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UNIT FRACTIONS INSTRUCTIONS

A UNIT fraction is a fraction with a numerator one(1). Instead of 3/4, the Ancient Egyptians would use the unit fractions 1/2 + 1/4. Fractions notation as we know it was not in use generally until after 1500 AD.

Think of slicing 3 cakes among 4 workers. We would slice each cake into 4 pieces and give each worker 3 of them. Each worker would get three 1/4 pieces. That means you will slice and move 12 pieces of cake.

With unit fractions you will slice two of the cakes into 2 pieces and the third into 4 pieces.Each worker would get one 1/2 piece and one 1/4 piece. Here you will slice and move 8 pieces. The unit fraction method could be a faster way to solve the problem.

The object of this game is to pay as many workers in five minutes as possible with barley cakes. You are to slice the barley cakes so that each worker gets the same amount.

After pressing the <START> button a timer will start. You will see a number of cakes and a number of buckets. First, press the OK button to choose the number of slices for each cake. Then press the <SLICE> button to move the slicer onto the cake. You can change the number of cake cutters before you slice. Then can drag each piece into a worker basket.

Do not put the same size slice into one basket. Each basket will contain one UNIT fraction in one size. For example, you should not put two 1/5 pieces into the same basket. If you do, the game will stop and ask you to press the <RESET EXAMPLE>.

If you make a mistake, just press the <RESET EXAMPLE> button to do the same example over.

The slicer will move over to the last piece when you have one slice left. This will allow you to slice the final piece into more parts, if necessary.

When all barley are given, a scroll will open showing Ancient Egyptian script for the fractional pieces of bread. The 1/15 Egypt Add image in this image shows that the denominator 15 is below a mouth image. The inverted U is used for the number 10, and each of the 5 bottom marks represent one(1). Because the numerator is always one(1) only the denominator needs to be shown.

How the ancient Egyptians solved the problem of 2 cakes to 5 workers.

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First, cut the cake into the largest unit fraction that we can take from 2/5.

Can you take 1/2 from 2/5? No.

Can you take 1/3 from 2/5? Yes.

So you will slice the cakes into 3 parts so that each workers will get a 1/3 piece. This will leave one 1/3 piece. This 1/3 piece can be sliced into 5 parts. Each part is actually 1/15 slice (1/5X1/3). You will drag each smaller pieces into each worker basket. That gives each worker 1/3 slice and 1/15 slice. If you add 1/3 and 1/15 you will get 2/5. This way you would have to move 10 pieces.

Compare this to a more modern way of slicing each cake into 5 pieces and giving each worker 2 of them. To do this you would also move 10 pieces.

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The clock will start when you press the <START> button. The clock will stop when you give out all the cakes. If correct, the scroll will open so that you can see the results. The timer will also stop and you will be warned if you put more than one of the same slice into the same basket. You will then redo the example by pressing the <RESET EXAMPLE> button.

You will have 5 minutes to pay the workers. The game will then stop when you complete the example you are working on. You will be given a score of the number of workers paid per minute. Press the REPORT button to get a printable report of your score.

Press the <START> button to start over with a new game.

Try to achieve a score of 5 workers per minute or better.

If you receive a score of 10 workers per minute or better you qualify for the title of ANCIENT EGYPTIAN PAYMASTER. This will show on your score report.

For the experts:

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Any unit fraction can be cut into smaller unit fractions. For example:

To give 4 cakes to 6 workers you would give each worker 1/2 cake and 1/6 cake. The 1/6 cake can can be sliced into 1/7 + 1/42. So instead of giving each worker 1/2 + 1/6 you could give 1/2 + 1/7+ 1/42.

If you get a piece equal to or smaller than 1/100 the scroll will show a coiled rope for each hundred. Be careful, though, the computer cannot show pieces smaller than 1/300. You will have to <RESET> the example and cut into larger pieces. What worker is going to want such a small piece anyway? These crumbs probably were eaten by the Ancient Egyptian Paymaster.

WHAT WILL I LEARN?

  1. Ancient Egyptians used unit fractions - with a numerator of one.
  2. A unit fraction can be broken down into a sum of smaller unit fractions.
  3. Unit fractions can be useful for purposes similar to this game.