth, and one-tenth].

Brad shared ' with each person

Each person now had ' + ' of a cookie

Brad had 3-1/4 pieces left over.

He then, as usual, cut each left-over '  into 2 equal pieces and ended up with 6 pieces.

How many of these pieces make a whole cookie? 8. What was the name of each piece? 1/8, yes. But Brad said the 6 pieces weren't enough to share between 7 people, so no one gets an eighth.

Each person now had ' + '  +  0/8 of a cookie.

[Notice: we wrote 0/8 to help see a pattern when it shows up- soon!]

Brad then, as usual, cut each left-over eighth into 2 equal pieces, getting 12 pieces (notice we had 12 pieces once before!)

How many of these pieces make a whole cookie? 16. What was the name of each piece? 1/16, yes.

Brad was able to share 1/16 of a cookie with each of the 7 people.

Each person now had ' + '  +  0/8 + 1/16 of a cookie.

And there were 5 of the 1/16's left over.

Brad then, as usual, cut each left-over sixteenth into 2 equal pieces, getting 10 pieces (notice we had 10 pieces once before!). Brad saw a pattern in what the bottom of the next fraction, 2, 4, 8, 16, thirty-twoths,

and these 10 can be shared between 7 people.

Each person now had ' + '  +  0/8 + 1/16 +  1/32 of a cookie.

with 3 of the 1/32's left over. When cut into 2 pieces, that will make 6- sixty-fourths, so we can't share 6 sixty-fourths between 7 people.

This pattern continues ' + '  +  0/8 + 1/16 +  1/32 + 0/64 + ' and this is an infinite series.

So each person gets  ' + '  +  0/8  + 1/16 + 1/32 + 0/64 + ' of a cookie

We can write each share as an infinite bimal: 0.110110' In a decimal the places are 1/10, 1/100, 1/1000,...whereas in a bimal the places instead are 1/2, 1/4, 1/8, ...  

Fantastic job, Brad!!

Try some cookie-sharing as Brad did, or cut into 3 equal pieces instead of 2, or 10  equal pieces; see what happens!

See "share any way you like" on the MAP, for other interesting cookie-sharing  that kids and adults have done .

 Have fun!!