Fractions Through Feeling and Touching
The Egyptians had an elaborate system for adding fractions. Who would have thought they even knew what these were? No one knows for sure why they used a summation of unit fractions so that a unit fraction was the correct answer. I have a theory to throw into the ring. It seems to me that it was an accounting method. I mean nothing in life is free. There wasn’t money at this time. How did one buy what they desired? You had Pharaohs who demanded a cut of harvests, smelted metals, goats, slaves and everything else they proclaimed as their own. Then of course the Pharaoh had to pay court functionaries even if all that meant was to feed them. The religious community demanded their share of resources too. The land owners demanded certain portions of food produced by their tenant farmers. There was a lot of buying and selling just as today. . . just no standardized coinage.
Here is where fractions come into human life. One would have to be able to add together or take apart physical property. There was a perceived value to this inventory. Product could be physically divided into equal quantity piles. Product value was more intimately embraced than today. People would know how much grain produced how many loaves of bread which would feed how many people for a certain number of days. So grain value as well as other commodities took a natural evolution towards the maintenance of society. What was needed was one standard to compare all other products to it.
This I think was the attempt of using unit fractions. It allows for different items to be valued independently. If these relative values could be related to an understood standard then accurate and fair trade could be established.
Looking at Fractions Differently
Fractions arose as a means to compare different products. My guess is that accounting, rudimentary economics and law led to rules where 1 part was “so and so-es” and the rest was for the “others”. It would be nice to believe that scholars were sitting around developing these intellectual identities. I believe there must have been a real and practical reason for these solutions be known. Perhaps it was to ensure the proper tribute was paid that a percentage system was determined and enforced. A way to compute division of property came into being. What arose from this was a table of equivalences. As a new identity was discovered it was recorded. It was memorized. It was passed from one generation to the next. Judge this speculation as you will. My opinion is that fractions provided a way to divide property along established law based on an accepted standard unit object.
So how can a home school parent or professional teacher use this natural “touchy feely” way to introduce fractions to children? The current methods of dividing pies into equal portions (or squares) is the best starting point. It is probably how our ancient ancestors first understood this process too. It is a defining moment when man can equate the concrete with the abstract. How the first person was able to know by looking at the parts of something and associate this with abstract values for physical objects was truly remarkable. It is a shame we cannot directly revere this early genius who paved the way for us to order our world.
Here is an example taken from the record known as the Egyptian Mathematical Leather Roll. Take the hyper link to the Wikipedia page that has a table of identities from this relic. I encourage those of you teaching to include your own unit identities in the comment section below. Today we are going to show this unit fraction identity: 1/3 + 1/6 = 1/2
Produce Your Own Unit Circle
You will need multiple copies of two different pie charts. You will need at least one of the 6 slice pie charts to stay black and white. The extra copies are for boo – boos and “start overs”. Here is a simple way to make any chart you want. There are of course ways to do this directly in Microsoft Word. I like to make charts in Excel. It is easier for me to manipulate.
1. Open an empty Excel worksheet.
2. In the upper left hand cell (called row 1 column A or simply 1A) type a number, say 1
3. In the cell directly below this (row 2 column A or 2A) type the same number you did in step 2.
4. In the cell directly below this (row 3 column A or 3A) type the same number a third time.
5. Click the function tab called INSERT that is on the row running from left to right at the top of your screen.
6. Click the PIE chart choice when this tab opens up.
7. Now click the first 2-D choice on the left that looks like your mom’s pie and you have sliced yourself a large serving.
8. You now have a pie chart that is divided into 3 equal parts.
Excel automatically pastes the pie chart on two pages. You can place your cursor inside of the area that popped up. Don’t choose a part of the chart itself to click on. Left click on the white area inside the square that is around the pie chart. Drag the image back to a spot inside of your page break area. Some spreadsheet applications automatically define cells that will appear on a page. If yours does not do this you will need to first create page breaks on your Excel program.
Now comes the part you will have to decide. The printed image may or may not be too small for the child to cut when the time comes along. Is the pie chart large enough for your child to handle? If you think it should be resized and made larger this is pretty easy to do. All you need to do is grab the handles in the center of each horizontal and vertical line of the image and drag the sides to fit the area you want. Make sure that you do not delete this sheet until you have made the second pie chart.
Now make a second pie chart. Follow the steps above except now we need a chart divided into 6 parts. So you will need to repeat steps 2, 3, 4. In other words fill the first 6 cells in the first column with the same number. As long as the six numbers are the same your pie chart will be what you need. Drag this pie chart into the page break like the first. Pull on your handles so this chart aligns on the same rows and columns as the first chart that you made. Both pie charts need to appear the same size.
How to Use These Pie Charts
Print multiple copies of these pie charts. Let the child color and cut their pie chart into pieces. Now have them place one of the 3 pie pieces on a black and white six piece pie chart. Have them place one of the 6 piece slices next to the first one. Together these two pieces of pie define half of a pie. This solves the identity above.
To demonstrate how the ancient genius tied this to every day commerce with the Pharaoh you will need some props. I looked around the house and I found that I had some black walnuts and hickory nuts. The problem then can be explained this way. Tell your student you are the Pharaoh and they are the Farmer. Tell them you will buy half a dime’s worth of nuts. If 3 walnuts are a dime and 6 hickories are also a dime, how many different ways can the Farmer sell these nuts to the Pharaoh? What if the Pharaoh changes their mind and now want to buy 15 cents worth of nuts? How would this work if it was still a problem in Egypt. What if 3 water buffalo were the same value as 6 gazelle? Making examples that hold your student's attention is how to involve them.
Be sure to have the pie charts handy so they can associate them with physically handling product. I’m sure that those ancient Kings relied heavily on their scribes to make sure they got their fair cut of merchandise. I imagine they wanted to make sure they were not swindled at a time when value was relative. To guarantee fairness the ingenious scribes produced known and memorized fractional methods they discovered through trial and error. Using unit fractions was an incredible first step to understanding and relating different product values. It led to rudimentary economics too. Older students may like the comparison to how the dollar value compares to other currencies. If the relationship stays the same for product count per dollar, what does this mean if our dollar devalues?
Sometimes presenting topics including alternative methods helps to cement topics in a student’s mind. I like to link history with Mathematical concepts. Sometimes how we understand something is directly related to a historical necessity. Too often they feel as though we are making things up just to torture them. Taking a bit of time to show there is real concrete reason for how we solve problems is enough to provide the relevance they need.