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Completing the square 307.PNG

Let's pretend that you are the person that invented agriculture.

Let's say that you figured out that when you put seeds in the ground in a square plot that is 10 paces by 10 paces you find that you have enough wheat to keep yourself fed while the next crop grows. Having done this you find that you have lots of time on your hands so you start thinking. Pretty soon you've discovered how to catch live wild birds. You find that if you feed your birds some of your grain they lay eggs for you which go nicely with the wheat that you've been eating. Unfortunately since you didn't increase the amount of wheat that you planted you find that you come up a little short at the end of the season and have to eat your birds. Since you're no dummy (you invented agriculture didn't you!) you land on the idea of using stones to keep track of the wheat you eat, and the amount of wheat a single bird eats. The next season you decide to plant wheat in a plot that is 11 X 11 paces. Your stones tell you that you eat 10 times as much wheat as your birds, and you have some wheat left over.

We know that the area of land to grow the amount of wheat you need is . We know that the amount of wheat that a bird needs is 1/10th what you need or 1/10 .

The amount of wheat you need is equal to 100 square paces. So

Plugging that into the equation we get:

where s=the square paces you need for you and your bird.

Since your last field of wheat was 11 x 11 square paces or 121 square paces you have 11 square paces worth of wheat left over. So you catch another bird.

Why did we use the the terms and above? is the term that we use to figure out the amount of land we need, but since our point of view is currently ourselves we can use variables like h for human, b for bird. In fact in the exercises below we will figure out values for other variables. Following the exercises we will change our point of view, and see how this change led to considerations like what are shown in the diagram on the right.

Exercises: 1) You decide to change the size of your field to 12 X 12. How many birds can you raise? 2) How many birds can you raise on a 13 X 13 field? 3) You can grow a new crop every 90 days in the spring, summer, and fall. Unfortunately in the winter your crop won't grow, so you need to save enough grain to get you and your birds through the winter. Try to write an equation that shows you how much grain you need to save for winter. 4) You can collect one egg a day from your birds. The birds live for 3 years, but you always eat the oldest bird because its the biggest. If you don't want to catch any more birds how many do you need to keep in order to have one egg every day and a bird for dinner every 9 days? 5) You decide to get a dog for company and protection. Your dog also needs meat as well as wheat. Feeding the dog requires 9 birds a season and 1/2 as much wheat as it takes to feed a person. How many additional birds do you need? How much additional land do you need ... don't forget to feed both the additional birds and the dog. 6)You use your dog to help you find birds. You can find 3 birds in the Spring, 6 birds in the Summer, 12 birds in the fall, and no birds in the Winter. How does this information affect your calculation above?

Imagine that you are a ruler in ancient Babylonia. It is your job to store and distribute grain to your various subjects and to say who can stay in the kingdom. In the spring you would have to give each subject enough grain to plant their fields.

Let's say that you know that you have 25,000 square paces of land, farmed by 100 subjects, but you don't know how much each subject farms. Because you don't want to end up deposed like the previous ruler you decide to keep track of the seed that you give out and the seed that comes back in. You find out that the previous king gave out 300 jars of seed to the farmers. You think this is easy, you will give each farmer 3 jars of seeds and everything will work out fine. Instead what happens is that 50 of the farmers come back to you for more seed because you didn't give them enough.

If you don't give your subjects enough grain they will come back and ask for more, but they will turn any extra grain that you give them into bread. This wouldn't be too bad, except that every seven years or so there is a drought and the fields don't produce enough grain to feed the farmers for the coming year. If you have extra grain in your granary to keep your subjects fed during the drought years they will think you're a wise king. If you don't have any extra grain your subjects may try to depose you and find a new king, so its in your best interest to store as much grain as you can.