The Hispaniola Water Shortage
Posted October 16, 2000 as a Math Forum EPoW

Introduction: In this problem, students were asked to use virtual containers of water, of definite sizes, and combine them in various ways to come up with as many different resulting volumes as possible.

Where’s the Math: This problem gives students an introduction to number theory by emphasizing the significance of starting with containers of different parity, as compared to containers of the same parity. There is some implied work with modular arithmetic, i.e. using the 3 oz. and 8 oz. containers, whenever one fills up the 8 oz. with the 3 oz., 1 oz. will be left in the 3 oz. container because 8 is congruent to 2 (mod 3). Questions encourage students to find a pattern involving parity. Standards: Number & operations Role of Components: A single PEN component is configured with a PEN program that allows users to drag the cups and draws the current amount of water in each container. The amounts are wired to an invisible custom component that turns the water amounts into HTML. The HTML is then wired to a customized Java editor pane where users can view the history of pours.

Try the first applet!

Try the second applet!

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Sample submitted solutions:

From:  Monica, age 12
Long, age 13
School:  Caroline Davis Intermediate School, San Jose, California

1. Do you notice any patterns in Table 1? Explain.
I could get all the numbers in the first set by adding or subtracting one cup by the other. That is possible, because one cup is an even number and the other is odd, 3 and 8.

2. Do you notice any patterns in Table 2? Explain.
In the second set there was a little problem, both cups were even numbered size cups. I managed to get all the even numbers, 12, 10, 8, 6, 4, and 2, but I couldn't get any odd numbers.

3. Compare the patterns in your two tables. Think about the combinations you used. What do you notice about even and odd numbers?
In the first set when I add the odd with the even I got all the numbers, because when you are trying to get a number that is even or odd and you have an even number and an odd number to use, it is possible that you could get the desired number. However, if you only got two numbers that is both evens then you got a problem. That is because you could only get the even numbers, not the odds. Even plus even is even. If you got two numbers that is both odd, there is certain numbers that you can't get.

4. What are your conclusions about the amounts that can (or cannot) be measured using the two different sets of measuring cups?
In set one I found that it is difficult to get the amount of 1, 4, 7, and 10. Because I have to do all this pouring back and forth a lot. The other numbers were easy to get. On set two I think it is hard to get 2, because of the same reason as above. Plus I couldn't get any odd numbers.

5. Which set of cups would you recommend that Rodolfo buy?
I definitely would recommend that he should the first set, 3 and 8. Because it is (versatile?) a set that could be change to the amount he wanted to use to cook. He could get any amount if he tried hard to figure out. And there is no need to spill out water.

1. Do you notice any patterns in Table 1? Explain.
Yes, every thing was yes, because if you think about it, you don't have to put everything into the recipe at the same time. Since you can get 1 oz. of water with the 3 oz. and 8 oz. measuring cups, one can just put one cup at a time in to the recipe. (Please accept this idea, for all my other answers are based on this.)

2. Do you notice any patterns in Table 2? Explain.
Yeah,the only amounts of water that the 4 and 10 oz. measuring cups can hold, are the even amounts of ounces because they can put 2 cups of water at a time into the recipe.

3. Compare the patterns in your two tables. Think about the
combinations you used. What do you notice about even and odd numbers?
I didn't notice anything about the "odd or even numbers"

4. What are your conclusions about the amounts that can (or cannot) be measured using the two different sets of measuring cups?
The 3 and 8 ounce measuring cup can measure all the amounts of water while the 4 and 10 ounce measuring cups can only measure even amounts of ounces, therefore, the 3 and 8 measuring cups are better.

5. Which set of cups would you recommend that Rodolfo buy?
I recommend the 3 oz. and 8 oz. cups because Rodolfo can get all the certain amounts of water he wants.

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