Wednesday, November 29, 2017

11/28 Egyptian Fractions

Brainstorming this week I became interested in Egyptian Fractions because they dovetail nicely with the math history from last time.  Here's a topic that is both historical and mathematically interesting. I was going to originally title this week Funny Fractions and do a unit on both Egyptian Fractions and Farey Sequences but on consideration I decided there was enough to deal with just focusing on the first idea.  That was right decision to make based on actual time management.  As I discovered also over the hour, these provide a great platform for practicing other more basic skills,

To start off I had everyone guess when fractions were first documented as being used. I mentioned the late entry of decimals as a starting point. I was pleased someone remembered the Babylonian base 60 fractions from last week.  I then did a quick read of the background of the  Rhind papyrus with some information and a printout of the scroll from:

I then used a modified version of the series of questions and activities from here:   I particularly focused on finding ways to break Egyptian fractions apart into sums of other Egyptian fractions and discovering algorithms to find an Egyptian fraction sum for a regular fraction.   Once kids started to brainstorm on the whiteboard I started feeding further problems as different groups progressed:

Further Problems:

1. The Mullah's horse: The former Grand Wizier, Mullah Nasrudin was approached by three men with 19 horses. The men asked him to adjudicate the will of their recently dead father which required that his horses be divided among his three sons so that the oldest son receives 1/2, the middle son gets 1/3, and the youngest son would get 1/7. With little hesitation Nasrudin added his own horse to the herd and said, "What is half of 20, 1/4 of 20, and 1/5 of 20" After some time the men replied, "10, 5, and 4". The eldest son then took 10 of the horses, the middle son took 5 of the horses, and the youngest son took 4 of the horses. The Mullah Nasrudin, then took the remaining horse and rode home. Can you explain what occured?

2. Find all the solutions (there are less than 10) to the problem (n-1)/n = 1/a + 1/b + 1/c, where a < b,
b < c, a, b, and c are positive integers with least common multiple n. Note. a = 2, b = 4, c = 6, and n = 12 gives one solution.

3. How many different egyptian fractions can be used to describe 2/3? Two of them are 1/2 + 1/3 + 1/6 and 1/3 + 1/10 + 1/15.

4. Want to solve an unsolved problem? One of the most famous problems on Egyptian Fractions asks, "Can every proper fraction of the form 4/q be expressed with an egyptian fraction with less than 4 terms?" Can every proper fraction of the form 5/q be expressed with an egyptian fraction with less than 4 terms?

5. The sailor, coconut, and monkey problem: Five sailors were abandoned on an island. To provide food, they collected all the coconuts they could find. During the night one of the sailors awoke and decided to take his share of the coconuts. He divided the nuts into five equal piles and discovered that one was left over, so he threw the extra coconut to the monkies. He then hid his share and went back to sleep. A little later a second sailor awoke and had the same idea as the first. He divided the remainder of the nuts into five equal piles, discovered also that one was left over, and through it to the monkies before hiding his share. In turn each of the other three sailors did the same - dividing the observable amount into five equal piles, hiding one, throwing one left over to the monkies. The next morning the sailors, looking innocent, divided the remaining nuts into five piles with none left over. Find the smallest number of nuts in the original pile.

6. find 1/a + 1/b + 1/c + 1/d + 1/e  = 1

7. 355/113 approximates to 6 places. (355/113) - 3 = 16/113. Find an egyptian fraction whose sum is 16/113
I also printed out a fun geometric puzzle for tired students to relax with when they needed a break.

Overall, this went well but I'd improve several things if repeating:

  • This time too many kids went over to the puzzle a little too quickly. I have to think of way to keep everyone on task longer. 
  • I also wanted to do some notice/wonder activities around patterns in the puzzle but that was not possible while focusing on the main activity.  
  • I needed one or two more problems in the set to fully round things out. Several were of the type that kids could become stuck on. So a few more easier warm ups would work well. I'd have kids work out a variety of easy equivalent fractions next time i.e. find 3/4, 2/7 etc.
  • I had one student who out of character just wanted to read and not work on math today. Given the other needs of the kids I let her do that but I want to make sure next week she's engaged.

Also during the time I noticed a lot of fluency issues while the kids worked on the math. 
  • Adding fractions like 1/4 + 1/5.
  • Long division.
  • Mental math for fairly easy computations like 84 divided by 4.
In each of these cases I ended up doing mini walk-throughs  and I think the session acted as a way to review rusty skills.  But overall, I'm toying with the idea of  finding other activities that also stress these again. 


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