Answers for Chapter 2 Sample Problems

Here are some solutions to various cookie-sharing problems:

1. Share 3 cookies with 2 people. Stephanie, age 6 3/4 (she told me), did this:
She cut each cookie into 4 pieces.

We talked about the name of each of these pieces is 1/4 or one-fourth or one quarter (as in money) because 4 pieces make a whole cookie.

She gave each person 4/4, then 2/4 of a cookie. So each person got 4/4 + 2/4 = 1 + 2/4 = 6/4 cookies. Stephanie wasn't sure how to write these, so I wrote them, then she copied them. (Note, I write the fractions with a horizontal line separating the top and bottom numbers, but this is too hard to write in html).

2. Share 5 cookies with 4 people. Robin, age 7, shared them this way: She shared 1 cookie with each person, leaving one cookie left over. She then cut this cookie into 5 pieces, each 1/5 and shared these. This left 1/5 of a cookie left over. She proceeded to cut this 1/5 into 4 pieces. It took 20 of these small pieces to make a whole cookie, so each of these is 1/20 of a cookie. She gave each person one of them. So each person got 1 + 1/5 + 1/20 cookies!

3. Share 2 cookies with 5 people. John, a 4th grader, did this:
First he asked if he could have a ruler. I said why not do the best you can, don't worry about making perfect cuts. He thought about his cutting a while. Then he cut each cookie into 3 pieces, but not all equal pieces. He cut 2 pieces the same size and one piece 1/2 the size of the other two! I asked him what the size of the larger piece was and he immediately said 1/(2 1/2), one two and a halfth! He shared one of each of these larger pieces with each person, with 2 of the smaller pieces left over. "What is the name of the smaller piece?", I asked. He thought about this and saw that these two smaller pieces was as large as one of the bigger pieces and 5 would make a whole cookie, so each was 1/5 of a cookie. So each person got 1/(2 1/2) or 2/5 of a cookie! In all the years I've been doing the cookie-sharing, no one did what John did; that was exciting!

4. Share 2 cookies with 3 people. (In my Worksheet book I use lots of pictures, I don't know how to do that here, sorry). This is how Brad (finished 2nd grade) would have done this:
He shared when he could, then always cut each piece left into two equal pieces! After giving Brad the problem, I went to talk with other students. When I returned there was a bunch of larger, then smaller and even smaller pieces (crumbs!). I asked him what he had done and this exciting method and result emerged. He couldn't share 2 cookies with 3 people, so he cut each cookie into 2 equal pieces. (We always had discussions about the name of the pieces as we went along). He now had 4 pieces, each 1/2 of a cookie. He shared each 1/2 with the 3 people, with 1/2 leftover.

[Each person now had 1/2 of a cookie].

The 1/2 of a cookie he cut into 2 equal pieces, each piece 1/4 of a cookie. He couldn't share these 1/4's because there were only 2 pieces and we had 3 people to share with.

[Each person now had 1/2 + 0/4 cookies].

He cut the 2-1/4's into 2 equal pieces, getting 4 pieces, each 1/8 of a cookie. He shared three of these, with one left over.

[Each person now had 1/2 + 0/4 + 1/8 cookies].

He continued this to sixteenths, but he saw patterns and was able to write further terms.

[Each person ended up with 1/2 + 0/4 + 1/8 + 0/16 + 1/32 + 0/64 + 1/128 + ... cookies!].

The 3 dots means it goes on forever! This is known as an infinite series. What simple fraction is this equal to?
Notice that I had to help him name the pieces, which he caught on to, and helped him to write the fraction names, which he caught on to also. It was Brad's idea to cut the cookies this way. It lead to others getting infinite series by sharing 2 cookies with 5 people and then I varied the special scissors to cut into only 3 equal pieces or 10 equal pieces! (This last one gets us to decimals). There are so many interesting things to do!

A program in Basic to do the cookie sharing using Brad's method and get the bimal (the numerators only, the denominators are powers of 2).

This starts with such a simple idea, gets into important mathematics (division, sharing, fractions, infinite series), and most young people can and should do it!

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